LMFDB - Embedded newform 4842.2.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4842,2,Mod(1,4842)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4842, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4842.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4842 = 2 \cdot 3^{2} \cdot 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4842.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:	$38.6635646587$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 33x^{6} + 352x^{4} - 18x^{3} - 1229x^{2} + 178x + 108 $ x^8 - 33x^6 + 352x^4 - 18x^3 - 1229x^2 + 178*x + 108
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1614)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.17461$ of defining polynomial
Character	$\chi$	$=$	4842.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{4} -3.17461 q^{5} -4.10776 q^{7} +1.00000 q^{8} -3.17461 q^{10} -5.68264 q^{11} -6.22695 q^{13} -4.10776 q^{14} +1.00000 q^{16} -5.79856 q^{17} +2.64964 q^{19} -3.17461 q^{20} -5.68264 q^{22} +6.44820 q^{23} +5.07816 q^{25} -6.22695 q^{26} -4.10776 q^{28} -3.80603 q^{29} +2.24492 q^{31} +1.00000 q^{32} -5.79856 q^{34} +13.0405 q^{35} +7.14892 q^{37} +2.64964 q^{38} -3.17461 q^{40} -6.53055 q^{41} +6.32900 q^{43} -5.68264 q^{44} +6.44820 q^{46} -3.49047 q^{47} +9.87371 q^{49} +5.07816 q^{50} -6.22695 q^{52} -7.70122 q^{53} +18.0402 q^{55} -4.10776 q^{56} -3.80603 q^{58} -5.61944 q^{59} -6.40288 q^{61} +2.24492 q^{62} +1.00000 q^{64} +19.7682 q^{65} -4.28369 q^{67} -5.79856 q^{68} +13.0405 q^{70} +13.7436 q^{71} -6.85491 q^{73} +7.14892 q^{74} +2.64964 q^{76} +23.3429 q^{77} +8.62991 q^{79} -3.17461 q^{80} -6.53055 q^{82} +2.78811 q^{83} +18.4082 q^{85} +6.32900 q^{86} -5.68264 q^{88} -0.930456 q^{89} +25.5788 q^{91} +6.44820 q^{92} -3.49047 q^{94} -8.41157 q^{95} +6.12189 q^{97} +9.87371 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 8 q^{4} + 9 q^{7} + 8 q^{8} - 8 q^{11} - 3 q^{13} + 9 q^{14} + 8 q^{16} - 8 q^{17} + 18 q^{19} - 8 q^{22} + 10 q^{23} + 26 q^{25} - 3 q^{26} + 9 q^{28} + 9 q^{29} + 18 q^{31} + 8 q^{32}+ \cdots + 29 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 + 8 * q^4 + 9 * q^7 + 8 * q^8 - 8 * q^11 - 3 * q^13 + 9 * q^14 + 8 * q^16 - 8 * q^17 + 18 * q^19 - 8 * q^22 + 10 * q^23 + 26 * q^25 - 3 * q^26 + 9 * q^28 + 9 * q^29 + 18 * q^31 + 8 * q^32 - 8 * q^34 - 9 * q^35 + 22 * q^37 + 18 * q^38 + q^41 + 14 * q^43 - 8 * q^44 + 10 * q^46 - 6 * q^47 + 29 * q^49 + 26 * q^50 - 3 * q^52 + 3 * q^53 + 17 * q^55 + 9 * q^56 + 9 * q^58 - q^59 + 18 * q^61 + 18 * q^62 + 8 * q^64 + 14 * q^65 + 30 * q^67 - 8 * q^68 - 9 * q^70 + 5 * q^71 + 15 * q^73 + 22 * q^74 + 18 * q^76 + 20 * q^77 + 21 * q^79 + q^82 + 11 * q^83 + 18 * q^85 + 14 * q^86 - 8 * q^88 - 22 * q^89 + 53 * q^91 + 10 * q^92 - 6 * q^94 + 9 * q^95 + 13 * q^97 + 29 * q^98

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−3.17461	−1.41973	−0.709865	−	0.704338i	$-0.751244\pi$
$5$	−3.17461	−1.41973	−0.709865	+	0.704338i	$0.751244\pi$
$6$	0	0
$7$	−4.10776	−1.55259	−0.776294	−	0.630371i	$-0.782903\pi$
$7$	−4.10776	−1.55259	−0.776294	+	0.630371i	$0.782903\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−3.17461	−1.00390
$11$	−5.68264	−1.71338	−0.856690	−	0.515832i	$-0.827483\pi$
$11$	−5.68264	−1.71338	−0.856690	+	0.515832i	$0.827483\pi$
$12$	0	0
$13$	−6.22695	−1.72705	−0.863523	−	0.504309i	$-0.831747\pi$
$13$	−6.22695	−1.72705	−0.863523	+	0.504309i	$0.831747\pi$
$14$	−4.10776	−1.09785
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.79856	−1.40636	−0.703179	−	0.711013i	$-0.748237\pi$
$17$	−5.79856	−1.40636	−0.703179	+	0.711013i	$0.748237\pi$
$18$	0	0
$19$	2.64964	0.607869	0.303934	−	0.952693i	$-0.401700\pi$
$19$	2.64964	0.607869	0.303934	+	0.952693i	$0.401700\pi$
$20$	−3.17461	−0.709865
$21$	0	0
$22$	−5.68264	−1.21154
$23$	6.44820	1.34454	0.672271	−	0.740305i	$-0.265319\pi$
$23$	6.44820	1.34454	0.672271	+	0.740305i	$0.265319\pi$
$24$	0	0
$25$	5.07816	1.01563
$26$	−6.22695	−1.22121
$27$	0	0
$28$	−4.10776	−0.776294
$29$	−3.80603	−0.706762	−0.353381	−	0.935480i	$-0.614968\pi$
$29$	−3.80603	−0.706762	−0.353381	+	0.935480i	$0.614968\pi$
$30$	0	0
$31$	2.24492	0.403199	0.201599	−	0.979468i	$-0.435386\pi$
$31$	2.24492	0.403199	0.201599	+	0.979468i	$0.435386\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−5.79856	−0.994445
$35$	13.0405	2.20425
$36$	0	0
$37$	7.14892	1.17528	0.587638	−	0.809124i	$-0.300058\pi$
$37$	7.14892	1.17528	0.587638	+	0.809124i	$0.300058\pi$
$38$	2.64964	0.429828
$39$	0	0
$40$	−3.17461	−0.501950
$41$	−6.53055	−1.01990	−0.509950	−	0.860204i	$-0.670336\pi$
$41$	−6.53055	−1.01990	−0.509950	+	0.860204i	$0.670336\pi$
$42$	0	0
$43$	6.32900	0.965164	0.482582	−	0.875851i	$-0.339699\pi$
$43$	6.32900	0.965164	0.482582	+	0.875851i	$0.339699\pi$
$44$	−5.68264	−0.856690
$45$	0	0
$46$	6.44820	0.950735
$47$	−3.49047	−0.509137	−0.254569	−	0.967055i	$-0.581934\pi$
$47$	−3.49047	−0.509137	−0.254569	+	0.967055i	$0.581934\pi$
$48$	0	0
$49$	9.87371	1.41053
$50$	5.07816	0.718160
$51$	0	0
$52$	−6.22695	−0.863523
$53$	−7.70122	−1.05784	−0.528922	−	0.848670i	$-0.677404\pi$
$53$	−7.70122	−1.05784	−0.528922	+	0.848670i	$0.677404\pi$
$54$	0	0
$55$	18.0402	2.43254
$56$	−4.10776	−0.548923
$57$	0	0
$58$	−3.80603	−0.499756
$59$	−5.61944	−0.731589	−0.365794	−	0.930696i	$-0.619203\pi$
$59$	−5.61944	−0.731589	−0.365794	+	0.930696i	$0.619203\pi$
$60$	0	0
$61$	−6.40288	−0.819805	−0.409902	−	0.912129i	$-0.634437\pi$
$61$	−6.40288	−0.819805	−0.409902	+	0.912129i	$0.634437\pi$
$62$	2.24492	0.285105
$63$	0	0
$64$	1.00000	0.125000
$65$	19.7682	2.45194
$66$	0	0
$67$	−4.28369	−0.523336	−0.261668	−	0.965158i	$-0.584273\pi$
$67$	−4.28369	−0.523336	−0.261668	+	0.965158i	$0.584273\pi$
$68$	−5.79856	−0.703179
$69$	0	0
$70$	13.0405	1.55864
$71$	13.7436	1.63106	0.815531	−	0.578713i	$-0.196445\pi$
$71$	13.7436	1.63106	0.815531	+	0.578713i	$0.196445\pi$
$72$	0	0
$73$	−6.85491	−0.802307	−0.401154	−	0.916011i	$-0.631391\pi$
$73$	−6.85491	−0.802307	−0.401154	+	0.916011i	$0.631391\pi$
$74$	7.14892	0.831045
$75$	0	0
$76$	2.64964	0.303934
$77$	23.3429	2.66017
$78$	0	0
$79$	8.62991	0.970941	0.485471	−	0.874253i	$-0.338648\pi$
$79$	8.62991	0.970941	0.485471	+	0.874253i	$0.338648\pi$
$80$	−3.17461	−0.354932
$81$	0	0
$82$	−6.53055	−0.721179
$83$	2.78811	0.306035	0.153017	−	0.988223i	$-0.451101\pi$
$83$	2.78811	0.306035	0.153017	+	0.988223i	$0.451101\pi$
$84$	0	0
$85$	18.4082	1.99665
$86$	6.32900	0.682474
$87$	0	0
$88$	−5.68264	−0.605771
$89$	−0.930456	−0.0986281	−0.0493140	−	0.998783i	$-0.515704\pi$
$89$	−0.930456	−0.0986281	−0.0493140	+	0.998783i	$0.515704\pi$
$90$	0	0
$91$	25.5788	2.68139
$92$	6.44820	0.672271
$93$	0	0
$94$	−3.49047	−0.360015
$95$	−8.41157	−0.863009
$96$	0	0
$97$	6.12189	0.621583	0.310792	−	0.950478i	$-0.399406\pi$
$97$	6.12189	0.621583	0.310792	+	0.950478i	$0.399406\pi$
$98$	9.87371	0.997395
$99$	0	0
$100$	5.07816	0.507816
$101$	−0.943820	−0.0939136	−0.0469568	−	0.998897i	$-0.514952\pi$
$101$	−0.943820	−0.0939136	−0.0469568	+	0.998897i	$0.514952\pi$
$102$	0	0
$103$	6.25408	0.616233	0.308117	−	0.951349i	$-0.400301\pi$
$103$	6.25408	0.616233	0.308117	+	0.951349i	$0.400301\pi$
$104$	−6.22695	−0.610603
$105$	0	0
$106$	−7.70122	−0.748009
$107$	14.4252	1.39453	0.697267	−	0.716811i	$-0.254399\pi$
$107$	14.4252	1.39453	0.697267	+	0.716811i	$0.254399\pi$
$108$	0	0
$109$	−6.31130	−0.604513	−0.302256	−	0.953227i	$-0.597740\pi$
$109$	−6.31130	−0.604513	−0.302256	+	0.953227i	$0.597740\pi$
$110$	18.0402	1.72006
$111$	0	0
$112$	−4.10776	−0.388147
$113$	−0.989886	−0.0931206	−0.0465603	−	0.998915i	$-0.514826\pi$
$113$	−0.989886	−0.0931206	−0.0465603	+	0.998915i	$0.514826\pi$
$114$	0	0
$115$	−20.4705	−1.90889
$116$	−3.80603	−0.353381
$117$	0	0
$118$	−5.61944	−0.517311
$119$	23.8191	2.18349
$120$	0	0
$121$	21.2924	1.93567
$122$	−6.40288	−0.579690
$123$	0	0
$124$	2.24492	0.201599
$125$	−0.248117	−0.0221922
$126$	0	0
$127$	−11.8381	−1.05046	−0.525230	−	0.850961i	$-0.676021\pi$
$127$	−11.8381	−1.05046	−0.525230	+	0.850961i	$0.676021\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	19.7682	1.73378
$131$	−14.1341	−1.23490	−0.617451	−	0.786609i	$-0.711835\pi$
$131$	−14.1341	−1.23490	−0.617451	+	0.786609i	$0.711835\pi$
$132$	0	0
$133$	−10.8841	−0.943770
$134$	−4.28369	−0.370054
$135$	0	0
$136$	−5.79856	−0.497222
$137$	−14.1532	−1.20919	−0.604596	−	0.796532i	$-0.706665\pi$
$137$	−14.1532	−1.20919	−0.604596	+	0.796532i	$0.706665\pi$
$138$	0	0
$139$	−19.4857	−1.65275	−0.826376	−	0.563118i	$-0.809601\pi$
$139$	−19.4857	−1.65275	−0.826376	+	0.563118i	$0.809601\pi$
$140$	13.0405	1.10213
$141$	0	0
$142$	13.7436	1.15334
$143$	35.3855	2.95909
$144$	0	0
$145$	12.0827	1.00341
$146$	−6.85491	−0.567317
$147$	0	0
$148$	7.14892	0.587638
$149$	−6.42565	−0.526410	−0.263205	−	0.964740i	$-0.584780\pi$
$149$	−6.42565	−0.526410	−0.263205	+	0.964740i	$0.584780\pi$
$150$	0	0
$151$	−9.04131	−0.735771	−0.367886	−	0.929871i	$-0.619918\pi$
$151$	−9.04131	−0.735771	−0.367886	+	0.929871i	$0.619918\pi$
$152$	2.64964	0.214914
$153$	0	0
$154$	23.3429	1.88103
$155$	−7.12674	−0.572433
$156$	0	0
$157$	−9.57094	−0.763844	−0.381922	−	0.924194i	$-0.624738\pi$
$157$	−9.57094	−0.763844	−0.381922	+	0.924194i	$0.624738\pi$
$158$	8.62991	0.686559
$159$	0	0
$160$	−3.17461	−0.250975
$161$	−26.4877	−2.08752
$162$	0	0
$163$	16.7227	1.30983	0.654913	−	0.755705i	$-0.272705\pi$
$163$	16.7227	1.30983	0.654913	+	0.755705i	$0.272705\pi$
$164$	−6.53055	−0.509950
$165$	0	0
$166$	2.78811	0.216399
$167$	18.6623	1.44413	0.722067	−	0.691823i	$-0.243192\pi$
$167$	18.6623	1.44413	0.722067	+	0.691823i	$0.243192\pi$
$168$	0	0
$169$	25.7750	1.98269
$170$	18.4082	1.41184
$171$	0	0
$172$	6.32900	0.482582
$173$	−12.0255	−0.914285	−0.457142	−	0.889394i	$-0.651127\pi$
$173$	−12.0255	−0.914285	−0.457142	+	0.889394i	$0.651127\pi$
$174$	0	0
$175$	−20.8599	−1.57686
$176$	−5.68264	−0.428345
$177$	0	0
$178$	−0.930456	−0.0697406
$179$	−23.4170	−1.75027	−0.875135	−	0.483878i	$-0.839228\pi$
$179$	−23.4170	−1.75027	−0.875135	+	0.483878i	$0.839228\pi$
$180$	0	0
$181$	17.3952	1.29298	0.646489	−	0.762923i	$-0.276237\pi$
$181$	17.3952	1.29298	0.646489	+	0.762923i	$0.276237\pi$
$182$	25.5788	1.89603
$183$	0	0
$184$	6.44820	0.475367
$185$	−22.6950	−1.66857
$186$	0	0
$187$	32.9511	2.40962
$188$	−3.49047	−0.254569
$189$	0	0
$190$	−8.41157	−0.610239
$191$	−15.3069	−1.10757	−0.553783	−	0.832661i	$-0.686816\pi$
$191$	−15.3069	−1.10757	−0.553783	+	0.832661i	$0.686816\pi$
$192$	0	0
$193$	−4.91872	−0.354057	−0.177028	−	0.984206i	$-0.556648\pi$
$193$	−4.91872	−0.354057	−0.177028	+	0.984206i	$0.556648\pi$
$194$	6.12189	0.439526
$195$	0	0
$196$	9.87371	0.705265
$197$	4.81884	0.343328	0.171664	−	0.985156i	$-0.445086\pi$
$197$	4.81884	0.343328	0.171664	+	0.985156i	$0.445086\pi$
$198$	0	0
$199$	−25.2725	−1.79152	−0.895760	−	0.444538i	$-0.853368\pi$
$199$	−25.2725	−1.79152	−0.895760	+	0.444538i	$0.853368\pi$
$200$	5.07816	0.359080
$201$	0	0
$202$	−0.943820	−0.0664070
$203$	15.6343	1.09731
$204$	0	0
$205$	20.7320	1.44798
$206$	6.25408	0.435743
$207$	0	0
$208$	−6.22695	−0.431762
$209$	−15.0569	−1.04151
$210$	0	0
$211$	−26.6369	−1.83376	−0.916879	−	0.399164i	$-0.869300\pi$
$211$	−26.6369	−1.83376	−0.916879	+	0.399164i	$0.869300\pi$
$212$	−7.70122	−0.528922
$213$	0	0
$214$	14.4252	0.986085
$215$	−20.0921	−1.37027
$216$	0	0
$217$	−9.22158	−0.626002
$218$	−6.31130	−0.427455
$219$	0	0
$220$	18.0402	1.21627
$221$	36.1074	2.42884
$222$	0	0
$223$	−14.2328	−0.953101	−0.476551	−	0.879147i	$-0.658113\pi$
$223$	−14.2328	−0.953101	−0.476551	+	0.879147i	$0.658113\pi$
$224$	−4.10776	−0.274461
$225$	0	0
$226$	−0.989886	−0.0658462
$227$	−12.1549	−0.806752	−0.403376	−	0.915034i	$-0.632163\pi$
$227$	−12.1549	−0.806752	−0.403376	+	0.915034i	$0.632163\pi$
$228$	0	0
$229$	22.1427	1.46323	0.731614	−	0.681719i	$-0.238767\pi$
$229$	22.1427	1.46323	0.731614	+	0.681719i	$0.238767\pi$
$230$	−20.4705	−1.34979
$231$	0	0
$232$	−3.80603	−0.249878
$233$	26.0939	1.70947	0.854733	−	0.519068i	$-0.173721\pi$
$233$	26.0939	1.70947	0.854733	+	0.519068i	$0.173721\pi$
$234$	0	0
$235$	11.0809	0.722837
$236$	−5.61944	−0.365794
$237$	0	0
$238$	23.8191	1.54396
$239$	8.53022	0.551774	0.275887	−	0.961190i	$-0.411028\pi$
$239$	8.53022	0.551774	0.275887	+	0.961190i	$0.411028\pi$
$240$	0	0
$241$	−6.10413	−0.393202	−0.196601	−	0.980484i	$-0.562990\pi$
$241$	−6.10413	−0.393202	−0.196601	+	0.980484i	$0.562990\pi$
$242$	21.2924	1.36873
$243$	0	0
$244$	−6.40288	−0.409902
$245$	−31.3452	−2.00257
$246$	0	0
$247$	−16.4992	−1.04982
$248$	2.24492	0.142552
$249$	0	0
$250$	−0.248117	−0.0156923
$251$	14.9750	0.945212	0.472606	−	0.881274i	$-0.343313\pi$
$251$	14.9750	0.945212	0.472606	+	0.881274i	$0.343313\pi$
$252$	0	0
$253$	−36.6428	−2.30371
$254$	−11.8381	−0.742787
$255$	0	0
$256$	1.00000	0.0625000
$257$	−5.54581	−0.345938	−0.172969	−	0.984927i	$-0.555336\pi$
$257$	−5.54581	−0.345938	−0.172969	+	0.984927i	$0.555336\pi$
$258$	0	0
$259$	−29.3661	−1.82472
$260$	19.7682	1.22597
$261$	0	0
$262$	−14.1341	−0.873208
$263$	−18.6570	−1.15044	−0.575219	−	0.818000i	$-0.695083\pi$
$263$	−18.6570	−1.15044	−0.575219	+	0.818000i	$0.695083\pi$
$264$	0	0
$265$	24.4484	1.50185
$266$	−10.8841	−0.667346
$267$	0	0
$268$	−4.28369	−0.261668
$269$	1.00000	0.0609711
$269$	1.00000	0.0609711
$270$	0	0
$271$	10.1487	0.616491	0.308245	−	0.951307i	$-0.400258\pi$
$271$	10.1487	0.616491	0.308245	+	0.951307i	$0.400258\pi$
$272$	−5.79856	−0.351589
$273$	0	0
$274$	−14.1532	−0.855028
$275$	−28.8573	−1.74016
$276$	0	0
$277$	21.7072	1.30426	0.652129	−	0.758108i	$-0.273876\pi$
$277$	21.7072	1.30426	0.652129	+	0.758108i	$0.273876\pi$
$278$	−19.4857	−1.16867
$279$	0	0
$280$	13.0405	0.779322
$281$	28.4931	1.69975	0.849877	−	0.526982i	$-0.176676\pi$
$281$	28.4931	1.69975	0.849877	+	0.526982i	$0.176676\pi$
$282$	0	0
$283$	−16.2627	−0.966717	−0.483359	−	0.875422i	$-0.660583\pi$
$283$	−16.2627	−0.966717	−0.483359	+	0.875422i	$0.660583\pi$
$284$	13.7436	0.815531
$285$	0	0
$286$	35.3855	2.09239
$287$	26.8259	1.58349
$288$	0	0
$289$	16.6233	0.977840
$290$	12.0827	0.709518
$291$	0	0
$292$	−6.85491	−0.401154
$293$	−2.48839	−0.145374	−0.0726868	−	0.997355i	$-0.523157\pi$
$293$	−2.48839	−0.145374	−0.0726868	+	0.997355i	$0.523157\pi$
$294$	0	0
$295$	17.8395	1.03866
$296$	7.14892	0.415523
$297$	0	0
$298$	−6.42565	−0.372228
$299$	−40.1526	−2.32209
$300$	0	0
$301$	−25.9980	−1.49850
$302$	−9.04131	−0.520269
$303$	0	0
$304$	2.64964	0.151967
$305$	20.3267	1.16390
$306$	0	0
$307$	4.92490	0.281079	0.140539	−	0.990075i	$-0.455116\pi$
$307$	4.92490	0.281079	0.140539	+	0.990075i	$0.455116\pi$
$308$	23.3429	1.33009
$309$	0	0
$310$	−7.12674	−0.404771
$311$	26.4492	1.49980	0.749899	−	0.661552i	$-0.230102\pi$
$311$	26.4492	1.49980	0.749899	+	0.661552i	$0.230102\pi$
$312$	0	0
$313$	−20.8073	−1.17610	−0.588048	−	0.808826i	$-0.700104\pi$
$313$	−20.8073	−1.17610	−0.588048	+	0.808826i	$0.700104\pi$
$314$	−9.57094	−0.540120
$315$	0	0
$316$	8.62991	0.485471
$317$	26.7500	1.50243	0.751215	−	0.660057i	$-0.229468\pi$
$317$	26.7500	1.50243	0.751215	+	0.660057i	$0.229468\pi$
$318$	0	0
$319$	21.6283	1.21095
$320$	−3.17461	−0.177466
$321$	0	0
$322$	−26.4877	−1.47610
$323$	−15.3641	−0.854880
$324$	0	0
$325$	−31.6214	−1.75404
$326$	16.7227	0.926186
$327$	0	0
$328$	−6.53055	−0.360589
$329$	14.3380	0.790481
$330$	0	0
$331$	−20.8203	−1.14439	−0.572193	−	0.820119i	$-0.693907\pi$
$331$	−20.8203	−1.14439	−0.572193	+	0.820119i	$0.693907\pi$
$332$	2.78811	0.153017
$333$	0	0
$334$	18.6623	1.02116
$335$	13.5990	0.742995
$336$	0	0
$337$	−15.9020	−0.866238	−0.433119	−	0.901337i	$-0.642587\pi$
$337$	−15.9020	−0.866238	−0.433119	+	0.901337i	$0.642587\pi$
$338$	25.7750	1.40197
$339$	0	0
$340$	18.4082	0.998323
$341$	−12.7570	−0.690833
$342$	0	0
$343$	−11.8045	−0.637384
$344$	6.32900	0.341237
$345$	0	0
$346$	−12.0255	−0.646497
$347$	5.82986	0.312963	0.156482	−	0.987681i	$-0.449985\pi$
$347$	5.82986	0.312963	0.156482	+	0.987681i	$0.449985\pi$
$348$	0	0
$349$	17.4102	0.931948	0.465974	−	0.884798i	$-0.345704\pi$
$349$	17.4102	0.931948	0.465974	+	0.884798i	$0.345704\pi$
$350$	−20.8599	−1.11501
$351$	0	0
$352$	−5.68264	−0.302886
$353$	6.12122	0.325800	0.162900	−	0.986643i	$-0.447915\pi$
$353$	6.12122	0.325800	0.162900	+	0.986643i	$0.447915\pi$
$354$	0	0
$355$	−43.6305	−2.31567
$356$	−0.930456	−0.0493140
$357$	0	0
$358$	−23.4170	−1.23763
$359$	20.4757	1.08067	0.540333	−	0.841451i	$-0.318298\pi$
$359$	20.4757	1.08067	0.540333	+	0.841451i	$0.318298\pi$
$360$	0	0
$361$	−11.9794	−0.630496
$362$	17.3952	0.914274
$363$	0	0
$364$	25.5788	1.34070
$365$	21.7617	1.13906
$366$	0	0
$367$	33.9483	1.77209	0.886043	−	0.463603i	$-0.153444\pi$
$367$	33.9483	1.77209	0.886043	+	0.463603i	$0.153444\pi$
$368$	6.44820	0.336135
$369$	0	0
$370$	−22.6950	−1.17986
$371$	31.6348	1.64240
$372$	0	0
$373$	18.5917	0.962644	0.481322	−	0.876544i	$-0.340157\pi$
$373$	18.5917	0.962644	0.481322	+	0.876544i	$0.340157\pi$
$374$	32.9511	1.70386
$375$	0	0
$376$	−3.49047	−0.180007
$377$	23.7000	1.22061
$378$	0	0
$379$	26.0647	1.33885	0.669427	−	0.742878i	$-0.266540\pi$
$379$	26.0647	1.33885	0.669427	+	0.742878i	$0.266540\pi$
$380$	−8.41157	−0.431504
$381$	0	0
$382$	−15.3069	−0.783168
$383$	7.56172	0.386386	0.193193	−	0.981161i	$-0.438116\pi$
$383$	7.56172	0.386386	0.193193	+	0.981161i	$0.438116\pi$
$384$	0	0
$385$	−74.1047	−3.77673
$386$	−4.91872	−0.250356
$387$	0	0
$388$	6.12189	0.310792
$389$	−28.7906	−1.45974	−0.729870	−	0.683586i	$-0.760419\pi$
$389$	−28.7906	−1.45974	−0.729870	+	0.683586i	$0.760419\pi$
$390$	0	0
$391$	−37.3902	−1.89091
$392$	9.87371	0.498698
$393$	0	0
$394$	4.81884	0.242769
$395$	−27.3966	−1.37847
$396$	0	0
$397$	32.7879	1.64558	0.822789	−	0.568347i	$-0.192417\pi$
$397$	32.7879	1.64558	0.822789	+	0.568347i	$0.192417\pi$
$398$	−25.2725	−1.26680
$399$	0	0
$400$	5.07816	0.253908
$401$	9.95075	0.496917	0.248458	−	0.968643i	$-0.420076\pi$
$401$	9.95075	0.496917	0.248458	+	0.968643i	$0.420076\pi$
$402$	0	0
$403$	−13.9790	−0.696343
$404$	−0.943820	−0.0469568
$405$	0	0
$406$	15.6343	0.775915
$407$	−40.6247	−2.01369
$408$	0	0
$409$	6.24500	0.308795	0.154398	−	0.988009i	$-0.450656\pi$
$409$	6.24500	0.308795	0.154398	+	0.988009i	$0.450656\pi$
$410$	20.7320	1.02388
$411$	0	0
$412$	6.25408	0.308117
$413$	23.0833	1.13586
$414$	0	0
$415$	−8.85117	−0.434487
$416$	−6.22695	−0.305302
$417$	0	0
$418$	−15.0569	−0.736459
$419$	3.51883	0.171906	0.0859531	−	0.996299i	$-0.472606\pi$
$419$	3.51883	0.171906	0.0859531	+	0.996299i	$0.472606\pi$
$420$	0	0
$421$	8.99561	0.438419	0.219210	−	0.975678i	$-0.429652\pi$
$421$	8.99561	0.438419	0.219210	+	0.975678i	$0.429652\pi$
$422$	−26.6369	−1.29666
$423$	0	0
$424$	−7.70122	−0.374004
$425$	−29.4460	−1.42834
$426$	0	0
$427$	26.3015	1.27282
$428$	14.4252	0.697267
$429$	0	0
$430$	−20.0921	−0.968929
$431$	19.6263	0.945364	0.472682	−	0.881233i	$-0.343286\pi$
$431$	19.6263	0.945364	0.472682	+	0.881233i	$0.343286\pi$
$432$	0	0
$433$	−26.1058	−1.25456	−0.627282	−	0.778792i	$-0.715833\pi$
$433$	−26.1058	−1.25456	−0.627282	+	0.778792i	$0.715833\pi$
$434$	−9.22158	−0.442650
$435$	0	0
$436$	−6.31130	−0.302256
$437$	17.0854	0.817305
$438$	0	0
$439$	−12.1949	−0.582030	−0.291015	−	0.956718i	$-0.593993\pi$
$439$	−12.1949	−0.582030	−0.291015	+	0.956718i	$0.593993\pi$
$440$	18.0402	0.860031
$441$	0	0
$442$	36.1074	1.71745
$443$	−3.86811	−0.183779	−0.0918896	−	0.995769i	$-0.529291\pi$
$443$	−3.86811	−0.183779	−0.0918896	+	0.995769i	$0.529291\pi$
$444$	0	0
$445$	2.95383	0.140025
$446$	−14.2328	−0.673944
$447$	0	0
$448$	−4.10776	−0.194074
$449$	37.2923	1.75993	0.879966	−	0.475037i	$-0.157565\pi$
$449$	37.2923	1.75993	0.879966	+	0.475037i	$0.157565\pi$
$450$	0	0
$451$	37.1108	1.74748
$452$	−0.989886	−0.0465603
$453$	0	0
$454$	−12.1549	−0.570460
$455$	−81.2029	−3.80685
$456$	0	0
$457$	20.9622	0.980572	0.490286	−	0.871562i	$-0.336892\pi$
$457$	20.9622	0.980572	0.490286	+	0.871562i	$0.336892\pi$
$458$	22.1427	1.03466
$459$	0	0
$460$	−20.4705	−0.954443
$461$	−26.5604	−1.23704	−0.618520	−	0.785769i	$-0.712267\pi$
$461$	−26.5604	−1.23704	−0.618520	+	0.785769i	$0.712267\pi$
$462$	0	0
$463$	−35.9385	−1.67020	−0.835101	−	0.550097i	$-0.814591\pi$
$463$	−35.9385	−1.67020	−0.835101	+	0.550097i	$0.814591\pi$
$464$	−3.80603	−0.176690
$465$	0	0
$466$	26.0939	1.20877
$467$	26.4519	1.22405	0.612023	−	0.790840i	$-0.290356\pi$
$467$	26.4519	1.22405	0.612023	+	0.790840i	$0.290356\pi$
$468$	0	0
$469$	17.5964	0.812525
$470$	11.0809	0.511123
$471$	0	0
$472$	−5.61944	−0.258656
$473$	−35.9654	−1.65369
$474$	0	0
$475$	13.4553	0.617370
$476$	23.8191	1.09175
$477$	0	0
$478$	8.53022	0.390163
$479$	39.6800	1.81303	0.906513	−	0.422177i	$-0.138734\pi$
$479$	39.6800	1.81303	0.906513	+	0.422177i	$0.138734\pi$
$480$	0	0
$481$	−44.5160	−2.02976
$482$	−6.10413	−0.278036
$483$	0	0
$484$	21.2924	0.967836
$485$	−19.4346	−0.882480
$486$	0	0
$487$	−20.1443	−0.912824	−0.456412	−	0.889768i	$-0.650866\pi$
$487$	−20.1443	−0.912824	−0.456412	+	0.889768i	$0.650866\pi$
$488$	−6.40288	−0.289845
$489$	0	0
$490$	−31.3452	−1.41603
$491$	7.53330	0.339973	0.169986	−	0.985446i	$-0.445628\pi$
$491$	7.53330	0.339973	0.169986	+	0.985446i	$0.445628\pi$
$492$	0	0
$493$	22.0695	0.993959
$494$	−16.4992	−0.742333
$495$	0	0
$496$	2.24492	0.100800
$497$	−56.4553	−2.53237
$498$	0	0
$499$	−36.3914	−1.62910	−0.814551	−	0.580092i	$-0.803017\pi$
$499$	−36.3914	−1.62910	−0.814551	+	0.580092i	$0.803017\pi$
$500$	−0.248117	−0.0110961
$501$	0	0
$502$	14.9750	0.668366
$503$	26.0667	1.16226	0.581128	−	0.813812i	$-0.302611\pi$
$503$	26.0667	1.16226	0.581128	+	0.813812i	$0.302611\pi$
$504$	0	0
$505$	2.99626	0.133332
$506$	−36.6428	−1.62897
$507$	0	0
$508$	−11.8381	−0.525230
$509$	−16.2672	−0.721030	−0.360515	−	0.932754i	$-0.617399\pi$
$509$	−16.2672	−0.721030	−0.360515	+	0.932754i	$0.617399\pi$
$510$	0	0
$511$	28.1584	1.24565
$512$	1.00000	0.0441942
$513$	0	0
$514$	−5.54581	−0.244615
$515$	−19.8543	−0.874884
$516$	0	0
$517$	19.8351	0.872346
$518$	−29.3661	−1.29027
$519$	0	0
$520$	19.7682	0.866891
$521$	30.2303	1.32441	0.662207	−	0.749321i	$-0.269620\pi$
$521$	30.2303	1.32441	0.662207	+	0.749321i	$0.269620\pi$
$522$	0	0
$523$	−21.0812	−0.921819	−0.460909	−	0.887447i	$-0.652477\pi$
$523$	−21.0812	−0.921819	−0.460909	+	0.887447i	$0.652477\pi$
$524$	−14.1341	−0.617451
$525$	0	0
$526$	−18.6570	−0.813482
$527$	−13.0173	−0.567042
$528$	0	0
$529$	18.5792	0.807793
$530$	24.4484	1.06197
$531$	0	0
$532$	−10.8841	−0.471885
$533$	40.6654	1.76142
$534$	0	0
$535$	−45.7943	−1.97986
$536$	−4.28369	−0.185027
$537$	0	0
$538$	1.00000	0.0431131
$539$	−56.1087	−2.41677
$540$	0	0
$541$	−31.4624	−1.35268	−0.676338	−	0.736592i	$-0.736434\pi$
$541$	−31.4624	−1.35268	−0.676338	+	0.736592i	$0.736434\pi$
$542$	10.1487	0.435925
$543$	0	0
$544$	−5.79856	−0.248611
$545$	20.0359	0.858244
$546$	0	0
$547$	17.0027	0.726981	0.363491	−	0.931598i	$-0.381585\pi$
$547$	17.0027	0.726981	0.363491	+	0.931598i	$0.381585\pi$
$548$	−14.1532	−0.604596
$549$	0	0
$550$	−28.8573	−1.23048
$551$	−10.0846	−0.429618
$552$	0	0
$553$	−35.4496	−1.50747
$554$	21.7072	0.922250
$555$	0	0
$556$	−19.4857	−0.826376
$557$	19.9617	0.845803	0.422902	−	0.906176i	$-0.361012\pi$
$557$	19.9617	0.845803	0.422902	+	0.906176i	$0.361012\pi$
$558$	0	0
$559$	−39.4104	−1.66688
$560$	13.0405	0.551064
$561$	0	0
$562$	28.4931	1.20191
$563$	7.86403	0.331429	0.165715	−	0.986174i	$-0.447007\pi$
$563$	7.86403	0.331429	0.165715	+	0.986174i	$0.447007\pi$
$564$	0	0
$565$	3.14250	0.132206
$566$	−16.2627	−0.683572
$567$	0	0
$568$	13.7436	0.576668
$569$	−5.29975	−0.222177	−0.111089	−	0.993811i	$-0.535434\pi$
$569$	−5.29975	−0.222177	−0.111089	+	0.993811i	$0.535434\pi$
$570$	0	0
$571$	−27.4951	−1.15064	−0.575318	−	0.817930i	$-0.695122\pi$
$571$	−27.4951	−1.15064	−0.575318	+	0.817930i	$0.695122\pi$
$572$	35.3855	1.47954
$573$	0	0
$574$	26.8259	1.11969
$575$	32.7450	1.36556
$576$	0	0
$577$	−3.74977	−0.156105	−0.0780525	−	0.996949i	$-0.524870\pi$
$577$	−3.74977	−0.156105	−0.0780525	+	0.996949i	$0.524870\pi$
$578$	16.6233	0.691438
$579$	0	0
$580$	12.0827	0.501705
$581$	−11.4529	−0.475146
$582$	0	0
$583$	43.7633	1.81249
$584$	−6.85491	−0.283658
$585$	0	0
$586$	−2.48839	−0.102795
$587$	−8.94970	−0.369394	−0.184697	−	0.982796i	$-0.559130\pi$
$587$	−8.94970	−0.369394	−0.184697	+	0.982796i	$0.559130\pi$
$588$	0	0
$589$	5.94821	0.245092
$590$	17.8395	0.734442
$591$	0	0
$592$	7.14892	0.293819
$593$	−16.6627	−0.684255	−0.342127	−	0.939654i	$-0.611147\pi$
$593$	−16.6627	−0.684255	−0.342127	+	0.939654i	$0.611147\pi$
$594$	0	0
$595$	−75.6164	−3.09997
$596$	−6.42565	−0.263205
$597$	0	0
$598$	−40.1526	−1.64196
$599$	−25.3367	−1.03523	−0.517615	−	0.855614i	$-0.673180\pi$
$599$	−25.3367	−1.03523	−0.517615	+	0.855614i	$0.673180\pi$
$600$	0	0
$601$	−24.0671	−0.981719	−0.490859	−	0.871239i	$-0.663317\pi$
$601$	−24.0671	−0.981719	−0.490859	+	0.871239i	$0.663317\pi$
$602$	−25.9980	−1.05960
$603$	0	0
$604$	−9.04131	−0.367886
$605$	−67.5950	−2.74813
$606$	0	0
$607$	19.8247	0.804658	0.402329	−	0.915495i	$-0.368201\pi$
$607$	19.8247	0.804658	0.402329	+	0.915495i	$0.368201\pi$
$608$	2.64964	0.107457
$609$	0	0
$610$	20.3267	0.823002
$611$	21.7350	0.879304
$612$	0	0
$613$	−7.63056	−0.308195	−0.154098	−	0.988056i	$-0.549247\pi$
$613$	−7.63056	−0.308195	−0.154098	+	0.988056i	$0.549247\pi$
$614$	4.92490	0.198753
$615$	0	0
$616$	23.3429	0.940513
$617$	24.7901	0.998014	0.499007	−	0.866598i	$-0.333698\pi$
$617$	24.7901	0.998014	0.499007	+	0.866598i	$0.333698\pi$
$618$	0	0
$619$	−22.5092	−0.904722	−0.452361	−	0.891835i	$-0.649418\pi$
$619$	−22.5092	−0.904722	−0.452361	+	0.891835i	$0.649418\pi$
$620$	−7.12674	−0.286217
$621$	0	0
$622$	26.4492	1.06052
$623$	3.82209	0.153129
$624$	0	0
$625$	−24.6031	−0.984124
$626$	−20.8073	−0.831626
$627$	0	0
$628$	−9.57094	−0.381922
$629$	−41.4534	−1.65286
$630$	0	0
$631$	34.5121	1.37391	0.686953	−	0.726702i	$-0.258948\pi$
$631$	34.5121	1.37391	0.686953	+	0.726702i	$0.258948\pi$
$632$	8.62991	0.343280
$633$	0	0
$634$	26.7500	1.06238
$635$	37.5813	1.49137
$636$	0	0
$637$	−61.4831	−2.43605
$638$	21.6283	0.856272
$639$	0	0
$640$	−3.17461	−0.125488
$641$	−5.98465	−0.236379	−0.118190	−	0.992991i	$-0.537709\pi$
$641$	−5.98465	−0.236379	−0.118190	+	0.992991i	$0.537709\pi$
$642$	0	0
$643$	41.7247	1.64546	0.822730	−	0.568432i	$-0.192450\pi$
$643$	41.7247	1.64546	0.822730	+	0.568432i	$0.192450\pi$
$644$	−26.4877	−1.04376
$645$	0	0
$646$	−15.3641	−0.604492
$647$	−44.2771	−1.74071	−0.870357	−	0.492421i	$-0.836112\pi$
$647$	−44.2771	−1.74071	−0.870357	+	0.492421i	$0.836112\pi$
$648$	0	0
$649$	31.9332	1.25349
$650$	−31.6214	−1.24030
$651$	0	0
$652$	16.7227	0.654913
$653$	−0.719353	−0.0281505	−0.0140752	−	0.999901i	$-0.504480\pi$
$653$	−0.719353	−0.0281505	−0.0140752	+	0.999901i	$0.504480\pi$
$654$	0	0
$655$	44.8703	1.75323
$656$	−6.53055	−0.254975
$657$	0	0
$658$	14.3380	0.558954
$659$	−45.8118	−1.78457	−0.892287	−	0.451469i	$-0.850900\pi$
$659$	−45.8118	−1.78457	−0.892287	+	0.451469i	$0.850900\pi$
$660$	0	0
$661$	8.24122	0.320546	0.160273	−	0.987073i	$-0.448762\pi$
$661$	8.24122	0.320546	0.160273	+	0.987073i	$0.448762\pi$
$662$	−20.8203	−0.809202
$663$	0	0
$664$	2.78811	0.108200
$665$	34.5527	1.33990
$666$	0	0
$667$	−24.5420	−0.950271
$668$	18.6623	0.722067
$669$	0	0
$670$	13.5990	0.525377
$671$	36.3853	1.40464
$672$	0	0
$673$	−8.63428	−0.332827	−0.166413	−	0.986056i	$-0.553219\pi$
$673$	−8.63428	−0.332827	−0.166413	+	0.986056i	$0.553219\pi$
$674$	−15.9020	−0.612522
$675$	0	0
$676$	25.7750	0.991344
$677$	−19.4130	−0.746102	−0.373051	−	0.927811i	$-0.621688\pi$
$677$	−19.4130	−0.746102	−0.373051	+	0.927811i	$0.621688\pi$
$678$	0	0
$679$	−25.1473	−0.965063
$680$	18.4082	0.705921
$681$	0	0
$682$	−12.7570	−0.488493
$683$	36.8946	1.41173	0.705866	−	0.708345i	$-0.250558\pi$
$683$	36.8946	1.41173	0.705866	+	0.708345i	$0.250558\pi$
$684$	0	0
$685$	44.9310	1.71673
$686$	−11.8045	−0.450699
$687$	0	0
$688$	6.32900	0.241291
$689$	47.9551	1.82695
$690$	0	0
$691$	17.6603	0.671829	0.335915	−	0.941892i	$-0.390955\pi$
$691$	17.6603	0.671829	0.335915	+	0.941892i	$0.390955\pi$
$692$	−12.0255	−0.457142
$693$	0	0
$694$	5.82986	0.221298
$695$	61.8594	2.34646
$696$	0	0
$697$	37.8678	1.43434
$698$	17.4102	0.658987
$699$	0	0
$700$	−20.8599	−0.788429
$701$	40.6474	1.53523	0.767616	−	0.640910i	$-0.221443\pi$
$701$	40.6474	1.53523	0.767616	+	0.640910i	$0.221443\pi$
$702$	0	0
$703$	18.9421	0.714413
$704$	−5.68264	−0.214173
$705$	0	0
$706$	6.12122	0.230375
$707$	3.87699	0.145809
$708$	0	0
$709$	−39.2019	−1.47226	−0.736129	−	0.676842i	$-0.763348\pi$
$709$	−39.2019	−1.47226	−0.736129	+	0.676842i	$0.763348\pi$
$710$	−43.6305	−1.63742
$711$	0	0
$712$	−0.930456	−0.0348703
$713$	14.4757	0.542118
$714$	0	0
$715$	−112.335	−4.20110
$716$	−23.4170	−0.875135
$717$	0	0
$718$	20.4757	0.764147
$719$	−34.0762	−1.27083	−0.635414	−	0.772172i	$-0.719170\pi$
$719$	−34.0762	−1.27083	−0.635414	+	0.772172i	$0.719170\pi$
$720$	0	0
$721$	−25.6903	−0.956756
$722$	−11.9794	−0.445828
$723$	0	0
$724$	17.3952	0.646489
$725$	−19.3276	−0.717809
$726$	0	0
$727$	4.96114	0.183998	0.0919992	−	0.995759i	$-0.470674\pi$
$727$	4.96114	0.183998	0.0919992	+	0.995759i	$0.470674\pi$
$728$	25.5788	0.948015
$729$	0	0
$730$	21.7617	0.805436
$731$	−36.6991	−1.35737
$732$	0	0
$733$	38.4899	1.42166	0.710828	−	0.703366i	$-0.248320\pi$
$733$	38.4899	1.42166	0.710828	+	0.703366i	$0.248320\pi$
$734$	33.9483	1.25305
$735$	0	0
$736$	6.44820	0.237684
$737$	24.3427	0.896673
$738$	0	0
$739$	−16.6642	−0.613001	−0.306501	−	0.951870i	$-0.599158\pi$
$739$	−16.6642	−0.613001	−0.306501	+	0.951870i	$0.599158\pi$
$740$	−22.6950	−0.834286
$741$	0	0
$742$	31.6348	1.16135
$743$	11.3693	0.417099	0.208549	−	0.978012i	$-0.433126\pi$
$743$	11.3693	0.417099	0.208549	+	0.978012i	$0.433126\pi$
$744$	0	0
$745$	20.3990	0.747360
$746$	18.5917	0.680692
$747$	0	0
$748$	32.9511	1.20481
$749$	−59.2552	−2.16514
$750$	0	0
$751$	−6.87518	−0.250879	−0.125439	−	0.992101i	$-0.540034\pi$
$751$	−6.87518	−0.250879	−0.125439	+	0.992101i	$0.540034\pi$
$752$	−3.49047	−0.127284
$753$	0	0
$754$	23.7000	0.863102
$755$	28.7026	1.04460
$756$	0	0
$757$	−8.44955	−0.307104	−0.153552	−	0.988141i	$-0.549071\pi$
$757$	−8.44955	−0.307104	−0.153552	+	0.988141i	$0.549071\pi$
$758$	26.0647	0.946713
$759$	0	0
$760$	−8.41157	−0.305120
$761$	12.8529	0.465918	0.232959	−	0.972487i	$-0.425159\pi$
$761$	12.8529	0.465918	0.232959	+	0.972487i	$0.425159\pi$
$762$	0	0
$763$	25.9253	0.938559
$764$	−15.3069	−0.553783
$765$	0	0
$766$	7.56172	0.273216
$767$	34.9920	1.26349
$768$	0	0
$769$	19.7023	0.710482	0.355241	−	0.934775i	$-0.384399\pi$
$769$	19.7023	0.710482	0.355241	+	0.934775i	$0.384399\pi$
$770$	−74.1047	−2.67055
$771$	0	0
$772$	−4.91872	−0.177028
$773$	−48.1533	−1.73195	−0.865976	−	0.500085i	$-0.833302\pi$
$773$	−48.1533	−1.73195	−0.865976	+	0.500085i	$0.833302\pi$
$774$	0	0
$775$	11.4000	0.409501
$776$	6.12189	0.219763
$777$	0	0
$778$	−28.7906	−1.03219
$779$	−17.3036	−0.619966
$780$	0	0
$781$	−78.0998	−2.79463
$782$	−37.3902	−1.33707
$783$	0	0
$784$	9.87371	0.352633
$785$	30.3840	1.08445
$786$	0	0
$787$	25.6145	0.913060	0.456530	−	0.889708i	$-0.349092\pi$
$787$	25.6145	0.913060	0.456530	+	0.889708i	$0.349092\pi$
$788$	4.81884	0.171664
$789$	0	0
$790$	−27.3966	−0.974728
$791$	4.06622	0.144578
$792$	0	0
$793$	39.8704	1.41584
$794$	32.7879	1.16360
$795$	0	0
$796$	−25.2725	−0.895760
$797$	37.2446	1.31927	0.659636	−	0.751585i	$-0.270710\pi$
$797$	37.2446	1.31927	0.659636	+	0.751585i	$0.270710\pi$
$798$	0	0
$799$	20.2397	0.716029
$800$	5.07816	0.179540
$801$	0	0
$802$	9.95075	0.351373
$803$	38.9540	1.37466
$804$	0	0
$805$	84.0880	2.96371
$806$	−13.9790	−0.492389
$807$	0	0
$808$	−0.943820	−0.0332035
$809$	−34.8028	−1.22360	−0.611800	−	0.791012i	$-0.709554\pi$
$809$	−34.8028	−1.22360	−0.611800	+	0.791012i	$0.709554\pi$
$810$	0	0
$811$	−16.4324	−0.577019	−0.288509	−	0.957477i	$-0.593160\pi$
$811$	−16.4324	−0.577019	−0.288509	+	0.957477i	$0.593160\pi$
$812$	15.6343	0.548655
$813$	0	0
$814$	−40.6247	−1.42390
$815$	−53.0882	−1.85960
$816$	0	0
$817$	16.7696	0.586693
$818$	6.24500	0.218351
$819$	0	0
$820$	20.7320	0.723991
$821$	−28.0742	−0.979795	−0.489898	−	0.871780i	$-0.662966\pi$
$821$	−28.0742	−0.979795	−0.489898	+	0.871780i	$0.662966\pi$
$822$	0	0
$823$	16.7051	0.582304	0.291152	−	0.956677i	$-0.405962\pi$
$823$	16.7051	0.582304	0.291152	+	0.956677i	$0.405962\pi$
$824$	6.25408	0.217871
$825$	0	0
$826$	23.0833	0.803172
$827$	−54.9525	−1.91089	−0.955444	−	0.295174i	$-0.904622\pi$
$827$	−54.9525	−1.91089	−0.955444	+	0.295174i	$0.904622\pi$
$828$	0	0
$829$	−16.4205	−0.570309	−0.285155	−	0.958482i	$-0.592045\pi$
$829$	−16.4205	−0.570309	−0.285155	+	0.958482i	$0.592045\pi$
$830$	−8.85117	−0.307229
$831$	0	0
$832$	−6.22695	−0.215881
$833$	−57.2533	−1.98371
$834$	0	0
$835$	−59.2456	−2.05028
$836$	−15.0569	−0.520755
$837$	0	0
$838$	3.51883	0.121556
$839$	23.6615	0.816886	0.408443	−	0.912784i	$-0.366072\pi$
$839$	23.6615	0.816886	0.408443	+	0.912784i	$0.366072\pi$
$840$	0	0
$841$	−14.5142	−0.500488
$842$	8.99561	0.310009
$843$	0	0
$844$	−26.6369	−0.916879
$845$	−81.8255	−2.81488
$846$	0	0
$847$	−87.4640	−3.00530
$848$	−7.70122	−0.264461
$849$	0	0
$850$	−29.4460	−1.00999
$851$	46.0977	1.58021
$852$	0	0
$853$	−29.2190	−1.00044	−0.500220	−	0.865898i	$-0.666748\pi$
$853$	−29.2190	−1.00044	−0.500220	+	0.865898i	$0.666748\pi$
$854$	26.3015	0.900019
$855$	0	0
$856$	14.4252	0.493042
$857$	−31.1311	−1.06342	−0.531709	−	0.846927i	$-0.678450\pi$
$857$	−31.1311	−1.06342	−0.531709	+	0.846927i	$0.678450\pi$
$858$	0	0
$859$	−11.6168	−0.396361	−0.198181	−	0.980166i	$-0.563503\pi$
$859$	−11.6168	−0.396361	−0.198181	+	0.980166i	$0.563503\pi$
$860$	−20.0921	−0.685136
$861$	0	0
$862$	19.6263	0.668473
$863$	25.2723	0.860278	0.430139	−	0.902763i	$-0.358465\pi$
$863$	25.2723	0.860278	0.430139	+	0.902763i	$0.358465\pi$
$864$	0	0
$865$	38.1764	1.29804
$866$	−26.1058	−0.887111
$867$	0	0
$868$	−9.22158	−0.313001
$869$	−49.0407	−1.66359
$870$	0	0
$871$	26.6743	0.903825
$872$	−6.31130	−0.213727
$873$	0	0
$874$	17.0854	0.577922
$875$	1.01920	0.0344554
$876$	0	0
$877$	41.5382	1.40265	0.701323	−	0.712843i	$-0.252593\pi$
$877$	41.5382	1.40265	0.701323	+	0.712843i	$0.252593\pi$
$878$	−12.1949	−0.411558
$879$	0	0
$880$	18.0402	0.608134
$881$	53.4725	1.80153	0.900766	−	0.434304i	$-0.143006\pi$
$881$	53.4725	1.80153	0.900766	+	0.434304i	$0.143006\pi$
$882$	0	0
$883$	44.6264	1.50180	0.750899	−	0.660417i	$-0.229620\pi$
$883$	44.6264	1.50180	0.750899	+	0.660417i	$0.229620\pi$
$884$	36.1074	1.21442
$885$	0	0
$886$	−3.86811	−0.129952
$887$	52.1674	1.75161	0.875805	−	0.482666i	$-0.160331\pi$
$887$	52.1674	1.75161	0.875805	+	0.482666i	$0.160331\pi$
$888$	0	0
$889$	48.6280	1.63093
$890$	2.95383	0.0990128
$891$	0	0
$892$	−14.2328	−0.476551
$893$	−9.24848	−0.309489
$894$	0	0
$895$	74.3399	2.48491
$896$	−4.10776	−0.137231
$897$	0	0
$898$	37.2923	1.24446
$899$	−8.54421	−0.284965
$900$	0	0
$901$	44.6560	1.48771
$902$	37.1108	1.23565
$903$	0	0
$904$	−0.989886	−0.0329231
$905$	−55.2231	−1.83568
$906$	0	0
$907$	−41.1364	−1.36591	−0.682957	−	0.730459i	$-0.739306\pi$
$907$	−41.1364	−1.36591	−0.682957	+	0.730459i	$0.739306\pi$
$908$	−12.1549	−0.403376
$909$	0	0
$910$	−81.2029	−2.69185
$911$	2.10324	0.0696834	0.0348417	−	0.999393i	$-0.488907\pi$
$911$	2.10324	0.0696834	0.0348417	+	0.999393i	$0.488907\pi$
$912$	0	0
$913$	−15.8438	−0.524354
$914$	20.9622	0.693369
$915$	0	0
$916$	22.1427	0.731614
$917$	58.0595	1.91730
$918$	0	0
$919$	9.72272	0.320723	0.160361	−	0.987058i	$-0.448734\pi$
$919$	9.72272	0.320723	0.160361	+	0.987058i	$0.448734\pi$
$920$	−20.4705	−0.674893
$921$	0	0
$922$	−26.5604	−0.874720
$923$	−85.5806	−2.81692
$924$	0	0
$925$	36.3033	1.19365
$926$	−35.9385	−1.18101
$927$	0	0
$928$	−3.80603	−0.124939
$929$	−16.8016	−0.551242	−0.275621	−	0.961266i	$-0.588883\pi$
$929$	−16.8016	−0.551242	−0.275621	+	0.961266i	$0.588883\pi$
$930$	0	0
$931$	26.1618	0.857417
$932$	26.0939	0.854733
$933$	0	0
$934$	26.4519	0.865531
$935$	−104.607	−3.42101
$936$	0	0
$937$	−3.05335	−0.0997487	−0.0498743	−	0.998756i	$-0.515882\pi$
$937$	−3.05335	−0.0997487	−0.0498743	+	0.998756i	$0.515882\pi$
$938$	17.5964	0.574542
$939$	0	0
$940$	11.0809	0.361419
$941$	17.8006	0.580284	0.290142	−	0.956984i	$-0.406298\pi$
$941$	17.8006	0.580284	0.290142	+	0.956984i	$0.406298\pi$
$942$	0	0
$943$	−42.1103	−1.37130
$944$	−5.61944	−0.182897
$945$	0	0
$946$	−35.9654	−1.16934
$947$	−36.0571	−1.17170	−0.585849	−	0.810420i	$-0.699239\pi$
$947$	−36.0571	−1.17170	−0.585849	+	0.810420i	$0.699239\pi$
$948$	0	0
$949$	42.6852	1.38562
$950$	13.4553	0.436547
$951$	0	0
$952$	23.8191	0.771982
$953$	41.9663	1.35942	0.679711	−	0.733480i	$-0.262105\pi$
$953$	41.9663	1.35942	0.679711	+	0.733480i	$0.262105\pi$
$954$	0	0
$955$	48.5934	1.57244
$956$	8.53022	0.275887
$957$	0	0
$958$	39.6800	1.28200
$959$	58.1381	1.87738
$960$	0	0
$961$	−25.9604	−0.837431
$962$	−44.5160	−1.43525
$963$	0	0
$964$	−6.10413	−0.196601
$965$	15.6150	0.502665
$966$	0	0
$967$	19.4429	0.625242	0.312621	−	0.949878i	$-0.398793\pi$
$967$	19.4429	0.625242	0.312621	+	0.949878i	$0.398793\pi$
$968$	21.2924	0.684363
$969$	0	0
$970$	−19.4346	−0.624008
$971$	2.98629	0.0958346	0.0479173	−	0.998851i	$-0.484742\pi$
$971$	2.98629	0.0958346	0.0479173	+	0.998851i	$0.484742\pi$
$972$	0	0
$973$	80.0425	2.56604
$974$	−20.1443	−0.645464
$975$	0	0
$976$	−6.40288	−0.204951
$977$	16.4922	0.527632	0.263816	−	0.964573i	$-0.415019\pi$
$977$	16.4922	0.527632	0.263816	+	0.964573i	$0.415019\pi$
$978$	0	0
$979$	5.28744	0.168987
$980$	−31.3452	−1.00129
$981$	0	0
$982$	7.53330	0.240397
$983$	50.6696	1.61611	0.808054	−	0.589108i	$-0.200521\pi$
$983$	50.6696	1.61611	0.808054	+	0.589108i	$0.200521\pi$
$984$	0	0
$985$	−15.2979	−0.487433
$986$	22.0695	0.702835
$987$	0	0
$988$	−16.4992	−0.524909
$989$	40.8107	1.29770
$990$	0	0
$991$	−32.1081	−1.01995	−0.509974	−	0.860190i	$-0.670345\pi$
$991$	−32.1081	−1.01995	−0.509974	+	0.860190i	$0.670345\pi$
$992$	2.24492	0.0712762
$993$	0	0
$994$	−56.4553	−1.79065
$995$	80.2304	2.54347
$996$	0	0
$997$	−9.48574	−0.300416	−0.150208	−	0.988654i	$-0.547994\pi$
$997$	−9.48574	−0.300416	−0.150208	+	0.988654i	$0.547994\pi$
$998$	−36.3914	−1.15195
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4842.2.a.q.1.2		8
3.2	odd	2		1614.2.a.i.1.7	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1614.2.a.i.1.7	✓	8	3.2	odd	2
4842.2.a.q.1.2		8	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:	\( N \)	\(=\)	\( 4842 = 2 \cdot 3^{2} \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4842.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.17461 q^{5} -4.10776 q^{7} +1.00000 q^{8} -3.17461 q^{10} -5.68264 q^{11} -6.22695 q^{13} -4.10776 q^{14} +1.00000 q^{16} -5.79856 q^{17} +2.64964 q^{19} -3.17461 q^{20} -5.68264 q^{22} +6.44820 q^{23} +5.07816 q^{25} -6.22695 q^{26} -4.10776 q^{28} -3.80603 q^{29} +2.24492 q^{31} +1.00000 q^{32} -5.79856 q^{34} +13.0405 q^{35} +7.14892 q^{37} +2.64964 q^{38} -3.17461 q^{40} -6.53055 q^{41} +6.32900 q^{43} -5.68264 q^{44} +6.44820 q^{46} -3.49047 q^{47} +9.87371 q^{49} +5.07816 q^{50} -6.22695 q^{52} -7.70122 q^{53} +18.0402 q^{55} -4.10776 q^{56} -3.80603 q^{58} -5.61944 q^{59} -6.40288 q^{61} +2.24492 q^{62} +1.00000 q^{64} +19.7682 q^{65} -4.28369 q^{67} -5.79856 q^{68} +13.0405 q^{70} +13.7436 q^{71} -6.85491 q^{73} +7.14892 q^{74} +2.64964 q^{76} +23.3429 q^{77} +8.62991 q^{79} -3.17461 q^{80} -6.53055 q^{82} +2.78811 q^{83} +18.4082 q^{85} +6.32900 q^{86} -5.68264 q^{88} -0.930456 q^{89} +25.5788 q^{91} +6.44820 q^{92} -3.49047 q^{94} -8.41157 q^{95} +6.12189 q^{97} +9.87371 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{4} + 9 q^{7} + 8 q^{8} - 8 q^{11} - 3 q^{13} + 9 q^{14} + 8 q^{16} - 8 q^{17} + 18 q^{19} - 8 q^{22} + 10 q^{23} + 26 q^{25} - 3 q^{26} + 9 q^{28} + 9 q^{29} + 18 q^{31} + 8 q^{32}+ \cdots + 29 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^4 + 9 * q^7 + 8 * q^8 - 8 * q^11 - 3 * q^13 + 9 * q^14 + 8 * q^16 - 8 * q^17 + 18 * q^19 - 8 * q^22 + 10 * q^23 + 26 * q^25 - 3 * q^26 + 9 * q^28 + 9 * q^29 + 18 * q^31 + 8 * q^32 - 8 * q^34 - 9 * q^35 + 22 * q^37 + 18 * q^38 + q^41 + 14 * q^43 - 8 * q^44 + 10 * q^46 - 6 * q^47 + 29 * q^49 + 26 * q^50 - 3 * q^52 + 3 * q^53 + 17 * q^55 + 9 * q^56 + 9 * q^58 - q^59 + 18 * q^61 + 18 * q^62 + 8 * q^64 + 14 * q^65 + 30 * q^67 - 8 * q^68 - 9 * q^70 + 5 * q^71 + 15 * q^73 + 22 * q^74 + 18 * q^76 + 20 * q^77 + 21 * q^79 + q^82 + 11 * q^83 + 18 * q^85 + 14 * q^86 - 8 * q^88 - 22 * q^89 + 53 * q^91 + 10 * q^92 - 6 * q^94 + 9 * q^95 + 13 * q^97 + 29 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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