LMFDB - Embedded newform 7448.2.a.bn.1.2

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:	$6$
Coefficient field:	6.6.98211824.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 6x^{4} + 15x^{3} + 8x^{2} - 9x + 1 $ x^6 - 3x^5 - 6x^4 + 15x^3 + 8x^2 - 9*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.15897$ 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)

$f(q)$	$=$	$q-1.15897 q^{3} +1.74190 q^{5} -1.65679 q^{9} +O(q^{10})$	$q-1.15897 q^{3} +1.74190 q^{5} -1.65679 q^{9} -5.45213 q^{11} -3.91531 q^{13} -2.01881 q^{15} -1.01881 q^{17} -1.00000 q^{19} +7.27914 q^{23} -1.96579 q^{25} +5.39708 q^{27} +3.52394 q^{29} -5.06969 q^{31} +6.31886 q^{33} -2.61756 q^{37} +4.53773 q^{39} -11.4218 q^{41} -2.55126 q^{43} -2.88596 q^{45} +0.536392 q^{47} +1.18077 q^{51} -1.33771 q^{53} -9.49706 q^{55} +1.15897 q^{57} +10.6368 q^{59} -3.90221 q^{61} -6.82008 q^{65} -13.9146 q^{67} -8.43630 q^{69} +7.41290 q^{71} +6.92561 q^{73} +2.27829 q^{75} +0.214460 q^{79} -1.28469 q^{81} +10.9949 q^{83} -1.77466 q^{85} -4.08414 q^{87} -5.26815 q^{89} +5.87562 q^{93} -1.74190 q^{95} +7.06281 q^{97} +9.03303 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{3} + q^{5} + 3 q^{9}+O(q^{10}) $ 6 * q + 3 * q^3 + q^5 + 3 * q^9	$ 6 q + 3 q^{3} + q^{5} + 3 q^{9} + 3 q^{11} + 6 q^{13} - 8 q^{15} - 2 q^{17} - 6 q^{19} + 2 q^{23} + 5 q^{25} + 6 q^{27} - q^{29} + 14 q^{31} + 19 q^{33} - 7 q^{37} + 22 q^{39} - 21 q^{41} + q^{43} - 4 q^{45} + 23 q^{47} + 2 q^{51} - 15 q^{53} + 2 q^{55} - 3 q^{57} + 25 q^{59} + 21 q^{61} + 6 q^{65} + 2 q^{67} - 20 q^{69} + 13 q^{71} - 2 q^{73} + 24 q^{75} - 17 q^{79} - 2 q^{81} + 4 q^{83} + 40 q^{85} + 30 q^{87} - q^{89} + 6 q^{93} - q^{95} - 13 q^{97} + 41 q^{99}+O(q^{100}) $ 6 * q + 3 * q^3 + q^5 + 3 * q^9 + 3 * q^11 + 6 * q^13 - 8 * q^15 - 2 * q^17 - 6 * q^19 + 2 * q^23 + 5 * q^25 + 6 * q^27 - q^29 + 14 * q^31 + 19 * q^33 - 7 * q^37 + 22 * q^39 - 21 * q^41 + q^43 - 4 * q^45 + 23 * q^47 + 2 * q^51 - 15 * q^53 + 2 * q^55 - 3 * q^57 + 25 * q^59 + 21 * q^61 + 6 * q^65 + 2 * q^67 - 20 * q^69 + 13 * q^71 - 2 * q^73 + 24 * q^75 - 17 * q^79 - 2 * q^81 + 4 * q^83 + 40 * q^85 + 30 * q^87 - q^89 + 6 * q^93 - q^95 - 13 * q^97 + 41 * 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
$3$	−1.15897	−0.669132	−0.334566	−	0.942372i	$-0.608590\pi$
$3$	−1.15897	−0.669132	−0.334566	+	0.942372i	$0.608590\pi$
$4$	0	0
$5$	1.74190	0.779001	0.389500	−	0.921026i	$-0.372648\pi$
$5$	1.74190	0.779001	0.389500	+	0.921026i	$0.372648\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.65679	−0.552263
$10$	0	0
$11$	−5.45213	−1.64388	−0.821940	−	0.569575i	$-0.807108\pi$
$11$	−5.45213	−1.64388	−0.821940	+	0.569575i	$0.807108\pi$
$12$	0	0
$13$	−3.91531	−1.08591	−0.542956	−	0.839761i	$-0.682695\pi$
$13$	−3.91531	−1.08591	−0.542956	+	0.839761i	$0.682695\pi$
$14$	0	0
$15$	−2.01881	−0.521254
$16$	0	0
$17$	−1.01881	−0.247097	−0.123549	−	0.992339i	$-0.539427\pi$
$17$	−1.01881	−0.247097	−0.123549	+	0.992339i	$0.539427\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.27914	1.51781	0.758903	−	0.651204i	$-0.225736\pi$
$23$	7.27914	1.51781	0.758903	+	0.651204i	$0.225736\pi$
$24$	0	0
$25$	−1.96579	−0.393158
$26$	0	0
$27$	5.39708	1.03867
$28$	0	0
$29$	3.52394	0.654379	0.327189	−	0.944959i	$-0.393898\pi$
$29$	3.52394	0.654379	0.327189	+	0.944959i	$0.393898\pi$
$30$	0	0
$31$	−5.06969	−0.910544	−0.455272	−	0.890352i	$-0.650458\pi$
$31$	−5.06969	−0.910544	−0.455272	+	0.890352i	$0.650458\pi$
$32$	0	0
$33$	6.31886	1.09997
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.61756	−0.430325	−0.215162	−	0.976578i	$-0.569028\pi$
$37$	−2.61756	−0.430325	−0.215162	+	0.976578i	$0.569028\pi$
$38$	0	0
$39$	4.53773	0.726619
$40$	0	0
$41$	−11.4218	−1.78379	−0.891896	−	0.452240i	$-0.850625\pi$
$41$	−11.4218	−1.78379	−0.891896	+	0.452240i	$0.850625\pi$
$42$	0	0
$43$	−2.55126	−0.389064	−0.194532	−	0.980896i	$-0.562319\pi$
$43$	−2.55126	−0.389064	−0.194532	+	0.980896i	$0.562319\pi$
$44$	0	0
$45$	−2.88596	−0.430213
$46$	0	0
$47$	0.536392	0.0782409	0.0391204	−	0.999235i	$-0.487544\pi$
$47$	0.536392	0.0782409	0.0391204	+	0.999235i	$0.487544\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	1.18077	0.165341
$52$	0	0
$53$	−1.33771	−0.183748	−0.0918740	−	0.995771i	$-0.529286\pi$
$53$	−1.33771	−0.183748	−0.0918740	+	0.995771i	$0.529286\pi$
$54$	0	0
$55$	−9.49706	−1.28058
$56$	0	0
$57$	1.15897	0.153509
$58$	0	0
$59$	10.6368	1.38479	0.692396	−	0.721517i	$-0.256555\pi$
$59$	10.6368	1.38479	0.692396	+	0.721517i	$0.256555\pi$
$60$	0	0
$61$	−3.90221	−0.499626	−0.249813	−	0.968294i	$-0.580369\pi$
$61$	−3.90221	−0.499626	−0.249813	+	0.968294i	$0.580369\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.82008	−0.845927
$66$	0	0
$67$	−13.9146	−1.69993	−0.849967	−	0.526837i	$-0.823378\pi$
$67$	−13.9146	−1.69993	−0.849967	+	0.526837i	$0.823378\pi$
$68$	0	0
$69$	−8.43630	−1.01561
$70$	0	0
$71$	7.41290	0.879750	0.439875	−	0.898059i	$-0.355023\pi$
$71$	7.41290	0.879750	0.439875	+	0.898059i	$0.355023\pi$
$72$	0	0
$73$	6.92561	0.810581	0.405290	−	0.914188i	$-0.367170\pi$
$73$	6.92561	0.810581	0.405290	+	0.914188i	$0.367170\pi$
$74$	0	0
$75$	2.27829	0.263074
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0.214460	0.0241287	0.0120643	−	0.999927i	$-0.496160\pi$
$79$	0.214460	0.0241287	0.0120643	+	0.999927i	$0.496160\pi$
$80$	0	0
$81$	−1.28469	−0.142743
$82$	0	0
$83$	10.9949	1.20684	0.603422	−	0.797422i	$-0.293803\pi$
$83$	10.9949	1.20684	0.603422	+	0.797422i	$0.293803\pi$
$84$	0	0
$85$	−1.77466	−0.192489
$86$	0	0
$87$	−4.08414	−0.437866
$88$	0	0
$89$	−5.26815	−0.558423	−0.279212	−	0.960230i	$-0.590073\pi$
$89$	−5.26815	−0.558423	−0.279212	+	0.960230i	$0.590073\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	5.87562	0.609274
$94$	0	0
$95$	−1.74190	−0.178715
$96$	0	0
$97$	7.06281	0.717119	0.358560	−	0.933507i	$-0.383268\pi$
$97$	7.06281	0.717119	0.358560	+	0.933507i	$0.383268\pi$
$98$	0	0
$99$	9.03303	0.907853
$100$	0	0
$101$	18.0830	1.79932	0.899661	−	0.436590i	$-0.143814\pi$
$101$	18.0830	1.79932	0.899661	+	0.436590i	$0.143814\pi$
$102$	0	0
$103$	11.3179	1.11519	0.557593	−	0.830115i	$-0.311725\pi$
$103$	11.3179	1.11519	0.557593	+	0.830115i	$0.311725\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.9021	1.44064	0.720322	−	0.693640i	$-0.243994\pi$
$107$	14.9021	1.44064	0.720322	+	0.693640i	$0.243994\pi$
$108$	0	0
$109$	−11.2146	−1.07417	−0.537084	−	0.843529i	$-0.680474\pi$
$109$	−11.2146	−1.07417	−0.537084	+	0.843529i	$0.680474\pi$
$110$	0	0
$111$	3.03368	0.287944
$112$	0	0
$113$	−10.7219	−1.00863	−0.504317	−	0.863519i	$-0.668256\pi$
$113$	−10.7219	−1.00863	−0.504317	+	0.863519i	$0.668256\pi$
$114$	0	0
$115$	12.6795	1.18237
$116$	0	0
$117$	6.48685	0.599709
$118$	0	0
$119$	0	0
$120$	0	0
$121$	18.7257	1.70234
$122$	0	0
$123$	13.2376	1.19359
$124$	0	0
$125$	−12.1337	−1.08527
$126$	0	0
$127$	−13.6962	−1.21534	−0.607670	−	0.794189i	$-0.707896\pi$
$127$	−13.6962	−1.21534	−0.607670	+	0.794189i	$0.707896\pi$
$128$	0	0
$129$	2.95684	0.260335
$130$	0	0
$131$	13.3514	1.16651	0.583257	−	0.812287i	$-0.301778\pi$
$131$	13.3514	1.16651	0.583257	+	0.812287i	$0.301778\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	9.40116	0.809123
$136$	0	0
$137$	−4.51449	−0.385699	−0.192850	−	0.981228i	$-0.561773\pi$
$137$	−4.51449	−0.385699	−0.192850	+	0.981228i	$0.561773\pi$
$138$	0	0
$139$	17.3162	1.46875	0.734373	−	0.678747i	$-0.237477\pi$
$139$	17.3162	1.46875	0.734373	+	0.678747i	$0.237477\pi$
$140$	0	0
$141$	−0.621663	−0.0523534
$142$	0	0
$143$	21.3468	1.78511
$144$	0	0
$145$	6.13834	0.509762
$146$	0	0
$147$	0	0
$148$	0	0
$149$	4.79098	0.392492	0.196246	−	0.980555i	$-0.437125\pi$
$149$	4.79098	0.392492	0.196246	+	0.980555i	$0.437125\pi$
$150$	0	0
$151$	−1.22460	−0.0996568	−0.0498284	−	0.998758i	$-0.515867\pi$
$151$	−1.22460	−0.0996568	−0.0498284	+	0.998758i	$0.515867\pi$
$152$	0	0
$153$	1.68795	0.136463
$154$	0	0
$155$	−8.83089	−0.709314
$156$	0	0
$157$	−10.6980	−0.853791	−0.426896	−	0.904301i	$-0.640393\pi$
$157$	−10.6980	−0.853791	−0.426896	+	0.904301i	$0.640393\pi$
$158$	0	0
$159$	1.55036	0.122952
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−6.22523	−0.487598	−0.243799	−	0.969826i	$-0.578394\pi$
$163$	−6.22523	−0.487598	−0.243799	+	0.969826i	$0.578394\pi$
$164$	0	0
$165$	11.0068	0.856879
$166$	0	0
$167$	−19.1481	−1.48172	−0.740861	−	0.671658i	$-0.765582\pi$
$167$	−19.1481	−1.48172	−0.740861	+	0.671658i	$0.765582\pi$
$168$	0	0
$169$	2.32968	0.179206
$170$	0	0
$171$	1.65679	0.126698
$172$	0	0
$173$	21.2360	1.61454	0.807271	−	0.590181i	$-0.200944\pi$
$173$	21.2360	1.61454	0.807271	+	0.590181i	$0.200944\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−12.3277	−0.926609
$178$	0	0
$179$	12.5546	0.938377	0.469189	−	0.883098i	$-0.344546\pi$
$179$	12.5546	0.938377	0.469189	+	0.883098i	$0.344546\pi$
$180$	0	0
$181$	18.2194	1.35424	0.677119	−	0.735874i	$-0.263228\pi$
$181$	18.2194	1.35424	0.677119	+	0.735874i	$0.263228\pi$
$182$	0	0
$183$	4.52254	0.334316
$184$	0	0
$185$	−4.55953	−0.335223
$186$	0	0
$187$	5.55468	0.406198
$188$	0	0
$189$	0	0
$190$	0	0
$191$	11.5319	0.834421	0.417211	−	0.908810i	$-0.363008\pi$
$191$	11.5319	0.834421	0.417211	+	0.908810i	$0.363008\pi$
$192$	0	0
$193$	−3.33145	−0.239803	−0.119902	−	0.992786i	$-0.538258\pi$
$193$	−3.33145	−0.239803	−0.119902	+	0.992786i	$0.538258\pi$
$194$	0	0
$195$	7.90427	0.566036
$196$	0	0
$197$	−16.1096	−1.14776	−0.573879	−	0.818940i	$-0.694562\pi$
$197$	−16.1096	−1.14776	−0.573879	+	0.818940i	$0.694562\pi$
$198$	0	0
$199$	15.3330	1.08693	0.543465	−	0.839432i	$-0.317112\pi$
$199$	15.3330	1.08693	0.543465	+	0.839432i	$0.317112\pi$
$200$	0	0
$201$	16.1265	1.13748
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−19.8957	−1.38958
$206$	0	0
$207$	−12.0600	−0.838227
$208$	0	0
$209$	5.45213	0.377132
$210$	0	0
$211$	16.6714	1.14771	0.573853	−	0.818959i	$-0.305448\pi$
$211$	16.6714	1.14771	0.573853	+	0.818959i	$0.305448\pi$
$212$	0	0
$213$	−8.59133	−0.588669
$214$	0	0
$215$	−4.44404	−0.303081
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−8.02657	−0.542385
$220$	0	0
$221$	3.98895	0.268326
$222$	0	0
$223$	−1.05554	−0.0706840	−0.0353420	−	0.999375i	$-0.511252\pi$
$223$	−1.05554	−0.0706840	−0.0353420	+	0.999375i	$0.511252\pi$
$224$	0	0
$225$	3.25690	0.217126
$226$	0	0
$227$	6.20712	0.411981	0.205990	−	0.978554i	$-0.433958\pi$
$227$	6.20712	0.411981	0.205990	+	0.978554i	$0.433958\pi$
$228$	0	0
$229$	1.44789	0.0956793	0.0478397	−	0.998855i	$-0.484766\pi$
$229$	1.44789	0.0956793	0.0478397	+	0.998855i	$0.484766\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−23.3805	−1.53170	−0.765852	−	0.643016i	$-0.777683\pi$
$233$	−23.3805	−1.53170	−0.765852	+	0.643016i	$0.777683\pi$
$234$	0	0
$235$	0.934341	0.0609497
$236$	0	0
$237$	−0.248553	−0.0161452
$238$	0	0
$239$	−12.2577	−0.792888	−0.396444	−	0.918059i	$-0.629756\pi$
$239$	−12.2577	−0.792888	−0.396444	+	0.918059i	$0.629756\pi$
$240$	0	0
$241$	6.60037	0.425167	0.212583	−	0.977143i	$-0.431812\pi$
$241$	6.60037	0.425167	0.212583	+	0.977143i	$0.431812\pi$
$242$	0	0
$243$	−14.7023	−0.943154
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.91531	0.249125
$248$	0	0
$249$	−12.7427	−0.807538
$250$	0	0
$251$	−19.6376	−1.23952	−0.619758	−	0.784793i	$-0.712769\pi$
$251$	−19.6376	−1.23952	−0.619758	+	0.784793i	$0.712769\pi$
$252$	0	0
$253$	−39.6868	−2.49509
$254$	0	0
$255$	2.05678	0.128800
$256$	0	0
$257$	9.88504	0.616612	0.308306	−	0.951287i	$-0.400238\pi$
$257$	9.88504	0.616612	0.308306	+	0.951287i	$0.400238\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−5.83842	−0.361389
$262$	0	0
$263$	−5.84626	−0.360496	−0.180248	−	0.983621i	$-0.557690\pi$
$263$	−5.84626	−0.360496	−0.180248	+	0.983621i	$0.557690\pi$
$264$	0	0
$265$	−2.33015	−0.143140
$266$	0	0
$267$	6.10563	0.373659
$268$	0	0
$269$	9.64384	0.587995	0.293998	−	0.955806i	$-0.405014\pi$
$269$	9.64384	0.587995	0.293998	+	0.955806i	$0.405014\pi$
$270$	0	0
$271$	17.5016	1.06315	0.531574	−	0.847012i	$-0.321601\pi$
$271$	17.5016	1.06315	0.531574	+	0.847012i	$0.321601\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	10.7177	0.646304
$276$	0	0
$277$	2.52578	0.151759	0.0758796	−	0.997117i	$-0.475824\pi$
$277$	2.52578	0.151759	0.0758796	+	0.997117i	$0.475824\pi$
$278$	0	0
$279$	8.39941	0.502859
$280$	0	0
$281$	25.7152	1.53404	0.767019	−	0.641625i	$-0.221739\pi$
$281$	25.7152	1.53404	0.767019	+	0.641625i	$0.221739\pi$
$282$	0	0
$283$	18.4402	1.09616	0.548078	−	0.836427i	$-0.315360\pi$
$283$	18.4402	1.09616	0.548078	+	0.836427i	$0.315360\pi$
$284$	0	0
$285$	2.01881	0.119584
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−15.9620	−0.938943
$290$	0	0
$291$	−8.18558	−0.479847
$292$	0	0
$293$	−25.9767	−1.51758	−0.758788	−	0.651338i	$-0.774208\pi$
$293$	−25.9767	−1.51758	−0.758788	+	0.651338i	$0.774208\pi$
$294$	0	0
$295$	18.5282	1.07875
$296$	0	0
$297$	−29.4256	−1.70744
$298$	0	0
$299$	−28.5001	−1.64820
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−20.9576	−1.20398
$304$	0	0
$305$	−6.79725	−0.389209
$306$	0	0
$307$	−0.121059	−0.00690918	−0.00345459	−	0.999994i	$-0.501100\pi$
$307$	−0.121059	−0.00690918	−0.00345459	+	0.999994i	$0.501100\pi$
$308$	0	0
$309$	−13.1171	−0.746206
$310$	0	0
$311$	7.14728	0.405285	0.202642	−	0.979253i	$-0.435047\pi$
$311$	7.14728	0.405285	0.202642	+	0.979253i	$0.435047\pi$
$312$	0	0
$313$	1.29570	0.0732372	0.0366186	−	0.999329i	$-0.488341\pi$
$313$	1.29570	0.0732372	0.0366186	+	0.999329i	$0.488341\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.93181	0.164667	0.0823333	−	0.996605i	$-0.473763\pi$
$317$	2.93181	0.164667	0.0823333	+	0.996605i	$0.473763\pi$
$318$	0	0
$319$	−19.2130	−1.07572
$320$	0	0
$321$	−17.2711	−0.963981
$322$	0	0
$323$	1.01881	0.0566880
$324$	0	0
$325$	7.69668	0.426935
$326$	0	0
$327$	12.9974	0.718760
$328$	0	0
$329$	0	0
$330$	0	0
$331$	5.47990	0.301203	0.150601	−	0.988595i	$-0.451879\pi$
$331$	5.47990	0.301203	0.150601	+	0.988595i	$0.451879\pi$
$332$	0	0
$333$	4.33675	0.237652
$334$	0	0
$335$	−24.2377	−1.32425
$336$	0	0
$337$	20.1683	1.09864	0.549319	−	0.835613i	$-0.314887\pi$
$337$	20.1683	1.09864	0.549319	+	0.835613i	$0.314887\pi$
$338$	0	0
$339$	12.4264	0.674909
$340$	0	0
$341$	27.6406	1.49682
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−14.6952	−0.791162
$346$	0	0
$347$	24.0352	1.29028	0.645139	−	0.764065i	$-0.276800\pi$
$347$	24.0352	1.29028	0.645139	+	0.764065i	$0.276800\pi$
$348$	0	0
$349$	−26.1496	−1.39976	−0.699879	−	0.714261i	$-0.746763\pi$
$349$	−26.1496	−1.39976	−0.699879	+	0.714261i	$0.746763\pi$
$350$	0	0
$351$	−21.1313	−1.12790
$352$	0	0
$353$	−3.97407	−0.211518	−0.105759	−	0.994392i	$-0.533727\pi$
$353$	−3.97407	−0.211518	−0.105759	+	0.994392i	$0.533727\pi$
$354$	0	0
$355$	12.9125	0.685326
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−30.7706	−1.62401	−0.812005	−	0.583650i	$-0.801624\pi$
$359$	−30.7706	−1.62401	−0.812005	+	0.583650i	$0.801624\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−21.7025	−1.13909
$364$	0	0
$365$	12.0637	0.631443
$366$	0	0
$367$	−2.77758	−0.144988	−0.0724942	−	0.997369i	$-0.523096\pi$
$367$	−2.77758	−0.144988	−0.0724942	+	0.997369i	$0.523096\pi$
$368$	0	0
$369$	18.9236	0.985122
$370$	0	0
$371$	0	0
$372$	0	0
$373$	31.5603	1.63413	0.817065	−	0.576546i	$-0.195600\pi$
$373$	31.5603	1.63413	0.817065	+	0.576546i	$0.195600\pi$
$374$	0	0
$375$	14.0626	0.726189
$376$	0	0
$377$	−13.7973	−0.710598
$378$	0	0
$379$	9.53802	0.489935	0.244968	−	0.969531i	$-0.421223\pi$
$379$	9.53802	0.489935	0.244968	+	0.969531i	$0.421223\pi$
$380$	0	0
$381$	15.8735	0.813223
$382$	0	0
$383$	1.54697	0.0790462	0.0395231	−	0.999219i	$-0.487416\pi$
$383$	1.54697	0.0790462	0.0395231	+	0.999219i	$0.487416\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.22690	0.214865
$388$	0	0
$389$	35.6869	1.80940	0.904699	−	0.426052i	$-0.140096\pi$
$389$	35.6869	1.80940	0.904699	+	0.426052i	$0.140096\pi$
$390$	0	0
$391$	−7.41605	−0.375046
$392$	0	0
$393$	−15.4738	−0.780552
$394$	0	0
$395$	0.373568	0.0187962
$396$	0	0
$397$	25.5296	1.28129	0.640647	−	0.767835i	$-0.278666\pi$
$397$	25.5296	1.28129	0.640647	+	0.767835i	$0.278666\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	34.2571	1.71072	0.855359	−	0.518035i	$-0.173336\pi$
$401$	34.2571	1.71072	0.855359	+	0.518035i	$0.173336\pi$
$402$	0	0
$403$	19.8494	0.988771
$404$	0	0
$405$	−2.23779	−0.111197
$406$	0	0
$407$	14.2713	0.707402
$408$	0	0
$409$	−29.3744	−1.45247	−0.726236	−	0.687445i	$-0.758732\pi$
$409$	−29.3744	−1.45247	−0.726236	+	0.687445i	$0.758732\pi$
$410$	0	0
$411$	5.23216	0.258083
$412$	0	0
$413$	0	0
$414$	0	0
$415$	19.1520	0.940133
$416$	0	0
$417$	−20.0690	−0.982784
$418$	0	0
$419$	28.7692	1.40547	0.702735	−	0.711452i	$-0.251962\pi$
$419$	28.7692	1.40547	0.702735	+	0.711452i	$0.251962\pi$
$420$	0	0
$421$	−33.3082	−1.62334	−0.811671	−	0.584114i	$-0.801442\pi$
$421$	−33.3082	−1.62334	−0.811671	+	0.584114i	$0.801442\pi$
$422$	0	0
$423$	−0.888689	−0.0432095
$424$	0	0
$425$	2.00276	0.0971482
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−24.7403	−1.19447
$430$	0	0
$431$	24.1705	1.16425	0.582126	−	0.813099i	$-0.302221\pi$
$431$	24.1705	1.16425	0.582126	+	0.813099i	$0.302221\pi$
$432$	0	0
$433$	29.0858	1.39778	0.698888	−	0.715232i	$-0.253679\pi$
$433$	29.0858	1.39778	0.698888	+	0.715232i	$0.253679\pi$
$434$	0	0
$435$	−7.11415	−0.341098
$436$	0	0
$437$	−7.27914	−0.348208
$438$	0	0
$439$	6.18898	0.295384	0.147692	−	0.989033i	$-0.452816\pi$
$439$	6.18898	0.295384	0.147692	+	0.989033i	$0.452816\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−37.8146	−1.79662	−0.898312	−	0.439358i	$-0.855206\pi$
$443$	−37.8146	−1.79662	−0.898312	+	0.439358i	$0.855206\pi$
$444$	0	0
$445$	−9.17659	−0.435012
$446$	0	0
$447$	−5.55260	−0.262629
$448$	0	0
$449$	−10.2594	−0.484171	−0.242085	−	0.970255i	$-0.577831\pi$
$449$	−10.2594	−0.484171	−0.242085	+	0.970255i	$0.577831\pi$
$450$	0	0
$451$	62.2734	2.93234
$452$	0	0
$453$	1.41928	0.0666836
$454$	0	0
$455$	0	0
$456$	0	0
$457$	20.8390	0.974809	0.487404	−	0.873176i	$-0.337944\pi$
$457$	20.8390	0.974809	0.487404	+	0.873176i	$0.337944\pi$
$458$	0	0
$459$	−5.49859	−0.256652
$460$	0	0
$461$	−35.7831	−1.66658	−0.833292	−	0.552833i	$-0.813547\pi$
$461$	−35.7831	−1.66658	−0.833292	+	0.552833i	$0.813547\pi$
$462$	0	0
$463$	6.54002	0.303941	0.151970	−	0.988385i	$-0.451438\pi$
$463$	6.54002	0.303941	0.151970	+	0.988385i	$0.451438\pi$
$464$	0	0
$465$	10.2347	0.474625
$466$	0	0
$467$	1.39826	0.0647037	0.0323518	−	0.999477i	$-0.489700\pi$
$467$	1.39826	0.0647037	0.0323518	+	0.999477i	$0.489700\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	12.3986	0.571299
$472$	0	0
$473$	13.9098	0.639574
$474$	0	0
$475$	1.96579	0.0901966
$476$	0	0
$477$	2.21630	0.101477
$478$	0	0
$479$	−11.6232	−0.531076	−0.265538	−	0.964100i	$-0.585549\pi$
$479$	−11.6232	−0.531076	−0.265538	+	0.964100i	$0.585549\pi$
$480$	0	0
$481$	10.2486	0.467295
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12.3027	0.558636
$486$	0	0
$487$	−25.4177	−1.15178	−0.575892	−	0.817526i	$-0.695345\pi$
$487$	−25.4177	−1.15178	−0.575892	+	0.817526i	$0.695345\pi$
$488$	0	0
$489$	7.21485	0.326267
$490$	0	0
$491$	3.49578	0.157762	0.0788812	−	0.996884i	$-0.474865\pi$
$491$	3.49578	0.157762	0.0788812	+	0.996884i	$0.474865\pi$
$492$	0	0
$493$	−3.59022	−0.161695
$494$	0	0
$495$	15.7346	0.707218
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−18.4993	−0.828143	−0.414072	−	0.910244i	$-0.635894\pi$
$499$	−18.4993	−0.828143	−0.414072	+	0.910244i	$0.635894\pi$
$500$	0	0
$501$	22.1920	0.991467
$502$	0	0
$503$	40.5993	1.81023	0.905116	−	0.425165i	$-0.139784\pi$
$503$	40.5993	1.81023	0.905116	+	0.425165i	$0.139784\pi$
$504$	0	0
$505$	31.4987	1.40167
$506$	0	0
$507$	−2.70003	−0.119913
$508$	0	0
$509$	8.37357	0.371152	0.185576	−	0.982630i	$-0.440585\pi$
$509$	8.37357	0.371152	0.185576	+	0.982630i	$0.440585\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−5.39708	−0.238287
$514$	0	0
$515$	19.7146	0.868730
$516$	0	0
$517$	−2.92448	−0.128619
$518$	0	0
$519$	−24.6119	−1.08034
$520$	0	0
$521$	−42.4863	−1.86136	−0.930679	−	0.365836i	$-0.880783\pi$
$521$	−42.4863	−1.86136	−0.930679	+	0.365836i	$0.880783\pi$
$522$	0	0
$523$	19.4417	0.850125	0.425063	−	0.905164i	$-0.360252\pi$
$523$	19.4417	0.850125	0.425063	+	0.905164i	$0.360252\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.16505	0.224993
$528$	0	0
$529$	29.9859	1.30373
$530$	0	0
$531$	−17.6229	−0.764770
$532$	0	0
$533$	44.7201	1.93704
$534$	0	0
$535$	25.9580	1.12226
$536$	0	0
$537$	−14.5504	−0.627898
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−46.1121	−1.98252	−0.991258	−	0.131938i	$-0.957880\pi$
$541$	−46.1121	−1.98252	−0.991258	+	0.131938i	$0.957880\pi$
$542$	0	0
$543$	−21.1157	−0.906163
$544$	0	0
$545$	−19.5348	−0.836778
$546$	0	0
$547$	4.29879	0.183803	0.0919016	−	0.995768i	$-0.470705\pi$
$547$	4.29879	0.183803	0.0919016	+	0.995768i	$0.470705\pi$
$548$	0	0
$549$	6.46513	0.275925
$550$	0	0
$551$	−3.52394	−0.150125
$552$	0	0
$553$	0	0
$554$	0	0
$555$	5.28436	0.224309
$556$	0	0
$557$	38.9723	1.65131	0.825656	−	0.564174i	$-0.190805\pi$
$557$	38.9723	1.65131	0.825656	+	0.564174i	$0.190805\pi$
$558$	0	0
$559$	9.98899	0.422489
$560$	0	0
$561$	−6.43770	−0.271800
$562$	0	0
$563$	27.0802	1.14130	0.570648	−	0.821195i	$-0.306692\pi$
$563$	27.0802	1.14130	0.570648	+	0.821195i	$0.306692\pi$
$564$	0	0
$565$	−18.6765	−0.785727
$566$	0	0
$567$	0	0
$568$	0	0
$569$	30.7718	1.29002	0.645011	−	0.764173i	$-0.276853\pi$
$569$	30.7718	1.29002	0.645011	+	0.764173i	$0.276853\pi$
$570$	0	0
$571$	−12.3229	−0.515696	−0.257848	−	0.966185i	$-0.583013\pi$
$571$	−12.3229	−0.515696	−0.257848	+	0.966185i	$0.583013\pi$
$572$	0	0
$573$	−13.3652	−0.558338
$574$	0	0
$575$	−14.3093	−0.596737
$576$	0	0
$577$	23.3163	0.970672	0.485336	−	0.874328i	$-0.338697\pi$
$577$	23.3163	0.970672	0.485336	+	0.874328i	$0.338697\pi$
$578$	0	0
$579$	3.86105	0.160460
$580$	0	0
$581$	0	0
$582$	0	0
$583$	7.29335	0.302059
$584$	0	0
$585$	11.2994	0.467174
$586$	0	0
$587$	11.1401	0.459803	0.229901	−	0.973214i	$-0.426160\pi$
$587$	11.1401	0.459803	0.229901	+	0.973214i	$0.426160\pi$
$588$	0	0
$589$	5.06969	0.208893
$590$	0	0
$591$	18.6705	0.768001
$592$	0	0
$593$	−2.14637	−0.0881409	−0.0440705	−	0.999028i	$-0.514033\pi$
$593$	−2.14637	−0.0881409	−0.0440705	+	0.999028i	$0.514033\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−17.7705	−0.727299
$598$	0	0
$599$	−21.3811	−0.873609	−0.436804	−	0.899556i	$-0.643890\pi$
$599$	−21.3811	−0.873609	−0.436804	+	0.899556i	$0.643890\pi$
$600$	0	0
$601$	13.7457	0.560701	0.280350	−	0.959898i	$-0.409549\pi$
$601$	13.7457	0.560701	0.280350	+	0.959898i	$0.409549\pi$
$602$	0	0
$603$	23.0535	0.938810
$604$	0	0
$605$	32.6183	1.32612
$606$	0	0
$607$	−21.0041	−0.852529	−0.426264	−	0.904599i	$-0.640171\pi$
$607$	−21.0041	−0.852529	−0.426264	+	0.904599i	$0.640171\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2.10014	−0.0849627
$612$	0	0
$613$	−19.7824	−0.799005	−0.399502	−	0.916732i	$-0.630817\pi$
$613$	−19.7824	−0.799005	−0.399502	+	0.916732i	$0.630817\pi$
$614$	0	0
$615$	23.0585	0.929809
$616$	0	0
$617$	10.8866	0.438278	0.219139	−	0.975694i	$-0.429675\pi$
$617$	10.8866	0.438278	0.219139	+	0.975694i	$0.429675\pi$
$618$	0	0
$619$	40.1046	1.61194	0.805971	−	0.591955i	$-0.201644\pi$
$619$	40.1046	1.61194	0.805971	+	0.591955i	$0.201644\pi$
$620$	0	0
$621$	39.2861	1.57650
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−11.3067	−0.452269
$626$	0	0
$627$	−6.31886	−0.252351
$628$	0	0
$629$	2.66680	0.106332
$630$	0	0
$631$	22.4234	0.892660	0.446330	−	0.894868i	$-0.352731\pi$
$631$	22.4234	0.892660	0.446330	+	0.894868i	$0.352731\pi$
$632$	0	0
$633$	−19.3216	−0.767966
$634$	0	0
$635$	−23.8574	−0.946751
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−12.2816	−0.485853
$640$	0	0
$641$	−17.9934	−0.710696	−0.355348	−	0.934734i	$-0.615638\pi$
$641$	−17.9934	−0.710696	−0.355348	+	0.934734i	$0.615638\pi$
$642$	0	0
$643$	16.8909	0.666114	0.333057	−	0.942907i	$-0.391920\pi$
$643$	16.8909	0.666114	0.333057	+	0.942907i	$0.391920\pi$
$644$	0	0
$645$	5.15051	0.202801
$646$	0	0
$647$	−16.6995	−0.656525	−0.328263	−	0.944587i	$-0.606463\pi$
$647$	−16.6995	−0.656525	−0.328263	+	0.944587i	$0.606463\pi$
$648$	0	0
$649$	−57.9932	−2.27643
$650$	0	0
$651$	0	0
$652$	0	0
$653$	17.2314	0.674315	0.337158	−	0.941448i	$-0.390534\pi$
$653$	17.2314	0.674315	0.337158	+	0.941448i	$0.390534\pi$
$654$	0	0
$655$	23.2567	0.908716
$656$	0	0
$657$	−11.4743	−0.447654
$658$	0	0
$659$	−30.0457	−1.17041	−0.585207	−	0.810884i	$-0.698987\pi$
$659$	−30.0457	−1.17041	−0.585207	+	0.810884i	$0.698987\pi$
$660$	0	0
$661$	−1.55903	−0.0606393	−0.0303196	−	0.999540i	$-0.509653\pi$
$661$	−1.55903	−0.0606393	−0.0303196	+	0.999540i	$0.509653\pi$
$662$	0	0
$663$	−4.62308	−0.179545
$664$	0	0
$665$	0	0
$666$	0	0
$667$	25.6512	0.993220
$668$	0	0
$669$	1.22334	0.0472969
$670$	0	0
$671$	21.2753	0.821326
$672$	0	0
$673$	−17.5500	−0.676504	−0.338252	−	0.941056i	$-0.609836\pi$
$673$	−17.5500	−0.676504	−0.338252	+	0.941056i	$0.609836\pi$
$674$	0	0
$675$	−10.6095	−0.408361
$676$	0	0
$677$	−37.4505	−1.43934	−0.719669	−	0.694317i	$-0.755707\pi$
$677$	−37.4505	−1.43934	−0.719669	+	0.694317i	$0.755707\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−7.19386	−0.275669
$682$	0	0
$683$	−11.2872	−0.431892	−0.215946	−	0.976405i	$-0.569283\pi$
$683$	−11.2872	−0.431892	−0.215946	+	0.976405i	$0.569283\pi$
$684$	0	0
$685$	−7.86379	−0.300460
$686$	0	0
$687$	−1.67806	−0.0640221
$688$	0	0
$689$	5.23754	0.199534
$690$	0	0
$691$	31.6804	1.20518	0.602589	−	0.798052i	$-0.294136\pi$
$691$	31.6804	1.20518	0.602589	+	0.798052i	$0.294136\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	30.1632	1.14415
$696$	0	0
$697$	11.6367	0.440770
$698$	0	0
$699$	27.0973	1.02491
$700$	0	0
$701$	−0.469405	−0.0177292	−0.00886459	−	0.999961i	$-0.502822\pi$
$701$	−0.469405	−0.0177292	−0.00886459	+	0.999961i	$0.502822\pi$
$702$	0	0
$703$	2.61756	0.0987233
$704$	0	0
$705$	−1.08287	−0.0407834
$706$	0	0
$707$	0	0
$708$	0	0
$709$	12.8409	0.482251	0.241126	−	0.970494i	$-0.422483\pi$
$709$	12.8409	0.482251	0.241126	+	0.970494i	$0.422483\pi$
$710$	0	0
$711$	−0.355315	−0.0133254
$712$	0	0
$713$	−36.9030	−1.38203
$714$	0	0
$715$	37.1840	1.39060
$716$	0	0
$717$	14.2064	0.530546
$718$	0	0
$719$	24.2164	0.903120	0.451560	−	0.892241i	$-0.350868\pi$
$719$	24.2164	0.903120	0.451560	+	0.892241i	$0.350868\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−7.64963	−0.284493
$724$	0	0
$725$	−6.92732	−0.257274
$726$	0	0
$727$	21.7727	0.807505	0.403753	−	0.914868i	$-0.367706\pi$
$727$	21.7727	0.807505	0.403753	+	0.914868i	$0.367706\pi$
$728$	0	0
$729$	20.8936	0.773837
$730$	0	0
$731$	2.59925	0.0961366
$732$	0	0
$733$	48.7308	1.79991	0.899956	−	0.435981i	$-0.143598\pi$
$733$	48.7308	1.79991	0.899956	+	0.435981i	$0.143598\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	75.8639	2.79448
$738$	0	0
$739$	38.9231	1.43181	0.715905	−	0.698197i	$-0.246014\pi$
$739$	38.9231	1.43181	0.715905	+	0.698197i	$0.246014\pi$
$740$	0	0
$741$	−4.53773	−0.166698
$742$	0	0
$743$	20.8849	0.766192	0.383096	−	0.923709i	$-0.374858\pi$
$743$	20.8849	0.766192	0.383096	+	0.923709i	$0.374858\pi$
$744$	0	0
$745$	8.34540	0.305752
$746$	0	0
$747$	−18.2162	−0.666495
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−29.0940	−1.06165	−0.530827	−	0.847480i	$-0.678119\pi$
$751$	−29.0940	−1.06165	−0.530827	+	0.847480i	$0.678119\pi$
$752$	0	0
$753$	22.7594	0.829399
$754$	0	0
$755$	−2.13314	−0.0776328
$756$	0	0
$757$	33.8924	1.23184	0.615920	−	0.787808i	$-0.288784\pi$
$757$	33.8924	1.23184	0.615920	+	0.787808i	$0.288784\pi$
$758$	0	0
$759$	45.9958	1.66954
$760$	0	0
$761$	5.46397	0.198069	0.0990344	−	0.995084i	$-0.468425\pi$
$761$	5.46397	0.198069	0.0990344	+	0.995084i	$0.468425\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	2.94024	0.106305
$766$	0	0
$767$	−41.6464	−1.50376
$768$	0	0
$769$	−31.6212	−1.14029	−0.570145	−	0.821544i	$-0.693113\pi$
$769$	−31.6212	−1.14029	−0.570145	+	0.821544i	$0.693113\pi$
$770$	0	0
$771$	−11.4565	−0.412595
$772$	0	0
$773$	−30.3820	−1.09277	−0.546383	−	0.837535i	$-0.683996\pi$
$773$	−30.3820	−1.09277	−0.546383	+	0.837535i	$0.683996\pi$
$774$	0	0
$775$	9.96595	0.357987
$776$	0	0
$777$	0	0
$778$	0	0
$779$	11.4218	0.409230
$780$	0	0
$781$	−40.4161	−1.44620
$782$	0	0
$783$	19.0190	0.679682
$784$	0	0
$785$	−18.6348	−0.665104
$786$	0	0
$787$	−49.8241	−1.77604	−0.888020	−	0.459806i	$-0.847919\pi$
$787$	−49.8241	−1.77604	−0.888020	+	0.459806i	$0.847919\pi$
$788$	0	0
$789$	6.77564	0.241219
$790$	0	0
$791$	0	0
$792$	0	0
$793$	15.2784	0.542551
$794$	0	0
$795$	2.70057	0.0957794
$796$	0	0
$797$	3.52663	0.124920	0.0624598	−	0.998047i	$-0.480105\pi$
$797$	3.52663	0.124920	0.0624598	+	0.998047i	$0.480105\pi$
$798$	0	0
$799$	−0.546481	−0.0193331
$800$	0	0
$801$	8.72821	0.308396
$802$	0	0
$803$	−37.7593	−1.33250
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−11.1769	−0.393446
$808$	0	0
$809$	−19.2422	−0.676519	−0.338259	−	0.941053i	$-0.609838\pi$
$809$	−19.2422	−0.676519	−0.338259	+	0.941053i	$0.609838\pi$
$810$	0	0
$811$	−42.7925	−1.50265	−0.751324	−	0.659934i	$-0.770584\pi$
$811$	−42.7925	−1.50265	−0.751324	+	0.659934i	$0.770584\pi$
$812$	0	0
$813$	−20.2839	−0.711386
$814$	0	0
$815$	−10.8437	−0.379839
$816$	0	0
$817$	2.55126	0.0892573
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−35.6682	−1.24483	−0.622414	−	0.782688i	$-0.713848\pi$
$821$	−35.6682	−1.24483	−0.622414	+	0.782688i	$0.713848\pi$
$822$	0	0
$823$	−26.3256	−0.917653	−0.458826	−	0.888526i	$-0.651730\pi$
$823$	−26.3256	−0.917653	−0.458826	+	0.888526i	$0.651730\pi$
$824$	0	0
$825$	−12.4215	−0.432462
$826$	0	0
$827$	24.1983	0.841458	0.420729	−	0.907186i	$-0.361774\pi$
$827$	24.1983	0.841458	0.420729	+	0.907186i	$0.361774\pi$
$828$	0	0
$829$	−2.91616	−0.101283	−0.0506413	−	0.998717i	$-0.516127\pi$
$829$	−2.91616	−0.101283	−0.0506413	+	0.998717i	$0.516127\pi$
$830$	0	0
$831$	−2.92730	−0.101547
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−33.3540	−1.15426
$836$	0	0
$837$	−27.3615	−0.945753
$838$	0	0
$839$	30.3903	1.04919	0.524595	−	0.851352i	$-0.324217\pi$
$839$	30.3903	1.04919	0.524595	+	0.851352i	$0.324217\pi$
$840$	0	0
$841$	−16.5819	−0.571788
$842$	0	0
$843$	−29.8031	−1.02647
$844$	0	0
$845$	4.05807	0.139602
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−21.3716	−0.733473
$850$	0	0
$851$	−19.0536	−0.653149
$852$	0	0
$853$	34.4884	1.18086	0.590430	−	0.807089i	$-0.298958\pi$
$853$	34.4884	1.18086	0.590430	+	0.807089i	$0.298958\pi$
$854$	0	0
$855$	2.88596	0.0986977
$856$	0	0
$857$	−40.2979	−1.37655	−0.688274	−	0.725451i	$-0.741631\pi$
$857$	−40.2979	−1.37655	−0.688274	+	0.725451i	$0.741631\pi$
$858$	0	0
$859$	1.45229	0.0495516	0.0247758	−	0.999693i	$-0.492113\pi$
$859$	1.45229	0.0495516	0.0247758	+	0.999693i	$0.492113\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−12.6326	−0.430020	−0.215010	−	0.976612i	$-0.568978\pi$
$863$	−12.6326	−0.430020	−0.215010	+	0.976612i	$0.568978\pi$
$864$	0	0
$865$	36.9909	1.25773
$866$	0	0
$867$	18.4995	0.628276
$868$	0	0
$869$	−1.16926	−0.0396646
$870$	0	0
$871$	54.4798	1.84598
$872$	0	0
$873$	−11.7016	−0.396038
$874$	0	0
$875$	0	0
$876$	0	0
$877$	7.16028	0.241786	0.120893	−	0.992666i	$-0.461424\pi$
$877$	7.16028	0.241786	0.120893	+	0.992666i	$0.461424\pi$
$878$	0	0
$879$	30.1062	1.01546
$880$	0	0
$881$	1.95742	0.0659472	0.0329736	−	0.999456i	$-0.489502\pi$
$881$	1.95742	0.0659472	0.0329736	+	0.999456i	$0.489502\pi$
$882$	0	0
$883$	−20.3489	−0.684796	−0.342398	−	0.939555i	$-0.611239\pi$
$883$	−20.3489	−0.684796	−0.342398	+	0.939555i	$0.611239\pi$
$884$	0	0
$885$	−21.4737	−0.721829
$886$	0	0
$887$	8.10888	0.272270	0.136135	−	0.990690i	$-0.456532\pi$
$887$	8.10888	0.272270	0.136135	+	0.990690i	$0.456532\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	7.00428	0.234652
$892$	0	0
$893$	−0.536392	−0.0179497
$894$	0	0
$895$	21.8689	0.730997
$896$	0	0
$897$	33.0308	1.10287
$898$	0	0
$899$	−17.8653	−0.595841
$900$	0	0
$901$	1.36287	0.0454036
$902$	0	0
$903$	0	0
$904$	0	0
$905$	31.7364	1.05495
$906$	0	0
$907$	29.7288	0.987129	0.493565	−	0.869709i	$-0.335694\pi$
$907$	29.7288	0.987129	0.493565	+	0.869709i	$0.335694\pi$
$908$	0	0
$909$	−29.9596	−0.993698
$910$	0	0
$911$	−30.1074	−0.997501	−0.498751	−	0.866746i	$-0.666208\pi$
$911$	−30.1074	−0.997501	−0.498751	+	0.866746i	$0.666208\pi$
$912$	0	0
$913$	−59.9455	−1.98391
$914$	0	0
$915$	7.87781	0.260432
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−4.12477	−0.136064	−0.0680318	−	0.997683i	$-0.521672\pi$
$919$	−4.12477	−0.136064	−0.0680318	+	0.997683i	$0.521672\pi$
$920$	0	0
$921$	0.140303	0.00462315
$922$	0	0
$923$	−29.0238	−0.955332
$924$	0	0
$925$	5.14558	0.169186
$926$	0	0
$927$	−18.7514	−0.615876
$928$	0	0
$929$	35.9094	1.17815	0.589075	−	0.808078i	$-0.299492\pi$
$929$	35.9094	1.17815	0.589075	+	0.808078i	$0.299492\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−8.28348	−0.271189
$934$	0	0
$935$	9.67568	0.316429
$936$	0	0
$937$	24.0332	0.785130	0.392565	−	0.919724i	$-0.371588\pi$
$937$	24.0332	0.785130	0.392565	+	0.919724i	$0.371588\pi$
$938$	0	0
$939$	−1.50168	−0.0490053
$940$	0	0
$941$	−7.78796	−0.253880	−0.126940	−	0.991910i	$-0.540516\pi$
$941$	−7.78796	−0.253880	−0.126940	+	0.991910i	$0.540516\pi$
$942$	0	0
$943$	−83.1412	−2.70745
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−20.1844	−0.655904	−0.327952	−	0.944694i	$-0.606358\pi$
$947$	−20.1844	−0.655904	−0.327952	+	0.944694i	$0.606358\pi$
$948$	0	0
$949$	−27.1159	−0.880220
$950$	0	0
$951$	−3.39788	−0.110184
$952$	0	0
$953$	2.08018	0.0673838	0.0336919	−	0.999432i	$-0.489274\pi$
$953$	2.08018	0.0673838	0.0336919	+	0.999432i	$0.489274\pi$
$954$	0	0
$955$	20.0875	0.650015
$956$	0	0
$957$	22.2673	0.719798
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−5.29821	−0.170910
$962$	0	0
$963$	−24.6897	−0.795614
$964$	0	0
$965$	−5.80305	−0.186807
$966$	0	0
$967$	21.0053	0.675484	0.337742	−	0.941239i	$-0.390337\pi$
$967$	21.0053	0.675484	0.337742	+	0.941239i	$0.390337\pi$
$968$	0	0
$969$	−1.18077	−0.0379317
$970$	0	0
$971$	34.2931	1.10052	0.550260	−	0.834994i	$-0.314529\pi$
$971$	34.2931	1.10052	0.550260	+	0.834994i	$0.314529\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−8.92022	−0.285676
$976$	0	0
$977$	8.43730	0.269933	0.134967	−	0.990850i	$-0.456907\pi$
$977$	8.43730	0.269933	0.134967	+	0.990850i	$0.456907\pi$
$978$	0	0
$979$	28.7226	0.917980
$980$	0	0
$981$	18.5803	0.593223
$982$	0	0
$983$	−51.7499	−1.65056	−0.825282	−	0.564721i	$-0.808984\pi$
$983$	−51.7499	−1.65056	−0.825282	+	0.564721i	$0.808984\pi$
$984$	0	0
$985$	−28.0612	−0.894104
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−18.5710	−0.590523
$990$	0	0
$991$	−6.15109	−0.195396	−0.0976979	−	0.995216i	$-0.531148\pi$
$991$	−6.15109	−0.195396	−0.0976979	+	0.995216i	$0.531148\pi$
$992$	0	0
$993$	−6.35104	−0.201544
$994$	0	0
$995$	26.7086	0.846719
$996$	0	0
$997$	26.0839	0.826085	0.413043	−	0.910712i	$-0.364466\pi$
$997$	26.0839	0.826085	0.413043	+	0.910712i	$0.364466\pi$
$998$	0	0
$999$	−14.1272	−0.446965

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.bn.1.2	yes	6
7.6	odd	2		7448.2.a.bm.1.5	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7448.2.a.bm.1.5	✓	6	7.6	odd	2
7448.2.a.bn.1.2	yes	6	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:	\( N \)	\(=\)	\( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7448.a (trivial)

\(f(q)\)	\(=\)	\(q-1.15897 q^{3} +1.74190 q^{5} -1.65679 q^{9} +O(q^{10})\)	\(q-1.15897 q^{3} +1.74190 q^{5} -1.65679 q^{9} -5.45213 q^{11} -3.91531 q^{13} -2.01881 q^{15} -1.01881 q^{17} -1.00000 q^{19} +7.27914 q^{23} -1.96579 q^{25} +5.39708 q^{27} +3.52394 q^{29} -5.06969 q^{31} +6.31886 q^{33} -2.61756 q^{37} +4.53773 q^{39} -11.4218 q^{41} -2.55126 q^{43} -2.88596 q^{45} +0.536392 q^{47} +1.18077 q^{51} -1.33771 q^{53} -9.49706 q^{55} +1.15897 q^{57} +10.6368 q^{59} -3.90221 q^{61} -6.82008 q^{65} -13.9146 q^{67} -8.43630 q^{69} +7.41290 q^{71} +6.92561 q^{73} +2.27829 q^{75} +0.214460 q^{79} -1.28469 q^{81} +10.9949 q^{83} -1.77466 q^{85} -4.08414 q^{87} -5.26815 q^{89} +5.87562 q^{93} -1.74190 q^{95} +7.06281 q^{97} +9.03303 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{3} + q^{5} + 3 q^{9}+O(q^{10}) \) 6 * q + 3 * q^3 + q^5 + 3 * q^9	\( 6 q + 3 q^{3} + q^{5} + 3 q^{9} + 3 q^{11} + 6 q^{13} - 8 q^{15} - 2 q^{17} - 6 q^{19} + 2 q^{23} + 5 q^{25} + 6 q^{27} - q^{29} + 14 q^{31} + 19 q^{33} - 7 q^{37} + 22 q^{39} - 21 q^{41} + q^{43} - 4 q^{45} + 23 q^{47} + 2 q^{51} - 15 q^{53} + 2 q^{55} - 3 q^{57} + 25 q^{59} + 21 q^{61} + 6 q^{65} + 2 q^{67} - 20 q^{69} + 13 q^{71} - 2 q^{73} + 24 q^{75} - 17 q^{79} - 2 q^{81} + 4 q^{83} + 40 q^{85} + 30 q^{87} - q^{89} + 6 q^{93} - q^{95} - 13 q^{97} + 41 q^{99}+O(q^{100}) \) 6 * q + 3 * q^3 + q^5 + 3 * q^9 + 3 * q^11 + 6 * q^13 - 8 * q^15 - 2 * q^17 - 6 * q^19 + 2 * q^23 + 5 * q^25 + 6 * q^27 - q^29 + 14 * q^31 + 19 * q^33 - 7 * q^37 + 22 * q^39 - 21 * q^41 + q^43 - 4 * q^45 + 23 * q^47 + 2 * q^51 - 15 * q^53 + 2 * q^55 - 3 * q^57 + 25 * q^59 + 21 * q^61 + 6 * q^65 + 2 * q^67 - 20 * q^69 + 13 * q^71 - 2 * q^73 + 24 * q^75 - 17 * q^79 - 2 * q^81 + 4 * q^83 + 40 * q^85 + 30 * q^87 - q^89 + 6 * q^93 - q^95 - 13 * q^97 + 41 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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