LMFDB - Embedded newform 7448.2.a.bq.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7448,2,Mod(1,7448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7448, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7448.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7448 = 2^{3} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7448.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:	$59.4725794254$
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} - 15x^{6} - x^{5} + 66x^{4} + 4x^{3} - 76x^{2} + 8x + 16 $ x^8 - 15x^6 - x^5 + 66x^4 + 4x^3 - 76x^2 + 8*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1064)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.912296$ of defining polynomial
Character	$\chi$	$=$	7448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.912296 q^{3} -1.66528 q^{5} -2.16772 q^{9} +O(q^{10})$	$q+0.912296 q^{3} -1.66528 q^{5} -2.16772 q^{9} -0.242241 q^{11} -5.76999 q^{13} -1.51923 q^{15} +5.76637 q^{17} +1.00000 q^{19} +6.61545 q^{23} -2.22684 q^{25} -4.71449 q^{27} +8.91381 q^{29} -5.93284 q^{31} -0.220995 q^{33} -9.80371 q^{37} -5.26394 q^{39} -2.00000 q^{41} -7.83237 q^{43} +3.60986 q^{45} +1.57918 q^{47} +5.26064 q^{51} +6.48146 q^{53} +0.403399 q^{55} +0.912296 q^{57} +1.07498 q^{59} +9.10907 q^{61} +9.60866 q^{65} +8.02541 q^{67} +6.03525 q^{69} +1.22955 q^{71} +0.801370 q^{73} -2.03153 q^{75} -7.58202 q^{79} +2.20214 q^{81} -8.91292 q^{83} -9.60264 q^{85} +8.13204 q^{87} -13.1146 q^{89} -5.41250 q^{93} -1.66528 q^{95} +1.11664 q^{97} +0.525109 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{5} + 6 q^{9}+O(q^{10}) $ 8 * q - q^5 + 6 * q^9	$ 8 q - q^{5} + 6 q^{9} + 9 q^{11} + 8 q^{15} + 4 q^{17} + 8 q^{19} + 25 q^{23} + 15 q^{25} + 3 q^{27} + 6 q^{29} + 8 q^{33} + 13 q^{37} - 11 q^{39} - 16 q^{41} + 17 q^{43} + 17 q^{45} - 24 q^{47} + 5 q^{51} + 2 q^{53} + 5 q^{55} + 2 q^{59} - 13 q^{61} - 26 q^{65} + 2 q^{67} - 11 q^{69} + 10 q^{71} + 5 q^{73} - 20 q^{75} + 16 q^{79} - 12 q^{81} - 43 q^{83} + 24 q^{85} + 20 q^{87} + 8 q^{89} + 2 q^{93} - q^{95} - 12 q^{97} + 37 q^{99}+O(q^{100}) $ 8 * q - q^5 + 6 * q^9 + 9 * q^11 + 8 * q^15 + 4 * q^17 + 8 * q^19 + 25 * q^23 + 15 * q^25 + 3 * q^27 + 6 * q^29 + 8 * q^33 + 13 * q^37 - 11 * q^39 - 16 * q^41 + 17 * q^43 + 17 * q^45 - 24 * q^47 + 5 * q^51 + 2 * q^53 + 5 * q^55 + 2 * q^59 - 13 * q^61 - 26 * q^65 + 2 * q^67 - 11 * q^69 + 10 * q^71 + 5 * q^73 - 20 * q^75 + 16 * q^79 - 12 * q^81 - 43 * q^83 + 24 * q^85 + 20 * q^87 + 8 * q^89 + 2 * q^93 - q^95 - 12 * q^97 + 37 * 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
$3$	0.912296	0.526714	0.263357	−	0.964698i	$-0.415170\pi$
$3$	0.912296	0.526714	0.263357	+	0.964698i	$0.415170\pi$
$4$	0	0
$5$	−1.66528	−0.744737	−0.372368	−	0.928085i	$-0.621454\pi$
$5$	−1.66528	−0.744737	−0.372368	+	0.928085i	$0.621454\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.16772	−0.722572
$10$	0	0
$11$	−0.242241	−0.0730384	−0.0365192	−	0.999333i	$-0.511627\pi$
$11$	−0.242241	−0.0730384	−0.0365192	+	0.999333i	$0.511627\pi$
$12$	0	0
$13$	−5.76999	−1.60031	−0.800154	−	0.599795i	$-0.795249\pi$
$13$	−5.76999	−1.60031	−0.800154	+	0.599795i	$0.795249\pi$
$14$	0	0
$15$	−1.51923	−0.392264
$16$	0	0
$17$	5.76637	1.39855	0.699275	−	0.714852i	$-0.253506\pi$
$17$	5.76637	1.39855	0.699275	+	0.714852i	$0.253506\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.61545	1.37942	0.689709	−	0.724087i	$-0.257739\pi$
$23$	6.61545	1.37942	0.689709	+	0.724087i	$0.257739\pi$
$24$	0	0
$25$	−2.22684	−0.445367
$26$	0	0
$27$	−4.71449	−0.907303
$28$	0	0
$29$	8.91381	1.65525	0.827627	−	0.561279i	$-0.189690\pi$
$29$	8.91381	1.65525	0.827627	+	0.561279i	$0.189690\pi$
$30$	0	0
$31$	−5.93284	−1.06557	−0.532785	−	0.846251i	$-0.678854\pi$
$31$	−5.93284	−1.06557	−0.532785	+	0.846251i	$0.678854\pi$
$32$	0	0
$33$	−0.220995	−0.0384704
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9.80371	−1.61172	−0.805860	−	0.592106i	$-0.798297\pi$
$37$	−9.80371	−1.61172	−0.805860	+	0.592106i	$0.798297\pi$
$38$	0	0
$39$	−5.26394	−0.842905
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	−7.83237	−1.19443	−0.597213	−	0.802083i	$-0.703725\pi$
$43$	−7.83237	−1.19443	−0.597213	+	0.802083i	$0.703725\pi$
$44$	0	0
$45$	3.60986	0.538126
$46$	0	0
$47$	1.57918	0.230347	0.115173	−	0.993345i	$-0.463258\pi$
$47$	1.57918	0.230347	0.115173	+	0.993345i	$0.463258\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	5.26064	0.736637
$52$	0	0
$53$	6.48146	0.890297	0.445148	−	0.895457i	$-0.353151\pi$
$53$	6.48146	0.890297	0.445148	+	0.895457i	$0.353151\pi$
$54$	0	0
$55$	0.403399	0.0543944
$56$	0	0
$57$	0.912296	0.120837
$58$	0	0
$59$	1.07498	0.139950	0.0699750	−	0.997549i	$-0.477708\pi$
$59$	1.07498	0.139950	0.0699750	+	0.997549i	$0.477708\pi$
$60$	0	0
$61$	9.10907	1.16630	0.583149	−	0.812366i	$-0.301821\pi$
$61$	9.10907	1.16630	0.583149	+	0.812366i	$0.301821\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	9.60866	1.19181
$66$	0	0
$67$	8.02541	0.980460	0.490230	−	0.871593i	$-0.336913\pi$
$67$	8.02541	0.980460	0.490230	+	0.871593i	$0.336913\pi$
$68$	0	0
$69$	6.03525	0.726559
$70$	0	0
$71$	1.22955	0.145921	0.0729606	−	0.997335i	$-0.476755\pi$
$71$	1.22955	0.145921	0.0729606	+	0.997335i	$0.476755\pi$
$72$	0	0
$73$	0.801370	0.0937933	0.0468966	−	0.998900i	$-0.485067\pi$
$73$	0.801370	0.0937933	0.0468966	+	0.998900i	$0.485067\pi$
$74$	0	0
$75$	−2.03153	−0.234581
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−7.58202	−0.853044	−0.426522	−	0.904477i	$-0.640261\pi$
$79$	−7.58202	−0.853044	−0.426522	+	0.904477i	$0.640261\pi$
$80$	0	0
$81$	2.20214	0.244682
$82$	0	0
$83$	−8.91292	−0.978320	−0.489160	−	0.872194i	$-0.662697\pi$
$83$	−8.91292	−0.978320	−0.489160	+	0.872194i	$0.662697\pi$
$84$	0	0
$85$	−9.60264	−1.04155
$86$	0	0
$87$	8.13204	0.871846
$88$	0	0
$89$	−13.1146	−1.39015	−0.695074	−	0.718938i	$-0.744628\pi$
$89$	−13.1146	−1.39015	−0.695074	+	0.718938i	$0.744628\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−5.41250	−0.561250
$94$	0	0
$95$	−1.66528	−0.170854
$96$	0	0
$97$	1.11664	0.113377	0.0566886	−	0.998392i	$-0.481946\pi$
$97$	1.11664	0.113377	0.0566886	+	0.998392i	$0.481946\pi$
$98$	0	0
$99$	0.525109	0.0527755
$100$	0	0
$101$	13.0276	1.29630	0.648149	−	0.761514i	$-0.275543\pi$
$101$	13.0276	1.29630	0.648149	+	0.761514i	$0.275543\pi$
$102$	0	0
$103$	−13.7259	−1.35245	−0.676226	−	0.736694i	$-0.736386\pi$
$103$	−13.7259	−1.35245	−0.676226	+	0.736694i	$0.736386\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.76847	0.364312	0.182156	−	0.983270i	$-0.441692\pi$
$107$	3.76847	0.364312	0.182156	+	0.983270i	$0.441692\pi$
$108$	0	0
$109$	−0.674254	−0.0645819	−0.0322909	−	0.999479i	$-0.510280\pi$
$109$	−0.674254	−0.0645819	−0.0322909	+	0.999479i	$0.510280\pi$
$110$	0	0
$111$	−8.94388	−0.848916
$112$	0	0
$113$	10.1480	0.954646	0.477323	−	0.878728i	$-0.341607\pi$
$113$	10.1480	0.954646	0.477323	+	0.878728i	$0.341607\pi$
$114$	0	0
$115$	−11.0166	−1.02730
$116$	0	0
$117$	12.5077	1.15634
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.9413	−0.994665
$122$	0	0
$123$	−1.82459	−0.164518
$124$	0	0
$125$	12.0347	1.07642
$126$	0	0
$127$	18.5974	1.65025	0.825126	−	0.564949i	$-0.191104\pi$
$127$	18.5974	1.65025	0.825126	+	0.564949i	$0.191104\pi$
$128$	0	0
$129$	−7.14544	−0.629121
$130$	0	0
$131$	7.97141	0.696465	0.348233	−	0.937408i	$-0.386782\pi$
$131$	7.97141	0.696465	0.348233	+	0.937408i	$0.386782\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	7.85095	0.675702
$136$	0	0
$137$	−0.318763	−0.0272338	−0.0136169	−	0.999907i	$-0.504335\pi$
$137$	−0.318763	−0.0272338	−0.0136169	+	0.999907i	$0.504335\pi$
$138$	0	0
$139$	−5.80115	−0.492047	−0.246024	−	0.969264i	$-0.579124\pi$
$139$	−5.80115	−0.492047	−0.246024	+	0.969264i	$0.579124\pi$
$140$	0	0
$141$	1.44068	0.121327
$142$	0	0
$143$	1.39773	0.116884
$144$	0	0
$145$	−14.8440	−1.23273
$146$	0	0
$147$	0	0
$148$	0	0
$149$	13.6701	1.11990	0.559948	−	0.828528i	$-0.310821\pi$
$149$	13.6701	1.11990	0.559948	+	0.828528i	$0.310821\pi$
$150$	0	0
$151$	11.0282	0.897464	0.448732	−	0.893666i	$-0.351876\pi$
$151$	11.0282	0.897464	0.448732	+	0.893666i	$0.351876\pi$
$152$	0	0
$153$	−12.4999	−1.01055
$154$	0	0
$155$	9.87985	0.793568
$156$	0	0
$157$	17.8566	1.42511	0.712556	−	0.701616i	$-0.247538\pi$
$157$	17.8566	1.42511	0.712556	+	0.701616i	$0.247538\pi$
$158$	0	0
$159$	5.91301	0.468932
$160$	0	0
$161$	0	0
$162$	0	0
$163$	11.1970	0.877018	0.438509	−	0.898727i	$-0.355507\pi$
$163$	11.1970	0.877018	0.438509	+	0.898727i	$0.355507\pi$
$164$	0	0
$165$	0.368020	0.0286503
$166$	0	0
$167$	1.44712	0.111981	0.0559906	−	0.998431i	$-0.482168\pi$
$167$	1.44712	0.111981	0.0559906	+	0.998431i	$0.482168\pi$
$168$	0	0
$169$	20.2928	1.56098
$170$	0	0
$171$	−2.16772	−0.165769
$172$	0	0
$173$	11.8941	0.904292	0.452146	−	0.891944i	$-0.350659\pi$
$173$	11.8941	0.904292	0.452146	+	0.891944i	$0.350659\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0.980697	0.0737137
$178$	0	0
$179$	25.1728	1.88150	0.940751	−	0.339098i	$-0.110122\pi$
$179$	25.1728	1.88150	0.940751	+	0.339098i	$0.110122\pi$
$180$	0	0
$181$	18.7410	1.39301	0.696503	−	0.717554i	$-0.254738\pi$
$181$	18.7410	1.39301	0.696503	+	0.717554i	$0.254738\pi$
$182$	0	0
$183$	8.31017	0.614305
$184$	0	0
$185$	16.3259	1.20031
$186$	0	0
$187$	−1.39685	−0.102148
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0.356002	0.0257594	0.0128797	−	0.999917i	$-0.495900\pi$
$191$	0.356002	0.0257594	0.0128797	+	0.999917i	$0.495900\pi$
$192$	0	0
$193$	17.0988	1.23080	0.615398	−	0.788216i	$-0.288995\pi$
$193$	17.0988	1.23080	0.615398	+	0.788216i	$0.288995\pi$
$194$	0	0
$195$	8.76594	0.627742
$196$	0	0
$197$	13.1088	0.933964	0.466982	−	0.884267i	$-0.345341\pi$
$197$	13.1088	0.933964	0.466982	+	0.884267i	$0.345341\pi$
$198$	0	0
$199$	4.76977	0.338120	0.169060	−	0.985606i	$-0.445927\pi$
$199$	4.76977	0.338120	0.169060	+	0.985606i	$0.445927\pi$
$200$	0	0
$201$	7.32155	0.516422
$202$	0	0
$203$	0	0
$204$	0	0
$205$	3.33056	0.232617
$206$	0	0
$207$	−14.3404	−0.996728
$208$	0	0
$209$	−0.242241	−0.0167561
$210$	0	0
$211$	22.0442	1.51759	0.758794	−	0.651331i	$-0.225789\pi$
$211$	22.0442	1.51759	0.758794	+	0.651331i	$0.225789\pi$
$212$	0	0
$213$	1.12172	0.0768588
$214$	0	0
$215$	13.0431	0.889533
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0.731087	0.0494023
$220$	0	0
$221$	−33.2719	−2.23811
$222$	0	0
$223$	19.5836	1.31142	0.655708	−	0.755014i	$-0.272370\pi$
$223$	19.5836	1.31142	0.655708	+	0.755014i	$0.272370\pi$
$224$	0	0
$225$	4.82715	0.321810
$226$	0	0
$227$	−5.78829	−0.384182	−0.192091	−	0.981377i	$-0.561527\pi$
$227$	−5.78829	−0.384182	−0.192091	+	0.981377i	$0.561527\pi$
$228$	0	0
$229$	−19.1074	−1.26266	−0.631328	−	0.775516i	$-0.717490\pi$
$229$	−19.1074	−1.26266	−0.631328	+	0.775516i	$0.717490\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.17125	−0.404292	−0.202146	−	0.979355i	$-0.564792\pi$
$233$	−6.17125	−0.404292	−0.202146	+	0.979355i	$0.564792\pi$
$234$	0	0
$235$	−2.62978	−0.171548
$236$	0	0
$237$	−6.91705	−0.449311
$238$	0	0
$239$	26.1425	1.69101	0.845507	−	0.533964i	$-0.179298\pi$
$239$	26.1425	1.69101	0.845507	+	0.533964i	$0.179298\pi$
$240$	0	0
$241$	−19.5916	−1.26201	−0.631003	−	0.775780i	$-0.717357\pi$
$241$	−19.5916	−1.26201	−0.631003	+	0.775780i	$0.717357\pi$
$242$	0	0
$243$	16.1525	1.03618
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5.76999	−0.367136
$248$	0	0
$249$	−8.13122	−0.515295
$250$	0	0
$251$	−6.39080	−0.403384	−0.201692	−	0.979449i	$-0.564644\pi$
$251$	−6.39080	−0.403384	−0.201692	+	0.979449i	$0.564644\pi$
$252$	0	0
$253$	−1.60253	−0.100750
$254$	0	0
$255$	−8.76045	−0.548600
$256$	0	0
$257$	21.8742	1.36448	0.682238	−	0.731130i	$-0.261007\pi$
$257$	21.8742	1.36448	0.682238	+	0.731130i	$0.261007\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−19.3226	−1.19604
$262$	0	0
$263$	−29.7237	−1.83284	−0.916420	−	0.400218i	$-0.868934\pi$
$263$	−29.7237	−1.83284	−0.916420	+	0.400218i	$0.868934\pi$
$264$	0	0
$265$	−10.7935	−0.663037
$266$	0	0
$267$	−11.9644	−0.732211
$268$	0	0
$269$	13.3856	0.816133	0.408067	−	0.912952i	$-0.366203\pi$
$269$	13.3856	0.816133	0.408067	+	0.912952i	$0.366203\pi$
$270$	0	0
$271$	−25.7732	−1.56561	−0.782805	−	0.622267i	$-0.786212\pi$
$271$	−25.7732	−1.56561	−0.782805	+	0.622267i	$0.786212\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.539431	0.0325289
$276$	0	0
$277$	−2.97402	−0.178692	−0.0893458	−	0.996001i	$-0.528478\pi$
$277$	−2.97402	−0.178692	−0.0893458	+	0.996001i	$0.528478\pi$
$278$	0	0
$279$	12.8607	0.769950
$280$	0	0
$281$	−18.8947	−1.12716	−0.563582	−	0.826060i	$-0.690577\pi$
$281$	−18.8947	−1.12716	−0.563582	+	0.826060i	$0.690577\pi$
$282$	0	0
$283$	22.8300	1.35710	0.678551	−	0.734554i	$-0.262608\pi$
$283$	22.8300	1.35710	0.678551	+	0.734554i	$0.262608\pi$
$284$	0	0
$285$	−1.51923	−0.0899914
$286$	0	0
$287$	0	0
$288$	0	0
$289$	16.2510	0.955944
$290$	0	0
$291$	1.01870	0.0597174
$292$	0	0
$293$	11.3710	0.664299	0.332150	−	0.943227i	$-0.392226\pi$
$293$	11.3710	0.664299	0.332150	+	0.943227i	$0.392226\pi$
$294$	0	0
$295$	−1.79014	−0.104226
$296$	0	0
$297$	1.14204	0.0662680
$298$	0	0
$299$	−38.1711	−2.20749
$300$	0	0
$301$	0	0
$302$	0	0
$303$	11.8851	0.682779
$304$	0	0
$305$	−15.1692	−0.868584
$306$	0	0
$307$	−3.87636	−0.221236	−0.110618	−	0.993863i	$-0.535283\pi$
$307$	−3.87636	−0.221236	−0.110618	+	0.993863i	$0.535283\pi$
$308$	0	0
$309$	−12.5221	−0.712356
$310$	0	0
$311$	22.0561	1.25069	0.625344	−	0.780349i	$-0.284959\pi$
$311$	22.0561	1.25069	0.625344	+	0.780349i	$0.284959\pi$
$312$	0	0
$313$	−31.0643	−1.75586	−0.877930	−	0.478790i	$-0.841076\pi$
$313$	−31.0643	−1.75586	−0.877930	+	0.478790i	$0.841076\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	33.2863	1.86954	0.934772	−	0.355249i	$-0.115604\pi$
$317$	33.2863	1.86954	0.934772	+	0.355249i	$0.115604\pi$
$318$	0	0
$319$	−2.15929	−0.120897
$320$	0	0
$321$	3.43796	0.191888
$322$	0	0
$323$	5.76637	0.320850
$324$	0	0
$325$	12.8488	0.712724
$326$	0	0
$327$	−0.615120	−0.0340162
$328$	0	0
$329$	0	0
$330$	0	0
$331$	8.37379	0.460265	0.230133	−	0.973159i	$-0.426084\pi$
$331$	8.37379	0.460265	0.230133	+	0.973159i	$0.426084\pi$
$332$	0	0
$333$	21.2517	1.16458
$334$	0	0
$335$	−13.3646	−0.730184
$336$	0	0
$337$	−1.87908	−0.102360	−0.0511799	−	0.998689i	$-0.516298\pi$
$337$	−1.87908	−0.102360	−0.0511799	+	0.998689i	$0.516298\pi$
$338$	0	0
$339$	9.25801	0.502826
$340$	0	0
$341$	1.43718	0.0778274
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−10.0504	−0.541095
$346$	0	0
$347$	−6.13726	−0.329465	−0.164733	−	0.986338i	$-0.552676\pi$
$347$	−6.13726	−0.329465	−0.164733	+	0.986338i	$0.552676\pi$
$348$	0	0
$349$	−18.9103	−1.01225	−0.506123	−	0.862461i	$-0.668922\pi$
$349$	−18.9103	−1.01225	−0.506123	+	0.862461i	$0.668922\pi$
$350$	0	0
$351$	27.2025	1.45196
$352$	0	0
$353$	−9.94453	−0.529294	−0.264647	−	0.964345i	$-0.585255\pi$
$353$	−9.94453	−0.529294	−0.264647	+	0.964345i	$0.585255\pi$
$354$	0	0
$355$	−2.04755	−0.108673
$356$	0	0
$357$	0	0
$358$	0	0
$359$	18.3900	0.970590	0.485295	−	0.874351i	$-0.338712\pi$
$359$	18.3900	0.970590	0.485295	+	0.874351i	$0.338712\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−9.98172	−0.523905
$364$	0	0
$365$	−1.33451	−0.0698513
$366$	0	0
$367$	−14.2152	−0.742029	−0.371014	−	0.928627i	$-0.620990\pi$
$367$	−14.2152	−0.742029	−0.371014	+	0.928627i	$0.620990\pi$
$368$	0	0
$369$	4.33543	0.225694
$370$	0	0
$371$	0	0
$372$	0	0
$373$	25.6016	1.32560	0.662800	−	0.748796i	$-0.269368\pi$
$373$	25.6016	1.32560	0.662800	+	0.748796i	$0.269368\pi$
$374$	0	0
$375$	10.9792	0.566965
$376$	0	0
$377$	−51.4326	−2.64891
$378$	0	0
$379$	−8.51685	−0.437481	−0.218741	−	0.975783i	$-0.570195\pi$
$379$	−8.51685	−0.437481	−0.218741	+	0.975783i	$0.570195\pi$
$380$	0	0
$381$	16.9663	0.869211
$382$	0	0
$383$	−10.0169	−0.511841	−0.255921	−	0.966698i	$-0.582379\pi$
$383$	−10.0169	−0.511841	−0.255921	+	0.966698i	$0.582379\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	16.9784	0.863059
$388$	0	0
$389$	−20.2802	−1.02825	−0.514123	−	0.857717i	$-0.671882\pi$
$389$	−20.2802	−1.02825	−0.514123	+	0.857717i	$0.671882\pi$
$390$	0	0
$391$	38.1472	1.92918
$392$	0	0
$393$	7.27229	0.366838
$394$	0	0
$395$	12.6262	0.635293
$396$	0	0
$397$	−26.8275	−1.34643	−0.673216	−	0.739446i	$-0.735088\pi$
$397$	−26.8275	−1.34643	−0.673216	+	0.739446i	$0.735088\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−13.8582	−0.692047	−0.346023	−	0.938226i	$-0.612468\pi$
$401$	−13.8582	−0.692047	−0.346023	+	0.938226i	$0.612468\pi$
$402$	0	0
$403$	34.2324	1.70524
$404$	0	0
$405$	−3.66719	−0.182224
$406$	0	0
$407$	2.37486	0.117717
$408$	0	0
$409$	−6.19430	−0.306288	−0.153144	−	0.988204i	$-0.548940\pi$
$409$	−6.19430	−0.306288	−0.153144	+	0.988204i	$0.548940\pi$
$410$	0	0
$411$	−0.290807	−0.0143444
$412$	0	0
$413$	0	0
$414$	0	0
$415$	14.8425	0.728591
$416$	0	0
$417$	−5.29237	−0.259168
$418$	0	0
$419$	14.4938	0.708068	0.354034	−	0.935233i	$-0.384810\pi$
$419$	14.4938	0.708068	0.354034	+	0.935233i	$0.384810\pi$
$420$	0	0
$421$	29.8753	1.45603	0.728016	−	0.685560i	$-0.240443\pi$
$421$	29.8753	1.45603	0.728016	+	0.685560i	$0.240443\pi$
$422$	0	0
$423$	−3.42321	−0.166442
$424$	0	0
$425$	−12.8408	−0.622868
$426$	0	0
$427$	0	0
$428$	0	0
$429$	1.27514	0.0615644
$430$	0	0
$431$	21.8609	1.05300	0.526501	−	0.850175i	$-0.323504\pi$
$431$	21.8609	1.05300	0.526501	+	0.850175i	$0.323504\pi$
$432$	0	0
$433$	−11.1847	−0.537500	−0.268750	−	0.963210i	$-0.586611\pi$
$433$	−11.1847	−0.537500	−0.268750	+	0.963210i	$0.586611\pi$
$434$	0	0
$435$	−13.5421	−0.649296
$436$	0	0
$437$	6.61545	0.316460
$438$	0	0
$439$	−15.4187	−0.735896	−0.367948	−	0.929846i	$-0.619940\pi$
$439$	−15.4187	−0.735896	−0.367948	+	0.929846i	$0.619940\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−0.939483	−0.0446362	−0.0223181	−	0.999751i	$-0.507105\pi$
$443$	−0.939483	−0.0446362	−0.0223181	+	0.999751i	$0.507105\pi$
$444$	0	0
$445$	21.8396	1.03529
$446$	0	0
$447$	12.4711	0.589865
$448$	0	0
$449$	−28.2058	−1.33112	−0.665558	−	0.746346i	$-0.731806\pi$
$449$	−28.2058	−1.33112	−0.665558	+	0.746346i	$0.731806\pi$
$450$	0	0
$451$	0.484482	0.0228134
$452$	0	0
$453$	10.0610	0.472707
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0.777826	0.0363852	0.0181926	−	0.999835i	$-0.494209\pi$
$457$	0.777826	0.0363852	0.0181926	+	0.999835i	$0.494209\pi$
$458$	0	0
$459$	−27.1855	−1.26891
$460$	0	0
$461$	−19.3614	−0.901751	−0.450875	−	0.892587i	$-0.648888\pi$
$461$	−19.3614	−0.901751	−0.450875	+	0.892587i	$0.648888\pi$
$462$	0	0
$463$	34.5688	1.60655	0.803275	−	0.595608i	$-0.203089\pi$
$463$	34.5688	1.60655	0.803275	+	0.595608i	$0.203089\pi$
$464$	0	0
$465$	9.01335	0.417984
$466$	0	0
$467$	−29.8545	−1.38150	−0.690750	−	0.723094i	$-0.742719\pi$
$467$	−29.8545	−1.38150	−0.690750	+	0.723094i	$0.742719\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	16.2905	0.750626
$472$	0	0
$473$	1.89732	0.0872389
$474$	0	0
$475$	−2.22684	−0.102174
$476$	0	0
$477$	−14.0500	−0.643303
$478$	0	0
$479$	−27.8865	−1.27416	−0.637082	−	0.770796i	$-0.719859\pi$
$479$	−27.8865	−1.27416	−0.637082	+	0.770796i	$0.719859\pi$
$480$	0	0
$481$	56.5673	2.57925
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.85951	−0.0844361
$486$	0	0
$487$	11.6224	0.526660	0.263330	−	0.964706i	$-0.415179\pi$
$487$	11.6224	0.526660	0.263330	+	0.964706i	$0.415179\pi$
$488$	0	0
$489$	10.2150	0.461938
$490$	0	0
$491$	−25.4037	−1.14645	−0.573225	−	0.819398i	$-0.694308\pi$
$491$	−25.4037	−1.14645	−0.573225	+	0.819398i	$0.694308\pi$
$492$	0	0
$493$	51.4004	2.31496
$494$	0	0
$495$	−0.874455	−0.0393038
$496$	0	0
$497$	0	0
$498$	0	0
$499$	2.67840	0.119902	0.0599509	−	0.998201i	$-0.480906\pi$
$499$	2.67840	0.119902	0.0599509	+	0.998201i	$0.480906\pi$
$500$	0	0
$501$	1.32020	0.0589821
$502$	0	0
$503$	−37.0893	−1.65373	−0.826866	−	0.562399i	$-0.809879\pi$
$503$	−37.0893	−1.65373	−0.826866	+	0.562399i	$0.809879\pi$
$504$	0	0
$505$	−21.6947	−0.965401
$506$	0	0
$507$	18.5130	0.822192
$508$	0	0
$509$	14.8243	0.657075	0.328537	−	0.944491i	$-0.393444\pi$
$509$	14.8243	0.657075	0.328537	+	0.944491i	$0.393444\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−4.71449	−0.208150
$514$	0	0
$515$	22.8575	1.00722
$516$	0	0
$517$	−0.382542	−0.0168242
$518$	0	0
$519$	10.8509	0.476303
$520$	0	0
$521$	−10.4922	−0.459670	−0.229835	−	0.973230i	$-0.573819\pi$
$521$	−10.4922	−0.459670	−0.229835	+	0.973230i	$0.573819\pi$
$522$	0	0
$523$	−15.4757	−0.676707	−0.338353	−	0.941019i	$-0.609870\pi$
$523$	−15.4757	−0.676707	−0.338353	+	0.941019i	$0.609870\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−34.2109	−1.49025
$528$	0	0
$529$	20.7642	0.902792
$530$	0	0
$531$	−2.33024	−0.101124
$532$	0	0
$533$	11.5400	0.499852
$534$	0	0
$535$	−6.27557	−0.271316
$536$	0	0
$537$	22.9650	0.991014
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−18.7927	−0.807961	−0.403980	−	0.914768i	$-0.632374\pi$
$541$	−18.7927	−0.807961	−0.403980	+	0.914768i	$0.632374\pi$
$542$	0	0
$543$	17.0973	0.733716
$544$	0	0
$545$	1.12282	0.0480965
$546$	0	0
$547$	−0.809060	−0.0345929	−0.0172965	−	0.999850i	$-0.505506\pi$
$547$	−0.809060	−0.0345929	−0.0172965	+	0.999850i	$0.505506\pi$
$548$	0	0
$549$	−19.7459	−0.842734
$550$	0	0
$551$	8.91381	0.379741
$552$	0	0
$553$	0	0
$554$	0	0
$555$	14.8941	0.632219
$556$	0	0
$557$	−17.4536	−0.739533	−0.369767	−	0.929125i	$-0.620562\pi$
$557$	−17.4536	−0.739533	−0.369767	+	0.929125i	$0.620562\pi$
$558$	0	0
$559$	45.1927	1.91145
$560$	0	0
$561$	−1.27434	−0.0538027
$562$	0	0
$563$	−3.68375	−0.155251	−0.0776257	−	0.996983i	$-0.524734\pi$
$563$	−3.68375	−0.155251	−0.0776257	+	0.996983i	$0.524734\pi$
$564$	0	0
$565$	−16.8993	−0.710960
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−9.13224	−0.382843	−0.191422	−	0.981508i	$-0.561310\pi$
$569$	−9.13224	−0.382843	−0.191422	+	0.981508i	$0.561310\pi$
$570$	0	0
$571$	21.1247	0.884039	0.442020	−	0.897005i	$-0.354262\pi$
$571$	21.1247	0.884039	0.442020	+	0.897005i	$0.354262\pi$
$572$	0	0
$573$	0.324779	0.0135678
$574$	0	0
$575$	−14.7315	−0.614347
$576$	0	0
$577$	5.24274	0.218258	0.109129	−	0.994028i	$-0.465194\pi$
$577$	5.24274	0.218258	0.109129	+	0.994028i	$0.465194\pi$
$578$	0	0
$579$	15.5991	0.648278
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−1.57007	−0.0650258
$584$	0	0
$585$	−20.8288	−0.861167
$586$	0	0
$587$	−3.26045	−0.134573	−0.0672865	−	0.997734i	$-0.521434\pi$
$587$	−3.26045	−0.134573	−0.0672865	+	0.997734i	$0.521434\pi$
$588$	0	0
$589$	−5.93284	−0.244458
$590$	0	0
$591$	11.9591	0.491932
$592$	0	0
$593$	19.9230	0.818141	0.409070	−	0.912503i	$-0.365853\pi$
$593$	19.9230	0.818141	0.409070	+	0.912503i	$0.365853\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.35144	0.178093
$598$	0	0
$599$	44.4649	1.81679	0.908394	−	0.418116i	$-0.137309\pi$
$599$	44.4649	1.81679	0.908394	+	0.418116i	$0.137309\pi$
$600$	0	0
$601$	22.3392	0.911233	0.455617	−	0.890176i	$-0.349419\pi$
$601$	22.3392	0.911233	0.455617	+	0.890176i	$0.349419\pi$
$602$	0	0
$603$	−17.3968	−0.708453
$604$	0	0
$605$	18.2204	0.740764
$606$	0	0
$607$	34.4089	1.39661	0.698306	−	0.715799i	$-0.253937\pi$
$607$	34.4089	1.39661	0.698306	+	0.715799i	$0.253937\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−9.11184	−0.368626
$612$	0	0
$613$	−24.2940	−0.981226	−0.490613	−	0.871378i	$-0.663227\pi$
$613$	−24.2940	−0.981226	−0.490613	+	0.871378i	$0.663227\pi$
$614$	0	0
$615$	3.03846	0.122523
$616$	0	0
$617$	−17.0255	−0.685422	−0.342711	−	0.939441i	$-0.611345\pi$
$617$	−17.0255	−0.685422	−0.342711	+	0.939441i	$0.611345\pi$
$618$	0	0
$619$	28.6795	1.15273	0.576363	−	0.817194i	$-0.304472\pi$
$619$	28.6795	1.15273	0.576363	+	0.817194i	$0.304472\pi$
$620$	0	0
$621$	−31.1885	−1.25155
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−8.90703	−0.356281
$626$	0	0
$627$	−0.220995	−0.00882570
$628$	0	0
$629$	−56.5318	−2.25407
$630$	0	0
$631$	−29.4675	−1.17308	−0.586542	−	0.809919i	$-0.699511\pi$
$631$	−29.4675	−1.17308	−0.586542	+	0.809919i	$0.699511\pi$
$632$	0	0
$633$	20.1109	0.799335
$634$	0	0
$635$	−30.9699	−1.22900
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−2.66532	−0.105439
$640$	0	0
$641$	−16.1367	−0.637363	−0.318681	−	0.947862i	$-0.603240\pi$
$641$	−16.1367	−0.637363	−0.318681	+	0.947862i	$0.603240\pi$
$642$	0	0
$643$	22.8353	0.900535	0.450267	−	0.892894i	$-0.351329\pi$
$643$	22.8353	0.900535	0.450267	+	0.892894i	$0.351329\pi$
$644$	0	0
$645$	11.8992	0.468530
$646$	0	0
$647$	20.7399	0.815370	0.407685	−	0.913123i	$-0.366336\pi$
$647$	20.7399	0.815370	0.407685	+	0.913123i	$0.366336\pi$
$648$	0	0
$649$	−0.260403	−0.0102217
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−19.3767	−0.758268	−0.379134	−	0.925342i	$-0.623778\pi$
$653$	−19.3767	−0.758268	−0.379134	+	0.925342i	$0.623778\pi$
$654$	0	0
$655$	−13.2746	−0.518683
$656$	0	0
$657$	−1.73714	−0.0677724
$658$	0	0
$659$	7.33347	0.285672	0.142836	−	0.989746i	$-0.454378\pi$
$659$	7.33347	0.285672	0.142836	+	0.989746i	$0.454378\pi$
$660$	0	0
$661$	−37.7069	−1.46663	−0.733315	−	0.679889i	$-0.762028\pi$
$661$	−37.7069	−1.46663	−0.733315	+	0.679889i	$0.762028\pi$
$662$	0	0
$663$	−30.3538	−1.17884
$664$	0	0
$665$	0	0
$666$	0	0
$667$	58.9689	2.28329
$668$	0	0
$669$	17.8661	0.690742
$670$	0	0
$671$	−2.20659	−0.0851844
$672$	0	0
$673$	48.4005	1.86570	0.932851	−	0.360263i	$-0.117313\pi$
$673$	48.4005	1.86570	0.932851	+	0.360263i	$0.117313\pi$
$674$	0	0
$675$	10.4984	0.404083
$676$	0	0
$677$	15.8658	0.609772	0.304886	−	0.952389i	$-0.401382\pi$
$677$	15.8658	0.609772	0.304886	+	0.952389i	$0.401382\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−5.28063	−0.202354
$682$	0	0
$683$	17.2671	0.660707	0.330353	−	0.943857i	$-0.392832\pi$
$683$	17.2671	0.660707	0.330353	+	0.943857i	$0.392832\pi$
$684$	0	0
$685$	0.530831	0.0202820
$686$	0	0
$687$	−17.4316	−0.665059
$688$	0	0
$689$	−37.3979	−1.42475
$690$	0	0
$691$	−49.2612	−1.87398	−0.936992	−	0.349351i	$-0.886402\pi$
$691$	−49.2612	−1.87398	−0.936992	+	0.349351i	$0.886402\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	9.66056	0.366446
$696$	0	0
$697$	−11.5327	−0.436834
$698$	0	0
$699$	−5.63001	−0.212947
$700$	0	0
$701$	10.1609	0.383770	0.191885	−	0.981417i	$-0.438540\pi$
$701$	10.1609	0.383770	0.191885	+	0.981417i	$0.438540\pi$
$702$	0	0
$703$	−9.80371	−0.369754
$704$	0	0
$705$	−2.39914	−0.0903567
$706$	0	0
$707$	0	0
$708$	0	0
$709$	7.73019	0.290313	0.145157	−	0.989409i	$-0.453631\pi$
$709$	7.73019	0.290313	0.145157	+	0.989409i	$0.453631\pi$
$710$	0	0
$711$	16.4357	0.616386
$712$	0	0
$713$	−39.2484	−1.46986
$714$	0	0
$715$	−2.32761	−0.0870477
$716$	0	0
$717$	23.8497	0.890682
$718$	0	0
$719$	25.3207	0.944303	0.472152	−	0.881517i	$-0.343477\pi$
$719$	25.3207	0.944303	0.472152	+	0.881517i	$0.343477\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−17.8733	−0.664717
$724$	0	0
$725$	−19.8496	−0.737196
$726$	0	0
$727$	24.5601	0.910884	0.455442	−	0.890266i	$-0.349481\pi$
$727$	24.5601	0.910884	0.455442	+	0.890266i	$0.349481\pi$
$728$	0	0
$729$	8.12941	0.301089
$730$	0	0
$731$	−45.1644	−1.67046
$732$	0	0
$733$	−27.3033	−1.00847	−0.504235	−	0.863567i	$-0.668225\pi$
$733$	−27.3033	−1.00847	−0.504235	+	0.863567i	$0.668225\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.94408	−0.0716112
$738$	0	0
$739$	48.0375	1.76709	0.883544	−	0.468348i	$-0.155151\pi$
$739$	48.0375	1.76709	0.883544	+	0.468348i	$0.155151\pi$
$740$	0	0
$741$	−5.26394	−0.193376
$742$	0	0
$743$	10.8539	0.398190	0.199095	−	0.979980i	$-0.436200\pi$
$743$	10.8539	0.398190	0.199095	+	0.979980i	$0.436200\pi$
$744$	0	0
$745$	−22.7645	−0.834027
$746$	0	0
$747$	19.3207	0.706906
$748$	0	0
$749$	0	0
$750$	0	0
$751$	17.9144	0.653705	0.326852	−	0.945075i	$-0.394012\pi$
$751$	17.9144	0.653705	0.326852	+	0.945075i	$0.394012\pi$
$752$	0	0
$753$	−5.83030	−0.212468
$754$	0	0
$755$	−18.3651	−0.668374
$756$	0	0
$757$	19.7888	0.719235	0.359617	−	0.933100i	$-0.382907\pi$
$757$	19.7888	0.719235	0.359617	+	0.933100i	$0.382907\pi$
$758$	0	0
$759$	−1.46198	−0.0530667
$760$	0	0
$761$	51.4603	1.86543	0.932717	−	0.360609i	$-0.117431\pi$
$761$	51.4603	1.86543	0.932717	+	0.360609i	$0.117431\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	20.8158	0.752596
$766$	0	0
$767$	−6.20260	−0.223963
$768$	0	0
$769$	−26.2468	−0.946485	−0.473243	−	0.880932i	$-0.656917\pi$
$769$	−26.2468	−0.946485	−0.473243	+	0.880932i	$0.656917\pi$
$770$	0	0
$771$	19.9558	0.718689
$772$	0	0
$773$	6.05223	0.217684	0.108842	−	0.994059i	$-0.465286\pi$
$773$	6.05223	0.217684	0.108842	+	0.994059i	$0.465286\pi$
$774$	0	0
$775$	13.2115	0.474569
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.00000	−0.0716574
$780$	0	0
$781$	−0.297848	−0.0106578
$782$	0	0
$783$	−42.0241	−1.50182
$784$	0	0
$785$	−29.7363	−1.06133
$786$	0	0
$787$	−12.4576	−0.444064	−0.222032	−	0.975039i	$-0.571269\pi$
$787$	−12.4576	−0.444064	−0.222032	+	0.975039i	$0.571269\pi$
$788$	0	0
$789$	−27.1168	−0.965383
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−52.5592	−1.86643
$794$	0	0
$795$	−9.84682	−0.349231
$796$	0	0
$797$	7.13423	0.252707	0.126354	−	0.991985i	$-0.459673\pi$
$797$	7.13423	0.252707	0.126354	+	0.991985i	$0.459673\pi$
$798$	0	0
$799$	9.10613	0.322152
$800$	0	0
$801$	28.4288	1.00448
$802$	0	0
$803$	−0.194125	−0.00685051
$804$	0	0
$805$	0	0
$806$	0	0
$807$	12.2116	0.429869
$808$	0	0
$809$	9.75801	0.343074	0.171537	−	0.985178i	$-0.445127\pi$
$809$	9.75801	0.343074	0.171537	+	0.985178i	$0.445127\pi$
$810$	0	0
$811$	−1.23879	−0.0434998	−0.0217499	−	0.999763i	$-0.506924\pi$
$811$	−1.23879	−0.0434998	−0.0217499	+	0.999763i	$0.506924\pi$
$812$	0	0
$813$	−23.5128	−0.824630
$814$	0	0
$815$	−18.6462	−0.653148
$816$	0	0
$817$	−7.83237	−0.274020
$818$	0	0
$819$	0	0
$820$	0	0
$821$	42.8811	1.49656	0.748281	−	0.663382i	$-0.230880\pi$
$821$	42.8811	1.49656	0.748281	+	0.663382i	$0.230880\pi$
$822$	0	0
$823$	−12.7541	−0.444579	−0.222290	−	0.974981i	$-0.571353\pi$
$823$	−12.7541	−0.444579	−0.222290	+	0.974981i	$0.571353\pi$
$824$	0	0
$825$	0.492120	0.0171334
$826$	0	0
$827$	52.6536	1.83095	0.915473	−	0.402380i	$-0.131817\pi$
$827$	52.6536	1.83095	0.915473	+	0.402380i	$0.131817\pi$
$828$	0	0
$829$	−0.121544	−0.00422139	−0.00211070	−	0.999998i	$-0.500672\pi$
$829$	−0.121544	−0.00422139	−0.00211070	+	0.999998i	$0.500672\pi$
$830$	0	0
$831$	−2.71319	−0.0941194
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−2.40986	−0.0833966
$836$	0	0
$837$	27.9703	0.966794
$838$	0	0
$839$	45.8707	1.58363	0.791816	−	0.610759i	$-0.209136\pi$
$839$	45.8707	1.58363	0.791816	+	0.610759i	$0.209136\pi$
$840$	0	0
$841$	50.4561	1.73987
$842$	0	0
$843$	−17.2376	−0.593693
$844$	0	0
$845$	−33.7932	−1.16252
$846$	0	0
$847$	0	0
$848$	0	0
$849$	20.8277	0.714805
$850$	0	0
$851$	−64.8560	−2.22323
$852$	0	0
$853$	−18.8106	−0.644064	−0.322032	−	0.946729i	$-0.604366\pi$
$853$	−18.8106	−0.644064	−0.322032	+	0.946729i	$0.604366\pi$
$854$	0	0
$855$	3.60986	0.123455
$856$	0	0
$857$	−21.9936	−0.751288	−0.375644	−	0.926764i	$-0.622578\pi$
$857$	−21.9936	−0.751288	−0.375644	+	0.926764i	$0.622578\pi$
$858$	0	0
$859$	33.9757	1.15924	0.579619	−	0.814888i	$-0.303202\pi$
$859$	33.9757	1.15924	0.579619	+	0.814888i	$0.303202\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−33.0442	−1.12484	−0.562419	−	0.826852i	$-0.690129\pi$
$863$	−33.0442	−1.12484	−0.562419	+	0.826852i	$0.690129\pi$
$864$	0	0
$865$	−19.8070	−0.673459
$866$	0	0
$867$	14.8258	0.503509
$868$	0	0
$869$	1.83668	0.0623049
$870$	0	0
$871$	−46.3065	−1.56904
$872$	0	0
$873$	−2.42055	−0.0819232
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−26.7266	−0.902494	−0.451247	−	0.892399i	$-0.649021\pi$
$877$	−26.7266	−0.902494	−0.451247	+	0.892399i	$0.649021\pi$
$878$	0	0
$879$	10.3737	0.349896
$880$	0	0
$881$	33.6752	1.13454	0.567272	−	0.823530i	$-0.307999\pi$
$881$	33.6752	1.13454	0.567272	+	0.823530i	$0.307999\pi$
$882$	0	0
$883$	−15.6753	−0.527516	−0.263758	−	0.964589i	$-0.584962\pi$
$883$	−15.6753	−0.527516	−0.263758	+	0.964589i	$0.584962\pi$
$884$	0	0
$885$	−1.63314	−0.0548973
$886$	0	0
$887$	5.19711	0.174502	0.0872510	−	0.996186i	$-0.472192\pi$
$887$	5.19711	0.174502	0.0872510	+	0.996186i	$0.472192\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−0.533448	−0.0178712
$892$	0	0
$893$	1.57918	0.0528452
$894$	0	0
$895$	−41.9198	−1.40122
$896$	0	0
$897$	−34.8233	−1.16272
$898$	0	0
$899$	−52.8842	−1.76379
$900$	0	0
$901$	37.3745	1.24512
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−31.2090	−1.03742
$906$	0	0
$907$	−37.5830	−1.24792	−0.623962	−	0.781455i	$-0.714478\pi$
$907$	−37.5830	−1.24792	−0.623962	+	0.781455i	$0.714478\pi$
$908$	0	0
$909$	−28.2402	−0.936669
$910$	0	0
$911$	19.9982	0.662571	0.331286	−	0.943530i	$-0.392518\pi$
$911$	19.9982	0.662571	0.331286	+	0.943530i	$0.392518\pi$
$912$	0	0
$913$	2.15907	0.0714549
$914$	0	0
$915$	−13.8388	−0.457496
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−41.6251	−1.37309	−0.686543	−	0.727089i	$-0.740873\pi$
$919$	−41.6251	−1.37309	−0.686543	+	0.727089i	$0.740873\pi$
$920$	0	0
$921$	−3.53639	−0.116528
$922$	0	0
$923$	−7.09451	−0.233519
$924$	0	0
$925$	21.8312	0.717807
$926$	0	0
$927$	29.7538	0.977244
$928$	0	0
$929$	−32.3223	−1.06046	−0.530229	−	0.847854i	$-0.677894\pi$
$929$	−32.3223	−1.06046	−0.530229	+	0.847854i	$0.677894\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	20.1217	0.658755
$934$	0	0
$935$	2.32615	0.0760733
$936$	0	0
$937$	−23.8754	−0.779975	−0.389988	−	0.920820i	$-0.627521\pi$
$937$	−23.8754	−0.779975	−0.389988	+	0.920820i	$0.627521\pi$
$938$	0	0
$939$	−28.3399	−0.924836
$940$	0	0
$941$	−30.6962	−1.00067	−0.500334	−	0.865833i	$-0.666789\pi$
$941$	−30.6962	−1.00067	−0.500334	+	0.865833i	$0.666789\pi$
$942$	0	0
$943$	−13.2309	−0.430858
$944$	0	0
$945$	0	0
$946$	0	0
$947$	27.7378	0.901357	0.450678	−	0.892686i	$-0.351182\pi$
$947$	27.7378	0.901357	0.450678	+	0.892686i	$0.351182\pi$
$948$	0	0
$949$	−4.62390	−0.150098
$950$	0	0
$951$	30.3669	0.984715
$952$	0	0
$953$	−29.9634	−0.970611	−0.485305	−	0.874345i	$-0.661292\pi$
$953$	−29.9634	−0.970611	−0.485305	+	0.874345i	$0.661292\pi$
$954$	0	0
$955$	−0.592844	−0.0191840
$956$	0	0
$957$	−1.96991	−0.0636782
$958$	0	0
$959$	0	0
$960$	0	0
$961$	4.19856	0.135437
$962$	0	0
$963$	−8.16897	−0.263241
$964$	0	0
$965$	−28.4743	−0.916619
$966$	0	0
$967$	−19.2270	−0.618299	−0.309150	−	0.951013i	$-0.600044\pi$
$967$	−19.2270	−0.618299	−0.309150	+	0.951013i	$0.600044\pi$
$968$	0	0
$969$	5.26064	0.168996
$970$	0	0
$971$	16.2936	0.522887	0.261443	−	0.965219i	$-0.415802\pi$
$971$	16.2936	0.522887	0.261443	+	0.965219i	$0.415802\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	11.7219	0.375402
$976$	0	0
$977$	−21.3065	−0.681655	−0.340828	−	0.940126i	$-0.610707\pi$
$977$	−21.3065	−0.681655	−0.340828	+	0.940126i	$0.610707\pi$
$978$	0	0
$979$	3.17690	0.101534
$980$	0	0
$981$	1.46159	0.0466650
$982$	0	0
$983$	−7.27409	−0.232007	−0.116004	−	0.993249i	$-0.537008\pi$
$983$	−7.27409	−0.232007	−0.116004	+	0.993249i	$0.537008\pi$
$984$	0	0
$985$	−21.8299	−0.695558
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−51.8147	−1.64761
$990$	0	0
$991$	−39.2758	−1.24764	−0.623818	−	0.781569i	$-0.714419\pi$
$991$	−39.2758	−1.24764	−0.623818	+	0.781569i	$0.714419\pi$
$992$	0	0
$993$	7.63938	0.242428
$994$	0	0
$995$	−7.94301	−0.251811
$996$	0	0
$997$	8.97292	0.284175	0.142088	−	0.989854i	$-0.454619\pi$
$997$	8.97292	0.284175	0.142088	+	0.989854i	$0.454619\pi$
$998$	0	0
$999$	46.2195	1.46232

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.bq.1.6		8
7.2	even	3		1064.2.q.n.305.3	✓	16
7.4	even	3		1064.2.q.n.457.3	yes	16
7.6	odd	2		7448.2.a.br.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1064.2.q.n.305.3	✓	16	7.2	even	3
1064.2.q.n.457.3	yes	16	7.4	even	3
7448.2.a.bq.1.6		8	1.1	even	1	trivial
7448.2.a.br.1.3		8	7.6	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7448.a (trivial)

\(f(q)\)	\(=\)	\(q+0.912296 q^{3} -1.66528 q^{5} -2.16772 q^{9} +O(q^{10})\)	\(q+0.912296 q^{3} -1.66528 q^{5} -2.16772 q^{9} -0.242241 q^{11} -5.76999 q^{13} -1.51923 q^{15} +5.76637 q^{17} +1.00000 q^{19} +6.61545 q^{23} -2.22684 q^{25} -4.71449 q^{27} +8.91381 q^{29} -5.93284 q^{31} -0.220995 q^{33} -9.80371 q^{37} -5.26394 q^{39} -2.00000 q^{41} -7.83237 q^{43} +3.60986 q^{45} +1.57918 q^{47} +5.26064 q^{51} +6.48146 q^{53} +0.403399 q^{55} +0.912296 q^{57} +1.07498 q^{59} +9.10907 q^{61} +9.60866 q^{65} +8.02541 q^{67} +6.03525 q^{69} +1.22955 q^{71} +0.801370 q^{73} -2.03153 q^{75} -7.58202 q^{79} +2.20214 q^{81} -8.91292 q^{83} -9.60264 q^{85} +8.13204 q^{87} -13.1146 q^{89} -5.41250 q^{93} -1.66528 q^{95} +1.11664 q^{97} +0.525109 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{5} + 6 q^{9}+O(q^{10}) \) 8 * q - q^5 + 6 * q^9	\( 8 q - q^{5} + 6 q^{9} + 9 q^{11} + 8 q^{15} + 4 q^{17} + 8 q^{19} + 25 q^{23} + 15 q^{25} + 3 q^{27} + 6 q^{29} + 8 q^{33} + 13 q^{37} - 11 q^{39} - 16 q^{41} + 17 q^{43} + 17 q^{45} - 24 q^{47} + 5 q^{51} + 2 q^{53} + 5 q^{55} + 2 q^{59} - 13 q^{61} - 26 q^{65} + 2 q^{67} - 11 q^{69} + 10 q^{71} + 5 q^{73} - 20 q^{75} + 16 q^{79} - 12 q^{81} - 43 q^{83} + 24 q^{85} + 20 q^{87} + 8 q^{89} + 2 q^{93} - q^{95} - 12 q^{97} + 37 q^{99}+O(q^{100}) \) 8 * q - q^5 + 6 * q^9 + 9 * q^11 + 8 * q^15 + 4 * q^17 + 8 * q^19 + 25 * q^23 + 15 * q^25 + 3 * q^27 + 6 * q^29 + 8 * q^33 + 13 * q^37 - 11 * q^39 - 16 * q^41 + 17 * q^43 + 17 * q^45 - 24 * q^47 + 5 * q^51 + 2 * q^53 + 5 * q^55 + 2 * q^59 - 13 * q^61 - 26 * q^65 + 2 * q^67 - 11 * q^69 + 10 * q^71 + 5 * q^73 - 20 * q^75 + 16 * q^79 - 12 * q^81 - 43 * q^83 + 24 * q^85 + 20 * q^87 + 8 * q^89 + 2 * q^93 - q^95 - 12 * q^97 + 37 * q^99

Introduction

L-functions

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