LMFDB - Embedded newform 2224.2.a.o.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2224,2,Mod(1,2224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2224, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2224.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2224 = 2^{4} \cdot 139 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2224.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:	$17.7587294095$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 11x^{5} + 8x^{4} + 35x^{3} - 10x^{2} - 32x - 8 $ x^7 - x^6 - 11x^5 + 8x^4 + 35x^3 - 10x^2 - 32*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 139)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.47985$ of defining polynomial
Character	$\chi$	$=$	2224.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.245836 q^{3} -2.31250 q^{5} +0.326651 q^{7} -2.93956 q^{9} +O(q^{10})$	$q+0.245836 q^{3} -2.31250 q^{5} +0.326651 q^{7} -2.93956 q^{9} -0.570671 q^{11} +0.655030 q^{13} -0.568496 q^{15} +1.14453 q^{17} +2.77067 q^{19} +0.0803026 q^{21} -7.23464 q^{23} +0.347657 q^{25} -1.46016 q^{27} +7.86393 q^{29} +8.91763 q^{31} -0.140292 q^{33} -0.755380 q^{35} -4.69099 q^{37} +0.161030 q^{39} +9.63165 q^{41} -11.2545 q^{43} +6.79774 q^{45} +8.96119 q^{47} -6.89330 q^{49} +0.281366 q^{51} +7.21477 q^{53} +1.31968 q^{55} +0.681130 q^{57} +5.58094 q^{59} +10.8663 q^{61} -0.960211 q^{63} -1.51476 q^{65} +3.89682 q^{67} -1.77853 q^{69} +4.74151 q^{71} -5.98781 q^{73} +0.0854666 q^{75} -0.186410 q^{77} -6.17739 q^{79} +8.45973 q^{81} +14.5135 q^{83} -2.64672 q^{85} +1.93324 q^{87} +11.5467 q^{89} +0.213966 q^{91} +2.19228 q^{93} -6.40717 q^{95} -5.82160 q^{97} +1.67752 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{3} + 11 q^{5} + 5 q^{7} + 13 q^{9}+O(q^{10}) $ 7 * q + 2 * q^3 + 11 * q^5 + 5 * q^7 + 13 * q^9	$ 7 q + 2 q^{3} + 11 q^{5} + 5 q^{7} + 13 q^{9} - 2 q^{11} + 6 q^{13} + 3 q^{15} + 5 q^{17} + 10 q^{19} - 5 q^{21} + q^{23} + 14 q^{25} + 11 q^{27} + 30 q^{29} + 20 q^{31} - 20 q^{33} + 7 q^{35} + 6 q^{37} - 11 q^{39} + 19 q^{41} + 12 q^{43} + 27 q^{45} + 3 q^{47} - 8 q^{49} - 23 q^{51} + 38 q^{53} - 7 q^{55} - 19 q^{57} + 14 q^{59} + 4 q^{61} + 18 q^{63} + 10 q^{65} - 9 q^{67} + 9 q^{69} - 24 q^{71} - 5 q^{73} + 21 q^{75} - 13 q^{77} - 8 q^{79} + 39 q^{81} + 9 q^{83} - 22 q^{85} + 25 q^{87} + 10 q^{89} - 7 q^{91} - 15 q^{93} + 21 q^{95} - 5 q^{97} + 31 q^{99}+O(q^{100}) $ 7 * q + 2 * q^3 + 11 * q^5 + 5 * q^7 + 13 * q^9 - 2 * q^11 + 6 * q^13 + 3 * q^15 + 5 * q^17 + 10 * q^19 - 5 * q^21 + q^23 + 14 * q^25 + 11 * q^27 + 30 * q^29 + 20 * q^31 - 20 * q^33 + 7 * q^35 + 6 * q^37 - 11 * q^39 + 19 * q^41 + 12 * q^43 + 27 * q^45 + 3 * q^47 - 8 * q^49 - 23 * q^51 + 38 * q^53 - 7 * q^55 - 19 * q^57 + 14 * q^59 + 4 * q^61 + 18 * q^63 + 10 * q^65 - 9 * q^67 + 9 * q^69 - 24 * q^71 - 5 * q^73 + 21 * q^75 - 13 * q^77 - 8 * q^79 + 39 * q^81 + 9 * q^83 - 22 * q^85 + 25 * q^87 + 10 * q^89 - 7 * q^91 - 15 * q^93 + 21 * q^95 - 5 * q^97 + 31 * 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
$3$	0.245836	0.141934	0.0709668	−	0.997479i	$-0.477392\pi$
$3$	0.245836	0.141934	0.0709668	+	0.997479i	$0.477392\pi$
$4$	0	0
$5$	−2.31250	−1.03418	−0.517091	−	0.855931i	$-0.672985\pi$
$5$	−2.31250	−1.03418	−0.517091	+	0.855931i	$0.672985\pi$
$6$	0	0
$7$	0.326651	0.123462	0.0617312	−	0.998093i	$-0.480338\pi$
$7$	0.326651	0.123462	0.0617312	+	0.998093i	$0.480338\pi$
$8$	0	0
$9$	−2.93956	−0.979855
$10$	0	0
$11$	−0.570671	−0.172064	−0.0860319	−	0.996292i	$-0.527419\pi$
$11$	−0.570671	−0.172064	−0.0860319	+	0.996292i	$0.527419\pi$
$12$	0	0
$13$	0.655030	0.181673	0.0908364	−	0.995866i	$-0.471046\pi$
$13$	0.655030	0.181673	0.0908364	+	0.995866i	$0.471046\pi$
$14$	0	0
$15$	−0.568496	−0.146785
$16$	0	0
$17$	1.14453	0.277589	0.138794	−	0.990321i	$-0.455677\pi$
$17$	1.14453	0.277589	0.138794	+	0.990321i	$0.455677\pi$
$18$	0	0
$19$	2.77067	0.635634	0.317817	−	0.948152i	$-0.397050\pi$
$19$	2.77067	0.635634	0.317817	+	0.948152i	$0.397050\pi$
$20$	0	0
$21$	0.0803026	0.0175235
$22$	0	0
$23$	−7.23464	−1.50853	−0.754263	−	0.656572i	$-0.772006\pi$
$23$	−7.23464	−1.50853	−0.754263	+	0.656572i	$0.772006\pi$
$24$	0	0
$25$	0.347657	0.0695314
$26$	0	0
$27$	−1.46016	−0.281008
$28$	0	0
$29$	7.86393	1.46029	0.730147	−	0.683290i	$-0.239451\pi$
$29$	7.86393	1.46029	0.730147	+	0.683290i	$0.239451\pi$
$30$	0	0
$31$	8.91763	1.60165	0.800827	−	0.598896i	$-0.204394\pi$
$31$	8.91763	1.60165	0.800827	+	0.598896i	$0.204394\pi$
$32$	0	0
$33$	−0.140292	−0.0244216
$34$	0	0
$35$	−0.755380	−0.127683
$36$	0	0
$37$	−4.69099	−0.771194	−0.385597	−	0.922667i	$-0.626004\pi$
$37$	−4.69099	−0.771194	−0.385597	+	0.922667i	$0.626004\pi$
$38$	0	0
$39$	0.161030	0.0257855
$40$	0	0
$41$	9.63165	1.50421	0.752105	−	0.659043i	$-0.229039\pi$
$41$	9.63165	1.50421	0.752105	+	0.659043i	$0.229039\pi$
$42$	0	0
$43$	−11.2545	−1.71630	−0.858148	−	0.513403i	$-0.828385\pi$
$43$	−11.2545	−1.71630	−0.858148	+	0.513403i	$0.828385\pi$
$44$	0	0
$45$	6.79774	1.01335
$46$	0	0
$47$	8.96119	1.30712	0.653562	−	0.756873i	$-0.273274\pi$
$47$	8.96119	1.30712	0.653562	+	0.756873i	$0.273274\pi$
$48$	0	0
$49$	−6.89330	−0.984757
$50$	0	0
$51$	0.281366	0.0393991
$52$	0	0
$53$	7.21477	0.991025	0.495512	−	0.868601i	$-0.334980\pi$
$53$	7.21477	0.991025	0.495512	+	0.868601i	$0.334980\pi$
$54$	0	0
$55$	1.31968	0.177945
$56$	0	0
$57$	0.681130	0.0902178
$58$	0	0
$59$	5.58094	0.726577	0.363288	−	0.931677i	$-0.381654\pi$
$59$	5.58094	0.726577	0.363288	+	0.931677i	$0.381654\pi$
$60$	0	0
$61$	10.8663	1.39129	0.695646	−	0.718385i	$-0.255118\pi$
$61$	10.8663	1.39129	0.695646	+	0.718385i	$0.255118\pi$
$62$	0	0
$63$	−0.960211	−0.120975
$64$	0	0
$65$	−1.51476	−0.187883
$66$	0	0
$67$	3.89682	0.476073	0.238036	−	0.971256i	$-0.423496\pi$
$67$	3.89682	0.476073	0.238036	+	0.971256i	$0.423496\pi$
$68$	0	0
$69$	−1.77853	−0.214110
$70$	0	0
$71$	4.74151	0.562713	0.281357	−	0.959603i	$-0.409216\pi$
$71$	4.74151	0.562713	0.281357	+	0.959603i	$0.409216\pi$
$72$	0	0
$73$	−5.98781	−0.700820	−0.350410	−	0.936596i	$-0.613958\pi$
$73$	−5.98781	−0.700820	−0.350410	+	0.936596i	$0.613958\pi$
$74$	0	0
$75$	0.0854666	0.00986883
$76$	0	0
$77$	−0.186410	−0.0212434
$78$	0	0
$79$	−6.17739	−0.695010	−0.347505	−	0.937678i	$-0.612971\pi$
$79$	−6.17739	−0.695010	−0.347505	+	0.937678i	$0.612971\pi$
$80$	0	0
$81$	8.45973	0.939970
$82$	0	0
$83$	14.5135	1.59306	0.796529	−	0.604600i	$-0.206667\pi$
$83$	14.5135	1.59306	0.796529	+	0.604600i	$0.206667\pi$
$84$	0	0
$85$	−2.64672	−0.287077
$86$	0	0
$87$	1.93324	0.207265
$88$	0	0
$89$	11.5467	1.22395	0.611975	−	0.790877i	$-0.290375\pi$
$89$	11.5467	1.22395	0.611975	+	0.790877i	$0.290375\pi$
$90$	0	0
$91$	0.213966	0.0224298
$92$	0	0
$93$	2.19228	0.227328
$94$	0	0
$95$	−6.40717	−0.657361
$96$	0	0
$97$	−5.82160	−0.591094	−0.295547	−	0.955328i	$-0.595502\pi$
$97$	−5.82160	−0.591094	−0.295547	+	0.955328i	$0.595502\pi$
$98$	0	0
$99$	1.67752	0.168598
$100$	0	0
$101$	10.2902	1.02392	0.511958	−	0.859011i	$-0.328920\pi$
$101$	10.2902	1.02392	0.511958	+	0.859011i	$0.328920\pi$
$102$	0	0
$103$	0.0520725	0.00513086	0.00256543	−	0.999997i	$-0.499183\pi$
$103$	0.0520725	0.00513086	0.00256543	+	0.999997i	$0.499183\pi$
$104$	0	0
$105$	−0.185700	−0.0181224
$106$	0	0
$107$	8.26648	0.799151	0.399575	−	0.916700i	$-0.369158\pi$
$107$	8.26648	0.799151	0.399575	+	0.916700i	$0.369158\pi$
$108$	0	0
$109$	−0.971231	−0.0930270	−0.0465135	−	0.998918i	$-0.514811\pi$
$109$	−0.971231	−0.0930270	−0.0465135	+	0.998918i	$0.514811\pi$
$110$	0	0
$111$	−1.15321	−0.109458
$112$	0	0
$113$	0.0781957	0.00735603	0.00367802	−	0.999993i	$-0.498829\pi$
$113$	0.0781957	0.00735603	0.00367802	+	0.999993i	$0.498829\pi$
$114$	0	0
$115$	16.7301	1.56009
$116$	0	0
$117$	−1.92550	−0.178013
$118$	0	0
$119$	0.373861	0.0342718
$120$	0	0
$121$	−10.6743	−0.970394
$122$	0	0
$123$	2.36781	0.213498
$124$	0	0
$125$	10.7585	0.962273
$126$	0	0
$127$	3.90704	0.346693	0.173347	−	0.984861i	$-0.444542\pi$
$127$	3.90704	0.346693	0.173347	+	0.984861i	$0.444542\pi$
$128$	0	0
$129$	−2.76676	−0.243600
$130$	0	0
$131$	3.91653	0.342189	0.171095	−	0.985255i	$-0.445270\pi$
$131$	3.91653	0.342189	0.171095	+	0.985255i	$0.445270\pi$
$132$	0	0
$133$	0.905041	0.0784770
$134$	0	0
$135$	3.37662	0.290613
$136$	0	0
$137$	−19.5648	−1.67153	−0.835766	−	0.549086i	$-0.814976\pi$
$137$	−19.5648	−1.67153	−0.835766	+	0.549086i	$0.814976\pi$
$138$	0	0
$139$	−1.00000	−0.0848189
$139$	−1.00000	−0.0848189
$140$	0	0
$141$	2.20298	0.185525
$142$	0	0
$143$	−0.373807	−0.0312593
$144$	0	0
$145$	−18.1853	−1.51021
$146$	0	0
$147$	−1.69462	−0.139770
$148$	0	0
$149$	4.30582	0.352747	0.176373	−	0.984323i	$-0.443563\pi$
$149$	4.30582	0.352747	0.176373	+	0.984323i	$0.443563\pi$
$150$	0	0
$151$	15.8273	1.28801	0.644003	−	0.765023i	$-0.277272\pi$
$151$	15.8273	1.28801	0.644003	+	0.765023i	$0.277272\pi$
$152$	0	0
$153$	−3.36441	−0.271997
$154$	0	0
$155$	−20.6220	−1.65640
$156$	0	0
$157$	1.14090	0.0910534	0.0455267	−	0.998963i	$-0.485503\pi$
$157$	1.14090	0.0910534	0.0455267	+	0.998963i	$0.485503\pi$
$158$	0	0
$159$	1.77365	0.140660
$160$	0	0
$161$	−2.36320	−0.186246
$162$	0	0
$163$	2.94076	0.230338	0.115169	−	0.993346i	$-0.463259\pi$
$163$	2.94076	0.230338	0.115169	+	0.993346i	$0.463259\pi$
$164$	0	0
$165$	0.324424	0.0252564
$166$	0	0
$167$	−4.64987	−0.359818	−0.179909	−	0.983683i	$-0.557580\pi$
$167$	−4.64987	−0.359818	−0.179909	+	0.983683i	$0.557580\pi$
$168$	0	0
$169$	−12.5709	−0.966995
$170$	0	0
$171$	−8.14455	−0.622830
$172$	0	0
$173$	2.04281	0.155312	0.0776559	−	0.996980i	$-0.475256\pi$
$173$	2.04281	0.155312	0.0776559	+	0.996980i	$0.475256\pi$
$174$	0	0
$175$	0.113562	0.00858451
$176$	0	0
$177$	1.37200	0.103126
$178$	0	0
$179$	18.4289	1.37744	0.688719	−	0.725029i	$-0.258173\pi$
$179$	18.4289	1.37744	0.688719	+	0.725029i	$0.258173\pi$
$180$	0	0
$181$	1.48337	0.110258	0.0551290	−	0.998479i	$-0.482443\pi$
$181$	1.48337	0.110258	0.0551290	+	0.998479i	$0.482443\pi$
$182$	0	0
$183$	2.67134	0.197471
$184$	0	0
$185$	10.8479	0.797554
$186$	0	0
$187$	−0.653149	−0.0477630
$188$	0	0
$189$	−0.476962	−0.0346939
$190$	0	0
$191$	−15.1176	−1.09387	−0.546937	−	0.837174i	$-0.684206\pi$
$191$	−15.1176	−1.09387	−0.546937	+	0.837174i	$0.684206\pi$
$192$	0	0
$193$	12.6498	0.910553	0.455276	−	0.890350i	$-0.349540\pi$
$193$	12.6498	0.910553	0.455276	+	0.890350i	$0.349540\pi$
$194$	0	0
$195$	−0.372382	−0.0266668
$196$	0	0
$197$	20.0054	1.42532	0.712662	−	0.701508i	$-0.247489\pi$
$197$	20.0054	1.42532	0.712662	+	0.701508i	$0.247489\pi$
$198$	0	0
$199$	11.0064	0.780220	0.390110	−	0.920768i	$-0.372437\pi$
$199$	11.0064	0.780220	0.390110	+	0.920768i	$0.372437\pi$
$200$	0	0
$201$	0.957980	0.0675707
$202$	0	0
$203$	2.56876	0.180292
$204$	0	0
$205$	−22.2732	−1.55563
$206$	0	0
$207$	21.2667	1.47814
$208$	0	0
$209$	−1.58114	−0.109370
$210$	0	0
$211$	−9.89782	−0.681394	−0.340697	−	0.940173i	$-0.610663\pi$
$211$	−9.89782	−0.681394	−0.340697	+	0.940173i	$0.610663\pi$
$212$	0	0
$213$	1.16563	0.0798679
$214$	0	0
$215$	26.0260	1.77496
$216$	0	0
$217$	2.91295	0.197744
$218$	0	0
$219$	−1.47202	−0.0994698
$220$	0	0
$221$	0.749700	0.0504303
$222$	0	0
$223$	−6.88052	−0.460753	−0.230377	−	0.973102i	$-0.573996\pi$
$223$	−6.88052	−0.460753	−0.230377	+	0.973102i	$0.573996\pi$
$224$	0	0
$225$	−1.02196	−0.0681307
$226$	0	0
$227$	−28.2205	−1.87306	−0.936529	−	0.350591i	$-0.885981\pi$
$227$	−28.2205	−1.87306	−0.936529	+	0.350591i	$0.885981\pi$
$228$	0	0
$229$	24.7429	1.63505	0.817527	−	0.575890i	$-0.195344\pi$
$229$	24.7429	1.63505	0.817527	+	0.575890i	$0.195344\pi$
$230$	0	0
$231$	−0.0458263	−0.00301515
$232$	0	0
$233$	−18.1608	−1.18975	−0.594877	−	0.803817i	$-0.702799\pi$
$233$	−18.1608	−1.18975	−0.594877	+	0.803817i	$0.702799\pi$
$234$	0	0
$235$	−20.7228	−1.35180
$236$	0	0
$237$	−1.51862	−0.0986452
$238$	0	0
$239$	−22.9960	−1.48748	−0.743742	−	0.668466i	$-0.766951\pi$
$239$	−22.9960	−1.48748	−0.743742	+	0.668466i	$0.766951\pi$
$240$	0	0
$241$	29.9171	1.92713	0.963564	−	0.267477i	$-0.0861900\pi$
$241$	29.9171	1.92713	0.963564	+	0.267477i	$0.0861900\pi$
$242$	0	0
$243$	6.46019	0.414421
$244$	0	0
$245$	15.9408	1.01842
$246$	0	0
$247$	1.81487	0.115477
$248$	0	0
$249$	3.56793	0.226108
$250$	0	0
$251$	−14.5346	−0.917416	−0.458708	−	0.888587i	$-0.651688\pi$
$251$	−14.5346	−0.917416	−0.458708	+	0.888587i	$0.651688\pi$
$252$	0	0
$253$	4.12860	0.259563
$254$	0	0
$255$	−0.650659	−0.0407459
$256$	0	0
$257$	0.986686	0.0615478	0.0307739	−	0.999526i	$-0.490203\pi$
$257$	0.986686	0.0615478	0.0307739	+	0.999526i	$0.490203\pi$
$258$	0	0
$259$	−1.53232	−0.0952134
$260$	0	0
$261$	−23.1165	−1.43088
$262$	0	0
$263$	9.74159	0.600692	0.300346	−	0.953830i	$-0.402898\pi$
$263$	9.74159	0.600692	0.300346	+	0.953830i	$0.402898\pi$
$264$	0	0
$265$	−16.6842	−1.02490
$266$	0	0
$267$	2.83860	0.173720
$268$	0	0
$269$	2.53822	0.154758	0.0773791	−	0.997002i	$-0.475345\pi$
$269$	2.53822	0.154758	0.0773791	+	0.997002i	$0.475345\pi$
$270$	0	0
$271$	−13.1783	−0.800527	−0.400263	−	0.916400i	$-0.631081\pi$
$271$	−13.1783	−0.800527	−0.400263	+	0.916400i	$0.631081\pi$
$272$	0	0
$273$	0.0526006	0.00318354
$274$	0	0
$275$	−0.198398	−0.0119638
$276$	0	0
$277$	14.6123	0.877967	0.438983	−	0.898495i	$-0.355339\pi$
$277$	14.6123	0.877967	0.438983	+	0.898495i	$0.355339\pi$
$278$	0	0
$279$	−26.2140	−1.56939
$280$	0	0
$281$	3.35534	0.200163	0.100081	−	0.994979i	$-0.468090\pi$
$281$	3.35534	0.200163	0.100081	+	0.994979i	$0.468090\pi$
$282$	0	0
$283$	−11.8499	−0.704403	−0.352202	−	0.935924i	$-0.614567\pi$
$283$	−11.8499	−0.704403	−0.352202	+	0.935924i	$0.614567\pi$
$284$	0	0
$285$	−1.57511	−0.0933016
$286$	0	0
$287$	3.14619	0.185713
$288$	0	0
$289$	−15.6901	−0.922945
$290$	0	0
$291$	−1.43116	−0.0838960
$292$	0	0
$293$	13.7232	0.801717	0.400859	−	0.916140i	$-0.368712\pi$
$293$	13.7232	0.801717	0.400859	+	0.916140i	$0.368712\pi$
$294$	0	0
$295$	−12.9059	−0.751412
$296$	0	0
$297$	0.833270	0.0483513
$298$	0	0
$299$	−4.73891	−0.274058
$300$	0	0
$301$	−3.67629	−0.211898
$302$	0	0
$303$	2.52971	0.145328
$304$	0	0
$305$	−25.1284	−1.43885
$306$	0	0
$307$	19.3132	1.10226	0.551131	−	0.834419i	$-0.314197\pi$
$307$	19.3132	1.10226	0.551131	+	0.834419i	$0.314197\pi$
$308$	0	0
$309$	0.0128013	0.000728241 0
$310$	0	0
$311$	−20.7756	−1.17807	−0.589037	−	0.808106i	$-0.700493\pi$
$311$	−20.7756	−1.17807	−0.589037	+	0.808106i	$0.700493\pi$
$312$	0	0
$313$	−27.5648	−1.55805	−0.779026	−	0.626991i	$-0.784286\pi$
$313$	−27.5648	−1.55805	−0.779026	+	0.626991i	$0.784286\pi$
$314$	0	0
$315$	2.22049	0.125110
$316$	0	0
$317$	−18.7677	−1.05410	−0.527049	−	0.849835i	$-0.676702\pi$
$317$	−18.7677	−1.05410	−0.527049	+	0.849835i	$0.676702\pi$
$318$	0	0
$319$	−4.48772	−0.251264
$320$	0	0
$321$	2.03220	0.113426
$322$	0	0
$323$	3.17110	0.176445
$324$	0	0
$325$	0.227726	0.0126320
$326$	0	0
$327$	−0.238764	−0.0132037
$328$	0	0
$329$	2.92718	0.161381
$330$	0	0
$331$	0.0825219	0.00453581	0.00226791	−	0.999997i	$-0.499278\pi$
$331$	0.0825219	0.00453581	0.00226791	+	0.999997i	$0.499278\pi$
$332$	0	0
$333$	13.7895	0.755658
$334$	0	0
$335$	−9.01141	−0.492346
$336$	0	0
$337$	13.5611	0.738718	0.369359	−	0.929287i	$-0.379577\pi$
$337$	13.5611	0.738718	0.369359	+	0.929287i	$0.379577\pi$
$338$	0	0
$339$	0.0192233	0.00104407
$340$	0	0
$341$	−5.08903	−0.275587
$342$	0	0
$343$	−4.53826	−0.245043
$344$	0	0
$345$	4.11286	0.221429
$346$	0	0
$347$	14.0479	0.754130	0.377065	−	0.926187i	$-0.376933\pi$
$347$	14.0479	0.754130	0.377065	+	0.926187i	$0.376933\pi$
$348$	0	0
$349$	29.7905	1.59465	0.797323	−	0.603553i	$-0.206249\pi$
$349$	29.7905	1.59465	0.797323	+	0.603553i	$0.206249\pi$
$350$	0	0
$351$	−0.956449	−0.0510515
$352$	0	0
$353$	1.66236	0.0884787	0.0442394	−	0.999021i	$-0.485914\pi$
$353$	1.66236	0.0884787	0.0442394	+	0.999021i	$0.485914\pi$
$354$	0	0
$355$	−10.9647	−0.581948
$356$	0	0
$357$	0.0919085	0.00486431
$358$	0	0
$359$	19.7653	1.04317	0.521586	−	0.853198i	$-0.325340\pi$
$359$	19.7653	1.04317	0.521586	+	0.853198i	$0.325340\pi$
$360$	0	0
$361$	−11.3234	−0.595969
$362$	0	0
$363$	−2.62414	−0.137731
$364$	0	0
$365$	13.8468	0.724775
$366$	0	0
$367$	20.9268	1.09237	0.546185	−	0.837665i	$-0.316080\pi$
$367$	20.9268	1.09237	0.546185	+	0.837665i	$0.316080\pi$
$368$	0	0
$369$	−28.3128	−1.47391
$370$	0	0
$371$	2.35671	0.122354
$372$	0	0
$373$	11.5715	0.599147	0.299574	−	0.954073i	$-0.403156\pi$
$373$	11.5715	0.599147	0.299574	+	0.954073i	$0.403156\pi$
$374$	0	0
$375$	2.64484	0.136579
$376$	0	0
$377$	5.15111	0.265296
$378$	0	0
$379$	−20.9839	−1.07787	−0.538935	−	0.842347i	$-0.681173\pi$
$379$	−20.9839	−1.07787	−0.538935	+	0.842347i	$0.681173\pi$
$380$	0	0
$381$	0.960490	0.0492074
$382$	0	0
$383$	15.1795	0.775639	0.387819	−	0.921735i	$-0.373228\pi$
$383$	15.1795	0.775639	0.387819	+	0.921735i	$0.373228\pi$
$384$	0	0
$385$	0.431074	0.0219695
$386$	0	0
$387$	33.0833	1.68172
$388$	0	0
$389$	2.64861	0.134290	0.0671450	−	0.997743i	$-0.478611\pi$
$389$	2.64861	0.134290	0.0671450	+	0.997743i	$0.478611\pi$
$390$	0	0
$391$	−8.28024	−0.418750
$392$	0	0
$393$	0.962825	0.0485681
$394$	0	0
$395$	14.2852	0.718767
$396$	0	0
$397$	−24.5634	−1.23280	−0.616400	−	0.787433i	$-0.711410\pi$
$397$	−24.5634	−1.23280	−0.616400	+	0.787433i	$0.711410\pi$
$398$	0	0
$399$	0.222492	0.0111385
$400$	0	0
$401$	36.8459	1.84000	0.919998	−	0.391923i	$-0.128190\pi$
$401$	36.8459	1.84000	0.919998	+	0.391923i	$0.128190\pi$
$402$	0	0
$403$	5.84132	0.290977
$404$	0	0
$405$	−19.5631	−0.972100
$406$	0	0
$407$	2.67701	0.132694
$408$	0	0
$409$	−6.96504	−0.344399	−0.172199	−	0.985062i	$-0.555087\pi$
$409$	−6.96504	−0.344399	−0.172199	+	0.985062i	$0.555087\pi$
$410$	0	0
$411$	−4.80973	−0.237246
$412$	0	0
$413$	1.82302	0.0897049
$414$	0	0
$415$	−33.5624	−1.64751
$416$	0	0
$417$	−0.245836	−0.0120386
$418$	0	0
$419$	1.96945	0.0962140	0.0481070	−	0.998842i	$-0.484681\pi$
$419$	1.96945	0.0962140	0.0481070	+	0.998842i	$0.484681\pi$
$420$	0	0
$421$	−35.7362	−1.74168	−0.870838	−	0.491570i	$-0.836423\pi$
$421$	−35.7362	−1.74168	−0.870838	+	0.491570i	$0.836423\pi$
$422$	0	0
$423$	−26.3420	−1.28079
$424$	0	0
$425$	0.397903	0.0193011
$426$	0	0
$427$	3.54950	0.171772
$428$	0	0
$429$	−0.0918952	−0.00443674
$430$	0	0
$431$	28.8284	1.38862	0.694308	−	0.719678i	$-0.255711\pi$
$431$	28.8284	1.38862	0.694308	+	0.719678i	$0.255711\pi$
$432$	0	0
$433$	−28.0446	−1.34774	−0.673869	−	0.738851i	$-0.735369\pi$
$433$	−28.0446	−1.34774	−0.673869	+	0.738851i	$0.735369\pi$
$434$	0	0
$435$	−4.47061	−0.214349
$436$	0	0
$437$	−20.0448	−0.958871
$438$	0	0
$439$	−27.0360	−1.29036	−0.645180	−	0.764031i	$-0.723218\pi$
$439$	−27.0360	−1.29036	−0.645180	+	0.764031i	$0.723218\pi$
$440$	0	0
$441$	20.2633	0.964919
$442$	0	0
$443$	8.67760	0.412285	0.206143	−	0.978522i	$-0.433909\pi$
$443$	8.67760	0.412285	0.206143	+	0.978522i	$0.433909\pi$
$444$	0	0
$445$	−26.7018	−1.26579
$446$	0	0
$447$	1.05853	0.0500666
$448$	0	0
$449$	−14.6027	−0.689142	−0.344571	−	0.938760i	$-0.611976\pi$
$449$	−14.6027	−0.689142	−0.344571	+	0.938760i	$0.611976\pi$
$450$	0	0
$451$	−5.49650	−0.258820
$452$	0	0
$453$	3.89092	0.182811
$454$	0	0
$455$	−0.494797	−0.0231964
$456$	0	0
$457$	13.1688	0.616010	0.308005	−	0.951385i	$-0.400339\pi$
$457$	13.1688	0.616010	0.308005	+	0.951385i	$0.400339\pi$
$458$	0	0
$459$	−1.67119	−0.0780046
$460$	0	0
$461$	−4.68827	−0.218354	−0.109177	−	0.994022i	$-0.534822\pi$
$461$	−4.68827	−0.218354	−0.109177	+	0.994022i	$0.534822\pi$
$462$	0	0
$463$	−25.1950	−1.17091	−0.585454	−	0.810705i	$-0.699084\pi$
$463$	−25.1950	−1.17091	−0.585454	+	0.810705i	$0.699084\pi$
$464$	0	0
$465$	−5.06964	−0.235099
$466$	0	0
$467$	24.2118	1.12039	0.560193	−	0.828362i	$-0.310727\pi$
$467$	24.2118	1.12039	0.560193	+	0.828362i	$0.310727\pi$
$468$	0	0
$469$	1.27290	0.0587771
$470$	0	0
$471$	0.280473	0.0129235
$472$	0	0
$473$	6.42262	0.295312
$474$	0	0
$475$	0.963241	0.0441965
$476$	0	0
$477$	−21.2083	−0.971060
$478$	0	0
$479$	22.0171	1.00599	0.502994	−	0.864290i	$-0.332232\pi$
$479$	22.0171	1.00599	0.502994	+	0.864290i	$0.332232\pi$
$480$	0	0
$481$	−3.07274	−0.140105
$482$	0	0
$483$	−0.580960	−0.0264346
$484$	0	0
$485$	13.4625	0.611298
$486$	0	0
$487$	35.4094	1.60455	0.802276	−	0.596953i	$-0.203622\pi$
$487$	35.4094	1.60455	0.802276	+	0.596953i	$0.203622\pi$
$488$	0	0
$489$	0.722944	0.0326927
$490$	0	0
$491$	−43.9214	−1.98214	−0.991072	−	0.133328i	$-0.957434\pi$
$491$	−43.9214	−1.98214	−0.991072	+	0.133328i	$0.957434\pi$
$492$	0	0
$493$	9.00048	0.405361
$494$	0	0
$495$	−3.87928	−0.174360
$496$	0	0
$497$	1.54882	0.0694740
$498$	0	0
$499$	−6.73789	−0.301630	−0.150815	−	0.988562i	$-0.548190\pi$
$499$	−6.73789	−0.301630	−0.150815	+	0.988562i	$0.548190\pi$
$500$	0	0
$501$	−1.14311	−0.0510702
$502$	0	0
$503$	0.982907	0.0438257	0.0219128	−	0.999760i	$-0.493024\pi$
$503$	0.982907	0.0438257	0.0219128	+	0.999760i	$0.493024\pi$
$504$	0	0
$505$	−23.7962	−1.05891
$506$	0	0
$507$	−3.09039	−0.137249
$508$	0	0
$509$	−32.6107	−1.44544	−0.722721	−	0.691140i	$-0.757109\pi$
$509$	−32.6107	−1.44544	−0.722721	+	0.691140i	$0.757109\pi$
$510$	0	0
$511$	−1.95592	−0.0865249
$512$	0	0
$513$	−4.04561	−0.178618
$514$	0	0
$515$	−0.120418	−0.00530624
$516$	0	0
$517$	−5.11389	−0.224909
$518$	0	0
$519$	0.502196	0.0220440
$520$	0	0
$521$	8.76803	0.384134	0.192067	−	0.981382i	$-0.438481\pi$
$521$	8.76803	0.384134	0.192067	+	0.981382i	$0.438481\pi$
$522$	0	0
$523$	28.4574	1.24436	0.622178	−	0.782876i	$-0.286248\pi$
$523$	28.4574	1.24436	0.622178	+	0.782876i	$0.286248\pi$
$524$	0	0
$525$	0.0279177	0.00121843
$526$	0	0
$527$	10.2065	0.444601
$528$	0	0
$529$	29.3400	1.27565
$530$	0	0
$531$	−16.4055	−0.711940
$532$	0	0
$533$	6.30902	0.273274
$534$	0	0
$535$	−19.1162	−0.826467
$536$	0	0
$537$	4.53048	0.195505
$538$	0	0
$539$	3.93381	0.169441
$540$	0	0
$541$	25.4795	1.09545	0.547725	−	0.836658i	$-0.315494\pi$
$541$	25.4795	1.09545	0.547725	+	0.836658i	$0.315494\pi$
$542$	0	0
$543$	0.364666	0.0156493
$544$	0	0
$545$	2.24597	0.0962068
$546$	0	0
$547$	10.7278	0.458689	0.229344	−	0.973345i	$-0.426342\pi$
$547$	10.7278	0.458689	0.229344	+	0.973345i	$0.426342\pi$
$548$	0	0
$549$	−31.9423	−1.36326
$550$	0	0
$551$	21.7883	0.928214
$552$	0	0
$553$	−2.01785	−0.0858077
$554$	0	0
$555$	2.66681	0.113200
$556$	0	0
$557$	31.2761	1.32521	0.662606	−	0.748968i	$-0.269450\pi$
$557$	31.2761	1.32521	0.662606	+	0.748968i	$0.269450\pi$
$558$	0	0
$559$	−7.37204	−0.311804
$560$	0	0
$561$	−0.160567	−0.00677916
$562$	0	0
$563$	−18.9842	−0.800090	−0.400045	−	0.916496i	$-0.631005\pi$
$563$	−18.9842	−0.800090	−0.400045	+	0.916496i	$0.631005\pi$
$564$	0	0
$565$	−0.180828	−0.00760747
$566$	0	0
$567$	2.76338	0.116051
$568$	0	0
$569$	20.9669	0.878978	0.439489	−	0.898248i	$-0.355159\pi$
$569$	20.9669	0.878978	0.439489	+	0.898248i	$0.355159\pi$
$570$	0	0
$571$	10.2145	0.427464	0.213732	−	0.976892i	$-0.431438\pi$
$571$	10.2145	0.427464	0.213732	+	0.976892i	$0.431438\pi$
$572$	0	0
$573$	−3.71646	−0.155257
$574$	0	0
$575$	−2.51517	−0.104890
$576$	0	0
$577$	−3.73780	−0.155607	−0.0778033	−	0.996969i	$-0.524791\pi$
$577$	−3.73780	−0.155607	−0.0778033	+	0.996969i	$0.524791\pi$
$578$	0	0
$579$	3.10978	0.129238
$580$	0	0
$581$	4.74083	0.196683
$582$	0	0
$583$	−4.11726	−0.170519
$584$	0	0
$585$	4.45273	0.184098
$586$	0	0
$587$	15.3677	0.634293	0.317147	−	0.948377i	$-0.397275\pi$
$587$	15.3677	0.634293	0.317147	+	0.948377i	$0.397275\pi$
$588$	0	0
$589$	24.7078	1.01807
$590$	0	0
$591$	4.91804	0.202301
$592$	0	0
$593$	−34.5495	−1.41878	−0.709389	−	0.704817i	$-0.751029\pi$
$593$	−34.5495	−1.41878	−0.709389	+	0.704817i	$0.751029\pi$
$594$	0	0
$595$	−0.864554	−0.0354432
$596$	0	0
$597$	2.70576	0.110739
$598$	0	0
$599$	−46.5366	−1.90143	−0.950717	−	0.310060i	$-0.899651\pi$
$599$	−46.5366	−1.90143	−0.950717	+	0.310060i	$0.899651\pi$
$600$	0	0
$601$	−30.4861	−1.24356	−0.621778	−	0.783194i	$-0.713589\pi$
$601$	−30.4861	−1.24356	−0.621778	+	0.783194i	$0.713589\pi$
$602$	0	0
$603$	−11.4550	−0.466482
$604$	0	0
$605$	24.6844	1.00356
$606$	0	0
$607$	13.2182	0.536511	0.268256	−	0.963348i	$-0.413553\pi$
$607$	13.2182	0.536511	0.268256	+	0.963348i	$0.413553\pi$
$608$	0	0
$609$	0.631494	0.0255894
$610$	0	0
$611$	5.86986	0.237469
$612$	0	0
$613$	−16.8823	−0.681869	−0.340935	−	0.940087i	$-0.610743\pi$
$613$	−16.8823	−0.681869	−0.340935	+	0.940087i	$0.610743\pi$
$614$	0	0
$615$	−5.47555	−0.220796
$616$	0	0
$617$	−16.5346	−0.665659	−0.332829	−	0.942987i	$-0.608003\pi$
$617$	−16.5346	−0.665659	−0.332829	+	0.942987i	$0.608003\pi$
$618$	0	0
$619$	−19.1064	−0.767952	−0.383976	−	0.923343i	$-0.625445\pi$
$619$	−19.1064	−0.767952	−0.383976	+	0.923343i	$0.625445\pi$
$620$	0	0
$621$	10.5637	0.423908
$622$	0	0
$623$	3.77175	0.151112
$624$	0	0
$625$	−26.6174	−1.06470
$626$	0	0
$627$	−0.388701	−0.0155232
$628$	0	0
$629$	−5.36896	−0.214075
$630$	0	0
$631$	20.1170	0.800844	0.400422	−	0.916331i	$-0.368864\pi$
$631$	20.1170	0.800844	0.400422	+	0.916331i	$0.368864\pi$
$632$	0	0
$633$	−2.43324	−0.0967127
$634$	0	0
$635$	−9.03502	−0.358544
$636$	0	0
$637$	−4.51532	−0.178904
$638$	0	0
$639$	−13.9380	−0.551378
$640$	0	0
$641$	−19.2938	−0.762061	−0.381031	−	0.924562i	$-0.624431\pi$
$641$	−19.2938	−0.762061	−0.381031	+	0.924562i	$0.624431\pi$
$642$	0	0
$643$	−29.2745	−1.15447	−0.577237	−	0.816577i	$-0.695869\pi$
$643$	−29.2745	−1.15447	−0.577237	+	0.816577i	$0.695869\pi$
$644$	0	0
$645$	6.39814	0.251926
$646$	0	0
$647$	−19.8266	−0.779466	−0.389733	−	0.920928i	$-0.627433\pi$
$647$	−19.8266	−0.779466	−0.389733	+	0.920928i	$0.627433\pi$
$648$	0	0
$649$	−3.18488	−0.125018
$650$	0	0
$651$	0.716109	0.0280665
$652$	0	0
$653$	−35.7848	−1.40037	−0.700183	−	0.713963i	$-0.746898\pi$
$653$	−35.7848	−1.40037	−0.700183	+	0.713963i	$0.746898\pi$
$654$	0	0
$655$	−9.05698	−0.353886
$656$	0	0
$657$	17.6015	0.686702
$658$	0	0
$659$	21.1886	0.825389	0.412695	−	0.910869i	$-0.364588\pi$
$659$	21.1886	0.825389	0.412695	+	0.910869i	$0.364588\pi$
$660$	0	0
$661$	8.67722	0.337505	0.168752	−	0.985658i	$-0.446026\pi$
$661$	8.67722	0.337505	0.168752	+	0.985658i	$0.446026\pi$
$662$	0	0
$663$	0.184303	0.00715775
$664$	0	0
$665$	−2.09291	−0.0811594
$666$	0	0
$667$	−56.8927	−2.20289
$668$	0	0
$669$	−1.69148	−0.0653964
$670$	0	0
$671$	−6.20110	−0.239391
$672$	0	0
$673$	−7.11219	−0.274155	−0.137077	−	0.990560i	$-0.543771\pi$
$673$	−7.11219	−0.274155	−0.137077	+	0.990560i	$0.543771\pi$
$674$	0	0
$675$	−0.507634	−0.0195389
$676$	0	0
$677$	−16.7747	−0.644704	−0.322352	−	0.946620i	$-0.604473\pi$
$677$	−16.7747	−0.644704	−0.322352	+	0.946620i	$0.604473\pi$
$678$	0	0
$679$	−1.90163	−0.0729779
$680$	0	0
$681$	−6.93760	−0.265850
$682$	0	0
$683$	−47.7431	−1.82684	−0.913419	−	0.407020i	$-0.866568\pi$
$683$	−47.7431	−1.82684	−0.913419	+	0.407020i	$0.866568\pi$
$684$	0	0
$685$	45.2435	1.72867
$686$	0	0
$687$	6.08269	0.232069
$688$	0	0
$689$	4.72589	0.180042
$690$	0	0
$691$	30.4574	1.15865	0.579327	−	0.815095i	$-0.303315\pi$
$691$	30.4574	1.15865	0.579327	+	0.815095i	$0.303315\pi$
$692$	0	0
$693$	0.547965	0.0208155
$694$	0	0
$695$	2.31250	0.0877181
$696$	0	0
$697$	11.0237	0.417552
$698$	0	0
$699$	−4.46458	−0.168866
$700$	0	0
$701$	−19.6196	−0.741020	−0.370510	−	0.928828i	$-0.620817\pi$
$701$	−19.6196	−0.741020	−0.370510	+	0.928828i	$0.620817\pi$
$702$	0	0
$703$	−12.9972	−0.490197
$704$	0	0
$705$	−5.09440	−0.191866
$706$	0	0
$707$	3.36131	0.126415
$708$	0	0
$709$	11.6038	0.435790	0.217895	−	0.975972i	$-0.430081\pi$
$709$	11.6038	0.435790	0.217895	+	0.975972i	$0.430081\pi$
$710$	0	0
$711$	18.1588	0.681009
$712$	0	0
$713$	−64.5158	−2.41614
$714$	0	0
$715$	0.864429	0.0323278
$716$	0	0
$717$	−5.65323	−0.211124
$718$	0	0
$719$	−26.2649	−0.979514	−0.489757	−	0.871859i	$-0.662915\pi$
$719$	−26.2649	−0.979514	−0.489757	+	0.871859i	$0.662915\pi$
$720$	0	0
$721$	0.0170095	0.000633468 0
$722$	0	0
$723$	7.35470	0.273524
$724$	0	0
$725$	2.73395	0.101536
$726$	0	0
$727$	−0.451096	−0.0167302	−0.00836512	−	0.999965i	$-0.502663\pi$
$727$	−0.451096	−0.0167302	−0.00836512	+	0.999965i	$0.502663\pi$
$728$	0	0
$729$	−23.7911	−0.881150
$730$	0	0
$731$	−12.8811	−0.476424
$732$	0	0
$733$	36.4971	1.34805	0.674025	−	0.738708i	$-0.264564\pi$
$733$	36.4971	1.34805	0.674025	+	0.738708i	$0.264564\pi$
$734$	0	0
$735$	3.91881	0.144548
$736$	0	0
$737$	−2.22380	−0.0819149
$738$	0	0
$739$	15.0007	0.551811	0.275905	−	0.961185i	$-0.411022\pi$
$739$	15.0007	0.551811	0.275905	+	0.961185i	$0.411022\pi$
$740$	0	0
$741$	0.446161	0.0163901
$742$	0	0
$743$	28.6822	1.05225	0.526125	−	0.850407i	$-0.323644\pi$
$743$	28.6822	1.05225	0.526125	+	0.850407i	$0.323644\pi$
$744$	0	0
$745$	−9.95722	−0.364804
$746$	0	0
$747$	−42.6632	−1.56097
$748$	0	0
$749$	2.70025	0.0986651
$750$	0	0
$751$	−30.4351	−1.11059	−0.555297	−	0.831652i	$-0.687395\pi$
$751$	−30.4351	−1.11059	−0.555297	+	0.831652i	$0.687395\pi$
$752$	0	0
$753$	−3.57313	−0.130212
$754$	0	0
$755$	−36.6006	−1.33203
$756$	0	0
$757$	51.7252	1.87998	0.939992	−	0.341197i	$-0.110832\pi$
$757$	51.7252	1.87998	0.939992	+	0.341197i	$0.110832\pi$
$758$	0	0
$759$	1.01496	0.0368406
$760$	0	0
$761$	29.6807	1.07593	0.537963	−	0.842969i	$-0.319194\pi$
$761$	29.6807	1.07593	0.537963	+	0.842969i	$0.319194\pi$
$762$	0	0
$763$	−0.317253	−0.0114853
$764$	0	0
$765$	7.78020	0.281294
$766$	0	0
$767$	3.65569	0.131999
$768$	0	0
$769$	−3.68318	−0.132819	−0.0664095	−	0.997792i	$-0.521154\pi$
$769$	−3.68318	−0.132819	−0.0664095	+	0.997792i	$0.521154\pi$
$770$	0	0
$771$	0.242563	0.00873570
$772$	0	0
$773$	24.8227	0.892809	0.446405	−	0.894831i	$-0.352704\pi$
$773$	24.8227	0.892809	0.446405	+	0.894831i	$0.352704\pi$
$774$	0	0
$775$	3.10028	0.111365
$776$	0	0
$777$	−0.376698	−0.0135140
$778$	0	0
$779$	26.6861	0.956128
$780$	0	0
$781$	−2.70584	−0.0968226
$782$	0	0
$783$	−11.4826	−0.410354
$784$	0	0
$785$	−2.63832	−0.0941657
$786$	0	0
$787$	21.0833	0.751538	0.375769	−	0.926713i	$-0.377379\pi$
$787$	21.0833	0.751538	0.375769	+	0.926713i	$0.377379\pi$
$788$	0	0
$789$	2.39483	0.0852583
$790$	0	0
$791$	0.0255427	0.000908194 0
$792$	0	0
$793$	7.11778	0.252760
$794$	0	0
$795$	−4.10157	−0.145468
$796$	0	0
$797$	−24.3975	−0.864204	−0.432102	−	0.901825i	$-0.642228\pi$
$797$	−24.3975	−0.864204	−0.432102	+	0.901825i	$0.642228\pi$
$798$	0	0
$799$	10.2563	0.362843
$800$	0	0
$801$	−33.9423	−1.19929
$802$	0	0
$803$	3.41707	0.120586
$804$	0	0
$805$	5.46490	0.192612
$806$	0	0
$807$	0.623987	0.0219654
$808$	0	0
$809$	9.84260	0.346048	0.173024	−	0.984918i	$-0.444646\pi$
$809$	9.84260	0.346048	0.173024	+	0.984918i	$0.444646\pi$
$810$	0	0
$811$	46.1782	1.62154	0.810769	−	0.585367i	$-0.199049\pi$
$811$	46.1782	1.62154	0.810769	+	0.585367i	$0.199049\pi$
$812$	0	0
$813$	−3.23971	−0.113622
$814$	0	0
$815$	−6.80050	−0.238211
$816$	0	0
$817$	−31.1825	−1.09094
$818$	0	0
$819$	−0.628968	−0.0219779
$820$	0	0
$821$	−3.25038	−0.113439	−0.0567195	−	0.998390i	$-0.518064\pi$
$821$	−3.25038	−0.113439	−0.0567195	+	0.998390i	$0.518064\pi$
$822$	0	0
$823$	1.80449	0.0629005	0.0314503	−	0.999505i	$-0.489987\pi$
$823$	1.80449	0.0629005	0.0314503	+	0.999505i	$0.489987\pi$
$824$	0	0
$825$	−0.0487733	−0.00169807
$826$	0	0
$827$	8.40455	0.292255	0.146127	−	0.989266i	$-0.453319\pi$
$827$	8.40455	0.292255	0.146127	+	0.989266i	$0.453319\pi$
$828$	0	0
$829$	−11.0909	−0.385205	−0.192602	−	0.981277i	$-0.561693\pi$
$829$	−11.0909	−0.385205	−0.192602	+	0.981277i	$0.561693\pi$
$830$	0	0
$831$	3.59223	0.124613
$832$	0	0
$833$	−7.88957	−0.273357
$834$	0	0
$835$	10.7528	0.372117
$836$	0	0
$837$	−13.0212	−0.450077
$838$	0	0
$839$	49.3189	1.70268	0.851339	−	0.524617i	$-0.175791\pi$
$839$	49.3189	1.70268	0.851339	+	0.524617i	$0.175791\pi$
$840$	0	0
$841$	32.8413	1.13246
$842$	0	0
$843$	0.824864	0.0284098
$844$	0	0
$845$	29.0703	1.00005
$846$	0	0
$847$	−3.48678	−0.119807
$848$	0	0
$849$	−2.91313	−0.0999784
$850$	0	0
$851$	33.9376	1.16337
$852$	0	0
$853$	−7.86042	−0.269136	−0.134568	−	0.990904i	$-0.542965\pi$
$853$	−7.86042	−0.269136	−0.134568	+	0.990904i	$0.542965\pi$
$854$	0	0
$855$	18.8343	0.644119
$856$	0	0
$857$	3.23243	0.110418	0.0552088	−	0.998475i	$-0.482418\pi$
$857$	3.23243	0.110418	0.0552088	+	0.998475i	$0.482418\pi$
$858$	0	0
$859$	3.07827	0.105029	0.0525146	−	0.998620i	$-0.483276\pi$
$859$	3.07827	0.105029	0.0525146	+	0.998620i	$0.483276\pi$
$860$	0	0
$861$	0.773446	0.0263590
$862$	0	0
$863$	50.4748	1.71818	0.859091	−	0.511822i	$-0.171029\pi$
$863$	50.4748	1.71818	0.859091	+	0.511822i	$0.171029\pi$
$864$	0	0
$865$	−4.72399	−0.160621
$866$	0	0
$867$	−3.85718	−0.130997
$868$	0	0
$869$	3.52525	0.119586
$870$	0	0
$871$	2.55254	0.0864895
$872$	0	0
$873$	17.1130	0.579186
$874$	0	0
$875$	3.51429	0.118805
$876$	0	0
$877$	−21.1020	−0.712563	−0.356281	−	0.934379i	$-0.615956\pi$
$877$	−21.1020	−0.712563	−0.356281	+	0.934379i	$0.615956\pi$
$878$	0	0
$879$	3.37366	0.113791
$880$	0	0
$881$	27.3881	0.922730	0.461365	−	0.887210i	$-0.347360\pi$
$881$	27.3881	0.922730	0.461365	+	0.887210i	$0.347360\pi$
$882$	0	0
$883$	−23.3136	−0.784567	−0.392283	−	0.919844i	$-0.628315\pi$
$883$	−23.3136	−0.784567	−0.392283	+	0.919844i	$0.628315\pi$
$884$	0	0
$885$	−3.17274	−0.106651
$886$	0	0
$887$	−30.1658	−1.01287	−0.506435	−	0.862278i	$-0.669037\pi$
$887$	−30.1658	−1.01287	−0.506435	+	0.862278i	$0.669037\pi$
$888$	0	0
$889$	1.27624	0.0428036
$890$	0	0
$891$	−4.82772	−0.161735
$892$	0	0
$893$	24.8285	0.830853
$894$	0	0
$895$	−42.6167	−1.42452
$896$	0	0
$897$	−1.16499	−0.0388980
$898$	0	0
$899$	70.1276	2.33889
$900$	0	0
$901$	8.25750	0.275097
$902$	0	0
$903$	−0.903766	−0.0300754
$904$	0	0
$905$	−3.43029	−0.114027
$906$	0	0
$907$	26.8143	0.890353	0.445176	−	0.895443i	$-0.353141\pi$
$907$	26.8143	0.890353	0.445176	+	0.895443i	$0.353141\pi$
$908$	0	0
$909$	−30.2488	−1.00329
$910$	0	0
$911$	−24.7438	−0.819798	−0.409899	−	0.912131i	$-0.634436\pi$
$911$	−24.7438	−0.819798	−0.409899	+	0.912131i	$0.634436\pi$
$912$	0	0
$913$	−8.28241	−0.274108
$914$	0	0
$915$	−6.17747	−0.204221
$916$	0	0
$917$	1.27934	0.0422475
$918$	0	0
$919$	27.2810	0.899916	0.449958	−	0.893050i	$-0.351439\pi$
$919$	27.2810	0.899916	0.449958	+	0.893050i	$0.351439\pi$
$920$	0	0
$921$	4.74788	0.156448
$922$	0	0
$923$	3.10583	0.102230
$924$	0	0
$925$	−1.63085	−0.0536221
$926$	0	0
$927$	−0.153071	−0.00502750
$928$	0	0
$929$	31.6186	1.03737	0.518687	−	0.854964i	$-0.326421\pi$
$929$	31.6186	1.03737	0.518687	+	0.854964i	$0.326421\pi$
$930$	0	0
$931$	−19.0990	−0.625946
$932$	0	0
$933$	−5.10738	−0.167208
$934$	0	0
$935$	1.51041	0.0493956
$936$	0	0
$937$	35.8451	1.17101	0.585504	−	0.810670i	$-0.300897\pi$
$937$	35.8451	1.17101	0.585504	+	0.810670i	$0.300897\pi$
$938$	0	0
$939$	−6.77641	−0.221140
$940$	0	0
$941$	49.3069	1.60736	0.803679	−	0.595063i	$-0.202873\pi$
$941$	49.3069	1.60736	0.803679	+	0.595063i	$0.202873\pi$
$942$	0	0
$943$	−69.6815	−2.26914
$944$	0	0
$945$	1.10298	0.0358798
$946$	0	0
$947$	43.3960	1.41018	0.705090	−	0.709118i	$-0.250907\pi$
$947$	43.3960	1.41018	0.705090	+	0.709118i	$0.250907\pi$
$948$	0	0
$949$	−3.92220	−0.127320
$950$	0	0
$951$	−4.61377	−0.149612
$952$	0	0
$953$	46.8528	1.51771	0.758856	−	0.651259i	$-0.225759\pi$
$953$	46.8528	1.51771	0.758856	+	0.651259i	$0.225759\pi$
$954$	0	0
$955$	34.9596	1.13126
$956$	0	0
$957$	−1.10324	−0.0356628
$958$	0	0
$959$	−6.39085	−0.206371
$960$	0	0
$961$	48.5242	1.56530
$962$	0	0
$963$	−24.2999	−0.783052
$964$	0	0
$965$	−29.2527	−0.941677
$966$	0	0
$967$	9.07659	0.291884	0.145942	−	0.989293i	$-0.453379\pi$
$967$	9.07659	0.291884	0.145942	+	0.989293i	$0.453379\pi$
$968$	0	0
$969$	0.779572	0.0250435
$970$	0	0
$971$	40.2761	1.29252	0.646260	−	0.763117i	$-0.276332\pi$
$971$	40.2761	1.29252	0.646260	+	0.763117i	$0.276332\pi$
$972$	0	0
$973$	−0.326651	−0.0104719
$974$	0	0
$975$	0.0559832	0.00179290
$976$	0	0
$977$	−22.5984	−0.722987	−0.361493	−	0.932375i	$-0.617733\pi$
$977$	−22.5984	−0.722987	−0.361493	+	0.932375i	$0.617733\pi$
$978$	0	0
$979$	−6.58938	−0.210598
$980$	0	0
$981$	2.85500	0.0911530
$982$	0	0
$983$	−36.5906	−1.16706	−0.583530	−	0.812092i	$-0.698329\pi$
$983$	−36.5906	−1.16706	−0.583530	+	0.812092i	$0.698329\pi$
$984$	0	0
$985$	−46.2624	−1.47404
$986$	0	0
$987$	0.719607	0.0229053
$988$	0	0
$989$	81.4222	2.58908
$990$	0	0
$991$	3.42163	0.108692	0.0543458	−	0.998522i	$-0.482693\pi$
$991$	3.42163	0.108692	0.0543458	+	0.998522i	$0.482693\pi$
$992$	0	0
$993$	0.0202868	0.000643784 0
$994$	0	0
$995$	−25.4522	−0.806889
$996$	0	0
$997$	57.7784	1.82986	0.914930	−	0.403613i	$-0.132246\pi$
$997$	57.7784	1.82986	0.914930	+	0.403613i	$0.132246\pi$
$998$	0	0
$999$	6.84959	0.216711

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2224.2.a.o.1.4		7
4.3	odd	2		139.2.a.c.1.7	✓	7
8.3	odd	2		8896.2.a.be.1.4		7
8.5	even	2		8896.2.a.bd.1.4		7
12.11	even	2		1251.2.a.k.1.1		7
20.19	odd	2		3475.2.a.e.1.1		7
28.27	even	2		6811.2.a.p.1.7		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
139.2.a.c.1.7	✓	7	4.3	odd	2
1251.2.a.k.1.1		7	12.11	even	2
2224.2.a.o.1.4		7	1.1	even	1	trivial
3475.2.a.e.1.1		7	20.19	odd	2
6811.2.a.p.1.7		7	28.27	even	2
8896.2.a.bd.1.4		7	8.5	even	2
8896.2.a.be.1.4		7	8.3	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:	\( N \)	\(=\)	\( 2224 = 2^{4} \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2224.a (trivial)

\(f(q)\)	\(=\)	\(q+0.245836 q^{3} -2.31250 q^{5} +0.326651 q^{7} -2.93956 q^{9} +O(q^{10})\)	\(q+0.245836 q^{3} -2.31250 q^{5} +0.326651 q^{7} -2.93956 q^{9} -0.570671 q^{11} +0.655030 q^{13} -0.568496 q^{15} +1.14453 q^{17} +2.77067 q^{19} +0.0803026 q^{21} -7.23464 q^{23} +0.347657 q^{25} -1.46016 q^{27} +7.86393 q^{29} +8.91763 q^{31} -0.140292 q^{33} -0.755380 q^{35} -4.69099 q^{37} +0.161030 q^{39} +9.63165 q^{41} -11.2545 q^{43} +6.79774 q^{45} +8.96119 q^{47} -6.89330 q^{49} +0.281366 q^{51} +7.21477 q^{53} +1.31968 q^{55} +0.681130 q^{57} +5.58094 q^{59} +10.8663 q^{61} -0.960211 q^{63} -1.51476 q^{65} +3.89682 q^{67} -1.77853 q^{69} +4.74151 q^{71} -5.98781 q^{73} +0.0854666 q^{75} -0.186410 q^{77} -6.17739 q^{79} +8.45973 q^{81} +14.5135 q^{83} -2.64672 q^{85} +1.93324 q^{87} +11.5467 q^{89} +0.213966 q^{91} +2.19228 q^{93} -6.40717 q^{95} -5.82160 q^{97} +1.67752 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{3} + 11 q^{5} + 5 q^{7} + 13 q^{9}+O(q^{10}) \) 7 * q + 2 * q^3 + 11 * q^5 + 5 * q^7 + 13 * q^9	\( 7 q + 2 q^{3} + 11 q^{5} + 5 q^{7} + 13 q^{9} - 2 q^{11} + 6 q^{13} + 3 q^{15} + 5 q^{17} + 10 q^{19} - 5 q^{21} + q^{23} + 14 q^{25} + 11 q^{27} + 30 q^{29} + 20 q^{31} - 20 q^{33} + 7 q^{35} + 6 q^{37} - 11 q^{39} + 19 q^{41} + 12 q^{43} + 27 q^{45} + 3 q^{47} - 8 q^{49} - 23 q^{51} + 38 q^{53} - 7 q^{55} - 19 q^{57} + 14 q^{59} + 4 q^{61} + 18 q^{63} + 10 q^{65} - 9 q^{67} + 9 q^{69} - 24 q^{71} - 5 q^{73} + 21 q^{75} - 13 q^{77} - 8 q^{79} + 39 q^{81} + 9 q^{83} - 22 q^{85} + 25 q^{87} + 10 q^{89} - 7 q^{91} - 15 q^{93} + 21 q^{95} - 5 q^{97} + 31 q^{99}+O(q^{100}) \) 7 * q + 2 * q^3 + 11 * q^5 + 5 * q^7 + 13 * q^9 - 2 * q^11 + 6 * q^13 + 3 * q^15 + 5 * q^17 + 10 * q^19 - 5 * q^21 + q^23 + 14 * q^25 + 11 * q^27 + 30 * q^29 + 20 * q^31 - 20 * q^33 + 7 * q^35 + 6 * q^37 - 11 * q^39 + 19 * q^41 + 12 * q^43 + 27 * q^45 + 3 * q^47 - 8 * q^49 - 23 * q^51 + 38 * q^53 - 7 * q^55 - 19 * q^57 + 14 * q^59 + 4 * q^61 + 18 * q^63 + 10 * q^65 - 9 * q^67 + 9 * q^69 - 24 * q^71 - 5 * q^73 + 21 * q^75 - 13 * q^77 - 8 * q^79 + 39 * q^81 + 9 * q^83 - 22 * q^85 + 25 * q^87 + 10 * q^89 - 7 * q^91 - 15 * q^93 + 21 * q^95 - 5 * q^97 + 31 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more