LMFDB - Embedded newform 8048.2.a.q.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, 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("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.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:	$64.2636035467$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 13 x^{10} + 94 x^{9} + 15 x^{8} - 616 x^{7} + 368 x^{6} + 1643 x^{5} - 1463 x^{4} + \cdots - 208 $ x^12 - 5x^11 - 13x^10 + 94x^9 + 15x^8 - 616x^7 + 368x^6 + 1643x^5 - 1463x^4 - 1556x^3 + 1284x^2 + 576*x - 208
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1006)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.264006$ of defining polynomial
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.264006 q^{3} +1.70058 q^{5} -2.77338 q^{7} -2.93030 q^{9} +O(q^{10})$	$q-0.264006 q^{3} +1.70058 q^{5} -2.77338 q^{7} -2.93030 q^{9} +0.489106 q^{11} +0.503543 q^{13} -0.448965 q^{15} +3.96665 q^{17} -5.91347 q^{19} +0.732189 q^{21} +4.83634 q^{23} -2.10801 q^{25} +1.56564 q^{27} +3.23806 q^{29} +1.77062 q^{31} -0.129127 q^{33} -4.71636 q^{35} +2.45223 q^{37} -0.132939 q^{39} +1.21510 q^{41} +10.2109 q^{43} -4.98322 q^{45} -9.83131 q^{47} +0.691625 q^{49} -1.04722 q^{51} -1.49671 q^{53} +0.831766 q^{55} +1.56119 q^{57} +3.72212 q^{59} -7.69201 q^{61} +8.12683 q^{63} +0.856318 q^{65} +2.43313 q^{67} -1.27682 q^{69} +0.608248 q^{71} -2.67504 q^{73} +0.556528 q^{75} -1.35648 q^{77} -4.84369 q^{79} +8.37756 q^{81} -8.43632 q^{83} +6.74563 q^{85} -0.854867 q^{87} -0.160559 q^{89} -1.39652 q^{91} -0.467455 q^{93} -10.0564 q^{95} +16.5610 q^{97} -1.43323 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 5 q^{3} + 5 q^{5} - 8 q^{7} + 15 q^{9}+O(q^{10}) $ 12 * q - 5 * q^3 + 5 * q^5 - 8 * q^7 + 15 * q^9	$ 12 q - 5 q^{3} + 5 q^{5} - 8 q^{7} + 15 q^{9} - 18 q^{11} + 4 q^{13} - 2 q^{15} - 2 q^{17} - 6 q^{19} + q^{21} - 13 q^{23} + q^{25} - 8 q^{27} + 20 q^{29} - 7 q^{31} - 8 q^{33} - q^{35} + 10 q^{37} - 7 q^{39} + 2 q^{41} - 8 q^{43} - 7 q^{45} - 12 q^{47} + 4 q^{49} - 2 q^{51} + 12 q^{53} - 8 q^{55} - 10 q^{57} - 6 q^{59} - 10 q^{61} - 7 q^{63} + 4 q^{65} - 7 q^{67} - 12 q^{69} - 22 q^{71} - 23 q^{73} + 34 q^{75} - 19 q^{77} - 13 q^{79} - 28 q^{81} - q^{83} - 28 q^{85} + 22 q^{87} - 3 q^{89} + 21 q^{91} - 33 q^{93} + 2 q^{95} - 70 q^{97} + 6 q^{99}+O(q^{100}) $ 12 * q - 5 * q^3 + 5 * q^5 - 8 * q^7 + 15 * q^9 - 18 * q^11 + 4 * q^13 - 2 * q^15 - 2 * q^17 - 6 * q^19 + q^21 - 13 * q^23 + q^25 - 8 * q^27 + 20 * q^29 - 7 * q^31 - 8 * q^33 - q^35 + 10 * q^37 - 7 * q^39 + 2 * q^41 - 8 * q^43 - 7 * q^45 - 12 * q^47 + 4 * q^49 - 2 * q^51 + 12 * q^53 - 8 * q^55 - 10 * q^57 - 6 * q^59 - 10 * q^61 - 7 * q^63 + 4 * q^65 - 7 * q^67 - 12 * q^69 - 22 * q^71 - 23 * q^73 + 34 * q^75 - 19 * q^77 - 13 * q^79 - 28 * q^81 - q^83 - 28 * q^85 + 22 * q^87 - 3 * q^89 + 21 * q^91 - 33 * q^93 + 2 * q^95 - 70 * q^97 + 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−0.264006	−0.152424	−0.0762121	−	0.997092i	$-0.524283\pi$
$3$	−0.264006	−0.152424	−0.0762121	+	0.997092i	$0.524283\pi$
$4$	0	0
$5$	1.70058	0.760525	0.380262	−	0.924879i	$-0.375834\pi$
$5$	1.70058	0.760525	0.380262	+	0.924879i	$0.375834\pi$
$6$	0	0
$7$	−2.77338	−1.04824	−0.524119	−	0.851645i	$-0.675605\pi$
$7$	−2.77338	−1.04824	−0.524119	+	0.851645i	$0.675605\pi$
$8$	0	0
$9$	−2.93030	−0.976767
$10$	0	0
$11$	0.489106	0.147471	0.0737355	−	0.997278i	$-0.476508\pi$
$11$	0.489106	0.147471	0.0737355	+	0.997278i	$0.476508\pi$
$12$	0	0
$13$	0.503543	0.139658	0.0698289	−	0.997559i	$-0.477755\pi$
$13$	0.503543	0.139658	0.0698289	+	0.997559i	$0.477755\pi$
$14$	0	0
$15$	−0.448965	−0.115922
$16$	0	0
$17$	3.96665	0.962055	0.481027	−	0.876706i	$-0.340264\pi$
$17$	3.96665	0.962055	0.481027	+	0.876706i	$0.340264\pi$
$18$	0	0
$19$	−5.91347	−1.35664	−0.678321	−	0.734765i	$-0.737292\pi$
$19$	−5.91347	−1.35664	−0.678321	+	0.734765i	$0.737292\pi$
$20$	0	0
$21$	0.732189	0.159777
$22$	0	0
$23$	4.83634	1.00845	0.504223	−	0.863573i	$-0.331779\pi$
$23$	4.83634	1.00845	0.504223	+	0.863573i	$0.331779\pi$
$24$	0	0
$25$	−2.10801	−0.421602
$26$	0	0
$27$	1.56564	0.301307
$28$	0	0
$29$	3.23806	0.601292	0.300646	−	0.953736i	$-0.402798\pi$
$29$	3.23806	0.601292	0.300646	+	0.953736i	$0.402798\pi$
$30$	0	0
$31$	1.77062	0.318013	0.159007	−	0.987278i	$-0.449171\pi$
$31$	1.77062	0.318013	0.159007	+	0.987278i	$0.449171\pi$
$32$	0	0
$33$	−0.129127	−0.0224781
$34$	0	0
$35$	−4.71636	−0.797211
$36$	0	0
$37$	2.45223	0.403145	0.201572	−	0.979474i	$-0.435395\pi$
$37$	2.45223	0.403145	0.201572	+	0.979474i	$0.435395\pi$
$38$	0	0
$39$	−0.132939	−0.0212872
$40$	0	0
$41$	1.21510	0.189767	0.0948837	−	0.995488i	$-0.469752\pi$
$41$	1.21510	0.189767	0.0948837	+	0.995488i	$0.469752\pi$
$42$	0	0
$43$	10.2109	1.55715	0.778577	−	0.627549i	$-0.215942\pi$
$43$	10.2109	1.55715	0.778577	+	0.627549i	$0.215942\pi$
$44$	0	0
$45$	−4.98322	−0.742855
$46$	0	0
$47$	−9.83131	−1.43404	−0.717022	−	0.697050i	$-0.754495\pi$
$47$	−9.83131	−1.43404	−0.717022	+	0.697050i	$0.754495\pi$
$48$	0	0
$49$	0.691625	0.0988036
$50$	0	0
$51$	−1.04722	−0.146640
$52$	0	0
$53$	−1.49671	−0.205589	−0.102794	−	0.994703i	$-0.532778\pi$
$53$	−1.49671	−0.205589	−0.102794	+	0.994703i	$0.532778\pi$
$54$	0	0
$55$	0.831766	0.112155
$56$	0	0
$57$	1.56119	0.206785
$58$	0	0
$59$	3.72212	0.484578	0.242289	−	0.970204i	$-0.422102\pi$
$59$	3.72212	0.484578	0.242289	+	0.970204i	$0.422102\pi$
$60$	0	0
$61$	−7.69201	−0.984860	−0.492430	−	0.870352i	$-0.663891\pi$
$61$	−7.69201	−0.984860	−0.492430	+	0.870352i	$0.663891\pi$
$62$	0	0
$63$	8.12683	1.02388
$64$	0	0
$65$	0.856318	0.106213
$66$	0	0
$67$	2.43313	0.297254	0.148627	−	0.988893i	$-0.452515\pi$
$67$	2.43313	0.297254	0.148627	+	0.988893i	$0.452515\pi$
$68$	0	0
$69$	−1.27682	−0.153712
$70$	0	0
$71$	0.608248	0.0721857	0.0360929	−	0.999348i	$-0.488509\pi$
$71$	0.608248	0.0721857	0.0360929	+	0.999348i	$0.488509\pi$
$72$	0	0
$73$	−2.67504	−0.313090	−0.156545	−	0.987671i	$-0.550036\pi$
$73$	−2.67504	−0.313090	−0.156545	+	0.987671i	$0.550036\pi$
$74$	0	0
$75$	0.556528	0.0642624
$76$	0	0
$77$	−1.35648	−0.154585
$78$	0	0
$79$	−4.84369	−0.544958	−0.272479	−	0.962162i	$-0.587843\pi$
$79$	−4.84369	−0.544958	−0.272479	+	0.962162i	$0.587843\pi$
$80$	0	0
$81$	8.37756	0.930840
$82$	0	0
$83$	−8.43632	−0.926007	−0.463003	−	0.886357i	$-0.653228\pi$
$83$	−8.43632	−0.926007	−0.463003	+	0.886357i	$0.653228\pi$
$84$	0	0
$85$	6.74563	0.731666
$86$	0	0
$87$	−0.854867	−0.0916514
$88$	0	0
$89$	−0.160559	−0.0170192	−0.00850959	−	0.999964i	$-0.502709\pi$
$89$	−0.160559	−0.0170192	−0.00850959	+	0.999964i	$0.502709\pi$
$90$	0	0
$91$	−1.39652	−0.146395
$92$	0	0
$93$	−0.467455	−0.0484729
$94$	0	0
$95$	−10.0564	−1.03176
$96$	0	0
$97$	16.5610	1.68151	0.840757	−	0.541413i	$-0.182111\pi$
$97$	16.5610	1.68151	0.840757	+	0.541413i	$0.182111\pi$
$98$	0	0
$99$	−1.43323	−0.144045
$100$	0	0
$101$	−2.45694	−0.244475	−0.122237	−	0.992501i	$-0.539007\pi$
$101$	−2.45694	−0.244475	−0.122237	+	0.992501i	$0.539007\pi$
$102$	0	0
$103$	−1.10031	−0.108416	−0.0542081	−	0.998530i	$-0.517263\pi$
$103$	−1.10031	−0.108416	−0.0542081	+	0.998530i	$0.517263\pi$
$104$	0	0
$105$	1.24515	0.121514
$106$	0	0
$107$	8.32925	0.805219	0.402609	−	0.915372i	$-0.368103\pi$
$107$	8.32925	0.805219	0.402609	+	0.915372i	$0.368103\pi$
$108$	0	0
$109$	−19.3568	−1.85405	−0.927024	−	0.375001i	$-0.877642\pi$
$109$	−19.3568	−1.85405	−0.927024	+	0.375001i	$0.877642\pi$
$110$	0	0
$111$	−0.647405	−0.0614490
$112$	0	0
$113$	−3.46055	−0.325541	−0.162771	−	0.986664i	$-0.552043\pi$
$113$	−3.46055	−0.325541	−0.162771	+	0.986664i	$0.552043\pi$
$114$	0	0
$115$	8.22461	0.766948
$116$	0	0
$117$	−1.47553	−0.136413
$118$	0	0
$119$	−11.0010	−1.00846
$120$	0	0
$121$	−10.7608	−0.978252
$122$	0	0
$123$	−0.320795	−0.0289251
$124$	0	0
$125$	−12.0878	−1.08116
$126$	0	0
$127$	10.6546	0.945441	0.472721	−	0.881212i	$-0.343272\pi$
$127$	10.6546	0.945441	0.472721	+	0.881212i	$0.343272\pi$
$128$	0	0
$129$	−2.69575	−0.237348
$130$	0	0
$131$	−8.12465	−0.709854	−0.354927	−	0.934894i	$-0.615494\pi$
$131$	−8.12465	−0.709854	−0.354927	+	0.934894i	$0.615494\pi$
$132$	0	0
$133$	16.4003	1.42208
$134$	0	0
$135$	2.66250	0.229151
$136$	0	0
$137$	−10.5532	−0.901623	−0.450812	−	0.892619i	$-0.648865\pi$
$137$	−10.5532	−0.901623	−0.450812	+	0.892619i	$0.648865\pi$
$138$	0	0
$139$	−10.6019	−0.899240	−0.449620	−	0.893220i	$-0.648441\pi$
$139$	−10.6019	−0.899240	−0.449620	+	0.893220i	$0.648441\pi$
$140$	0	0
$141$	2.59553	0.218583
$142$	0	0
$143$	0.246286	0.0205955
$144$	0	0
$145$	5.50659	0.457297
$146$	0	0
$147$	−0.182593	−0.0150601
$148$	0	0
$149$	−19.0043	−1.55690	−0.778448	−	0.627709i	$-0.783993\pi$
$149$	−19.0043	−1.55690	−0.778448	+	0.627709i	$0.783993\pi$
$150$	0	0
$151$	4.94327	0.402277	0.201139	−	0.979563i	$-0.435536\pi$
$151$	4.94327	0.402277	0.201139	+	0.979563i	$0.435536\pi$
$152$	0	0
$153$	−11.6235	−0.939703
$154$	0	0
$155$	3.01109	0.241857
$156$	0	0
$157$	−5.33027	−0.425402	−0.212701	−	0.977117i	$-0.568226\pi$
$157$	−5.33027	−0.425402	−0.212701	+	0.977117i	$0.568226\pi$
$158$	0	0
$159$	0.395140	0.0313367
$160$	0	0
$161$	−13.4130	−1.05709
$162$	0	0
$163$	4.66004	0.365002	0.182501	−	0.983206i	$-0.441581\pi$
$163$	4.66004	0.365002	0.182501	+	0.983206i	$0.441581\pi$
$164$	0	0
$165$	−0.219592	−0.0170952
$166$	0	0
$167$	−2.01289	−0.155762	−0.0778810	−	0.996963i	$-0.524815\pi$
$167$	−2.01289	−0.155762	−0.0778810	+	0.996963i	$0.524815\pi$
$168$	0	0
$169$	−12.7464	−0.980496
$170$	0	0
$171$	17.3282	1.32512
$172$	0	0
$173$	3.62807	0.275837	0.137919	−	0.990444i	$-0.455959\pi$
$173$	3.62807	0.275837	0.137919	+	0.990444i	$0.455959\pi$
$174$	0	0
$175$	5.84631	0.441940
$176$	0	0
$177$	−0.982663	−0.0738614
$178$	0	0
$179$	−26.2469	−1.96179	−0.980893	−	0.194549i	$-0.937676\pi$
$179$	−26.2469	−1.96179	−0.980893	+	0.194549i	$0.937676\pi$
$180$	0	0
$181$	−21.3292	−1.58539	−0.792693	−	0.609621i	$-0.791321\pi$
$181$	−21.3292	−1.58539	−0.792693	+	0.609621i	$0.791321\pi$
$182$	0	0
$183$	2.03074	0.150117
$184$	0	0
$185$	4.17023	0.306601
$186$	0	0
$187$	1.94011	0.141875
$188$	0	0
$189$	−4.34210	−0.315842
$190$	0	0
$191$	1.48768	0.107645	0.0538225	−	0.998551i	$-0.482859\pi$
$191$	1.48768	0.107645	0.0538225	+	0.998551i	$0.482859\pi$
$192$	0	0
$193$	−4.32190	−0.311097	−0.155549	−	0.987828i	$-0.549715\pi$
$193$	−4.32190	−0.311097	−0.155549	+	0.987828i	$0.549715\pi$
$194$	0	0
$195$	−0.226074	−0.0161895
$196$	0	0
$197$	−12.7020	−0.904982	−0.452491	−	0.891769i	$-0.649464\pi$
$197$	−12.7020	−0.904982	−0.452491	+	0.891769i	$0.649464\pi$
$198$	0	0
$199$	12.0140	0.851651	0.425825	−	0.904805i	$-0.359984\pi$
$199$	12.0140	0.851651	0.425825	+	0.904805i	$0.359984\pi$
$200$	0	0
$201$	−0.642362	−0.0453087
$202$	0	0
$203$	−8.98035	−0.630297
$204$	0	0
$205$	2.06639	0.144323
$206$	0	0
$207$	−14.1719	−0.985017
$208$	0	0
$209$	−2.89231	−0.200065
$210$	0	0
$211$	6.08035	0.418588	0.209294	−	0.977853i	$-0.432883\pi$
$211$	6.08035	0.418588	0.209294	+	0.977853i	$0.432883\pi$
$212$	0	0
$213$	−0.160581	−0.0110028
$214$	0	0
$215$	17.3646	1.18425
$216$	0	0
$217$	−4.91060	−0.333353
$218$	0	0
$219$	0.706229	0.0477225
$220$	0	0
$221$	1.99738	0.134358
$222$	0	0
$223$	−8.47648	−0.567627	−0.283814	−	0.958879i	$-0.591600\pi$
$223$	−8.47648	−0.567627	−0.283814	+	0.958879i	$0.591600\pi$
$224$	0	0
$225$	6.17711	0.411807
$226$	0	0
$227$	−6.45625	−0.428517	−0.214258	−	0.976777i	$-0.568733\pi$
$227$	−6.45625	−0.428517	−0.214258	+	0.976777i	$0.568733\pi$
$228$	0	0
$229$	16.7311	1.10562	0.552811	−	0.833307i	$-0.313555\pi$
$229$	16.7311	1.10562	0.552811	+	0.833307i	$0.313555\pi$
$230$	0	0
$231$	0.358118	0.0235624
$232$	0	0
$233$	−9.43703	−0.618240	−0.309120	−	0.951023i	$-0.600035\pi$
$233$	−9.43703	−0.618240	−0.309120	+	0.951023i	$0.600035\pi$
$234$	0	0
$235$	−16.7190	−1.09063
$236$	0	0
$237$	1.27877	0.0830647
$238$	0	0
$239$	6.75425	0.436896	0.218448	−	0.975849i	$-0.429901\pi$
$239$	6.75425	0.436896	0.218448	+	0.975849i	$0.429901\pi$
$240$	0	0
$241$	2.26912	0.146167	0.0730835	−	0.997326i	$-0.476716\pi$
$241$	2.26912	0.146167	0.0730835	+	0.997326i	$0.476716\pi$
$242$	0	0
$243$	−6.90864	−0.443190
$244$	0	0
$245$	1.17617	0.0751426
$246$	0	0
$247$	−2.97769	−0.189466
$248$	0	0
$249$	2.22724	0.141146
$250$	0	0
$251$	4.04477	0.255304	0.127652	−	0.991819i	$-0.459256\pi$
$251$	4.04477	0.255304	0.127652	+	0.991819i	$0.459256\pi$
$252$	0	0
$253$	2.36548	0.148717
$254$	0	0
$255$	−1.78089	−0.111524
$256$	0	0
$257$	−3.06557	−0.191225	−0.0956123	−	0.995419i	$-0.530481\pi$
$257$	−3.06557	−0.191225	−0.0956123	+	0.995419i	$0.530481\pi$
$258$	0	0
$259$	−6.80097	−0.422592
$260$	0	0
$261$	−9.48848	−0.587322
$262$	0	0
$263$	−29.0913	−1.79384	−0.896922	−	0.442188i	$-0.854202\pi$
$263$	−29.0913	−1.79384	−0.896922	+	0.442188i	$0.854202\pi$
$264$	0	0
$265$	−2.54528	−0.156355
$266$	0	0
$267$	0.0423885	0.00259413
$268$	0	0
$269$	−2.49302	−0.152002	−0.0760010	−	0.997108i	$-0.524215\pi$
$269$	−2.49302	−0.152002	−0.0760010	+	0.997108i	$0.524215\pi$
$270$	0	0
$271$	12.3646	0.751099	0.375549	−	0.926802i	$-0.377454\pi$
$271$	12.3646	0.751099	0.375549	+	0.926802i	$0.377454\pi$
$272$	0	0
$273$	0.368689	0.0223141
$274$	0	0
$275$	−1.03104	−0.0621741
$276$	0	0
$277$	17.8174	1.07054	0.535271	−	0.844680i	$-0.320210\pi$
$277$	17.8174	1.07054	0.535271	+	0.844680i	$0.320210\pi$
$278$	0	0
$279$	−5.18845	−0.310625
$280$	0	0
$281$	−7.87017	−0.469495	−0.234748	−	0.972056i	$-0.575426\pi$
$281$	−7.87017	−0.469495	−0.234748	+	0.972056i	$0.575426\pi$
$282$	0	0
$283$	−17.6378	−1.04846	−0.524228	−	0.851578i	$-0.675646\pi$
$283$	−17.6378	−1.04846	−0.524228	+	0.851578i	$0.675646\pi$
$284$	0	0
$285$	2.65494	0.157265
$286$	0	0
$287$	−3.36994	−0.198921
$288$	0	0
$289$	−1.26567	−0.0744511
$290$	0	0
$291$	−4.37221	−0.256303
$292$	0	0
$293$	18.7670	1.09638	0.548189	−	0.836354i	$-0.315317\pi$
$293$	18.7670	1.09638	0.548189	+	0.836354i	$0.315317\pi$
$294$	0	0
$295$	6.32978	0.368534
$296$	0	0
$297$	0.765762	0.0444340
$298$	0	0
$299$	2.43531	0.140837
$300$	0	0
$301$	−28.3188	−1.63227
$302$	0	0
$303$	0.648648	0.0372638
$304$	0	0
$305$	−13.0809	−0.749011
$306$	0	0
$307$	24.1411	1.37780	0.688901	−	0.724855i	$-0.258093\pi$
$307$	24.1411	1.37780	0.688901	+	0.724855i	$0.258093\pi$
$308$	0	0
$309$	0.290488	0.0165253
$310$	0	0
$311$	13.6874	0.776141	0.388071	−	0.921630i	$-0.373142\pi$
$311$	13.6874	0.776141	0.388071	+	0.921630i	$0.373142\pi$
$312$	0	0
$313$	−4.33949	−0.245282	−0.122641	−	0.992451i	$-0.539136\pi$
$313$	−4.33949	−0.245282	−0.122641	+	0.992451i	$0.539136\pi$
$314$	0	0
$315$	13.8204	0.778689
$316$	0	0
$317$	−11.9221	−0.669612	−0.334806	−	0.942287i	$-0.608671\pi$
$317$	−11.9221	−0.669612	−0.334806	+	0.942287i	$0.608671\pi$
$318$	0	0
$319$	1.58375	0.0886731
$320$	0	0
$321$	−2.19897	−0.122735
$322$	0	0
$323$	−23.4567	−1.30516
$324$	0	0
$325$	−1.06148	−0.0588801
$326$	0	0
$327$	5.11033	0.282602
$328$	0	0
$329$	27.2659	1.50322
$330$	0	0
$331$	−31.6226	−1.73813	−0.869066	−	0.494695i	$-0.835280\pi$
$331$	−31.6226	−1.73813	−0.869066	+	0.494695i	$0.835280\pi$
$332$	0	0
$333$	−7.18578	−0.393778
$334$	0	0
$335$	4.13775	0.226069
$336$	0	0
$337$	21.9225	1.19420	0.597098	−	0.802168i	$-0.296320\pi$
$337$	21.9225	1.19420	0.597098	+	0.802168i	$0.296320\pi$
$338$	0	0
$339$	0.913608	0.0496204
$340$	0	0
$341$	0.866021	0.0468977
$342$	0	0
$343$	17.4955	0.944669
$344$	0	0
$345$	−2.17135	−0.116901
$346$	0	0
$347$	−26.0383	−1.39781	−0.698903	−	0.715216i	$-0.746328\pi$
$347$	−26.0383	−1.39781	−0.698903	+	0.715216i	$0.746328\pi$
$348$	0	0
$349$	0.387367	0.0207353	0.0103677	−	0.999946i	$-0.496700\pi$
$349$	0.387367	0.0207353	0.0103677	+	0.999946i	$0.496700\pi$
$350$	0	0
$351$	0.788366	0.0420799
$352$	0	0
$353$	−13.6658	−0.727359	−0.363680	−	0.931524i	$-0.618480\pi$
$353$	−13.6658	−0.727359	−0.363680	+	0.931524i	$0.618480\pi$
$354$	0	0
$355$	1.03438	0.0548990
$356$	0	0
$357$	2.90434	0.153714
$358$	0	0
$359$	−10.9703	−0.578992	−0.289496	−	0.957179i	$-0.593488\pi$
$359$	−10.9703	−0.578992	−0.289496	+	0.957179i	$0.593488\pi$
$360$	0	0
$361$	15.9691	0.840480
$362$	0	0
$363$	2.84091	0.149109
$364$	0	0
$365$	−4.54914	−0.238113
$366$	0	0
$367$	−17.9231	−0.935579	−0.467789	−	0.883840i	$-0.654949\pi$
$367$	−17.9231	−0.935579	−0.467789	+	0.883840i	$0.654949\pi$
$368$	0	0
$369$	−3.56062	−0.185358
$370$	0	0
$371$	4.15093	0.215506
$372$	0	0
$373$	−7.01196	−0.363066	−0.181533	−	0.983385i	$-0.558106\pi$
$373$	−7.01196	−0.363066	−0.181533	+	0.983385i	$0.558106\pi$
$374$	0	0
$375$	3.19125	0.164795
$376$	0	0
$377$	1.63050	0.0839751
$378$	0	0
$379$	−14.0895	−0.723729	−0.361864	−	0.932231i	$-0.617860\pi$
$379$	−14.0895	−0.723729	−0.361864	+	0.932231i	$0.617860\pi$
$380$	0	0
$381$	−2.81288	−0.144108
$382$	0	0
$383$	−15.1366	−0.773444	−0.386722	−	0.922196i	$-0.626393\pi$
$383$	−15.1366	−0.773444	−0.386722	+	0.922196i	$0.626393\pi$
$384$	0	0
$385$	−2.30680	−0.117565
$386$	0	0
$387$	−29.9211	−1.52098
$388$	0	0
$389$	15.1457	0.767915	0.383958	−	0.923351i	$-0.374561\pi$
$389$	15.1457	0.767915	0.383958	+	0.923351i	$0.374561\pi$
$390$	0	0
$391$	19.1841	0.970181
$392$	0	0
$393$	2.14496	0.108199
$394$	0	0
$395$	−8.23711	−0.414454
$396$	0	0
$397$	20.0540	1.00648	0.503240	−	0.864147i	$-0.332141\pi$
$397$	20.0540	1.00648	0.503240	+	0.864147i	$0.332141\pi$
$398$	0	0
$399$	−4.32978	−0.216760
$400$	0	0
$401$	20.2804	1.01276	0.506378	−	0.862312i	$-0.330984\pi$
$401$	20.2804	1.01276	0.506378	+	0.862312i	$0.330984\pi$
$402$	0	0
$403$	0.891585	0.0444130
$404$	0	0
$405$	14.2468	0.707927
$406$	0	0
$407$	1.19940	0.0594521
$408$	0	0
$409$	−34.7373	−1.71765	−0.858825	−	0.512270i	$-0.828805\pi$
$409$	−34.7373	−1.71765	−0.858825	+	0.512270i	$0.828805\pi$
$410$	0	0
$411$	2.78612	0.137429
$412$	0	0
$413$	−10.3228	−0.507954
$414$	0	0
$415$	−14.3467	−0.704251
$416$	0	0
$417$	2.79896	0.137066
$418$	0	0
$419$	−11.9600	−0.584285	−0.292143	−	0.956375i	$-0.594368\pi$
$419$	−11.9600	−0.584285	−0.292143	+	0.956375i	$0.594368\pi$
$420$	0	0
$421$	9.12944	0.444942	0.222471	−	0.974939i	$-0.428588\pi$
$421$	9.12944	0.444942	0.222471	+	0.974939i	$0.428588\pi$
$422$	0	0
$423$	28.8087	1.40073
$424$	0	0
$425$	−8.36175	−0.405604
$426$	0	0
$427$	21.3328	1.03237
$428$	0	0
$429$	−0.0650211	−0.00313925
$430$	0	0
$431$	−20.3556	−0.980493	−0.490247	−	0.871584i	$-0.663093\pi$
$431$	−20.3556	−0.980493	−0.490247	+	0.871584i	$0.663093\pi$
$432$	0	0
$433$	10.5212	0.505615	0.252808	−	0.967517i	$-0.418646\pi$
$433$	10.5212	0.505615	0.252808	+	0.967517i	$0.418646\pi$
$434$	0	0
$435$	−1.45377	−0.0697031
$436$	0	0
$437$	−28.5995	−1.36810
$438$	0	0
$439$	14.6234	0.697939	0.348969	−	0.937134i	$-0.386532\pi$
$439$	14.6234	0.697939	0.348969	+	0.937134i	$0.386532\pi$
$440$	0	0
$441$	−2.02667	−0.0965081
$442$	0	0
$443$	−6.94640	−0.330033	−0.165017	−	0.986291i	$-0.552768\pi$
$443$	−6.94640	−0.330033	−0.165017	+	0.986291i	$0.552768\pi$
$444$	0	0
$445$	−0.273044	−0.0129435
$446$	0	0
$447$	5.01726	0.237308
$448$	0	0
$449$	13.6844	0.645805	0.322903	−	0.946432i	$-0.395341\pi$
$449$	13.6844	0.645805	0.322903	+	0.946432i	$0.395341\pi$
$450$	0	0
$451$	0.594315	0.0279852
$452$	0	0
$453$	−1.30505	−0.0613168
$454$	0	0
$455$	−2.37489	−0.111337
$456$	0	0
$457$	−8.93983	−0.418188	−0.209094	−	0.977896i	$-0.567051\pi$
$457$	−8.93983	−0.418188	−0.209094	+	0.977896i	$0.567051\pi$
$458$	0	0
$459$	6.21034	0.289874
$460$	0	0
$461$	32.8793	1.53134	0.765672	−	0.643231i	$-0.222407\pi$
$461$	32.8793	1.53134	0.765672	+	0.643231i	$0.222407\pi$
$462$	0	0
$463$	−11.0218	−0.512226	−0.256113	−	0.966647i	$-0.582442\pi$
$463$	−11.0218	−0.512226	−0.256113	+	0.966647i	$0.582442\pi$
$464$	0	0
$465$	−0.794948	−0.0368648
$466$	0	0
$467$	−24.7644	−1.14596	−0.572980	−	0.819570i	$-0.694213\pi$
$467$	−24.7644	−1.14596	−0.572980	+	0.819570i	$0.694213\pi$
$468$	0	0
$469$	−6.74799	−0.311593
$470$	0	0
$471$	1.40722	0.0648415
$472$	0	0
$473$	4.99423	0.229635
$474$	0	0
$475$	12.4657	0.571964
$476$	0	0
$477$	4.38580	0.200812
$478$	0	0
$479$	−15.1346	−0.691516	−0.345758	−	0.938324i	$-0.612378\pi$
$479$	−15.1346	−0.691516	−0.345758	+	0.938324i	$0.612378\pi$
$480$	0	0
$481$	1.23481	0.0563023
$482$	0	0
$483$	3.54112	0.161126
$484$	0	0
$485$	28.1634	1.27883
$486$	0	0
$487$	6.94483	0.314700	0.157350	−	0.987543i	$-0.449705\pi$
$487$	6.94483	0.314700	0.157350	+	0.987543i	$0.449705\pi$
$488$	0	0
$489$	−1.23028	−0.0556351
$490$	0	0
$491$	23.9100	1.07904	0.539522	−	0.841971i	$-0.318605\pi$
$491$	23.9100	1.07904	0.539522	+	0.841971i	$0.318605\pi$
$492$	0	0
$493$	12.8442	0.578475
$494$	0	0
$495$	−2.43732	−0.109550
$496$	0	0
$497$	−1.68690	−0.0756678
$498$	0	0
$499$	−29.0093	−1.29864	−0.649318	−	0.760517i	$-0.724945\pi$
$499$	−29.0093	−1.29864	−0.649318	+	0.760517i	$0.724945\pi$
$500$	0	0
$501$	0.531415	0.0237419
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−4.17823	−0.185929
$506$	0	0
$507$	3.36514	0.149451
$508$	0	0
$509$	7.76339	0.344106	0.172053	−	0.985088i	$-0.444960\pi$
$509$	7.76339	0.344106	0.172053	+	0.985088i	$0.444960\pi$
$510$	0	0
$511$	7.41891	0.328193
$512$	0	0
$513$	−9.25835	−0.408766
$514$	0	0
$515$	−1.87116	−0.0824533
$516$	0	0
$517$	−4.80855	−0.211480
$518$	0	0
$519$	−0.957834	−0.0420443
$520$	0	0
$521$	−7.71912	−0.338181	−0.169090	−	0.985601i	$-0.554083\pi$
$521$	−7.71912	−0.338181	−0.169090	+	0.985601i	$0.554083\pi$
$522$	0	0
$523$	19.6173	0.857804	0.428902	−	0.903351i	$-0.358901\pi$
$523$	19.6173	0.857804	0.428902	+	0.903351i	$0.358901\pi$
$524$	0	0
$525$	−1.54346	−0.0673623
$526$	0	0
$527$	7.02344	0.305946
$528$	0	0
$529$	0.390183	0.0169645
$530$	0	0
$531$	−10.9069	−0.473320
$532$	0	0
$533$	0.611858	0.0265025
$534$	0	0
$535$	14.1646	0.612389
$536$	0	0
$537$	6.92935	0.299024
$538$	0	0
$539$	0.338278	0.0145707
$540$	0	0
$541$	−1.11159	−0.0477908	−0.0238954	−	0.999714i	$-0.507607\pi$
$541$	−1.11159	−0.0477908	−0.0238954	+	0.999714i	$0.507607\pi$
$542$	0	0
$543$	5.63104	0.241651
$544$	0	0
$545$	−32.9179	−1.41005
$546$	0	0
$547$	34.6931	1.48337	0.741685	−	0.670748i	$-0.234027\pi$
$547$	34.6931	1.48337	0.741685	+	0.670748i	$0.234027\pi$
$548$	0	0
$549$	22.5399	0.961979
$550$	0	0
$551$	−19.1481	−0.815738
$552$	0	0
$553$	13.4334	0.571246
$554$	0	0
$555$	−1.10097	−0.0467335
$556$	0	0
$557$	−8.52654	−0.361281	−0.180641	−	0.983549i	$-0.557817\pi$
$557$	−8.52654	−0.361281	−0.180641	+	0.983549i	$0.557817\pi$
$558$	0	0
$559$	5.14165	0.217469
$560$	0	0
$561$	−0.512202	−0.0216252
$562$	0	0
$563$	−9.84244	−0.414809	−0.207405	−	0.978255i	$-0.566502\pi$
$563$	−9.84244	−0.414809	−0.207405	+	0.978255i	$0.566502\pi$
$564$	0	0
$565$	−5.88496	−0.247582
$566$	0	0
$567$	−23.2342	−0.975743
$568$	0	0
$569$	34.1793	1.43287	0.716435	−	0.697654i	$-0.245773\pi$
$569$	34.1793	1.43287	0.716435	+	0.697654i	$0.245773\pi$
$570$	0	0
$571$	6.62723	0.277341	0.138670	−	0.990339i	$-0.455717\pi$
$571$	6.62723	0.277341	0.138670	+	0.990339i	$0.455717\pi$
$572$	0	0
$573$	−0.392758	−0.0164077
$574$	0	0
$575$	−10.1951	−0.425163
$576$	0	0
$577$	−33.9412	−1.41299	−0.706495	−	0.707718i	$-0.749725\pi$
$577$	−33.9412	−1.41299	−0.706495	+	0.707718i	$0.749725\pi$
$578$	0	0
$579$	1.14101	0.0474187
$580$	0	0
$581$	23.3971	0.970676
$582$	0	0
$583$	−0.732048	−0.0303183
$584$	0	0
$585$	−2.50927	−0.103746
$586$	0	0
$587$	44.4015	1.83265	0.916323	−	0.400439i	$-0.131142\pi$
$587$	44.4015	1.83265	0.916323	+	0.400439i	$0.131142\pi$
$588$	0	0
$589$	−10.4705	−0.431430
$590$	0	0
$591$	3.35341	0.137941
$592$	0	0
$593$	−11.2137	−0.460492	−0.230246	−	0.973132i	$-0.573953\pi$
$593$	−11.2137	−0.460492	−0.230246	+	0.973132i	$0.573953\pi$
$594$	0	0
$595$	−18.7082	−0.766961
$596$	0	0
$597$	−3.17178	−0.129812
$598$	0	0
$599$	−31.0359	−1.26809	−0.634045	−	0.773296i	$-0.718607\pi$
$599$	−31.0359	−1.26809	−0.634045	+	0.773296i	$0.718607\pi$
$600$	0	0
$601$	34.5547	1.40952	0.704758	−	0.709448i	$-0.251056\pi$
$601$	34.5547	1.40952	0.704758	+	0.709448i	$0.251056\pi$
$602$	0	0
$603$	−7.12981	−0.290348
$604$	0	0
$605$	−18.2996	−0.743985
$606$	0	0
$607$	2.44624	0.0992896	0.0496448	−	0.998767i	$-0.484191\pi$
$607$	2.44624	0.0992896	0.0496448	+	0.998767i	$0.484191\pi$
$608$	0	0
$609$	2.37087	0.0960725
$610$	0	0
$611$	−4.95049	−0.200275
$612$	0	0
$613$	−27.4211	−1.10753	−0.553764	−	0.832674i	$-0.686809\pi$
$613$	−27.4211	−1.10753	−0.553764	+	0.832674i	$0.686809\pi$
$614$	0	0
$615$	−0.545539	−0.0219983
$616$	0	0
$617$	48.5663	1.95520	0.977602	−	0.210462i	$-0.0674969\pi$
$617$	48.5663	1.95520	0.977602	+	0.210462i	$0.0674969\pi$
$618$	0	0
$619$	−22.6758	−0.911417	−0.455709	−	0.890129i	$-0.650614\pi$
$619$	−22.6758	−0.911417	−0.455709	+	0.890129i	$0.650614\pi$
$620$	0	0
$621$	7.57195	0.303852
$622$	0	0
$623$	0.445290	0.0178402
$624$	0	0
$625$	−10.0162	−0.400649
$626$	0	0
$627$	0.763589	0.0304948
$628$	0	0
$629$	9.72716	0.387847
$630$	0	0
$631$	−19.8475	−0.790116	−0.395058	−	0.918656i	$-0.629275\pi$
$631$	−19.8475	−0.790116	−0.395058	+	0.918656i	$0.629275\pi$
$632$	0	0
$633$	−1.60525	−0.0638030
$634$	0	0
$635$	18.1190	0.719031
$636$	0	0
$637$	0.348263	0.0137987
$638$	0	0
$639$	−1.78235	−0.0705086
$640$	0	0
$641$	−20.1430	−0.795602	−0.397801	−	0.917472i	$-0.630226\pi$
$641$	−20.1430	−0.795602	−0.397801	+	0.917472i	$0.630226\pi$
$642$	0	0
$643$	0.0146734	0.000578662 0	0.000289331	−	1.00000i	$-0.499908\pi$
$643$	0.0146734	0.000578662 0	0.000289331		1.00000i	$0.499908\pi$
$644$	0	0
$645$	−4.58436	−0.180509
$646$	0	0
$647$	−39.9997	−1.57255	−0.786275	−	0.617876i	$-0.787993\pi$
$647$	−39.9997	−1.57255	−0.786275	+	0.617876i	$0.787993\pi$
$648$	0	0
$649$	1.82051	0.0714612
$650$	0	0
$651$	1.29643	0.0508111
$652$	0	0
$653$	−2.14083	−0.0837771	−0.0418886	−	0.999122i	$-0.513337\pi$
$653$	−2.14083	−0.0837771	−0.0418886	+	0.999122i	$0.513337\pi$
$654$	0	0
$655$	−13.8167	−0.539862
$656$	0	0
$657$	7.83868	0.305816
$658$	0	0
$659$	−28.1414	−1.09623	−0.548117	−	0.836402i	$-0.684655\pi$
$659$	−28.1414	−1.09623	−0.548117	+	0.836402i	$0.684655\pi$
$660$	0	0
$661$	4.41391	0.171681	0.0858406	−	0.996309i	$-0.472642\pi$
$661$	4.41391	0.171681	0.0858406	+	0.996309i	$0.472642\pi$
$662$	0	0
$663$	−0.527322	−0.0204795
$664$	0	0
$665$	27.8901	1.08153
$666$	0	0
$667$	15.6603	0.606371
$668$	0	0
$669$	2.23785	0.0865201
$670$	0	0
$671$	−3.76221	−0.145238
$672$	0	0
$673$	−19.4693	−0.750485	−0.375242	−	0.926927i	$-0.622441\pi$
$673$	−19.4693	−0.750485	−0.375242	+	0.926927i	$0.622441\pi$
$674$	0	0
$675$	−3.30038	−0.127032
$676$	0	0
$677$	35.8642	1.37837	0.689187	−	0.724583i	$-0.257968\pi$
$677$	35.8642	1.37837	0.689187	+	0.724583i	$0.257968\pi$
$678$	0	0
$679$	−45.9299	−1.76263
$680$	0	0
$681$	1.70449	0.0653163
$682$	0	0
$683$	8.47358	0.324233	0.162116	−	0.986772i	$-0.448168\pi$
$683$	8.47358	0.324233	0.162116	+	0.986772i	$0.448168\pi$
$684$	0	0
$685$	−17.9467	−0.685707
$686$	0	0
$687$	−4.41712	−0.168524
$688$	0	0
$689$	−0.753657	−0.0287120
$690$	0	0
$691$	38.3402	1.45853	0.729264	−	0.684232i	$-0.239863\pi$
$691$	38.3402	1.45853	0.729264	+	0.684232i	$0.239863\pi$
$692$	0	0
$693$	3.97488	0.150993
$694$	0	0
$695$	−18.0294	−0.683894
$696$	0	0
$697$	4.81989	0.182567
$698$	0	0
$699$	2.49144	0.0942347
$700$	0	0
$701$	37.8783	1.43064	0.715322	−	0.698795i	$-0.246280\pi$
$701$	37.8783	1.43064	0.715322	+	0.698795i	$0.246280\pi$
$702$	0	0
$703$	−14.5012	−0.546923
$704$	0	0
$705$	4.41392	0.166238
$706$	0	0
$707$	6.81402	0.256268
$708$	0	0
$709$	17.2791	0.648930	0.324465	−	0.945898i	$-0.394816\pi$
$709$	17.2791	0.648930	0.324465	+	0.945898i	$0.394816\pi$
$710$	0	0
$711$	14.1935	0.532297
$712$	0	0
$713$	8.56333	0.320699
$714$	0	0
$715$	0.418830	0.0156634
$716$	0	0
$717$	−1.78317	−0.0665935
$718$	0	0
$719$	−44.6685	−1.66585	−0.832927	−	0.553383i	$-0.813337\pi$
$719$	−44.6685	−1.66585	−0.832927	+	0.553383i	$0.813337\pi$
$720$	0	0
$721$	3.05156	0.113646
$722$	0	0
$723$	−0.599063	−0.0222794
$724$	0	0
$725$	−6.82586	−0.253506
$726$	0	0
$727$	−27.3206	−1.01327	−0.506633	−	0.862162i	$-0.669110\pi$
$727$	−27.3206	−1.01327	−0.506633	+	0.862162i	$0.669110\pi$
$728$	0	0
$729$	−23.3088	−0.863288
$730$	0	0
$731$	40.5033	1.49807
$732$	0	0
$733$	48.7084	1.79908	0.899542	−	0.436834i	$-0.143900\pi$
$733$	48.7084	1.79908	0.899542	+	0.436834i	$0.143900\pi$
$734$	0	0
$735$	−0.310516	−0.0114535
$736$	0	0
$737$	1.19006	0.0438364
$738$	0	0
$739$	−33.1296	−1.21869	−0.609345	−	0.792905i	$-0.708568\pi$
$739$	−33.1296	−1.21869	−0.609345	+	0.792905i	$0.708568\pi$
$740$	0	0
$741$	0.786129	0.0288792
$742$	0	0
$743$	46.0968	1.69113	0.845563	−	0.533875i	$-0.179265\pi$
$743$	46.0968	1.69113	0.845563	+	0.533875i	$0.179265\pi$
$744$	0	0
$745$	−32.3185	−1.18406
$746$	0	0
$747$	24.7210	0.904493
$748$	0	0
$749$	−23.1002	−0.844061
$750$	0	0
$751$	0.237417	0.00866346	0.00433173	−	0.999991i	$-0.498621\pi$
$751$	0.237417	0.00866346	0.00433173	+	0.999991i	$0.498621\pi$
$752$	0	0
$753$	−1.06785	−0.0389145
$754$	0	0
$755$	8.40644	0.305942
$756$	0	0
$757$	−20.6409	−0.750206	−0.375103	−	0.926983i	$-0.622393\pi$
$757$	−20.6409	−0.750206	−0.375103	+	0.926983i	$0.622393\pi$
$758$	0	0
$759$	−0.624502	−0.0226680
$760$	0	0
$761$	−27.2109	−0.986396	−0.493198	−	0.869917i	$-0.664172\pi$
$761$	−27.2109	−0.986396	−0.493198	+	0.869917i	$0.664172\pi$
$762$	0	0
$763$	53.6838	1.94349
$764$	0	0
$765$	−19.7667	−0.714667
$766$	0	0
$767$	1.87425	0.0676752
$768$	0	0
$769$	−30.8623	−1.11292	−0.556462	−	0.830873i	$-0.687842\pi$
$769$	−30.8623	−1.11292	−0.556462	+	0.830873i	$0.687842\pi$
$770$	0	0
$771$	0.809329	0.0291473
$772$	0	0
$773$	34.7390	1.24947	0.624737	−	0.780835i	$-0.285206\pi$
$773$	34.7390	1.24947	0.624737	+	0.780835i	$0.285206\pi$
$774$	0	0
$775$	−3.73249	−0.134075
$776$	0	0
$777$	1.79550	0.0644132
$778$	0	0
$779$	−7.18548	−0.257446
$780$	0	0
$781$	0.297498	0.0106453
$782$	0	0
$783$	5.06962	0.181173
$784$	0	0
$785$	−9.06457	−0.323528
$786$	0	0
$787$	−14.2937	−0.509516	−0.254758	−	0.967005i	$-0.581996\pi$
$787$	−14.2937	−0.509516	−0.254758	+	0.967005i	$0.581996\pi$
$788$	0	0
$789$	7.68028	0.273425
$790$	0	0
$791$	9.59742	0.341245
$792$	0	0
$793$	−3.87326	−0.137543
$794$	0	0
$795$	0.671969	0.0238323
$796$	0	0
$797$	−1.85845	−0.0658296	−0.0329148	−	0.999458i	$-0.510479\pi$
$797$	−1.85845	−0.0658296	−0.0329148	+	0.999458i	$0.510479\pi$
$798$	0	0
$799$	−38.9974	−1.37963
$800$	0	0
$801$	0.470485	0.0166238
$802$	0	0
$803$	−1.30838	−0.0461717
$804$	0	0
$805$	−22.8099	−0.803945
$806$	0	0
$807$	0.658173	0.0231688
$808$	0	0
$809$	−9.11097	−0.320325	−0.160162	−	0.987091i	$-0.551202\pi$
$809$	−9.11097	−0.320325	−0.160162	+	0.987091i	$0.551202\pi$
$810$	0	0
$811$	−0.0817270	−0.00286982	−0.00143491	−	0.999999i	$-0.500457\pi$
$811$	−0.0817270	−0.00286982	−0.00143491	+	0.999999i	$0.500457\pi$
$812$	0	0
$813$	−3.26434	−0.114486
$814$	0	0
$815$	7.92479	0.277593
$816$	0	0
$817$	−60.3821	−2.11250
$818$	0	0
$819$	4.09221	0.142993
$820$	0	0
$821$	16.0958	0.561747	0.280873	−	0.959745i	$-0.409376\pi$
$821$	16.0958	0.561747	0.280873	+	0.959745i	$0.409376\pi$
$822$	0	0
$823$	33.0078	1.15058	0.575289	−	0.817950i	$-0.304889\pi$
$823$	33.0078	1.15058	0.575289	+	0.817950i	$0.304889\pi$
$824$	0	0
$825$	0.272201	0.00947683
$826$	0	0
$827$	3.80801	0.132418	0.0662088	−	0.997806i	$-0.478910\pi$
$827$	3.80801	0.132418	0.0662088	+	0.997806i	$0.478910\pi$
$828$	0	0
$829$	14.8116	0.514428	0.257214	−	0.966355i	$-0.417196\pi$
$829$	14.8116	0.514428	0.257214	+	0.966355i	$0.417196\pi$
$830$	0	0
$831$	−4.70390	−0.163176
$832$	0	0
$833$	2.74344	0.0950544
$834$	0	0
$835$	−3.42309	−0.118461
$836$	0	0
$837$	2.77215	0.0958196
$838$	0	0
$839$	−8.07929	−0.278928	−0.139464	−	0.990227i	$-0.544538\pi$
$839$	−8.07929	−0.278928	−0.139464	+	0.990227i	$0.544538\pi$
$840$	0	0
$841$	−18.5150	−0.638448
$842$	0	0
$843$	2.07778	0.0715624
$844$	0	0
$845$	−21.6764	−0.745691
$846$	0	0
$847$	29.8437	1.02544
$848$	0	0
$849$	4.65648	0.159810
$850$	0	0
$851$	11.8598	0.406550
$852$	0	0
$853$	−26.5131	−0.907790	−0.453895	−	0.891055i	$-0.649966\pi$
$853$	−26.5131	−0.907790	−0.453895	+	0.891055i	$0.649966\pi$
$854$	0	0
$855$	29.4681	1.00779
$856$	0	0
$857$	−8.21804	−0.280723	−0.140361	−	0.990100i	$-0.544826\pi$
$857$	−8.21804	−0.280723	−0.140361	+	0.990100i	$0.544826\pi$
$858$	0	0
$859$	−46.9731	−1.60270	−0.801351	−	0.598195i	$-0.795885\pi$
$859$	−46.9731	−1.60270	−0.801351	+	0.598195i	$0.795885\pi$
$860$	0	0
$861$	0.889686	0.0303204
$862$	0	0
$863$	−12.2051	−0.415467	−0.207734	−	0.978185i	$-0.566609\pi$
$863$	−12.2051	−0.415467	−0.207734	+	0.978185i	$0.566609\pi$
$864$	0	0
$865$	6.16985	0.209781
$866$	0	0
$867$	0.334145	0.0113481
$868$	0	0
$869$	−2.36908	−0.0803655
$870$	0	0
$871$	1.22519	0.0415139
$872$	0	0
$873$	−48.5287	−1.64245
$874$	0	0
$875$	33.5240	1.13332
$876$	0	0
$877$	−18.4995	−0.624682	−0.312341	−	0.949970i	$-0.601113\pi$
$877$	−18.4995	−0.624682	−0.312341	+	0.949970i	$0.601113\pi$
$878$	0	0
$879$	−4.95460	−0.167115
$880$	0	0
$881$	−40.6198	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$881$	−40.6198	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$882$	0	0
$883$	−13.4724	−0.453384	−0.226692	−	0.973966i	$-0.572791\pi$
$883$	−13.4724	−0.453384	−0.226692	+	0.973966i	$0.572791\pi$
$884$	0	0
$885$	−1.67110	−0.0561735
$886$	0	0
$887$	−9.72980	−0.326695	−0.163347	−	0.986569i	$-0.552229\pi$
$887$	−9.72980	−0.326695	−0.163347	+	0.986569i	$0.552229\pi$
$888$	0	0
$889$	−29.5492	−0.991048
$890$	0	0
$891$	4.09752	0.137272
$892$	0	0
$893$	58.1372	1.94549
$894$	0	0
$895$	−44.6351	−1.49199
$896$	0	0
$897$	−0.642937	−0.0214670
$898$	0	0
$899$	5.73337	0.191219
$900$	0	0
$901$	−5.93692	−0.197787
$902$	0	0
$903$	7.47634	0.248797
$904$	0	0
$905$	−36.2721	−1.20572
$906$	0	0
$907$	18.2977	0.607564	0.303782	−	0.952742i	$-0.401751\pi$
$907$	18.2977	0.607564	0.303782	+	0.952742i	$0.401751\pi$
$908$	0	0
$909$	7.19957	0.238795
$910$	0	0
$911$	−8.32911	−0.275956	−0.137978	−	0.990435i	$-0.544060\pi$
$911$	−8.32911	−0.275956	−0.137978	+	0.990435i	$0.544060\pi$
$912$	0	0
$913$	−4.12626	−0.136559
$914$	0	0
$915$	3.45344	0.114167
$916$	0	0
$917$	22.5327	0.744096
$918$	0	0
$919$	33.3594	1.10042	0.550212	−	0.835025i	$-0.314547\pi$
$919$	33.3594	1.10042	0.550212	+	0.835025i	$0.314547\pi$
$920$	0	0
$921$	−6.37339	−0.210010
$922$	0	0
$923$	0.306279	0.0100813
$924$	0	0
$925$	−5.16934	−0.169967
$926$	0	0
$927$	3.22423	0.105897
$928$	0	0
$929$	−6.26346	−0.205497	−0.102749	−	0.994707i	$-0.532764\pi$
$929$	−6.26346	−0.205497	−0.102749	+	0.994707i	$0.532764\pi$
$930$	0	0
$931$	−4.08990	−0.134041
$932$	0	0
$933$	−3.61356	−0.118303
$934$	0	0
$935$	3.29933	0.107900
$936$	0	0
$937$	−51.8247	−1.69304	−0.846520	−	0.532356i	$-0.821307\pi$
$937$	−51.8247	−1.69304	−0.846520	+	0.532356i	$0.821307\pi$
$938$	0	0
$939$	1.14565	0.0373870
$940$	0	0
$941$	−14.8476	−0.484018	−0.242009	−	0.970274i	$-0.577806\pi$
$941$	−14.8476	−0.484018	−0.242009	+	0.970274i	$0.577806\pi$
$942$	0	0
$943$	5.87666	0.191370
$944$	0	0
$945$	−7.38412	−0.240205
$946$	0	0
$947$	−6.66651	−0.216632	−0.108316	−	0.994116i	$-0.534546\pi$
$947$	−6.66651	−0.216632	−0.108316	+	0.994116i	$0.534546\pi$
$948$	0	0
$949$	−1.34700	−0.0437255
$950$	0	0
$951$	3.14751	0.102065
$952$	0	0
$953$	32.6563	1.05784	0.528921	−	0.848671i	$-0.322597\pi$
$953$	32.6563	1.05784	0.528921	+	0.848671i	$0.322597\pi$
$954$	0	0
$955$	2.52993	0.0818666
$956$	0	0
$957$	−0.418121	−0.0135159
$958$	0	0
$959$	29.2681	0.945116
$960$	0	0
$961$	−27.8649	−0.898868
$962$	0	0
$963$	−24.4072	−0.786511
$964$	0	0
$965$	−7.34976	−0.236597
$966$	0	0
$967$	1.87518	0.0603016	0.0301508	−	0.999545i	$-0.490401\pi$
$967$	1.87518	0.0603016	0.0301508	+	0.999545i	$0.490401\pi$
$968$	0	0
$969$	6.19271	0.198939
$970$	0	0
$971$	−38.2018	−1.22596	−0.612978	−	0.790100i	$-0.710029\pi$
$971$	−38.2018	−1.22596	−0.612978	+	0.790100i	$0.710029\pi$
$972$	0	0
$973$	29.4030	0.942618
$974$	0	0
$975$	0.280236	0.00897474
$976$	0	0
$977$	−59.8224	−1.91389	−0.956944	−	0.290274i	$-0.906254\pi$
$977$	−59.8224	−1.91389	−0.956944	+	0.290274i	$0.906254\pi$
$978$	0	0
$979$	−0.0785302	−0.00250984
$980$	0	0
$981$	56.7214	1.81097
$982$	0	0
$983$	45.3873	1.44763	0.723816	−	0.689993i	$-0.242387\pi$
$983$	45.3873	1.44763	0.723816	+	0.689993i	$0.242387\pi$
$984$	0	0
$985$	−21.6009	−0.688261
$986$	0	0
$987$	−7.19838	−0.229127
$988$	0	0
$989$	49.3836	1.57031
$990$	0	0
$991$	−44.1209	−1.40155	−0.700773	−	0.713384i	$-0.747162\pi$
$991$	−44.1209	−1.40155	−0.700773	+	0.713384i	$0.747162\pi$
$992$	0	0
$993$	8.34856	0.264933
$994$	0	0
$995$	20.4308	0.647701
$996$	0	0
$997$	−15.4025	−0.487802	−0.243901	−	0.969800i	$-0.578427\pi$
$997$	−15.4025	−0.487802	−0.243901	+	0.969800i	$0.578427\pi$
$998$	0	0
$999$	3.83931	0.121470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.q.1.7		12
4.3	odd	2		1006.2.a.j.1.6	✓	12
12.11	even	2		9054.2.a.bi.1.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1006.2.a.j.1.6	✓	12	4.3	odd	2
8048.2.a.q.1.7		12	1.1	even	1	trivial
9054.2.a.bi.1.5		12	12.11	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q-0.264006 q^{3} +1.70058 q^{5} -2.77338 q^{7} -2.93030 q^{9} +O(q^{10})\)	\(q-0.264006 q^{3} +1.70058 q^{5} -2.77338 q^{7} -2.93030 q^{9} +0.489106 q^{11} +0.503543 q^{13} -0.448965 q^{15} +3.96665 q^{17} -5.91347 q^{19} +0.732189 q^{21} +4.83634 q^{23} -2.10801 q^{25} +1.56564 q^{27} +3.23806 q^{29} +1.77062 q^{31} -0.129127 q^{33} -4.71636 q^{35} +2.45223 q^{37} -0.132939 q^{39} +1.21510 q^{41} +10.2109 q^{43} -4.98322 q^{45} -9.83131 q^{47} +0.691625 q^{49} -1.04722 q^{51} -1.49671 q^{53} +0.831766 q^{55} +1.56119 q^{57} +3.72212 q^{59} -7.69201 q^{61} +8.12683 q^{63} +0.856318 q^{65} +2.43313 q^{67} -1.27682 q^{69} +0.608248 q^{71} -2.67504 q^{73} +0.556528 q^{75} -1.35648 q^{77} -4.84369 q^{79} +8.37756 q^{81} -8.43632 q^{83} +6.74563 q^{85} -0.854867 q^{87} -0.160559 q^{89} -1.39652 q^{91} -0.467455 q^{93} -10.0564 q^{95} +16.5610 q^{97} -1.43323 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 5 q^{3} + 5 q^{5} - 8 q^{7} + 15 q^{9}+O(q^{10}) \) 12 * q - 5 * q^3 + 5 * q^5 - 8 * q^7 + 15 * q^9	\( 12 q - 5 q^{3} + 5 q^{5} - 8 q^{7} + 15 q^{9} - 18 q^{11} + 4 q^{13} - 2 q^{15} - 2 q^{17} - 6 q^{19} + q^{21} - 13 q^{23} + q^{25} - 8 q^{27} + 20 q^{29} - 7 q^{31} - 8 q^{33} - q^{35} + 10 q^{37} - 7 q^{39} + 2 q^{41} - 8 q^{43} - 7 q^{45} - 12 q^{47} + 4 q^{49} - 2 q^{51} + 12 q^{53} - 8 q^{55} - 10 q^{57} - 6 q^{59} - 10 q^{61} - 7 q^{63} + 4 q^{65} - 7 q^{67} - 12 q^{69} - 22 q^{71} - 23 q^{73} + 34 q^{75} - 19 q^{77} - 13 q^{79} - 28 q^{81} - q^{83} - 28 q^{85} + 22 q^{87} - 3 q^{89} + 21 q^{91} - 33 q^{93} + 2 q^{95} - 70 q^{97} + 6 q^{99}+O(q^{100}) \) 12 * q - 5 * q^3 + 5 * q^5 - 8 * q^7 + 15 * q^9 - 18 * q^11 + 4 * q^13 - 2 * q^15 - 2 * q^17 - 6 * q^19 + q^21 - 13 * q^23 + q^25 - 8 * q^27 + 20 * q^29 - 7 * q^31 - 8 * q^33 - q^35 + 10 * q^37 - 7 * q^39 + 2 * q^41 - 8 * q^43 - 7 * q^45 - 12 * q^47 + 4 * q^49 - 2 * q^51 + 12 * q^53 - 8 * q^55 - 10 * q^57 - 6 * q^59 - 10 * q^61 - 7 * q^63 + 4 * q^65 - 7 * q^67 - 12 * q^69 - 22 * q^71 - 23 * q^73 + 34 * q^75 - 19 * q^77 - 13 * q^79 - 28 * q^81 - q^83 - 28 * q^85 + 22 * q^87 - 3 * q^89 + 21 * q^91 - 33 * q^93 + 2 * q^95 - 70 * q^97 + 6 * q^99

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