LMFDB - Embedded newform 7448.2.a.bx.1.3

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:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 6 x^{13} - 11 x^{12} + 114 x^{11} - 10 x^{10} - 806 x^{9} + 523 x^{8} + 2586 x^{7} - 2226 x^{6} + \cdots + 8 $ x^14 - 6x^13 - 11x^12 + 114x^11 - 10x^10 - 806x^9 + 523x^8 + 2586x^7 - 2226x^6 - 3618x^5 + 3397x^4 + 1570x^3 - 1529x^2 + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.76077$ 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)

$f(q)$	$=$	$q-1.76077 q^{3} +2.55769 q^{5} +0.100310 q^{9} +O(q^{10})$	$q-1.76077 q^{3} +2.55769 q^{5} +0.100310 q^{9} +4.25895 q^{11} -0.715655 q^{13} -4.50350 q^{15} -1.47307 q^{17} +1.00000 q^{19} -5.33686 q^{23} +1.54178 q^{25} +5.10569 q^{27} +2.38490 q^{29} -0.259366 q^{31} -7.49904 q^{33} -3.90654 q^{37} +1.26010 q^{39} +5.32470 q^{41} +0.379131 q^{43} +0.256563 q^{45} +13.5398 q^{47} +2.59374 q^{51} -4.69089 q^{53} +10.8931 q^{55} -1.76077 q^{57} -7.73675 q^{59} -2.13425 q^{61} -1.83042 q^{65} +9.11553 q^{67} +9.39698 q^{69} +7.16740 q^{71} -1.97217 q^{73} -2.71472 q^{75} +4.41668 q^{79} -9.29087 q^{81} -0.547536 q^{83} -3.76766 q^{85} -4.19926 q^{87} +0.672532 q^{89} +0.456684 q^{93} +2.55769 q^{95} +17.9530 q^{97} +0.427217 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 6 q^{3} + 2 q^{5} + 16 q^{9}+O(q^{10}) $ 14 * q + 6 * q^3 + 2 * q^5 + 16 * q^9	$ 14 q + 6 q^{3} + 2 q^{5} + 16 q^{9} - 6 q^{11} + 16 q^{13} + 4 q^{15} - 4 q^{17} + 14 q^{19} - 4 q^{23} + 16 q^{25} + 36 q^{27} - 6 q^{29} + 16 q^{31} - 10 q^{33} + 6 q^{37} + 16 q^{39} - 14 q^{41} - 2 q^{43} + 30 q^{47} + 20 q^{51} - 6 q^{53} + 44 q^{55} + 6 q^{57} + 22 q^{59} + 10 q^{61} - 16 q^{65} + 4 q^{67} + 48 q^{69} + 6 q^{71} + 4 q^{73} + 64 q^{75} + 26 q^{79} + 30 q^{81} + 32 q^{83} - 8 q^{85} + 32 q^{87} - 54 q^{89} - 32 q^{93} + 2 q^{95} + 18 q^{97} - 38 q^{99}+O(q^{100}) $ 14 * q + 6 * q^3 + 2 * q^5 + 16 * q^9 - 6 * q^11 + 16 * q^13 + 4 * q^15 - 4 * q^17 + 14 * q^19 - 4 * q^23 + 16 * q^25 + 36 * q^27 - 6 * q^29 + 16 * q^31 - 10 * q^33 + 6 * q^37 + 16 * q^39 - 14 * q^41 - 2 * q^43 + 30 * q^47 + 20 * q^51 - 6 * q^53 + 44 * q^55 + 6 * q^57 + 22 * q^59 + 10 * q^61 - 16 * q^65 + 4 * q^67 + 48 * q^69 + 6 * q^71 + 4 * q^73 + 64 * q^75 + 26 * q^79 + 30 * q^81 + 32 * q^83 - 8 * q^85 + 32 * q^87 - 54 * q^89 - 32 * q^93 + 2 * q^95 + 18 * q^97 - 38 * 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
$3$	−1.76077	−1.01658	−0.508290	−	0.861186i	$-0.669722\pi$
$3$	−1.76077	−1.01658	−0.508290	+	0.861186i	$0.669722\pi$
$4$	0	0
$5$	2.55769	1.14383	0.571917	−	0.820312i	$-0.306200\pi$
$5$	2.55769	1.14383	0.571917	+	0.820312i	$0.306200\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0.100310	0.0334368
$10$	0	0
$11$	4.25895	1.28412	0.642061	−	0.766653i	$-0.278079\pi$
$11$	4.25895	1.28412	0.642061	+	0.766653i	$0.278079\pi$
$12$	0	0
$13$	−0.715655	−0.198487	−0.0992436	−	0.995063i	$-0.531642\pi$
$13$	−0.715655	−0.198487	−0.0992436	+	0.995063i	$0.531642\pi$
$14$	0	0
$15$	−4.50350	−1.16280
$16$	0	0
$17$	−1.47307	−0.357273	−0.178636	−	0.983915i	$-0.557169\pi$
$17$	−1.47307	−0.357273	−0.178636	+	0.983915i	$0.557169\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.33686	−1.11281	−0.556406	−	0.830911i	$-0.687820\pi$
$23$	−5.33686	−1.11281	−0.556406	+	0.830911i	$0.687820\pi$
$24$	0	0
$25$	1.54178	0.308356
$26$	0	0
$27$	5.10569	0.982590
$28$	0	0
$29$	2.38490	0.442865	0.221433	−	0.975176i	$-0.428927\pi$
$29$	2.38490	0.442865	0.221433	+	0.975176i	$0.428927\pi$
$30$	0	0
$31$	−0.259366	−0.0465835	−0.0232917	−	0.999729i	$-0.507415\pi$
$31$	−0.259366	−0.0465835	−0.0232917	+	0.999729i	$0.507415\pi$
$32$	0	0
$33$	−7.49904	−1.30541
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.90654	−0.642231	−0.321115	−	0.947040i	$-0.604058\pi$
$37$	−3.90654	−0.642231	−0.321115	+	0.947040i	$0.604058\pi$
$38$	0	0
$39$	1.26010	0.201778
$40$	0	0
$41$	5.32470	0.831578	0.415789	−	0.909461i	$-0.363505\pi$
$41$	5.32470	0.831578	0.415789	+	0.909461i	$0.363505\pi$
$42$	0	0
$43$	0.379131	0.0578170	0.0289085	−	0.999582i	$-0.490797\pi$
$43$	0.379131	0.0578170	0.0289085	+	0.999582i	$0.490797\pi$
$44$	0	0
$45$	0.256563	0.0382461
$46$	0	0
$47$	13.5398	1.97499	0.987495	−	0.157652i	$-0.0503924\pi$
$47$	13.5398	1.97499	0.987495	+	0.157652i	$0.0503924\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	2.59374	0.363196
$52$	0	0
$53$	−4.69089	−0.644343	−0.322171	−	0.946681i	$-0.604413\pi$
$53$	−4.69089	−0.644343	−0.322171	+	0.946681i	$0.604413\pi$
$54$	0	0
$55$	10.8931	1.46882
$56$	0	0
$57$	−1.76077	−0.233220
$58$	0	0
$59$	−7.73675	−1.00724	−0.503620	−	0.863925i	$-0.667999\pi$
$59$	−7.73675	−1.00724	−0.503620	+	0.863925i	$0.667999\pi$
$60$	0	0
$61$	−2.13425	−0.273263	−0.136631	−	0.990622i	$-0.543628\pi$
$61$	−2.13425	−0.273263	−0.136631	+	0.990622i	$0.543628\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.83042	−0.227036
$66$	0	0
$67$	9.11553	1.11364	0.556819	−	0.830634i	$-0.312022\pi$
$67$	9.11553	1.11364	0.556819	+	0.830634i	$0.312022\pi$
$68$	0	0
$69$	9.39698	1.13126
$70$	0	0
$71$	7.16740	0.850614	0.425307	−	0.905049i	$-0.360166\pi$
$71$	7.16740	0.850614	0.425307	+	0.905049i	$0.360166\pi$
$72$	0	0
$73$	−1.97217	−0.230825	−0.115413	−	0.993318i	$-0.536819\pi$
$73$	−1.97217	−0.230825	−0.115413	+	0.993318i	$0.536819\pi$
$74$	0	0
$75$	−2.71472	−0.313468
$76$	0	0
$77$	0	0
$78$	0	0
$79$	4.41668	0.496915	0.248457	−	0.968643i	$-0.420076\pi$
$79$	4.41668	0.496915	0.248457	+	0.968643i	$0.420076\pi$
$80$	0	0
$81$	−9.29087	−1.03232
$82$	0	0
$83$	−0.547536	−0.0600999	−0.0300500	−	0.999548i	$-0.509567\pi$
$83$	−0.547536	−0.0600999	−0.0300500	+	0.999548i	$0.509567\pi$
$84$	0	0
$85$	−3.76766	−0.408660
$86$	0	0
$87$	−4.19926	−0.450208
$88$	0	0
$89$	0.672532	0.0712882	0.0356441	−	0.999365i	$-0.488652\pi$
$89$	0.672532	0.0712882	0.0356441	+	0.999365i	$0.488652\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0.456684	0.0473559
$94$	0	0
$95$	2.55769	0.262413
$96$	0	0
$97$	17.9530	1.82285	0.911423	−	0.411470i	$-0.134984\pi$
$97$	17.9530	1.82285	0.911423	+	0.411470i	$0.134984\pi$
$98$	0	0
$99$	0.427217	0.0429369
$100$	0	0
$101$	2.64585	0.263271	0.131636	−	0.991298i	$-0.457977\pi$
$101$	2.64585	0.263271	0.131636	+	0.991298i	$0.457977\pi$
$102$	0	0
$103$	13.3570	1.31610	0.658051	−	0.752973i	$-0.271381\pi$
$103$	13.3570	1.31610	0.658051	+	0.752973i	$0.271381\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.8904	1.53618	0.768090	−	0.640342i	$-0.221207\pi$
$107$	15.8904	1.53618	0.768090	+	0.640342i	$0.221207\pi$
$108$	0	0
$109$	−19.7847	−1.89503	−0.947517	−	0.319704i	$-0.896417\pi$
$109$	−19.7847	−1.89503	−0.947517	+	0.319704i	$0.896417\pi$
$110$	0	0
$111$	6.87851	0.652880
$112$	0	0
$113$	1.57336	0.148009	0.0740046	−	0.997258i	$-0.476422\pi$
$113$	1.57336	0.148009	0.0740046	+	0.997258i	$0.476422\pi$
$114$	0	0
$115$	−13.6500	−1.27287
$116$	0	0
$117$	−0.0717876	−0.00663677
$118$	0	0
$119$	0	0
$120$	0	0
$121$	7.13868	0.648971
$122$	0	0
$123$	−9.37557	−0.845367
$124$	0	0
$125$	−8.84506	−0.791126
$126$	0	0
$127$	−3.48386	−0.309143	−0.154571	−	0.987982i	$-0.549400\pi$
$127$	−3.48386	−0.309143	−0.154571	+	0.987982i	$0.549400\pi$
$128$	0	0
$129$	−0.667563	−0.0587757
$130$	0	0
$131$	−1.89563	−0.165622	−0.0828111	−	0.996565i	$-0.526390\pi$
$131$	−1.89563	−0.165622	−0.0828111	+	0.996565i	$0.526390\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	13.0588	1.12392
$136$	0	0
$137$	2.18529	0.186702	0.0933510	−	0.995633i	$-0.470242\pi$
$137$	2.18529	0.186702	0.0933510	+	0.995633i	$0.470242\pi$
$138$	0	0
$139$	−1.50557	−0.127700	−0.0638502	−	0.997959i	$-0.520338\pi$
$139$	−1.50557	−0.127700	−0.0638502	+	0.997959i	$0.520338\pi$
$140$	0	0
$141$	−23.8406	−2.00774
$142$	0	0
$143$	−3.04794	−0.254882
$144$	0	0
$145$	6.09984	0.506564
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0.123769	0.0101395	0.00506977	−	0.999987i	$-0.498386\pi$
$149$	0.123769	0.0101395	0.00506977	+	0.999987i	$0.498386\pi$
$150$	0	0
$151$	8.66796	0.705389	0.352694	−	0.935739i	$-0.385266\pi$
$151$	8.66796	0.705389	0.352694	+	0.935739i	$0.385266\pi$
$152$	0	0
$153$	−0.147764	−0.0119460
$154$	0	0
$155$	−0.663378	−0.0532838
$156$	0	0
$157$	4.21034	0.336022	0.168011	−	0.985785i	$-0.446266\pi$
$157$	4.21034	0.336022	0.168011	+	0.985785i	$0.446266\pi$
$158$	0	0
$159$	8.25957	0.655027
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−18.3234	−1.43520	−0.717599	−	0.696457i	$-0.754759\pi$
$163$	−18.3234	−1.43520	−0.717599	+	0.696457i	$0.754759\pi$
$164$	0	0
$165$	−19.1802	−1.49318
$166$	0	0
$167$	12.3979	0.959380	0.479690	−	0.877438i	$-0.340749\pi$
$167$	12.3979	0.959380	0.479690	+	0.877438i	$0.340749\pi$
$168$	0	0
$169$	−12.4878	−0.960603
$170$	0	0
$171$	0.100310	0.00767092
$172$	0	0
$173$	19.4483	1.47863	0.739314	−	0.673361i	$-0.235150\pi$
$173$	19.4483	1.47863	0.739314	+	0.673361i	$0.235150\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	13.6226	1.02394
$178$	0	0
$179$	−11.3711	−0.849918	−0.424959	−	0.905213i	$-0.639712\pi$
$179$	−11.3711	−0.849918	−0.424959	+	0.905213i	$0.639712\pi$
$180$	0	0
$181$	0.688190	0.0511528	0.0255764	−	0.999673i	$-0.491858\pi$
$181$	0.688190	0.0511528	0.0255764	+	0.999673i	$0.491858\pi$
$182$	0	0
$183$	3.75792	0.277794
$184$	0	0
$185$	−9.99171	−0.734605
$186$	0	0
$187$	−6.27375	−0.458782
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.68377	0.121834	0.0609168	−	0.998143i	$-0.480598\pi$
$191$	1.68377	0.121834	0.0609168	+	0.998143i	$0.480598\pi$
$192$	0	0
$193$	−12.9679	−0.933449	−0.466725	−	0.884403i	$-0.654566\pi$
$193$	−12.9679	−0.933449	−0.466725	+	0.884403i	$0.654566\pi$
$194$	0	0
$195$	3.22296	0.230801
$196$	0	0
$197$	14.6807	1.04595	0.522977	−	0.852346i	$-0.324821\pi$
$197$	14.6807	1.04595	0.522977	+	0.852346i	$0.324821\pi$
$198$	0	0
$199$	−12.9356	−0.916978	−0.458489	−	0.888700i	$-0.651609\pi$
$199$	−12.9356	−0.916978	−0.458489	+	0.888700i	$0.651609\pi$
$200$	0	0
$201$	−16.0503	−1.13210
$202$	0	0
$203$	0	0
$204$	0	0
$205$	13.6189	0.951187
$206$	0	0
$207$	−0.535342	−0.0372088
$208$	0	0
$209$	4.25895	0.294598
$210$	0	0
$211$	3.17170	0.218349	0.109174	−	0.994023i	$-0.465179\pi$
$211$	3.17170	0.218349	0.109174	+	0.994023i	$0.465179\pi$
$212$	0	0
$213$	−12.6201	−0.864718
$214$	0	0
$215$	0.969701	0.0661330
$216$	0	0
$217$	0	0
$218$	0	0
$219$	3.47254	0.234653
$220$	0	0
$221$	1.05421	0.0709140
$222$	0	0
$223$	20.9795	1.40489	0.702446	−	0.711737i	$-0.252091\pi$
$223$	20.9795	1.40489	0.702446	+	0.711737i	$0.252091\pi$
$224$	0	0
$225$	0.154656	0.0103104
$226$	0	0
$227$	18.0959	1.20106	0.600532	−	0.799601i	$-0.294956\pi$
$227$	18.0959	1.20106	0.600532	+	0.799601i	$0.294956\pi$
$228$	0	0
$229$	7.15517	0.472827	0.236413	−	0.971653i	$-0.424028\pi$
$229$	7.15517	0.472827	0.236413	+	0.971653i	$0.424028\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−12.3065	−0.806229	−0.403114	−	0.915150i	$-0.632072\pi$
$233$	−12.3065	−0.806229	−0.403114	+	0.915150i	$0.632072\pi$
$234$	0	0
$235$	34.6307	2.25906
$236$	0	0
$237$	−7.77675	−0.505154
$238$	0	0
$239$	9.18450	0.594096	0.297048	−	0.954863i	$-0.403998\pi$
$239$	9.18450	0.594096	0.297048	+	0.954863i	$0.403998\pi$
$240$	0	0
$241$	−10.5515	−0.679682	−0.339841	−	0.940483i	$-0.610373\pi$
$241$	−10.5515	−0.679682	−0.339841	+	0.940483i	$0.610373\pi$
$242$	0	0
$243$	1.04202	0.0668458
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−0.715655	−0.0455361
$248$	0	0
$249$	0.964085	0.0610964
$250$	0	0
$251$	−0.596416	−0.0376454	−0.0188227	−	0.999823i	$-0.505992\pi$
$251$	−0.596416	−0.0376454	−0.0188227	+	0.999823i	$0.505992\pi$
$252$	0	0
$253$	−22.7294	−1.42899
$254$	0	0
$255$	6.63399	0.415436
$256$	0	0
$257$	−12.1975	−0.760858	−0.380429	−	0.924810i	$-0.624224\pi$
$257$	−12.1975	−0.760858	−0.380429	+	0.924810i	$0.624224\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0.239230	0.0148080
$262$	0	0
$263$	−12.3014	−0.758539	−0.379270	−	0.925286i	$-0.623825\pi$
$263$	−12.3014	−0.758539	−0.379270	+	0.925286i	$0.623825\pi$
$264$	0	0
$265$	−11.9978	−0.737021
$266$	0	0
$267$	−1.18417	−0.0724702
$268$	0	0
$269$	18.6990	1.14010	0.570048	−	0.821611i	$-0.306925\pi$
$269$	18.6990	1.14010	0.570048	+	0.821611i	$0.306925\pi$
$270$	0	0
$271$	12.0223	0.730301	0.365150	−	0.930949i	$-0.381018\pi$
$271$	12.0223	0.730301	0.365150	+	0.930949i	$0.381018\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.56636	0.395966
$276$	0	0
$277$	−14.8696	−0.893427	−0.446714	−	0.894677i	$-0.647406\pi$
$277$	−14.8696	−0.893427	−0.446714	+	0.894677i	$0.647406\pi$
$278$	0	0
$279$	−0.0260171	−0.00155760
$280$	0	0
$281$	−4.93497	−0.294395	−0.147198	−	0.989107i	$-0.547025\pi$
$281$	−4.93497	−0.294395	−0.147198	+	0.989107i	$0.547025\pi$
$282$	0	0
$283$	19.1320	1.13728	0.568639	−	0.822587i	$-0.307470\pi$
$283$	19.1320	1.13728	0.568639	+	0.822587i	$0.307470\pi$
$284$	0	0
$285$	−4.50350	−0.266765
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−14.8301	−0.872356
$290$	0	0
$291$	−31.6110	−1.85307
$292$	0	0
$293$	17.7103	1.03464	0.517322	−	0.855791i	$-0.326929\pi$
$293$	17.7103	1.03464	0.517322	+	0.855791i	$0.326929\pi$
$294$	0	0
$295$	−19.7882	−1.15211
$296$	0	0
$297$	21.7449	1.26177
$298$	0	0
$299$	3.81935	0.220879
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−4.65873	−0.267637
$304$	0	0
$305$	−5.45875	−0.312567
$306$	0	0
$307$	−3.51306	−0.200501	−0.100251	−	0.994962i	$-0.531964\pi$
$307$	−3.51306	−0.200501	−0.100251	+	0.994962i	$0.531964\pi$
$308$	0	0
$309$	−23.5186	−1.33792
$310$	0	0
$311$	9.20067	0.521722	0.260861	−	0.965376i	$-0.415994\pi$
$311$	9.20067	0.521722	0.260861	+	0.965376i	$0.415994\pi$
$312$	0	0
$313$	13.1161	0.741365	0.370683	−	0.928760i	$-0.379124\pi$
$313$	13.1161	0.741365	0.370683	+	0.928760i	$0.379124\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.5322	0.591546	0.295773	−	0.955258i	$-0.404423\pi$
$317$	10.5322	0.591546	0.295773	+	0.955258i	$0.404423\pi$
$318$	0	0
$319$	10.1572	0.568693
$320$	0	0
$321$	−27.9793	−1.56165
$322$	0	0
$323$	−1.47307	−0.0819639
$324$	0	0
$325$	−1.10338	−0.0612046
$326$	0	0
$327$	34.8364	1.92646
$328$	0	0
$329$	0	0
$330$	0	0
$331$	15.2941	0.840642	0.420321	−	0.907376i	$-0.361918\pi$
$331$	15.2941	0.840642	0.420321	+	0.907376i	$0.361918\pi$
$332$	0	0
$333$	−0.391866	−0.0214741
$334$	0	0
$335$	23.3147	1.27382
$336$	0	0
$337$	−3.59134	−0.195633	−0.0978163	−	0.995204i	$-0.531186\pi$
$337$	−3.59134	−0.195633	−0.0978163	+	0.995204i	$0.531186\pi$
$338$	0	0
$339$	−2.77032	−0.150463
$340$	0	0
$341$	−1.10463	−0.0598189
$342$	0	0
$343$	0	0
$344$	0	0
$345$	24.0346	1.29398
$346$	0	0
$347$	−0.506913	−0.0272125	−0.0136063	−	0.999907i	$-0.504331\pi$
$347$	−0.506913	−0.0272125	−0.0136063	+	0.999907i	$0.504331\pi$
$348$	0	0
$349$	−16.8807	−0.903603	−0.451802	−	0.892118i	$-0.649219\pi$
$349$	−16.8807	−0.903603	−0.451802	+	0.892118i	$0.649219\pi$
$350$	0	0
$351$	−3.65391	−0.195031
$352$	0	0
$353$	13.3672	0.711463	0.355731	−	0.934588i	$-0.384232\pi$
$353$	13.3672	0.711463	0.355731	+	0.934588i	$0.384232\pi$
$354$	0	0
$355$	18.3320	0.972961
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−13.4841	−0.711664	−0.355832	−	0.934550i	$-0.615802\pi$
$359$	−13.4841	−0.711664	−0.355832	+	0.934550i	$0.615802\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−12.5696	−0.659731
$364$	0	0
$365$	−5.04420	−0.264026
$366$	0	0
$367$	30.2608	1.57960	0.789799	−	0.613365i	$-0.210185\pi$
$367$	30.2608	1.57960	0.789799	+	0.613365i	$0.210185\pi$
$368$	0	0
$369$	0.534122	0.0278053
$370$	0	0
$371$	0	0
$372$	0	0
$373$	33.5833	1.73888	0.869439	−	0.494041i	$-0.164481\pi$
$373$	33.5833	1.73888	0.869439	+	0.494041i	$0.164481\pi$
$374$	0	0
$375$	15.5741	0.804244
$376$	0	0
$377$	−1.70677	−0.0879030
$378$	0	0
$379$	−5.43782	−0.279322	−0.139661	−	0.990199i	$-0.544601\pi$
$379$	−5.43782	−0.279322	−0.139661	+	0.990199i	$0.544601\pi$
$380$	0	0
$381$	6.13428	0.314269
$382$	0	0
$383$	30.5579	1.56144	0.780718	−	0.624884i	$-0.214854\pi$
$383$	30.5579	1.56144	0.780718	+	0.624884i	$0.214854\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.0380308	0.00193321
$388$	0	0
$389$	−33.8747	−1.71751	−0.858757	−	0.512383i	$-0.828763\pi$
$389$	−33.8747	−1.71751	−0.858757	+	0.512383i	$0.828763\pi$
$390$	0	0
$391$	7.86158	0.397577
$392$	0	0
$393$	3.33777	0.168368
$394$	0	0
$395$	11.2965	0.568388
$396$	0	0
$397$	9.21544	0.462510	0.231255	−	0.972893i	$-0.425717\pi$
$397$	9.21544	0.462510	0.231255	+	0.972893i	$0.425717\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−8.85414	−0.442155	−0.221077	−	0.975256i	$-0.570957\pi$
$401$	−8.85414	−0.442155	−0.221077	+	0.975256i	$0.570957\pi$
$402$	0	0
$403$	0.185617	0.00924622
$404$	0	0
$405$	−23.7632	−1.18080
$406$	0	0
$407$	−16.6378	−0.824703
$408$	0	0
$409$	−1.98589	−0.0981961	−0.0490980	−	0.998794i	$-0.515635\pi$
$409$	−1.98589	−0.0981961	−0.0490980	+	0.998794i	$0.515635\pi$
$410$	0	0
$411$	−3.84779	−0.189798
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−1.40043	−0.0687443
$416$	0	0
$417$	2.65095	0.129818
$418$	0	0
$419$	34.7368	1.69700	0.848502	−	0.529193i	$-0.177505\pi$
$419$	34.7368	1.69700	0.848502	+	0.529193i	$0.177505\pi$
$420$	0	0
$421$	7.45798	0.363480	0.181740	−	0.983347i	$-0.441827\pi$
$421$	7.45798	0.363480	0.181740	+	0.983347i	$0.441827\pi$
$422$	0	0
$423$	1.35819	0.0660373
$424$	0	0
$425$	−2.27115	−0.110167
$426$	0	0
$427$	0	0
$428$	0	0
$429$	5.36673	0.259108
$430$	0	0
$431$	20.1702	0.971564	0.485782	−	0.874080i	$-0.338535\pi$
$431$	20.1702	0.971564	0.485782	+	0.874080i	$0.338535\pi$
$432$	0	0
$433$	21.1955	1.01859	0.509295	−	0.860592i	$-0.329906\pi$
$433$	21.1955	1.01859	0.509295	+	0.860592i	$0.329906\pi$
$434$	0	0
$435$	−10.7404	−0.514963
$436$	0	0
$437$	−5.33686	−0.255296
$438$	0	0
$439$	−32.6968	−1.56053	−0.780267	−	0.625447i	$-0.784917\pi$
$439$	−32.6968	−1.56053	−0.780267	+	0.625447i	$0.784917\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	39.3131	1.86782	0.933912	−	0.357504i	$-0.116372\pi$
$443$	39.3131	1.86782	0.933912	+	0.357504i	$0.116372\pi$
$444$	0	0
$445$	1.72013	0.0815419
$446$	0	0
$447$	−0.217928	−0.0103077
$448$	0	0
$449$	29.6782	1.40060	0.700300	−	0.713849i	$-0.253050\pi$
$449$	29.6782	1.40060	0.700300	+	0.713849i	$0.253050\pi$
$450$	0	0
$451$	22.6776	1.06785
$452$	0	0
$453$	−15.2623	−0.717085
$454$	0	0
$455$	0	0
$456$	0	0
$457$	37.9334	1.77445	0.887224	−	0.461338i	$-0.152631\pi$
$457$	37.9334	1.77445	0.887224	+	0.461338i	$0.152631\pi$
$458$	0	0
$459$	−7.52104	−0.351052
$460$	0	0
$461$	−11.4458	−0.533085	−0.266542	−	0.963823i	$-0.585881\pi$
$461$	−11.4458	−0.533085	−0.266542	+	0.963823i	$0.585881\pi$
$462$	0	0
$463$	−29.8406	−1.38681	−0.693404	−	0.720549i	$-0.743890\pi$
$463$	−29.8406	−1.38681	−0.693404	+	0.720549i	$0.743890\pi$
$464$	0	0
$465$	1.16806	0.0541673
$466$	0	0
$467$	13.9546	0.645742	0.322871	−	0.946443i	$-0.395352\pi$
$467$	13.9546	0.645742	0.322871	+	0.946443i	$0.395352\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−7.41344	−0.341593
$472$	0	0
$473$	1.61470	0.0742441
$474$	0	0
$475$	1.54178	0.0707416
$476$	0	0
$477$	−0.470544	−0.0215447
$478$	0	0
$479$	−26.6540	−1.21785	−0.608925	−	0.793228i	$-0.708399\pi$
$479$	−26.6540	−1.21785	−0.608925	+	0.793228i	$0.708399\pi$
$480$	0	0
$481$	2.79574	0.127475
$482$	0	0
$483$	0	0
$484$	0	0
$485$	45.9181	2.08503
$486$	0	0
$487$	10.4461	0.473360	0.236680	−	0.971588i	$-0.423941\pi$
$487$	10.4461	0.473360	0.236680	+	0.971588i	$0.423941\pi$
$488$	0	0
$489$	32.2632	1.45899
$490$	0	0
$491$	27.7543	1.25253	0.626266	−	0.779609i	$-0.284582\pi$
$491$	27.7543	1.25253	0.626266	+	0.779609i	$0.284582\pi$
$492$	0	0
$493$	−3.51313	−0.158223
$494$	0	0
$495$	1.09269	0.0491127
$496$	0	0
$497$	0	0
$498$	0	0
$499$	15.5576	0.696456	0.348228	−	0.937410i	$-0.386784\pi$
$499$	15.5576	0.696456	0.348228	+	0.937410i	$0.386784\pi$
$500$	0	0
$501$	−21.8299	−0.975287
$502$	0	0
$503$	−2.96496	−0.132201	−0.0661005	−	0.997813i	$-0.521056\pi$
$503$	−2.96496	−0.132201	−0.0661005	+	0.997813i	$0.521056\pi$
$504$	0	0
$505$	6.76725	0.301139
$506$	0	0
$507$	21.9882	0.976531
$508$	0	0
$509$	−7.13130	−0.316089	−0.158045	−	0.987432i	$-0.550519\pi$
$509$	−7.13130	−0.316089	−0.158045	+	0.987432i	$0.550519\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	5.10569	0.225422
$514$	0	0
$515$	34.1630	1.50540
$516$	0	0
$517$	57.6656	2.53613
$518$	0	0
$519$	−34.2440	−1.50315
$520$	0	0
$521$	−28.9311	−1.26749	−0.633746	−	0.773541i	$-0.718484\pi$
$521$	−28.9311	−1.26749	−0.633746	+	0.773541i	$0.718484\pi$
$522$	0	0
$523$	−22.5891	−0.987751	−0.493875	−	0.869533i	$-0.664420\pi$
$523$	−22.5891	−0.987751	−0.493875	+	0.869533i	$0.664420\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.382065	0.0166430
$528$	0	0
$529$	5.48204	0.238350
$530$	0	0
$531$	−0.776076	−0.0336788
$532$	0	0
$533$	−3.81065	−0.165058
$534$	0	0
$535$	40.6426	1.75713
$536$	0	0
$537$	20.0219	0.864011
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−19.9948	−0.859642	−0.429821	−	0.902914i	$-0.641423\pi$
$541$	−19.9948	−0.859642	−0.429821	+	0.902914i	$0.641423\pi$
$542$	0	0
$543$	−1.21174	−0.0520009
$544$	0	0
$545$	−50.6032	−2.16760
$546$	0	0
$547$	0.883938	0.0377945	0.0188972	−	0.999821i	$-0.493984\pi$
$547$	0.883938	0.0377945	0.0188972	+	0.999821i	$0.493984\pi$
$548$	0	0
$549$	−0.214087	−0.00913702
$550$	0	0
$551$	2.38490	0.101600
$552$	0	0
$553$	0	0
$554$	0	0
$555$	17.5931	0.746786
$556$	0	0
$557$	3.02785	0.128294	0.0641470	−	0.997940i	$-0.479567\pi$
$557$	3.02785	0.128294	0.0641470	+	0.997940i	$0.479567\pi$
$558$	0	0
$559$	−0.271327	−0.0114759
$560$	0	0
$561$	11.0466	0.466389
$562$	0	0
$563$	24.2863	1.02355	0.511774	−	0.859120i	$-0.328989\pi$
$563$	24.2863	1.02355	0.511774	+	0.859120i	$0.328989\pi$
$564$	0	0
$565$	4.02417	0.169298
$566$	0	0
$567$	0	0
$568$	0	0
$569$	38.1438	1.59907	0.799535	−	0.600619i	$-0.205079\pi$
$569$	38.1438	1.59907	0.799535	+	0.600619i	$0.205079\pi$
$570$	0	0
$571$	44.8768	1.87804	0.939018	−	0.343868i	$-0.111737\pi$
$571$	44.8768	1.87804	0.939018	+	0.343868i	$0.111737\pi$
$572$	0	0
$573$	−2.96474	−0.123854
$574$	0	0
$575$	−8.22825	−0.343142
$576$	0	0
$577$	−25.7758	−1.07306	−0.536531	−	0.843880i	$-0.680266\pi$
$577$	−25.7758	−1.07306	−0.536531	+	0.843880i	$0.680266\pi$
$578$	0	0
$579$	22.8335	0.948927
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−19.9783	−0.827415
$584$	0	0
$585$	−0.183610	−0.00759136
$586$	0	0
$587$	0.370372	0.0152869	0.00764345	−	0.999971i	$-0.497567\pi$
$587$	0.370372	0.0152869	0.00764345	+	0.999971i	$0.497567\pi$
$588$	0	0
$589$	−0.259366	−0.0106870
$590$	0	0
$591$	−25.8493	−1.06330
$592$	0	0
$593$	−17.9136	−0.735623	−0.367811	−	0.929900i	$-0.619893\pi$
$593$	−17.9136	−0.735623	−0.367811	+	0.929900i	$0.619893\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	22.7765	0.932182
$598$	0	0
$599$	−19.8153	−0.809633	−0.404816	−	0.914398i	$-0.632665\pi$
$599$	−19.8153	−0.809633	−0.404816	+	0.914398i	$0.632665\pi$
$600$	0	0
$601$	15.0402	0.613504	0.306752	−	0.951789i	$-0.400758\pi$
$601$	15.0402	0.613504	0.306752	+	0.951789i	$0.400758\pi$
$602$	0	0
$603$	0.914381	0.0372365
$604$	0	0
$605$	18.2585	0.742315
$606$	0	0
$607$	−23.1751	−0.940649	−0.470324	−	0.882494i	$-0.655863\pi$
$607$	−23.1751	−0.940649	−0.470324	+	0.882494i	$0.655863\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−9.68986	−0.392010
$612$	0	0
$613$	−19.2797	−0.778698	−0.389349	−	0.921090i	$-0.627300\pi$
$613$	−19.2797	−0.778698	−0.389349	+	0.921090i	$0.627300\pi$
$614$	0	0
$615$	−23.9798	−0.966959
$616$	0	0
$617$	18.9960	0.764749	0.382374	−	0.924008i	$-0.375107\pi$
$617$	18.9960	0.764749	0.382374	+	0.924008i	$0.375107\pi$
$618$	0	0
$619$	9.49719	0.381724	0.190862	−	0.981617i	$-0.438872\pi$
$619$	9.49719	0.381724	0.190862	+	0.981617i	$0.438872\pi$
$620$	0	0
$621$	−27.2483	−1.09344
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−30.3318	−1.21327
$626$	0	0
$627$	−7.49904	−0.299483
$628$	0	0
$629$	5.75461	0.229451
$630$	0	0
$631$	−15.9877	−0.636460	−0.318230	−	0.948013i	$-0.603089\pi$
$631$	−15.9877	−0.636460	−0.318230	+	0.948013i	$0.603089\pi$
$632$	0	0
$633$	−5.58463	−0.221969
$634$	0	0
$635$	−8.91064	−0.353608
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0.718964	0.0284418
$640$	0	0
$641$	−23.7942	−0.939814	−0.469907	−	0.882716i	$-0.655713\pi$
$641$	−23.7942	−0.939814	−0.469907	+	0.882716i	$0.655713\pi$
$642$	0	0
$643$	−18.5039	−0.729723	−0.364861	−	0.931062i	$-0.618884\pi$
$643$	−18.5039	−0.729723	−0.364861	+	0.931062i	$0.618884\pi$
$644$	0	0
$645$	−1.70742	−0.0672296
$646$	0	0
$647$	−33.0700	−1.30012	−0.650058	−	0.759885i	$-0.725255\pi$
$647$	−33.0700	−1.30012	−0.650058	+	0.759885i	$0.725255\pi$
$648$	0	0
$649$	−32.9505	−1.29342
$650$	0	0
$651$	0	0
$652$	0	0
$653$	34.6869	1.35740	0.678702	−	0.734414i	$-0.262543\pi$
$653$	34.6869	1.35740	0.678702	+	0.734414i	$0.262543\pi$
$654$	0	0
$655$	−4.84844	−0.189444
$656$	0	0
$657$	−0.197829	−0.00771805
$658$	0	0
$659$	−14.1716	−0.552047	−0.276024	−	0.961151i	$-0.589017\pi$
$659$	−14.1716	−0.552047	−0.276024	+	0.961151i	$0.589017\pi$
$660$	0	0
$661$	14.4761	0.563053	0.281527	−	0.959553i	$-0.409159\pi$
$661$	14.4761	0.563053	0.281527	+	0.959553i	$0.409159\pi$
$662$	0	0
$663$	−1.85623	−0.0720898
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−12.7279	−0.492825
$668$	0	0
$669$	−36.9401	−1.42819
$670$	0	0
$671$	−9.08967	−0.350903
$672$	0	0
$673$	37.8900	1.46055	0.730276	−	0.683152i	$-0.239391\pi$
$673$	37.8900	1.46055	0.730276	+	0.683152i	$0.239391\pi$
$674$	0	0
$675$	7.87183	0.302987
$676$	0	0
$677$	49.3270	1.89579	0.947894	−	0.318585i	$-0.103207\pi$
$677$	49.3270	1.89579	0.947894	+	0.318585i	$0.103207\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−31.8626	−1.22098
$682$	0	0
$683$	−6.74750	−0.258186	−0.129093	−	0.991632i	$-0.541207\pi$
$683$	−6.74750	−0.258186	−0.129093	+	0.991632i	$0.541207\pi$
$684$	0	0
$685$	5.58930	0.213556
$686$	0	0
$687$	−12.5986	−0.480667
$688$	0	0
$689$	3.35706	0.127894
$690$	0	0
$691$	−26.1288	−0.993985	−0.496993	−	0.867755i	$-0.665562\pi$
$691$	−26.1288	−0.993985	−0.496993	+	0.867755i	$0.665562\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3.85077	−0.146068
$696$	0	0
$697$	−7.84367	−0.297100
$698$	0	0
$699$	21.6690	0.819597
$700$	0	0
$701$	31.7362	1.19866	0.599331	−	0.800502i	$-0.295433\pi$
$701$	31.7362	1.19866	0.599331	+	0.800502i	$0.295433\pi$
$702$	0	0
$703$	−3.90654	−0.147338
$704$	0	0
$705$	−60.9767	−2.29652
$706$	0	0
$707$	0	0
$708$	0	0
$709$	8.82658	0.331489	0.165745	−	0.986169i	$-0.446997\pi$
$709$	8.82658	0.331489	0.165745	+	0.986169i	$0.446997\pi$
$710$	0	0
$711$	0.443038	0.0166152
$712$	0	0
$713$	1.38420	0.0518386
$714$	0	0
$715$	−7.79569	−0.291542
$716$	0	0
$717$	−16.1718	−0.603946
$718$	0	0
$719$	−7.70355	−0.287294	−0.143647	−	0.989629i	$-0.545883\pi$
$719$	−7.70355	−0.287294	−0.143647	+	0.989629i	$0.545883\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	18.5788	0.690951
$724$	0	0
$725$	3.67699	0.136560
$726$	0	0
$727$	13.4330	0.498204	0.249102	−	0.968477i	$-0.419865\pi$
$727$	13.4330	0.498204	0.249102	+	0.968477i	$0.419865\pi$
$728$	0	0
$729$	26.0378	0.964365
$730$	0	0
$731$	−0.558488	−0.0206564
$732$	0	0
$733$	11.4647	0.423457	0.211729	−	0.977328i	$-0.432091\pi$
$733$	11.4647	0.423457	0.211729	+	0.977328i	$0.432091\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	38.8226	1.43005
$738$	0	0
$739$	39.4664	1.45180	0.725898	−	0.687802i	$-0.241424\pi$
$739$	39.4664	1.45180	0.725898	+	0.687802i	$0.241424\pi$
$740$	0	0
$741$	1.26010	0.0462911
$742$	0	0
$743$	44.8154	1.64412	0.822059	−	0.569403i	$-0.192825\pi$
$743$	44.8154	1.64412	0.822059	+	0.569403i	$0.192825\pi$
$744$	0	0
$745$	0.316562	0.0115979
$746$	0	0
$747$	−0.0549235	−0.00200955
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−36.1684	−1.31980	−0.659901	−	0.751352i	$-0.729402\pi$
$751$	−36.1684	−1.31980	−0.659901	+	0.751352i	$0.729402\pi$
$752$	0	0
$753$	1.05015	0.0382696
$754$	0	0
$755$	22.1700	0.806847
$756$	0	0
$757$	43.1456	1.56815	0.784076	−	0.620664i	$-0.213137\pi$
$757$	43.1456	1.56815	0.784076	+	0.620664i	$0.213137\pi$
$758$	0	0
$759$	40.0213	1.45268
$760$	0	0
$761$	−13.1769	−0.477661	−0.238830	−	0.971061i	$-0.576764\pi$
$761$	−13.1769	−0.477661	−0.238830	+	0.971061i	$0.576764\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−0.377935	−0.0136643
$766$	0	0
$767$	5.53685	0.199924
$768$	0	0
$769$	28.9761	1.04491	0.522453	−	0.852668i	$-0.325017\pi$
$769$	28.9761	1.04491	0.522453	+	0.852668i	$0.325017\pi$
$770$	0	0
$771$	21.4770	0.773474
$772$	0	0
$773$	4.93581	0.177529	0.0887644	−	0.996053i	$-0.471708\pi$
$773$	4.93581	0.177529	0.0887644	+	0.996053i	$0.471708\pi$
$774$	0	0
$775$	−0.399885	−0.0143643
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5.32470	0.190777
$780$	0	0
$781$	30.5256	1.09229
$782$	0	0
$783$	12.1766	0.435155
$784$	0	0
$785$	10.7687	0.384353
$786$	0	0
$787$	−8.34346	−0.297412	−0.148706	−	0.988881i	$-0.547511\pi$
$787$	−8.34346	−0.297412	−0.148706	+	0.988881i	$0.547511\pi$
$788$	0	0
$789$	21.6600	0.771117
$790$	0	0
$791$	0	0
$792$	0	0
$793$	1.52739	0.0542391
$794$	0	0
$795$	21.1254	0.749241
$796$	0	0
$797$	−15.4942	−0.548834	−0.274417	−	0.961611i	$-0.588485\pi$
$797$	−15.4942	−0.548834	−0.274417	+	0.961611i	$0.588485\pi$
$798$	0	0
$799$	−19.9452	−0.705609
$800$	0	0
$801$	0.0674619	0.00238365
$802$	0	0
$803$	−8.39939	−0.296408
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−32.9246	−1.15900
$808$	0	0
$809$	52.2463	1.83688	0.918440	−	0.395559i	$-0.129449\pi$
$809$	52.2463	1.83688	0.918440	+	0.395559i	$0.129449\pi$
$810$	0	0
$811$	−16.0532	−0.563705	−0.281853	−	0.959458i	$-0.590949\pi$
$811$	−16.0532	−0.563705	−0.281853	+	0.959458i	$0.590949\pi$
$812$	0	0
$813$	−21.1684	−0.742410
$814$	0	0
$815$	−46.8655	−1.64163
$816$	0	0
$817$	0.379131	0.0132641
$818$	0	0
$819$	0	0
$820$	0	0
$821$	34.3346	1.19829	0.599143	−	0.800642i	$-0.295508\pi$
$821$	34.3346	1.19829	0.599143	+	0.800642i	$0.295508\pi$
$822$	0	0
$823$	−8.15502	−0.284266	−0.142133	−	0.989848i	$-0.545396\pi$
$823$	−8.15502	−0.284266	−0.142133	+	0.989848i	$0.545396\pi$
$824$	0	0
$825$	−11.5618	−0.402532
$826$	0	0
$827$	−22.8513	−0.794616	−0.397308	−	0.917685i	$-0.630056\pi$
$827$	−22.8513	−0.794616	−0.397308	+	0.917685i	$0.630056\pi$
$828$	0	0
$829$	17.3913	0.604024	0.302012	−	0.953304i	$-0.402342\pi$
$829$	17.3913	0.604024	0.302012	+	0.953304i	$0.402342\pi$
$830$	0	0
$831$	26.1819	0.908241
$832$	0	0
$833$	0	0
$834$	0	0
$835$	31.7100	1.09737
$836$	0	0
$837$	−1.32424	−0.0457725
$838$	0	0
$839$	20.9636	0.723743	0.361871	−	0.932228i	$-0.382138\pi$
$839$	20.9636	0.723743	0.361871	+	0.932228i	$0.382138\pi$
$840$	0	0
$841$	−23.3122	−0.803871
$842$	0	0
$843$	8.68934	0.299277
$844$	0	0
$845$	−31.9400	−1.09877
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−33.6870	−1.15614
$850$	0	0
$851$	20.8486	0.714682
$852$	0	0
$853$	−23.9073	−0.818569	−0.409285	−	0.912407i	$-0.634222\pi$
$853$	−23.9073	−0.818569	−0.409285	+	0.912407i	$0.634222\pi$
$854$	0	0
$855$	0.256563	0.00877426
$856$	0	0
$857$	−51.2831	−1.75180	−0.875899	−	0.482495i	$-0.839731\pi$
$857$	−51.2831	−1.75180	−0.875899	+	0.482495i	$0.839731\pi$
$858$	0	0
$859$	37.8176	1.29032	0.645159	−	0.764048i	$-0.276791\pi$
$859$	37.8176	1.29032	0.645159	+	0.764048i	$0.276791\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−3.98881	−0.135781	−0.0678903	−	0.997693i	$-0.521627\pi$
$863$	−3.98881	−0.135781	−0.0678903	+	0.997693i	$0.521627\pi$
$864$	0	0
$865$	49.7428	1.69130
$866$	0	0
$867$	26.1123	0.886821
$868$	0	0
$869$	18.8104	0.638100
$870$	0	0
$871$	−6.52358	−0.221043
$872$	0	0
$873$	1.80087	0.0609501
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−12.4648	−0.420906	−0.210453	−	0.977604i	$-0.567494\pi$
$877$	−12.4648	−0.420906	−0.210453	+	0.977604i	$0.567494\pi$
$878$	0	0
$879$	−31.1837	−1.05180
$880$	0	0
$881$	42.3737	1.42761	0.713803	−	0.700346i	$-0.246971\pi$
$881$	42.3737	1.42761	0.713803	+	0.700346i	$0.246971\pi$
$882$	0	0
$883$	−27.1031	−0.912093	−0.456047	−	0.889956i	$-0.650735\pi$
$883$	−27.1031	−0.912093	−0.456047	+	0.889956i	$0.650735\pi$
$884$	0	0
$885$	34.8425	1.17122
$886$	0	0
$887$	30.0590	1.00928	0.504642	−	0.863329i	$-0.331625\pi$
$887$	30.0590	1.00928	0.504642	+	0.863329i	$0.331625\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−39.5694	−1.32562
$892$	0	0
$893$	13.5398	0.453094
$894$	0	0
$895$	−29.0838	−0.972165
$896$	0	0
$897$	−6.72500	−0.224541
$898$	0	0
$899$	−0.618562	−0.0206302
$900$	0	0
$901$	6.91001	0.230206
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.76018	0.0585102
$906$	0	0
$907$	21.2878	0.706849	0.353424	−	0.935463i	$-0.385017\pi$
$907$	21.2878	0.706849	0.353424	+	0.935463i	$0.385017\pi$
$908$	0	0
$909$	0.265406	0.00880295
$910$	0	0
$911$	11.8338	0.392070	0.196035	−	0.980597i	$-0.437193\pi$
$911$	11.8338	0.392070	0.196035	+	0.980597i	$0.437193\pi$
$912$	0	0
$913$	−2.33193	−0.0771757
$914$	0	0
$915$	9.61160	0.317750
$916$	0	0
$917$	0	0
$918$	0	0
$919$	17.3944	0.573787	0.286894	−	0.957962i	$-0.407377\pi$
$919$	17.3944	0.573787	0.286894	+	0.957962i	$0.407377\pi$
$920$	0	0
$921$	6.18569	0.203826
$922$	0	0
$923$	−5.12939	−0.168836
$924$	0	0
$925$	−6.02301	−0.198036
$926$	0	0
$927$	1.33984	0.0440062
$928$	0	0
$929$	−58.1037	−1.90632	−0.953160	−	0.302468i	$-0.902189\pi$
$929$	−58.1037	−1.90632	−0.953160	+	0.302468i	$0.902189\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−16.2003	−0.530373
$934$	0	0
$935$	−16.0463	−0.524770
$936$	0	0
$937$	−4.55661	−0.148858	−0.0744290	−	0.997226i	$-0.523713\pi$
$937$	−4.55661	−0.148858	−0.0744290	+	0.997226i	$0.523713\pi$
$938$	0	0
$939$	−23.0944	−0.753658
$940$	0	0
$941$	−59.0949	−1.92644	−0.963219	−	0.268716i	$-0.913401\pi$
$941$	−59.0949	−1.92644	−0.963219	+	0.268716i	$0.913401\pi$
$942$	0	0
$943$	−28.4172	−0.925390
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−47.7920	−1.55303	−0.776516	−	0.630098i	$-0.783015\pi$
$947$	−47.7920	−1.55303	−0.776516	+	0.630098i	$0.783015\pi$
$948$	0	0
$949$	1.41140	0.0458158
$950$	0	0
$951$	−18.5447	−0.601354
$952$	0	0
$953$	43.9234	1.42282	0.711409	−	0.702779i	$-0.248058\pi$
$953$	43.9234	1.42282	0.711409	+	0.702779i	$0.248058\pi$
$954$	0	0
$955$	4.30657	0.139357
$956$	0	0
$957$	−17.8845	−0.578122
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.9327	−0.997830
$962$	0	0
$963$	1.59397	0.0513649
$964$	0	0
$965$	−33.1678	−1.06771
$966$	0	0
$967$	29.5455	0.950119	0.475060	−	0.879954i	$-0.342427\pi$
$967$	29.5455	0.950119	0.475060	+	0.879954i	$0.342427\pi$
$968$	0	0
$969$	2.59374	0.0833230
$970$	0	0
$971$	20.1271	0.645910	0.322955	−	0.946414i	$-0.395324\pi$
$971$	20.1271	0.645910	0.322955	+	0.946414i	$0.395324\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	1.94280	0.0622194
$976$	0	0
$977$	−9.88330	−0.316195	−0.158097	−	0.987424i	$-0.550536\pi$
$977$	−9.88330	−0.316195	−0.158097	+	0.987424i	$0.550536\pi$
$978$	0	0
$979$	2.86428	0.0915428
$980$	0	0
$981$	−1.98461	−0.0633638
$982$	0	0
$983$	−11.1192	−0.354647	−0.177323	−	0.984153i	$-0.556744\pi$
$983$	−11.1192	−0.354647	−0.177323	+	0.984153i	$0.556744\pi$
$984$	0	0
$985$	37.5486	1.19640
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.02337	−0.0643394
$990$	0	0
$991$	−11.2363	−0.356931	−0.178466	−	0.983946i	$-0.557113\pi$
$991$	−11.2363	−0.356931	−0.178466	+	0.983946i	$0.557113\pi$
$992$	0	0
$993$	−26.9295	−0.854580
$994$	0	0
$995$	−33.0852	−1.04887
$996$	0	0
$997$	−0.451385	−0.0142955	−0.00714775	−	0.999974i	$-0.502275\pi$
$997$	−0.451385	−0.0142955	−0.00714775	+	0.999974i	$0.502275\pi$
$998$	0	0
$999$	−19.9456	−0.631050

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.bx.1.3	yes	14
7.6	odd	2		7448.2.a.bu.1.12	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7448.2.a.bu.1.12	✓	14	7.6	odd	2
7448.2.a.bx.1.3	yes	14	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)

\(f(q)\)	\(=\)	\(q-1.76077 q^{3} +2.55769 q^{5} +0.100310 q^{9} +O(q^{10})\)	\(q-1.76077 q^{3} +2.55769 q^{5} +0.100310 q^{9} +4.25895 q^{11} -0.715655 q^{13} -4.50350 q^{15} -1.47307 q^{17} +1.00000 q^{19} -5.33686 q^{23} +1.54178 q^{25} +5.10569 q^{27} +2.38490 q^{29} -0.259366 q^{31} -7.49904 q^{33} -3.90654 q^{37} +1.26010 q^{39} +5.32470 q^{41} +0.379131 q^{43} +0.256563 q^{45} +13.5398 q^{47} +2.59374 q^{51} -4.69089 q^{53} +10.8931 q^{55} -1.76077 q^{57} -7.73675 q^{59} -2.13425 q^{61} -1.83042 q^{65} +9.11553 q^{67} +9.39698 q^{69} +7.16740 q^{71} -1.97217 q^{73} -2.71472 q^{75} +4.41668 q^{79} -9.29087 q^{81} -0.547536 q^{83} -3.76766 q^{85} -4.19926 q^{87} +0.672532 q^{89} +0.456684 q^{93} +2.55769 q^{95} +17.9530 q^{97} +0.427217 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 6 q^{3} + 2 q^{5} + 16 q^{9}+O(q^{10}) \) 14 * q + 6 * q^3 + 2 * q^5 + 16 * q^9	\( 14 q + 6 q^{3} + 2 q^{5} + 16 q^{9} - 6 q^{11} + 16 q^{13} + 4 q^{15} - 4 q^{17} + 14 q^{19} - 4 q^{23} + 16 q^{25} + 36 q^{27} - 6 q^{29} + 16 q^{31} - 10 q^{33} + 6 q^{37} + 16 q^{39} - 14 q^{41} - 2 q^{43} + 30 q^{47} + 20 q^{51} - 6 q^{53} + 44 q^{55} + 6 q^{57} + 22 q^{59} + 10 q^{61} - 16 q^{65} + 4 q^{67} + 48 q^{69} + 6 q^{71} + 4 q^{73} + 64 q^{75} + 26 q^{79} + 30 q^{81} + 32 q^{83} - 8 q^{85} + 32 q^{87} - 54 q^{89} - 32 q^{93} + 2 q^{95} + 18 q^{97} - 38 q^{99}+O(q^{100}) \) 14 * q + 6 * q^3 + 2 * q^5 + 16 * q^9 - 6 * q^11 + 16 * q^13 + 4 * q^15 - 4 * q^17 + 14 * q^19 - 4 * q^23 + 16 * q^25 + 36 * q^27 - 6 * q^29 + 16 * q^31 - 10 * q^33 + 6 * q^37 + 16 * q^39 - 14 * q^41 - 2 * q^43 + 30 * q^47 + 20 * q^51 - 6 * q^53 + 44 * q^55 + 6 * q^57 + 22 * q^59 + 10 * q^61 - 16 * q^65 + 4 * q^67 + 48 * q^69 + 6 * q^71 + 4 * q^73 + 64 * q^75 + 26 * q^79 + 30 * q^81 + 32 * q^83 - 8 * q^85 + 32 * q^87 - 54 * q^89 - 32 * q^93 + 2 * q^95 + 18 * q^97 - 38 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more