LMFDB - Embedded newform 1251.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1251,2,Mod(1,1251)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1251, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1251.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1251 = 3^{2} \cdot 139 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1251.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:	$9.98928529286$
Analytic rank:	$1$
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			1.1
Root			$2.47985$ of defining polynomial
Character	$\chi$	$=$	1251.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.47985 q^{2} +4.14965 q^{4} +2.31250 q^{5} -0.326651 q^{7} -5.33082 q^{8} +O(q^{10})$	\(q-2.47985 q^{2} +4.14965 q^{4} +2.31250 q^{5} -0.326651 q^{7} -5.33082 q^{8} -5.73465 q^{10} -0.570671 q^{11} +0.655030 q^{13} +0.810045 q^{14} +4.92032 q^{16} -1.14453 q^{17} -2.77067 q^{19} +9.59608 q^{20} +1.41518 q^{22} -7.23464 q^{23} +0.347657 q^{25} -1.62438 q^{26} -1.35549 q^{28} -7.86393 q^{29} -8.91763 q^{31} -1.54002 q^{32} +2.83826 q^{34} -0.755380 q^{35} -4.69099 q^{37} +6.87084 q^{38} -12.3275 q^{40} -9.63165 q^{41} +11.2545 q^{43} -2.36809 q^{44} +17.9408 q^{46} +8.96119 q^{47} -6.89330 q^{49} -0.862137 q^{50} +2.71815 q^{52} -7.21477 q^{53} -1.31968 q^{55} +1.74132 q^{56} +19.5014 q^{58} +5.58094 q^{59} +10.8663 q^{61} +22.1144 q^{62} -6.02163 q^{64} +1.51476 q^{65} -3.89682 q^{67} -4.74939 q^{68} +1.87323 q^{70} +4.74151 q^{71} -5.98781 q^{73} +11.6329 q^{74} -11.4973 q^{76} +0.186410 q^{77} +6.17739 q^{79} +11.3782 q^{80} +23.8850 q^{82} +14.5135 q^{83} -2.64672 q^{85} -27.9095 q^{86} +3.04214 q^{88} -11.5467 q^{89} -0.213966 q^{91} -30.0212 q^{92} -22.2224 q^{94} -6.40717 q^{95} -5.82160 q^{97} +17.0943 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{2} + 9 q^{4} - 11 q^{5} - 5 q^{7} - 6 q^{8}+O(q^{10}) $ 7 * q - q^2 + 9 * q^4 - 11 * q^5 - 5 * q^7 - 6 * q^8	$ 7 q - q^{2} + 9 q^{4} - 11 q^{5} - 5 q^{7} - 6 q^{8} - 4 q^{10} - 2 q^{11} + 6 q^{13} - 7 q^{14} + 5 q^{16} - 5 q^{17} - 10 q^{19} - 12 q^{20} - 18 q^{22} + q^{23} + 14 q^{25} + 8 q^{26} - 28 q^{28} - 30 q^{29} - 20 q^{31} + 12 q^{32} - 17 q^{34} + 7 q^{35} + 6 q^{37} - 6 q^{38} - 22 q^{40} - 19 q^{41} - 12 q^{43} - 25 q^{44} + 22 q^{46} + 3 q^{47} - 8 q^{49} - 12 q^{50} - 8 q^{52} - 38 q^{53} + 7 q^{55} - 21 q^{56} - 21 q^{58} + 14 q^{59} + 4 q^{61} + q^{62} - 16 q^{64} - 10 q^{65} + 9 q^{67} + 25 q^{68} + 20 q^{70} - 24 q^{71} - 5 q^{73} - 9 q^{74} + 3 q^{76} + 13 q^{77} + 8 q^{79} - 11 q^{80} + 56 q^{82} + 9 q^{83} - 22 q^{85} - 39 q^{86} - 29 q^{88} - 10 q^{89} + 7 q^{91} - 29 q^{92} - 36 q^{94} + 21 q^{95} - 5 q^{97} + 49 q^{98}+O(q^{100}) $ 7 * q - q^2 + 9 * q^4 - 11 * q^5 - 5 * q^7 - 6 * q^8 - 4 * q^10 - 2 * q^11 + 6 * q^13 - 7 * q^14 + 5 * q^16 - 5 * q^17 - 10 * q^19 - 12 * q^20 - 18 * q^22 + q^23 + 14 * q^25 + 8 * q^26 - 28 * q^28 - 30 * q^29 - 20 * q^31 + 12 * q^32 - 17 * q^34 + 7 * q^35 + 6 * q^37 - 6 * q^38 - 22 * q^40 - 19 * q^41 - 12 * q^43 - 25 * q^44 + 22 * q^46 + 3 * q^47 - 8 * q^49 - 12 * q^50 - 8 * q^52 - 38 * q^53 + 7 * q^55 - 21 * q^56 - 21 * q^58 + 14 * q^59 + 4 * q^61 + q^62 - 16 * q^64 - 10 * q^65 + 9 * q^67 + 25 * q^68 + 20 * q^70 - 24 * q^71 - 5 * q^73 - 9 * q^74 + 3 * q^76 + 13 * q^77 + 8 * q^79 - 11 * q^80 + 56 * q^82 + 9 * q^83 - 22 * q^85 - 39 * q^86 - 29 * q^88 - 10 * q^89 + 7 * q^91 - 29 * q^92 - 36 * q^94 + 21 * q^95 - 5 * q^97 + 49 * q^98

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 $

$2$	−2.47985	−1.75352	−0.876759	−	0.480930i	$-0.840299\pi$
$2$	−2.47985	−1.75352	−0.876759	+	0.480930i	$0.840299\pi$
$3$	0	0
$3$	0	0
$4$	4.14965	2.07483
$5$	2.31250	1.03418	0.517091	−	0.855931i	$-0.327015\pi$
$5$	2.31250	1.03418	0.517091	+	0.855931i	$0.327015\pi$
$6$	0	0
$7$	−0.326651	−0.123462	−0.0617312	−	0.998093i	$-0.519662\pi$
$7$	−0.326651	−0.123462	−0.0617312	+	0.998093i	$0.519662\pi$
$8$	−5.33082	−1.88473
$9$	0	0
$10$	−5.73465	−1.81346
$11$	−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	$0.471046\pi$
$14$	0.810045	0.216494
$15$	0	0
$16$	4.92032	1.23008
$17$	−1.14453	−0.277589	−0.138794	−	0.990321i	$-0.544323\pi$
$17$	−1.14453	−0.277589	−0.138794	+	0.990321i	$0.544323\pi$
$18$	0	0
$19$	−2.77067	−0.635634	−0.317817	−	0.948152i	$-0.602950\pi$
$19$	−2.77067	−0.635634	−0.317817	+	0.948152i	$0.602950\pi$
$20$	9.59608	2.14575
$21$	0	0
$22$	1.41518	0.301717
$23$	−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
$26$	−1.62438	−0.318567
$27$	0	0
$28$	−1.35549	−0.256163
$29$	−7.86393	−1.46029	−0.730147	−	0.683290i	$-0.760549\pi$
$29$	−7.86393	−1.46029	−0.730147	+	0.683290i	$0.760549\pi$
$30$	0	0
$31$	−8.91763	−1.60165	−0.800827	−	0.598896i	$-0.795606\pi$
$31$	−8.91763	−1.60165	−0.800827	+	0.598896i	$0.795606\pi$
$32$	−1.54002	−0.272239
$33$	0	0
$34$	2.83826	0.486757
$35$	−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	$0.626004\pi$
$38$	6.87084	1.11460
$39$	0	0
$40$	−12.3275	−1.94915
$41$	−9.63165	−1.50421	−0.752105	−	0.659043i	$-0.770961\pi$
$41$	−9.63165	−1.50421	−0.752105	+	0.659043i	$0.770961\pi$
$42$	0	0
$43$	11.2545	1.71630	0.858148	−	0.513403i	$-0.171615\pi$
$43$	11.2545	1.71630	0.858148	+	0.513403i	$0.171615\pi$
$44$	−2.36809	−0.357003
$45$	0	0
$46$	17.9408	2.64523
$47$	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
$50$	−0.862137	−0.121925
$51$	0	0
$52$	2.71815	0.376940
$53$	−7.21477	−0.991025	−0.495512	−	0.868601i	$-0.665020\pi$
$53$	−7.21477	−0.991025	−0.495512	+	0.868601i	$0.665020\pi$
$54$	0	0
$55$	−1.31968	−0.177945
$56$	1.74132	0.232693
$57$	0	0
$58$	19.5014	2.56065
$59$	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	$0.255118\pi$
$62$	22.1144	2.80853
$63$	0	0
$64$	−6.02163	−0.752704
$65$	1.51476	0.187883
$66$	0	0
$67$	−3.89682	−0.476073	−0.238036	−	0.971256i	$-0.576504\pi$
$67$	−3.89682	−0.476073	−0.238036	+	0.971256i	$0.576504\pi$
$68$	−4.74939	−0.575949
$69$	0	0
$70$	1.87323	0.223894
$71$	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	$0.613958\pi$
$74$	11.6329	1.35230
$75$	0	0
$76$	−11.4973	−1.31883
$77$	0.186410	0.0212434
$78$	0	0
$79$	6.17739	0.695010	0.347505	−	0.937678i	$-0.387029\pi$
$79$	6.17739	0.695010	0.347505	+	0.937678i	$0.387029\pi$
$80$	11.3782	1.27213
$81$	0	0
$82$	23.8850	2.63766
$83$	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
$86$	−27.9095	−3.00956
$87$	0	0
$88$	3.04214	0.324294
$89$	−11.5467	−1.22395	−0.611975	−	0.790877i	$-0.709625\pi$
$89$	−11.5467	−1.22395	−0.611975	+	0.790877i	$0.709625\pi$
$90$	0	0
$91$	−0.213966	−0.0224298
$92$	−30.0212	−3.12993
$93$	0	0
$94$	−22.2224	−2.29207
$95$	−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	$0.595502\pi$
$98$	17.0943	1.72679
$99$	0	0
$100$	1.44266	0.144266
$101$	−10.2902	−1.02392	−0.511958	−	0.859011i	$-0.671080\pi$
$101$	−10.2902	−1.02392	−0.511958	+	0.859011i	$0.671080\pi$
$102$	0	0
$103$	−0.0520725	−0.00513086	−0.00256543	−	0.999997i	$-0.500817\pi$
$103$	−0.0520725	−0.00513086	−0.00256543	+	0.999997i	$0.500817\pi$
$104$	−3.49185	−0.342404
$105$	0	0
$106$	17.8915	1.73778
$107$	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	$0.514811\pi$
$110$	3.27260	0.312030
$111$	0	0
$112$	−1.60723	−0.151869
$113$	−0.0781957	−0.00735603	−0.00367802	−	0.999993i	$-0.501171\pi$
$113$	−0.0781957	−0.00735603	−0.00367802	+	0.999993i	$0.501171\pi$
$114$	0	0
$115$	−16.7301	−1.56009
$116$	−32.6326	−3.02986
$117$	0	0
$118$	−13.8399	−1.27407
$119$	0.373861	0.0342718
$120$	0	0
$121$	−10.6743	−0.970394
$122$	−26.9469	−2.43966
$123$	0	0
$124$	−37.0051	−3.32316
$125$	−10.7585	−0.962273
$126$	0	0
$127$	−3.90704	−0.346693	−0.173347	−	0.984861i	$-0.555458\pi$
$127$	−3.90704	−0.346693	−0.173347	+	0.984861i	$0.555458\pi$
$128$	18.0128	1.59212
$129$	0	0
$130$	−3.75637	−0.329456
$131$	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
$134$	9.66354	0.834803
$135$	0	0
$136$	6.10127	0.523179
$137$	19.5648	1.67153	0.835766	−	0.549086i	$-0.185024\pi$
$137$	19.5648	1.67153	0.835766	+	0.549086i	$0.185024\pi$
$138$	0	0
$139$	1.00000	0.0848189
$139$	1.00000	0.0848189
$140$	−3.13457	−0.264919
$141$	0	0
$142$	−11.7582	−0.986728
$143$	−0.373807	−0.0312593
$144$	0	0
$145$	−18.1853	−1.51021
$146$	14.8489	1.22890
$147$	0	0
$148$	−19.4660	−1.60009
$149$	−4.30582	−0.352747	−0.176373	−	0.984323i	$-0.556437\pi$
$149$	−4.30582	−0.352747	−0.176373	+	0.984323i	$0.556437\pi$
$150$	0	0
$151$	−15.8273	−1.28801	−0.644003	−	0.765023i	$-0.722728\pi$
$151$	−15.8273	−1.28801	−0.644003	+	0.765023i	$0.722728\pi$
$152$	14.7699	1.19800
$153$	0	0
$154$	−0.462269	−0.0372507
$155$	−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	$0.485503\pi$
$158$	−15.3190	−1.21871
$159$	0	0
$160$	−3.56129	−0.281545
$161$	2.36320	0.186246
$162$	0	0
$163$	−2.94076	−0.230338	−0.115169	−	0.993346i	$-0.536741\pi$
$163$	−2.94076	−0.230338	−0.115169	+	0.993346i	$0.536741\pi$
$164$	−39.9680	−3.12098
$165$	0	0
$166$	−35.9912	−2.79346
$167$	−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
$170$	6.56347	0.503395
$171$	0	0
$172$	46.7023	3.56102
$173$	−2.04281	−0.155312	−0.0776559	−	0.996980i	$-0.524744\pi$
$173$	−2.04281	−0.155312	−0.0776559	+	0.996980i	$0.524744\pi$
$174$	0	0
$175$	−0.113562	−0.00858451
$176$	−2.80788	−0.211652
$177$	0	0
$178$	28.6341	2.14622
$179$	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	$0.482443\pi$
$182$	0.530604	0.0393310
$183$	0	0
$184$	38.5665	2.84316
$185$	−10.8479	−0.797554
$186$	0	0
$187$	0.653149	0.0477630
$188$	37.1859	2.71206
$189$	0	0
$190$	15.8888	1.15270
$191$	−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	$0.349540\pi$
$194$	14.4367	1.03649
$195$	0	0
$196$	−28.6048	−2.04320
$197$	−20.0054	−1.42532	−0.712662	−	0.701508i	$-0.752511\pi$
$197$	−20.0054	−1.42532	−0.712662	+	0.701508i	$0.752511\pi$
$198$	0	0
$199$	−11.0064	−0.780220	−0.390110	−	0.920768i	$-0.627563\pi$
$199$	−11.0064	−0.780220	−0.390110	+	0.920768i	$0.627563\pi$
$200$	−1.85330	−0.131048
$201$	0	0
$202$	25.5182	1.79546
$203$	2.56876	0.180292
$204$	0	0
$205$	−22.2732	−1.55563
$206$	0.129132	0.00899705
$207$	0	0
$208$	3.22296	0.223472
$209$	1.58114	0.109370
$210$	0	0
$211$	9.89782	0.681394	0.340697	−	0.940173i	$-0.389337\pi$
$211$	9.89782	0.681394	0.340697	+	0.940173i	$0.389337\pi$
$212$	−29.9388	−2.05621
$213$	0	0
$214$	−20.4996	−1.40133
$215$	26.0260	1.77496
$216$	0	0
$217$	2.91295	0.197744
$218$	2.40851	0.163125
$219$	0	0
$220$	−5.47620	−0.369205
$221$	−0.749700	−0.0504303
$222$	0	0
$223$	6.88052	0.460753	0.230377	−	0.973102i	$-0.426004\pi$
$223$	6.88052	0.460753	0.230377	+	0.973102i	$0.426004\pi$
$224$	0.503048	0.0336113
$225$	0	0
$226$	0.193914	0.0128989
$227$	−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	$0.195344\pi$
$230$	41.4881	2.73565
$231$	0	0
$232$	41.9212	2.75226
$233$	18.1608	1.18975	0.594877	−	0.803817i	$-0.297201\pi$
$233$	18.1608	1.18975	0.594877	+	0.803817i	$0.297201\pi$
$234$	0	0
$235$	20.7228	1.35180
$236$	23.1590	1.50752
$237$	0	0
$238$	−0.927119	−0.0600962
$239$	−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	$0.0861900\pi$
$242$	26.4707	1.70160
$243$	0	0
$244$	45.0915	2.88669
$245$	−15.9408	−1.01842
$246$	0	0
$247$	−1.81487	−0.115477
$248$	47.5383	3.01868
$249$	0	0
$250$	26.6796	1.68736
$251$	−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
$254$	9.68886	0.607933
$255$	0	0
$256$	−32.6257	−2.03911
$257$	−0.986686	−0.0615478	−0.0307739	−	0.999526i	$-0.509797\pi$
$257$	−0.986686	−0.0615478	−0.0307739	+	0.999526i	$0.509797\pi$
$258$	0	0
$259$	1.53232	0.0952134
$260$	6.28572	0.389824
$261$	0	0
$262$	−9.71241	−0.600035
$263$	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
$266$	−2.24437	−0.137611
$267$	0	0
$268$	−16.1705	−0.987769
$269$	−2.53822	−0.154758	−0.0773791	−	0.997002i	$-0.524655\pi$
$269$	−2.53822	−0.154758	−0.0773791	+	0.997002i	$0.524655\pi$
$270$	0	0
$271$	13.1783	0.800527	0.400263	−	0.916400i	$-0.368919\pi$
$271$	13.1783	0.800527	0.400263	+	0.916400i	$0.368919\pi$
$272$	−5.63144	−0.341456
$273$	0	0
$274$	−48.5177	−2.93106
$275$	−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	$0.355339\pi$
$278$	−2.47985	−0.148731
$279$	0	0
$280$	4.02680	0.240647
$281$	−3.35534	−0.200163	−0.100081	−	0.994979i	$-0.531910\pi$
$281$	−3.35534	−0.200163	−0.100081	+	0.994979i	$0.531910\pi$
$282$	0	0
$283$	11.8499	0.704403	0.352202	−	0.935924i	$-0.385433\pi$
$283$	11.8499	0.704403	0.352202	+	0.935924i	$0.385433\pi$
$284$	19.6756	1.16753
$285$	0	0
$286$	0.926985	0.0548138
$287$	3.14619	0.185713
$288$	0	0
$289$	−15.6901	−0.922945
$290$	45.0969	2.64818
$291$	0	0
$292$	−24.8473	−1.45408
$293$	−13.7232	−0.801717	−0.400859	−	0.916140i	$-0.631288\pi$
$293$	−13.7232	−0.801717	−0.400859	+	0.916140i	$0.631288\pi$
$294$	0	0
$295$	12.9059	0.751412
$296$	25.0068	1.45349
$297$	0	0
$298$	10.6778	0.618548
$299$	−4.73891	−0.274058
$300$	0	0
$301$	−3.67629	−0.211898
$302$	39.2493	2.25854
$303$	0	0
$304$	−13.6326	−0.781881
$305$	25.1284	1.43885
$306$	0	0
$307$	−19.3132	−1.10226	−0.551131	−	0.834419i	$-0.685803\pi$
$307$	−19.3132	−1.10226	−0.551131	+	0.834419i	$0.685803\pi$
$308$	0.773538	0.0440764
$309$	0	0
$310$	51.1395	2.90453
$311$	−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	$0.784286\pi$
$314$	−2.82925	−0.159664
$315$	0	0
$316$	25.6340	1.44203
$317$	18.7677	1.05410	0.527049	−	0.849835i	$-0.323298\pi$
$317$	18.7677	1.05410	0.527049	+	0.849835i	$0.323298\pi$
$318$	0	0
$319$	4.48772	0.251264
$320$	−13.9250	−0.778432
$321$	0	0
$322$	−5.86038	−0.326586
$323$	3.17110	0.176445
$324$	0	0
$325$	0.227726	0.0126320
$326$	7.29264	0.403902
$327$	0	0
$328$	51.3446	2.83503
$329$	−2.92718	−0.161381
$330$	0	0
$331$	−0.0825219	−0.00453581	−0.00226791	−	0.999997i	$-0.500722\pi$
$331$	−0.0825219	−0.00453581	−0.00226791	+	0.999997i	$0.500722\pi$
$332$	60.2258	3.30532
$333$	0	0
$334$	11.5310	0.630947
$335$	−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	$0.379577\pi$
$338$	31.1740	1.69564
$339$	0	0
$340$	−10.9830	−0.595635
$341$	5.08903	0.275587
$342$	0	0
$343$	4.53826	0.245043
$344$	−59.9957	−3.23475
$345$	0	0
$346$	5.06586	0.272342
$347$	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	$0.206249\pi$
$350$	0.281618	0.0150531
$351$	0	0
$352$	0.878843	0.0468425
$353$	−1.66236	−0.0884787	−0.0442394	−	0.999021i	$-0.514086\pi$
$353$	−1.66236	−0.0884787	−0.0442394	+	0.999021i	$0.514086\pi$
$354$	0	0
$355$	10.9647	0.581948
$356$	−47.9149	−2.53949
$357$	0	0
$358$	−45.7008	−2.41536
$359$	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
$362$	−3.67854	−0.193340
$363$	0	0
$364$	−0.887886	−0.0465379
$365$	−13.8468	−0.724775
$366$	0	0
$367$	−20.9268	−1.09237	−0.546185	−	0.837665i	$-0.683920\pi$
$367$	−20.9268	−1.09237	−0.546185	+	0.837665i	$0.683920\pi$
$368$	−35.5967	−1.85561
$369$	0	0
$370$	26.9012	1.39853
$371$	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	$0.403156\pi$
$374$	−1.61971	−0.0837532
$375$	0	0
$376$	−47.7705	−2.46358
$377$	−5.15111	−0.265296
$378$	0	0
$379$	20.9839	1.07787	0.538935	−	0.842347i	$-0.318827\pi$
$379$	20.9839	1.07787	0.538935	+	0.842347i	$0.318827\pi$
$380$	−26.5875	−1.36391
$381$	0	0
$382$	37.4895	1.91813
$383$	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
$386$	−31.3696	−1.59667
$387$	0	0
$388$	−24.1576	−1.22642
$389$	−2.64861	−0.134290	−0.0671450	−	0.997743i	$-0.521389\pi$
$389$	−2.64861	−0.134290	−0.0671450	+	0.997743i	$0.521389\pi$
$390$	0	0
$391$	8.28024	0.418750
$392$	36.7469	1.85600
$393$	0	0
$394$	49.6103	2.49933
$395$	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	$0.711410\pi$
$398$	27.2941	1.36813
$399$	0	0
$400$	1.71058	0.0855292
$401$	−36.8459	−1.84000	−0.919998	−	0.391923i	$-0.871810\pi$
$401$	−36.8459	−1.84000	−0.919998	+	0.391923i	$0.871810\pi$
$402$	0	0
$403$	−5.84132	−0.290977
$404$	−42.7009	−2.12445
$405$	0	0
$406$	−6.37014	−0.316145
$407$	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	$0.555087\pi$
$410$	55.2341	2.72782
$411$	0	0
$412$	−0.216083	−0.0106456
$413$	−1.82302	−0.0897049
$414$	0	0
$415$	33.5624	1.64751
$416$	−1.00876	−0.0494585
$417$	0	0
$418$	−3.92099	−0.191782
$419$	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	$0.836423\pi$
$422$	−24.5451	−1.19484
$423$	0	0
$424$	38.4606	1.86781
$425$	−0.397903	−0.0193011
$426$	0	0
$427$	−3.54950	−0.171772
$428$	34.3030	1.65810
$429$	0	0
$430$	−64.5407	−3.11243
$431$	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	$0.735369\pi$
$434$	−7.22369	−0.346748
$435$	0	0
$436$	−4.03027	−0.193015
$437$	20.0448	0.958871
$438$	0	0
$439$	27.0360	1.29036	0.645180	−	0.764031i	$-0.276782\pi$
$439$	27.0360	1.29036	0.645180	+	0.764031i	$0.276782\pi$
$440$	7.03496	0.335378
$441$	0	0
$442$	1.85914	0.0884305
$443$	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
$446$	−17.0626	−0.807940
$447$	0	0
$448$	1.96697	0.0929306
$449$	14.6027	0.689142	0.344571	−	0.938760i	$-0.388024\pi$
$449$	14.6027	0.689142	0.344571	+	0.938760i	$0.388024\pi$
$450$	0	0
$451$	5.49650	0.258820
$452$	−0.324485	−0.0152625
$453$	0	0
$454$	69.9825	3.28444
$455$	−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	$0.400339\pi$
$458$	−61.3586	−2.86710
$459$	0	0
$460$	−69.4241	−3.23692
$461$	4.68827	0.218354	0.109177	−	0.994022i	$-0.465178\pi$
$461$	4.68827	0.218354	0.109177	+	0.994022i	$0.465178\pi$
$462$	0	0
$463$	25.1950	1.17091	0.585454	−	0.810705i	$-0.300916\pi$
$463$	25.1950	1.17091	0.585454	+	0.810705i	$0.300916\pi$
$464$	−38.6930	−1.79628
$465$	0	0
$466$	−45.0360	−2.08625
$467$	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
$470$	−51.3893	−2.37041
$471$	0	0
$472$	−29.7510	−1.36940
$473$	−6.42262	−0.295312
$474$	0	0
$475$	−0.963241	−0.0441965
$476$	1.55139	0.0711080
$477$	0	0
$478$	57.0265	2.60833
$479$	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
$482$	−74.1899	−3.37926
$483$	0	0
$484$	−44.2948	−2.01340
$485$	−13.4625	−0.611298
$486$	0	0
$487$	−35.4094	−1.60455	−0.802276	−	0.596953i	$-0.796378\pi$
$487$	−35.4094	−1.60455	−0.802276	+	0.596953i	$0.796378\pi$
$488$	−57.9265	−2.62221
$489$	0	0
$490$	39.5307	1.78581
$491$	−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
$494$	4.50061	0.202492
$495$	0	0
$496$	−43.8776	−1.97016
$497$	−1.54882	−0.0694740
$498$	0	0
$499$	6.73789	0.301630	0.150815	−	0.988562i	$-0.451810\pi$
$499$	6.73789	0.301630	0.150815	+	0.988562i	$0.451810\pi$
$500$	−44.6442	−1.99655
$501$	0	0
$502$	36.0436	1.60871
$503$	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
$506$	−10.2383	−0.455148
$507$	0	0
$508$	−16.2129	−0.719329
$509$	32.6107	1.44544	0.722721	−	0.691140i	$-0.242891\pi$
$509$	32.6107	1.44544	0.722721	+	0.691140i	$0.242891\pi$
$510$	0	0
$511$	1.95592	0.0865249
$512$	44.8813	1.98349
$513$	0	0
$514$	2.44683	0.107925
$515$	−0.120418	−0.00530624
$516$	0	0
$517$	−5.11389	−0.224909
$518$	−3.79991	−0.166959
$519$	0	0
$520$	−8.07490	−0.354108
$521$	−8.76803	−0.384134	−0.192067	−	0.981382i	$-0.561519\pi$
$521$	−8.76803	−0.384134	−0.192067	+	0.981382i	$0.561519\pi$
$522$	0	0
$523$	−28.4574	−1.24436	−0.622178	−	0.782876i	$-0.713752\pi$
$523$	−28.4574	−1.24436	−0.622178	+	0.782876i	$0.713752\pi$
$524$	16.2523	0.709983
$525$	0	0
$526$	−24.1577	−1.05332
$527$	10.2065	0.444601
$528$	0	0
$529$	29.3400	1.27565
$530$	41.3742	1.79718
$531$	0	0
$532$	3.75561	0.162826
$533$	−6.30902	−0.273274
$534$	0	0
$535$	19.1162	0.826467
$536$	20.7733	0.897268
$537$	0	0
$538$	6.29441	0.271371
$539$	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	$0.315494\pi$
$542$	−32.6803	−1.40374
$543$	0	0
$544$	1.76259	0.0755705
$545$	−2.24597	−0.0962068
$546$	0	0
$547$	−10.7278	−0.458689	−0.229344	−	0.973345i	$-0.573658\pi$
$547$	−10.7278	−0.458689	−0.229344	+	0.973345i	$0.573658\pi$
$548$	81.1870	3.46814
$549$	0	0
$550$	0.491996	0.0209788
$551$	21.7883	0.928214
$552$	0	0
$553$	−2.01785	−0.0858077
$554$	−36.2363	−1.53953
$555$	0	0
$556$	4.14965	0.175985
$557$	−31.2761	−1.32521	−0.662606	−	0.748968i	$-0.730550\pi$
$557$	−31.2761	−1.32521	−0.662606	+	0.748968i	$0.730550\pi$
$558$	0	0
$559$	7.37204	0.311804
$560$	−3.71671	−0.157060
$561$	0	0
$562$	8.32074	0.350989
$563$	−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
$566$	−29.3860	−1.23518
$567$	0	0
$568$	−25.2761	−1.06056
$569$	−20.9669	−0.878978	−0.439489	−	0.898248i	$-0.644841\pi$
$569$	−20.9669	−0.878978	−0.439489	+	0.898248i	$0.644841\pi$
$570$	0	0
$571$	−10.2145	−0.427464	−0.213732	−	0.976892i	$-0.568562\pi$
$571$	−10.2145	−0.427464	−0.213732	+	0.976892i	$0.568562\pi$
$572$	−1.55117	−0.0648576
$573$	0	0
$574$	−7.80207	−0.325652
$575$	−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	$0.524791\pi$
$578$	38.9090	1.61840
$579$	0	0
$580$	−75.4628	−3.13342
$581$	−4.74083	−0.196683
$582$	0	0
$583$	4.11726	0.170519
$584$	31.9199	1.32086
$585$	0	0
$586$	34.0315	1.40583
$587$	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
$590$	−32.0048	−1.31762
$591$	0	0
$592$	−23.0812	−0.948630
$593$	34.5495	1.41878	0.709389	−	0.704817i	$-0.248971\pi$
$593$	34.5495	1.41878	0.709389	+	0.704817i	$0.248971\pi$
$594$	0	0
$595$	0.864554	0.0354432
$596$	−17.8677	−0.731889
$597$	0	0
$598$	11.7518	0.480566
$599$	−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	$0.713589\pi$
$602$	9.11666	0.371567
$603$	0	0
$604$	−65.6778	−2.67239
$605$	−24.6844	−1.00356
$606$	0	0
$607$	−13.2182	−0.536511	−0.268256	−	0.963348i	$-0.586447\pi$
$607$	−13.2182	−0.536511	−0.268256	+	0.963348i	$0.586447\pi$
$608$	4.26688	0.173045
$609$	0	0
$610$	−62.3146	−2.52305
$611$	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	$0.610743\pi$
$614$	47.8938	1.93284
$615$	0	0
$616$	−0.993719	−0.0400381
$617$	16.5346	0.665659	0.332829	−	0.942987i	$-0.391997\pi$
$617$	16.5346	0.665659	0.332829	+	0.942987i	$0.391997\pi$
$618$	0	0
$619$	19.1064	0.767952	0.383976	−	0.923343i	$-0.374555\pi$
$619$	19.1064	0.767952	0.383976	+	0.923343i	$0.374555\pi$
$620$	−85.5743	−3.43675
$621$	0	0
$622$	51.5202	2.06577
$623$	3.77175	0.151112
$624$	0	0
$625$	−26.6174	−1.06470
$626$	68.3565	2.73207
$627$	0	0
$628$	4.73432	0.188920
$629$	5.36896	0.214075
$630$	0	0
$631$	−20.1170	−0.800844	−0.400422	−	0.916331i	$-0.631136\pi$
$631$	−20.1170	−0.800844	−0.400422	+	0.916331i	$0.631136\pi$
$632$	−32.9305	−1.30991
$633$	0	0
$634$	−46.5410	−1.84838
$635$	−9.03502	−0.358544
$636$	0	0
$637$	−4.51532	−0.178904
$638$	−11.1289	−0.440596
$639$	0	0
$640$	41.6545	1.64654
$641$	19.2938	0.762061	0.381031	−	0.924562i	$-0.375569\pi$
$641$	19.2938	0.762061	0.381031	+	0.924562i	$0.375569\pi$
$642$	0	0
$643$	29.2745	1.15447	0.577237	−	0.816577i	$-0.304131\pi$
$643$	29.2745	1.15447	0.577237	+	0.816577i	$0.304131\pi$
$644$	9.80647	0.386429
$645$	0	0
$646$	−7.86386	−0.309399
$647$	−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
$650$	−0.564726	−0.0221504
$651$	0	0
$652$	−12.2031	−0.477911
$653$	35.7848	1.40037	0.700183	−	0.713963i	$-0.253102\pi$
$653$	35.7848	1.40037	0.700183	+	0.713963i	$0.253102\pi$
$654$	0	0
$655$	9.05698	0.353886
$656$	−47.3908	−1.85030
$657$	0	0
$658$	7.25897	0.282984
$659$	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	$0.446026\pi$
$662$	0.204642	0.00795363
$663$	0	0
$664$	−77.3686	−3.00248
$665$	2.09291	0.0811594
$666$	0	0
$667$	56.8927	2.20289
$668$	−19.2953	−0.746559
$669$	0	0
$670$	22.3469	0.863337
$671$	−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	$0.543771\pi$
$674$	−33.6294	−1.29536
$675$	0	0
$676$	−52.1650	−2.00635
$677$	16.7747	0.644704	0.322352	−	0.946620i	$-0.395527\pi$
$677$	16.7747	0.644704	0.322352	+	0.946620i	$0.395527\pi$
$678$	0	0
$679$	1.90163	0.0729779
$680$	14.1092	0.541063
$681$	0	0
$682$	−12.6200	−0.483246
$683$	−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
$686$	−11.2542	−0.429687
$687$	0	0
$688$	55.3758	2.11118
$689$	−4.72589	−0.180042
$690$	0	0
$691$	−30.4574	−1.15865	−0.579327	−	0.815095i	$-0.696685\pi$
$691$	−30.4574	−1.15865	−0.579327	+	0.815095i	$0.696685\pi$
$692$	−8.47695	−0.322245
$693$	0	0
$694$	−34.8367	−1.32238
$695$	2.31250	0.0877181
$696$	0	0
$697$	11.0237	0.417552
$698$	−73.8758	−2.79624
$699$	0	0
$700$	−0.471245	−0.0178114
$701$	19.6196	0.741020	0.370510	−	0.928828i	$-0.379183\pi$
$701$	19.6196	0.741020	0.370510	+	0.928828i	$0.379183\pi$
$702$	0	0
$703$	12.9972	0.490197
$704$	3.43637	0.129513
$705$	0	0
$706$	4.12241	0.155149
$707$	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	$0.430081\pi$
$710$	−27.1909	−1.02046
$711$	0	0
$712$	61.5535	2.30681
$713$	64.5158	2.41614
$714$	0	0
$715$	−0.864429	−0.0323278
$716$	76.4734	2.85794
$717$	0	0
$718$	−49.0150	−1.82922
$719$	−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
$722$	28.0803	1.04504
$723$	0	0
$724$	6.15548	0.228766
$725$	−2.73395	−0.101536
$726$	0	0
$727$	0.451096	0.0167302	0.00836512	−	0.999965i	$-0.497337\pi$
$727$	0.451096	0.0167302	0.00836512	+	0.999965i	$0.497337\pi$
$728$	1.14062	0.0422740
$729$	0	0
$730$	34.3380	1.27091
$731$	−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	$0.264564\pi$
$734$	51.8953	1.91549
$735$	0	0
$736$	11.1415	0.410680
$737$	2.22380	0.0819149
$738$	0	0
$739$	−15.0007	−0.551811	−0.275905	−	0.961185i	$-0.588978\pi$
$739$	−15.0007	−0.551811	−0.275905	+	0.961185i	$0.588978\pi$
$740$	−45.0151	−1.65479
$741$	0	0
$742$	−5.84429	−0.214551
$743$	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
$746$	−28.6955	−1.05062
$747$	0	0
$748$	2.71034	0.0990999
$749$	−2.70025	−0.0986651
$750$	0	0
$751$	30.4351	1.11059	0.555297	−	0.831652i	$-0.312605\pi$
$751$	30.4351	1.11059	0.555297	+	0.831652i	$0.312605\pi$
$752$	44.0919	1.60787
$753$	0	0
$754$	12.7740	0.465201
$755$	−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	$0.110832\pi$
$758$	−52.0369	−1.89007
$759$	0	0
$760$	34.1554	1.23895
$761$	−29.6807	−1.07593	−0.537963	−	0.842969i	$-0.680806\pi$
$761$	−29.6807	−1.07593	−0.537963	+	0.842969i	$0.680806\pi$
$762$	0	0
$763$	0.317253	0.0114853
$764$	−62.7330	−2.26960
$765$	0	0
$766$	−37.6430	−1.36010
$767$	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	$0.521154\pi$
$770$	−1.06900	−0.0385240
$771$	0	0
$772$	52.4923	1.88924
$773$	−24.8227	−0.892809	−0.446405	−	0.894831i	$-0.647296\pi$
$773$	−24.8227	−0.892809	−0.446405	+	0.894831i	$0.647296\pi$
$774$	0	0
$775$	−3.10028	−0.111365
$776$	31.0339	1.11405
$777$	0	0
$778$	6.56816	0.235480
$779$	26.6861	0.956128
$780$	0	0
$781$	−2.70584	−0.0968226
$782$	−20.5338	−0.734285
$783$	0	0
$784$	−33.9172	−1.21133
$785$	2.63832	0.0941657
$786$	0	0
$787$	−21.0833	−0.751538	−0.375769	−	0.926713i	$-0.622621\pi$
$787$	−21.0833	−0.751538	−0.375769	+	0.926713i	$0.622621\pi$
$788$	−83.0154	−2.95730
$789$	0	0
$790$	−35.4252	−1.26037
$791$	0.0255427	0.000908194 0
$792$	0	0
$793$	7.11778	0.252760
$794$	60.9135	2.16174
$795$	0	0
$796$	−45.6726	−1.61882
$797$	24.3975	0.864204	0.432102	−	0.901825i	$-0.357772\pi$
$797$	24.3975	0.864204	0.432102	+	0.901825i	$0.357772\pi$
$798$	0	0
$799$	−10.2563	−0.362843
$800$	−0.535398	−0.0189292
$801$	0	0
$802$	91.3723	3.22647
$803$	3.41707	0.120586
$804$	0	0
$805$	5.46490	0.192612
$806$	14.4856	0.510233
$807$	0	0
$808$	54.8553	1.92980
$809$	−9.84260	−0.346048	−0.173024	−	0.984918i	$-0.555354\pi$
$809$	−9.84260	−0.346048	−0.173024	+	0.984918i	$0.555354\pi$
$810$	0	0
$811$	−46.1782	−1.62154	−0.810769	−	0.585367i	$-0.800951\pi$
$811$	−46.1782	−1.62154	−0.810769	+	0.585367i	$0.800951\pi$
$812$	10.6595	0.374074
$813$	0	0
$814$	−6.63858	−0.232682
$815$	−6.80050	−0.238211
$816$	0	0
$817$	−31.1825	−1.09094
$818$	17.2722	0.603910
$819$	0	0
$820$	−92.4260	−3.22766
$821$	3.25038	0.113439	0.0567195	−	0.998390i	$-0.481936\pi$
$821$	3.25038	0.113439	0.0567195	+	0.998390i	$0.481936\pi$
$822$	0	0
$823$	−1.80449	−0.0629005	−0.0314503	−	0.999505i	$-0.510013\pi$
$823$	−1.80449	−0.0629005	−0.0314503	+	0.999505i	$0.510013\pi$
$824$	0.277589	0.00967028
$825$	0	0
$826$	4.52081	0.157299
$827$	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	$0.561693\pi$
$830$	−83.2296	−2.88894
$831$	0	0
$832$	−3.94435	−0.136746
$833$	7.88957	0.273357
$834$	0	0
$835$	−10.7528	−0.372117
$836$	6.56118	0.226923
$837$	0	0
$838$	−4.88394	−0.168713
$839$	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
$842$	88.6204	3.05406
$843$	0	0
$844$	41.0725	1.41378
$845$	−29.0703	−1.00005
$846$	0	0
$847$	3.48678	0.119807
$848$	−35.4990	−1.21904
$849$	0	0
$850$	0.986739	0.0338449
$851$	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	$0.542965\pi$
$854$	8.80222	0.301206
$855$	0	0
$856$	−44.0671	−1.50618
$857$	−3.23243	−0.110418	−0.0552088	−	0.998475i	$-0.517582\pi$
$857$	−3.23243	−0.110418	−0.0552088	+	0.998475i	$0.517582\pi$
$858$	0	0
$859$	−3.07827	−0.105029	−0.0525146	−	0.998620i	$-0.516724\pi$
$859$	−3.07827	−0.105029	−0.0525146	+	0.998620i	$0.516724\pi$
$860$	107.999	3.68274
$861$	0	0
$862$	−71.4901	−2.43496
$863$	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
$866$	69.5464	2.36328
$867$	0	0
$868$	12.0877	0.410285
$869$	−3.52525	−0.119586
$870$	0	0
$871$	−2.55254	−0.0864895
$872$	5.17746	0.175331
$873$	0	0
$874$	−49.7080	−1.68140
$875$	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	$0.615956\pi$
$878$	−67.0453	−2.26267
$879$	0	0
$880$	−6.49323	−0.218887
$881$	−27.3881	−0.922730	−0.461365	−	0.887210i	$-0.652640\pi$
$881$	−27.3881	−0.922730	−0.461365	+	0.887210i	$0.652640\pi$
$882$	0	0
$883$	23.3136	0.784567	0.392283	−	0.919844i	$-0.371685\pi$
$883$	23.3136	0.784567	0.392283	+	0.919844i	$0.371685\pi$
$884$	−3.11100	−0.104634
$885$	0	0
$886$	−21.5191	−0.722950
$887$	−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
$890$	66.2164	2.21958
$891$	0	0
$892$	28.5518	0.955984
$893$	−24.8285	−0.830853
$894$	0	0
$895$	42.6167	1.42452
$896$	−5.88389	−0.196567
$897$	0	0
$898$	−36.2124	−1.20842
$899$	70.1276	2.33889
$900$	0	0
$901$	8.25750	0.275097
$902$	−13.6305	−0.453846
$903$	0	0
$904$	0.416847	0.0138641
$905$	3.43029	0.114027
$906$	0	0
$907$	−26.8143	−0.890353	−0.445176	−	0.895443i	$-0.646859\pi$
$907$	−26.8143	−0.890353	−0.445176	+	0.895443i	$0.646859\pi$
$908$	−117.105	−3.88627
$909$	0	0
$910$	1.22702	0.0406754
$911$	−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
$914$	−32.6566	−1.08018
$915$	0	0
$916$	102.674	3.39246
$917$	−1.27934	−0.0422475
$918$	0	0
$919$	−27.2810	−0.899916	−0.449958	−	0.893050i	$-0.648561\pi$
$919$	−27.2810	−0.899916	−0.449958	+	0.893050i	$0.648561\pi$
$920$	89.1851	2.94035
$921$	0	0
$922$	−11.6262	−0.382888
$923$	3.10583	0.102230
$924$	0	0
$925$	−1.63085	−0.0536221
$926$	−62.4797	−2.05321
$927$	0	0
$928$	12.1106	0.397549
$929$	−31.6186	−1.03737	−0.518687	−	0.854964i	$-0.673579\pi$
$929$	−31.6186	−1.03737	−0.518687	+	0.854964i	$0.673579\pi$
$930$	0	0
$931$	19.0990	0.625946
$932$	75.3610	2.46853
$933$	0	0
$934$	−60.0415	−1.96462
$935$	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	$0.300897\pi$
$938$	−3.15660	−0.103067
$939$	0	0
$940$	85.9923	2.80476
$941$	−49.3069	−1.60736	−0.803679	−	0.595063i	$-0.797127\pi$
$941$	−49.3069	−1.60736	−0.803679	+	0.595063i	$0.797127\pi$
$942$	0	0
$943$	69.6815	2.26914
$944$	27.4600	0.893748
$945$	0	0
$946$	15.9271	0.517836
$947$	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
$950$	2.38869	0.0774995
$951$	0	0
$952$	−1.99299	−0.0645930
$953$	−46.8528	−1.51771	−0.758856	−	0.651259i	$-0.774241\pi$
$953$	−46.8528	−1.51771	−0.758856	+	0.651259i	$0.774241\pi$
$954$	0	0
$955$	−34.9596	−1.13126
$956$	−95.4253	−3.08627
$957$	0	0
$958$	−54.5991	−1.76402
$959$	−6.39085	−0.206371
$960$	0	0
$961$	48.5242	1.56530
$962$	7.61993	0.245676
$963$	0	0
$964$	124.146	3.99846
$965$	29.2527	0.941677
$966$	0	0
$967$	−9.07659	−0.291884	−0.145942	−	0.989293i	$-0.546621\pi$
$967$	−9.07659	−0.291884	−0.145942	+	0.989293i	$0.546621\pi$
$968$	56.9029	1.82893
$969$	0	0
$970$	33.3849	1.07192
$971$	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
$974$	87.8099	2.81361
$975$	0	0
$976$	53.4658	1.71140
$977$	22.5984	0.722987	0.361493	−	0.932375i	$-0.382267\pi$
$977$	22.5984	0.722987	0.361493	+	0.932375i	$0.382267\pi$
$978$	0	0
$979$	6.58938	0.210598
$980$	−66.1486	−2.11304
$981$	0	0
$982$	108.918	3.47573
$983$	−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
$986$	−22.3198	−0.710808
$987$	0	0
$988$	−7.53109	−0.239596
$989$	−81.4222	−2.58908
$990$	0	0
$991$	−3.42163	−0.108692	−0.0543458	−	0.998522i	$-0.517307\pi$
$991$	−3.42163	−0.108692	−0.0543458	+	0.998522i	$0.517307\pi$
$992$	13.7333	0.436033
$993$	0	0
$994$	3.84084	0.121824
$995$	−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	$0.132246\pi$
$998$	−16.7090	−0.528913
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1251.2.a.k.1.1		7
3.2	odd	2		139.2.a.c.1.7	✓	7
12.11	even	2		2224.2.a.o.1.4		7
15.14	odd	2		3475.2.a.e.1.1		7
21.20	even	2		6811.2.a.p.1.7		7
24.5	odd	2		8896.2.a.be.1.4		7
24.11	even	2		8896.2.a.bd.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
139.2.a.c.1.7	✓	7	3.2	odd	2
1251.2.a.k.1.1		7	1.1	even	1	trivial
2224.2.a.o.1.4		7	12.11	even	2
3475.2.a.e.1.1		7	15.14	odd	2
6811.2.a.p.1.7		7	21.20	even	2
8896.2.a.bd.1.4		7	24.11	even	2
8896.2.a.be.1.4		7	24.5	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:	\( N \)	\(=\)	\( 1251 = 3^{2} \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1251.a (trivial)

\(f(q)\)	\(=\)	\(q-2.47985 q^{2} +4.14965 q^{4} +2.31250 q^{5} -0.326651 q^{7} -5.33082 q^{8} +O(q^{10})\)	\(q-2.47985 q^{2} +4.14965 q^{4} +2.31250 q^{5} -0.326651 q^{7} -5.33082 q^{8} -5.73465 q^{10} -0.570671 q^{11} +0.655030 q^{13} +0.810045 q^{14} +4.92032 q^{16} -1.14453 q^{17} -2.77067 q^{19} +9.59608 q^{20} +1.41518 q^{22} -7.23464 q^{23} +0.347657 q^{25} -1.62438 q^{26} -1.35549 q^{28} -7.86393 q^{29} -8.91763 q^{31} -1.54002 q^{32} +2.83826 q^{34} -0.755380 q^{35} -4.69099 q^{37} +6.87084 q^{38} -12.3275 q^{40} -9.63165 q^{41} +11.2545 q^{43} -2.36809 q^{44} +17.9408 q^{46} +8.96119 q^{47} -6.89330 q^{49} -0.862137 q^{50} +2.71815 q^{52} -7.21477 q^{53} -1.31968 q^{55} +1.74132 q^{56} +19.5014 q^{58} +5.58094 q^{59} +10.8663 q^{61} +22.1144 q^{62} -6.02163 q^{64} +1.51476 q^{65} -3.89682 q^{67} -4.74939 q^{68} +1.87323 q^{70} +4.74151 q^{71} -5.98781 q^{73} +11.6329 q^{74} -11.4973 q^{76} +0.186410 q^{77} +6.17739 q^{79} +11.3782 q^{80} +23.8850 q^{82} +14.5135 q^{83} -2.64672 q^{85} -27.9095 q^{86} +3.04214 q^{88} -11.5467 q^{89} -0.213966 q^{91} -30.0212 q^{92} -22.2224 q^{94} -6.40717 q^{95} -5.82160 q^{97} +17.0943 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{2} + 9 q^{4} - 11 q^{5} - 5 q^{7} - 6 q^{8}+O(q^{10}) \) 7 * q - q^2 + 9 * q^4 - 11 * q^5 - 5 * q^7 - 6 * q^8	\( 7 q - q^{2} + 9 q^{4} - 11 q^{5} - 5 q^{7} - 6 q^{8} - 4 q^{10} - 2 q^{11} + 6 q^{13} - 7 q^{14} + 5 q^{16} - 5 q^{17} - 10 q^{19} - 12 q^{20} - 18 q^{22} + q^{23} + 14 q^{25} + 8 q^{26} - 28 q^{28} - 30 q^{29} - 20 q^{31} + 12 q^{32} - 17 q^{34} + 7 q^{35} + 6 q^{37} - 6 q^{38} - 22 q^{40} - 19 q^{41} - 12 q^{43} - 25 q^{44} + 22 q^{46} + 3 q^{47} - 8 q^{49} - 12 q^{50} - 8 q^{52} - 38 q^{53} + 7 q^{55} - 21 q^{56} - 21 q^{58} + 14 q^{59} + 4 q^{61} + q^{62} - 16 q^{64} - 10 q^{65} + 9 q^{67} + 25 q^{68} + 20 q^{70} - 24 q^{71} - 5 q^{73} - 9 q^{74} + 3 q^{76} + 13 q^{77} + 8 q^{79} - 11 q^{80} + 56 q^{82} + 9 q^{83} - 22 q^{85} - 39 q^{86} - 29 q^{88} - 10 q^{89} + 7 q^{91} - 29 q^{92} - 36 q^{94} + 21 q^{95} - 5 q^{97} + 49 q^{98}+O(q^{100}) \) 7 * q - q^2 + 9 * q^4 - 11 * q^5 - 5 * q^7 - 6 * q^8 - 4 * q^10 - 2 * q^11 + 6 * q^13 - 7 * q^14 + 5 * q^16 - 5 * q^17 - 10 * q^19 - 12 * q^20 - 18 * q^22 + q^23 + 14 * q^25 + 8 * q^26 - 28 * q^28 - 30 * q^29 - 20 * q^31 + 12 * q^32 - 17 * q^34 + 7 * q^35 + 6 * q^37 - 6 * q^38 - 22 * q^40 - 19 * q^41 - 12 * q^43 - 25 * q^44 + 22 * q^46 + 3 * q^47 - 8 * q^49 - 12 * q^50 - 8 * q^52 - 38 * q^53 + 7 * q^55 - 21 * q^56 - 21 * q^58 + 14 * q^59 + 4 * q^61 + q^62 - 16 * q^64 - 10 * q^65 + 9 * q^67 + 25 * q^68 + 20 * q^70 - 24 * q^71 - 5 * q^73 - 9 * q^74 + 3 * q^76 + 13 * q^77 + 8 * q^79 - 11 * q^80 + 56 * q^82 + 9 * q^83 - 22 * q^85 - 39 * q^86 - 29 * q^88 - 10 * q^89 + 7 * q^91 - 29 * q^92 - 36 * q^94 + 21 * q^95 - 5 * q^97 + 49 * q^98

Introduction

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