LMFDB - Embedded newform 8649.2.a.be.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8649,2,Mod(1,8649)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8649, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8649.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8649 = 3^{2} \cdot 31^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8649.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,-2,0,8,-3,0,-2,9,0,-13,-18,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$69.0626127082$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.2051578125.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 9x^{6} + 19x^{5} + 14x^{4} - 28x^{3} - 11x^{2} + 6x + 1 $ x^8 - 2x^7 - 9x^6 + 19x^5 + 14x^4 - 28x^3 - 11x^2 + 6*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 31)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.431370$ of defining polynomial
Character	$\chi$	$=$	8649.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.26660 q^{2} -0.395721 q^{4} +3.80032 q^{5} +2.18899 q^{7} +3.03442 q^{8} -4.81349 q^{10} -0.950596 q^{11} -0.168142 q^{13} -2.77258 q^{14} -3.05196 q^{16} -6.57666 q^{17} -1.15284 q^{19} -1.50387 q^{20} +1.20403 q^{22} -4.62850 q^{23} +9.44244 q^{25} +0.212969 q^{26} -0.866228 q^{28} +1.33672 q^{29} -2.20322 q^{32} +8.33000 q^{34} +8.31886 q^{35} +3.87165 q^{37} +1.46019 q^{38} +11.5318 q^{40} -0.328203 q^{41} -9.63057 q^{43} +0.376170 q^{44} +5.86247 q^{46} -5.63858 q^{47} -2.20832 q^{49} -11.9598 q^{50} +0.0665374 q^{52} -7.33294 q^{53} -3.61257 q^{55} +6.64232 q^{56} -1.69309 q^{58} +2.65312 q^{59} +1.74967 q^{61} +8.89454 q^{64} -0.638995 q^{65} +0.552007 q^{67} +2.60252 q^{68} -10.5367 q^{70} -1.13699 q^{71} +7.92260 q^{73} -4.90384 q^{74} +0.456203 q^{76} -2.08084 q^{77} -4.54228 q^{79} -11.5984 q^{80} +0.415702 q^{82} +0.326396 q^{83} -24.9934 q^{85} +12.1981 q^{86} -2.88451 q^{88} -14.7102 q^{89} -0.368062 q^{91} +1.83159 q^{92} +7.14183 q^{94} -4.38117 q^{95} +15.5192 q^{97} +2.79707 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} + 8 q^{4} - 3 q^{5} - 2 q^{7} + 9 q^{8} - 13 q^{10} - 18 q^{11} + 8 q^{13} + 9 q^{14} + 4 q^{16} - 14 q^{17} - 6 q^{19} + 7 q^{20} + 4 q^{22} - 22 q^{23} + 13 q^{25} - 9 q^{26} - 5 q^{28}+ \cdots - 10 q^{98}+O(q^{100}) $ 8 * q - 2 * q^2 + 8 * q^4 - 3 * q^5 - 2 * q^7 + 9 * q^8 - 13 * q^10 - 18 * q^11 + 8 * q^13 + 9 * q^14 + 4 * q^16 - 14 * q^17 - 6 * q^19 + 7 * q^20 + 4 * q^22 - 22 * q^23 + 13 * q^25 - 9 * q^26 - 5 * q^28 - 12 * q^29 + 21 * q^32 - 7 * q^34 + 12 * q^35 - 8 * q^37 - 7 * q^38 + 11 * q^40 + 22 * q^41 - 2 * q^43 - 4 * q^44 + 18 * q^46 + 18 * q^47 - 2 * q^49 - 27 * q^50 - q^52 - 6 * q^53 + 28 * q^55 - 30 * q^56 - 5 * q^58 + 4 * q^59 + 30 * q^61 + 9 * q^64 - 3 * q^65 - 13 * q^67 - 30 * q^68 - 39 * q^70 + q^71 + 2 * q^73 - 8 * q^74 - 33 * q^76 + 24 * q^77 + 8 * q^79 - 24 * q^80 - 19 * q^82 - 39 * q^83 - 26 * q^85 + 16 * q^86 - 17 * q^88 - 27 * q^89 + 16 * q^91 - 32 * q^92 + 22 * q^94 + 6 * q^95 + 34 * q^97 - 10 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.26660	−0.895623	−0.447811	−	0.894128i	$-0.647796\pi$
$2$	−1.26660	−0.895623	−0.447811	+	0.894128i	$0.647796\pi$
$3$	0	0
$3$	0	0
$4$	−0.395721	−0.197860
$5$	3.80032	1.69956	0.849778	−	0.527141i	$-0.176736\pi$
$5$	3.80032	1.69956	0.849778	+	0.527141i	$0.176736\pi$
$6$	0	0
$7$	2.18899	0.827360	0.413680	−	0.910422i	$-0.364243\pi$
$7$	2.18899	0.827360	0.413680	+	0.910422i	$0.364243\pi$
$8$	3.03442	1.07283
$9$	0	0
$10$	−4.81349	−1.52216
$11$	−0.950596	−0.286615	−0.143308	−	0.989678i	$-0.545774\pi$
$11$	−0.950596	−0.286615	−0.143308	+	0.989678i	$0.545774\pi$
$12$	0	0
$13$	−0.168142	−0.0466343	−0.0233172	−	0.999728i	$-0.507423\pi$
$13$	−0.168142	−0.0466343	−0.0233172	+	0.999728i	$0.507423\pi$
$14$	−2.77258	−0.741003
$15$	0	0
$16$	−3.05196	−0.762991
$17$	−6.57666	−1.59507	−0.797537	−	0.603271i	$-0.793864\pi$
$17$	−6.57666	−1.59507	−0.797537	+	0.603271i	$0.793864\pi$
$18$	0	0
$19$	−1.15284	−0.264480	−0.132240	−	0.991218i	$-0.542217\pi$
$19$	−1.15284	−0.264480	−0.132240	+	0.991218i	$0.542217\pi$
$20$	−1.50387	−0.336274
$21$	0	0
$22$	1.20403	0.256699
$23$	−4.62850	−0.965109	−0.482554	−	0.875866i	$-0.660291\pi$
$23$	−4.62850	−0.965109	−0.482554	+	0.875866i	$0.660291\pi$
$24$	0	0
$25$	9.44244	1.88849
$26$	0.212969	0.0417667
$27$	0	0
$28$	−0.866228	−0.163702
$29$	1.33672	0.248222	0.124111	−	0.992268i	$-0.460392\pi$
$29$	1.33672	0.248222	0.124111	+	0.992268i	$0.460392\pi$
$30$	0	0
$31$	0	0
$31$	0	0
$32$	−2.20322	−0.389479
$33$	0	0
$34$	8.33000	1.42858
$35$	8.31886	1.40614
$36$	0	0
$37$	3.87165	0.636495	0.318248	−	0.948008i	$-0.396906\pi$
$37$	3.87165	0.636495	0.318248	+	0.948008i	$0.396906\pi$
$38$	1.46019	0.236874
$39$	0	0
$40$	11.5318	1.82333
$41$	−0.328203	−0.0512566	−0.0256283	−	0.999672i	$-0.508159\pi$
$41$	−0.328203	−0.0512566	−0.0256283	+	0.999672i	$0.508159\pi$
$42$	0	0
$43$	−9.63057	−1.46865	−0.734324	−	0.678799i	$-0.762501\pi$
$43$	−9.63057	−1.46865	−0.734324	+	0.678799i	$0.762501\pi$
$44$	0.376170	0.0567098
$45$	0	0
$46$	5.86247	0.864373
$47$	−5.63858	−0.822471	−0.411235	−	0.911529i	$-0.634903\pi$
$47$	−5.63858	−0.822471	−0.411235	+	0.911529i	$0.634903\pi$
$48$	0	0
$49$	−2.20832	−0.315475
$50$	−11.9598	−1.69137
$51$	0	0
$52$	0.0665374	0.00922708
$53$	−7.33294	−1.00726	−0.503629	−	0.863920i	$-0.668002\pi$
$53$	−7.33294	−1.00726	−0.503629	+	0.863920i	$0.668002\pi$
$54$	0	0
$55$	−3.61257	−0.487119
$56$	6.64232	0.887617
$57$	0	0
$58$	−1.69309	−0.222313
$59$	2.65312	0.345407	0.172703	−	0.984974i	$-0.444750\pi$
$59$	2.65312	0.345407	0.172703	+	0.984974i	$0.444750\pi$
$60$	0	0
$61$	1.74967	0.224023	0.112011	−	0.993707i	$-0.464271\pi$
$61$	1.74967	0.224023	0.112011	+	0.993707i	$0.464271\pi$
$62$	0	0
$63$	0	0
$64$	8.89454	1.11182
$65$	−0.638995	−0.0792576
$66$	0	0
$67$	0.552007	0.0674383	0.0337192	−	0.999431i	$-0.489265\pi$
$67$	0.552007	0.0674383	0.0337192	+	0.999431i	$0.489265\pi$
$68$	2.60252	0.315602
$69$	0	0
$70$	−10.5367	−1.25937
$71$	−1.13699	−0.134936	−0.0674679	−	0.997721i	$-0.521492\pi$
$71$	−1.13699	−0.134936	−0.0674679	+	0.997721i	$0.521492\pi$
$72$	0	0
$73$	7.92260	0.927270	0.463635	−	0.886026i	$-0.346545\pi$
$73$	7.92260	0.927270	0.463635	+	0.886026i	$0.346545\pi$
$74$	−4.90384	−0.570059
$75$	0	0
$76$	0.456203	0.0523301
$77$	−2.08084	−0.237134
$78$	0	0
$79$	−4.54228	−0.511046	−0.255523	−	0.966803i	$-0.582248\pi$
$79$	−4.54228	−0.511046	−0.255523	+	0.966803i	$0.582248\pi$
$80$	−11.5984	−1.29675
$81$	0	0
$82$	0.415702	0.0459066
$83$	0.326396	0.0358266	0.0179133	−	0.999840i	$-0.494298\pi$
$83$	0.326396	0.0358266	0.0179133	+	0.999840i	$0.494298\pi$
$84$	0	0
$85$	−24.9934	−2.71091
$86$	12.1981	1.31535
$87$	0	0
$88$	−2.88451	−0.307490
$89$	−14.7102	−1.55927	−0.779637	−	0.626232i	$-0.784596\pi$
$89$	−14.7102	−1.55927	−0.779637	+	0.626232i	$0.784596\pi$
$90$	0	0
$91$	−0.368062	−0.0385834
$92$	1.83159	0.190957
$93$	0	0
$94$	7.14183	0.736623
$95$	−4.38117	−0.449498
$96$	0	0
$97$	15.5192	1.57573	0.787867	−	0.615845i	$-0.211185\pi$
$97$	15.5192	1.57573	0.787867	+	0.615845i	$0.211185\pi$
$98$	2.79707	0.282546
$99$	0	0
$100$	−3.73657	−0.373657
$101$	−4.87360	−0.484941	−0.242471	−	0.970159i	$-0.577958\pi$
$101$	−4.87360	−0.484941	−0.242471	+	0.970159i	$0.577958\pi$
$102$	0	0
$103$	1.93062	0.190230	0.0951148	−	0.995466i	$-0.469678\pi$
$103$	1.93062	0.190230	0.0951148	+	0.995466i	$0.469678\pi$
$104$	−0.510215	−0.0500307
$105$	0	0
$106$	9.28792	0.902122
$107$	−12.5275	−1.21108	−0.605540	−	0.795815i	$-0.707043\pi$
$107$	−12.5275	−1.21108	−0.605540	+	0.795815i	$0.707043\pi$
$108$	0	0
$109$	−10.2625	−0.982968	−0.491484	−	0.870887i	$-0.663545\pi$
$109$	−10.2625	−0.982968	−0.491484	+	0.870887i	$0.663545\pi$
$110$	4.57568	0.436274
$111$	0	0
$112$	−6.68072	−0.631268
$113$	3.84294	0.361514	0.180757	−	0.983528i	$-0.442145\pi$
$113$	3.84294	0.361514	0.180757	+	0.983528i	$0.442145\pi$
$114$	0	0
$115$	−17.5898	−1.64026
$116$	−0.528966	−0.0491132
$117$	0	0
$118$	−3.36045	−0.309354
$119$	−14.3962	−1.31970
$120$	0	0
$121$	−10.0964	−0.917852
$122$	−2.21614	−0.200640
$123$	0	0
$124$	0	0
$125$	16.8827	1.51003
$126$	0	0
$127$	−7.34389	−0.651665	−0.325832	−	0.945428i	$-0.605644\pi$
$127$	−7.34389	−0.651665	−0.325832	+	0.945428i	$0.605644\pi$
$128$	−6.85938	−0.606290
$129$	0	0
$130$	0.809352	0.0709849
$131$	−13.0337	−1.13876	−0.569379	−	0.822075i	$-0.692816\pi$
$131$	−13.0337	−1.13876	−0.569379	+	0.822075i	$0.692816\pi$
$132$	0	0
$133$	−2.52356	−0.218820
$134$	−0.699172	−0.0603993
$135$	0	0
$136$	−19.9564	−1.71124
$137$	−0.134299	−0.0114740	−0.00573698	−	0.999984i	$-0.501826\pi$
$137$	−0.134299	−0.0114740	−0.00573698	+	0.999984i	$0.501826\pi$
$138$	0	0
$139$	−11.5275	−0.977749	−0.488874	−	0.872354i	$-0.662592\pi$
$139$	−11.5275	−0.977749	−0.488874	+	0.872354i	$0.662592\pi$
$140$	−3.29195	−0.278220
$141$	0	0
$142$	1.44011	0.120851
$143$	0.159835	0.0133661
$144$	0	0
$145$	5.07995	0.421867
$146$	−10.0348	−0.830484
$147$	0	0
$148$	−1.53209	−0.125937
$149$	5.44108	0.445751	0.222875	−	0.974847i	$-0.428456\pi$
$149$	5.44108	0.445751	0.222875	+	0.974847i	$0.428456\pi$
$150$	0	0
$151$	−13.6892	−1.11401	−0.557005	−	0.830509i	$-0.688050\pi$
$151$	−13.6892	−1.11401	−0.557005	+	0.830509i	$0.688050\pi$
$152$	−3.49821	−0.283742
$153$	0	0
$154$	2.63560	0.212383
$155$	0	0
$156$	0	0
$157$	−15.0553	−1.20155	−0.600774	−	0.799419i	$-0.705141\pi$
$157$	−15.0553	−1.20155	−0.600774	+	0.799419i	$0.705141\pi$
$158$	5.75326	0.457705
$159$	0	0
$160$	−8.37296	−0.661941
$161$	−10.1317	−0.798493
$162$	0	0
$163$	17.0381	1.33453	0.667263	−	0.744822i	$-0.267466\pi$
$163$	17.0381	1.33453	0.667263	+	0.744822i	$0.267466\pi$
$164$	0.129877	0.0101417
$165$	0	0
$166$	−0.413414	−0.0320871
$167$	−16.1982	−1.25346	−0.626729	−	0.779238i	$-0.715607\pi$
$167$	−16.1982	−1.25346	−0.626729	+	0.779238i	$0.715607\pi$
$168$	0	0
$169$	−12.9717	−0.997825
$170$	31.6567	2.42796
$171$	0	0
$172$	3.81102	0.290587
$173$	2.33122	0.177239	0.0886196	−	0.996066i	$-0.471754\pi$
$173$	2.33122	0.177239	0.0886196	+	0.996066i	$0.471754\pi$
$174$	0	0
$175$	20.6694	1.56246
$176$	2.90118	0.218685
$177$	0	0
$178$	18.6319	1.39652
$179$	−16.9706	−1.26844	−0.634222	−	0.773151i	$-0.718679\pi$
$179$	−16.9706	−1.26844	−0.634222	+	0.773151i	$0.718679\pi$
$180$	0	0
$181$	7.33576	0.545263	0.272631	−	0.962119i	$-0.412106\pi$
$181$	7.33576	0.545263	0.272631	+	0.962119i	$0.412106\pi$
$182$	0.466188	0.0345561
$183$	0	0
$184$	−14.0448	−1.03540
$185$	14.7135	1.08176
$186$	0	0
$187$	6.25174	0.457173
$188$	2.23130	0.162734
$189$	0	0
$190$	5.54919	0.402581
$191$	−7.82276	−0.566035	−0.283018	−	0.959115i	$-0.591335\pi$
$191$	−7.82276	−0.566035	−0.283018	+	0.959115i	$0.591335\pi$
$192$	0	0
$193$	−4.63151	−0.333383	−0.166692	−	0.986009i	$-0.553308\pi$
$193$	−4.63151	−0.333383	−0.166692	+	0.986009i	$0.553308\pi$
$194$	−19.6566	−1.41126
$195$	0	0
$196$	0.873880	0.0624200
$197$	−22.3844	−1.59482	−0.797411	−	0.603437i	$-0.793797\pi$
$197$	−22.3844	−1.59482	−0.797411	+	0.603437i	$0.793797\pi$
$198$	0	0
$199$	26.6038	1.88589	0.942945	−	0.332948i	$-0.108043\pi$
$199$	26.6038	1.88589	0.942945	+	0.332948i	$0.108043\pi$
$200$	28.6524	2.02603
$201$	0	0
$202$	6.17291	0.434324
$203$	2.92606	0.205369
$204$	0	0
$205$	−1.24728	−0.0871135
$206$	−2.44533	−0.170374
$207$	0	0
$208$	0.513165	0.0355816
$209$	1.09589	0.0758040
$210$	0	0
$211$	1.32614	0.0912951	0.0456476	−	0.998958i	$-0.485465\pi$
$211$	1.32614	0.0912951	0.0456476	+	0.998958i	$0.485465\pi$
$212$	2.90180	0.199296
$213$	0	0
$214$	15.8674	1.08467
$215$	−36.5993	−2.49605
$216$	0	0
$217$	0	0
$218$	12.9985	0.880368
$219$	0	0
$220$	1.42957	0.0963814
$221$	1.10581	0.0743851
$222$	0	0
$223$	12.1199	0.811612	0.405806	−	0.913959i	$-0.366991\pi$
$223$	12.1199	0.811612	0.405806	+	0.913959i	$0.366991\pi$
$224$	−4.82284	−0.322239
$225$	0	0
$226$	−4.86748	−0.323780
$227$	15.9022	1.05546	0.527732	−	0.849411i	$-0.323043\pi$
$227$	15.9022	1.05546	0.527732	+	0.849411i	$0.323043\pi$
$228$	0	0
$229$	16.2986	1.07704	0.538521	−	0.842612i	$-0.318983\pi$
$229$	16.2986	1.07704	0.538521	+	0.842612i	$0.318983\pi$
$230$	22.2792	1.46905
$231$	0	0
$232$	4.05616	0.266300
$233$	5.12296	0.335616	0.167808	−	0.985820i	$-0.446331\pi$
$233$	5.12296	0.335616	0.167808	+	0.985820i	$0.446331\pi$
$234$	0	0
$235$	−21.4284	−1.39783
$236$	−1.04989	−0.0683423
$237$	0	0
$238$	18.2343	1.18195
$239$	8.56800	0.554218	0.277109	−	0.960839i	$-0.410624\pi$
$239$	8.56800	0.554218	0.277109	+	0.960839i	$0.410624\pi$
$240$	0	0
$241$	6.84676	0.441039	0.220519	−	0.975383i	$-0.429225\pi$
$241$	6.84676	0.441039	0.220519	+	0.975383i	$0.429225\pi$
$242$	12.7881	0.822049
$243$	0	0
$244$	−0.692382	−0.0443252
$245$	−8.39234	−0.536167
$246$	0	0
$247$	0.193842	0.0123338
$248$	0	0
$249$	0	0
$250$	−21.3836	−1.35242
$251$	22.6661	1.43067	0.715336	−	0.698781i	$-0.246274\pi$
$251$	22.6661	1.43067	0.715336	+	0.698781i	$0.246274\pi$
$252$	0	0
$253$	4.39983	0.276615
$254$	9.30178	0.583645
$255$	0	0
$256$	−9.10097	−0.568810
$257$	16.6584	1.03912	0.519560	−	0.854434i	$-0.326096\pi$
$257$	16.6584	1.03912	0.519560	+	0.854434i	$0.326096\pi$
$258$	0	0
$259$	8.47500	0.526611
$260$	0.252864	0.0156819
$261$	0	0
$262$	16.5085	1.01990
$263$	−24.5864	−1.51606	−0.758030	−	0.652220i	$-0.773838\pi$
$263$	−24.5864	−1.51606	−0.758030	+	0.652220i	$0.773838\pi$
$264$	0	0
$265$	−27.8675	−1.71189
$266$	3.19634	0.195980
$267$	0	0
$268$	−0.218440	−0.0133434
$269$	12.5141	0.763001	0.381501	−	0.924369i	$-0.375407\pi$
$269$	12.5141	0.763001	0.381501	+	0.924369i	$0.375407\pi$
$270$	0	0
$271$	27.1673	1.65030	0.825148	−	0.564917i	$-0.191092\pi$
$271$	27.1673	1.65030	0.825148	+	0.564917i	$0.191092\pi$
$272$	20.0717	1.21703
$273$	0	0
$274$	0.170104	0.0102763
$275$	−8.97594	−0.541270
$276$	0	0
$277$	15.1573	0.910716	0.455358	−	0.890308i	$-0.349511\pi$
$277$	15.1573	0.910716	0.455358	+	0.890308i	$0.349511\pi$
$278$	14.6007	0.875694
$279$	0	0
$280$	25.2430	1.50855
$281$	−30.2477	−1.80443	−0.902214	−	0.431289i	$-0.858059\pi$
$281$	−30.2477	−1.80443	−0.902214	+	0.431289i	$0.858059\pi$
$282$	0	0
$283$	3.14532	0.186970	0.0934850	−	0.995621i	$-0.470199\pi$
$283$	3.14532	0.186970	0.0934850	+	0.995621i	$0.470199\pi$
$284$	0.449930	0.0266984
$285$	0	0
$286$	−0.202448	−0.0119710
$287$	−0.718432	−0.0424077
$288$	0	0
$289$	26.2524	1.54426
$290$	−6.43427	−0.377833
$291$	0	0
$292$	−3.13514	−0.183470
$293$	−1.90008	−0.111004	−0.0555020	−	0.998459i	$-0.517676\pi$
$293$	−1.90008	−0.111004	−0.0555020	+	0.998459i	$0.517676\pi$
$294$	0	0
$295$	10.0827	0.587038
$296$	11.7482	0.682851
$297$	0	0
$298$	−6.89168	−0.399224
$299$	0.778247	0.0450072
$300$	0	0
$301$	−21.0812	−1.21510
$302$	17.3387	0.997733
$303$	0	0
$304$	3.51843	0.201796
$305$	6.64932	0.380739
$306$	0	0
$307$	22.7281	1.29716	0.648580	−	0.761146i	$-0.275363\pi$
$307$	22.7281	1.29716	0.648580	+	0.761146i	$0.275363\pi$
$308$	0.823433	0.0469194
$309$	0	0
$310$	0	0
$311$	−15.8754	−0.900213	−0.450106	−	0.892975i	$-0.648614\pi$
$311$	−15.8754	−0.900213	−0.450106	+	0.892975i	$0.648614\pi$
$312$	0	0
$313$	−6.34935	−0.358886	−0.179443	−	0.983768i	$-0.557430\pi$
$313$	−6.34935	−0.358886	−0.179443	+	0.983768i	$0.557430\pi$
$314$	19.0691	1.07613
$315$	0	0
$316$	1.79747	0.101116
$317$	15.3010	0.859390	0.429695	−	0.902974i	$-0.358621\pi$
$317$	15.3010	0.859390	0.429695	+	0.902974i	$0.358621\pi$
$318$	0	0
$319$	−1.27068	−0.0711442
$320$	33.8021	1.88959
$321$	0	0
$322$	12.8329	0.715148
$323$	7.58184	0.421865
$324$	0	0
$325$	−1.58767	−0.0880683
$326$	−21.5805	−1.19523
$327$	0	0
$328$	−0.995906	−0.0549897
$329$	−12.3428	−0.680480
$330$	0	0
$331$	8.96795	0.492923	0.246462	−	0.969153i	$-0.420732\pi$
$331$	8.96795	0.492923	0.246462	+	0.969153i	$0.420732\pi$
$332$	−0.129162	−0.00708866
$333$	0	0
$334$	20.5167	1.12262
$335$	2.09780	0.114615
$336$	0	0
$337$	4.36877	0.237982	0.118991	−	0.992895i	$-0.462034\pi$
$337$	4.36877	0.237982	0.118991	+	0.992895i	$0.462034\pi$
$338$	16.4300	0.893675
$339$	0	0
$340$	9.89040	0.536382
$341$	0	0
$342$	0	0
$343$	−20.1569	−1.08837
$344$	−29.2232	−1.57561
$345$	0	0
$346$	−2.95272	−0.158739
$347$	24.4053	1.31015	0.655073	−	0.755566i	$-0.272638\pi$
$347$	24.4053	1.31015	0.655073	+	0.755566i	$0.272638\pi$
$348$	0	0
$349$	13.0439	0.698224	0.349112	−	0.937081i	$-0.386483\pi$
$349$	13.0439	0.698224	0.349112	+	0.937081i	$0.386483\pi$
$350$	−26.1799	−1.39937
$351$	0	0
$352$	2.09438	0.111631
$353$	10.8248	0.576146	0.288073	−	0.957608i	$-0.406985\pi$
$353$	10.8248	0.576146	0.288073	+	0.957608i	$0.406985\pi$
$354$	0	0
$355$	−4.32092	−0.229331
$356$	5.82111	0.308518
$357$	0	0
$358$	21.4950	1.13605
$359$	−33.7834	−1.78302	−0.891510	−	0.453001i	$-0.850353\pi$
$359$	−33.7834	−1.78302	−0.891510	+	0.453001i	$0.850353\pi$
$360$	0	0
$361$	−17.6710	−0.930050
$362$	−9.29149	−0.488350
$363$	0	0
$364$	0.145650	0.00763412
$365$	30.1084	1.57595
$366$	0	0
$367$	−22.6313	−1.18135	−0.590673	−	0.806911i	$-0.701138\pi$
$367$	−22.6313	−1.18135	−0.590673	+	0.806911i	$0.701138\pi$
$368$	14.1260	0.736369
$369$	0	0
$370$	−18.6362	−0.968847
$371$	−16.0517	−0.833365
$372$	0	0
$373$	−32.9720	−1.70723	−0.853613	−	0.520908i	$-0.825593\pi$
$373$	−32.9720	−1.70723	−0.853613	+	0.520908i	$0.825593\pi$
$374$	−7.91846	−0.409454
$375$	0	0
$376$	−17.1098	−0.882372
$377$	−0.224759	−0.0115757
$378$	0	0
$379$	−32.4919	−1.66900	−0.834498	−	0.551012i	$-0.814242\pi$
$379$	−32.4919	−1.66900	−0.834498	+	0.551012i	$0.814242\pi$
$380$	1.73372	0.0889379
$381$	0	0
$382$	9.90832	0.506954
$383$	−26.7804	−1.36842	−0.684208	−	0.729287i	$-0.739852\pi$
$383$	−26.7804	−1.36842	−0.684208	+	0.729287i	$0.739852\pi$
$384$	0	0
$385$	−7.90787	−0.403023
$386$	5.86627	0.298585
$387$	0	0
$388$	−6.14126	−0.311775
$389$	−17.7375	−0.899327	−0.449663	−	0.893198i	$-0.648456\pi$
$389$	−17.7375	−0.899327	−0.449663	+	0.893198i	$0.648456\pi$
$390$	0	0
$391$	30.4400	1.53942
$392$	−6.70099	−0.338451
$393$	0	0
$394$	28.3521	1.42836
$395$	−17.2621	−0.868552
$396$	0	0
$397$	9.94953	0.499352	0.249676	−	0.968329i	$-0.419676\pi$
$397$	9.94953	0.499352	0.249676	+	0.968329i	$0.419676\pi$
$398$	−33.6964	−1.68905
$399$	0	0
$400$	−28.8180	−1.44090
$401$	19.8956	0.993538	0.496769	−	0.867883i	$-0.334520\pi$
$401$	19.8956	0.993538	0.496769	+	0.867883i	$0.334520\pi$
$402$	0	0
$403$	0	0
$404$	1.92858	0.0959506
$405$	0	0
$406$	−3.70615	−0.183933
$407$	−3.68037	−0.182429
$408$	0	0
$409$	24.3255	1.20282	0.601410	−	0.798941i	$-0.294606\pi$
$409$	24.3255	1.20282	0.601410	+	0.798941i	$0.294606\pi$
$410$	1.57980	0.0780208
$411$	0	0
$412$	−0.763986	−0.0376389
$413$	5.80765	0.285776
$414$	0	0
$415$	1.24041	0.0608893
$416$	0.370456	0.0181631
$417$	0	0
$418$	−1.38805	−0.0678918
$419$	4.40675	0.215284	0.107642	−	0.994190i	$-0.465670\pi$
$419$	4.40675	0.215284	0.107642	+	0.994190i	$0.465670\pi$
$420$	0	0
$421$	−12.4237	−0.605493	−0.302746	−	0.953071i	$-0.597904\pi$
$421$	−12.4237	−0.605493	−0.302746	+	0.953071i	$0.597904\pi$
$422$	−1.67969	−0.0817660
$423$	0	0
$424$	−22.2513	−1.08062
$425$	−62.0997	−3.01228
$426$	0	0
$427$	3.83002	0.185348
$428$	4.95739	0.239625
$429$	0	0
$430$	46.3567	2.23552
$431$	18.5653	0.894259	0.447130	−	0.894469i	$-0.352446\pi$
$431$	18.5653	0.894259	0.447130	+	0.894469i	$0.352446\pi$
$432$	0	0
$433$	36.1204	1.73584	0.867918	−	0.496708i	$-0.165458\pi$
$433$	36.1204	1.73584	0.867918	+	0.496708i	$0.165458\pi$
$434$	0	0
$435$	0	0
$436$	4.06108	0.194490
$437$	5.33593	0.255252
$438$	0	0
$439$	−19.0094	−0.907268	−0.453634	−	0.891188i	$-0.649873\pi$
$439$	−19.0094	−0.907268	−0.453634	+	0.891188i	$0.649873\pi$
$440$	−10.9621	−0.522596
$441$	0	0
$442$	−1.40063	−0.0666210
$443$	−12.4540	−0.591709	−0.295854	−	0.955233i	$-0.595604\pi$
$443$	−12.4540	−0.591709	−0.295854	+	0.955233i	$0.595604\pi$
$444$	0	0
$445$	−55.9033	−2.65007
$446$	−15.3511	−0.726898
$447$	0	0
$448$	19.4700	0.919873
$449$	25.4366	1.20043	0.600214	−	0.799840i	$-0.295082\pi$
$449$	25.4366	1.20043	0.600214	+	0.799840i	$0.295082\pi$
$450$	0	0
$451$	0.311988	0.0146909
$452$	−1.52073	−0.0715292
$453$	0	0
$454$	−20.1417	−0.945297
$455$	−1.39875	−0.0655746
$456$	0	0
$457$	−31.9740	−1.49568	−0.747839	−	0.663880i	$-0.768909\pi$
$457$	−31.9740	−1.49568	−0.747839	+	0.663880i	$0.768909\pi$
$458$	−20.6438	−0.964623
$459$	0	0
$460$	6.96064	0.324542
$461$	19.3501	0.901223	0.450612	−	0.892720i	$-0.351206\pi$
$461$	19.3501	0.901223	0.450612	+	0.892720i	$0.351206\pi$
$462$	0	0
$463$	−0.0414625	−0.00192693	−0.000963463	−	1.00000i	$-0.500307\pi$
$463$	−0.0414625	−0.00192693	−0.000963463		1.00000i	$0.500307\pi$
$464$	−4.07961	−0.189391
$465$	0	0
$466$	−6.48875	−0.300586
$467$	−25.3083	−1.17113	−0.585564	−	0.810626i	$-0.699127\pi$
$467$	−25.3083	−1.17113	−0.585564	+	0.810626i	$0.699127\pi$
$468$	0	0
$469$	1.20834	0.0557958
$470$	27.1412	1.25193
$471$	0	0
$472$	8.05069	0.370563
$473$	9.15478	0.420937
$474$	0	0
$475$	−10.8856	−0.499467
$476$	5.69688	0.261116
$477$	0	0
$478$	−10.8522	−0.496370
$479$	21.2411	0.970533	0.485266	−	0.874366i	$-0.338723\pi$
$479$	21.2411	0.970533	0.485266	+	0.874366i	$0.338723\pi$
$480$	0	0
$481$	−0.650988	−0.0296825
$482$	−8.67212	−0.395004
$483$	0	0
$484$	3.99534	0.181606
$485$	58.9779	2.67805
$486$	0	0
$487$	−26.7620	−1.21270	−0.606350	−	0.795198i	$-0.707367\pi$
$487$	−26.7620	−1.21270	−0.606350	+	0.795198i	$0.707367\pi$
$488$	5.30925	0.240338
$489$	0	0
$490$	10.6298	0.480203
$491$	−12.5664	−0.567113	−0.283557	−	0.958955i	$-0.591514\pi$
$491$	−12.5664	−0.567113	−0.283557	+	0.958955i	$0.591514\pi$
$492$	0	0
$493$	−8.79111	−0.395932
$494$	−0.245520	−0.0110465
$495$	0	0
$496$	0	0
$497$	−2.48886	−0.111640
$498$	0	0
$499$	23.0304	1.03098	0.515490	−	0.856895i	$-0.327610\pi$
$499$	23.0304	1.03098	0.515490	+	0.856895i	$0.327610\pi$
$500$	−6.68083	−0.298776
$501$	0	0
$502$	−28.7089	−1.28134
$503$	−0.661578	−0.0294983	−0.0147492	−	0.999891i	$-0.504695\pi$
$503$	−0.661578	−0.0294983	−0.0147492	+	0.999891i	$0.504695\pi$
$504$	0	0
$505$	−18.5212	−0.824184
$506$	−5.57283	−0.247743
$507$	0	0
$508$	2.90613	0.128939
$509$	−11.6674	−0.517149	−0.258574	−	0.965991i	$-0.583253\pi$
$509$	−11.6674	−0.517149	−0.258574	+	0.965991i	$0.583253\pi$
$510$	0	0
$511$	17.3425	0.767186
$512$	25.2461	1.11573
$513$	0	0
$514$	−21.0995	−0.930659
$515$	7.33697	0.323306
$516$	0	0
$517$	5.36001	0.235733
$518$	−10.7344	−0.471644
$519$	0	0
$520$	−1.93898	−0.0850300
$521$	31.9184	1.39837	0.699186	−	0.714940i	$-0.253546\pi$
$521$	31.9184	1.39837	0.699186	+	0.714940i	$0.253546\pi$
$522$	0	0
$523$	0.00415040	0.000181484 0	9.07421e−5	−	1.00000i	$-0.499971\pi$
$523$	0.00415040	0.000181484 0	9.07421e−5		1.00000i	$0.499971\pi$
$524$	5.15769	0.225315
$525$	0	0
$526$	31.1411	1.35782
$527$	0	0
$528$	0	0
$529$	−1.57699	−0.0685646
$530$	35.2971	1.53321
$531$	0	0
$532$	0.998624	0.0432958
$533$	0.0551848	0.00239032
$534$	0	0
$535$	−47.6086	−2.05830
$536$	1.67502	0.0723499
$537$	0	0
$538$	−15.8504	−0.683361
$539$	2.09922	0.0904200
$540$	0	0
$541$	29.2750	1.25863	0.629316	−	0.777150i	$-0.283335\pi$
$541$	29.2750	1.25863	0.629316	+	0.777150i	$0.283335\pi$
$542$	−34.4101	−1.47804
$543$	0	0
$544$	14.4898	0.621247
$545$	−39.0007	−1.67061
$546$	0	0
$547$	41.1653	1.76010	0.880050	−	0.474880i	$-0.157509\pi$
$547$	41.1653	1.76010	0.880050	+	0.474880i	$0.157509\pi$
$548$	0.0531450	0.00227024
$549$	0	0
$550$	11.3689	0.484773
$551$	−1.54102	−0.0656497
$552$	0	0
$553$	−9.94300	−0.422820
$554$	−19.1983	−0.815658
$555$	0	0
$556$	4.56167	0.193458
$557$	−5.73810	−0.243131	−0.121566	−	0.992583i	$-0.538791\pi$
$557$	−5.73810	−0.243131	−0.121566	+	0.992583i	$0.538791\pi$
$558$	0	0
$559$	1.61931	0.0684894
$560$	−25.3889	−1.07288
$561$	0	0
$562$	38.3118	1.61609
$563$	14.2942	0.602428	0.301214	−	0.953557i	$-0.402608\pi$
$563$	14.2942	0.602428	0.301214	+	0.953557i	$0.402608\pi$
$564$	0	0
$565$	14.6044	0.614412
$566$	−3.98387	−0.167455
$567$	0	0
$568$	−3.45011	−0.144763
$569$	14.3618	0.602078	0.301039	−	0.953612i	$-0.402667\pi$
$569$	14.3618	0.602078	0.301039	+	0.953612i	$0.402667\pi$
$570$	0	0
$571$	−33.3790	−1.39687	−0.698434	−	0.715674i	$-0.746119\pi$
$571$	−33.3790	−1.39687	−0.698434	+	0.715674i	$0.746119\pi$
$572$	−0.0632502	−0.00264462
$573$	0	0
$574$	0.909967	0.0379813
$575$	−43.7043	−1.82260
$576$	0	0
$577$	−22.4796	−0.935838	−0.467919	−	0.883771i	$-0.654996\pi$
$577$	−22.4796	−0.935838	−0.467919	+	0.883771i	$0.654996\pi$
$578$	−33.2513	−1.38307
$579$	0	0
$580$	−2.01024	−0.0834706
$581$	0.714477	0.0296415
$582$	0	0
$583$	6.97067	0.288695
$584$	24.0405	0.994804
$585$	0	0
$586$	2.40665	0.0994176
$587$	−6.41617	−0.264824	−0.132412	−	0.991195i	$-0.542272\pi$
$587$	−6.41617	−0.264824	−0.132412	+	0.991195i	$0.542272\pi$
$588$	0	0
$589$	0	0
$590$	−12.7708	−0.525764
$591$	0	0
$592$	−11.8161	−0.485640
$593$	20.8601	0.856623	0.428312	−	0.903631i	$-0.359109\pi$
$593$	20.8601	0.856623	0.428312	+	0.903631i	$0.359109\pi$
$594$	0	0
$595$	−54.7103	−2.24290
$596$	−2.15315	−0.0881963
$597$	0	0
$598$	−0.985729	−0.0403095
$599$	11.5584	0.472263	0.236131	−	0.971721i	$-0.424120\pi$
$599$	11.5584	0.472263	0.236131	+	0.971721i	$0.424120\pi$
$600$	0	0
$601$	−4.31742	−0.176111	−0.0880555	−	0.996116i	$-0.528065\pi$
$601$	−4.31742	−0.176111	−0.0880555	+	0.996116i	$0.528065\pi$
$602$	26.7015	1.08827
$603$	0	0
$604$	5.41709	0.220418
$605$	−38.3694	−1.55994
$606$	0	0
$607$	44.9092	1.82281	0.911405	−	0.411510i	$-0.134999\pi$
$607$	44.9092	1.82281	0.911405	+	0.411510i	$0.134999\pi$
$608$	2.53997	0.103009
$609$	0	0
$610$	−8.42204	−0.340998
$611$	0.948084	0.0383554
$612$	0	0
$613$	2.60516	0.105222	0.0526108	−	0.998615i	$-0.483246\pi$
$613$	2.60516	0.105222	0.0526108	+	0.998615i	$0.483246\pi$
$614$	−28.7874	−1.16177
$615$	0	0
$616$	−6.31416	−0.254405
$617$	24.9868	1.00593	0.502965	−	0.864307i	$-0.332242\pi$
$617$	24.9868	1.00593	0.502965	+	0.864307i	$0.332242\pi$
$618$	0	0
$619$	−31.9083	−1.28250	−0.641252	−	0.767330i	$-0.721585\pi$
$619$	−31.9083	−1.28250	−0.641252	+	0.767330i	$0.721585\pi$
$620$	0	0
$621$	0	0
$622$	20.1078	0.806251
$623$	−32.2004	−1.29008
$624$	0	0
$625$	16.9474	0.677898
$626$	8.04209	0.321427
$627$	0	0
$628$	5.95771	0.237739
$629$	−25.4625	−1.01526
$630$	0	0
$631$	0.359751	0.0143215	0.00716073	−	0.999974i	$-0.497721\pi$
$631$	0.359751	0.0143215	0.00716073	+	0.999974i	$0.497721\pi$
$632$	−13.7832	−0.548266
$633$	0	0
$634$	−19.3803	−0.769689
$635$	−27.9091	−1.10754
$636$	0	0
$637$	0.371313	0.0147120
$638$	1.60944	0.0637183
$639$	0	0
$640$	−26.0679	−1.03042
$641$	−30.8069	−1.21680	−0.608400	−	0.793630i	$-0.708188\pi$
$641$	−30.8069	−1.21680	−0.608400	+	0.793630i	$0.708188\pi$
$642$	0	0
$643$	1.75184	0.0690856	0.0345428	−	0.999403i	$-0.489002\pi$
$643$	1.75184	0.0690856	0.0345428	+	0.999403i	$0.489002\pi$
$644$	4.00934	0.157990
$645$	0	0
$646$	−9.60317	−0.377832
$647$	−24.6504	−0.969106	−0.484553	−	0.874762i	$-0.661018\pi$
$647$	−24.6504	−0.969106	−0.484553	+	0.874762i	$0.661018\pi$
$648$	0	0
$649$	−2.52204	−0.0989989
$650$	2.01095	0.0788760
$651$	0	0
$652$	−6.74233	−0.264050
$653$	0.285664	0.0111789	0.00558946	−	0.999984i	$-0.498221\pi$
$653$	0.285664	0.0111789	0.00558946	+	0.999984i	$0.498221\pi$
$654$	0	0
$655$	−49.5321	−1.93538
$656$	1.00166	0.0391083
$657$	0	0
$658$	15.6334	0.609453
$659$	−37.4247	−1.45786	−0.728929	−	0.684589i	$-0.759982\pi$
$659$	−37.4247	−1.45786	−0.728929	+	0.684589i	$0.759982\pi$
$660$	0	0
$661$	4.19157	0.163033	0.0815167	−	0.996672i	$-0.474024\pi$
$661$	4.19157	0.163033	0.0815167	+	0.996672i	$0.474024\pi$
$662$	−11.3588	−0.441473
$663$	0	0
$664$	0.990423	0.0384359
$665$	−9.59033	−0.371897
$666$	0	0
$667$	−6.18699	−0.239561
$668$	6.40998	0.248009
$669$	0	0
$670$	−2.65708	−0.102652
$671$	−1.66323	−0.0642084
$672$	0	0
$673$	−9.97267	−0.384418	−0.192209	−	0.981354i	$-0.561565\pi$
$673$	−9.97267	−0.384418	−0.192209	+	0.981354i	$0.561565\pi$
$674$	−5.53349	−0.213142
$675$	0	0
$676$	5.13318	0.197430
$677$	−47.7577	−1.83548	−0.917738	−	0.397186i	$-0.869987\pi$
$677$	−47.7577	−1.83548	−0.917738	+	0.397186i	$0.869987\pi$
$678$	0	0
$679$	33.9713	1.30370
$680$	−75.8406	−2.90835
$681$	0	0
$682$	0	0
$683$	−27.7600	−1.06221	−0.531104	−	0.847307i	$-0.678223\pi$
$683$	−27.7600	−1.06221	−0.531104	+	0.847307i	$0.678223\pi$
$684$	0	0
$685$	−0.510380	−0.0195006
$686$	25.5308	0.974770
$687$	0	0
$688$	29.3922	1.12057
$689$	1.23298	0.0469728
$690$	0	0
$691$	38.0310	1.44677	0.723383	−	0.690447i	$-0.242586\pi$
$691$	38.0310	1.44677	0.723383	+	0.690447i	$0.242586\pi$
$692$	−0.922511	−0.0350686
$693$	0	0
$694$	−30.9118	−1.17340
$695$	−43.8082	−1.66174
$696$	0	0
$697$	2.15848	0.0817581
$698$	−16.5214	−0.625346
$699$	0	0
$700$	−8.17931	−0.309149
$701$	−41.0562	−1.55067	−0.775336	−	0.631548i	$-0.782420\pi$
$701$	−41.0562	−1.55067	−0.775336	+	0.631548i	$0.782420\pi$
$702$	0	0
$703$	−4.46340	−0.168340
$704$	−8.45511	−0.318664
$705$	0	0
$706$	−13.7107	−0.516010
$707$	−10.6683	−0.401221
$708$	0	0
$709$	−42.0657	−1.57981	−0.789906	−	0.613229i	$-0.789870\pi$
$709$	−42.0657	−1.57981	−0.789906	+	0.613229i	$0.789870\pi$
$710$	5.47289	0.205394
$711$	0	0
$712$	−44.6368	−1.67284
$713$	0	0
$714$	0	0
$715$	0.607426	0.0227164
$716$	6.71563	0.250975
$717$	0	0
$718$	42.7901	1.59691
$719$	−33.2690	−1.24072	−0.620362	−	0.784316i	$-0.713014\pi$
$719$	−33.2690	−1.24072	−0.620362	+	0.784316i	$0.713014\pi$
$720$	0	0
$721$	4.22611	0.157388
$722$	22.3821	0.832974
$723$	0	0
$724$	−2.90291	−0.107886
$725$	12.6218	0.468764
$726$	0	0
$727$	−32.7229	−1.21363	−0.606813	−	0.794845i	$-0.707552\pi$
$727$	−32.7229	−1.21363	−0.606813	+	0.794845i	$0.707552\pi$
$728$	−1.11686	−0.0413934
$729$	0	0
$730$	−38.1354	−1.41145
$731$	63.3370	2.34260
$732$	0	0
$733$	−9.96391	−0.368025	−0.184013	−	0.982924i	$-0.558909\pi$
$733$	−9.96391	−0.368025	−0.184013	+	0.982924i	$0.558909\pi$
$734$	28.6649	1.05804
$735$	0	0
$736$	10.1976	0.375889
$737$	−0.524735	−0.0193289
$738$	0	0
$739$	−30.9584	−1.13882	−0.569412	−	0.822052i	$-0.692829\pi$
$739$	−30.9584	−1.13882	−0.569412	+	0.822052i	$0.692829\pi$
$740$	−5.82244	−0.214037
$741$	0	0
$742$	20.3312	0.746380
$743$	1.11003	0.0407231	0.0203615	−	0.999793i	$-0.493518\pi$
$743$	1.11003	0.0407231	0.0203615	+	0.999793i	$0.493518\pi$
$744$	0	0
$745$	20.6778	0.757578
$746$	41.7624	1.52903
$747$	0	0
$748$	−2.47394	−0.0904563
$749$	−27.4226	−1.00200
$750$	0	0
$751$	28.1553	1.02740	0.513701	−	0.857969i	$-0.328274\pi$
$751$	28.1553	1.02740	0.513701	+	0.857969i	$0.328274\pi$
$752$	17.2087	0.627538
$753$	0	0
$754$	0.284679	0.0103674
$755$	−52.0233	−1.89332
$756$	0	0
$757$	−38.7223	−1.40739	−0.703693	−	0.710504i	$-0.748467\pi$
$757$	−38.7223	−1.40739	−0.703693	+	0.710504i	$0.748467\pi$
$758$	41.1542	1.49479
$759$	0	0
$760$	−13.2943	−0.482236
$761$	43.4369	1.57459	0.787293	−	0.616580i	$-0.211482\pi$
$761$	43.4369	1.57459	0.787293	+	0.616580i	$0.211482\pi$
$762$	0	0
$763$	−22.4645	−0.813269
$764$	3.09563	0.111996
$765$	0	0
$766$	33.9201	1.22558
$767$	−0.446102	−0.0161078
$768$	0	0
$769$	−3.11019	−0.112156	−0.0560781	−	0.998426i	$-0.517860\pi$
$769$	−3.11019	−0.112156	−0.0560781	+	0.998426i	$0.517860\pi$
$770$	10.0161	0.360956
$771$	0	0
$772$	1.83278	0.0659633
$773$	21.6402	0.778343	0.389172	−	0.921165i	$-0.372761\pi$
$773$	21.6402	0.778343	0.389172	+	0.921165i	$0.372761\pi$
$774$	0	0
$775$	0	0
$776$	47.0918	1.69050
$777$	0	0
$778$	22.4663	0.805457
$779$	0.378366	0.0135564
$780$	0	0
$781$	1.08082	0.0386747
$782$	−38.5554	−1.37874
$783$	0	0
$784$	6.73973	0.240705
$785$	−57.2151	−2.04210
$786$	0	0
$787$	7.87653	0.280768	0.140384	−	0.990097i	$-0.455166\pi$
$787$	7.87653	0.280768	0.140384	+	0.990097i	$0.455166\pi$
$788$	8.85796	0.315552
$789$	0	0
$790$	21.8642	0.777894
$791$	8.41216	0.299102
$792$	0	0
$793$	−0.294194	−0.0104471
$794$	−12.6021	−0.447231
$795$	0	0
$796$	−10.5277	−0.373143
$797$	29.5182	1.04559	0.522793	−	0.852459i	$-0.324890\pi$
$797$	29.5182	1.04559	0.522793	+	0.852459i	$0.324890\pi$
$798$	0	0
$799$	37.0830	1.31190
$800$	−20.8038	−0.735526
$801$	0	0
$802$	−25.1998	−0.889835
$803$	−7.53119	−0.265770
$804$	0	0
$805$	−38.5039	−1.35708
$806$	0	0
$807$	0	0
$808$	−14.7886	−0.520260
$809$	−2.12753	−0.0747998	−0.0373999	−	0.999300i	$-0.511908\pi$
$809$	−2.12753	−0.0747998	−0.0373999	+	0.999300i	$0.511908\pi$
$810$	0	0
$811$	7.20504	0.253003	0.126502	−	0.991966i	$-0.459625\pi$
$811$	7.20504	0.253003	0.126502	+	0.991966i	$0.459625\pi$
$812$	−1.15790	−0.0406343
$813$	0	0
$814$	4.66157	0.163388
$815$	64.7502	2.26810
$816$	0	0
$817$	11.1025	0.388428
$818$	−30.8107	−1.07727
$819$	0	0
$820$	0.493572	0.0172363
$821$	42.0597	1.46789	0.733947	−	0.679206i	$-0.237676\pi$
$821$	42.0597	1.46789	0.733947	+	0.679206i	$0.237676\pi$
$822$	0	0
$823$	30.1730	1.05176	0.525882	−	0.850558i	$-0.323735\pi$
$823$	30.1730	1.05176	0.525882	+	0.850558i	$0.323735\pi$
$824$	5.85832	0.204084
$825$	0	0
$826$	−7.35598	−0.255947
$827$	−3.87046	−0.134589	−0.0672945	−	0.997733i	$-0.521437\pi$
$827$	−3.87046	−0.134589	−0.0672945	+	0.997733i	$0.521437\pi$
$828$	0	0
$829$	−13.1716	−0.457469	−0.228735	−	0.973489i	$-0.573459\pi$
$829$	−13.1716	−0.457469	−0.228735	+	0.973489i	$0.573459\pi$
$830$	−1.57110	−0.0545338
$831$	0	0
$832$	−1.49555	−0.0518488
$833$	14.5234	0.503206
$834$	0	0
$835$	−61.5585	−2.13032
$836$	−0.433665	−0.0149986
$837$	0	0
$838$	−5.58159	−0.192813
$839$	34.2490	1.18241	0.591204	−	0.806522i	$-0.298653\pi$
$839$	34.2490	1.18241	0.591204	+	0.806522i	$0.298653\pi$
$840$	0	0
$841$	−27.2132	−0.938386
$842$	15.7358	0.542293
$843$	0	0
$844$	−0.524780	−0.0180637
$845$	−49.2967	−1.69586
$846$	0	0
$847$	−22.1008	−0.759394
$848$	22.3799	0.768528
$849$	0	0
$850$	78.6555	2.69786
$851$	−17.9199	−0.614287
$852$	0	0
$853$	−14.4479	−0.494686	−0.247343	−	0.968928i	$-0.579557\pi$
$853$	−14.4479	−0.494686	−0.247343	+	0.968928i	$0.579557\pi$
$854$	−4.85111	−0.166001
$855$	0	0
$856$	−38.0138	−1.29928
$857$	−27.8331	−0.950759	−0.475379	−	0.879781i	$-0.657689\pi$
$857$	−27.8331	−0.950759	−0.475379	+	0.879781i	$0.657689\pi$
$858$	0	0
$859$	47.3202	1.61454	0.807272	−	0.590180i	$-0.200943\pi$
$859$	47.3202	1.61454	0.807272	+	0.590180i	$0.200943\pi$
$860$	14.4831	0.493869
$861$	0	0
$862$	−23.5148	−0.800919
$863$	26.0534	0.886867	0.443434	−	0.896307i	$-0.353760\pi$
$863$	26.0534	0.886867	0.443434	+	0.896307i	$0.353760\pi$
$864$	0	0
$865$	8.85937	0.301228
$866$	−45.7502	−1.55465
$867$	0	0
$868$	0	0
$869$	4.31787	0.146474
$870$	0	0
$871$	−0.0928157	−0.00314494
$872$	−31.1407	−1.05456
$873$	0	0
$874$	−6.75849	−0.228609
$875$	36.9560	1.24934
$876$	0	0
$877$	−34.3215	−1.15895	−0.579477	−	0.814989i	$-0.696743\pi$
$877$	−34.3215	−1.15895	−0.579477	+	0.814989i	$0.696743\pi$
$878$	24.0773	0.812570
$879$	0	0
$880$	11.0254	0.371667
$881$	−3.63467	−0.122455	−0.0612276	−	0.998124i	$-0.519502\pi$
$881$	−3.63467	−0.122455	−0.0612276	+	0.998124i	$0.519502\pi$
$882$	0	0
$883$	−39.4574	−1.32785	−0.663923	−	0.747801i	$-0.731110\pi$
$883$	−39.4574	−1.32785	−0.663923	+	0.747801i	$0.731110\pi$
$884$	−0.437594	−0.0147179
$885$	0	0
$886$	15.7743	0.529948
$887$	42.3027	1.42039	0.710194	−	0.704007i	$-0.248607\pi$
$887$	42.3027	1.42039	0.710194	+	0.704007i	$0.248607\pi$
$888$	0	0
$889$	−16.0757	−0.539161
$890$	70.8072	2.37346
$891$	0	0
$892$	−4.79611	−0.160586
$893$	6.50038	0.217527
$894$	0	0
$895$	−64.4938	−2.15579
$896$	−15.0151	−0.501620
$897$	0	0
$898$	−32.2180	−1.07513
$899$	0	0
$900$	0	0
$901$	48.2262	1.60665
$902$	−0.395164	−0.0131575
$903$	0	0
$904$	11.6611	0.387843
$905$	27.8782	0.926704
$906$	0	0
$907$	−32.3680	−1.07476	−0.537381	−	0.843339i	$-0.680586\pi$
$907$	−32.3680	−1.07476	−0.537381	+	0.843339i	$0.680586\pi$
$908$	−6.29281	−0.208834
$909$	0	0
$910$	1.77166	0.0587301
$911$	−27.0607	−0.896563	−0.448281	−	0.893893i	$-0.647964\pi$
$911$	−27.0607	−0.896563	−0.448281	+	0.893893i	$0.647964\pi$
$912$	0	0
$913$	−0.310271	−0.0102685
$914$	40.4983	1.33956
$915$	0	0
$916$	−6.44969	−0.213104
$917$	−28.5306	−0.942162
$918$	0	0
$919$	20.7649	0.684972	0.342486	−	0.939523i	$-0.388731\pi$
$919$	20.7649	0.684972	0.342486	+	0.939523i	$0.388731\pi$
$920$	−53.3749	−1.75972
$921$	0	0
$922$	−24.5088	−0.807156
$923$	0.191176	0.00629264
$924$	0	0
$925$	36.5578	1.20201
$926$	0.0525165	0.00172580
$927$	0	0
$928$	−2.94508	−0.0966771
$929$	−20.6589	−0.677798	−0.338899	−	0.940823i	$-0.610055\pi$
$929$	−20.6589	−0.677798	−0.338899	+	0.940823i	$0.610055\pi$
$930$	0	0
$931$	2.54585	0.0834368
$932$	−2.02726	−0.0664052
$933$	0	0
$934$	32.0555	1.04889
$935$	23.7586	0.776990
$936$	0	0
$937$	2.92060	0.0954119	0.0477059	−	0.998861i	$-0.484809\pi$
$937$	2.92060	0.0954119	0.0477059	+	0.998861i	$0.484809\pi$
$938$	−1.53048	−0.0499720
$939$	0	0
$940$	8.47966	0.276576
$941$	32.8501	1.07088	0.535441	−	0.844573i	$-0.320145\pi$
$941$	32.8501	1.07088	0.535441	+	0.844573i	$0.320145\pi$
$942$	0	0
$943$	1.51909	0.0494682
$944$	−8.09722	−0.263542
$945$	0	0
$946$	−11.5955	−0.377001
$947$	39.1247	1.27138	0.635692	−	0.771943i	$-0.280715\pi$
$947$	39.1247	1.27138	0.635692	+	0.771943i	$0.280715\pi$
$948$	0	0
$949$	−1.33212	−0.0432426
$950$	13.7878	0.447334
$951$	0	0
$952$	−43.6843	−1.41581
$953$	6.01975	0.194999	0.0974994	−	0.995236i	$-0.468916\pi$
$953$	6.01975	0.194999	0.0974994	+	0.995236i	$0.468916\pi$
$954$	0	0
$955$	−29.7290	−0.962008
$956$	−3.39053	−0.109658
$957$	0	0
$958$	−26.9041	−0.869231
$959$	−0.293980	−0.00949310
$960$	0	0
$961$	0	0
$962$	0.824543	0.0265843
$963$	0	0
$964$	−2.70941	−0.0872641
$965$	−17.6012	−0.566603
$966$	0	0
$967$	−26.3103	−0.846083	−0.423042	−	0.906110i	$-0.639038\pi$
$967$	−26.3103	−0.846083	−0.423042	+	0.906110i	$0.639038\pi$
$968$	−30.6367	−0.984699
$969$	0	0
$970$	−74.7015	−2.39852
$971$	4.73587	0.151981	0.0759906	−	0.997109i	$-0.475788\pi$
$971$	4.73587	0.151981	0.0759906	+	0.997109i	$0.475788\pi$
$972$	0	0
$973$	−25.2336	−0.808951
$974$	33.8967	1.08612
$975$	0	0
$976$	−5.33994	−0.170927
$977$	17.2800	0.552837	0.276419	−	0.961037i	$-0.410852\pi$
$977$	17.2800	0.552837	0.276419	+	0.961037i	$0.410852\pi$
$978$	0	0
$979$	13.9834	0.446912
$980$	3.32102	0.106086
$981$	0	0
$982$	15.9166	0.507920
$983$	−40.9665	−1.30663	−0.653315	−	0.757087i	$-0.726622\pi$
$983$	−40.9665	−1.30663	−0.653315	+	0.757087i	$0.726622\pi$
$984$	0	0
$985$	−85.0678	−2.71049
$986$	11.1348	0.354605
$987$	0	0
$988$	−0.0767071	−0.00244038
$989$	44.5751	1.41741
$990$	0	0
$991$	−14.1338	−0.448975	−0.224487	−	0.974477i	$-0.572071\pi$
$991$	−14.1338	−0.448975	−0.224487	+	0.974477i	$0.572071\pi$
$992$	0	0
$993$	0	0
$994$	3.15239	0.0999877
$995$	101.103	3.20517
$996$	0	0
$997$	15.4538	0.489428	0.244714	−	0.969595i	$-0.421306\pi$
$997$	15.4538	0.489428	0.244714	+	0.969595i	$0.421306\pi$
$998$	−29.1703	−0.923370
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8649.2.a.be.1.3		8
3.2	odd	2		961.2.a.j.1.6		8
31.11	odd	30		279.2.y.c.28.2		16
31.17	odd	30		279.2.y.c.10.2		16
31.30	odd	2		8649.2.a.bf.1.3		8
93.2	odd	10		961.2.d.q.531.3		16
93.5	odd	6		961.2.c.i.521.6		16
93.8	odd	10		961.2.d.n.374.2		16
93.11	even	30		31.2.g.a.28.1	yes	16
93.14	odd	30		961.2.g.l.816.1		16
93.17	even	30		31.2.g.a.10.1	✓	16
93.20	odd	30		961.2.g.l.338.1		16
93.23	even	10		961.2.d.o.374.2		16
93.26	even	6		961.2.c.j.521.6		16
93.29	even	10		961.2.d.p.531.3		16
93.35	odd	10		961.2.d.n.388.2		16
93.38	odd	30		961.2.g.j.235.1		16
93.41	odd	30		961.2.g.n.844.2		16
93.44	even	30		961.2.g.s.448.2		16
93.47	odd	10		961.2.d.q.628.3		16
93.50	odd	30		961.2.g.m.547.2		16
93.53	even	30		961.2.g.k.732.1		16
93.56	odd	6		961.2.c.i.439.6		16
93.59	odd	30		961.2.g.n.846.2		16
93.65	even	30		961.2.g.t.846.2		16
93.68	even	6		961.2.c.j.439.6		16
93.71	odd	30		961.2.g.j.732.1		16
93.74	even	30		961.2.g.s.547.2		16
93.77	even	10		961.2.d.p.628.3		16
93.80	odd	30		961.2.g.m.448.2		16
93.83	even	30		961.2.g.t.844.2		16
93.86	even	30		961.2.g.k.235.1		16
93.89	even	10		961.2.d.o.388.2		16
93.92	even	2		961.2.a.i.1.6		8
372.11	odd	30		496.2.bg.c.369.2		16
372.203	odd	30		496.2.bg.c.289.2		16
465.17	odd	60		775.2.ck.a.599.4		32
465.104	even	30		775.2.bl.a.276.2		16
465.197	odd	60		775.2.ck.a.524.1		32
465.203	odd	60		775.2.ck.a.599.1		32
465.383	odd	60		775.2.ck.a.524.4		32
465.389	even	30		775.2.bl.a.351.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.2.g.a.10.1	✓	16	93.17	even	30
31.2.g.a.28.1	yes	16	93.11	even	30
279.2.y.c.10.2		16	31.17	odd	30
279.2.y.c.28.2		16	31.11	odd	30
496.2.bg.c.289.2		16	372.203	odd	30
496.2.bg.c.369.2		16	372.11	odd	30
775.2.bl.a.276.2		16	465.104	even	30
775.2.bl.a.351.2		16	465.389	even	30
775.2.ck.a.524.1		32	465.197	odd	60
775.2.ck.a.524.4		32	465.383	odd	60
775.2.ck.a.599.1		32	465.203	odd	60
775.2.ck.a.599.4		32	465.17	odd	60
961.2.a.i.1.6		8	93.92	even	2
961.2.a.j.1.6		8	3.2	odd	2
961.2.c.i.439.6		16	93.56	odd	6
961.2.c.i.521.6		16	93.5	odd	6
961.2.c.j.439.6		16	93.68	even	6
961.2.c.j.521.6		16	93.26	even	6
961.2.d.n.374.2		16	93.8	odd	10
961.2.d.n.388.2		16	93.35	odd	10
961.2.d.o.374.2		16	93.23	even	10
961.2.d.o.388.2		16	93.89	even	10
961.2.d.p.531.3		16	93.29	even	10
961.2.d.p.628.3		16	93.77	even	10
961.2.d.q.531.3		16	93.2	odd	10
961.2.d.q.628.3		16	93.47	odd	10
961.2.g.j.235.1		16	93.38	odd	30
961.2.g.j.732.1		16	93.71	odd	30
961.2.g.k.235.1		16	93.86	even	30
961.2.g.k.732.1		16	93.53	even	30
961.2.g.l.338.1		16	93.20	odd	30
961.2.g.l.816.1		16	93.14	odd	30
961.2.g.m.448.2		16	93.80	odd	30
961.2.g.m.547.2		16	93.50	odd	30
961.2.g.n.844.2		16	93.41	odd	30
961.2.g.n.846.2		16	93.59	odd	30
961.2.g.s.448.2		16	93.44	even	30
961.2.g.s.547.2		16	93.74	even	30
961.2.g.t.844.2		16	93.83	even	30
961.2.g.t.846.2		16	93.65	even	30
8649.2.a.be.1.3		8	1.1	even	1	trivial
8649.2.a.bf.1.3		8	31.30	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8649 = 3^{2} \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8649.a (trivial)

\(f(q)\)	\(=\)	\(q-1.26660 q^{2} -0.395721 q^{4} +3.80032 q^{5} +2.18899 q^{7} +3.03442 q^{8} -4.81349 q^{10} -0.950596 q^{11} -0.168142 q^{13} -2.77258 q^{14} -3.05196 q^{16} -6.57666 q^{17} -1.15284 q^{19} -1.50387 q^{20} +1.20403 q^{22} -4.62850 q^{23} +9.44244 q^{25} +0.212969 q^{26} -0.866228 q^{28} +1.33672 q^{29} -2.20322 q^{32} +8.33000 q^{34} +8.31886 q^{35} +3.87165 q^{37} +1.46019 q^{38} +11.5318 q^{40} -0.328203 q^{41} -9.63057 q^{43} +0.376170 q^{44} +5.86247 q^{46} -5.63858 q^{47} -2.20832 q^{49} -11.9598 q^{50} +0.0665374 q^{52} -7.33294 q^{53} -3.61257 q^{55} +6.64232 q^{56} -1.69309 q^{58} +2.65312 q^{59} +1.74967 q^{61} +8.89454 q^{64} -0.638995 q^{65} +0.552007 q^{67} +2.60252 q^{68} -10.5367 q^{70} -1.13699 q^{71} +7.92260 q^{73} -4.90384 q^{74} +0.456203 q^{76} -2.08084 q^{77} -4.54228 q^{79} -11.5984 q^{80} +0.415702 q^{82} +0.326396 q^{83} -24.9934 q^{85} +12.1981 q^{86} -2.88451 q^{88} -14.7102 q^{89} -0.368062 q^{91} +1.83159 q^{92} +7.14183 q^{94} -4.38117 q^{95} +15.5192 q^{97} +2.79707 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 8 q^{4} - 3 q^{5} - 2 q^{7} + 9 q^{8} - 13 q^{10} - 18 q^{11} + 8 q^{13} + 9 q^{14} + 4 q^{16} - 14 q^{17} - 6 q^{19} + 7 q^{20} + 4 q^{22} - 22 q^{23} + 13 q^{25} - 9 q^{26} - 5 q^{28}+ \cdots - 10 q^{98}+O(q^{100}) \) 8 * q - 2 * q^2 + 8 * q^4 - 3 * q^5 - 2 * q^7 + 9 * q^8 - 13 * q^10 - 18 * q^11 + 8 * q^13 + 9 * q^14 + 4 * q^16 - 14 * q^17 - 6 * q^19 + 7 * q^20 + 4 * q^22 - 22 * q^23 + 13 * q^25 - 9 * q^26 - 5 * q^28 - 12 * q^29 + 21 * q^32 - 7 * q^34 + 12 * q^35 - 8 * q^37 - 7 * q^38 + 11 * q^40 + 22 * q^41 - 2 * q^43 - 4 * q^44 + 18 * q^46 + 18 * q^47 - 2 * q^49 - 27 * q^50 - q^52 - 6 * q^53 + 28 * q^55 - 30 * q^56 - 5 * q^58 + 4 * q^59 + 30 * q^61 + 9 * q^64 - 3 * q^65 - 13 * q^67 - 30 * q^68 - 39 * q^70 + q^71 + 2 * q^73 - 8 * q^74 - 33 * q^76 + 24 * q^77 + 8 * q^79 - 24 * q^80 - 19 * q^82 - 39 * q^83 - 26 * q^85 + 16 * q^86 - 17 * q^88 - 27 * q^89 + 16 * q^91 - 32 * q^92 + 22 * q^94 + 6 * q^95 + 34 * q^97 - 10 * q^98

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