LMFDB - Embedded newform 1053.2.b.j.649.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1053,2,Mod(649,1053)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1053, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("1053.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1053 = 3^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1053.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$8.40824733284$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 16x^{8} + 91x^{6} + 222x^{4} + 228x^{2} + 81 $ x^10 + 16x^8 + 91x^6 + 222x^4 + 228x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	no (minimal twist has level 117)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.8
Root			$1.63641i$ of defining polynomial
Character	$\chi$	$=$	1053.649
Dual form			1053.2.b.j.649.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.63641i q^{2} -0.677835 q^{4} -1.09738i q^{5} -3.20939i q^{7} +2.16360i q^{8} +1.79576 q^{10} -1.75848i q^{11} +(-3.57405 - 0.475605i) q^{13} +5.25188 q^{14} -4.89621 q^{16} +1.47360 q^{17} -3.61452i q^{19} +0.743842i q^{20} +2.87759 q^{22} +4.69197 q^{23} +3.79576 q^{25} +(0.778285 - 5.84860i) q^{26} +2.17544i q^{28} -1.91817 q^{29} -6.56129i q^{31} -3.68500i q^{32} +2.41141i q^{34} -3.52192 q^{35} -11.6237i q^{37} +5.91483 q^{38} +2.37429 q^{40} +5.40415i q^{41} -1.77828 q^{43} +1.19196i q^{44} +7.67798i q^{46} -10.2774i q^{47} -3.30020 q^{49} +6.21142i q^{50} +(2.42261 + 0.322382i) q^{52} +11.7738 q^{53} -1.92972 q^{55} +6.94385 q^{56} -3.13891i q^{58} -5.52622i q^{59} +1.97030 q^{61} +10.7370 q^{62} -3.76225 q^{64} +(-0.521919 + 3.92208i) q^{65} +8.26112i q^{67} -0.998855 q^{68} -5.76330i q^{70} -5.84860i q^{71} -1.24694i q^{73} +19.0212 q^{74} +2.45005i q^{76} -5.64365 q^{77} +0.485823 q^{79} +5.37300i q^{80} -8.84340 q^{82} +15.5135i q^{83} -1.61709i q^{85} -2.91000i q^{86} +3.80465 q^{88} +13.4245i q^{89} +(-1.52640 + 11.4705i) q^{91} -3.18038 q^{92} +16.8181 q^{94} -3.96649 q^{95} -5.95544i q^{97} -5.40048i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 12 q^{4} - 8 q^{10} + 4 q^{13} + 18 q^{14} - 4 q^{16} - 6 q^{17} + 10 q^{22} - 24 q^{23} + 12 q^{25} - 6 q^{26} - 12 q^{29} - 6 q^{35} - 12 q^{38} + 8 q^{40} - 4 q^{43} + 10 q^{49} + 54 q^{53} + 10 q^{55}+ \cdots - 24 q^{95}+O(q^{100}) $ 10 * q - 12 * q^4 - 8 * q^10 + 4 * q^13 + 18 * q^14 - 4 * q^16 - 6 * q^17 + 10 * q^22 - 24 * q^23 + 12 * q^25 - 6 * q^26 - 12 * q^29 - 6 * q^35 - 12 * q^38 + 8 * q^40 - 4 * q^43 + 10 * q^49 + 54 * q^53 + 10 * q^55 - 36 * q^56 + 2 * q^61 - 36 * q^62 + 4 * q^64 + 24 * q^65 - 24 * q^68 + 42 * q^74 + 6 * q^77 + 14 * q^79 - 2 * q^82 - 22 * q^88 - 36 * q^91 + 84 * q^92 - 20 * q^94 - 24 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1053\mathbb{Z}\right)^\times$.

$n$	$326$	$730$
$\chi(n)$	$1$	$-1$

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 $

$2$			1.63641i			1.15712i	0.815641	+	0.578558i	$0.196384\pi$
$2$			1.63641i			1.15712i	−0.815641	+	0.578558i	$0.803616\pi$
$3$		0			0
$3$		0			0
$4$	−0.677835			−0.338918
$5$		−	1.09738i		−	0.490763i	−0.969427	−	0.245381i	$-0.921087\pi$
$5$		−	1.09738i		−	0.490763i	0.969427	−	0.245381i	$-0.0789132\pi$
$6$		0			0
$7$		−	3.20939i		−	1.21304i	−0.795070	−	0.606518i	$-0.792566\pi$
$7$		−	3.20939i		−	1.21304i	0.795070	−	0.606518i	$-0.207434\pi$
$8$			2.16360i			0.764949i
$9$		0			0
$10$	1.79576			0.567869
$11$		−	1.75848i		−	0.530201i	−0.964221	−	0.265101i	$-0.914595\pi$
$11$		−	1.75848i		−	0.530201i	0.964221	−	0.265101i	$-0.0854051\pi$
$12$		0			0
$13$	−3.57405	−	0.475605i	−0.991262	−	0.131909i
$13$	−3.57405	−	0.475605i	−0.991262	−	0.131909i
$14$	5.25188			1.40362
$15$		0			0
$16$	−4.89621			−1.22405
$17$	1.47360			0.357399			0.178700	−	0.983904i	$-0.442811\pi$
$17$	1.47360			0.357399			0.178700	+	0.983904i	$0.442811\pi$
$18$		0			0
$19$		−	3.61452i		−	0.829227i	−0.909998	−	0.414614i	$-0.863917\pi$
$19$		−	3.61452i		−	0.829227i	0.909998	−	0.414614i	$-0.136083\pi$
$20$			0.743842i			0.166328i
$21$		0			0
$22$	2.87759			0.613504
$23$	4.69197			0.978343			0.489172	−	0.872187i	$-0.337299\pi$
$23$	4.69197			0.978343			0.489172	+	0.872187i	$0.337299\pi$
$24$		0			0
$25$	3.79576			0.759152
$26$	0.778285	−	5.84860i	0.152634	−	1.14700i
$27$		0			0
$28$			2.17544i			0.411120i
$29$	−1.91817			−0.356195			−0.178098	−	0.984013i	$-0.556994\pi$
$29$	−1.91817			−0.356195			−0.178098	+	0.984013i	$0.556994\pi$
$30$		0			0
$31$		−	6.56129i		−	1.17844i	−0.807972	−	0.589221i	$-0.799435\pi$
$31$		−	6.56129i		−	1.17844i	0.807972	−	0.589221i	$-0.200565\pi$
$32$		−	3.68500i		−	0.651422i
$33$		0			0
$34$			2.41141i			0.413553i
$35$	−3.52192			−0.595313
$36$		0			0
$37$		−	11.6237i		−	1.91093i	−0.295103	−	0.955465i	$-0.595354\pi$
$37$		−	11.6237i		−	1.91093i	0.295103	−	0.955465i	$-0.404646\pi$
$38$	5.91483			0.959512
$39$		0			0
$40$	2.37429			0.375408
$41$			5.40415i			0.843986i	0.906599	+	0.421993i	$0.138669\pi$
$41$			5.40415i			0.843986i	−0.906599	+	0.421993i	$0.861331\pi$
$42$		0			0
$43$	−1.77828			−0.271186			−0.135593	−	0.990765i	$-0.543294\pi$
$43$	−1.77828			−0.271186			−0.135593	+	0.990765i	$0.543294\pi$
$44$			1.19196i			0.179695i
$45$		0			0
$46$			7.67798i			1.13206i
$47$		−	10.2774i		−	1.49912i	−0.661937	−	0.749559i	$-0.730265\pi$
$47$		−	10.2774i		−	1.49912i	0.661937	−	0.749559i	$-0.269735\pi$
$48$		0			0
$49$	−3.30020			−0.471458
$50$			6.21142i			0.878427i
$51$		0			0
$52$	2.42261	+	0.322382i	0.335956	+	0.0447063i
$53$	11.7738			1.61726			0.808628	−	0.588320i	$-0.200211\pi$
$53$	11.7738			1.61726			0.808628	+	0.588320i	$0.200211\pi$
$54$		0			0
$55$	−1.92972			−0.260203
$56$	6.94385			0.927911
$57$		0			0
$58$		−	3.13891i		−	0.412159i
$59$		−	5.52622i		−	0.719452i	−0.933058	−	0.359726i	$-0.882870\pi$
$59$		−	5.52622i		−	0.719452i	0.933058	−	0.359726i	$-0.117130\pi$
$60$		0			0
$61$	1.97030			0.252271			0.126135	−	0.992013i	$-0.459743\pi$
$61$	1.97030			0.252271			0.126135	+	0.992013i	$0.459743\pi$
$62$	10.7370			1.36359
$63$		0			0
$64$	−3.76225			−0.470282
$65$	−0.521919	+	3.92208i	−0.0647361	+	0.486474i
$66$		0			0
$67$			8.26112i			1.00926i	0.863337	+	0.504628i	$0.168370\pi$
$67$			8.26112i			1.00926i	−0.863337	+	0.504628i	$0.831630\pi$
$68$	−0.998855			−0.121129
$69$		0			0
$70$		−	5.76330i		−	0.688846i
$71$		−	5.84860i		−	0.694101i	−0.937846	−	0.347051i	$-0.887183\pi$
$71$		−	5.84860i		−	0.694101i	0.937846	−	0.347051i	$-0.112817\pi$
$72$		0			0
$73$		−	1.24694i		−	0.145943i	−0.997334	−	0.0729714i	$-0.976752\pi$
$73$		−	1.24694i		−	0.145943i	0.997334	−	0.0729714i	$-0.0232482\pi$
$74$	19.0212			2.21117
$75$		0			0
$76$			2.45005i			0.281040i
$77$	−5.64365			−0.643153
$78$		0			0
$79$	0.485823			0.0546594			0.0273297	−	0.999626i	$-0.491300\pi$
$79$	0.485823			0.0546594			0.0273297	+	0.999626i	$0.491300\pi$
$80$			5.37300i			0.600719i
$81$		0			0
$82$	−8.84340			−0.976590
$83$			15.5135i			1.70283i	0.524490	+	0.851417i	$0.324256\pi$
$83$			15.5135i			1.70283i	−0.524490	+	0.851417i	$0.675744\pi$
$84$		0			0
$85$		−	1.61709i		−	0.175398i
$86$		−	2.91000i		−	0.313794i
$87$		0			0
$88$	3.80465			0.405577
$89$			13.4245i			1.42300i	0.702688	+	0.711498i	$0.251983\pi$
$89$			13.4245i			1.42300i	−0.702688	+	0.711498i	$0.748017\pi$
$90$		0			0
$91$	−1.52640	+	11.4705i	−0.160011	+	1.20244i
$92$	−3.18038			−0.331578
$93$		0			0
$94$	16.8181			1.73465
$95$	−3.96649			−0.406954
$96$		0			0
$97$		−	5.95544i		−	0.604684i	−0.953200	−	0.302342i	$-0.902232\pi$
$97$		−	5.95544i		−	0.604684i	0.953200	−	0.302342i	$-0.0977684\pi$
$98$		−	5.40048i		−	0.545531i
$99$		0			0
$100$	−2.57290			−0.257290
$101$	−15.8858			−1.58070			−0.790348	−	0.612658i	$-0.790100\pi$
$101$	−15.8858			−1.58070			−0.790348	+	0.612658i	$0.790100\pi$
$102$		0			0
$103$	−6.25188			−0.616016			−0.308008	−	0.951384i	$-0.599662\pi$
$103$	−6.25188			−0.616016			−0.308008	+	0.951384i	$0.599662\pi$
$104$	1.02902	−	7.73281i	0.100904	−	0.758265i
$105$		0			0
$106$			19.2668i			1.87135i
$107$	−10.6339			−1.02802			−0.514010	−	0.857784i	$-0.671840\pi$
$107$	−10.6339			−1.02802			−0.514010	+	0.857784i	$0.671840\pi$
$108$		0			0
$109$			16.0203i			1.53447i	0.641368	+	0.767234i	$0.278367\pi$
$109$			16.0203i			1.53447i	−0.641368	+	0.767234i	$0.721633\pi$
$110$		−	3.15781i		−	0.301085i
$111$		0			0
$112$			15.7139i			1.48482i
$113$	−2.97736			−0.280086			−0.140043	−	0.990145i	$-0.544724\pi$
$113$	−2.97736			−0.280086			−0.140043	+	0.990145i	$0.544724\pi$
$114$		0			0
$115$		−	5.14887i		−	0.480134i
$116$	1.30020			0.120721
$117$		0			0
$118$	9.04316			0.832490
$119$		−	4.72935i		−	0.433539i
$120$		0			0
$121$	7.90776			0.718887
$122$			3.22421i			0.291906i
$123$		0			0
$124$			4.44747i			0.399395i
$125$		−	9.65228i		−	0.863326i
$126$		0			0
$127$	6.12947			0.543903			0.271951	−	0.962311i	$-0.412331\pi$
$127$	6.12947			0.543903			0.271951	+	0.962311i	$0.412331\pi$
$128$		−	13.5266i		−	1.19559i
$129$		0			0
$130$	−6.41813	−	0.854073i	−0.562907	−	0.0749072i
$131$	8.42527			0.736119			0.368060	−	0.929802i	$-0.380022\pi$
$131$	8.42527			0.736119			0.368060	+	0.929802i	$0.380022\pi$
$132$		0			0
$133$	−11.6004			−1.00588
$134$	−13.5186			−1.16783
$135$		0			0
$136$			3.18828i			0.273392i
$137$			7.74397i			0.661612i	0.943699	+	0.330806i	$0.107321\pi$
$137$			7.74397i			0.661612i	−0.943699	+	0.330806i	$0.892679\pi$
$138$		0			0
$139$	12.5959			1.06837			0.534186	−	0.845367i	$-0.320618\pi$
$139$	12.5959			1.06837			0.534186	+	0.845367i	$0.320618\pi$
$140$	2.38728			0.201762
$141$		0			0
$142$	9.57070			0.803156
$143$	−0.836341	+	6.28488i	−0.0699384	+	0.525568i
$144$		0			0
$145$			2.10496i			0.174807i
$146$	2.04050			0.168873
$147$		0			0
$148$			7.87898i			0.647648i
$149$			7.81445i			0.640184i	0.947386	+	0.320092i	$0.103714\pi$
$149$			7.81445i			0.640184i	−0.947386	+	0.320092i	$0.896286\pi$
$150$		0			0
$151$			17.6763i			1.43848i	0.694762	+	0.719239i	$0.255509\pi$
$151$			17.6763i			1.43848i	−0.694762	+	0.719239i	$0.744491\pi$
$152$	7.82038			0.634317
$153$		0			0
$154$		−	9.23532i		−	0.744203i
$155$	−7.20022			−0.578335
$156$		0			0
$157$	−17.7995			−1.42055			−0.710277	−	0.703923i	$-0.751430\pi$
$157$	−17.7995			−1.42055			−0.710277	+	0.703923i	$0.751430\pi$
$158$			0.795006i			0.0632473i
$159$		0			0
$160$	−4.04384			−0.319693
$161$		−	15.0584i		−	1.18677i
$162$		0			0
$163$		−	16.7251i		−	1.31001i	−0.755624	−	0.655006i	$-0.772666\pi$
$163$		−	16.7251i		−	1.31001i	0.755624	−	0.655006i	$-0.227334\pi$
$164$		−	3.66312i		−	0.286042i
$165$		0			0
$166$	−25.3865			−1.97038
$167$		−	5.33479i		−	0.412818i	−0.978466	−	0.206409i	$-0.933822\pi$
$167$		−	5.33479i		−	0.412818i	0.978466	−	0.206409i	$-0.0661778\pi$
$168$		0			0
$169$	12.5476	+	3.39967i	0.965200	+	0.261513i
$170$	2.64622			0.202956
$171$		0			0
$172$	1.20538			0.0919097
$173$	−12.3066			−0.935653			−0.467827	−	0.883820i	$-0.654963\pi$
$173$	−12.3066			−0.935653			−0.467827	+	0.883820i	$0.654963\pi$
$174$		0			0
$175$		−	12.1821i		−	0.920879i
$176$			8.60988i			0.648994i
$177$		0			0
$178$	−21.9680			−1.64657
$179$	−18.8439			−1.40846			−0.704228	−	0.709974i	$-0.748707\pi$
$179$	−18.8439			−1.40846			−0.704228	+	0.709974i	$0.748707\pi$
$180$		0			0
$181$	10.8768			0.808469			0.404235	−	0.914655i	$-0.367538\pi$
$181$	10.8768			0.808469			0.404235	+	0.914655i	$0.367538\pi$
$182$	−18.7705	−	2.49782i	−1.39136	−	0.185151i
$183$		0			0
$184$			10.1516i			0.748383i
$185$	−12.7556			−0.937813
$186$		0			0
$187$		−	2.59129i		−	0.189494i
$188$			6.96641i			0.508078i
$189$		0			0
$190$		−	6.49081i		−	0.470893i
$191$	18.5971			1.34564			0.672818	−	0.739808i	$-0.265084\pi$
$191$	18.5971			1.34564			0.672818	+	0.739808i	$0.265084\pi$
$192$		0			0
$193$		−	12.9574i		−	0.932692i	−0.884602	−	0.466346i	$-0.845570\pi$
$193$		−	12.9574i		−	0.932692i	0.884602	−	0.466346i	$-0.154430\pi$
$194$	9.74554			0.699689
$195$		0			0
$196$	2.23699			0.159785
$197$			14.0963i			1.00432i	0.864774	+	0.502162i	$0.167462\pi$
$197$			14.0963i			1.00432i	−0.864774	+	0.502162i	$0.832538\pi$
$198$		0			0
$199$	16.6913			1.18321			0.591607	−	0.806226i	$-0.298494\pi$
$199$	16.6913			1.18321			0.591607	+	0.806226i	$0.298494\pi$
$200$			8.21252i			0.580713i
$201$		0			0
$202$		−	25.9957i		−	1.82905i
$203$			6.15616i			0.432078i
$204$		0			0
$205$	5.93040			0.414197
$206$		−	10.2306i		−	0.712802i
$207$		0			0
$208$	17.4993	+	2.32866i	1.21336	+	0.161464i
$209$	−6.35605			−0.439657
$210$		0			0
$211$	5.48134			0.377351			0.188675	−	0.982040i	$-0.439581\pi$
$211$	5.48134			0.377351			0.188675	+	0.982040i	$0.439581\pi$
$212$	−7.98070			−0.548117
$213$		0			0
$214$		−	17.4014i		−	1.18954i
$215$			1.95145i			0.133088i
$216$		0			0
$217$	−21.0578			−1.42949
$218$	−26.2158			−1.77556
$219$		0			0
$220$	1.30803			0.0881873
$221$	−5.26670	−	0.700850i	−0.354276	−	0.0471443i
$222$		0			0
$223$		−	9.17528i		−	0.614422i	−0.951641	−	0.307211i	$-0.900604\pi$
$223$		−	9.17528i		−	0.614422i	0.951641	−	0.307211i	$-0.0993958\pi$
$224$	−11.8266			−0.790198
$225$		0			0
$226$		−	4.87217i		−	0.324092i
$227$			3.44493i			0.228648i	0.993443	+	0.114324i	$0.0364703\pi$
$227$			3.44493i			0.228648i	−0.993443	+	0.114324i	$0.963530\pi$
$228$		0			0
$229$			12.4818i			0.824818i	0.910999	+	0.412409i	$0.135313\pi$
$229$			12.4818i			0.824818i	−0.910999	+	0.412409i	$0.864687\pi$
$230$	8.42565			0.555571
$231$		0			0
$232$		−	4.15016i		−	0.272471i
$233$	−5.08821			−0.333340			−0.166670	−	0.986013i	$-0.553301\pi$
$233$	−5.08821			−0.333340			−0.166670	+	0.986013i	$0.553301\pi$
$234$		0			0
$235$	−11.2782			−0.735711
$236$			3.74587i			0.243835i
$237$		0			0
$238$	7.73915			0.501655
$239$		−	1.51801i		−	0.0981917i	−0.998794	−	0.0490958i	$-0.984366\pi$
$239$		−	1.51801i		−	0.0981917i	0.998794	−	0.0490958i	$-0.0156340\pi$
$240$		0			0
$241$		−	19.1028i		−	1.23052i	−0.788324	−	0.615260i	$-0.789051\pi$
$241$		−	19.1028i		−	1.23052i	0.788324	−	0.615260i	$-0.210949\pi$
$242$			12.9403i			0.831836i
$243$		0			0
$244$	−1.33554			−0.0854990
$245$			3.62157i			0.231374i
$246$		0			0
$247$	−1.71908	+	12.9184i	−0.109383	+	0.821981i
$248$	14.1960			0.901448
$249$		0			0
$250$	15.7951			0.998968
$251$	−7.96535			−0.502768			−0.251384	−	0.967887i	$-0.580886\pi$
$251$	−7.96535			−0.502768			−0.251384	+	0.967887i	$0.580886\pi$
$252$		0			0
$253$		−	8.25073i		−	0.518719i
$254$			10.0303i			0.629358i
$255$		0			0
$256$	14.6105			0.913157
$257$	−21.0656			−1.31403			−0.657017	−	0.753875i	$-0.728182\pi$
$257$	−21.0656			−1.31403			−0.657017	+	0.753875i	$0.728182\pi$
$258$		0			0
$259$	−37.3051			−2.31803
$260$	0.353775	−	2.65852i	0.0219402	−	0.164875i
$261$		0			0
$262$			13.7872i			0.851776i
$263$	24.1049			1.48637			0.743187	−	0.669084i	$-0.233313\pi$
$263$	24.1049			1.48637			0.743187	+	0.669084i	$0.233313\pi$
$264$		0			0
$265$		−	12.9203i		−	0.793689i
$266$		−	18.9830i		−	1.16392i
$267$		0			0
$268$		−	5.59968i		−	0.342055i
$269$	−21.7780			−1.32783			−0.663913	−	0.747809i	$-0.731106\pi$
$269$	−21.7780			−1.32783			−0.663913	+	0.747809i	$0.731106\pi$
$270$		0			0
$271$		−	5.09950i		−	0.309773i	−0.987932	−	0.154886i	$-0.950499\pi$
$271$		−	5.09950i		−	0.309773i	0.987932	−	0.154886i	$-0.0495012\pi$
$272$	−7.21503			−0.437476
$273$		0			0
$274$	−12.6723			−0.765562
$275$		−	6.67476i		−	0.402503i
$276$		0			0
$277$	12.2776			0.737687			0.368844	−	0.929491i	$-0.379754\pi$
$277$	12.2776			0.737687			0.368844	+	0.929491i	$0.379754\pi$
$278$			20.6121i			1.23623i
$279$		0			0
$280$		−	7.62003i		−	0.455384i
$281$			7.31219i			0.436208i	0.975925	+	0.218104i	$0.0699873\pi$
$281$			7.31219i			0.436208i	−0.975925	+	0.218104i	$0.930013\pi$
$282$		0			0
$283$	12.7977			0.760742			0.380371	−	0.924834i	$-0.375796\pi$
$283$	12.7977			0.760742			0.380371	+	0.924834i	$0.375796\pi$
$284$			3.96439i			0.235243i
$285$		0			0
$286$	−10.2846	−	1.36860i	−0.608143	−	0.0809268i
$287$	17.3440			1.02379
$288$		0			0
$289$	−14.8285			−0.872266
$290$	−3.44457			−0.202272
$291$		0			0
$292$			0.845217i			0.0494626i
$293$		−	6.57424i		−	0.384071i	−0.981388	−	0.192036i	$-0.938491\pi$
$293$		−	6.57424i		−	0.384071i	0.981388	−	0.192036i	$-0.0615089\pi$
$294$		0			0
$295$	−6.06435			−0.353080
$296$	25.1492			1.46176
$297$		0			0
$298$	−12.7876			−0.740768
$299$	−16.7693	−	2.23153i	−0.969794	−	0.129052i
$300$		0			0
$301$			5.70722i			0.328958i
$302$	−28.9257			−1.66449
$303$		0			0
$304$			17.6974i			1.01502i
$305$		−	2.16216i		−	0.123805i
$306$		0			0
$307$			24.3307i			1.38863i	0.719672	+	0.694315i	$0.244292\pi$
$307$			24.3307i			1.38863i	−0.719672	+	0.694315i	$0.755708\pi$
$308$	3.82546			0.217976
$309$		0			0
$310$		−	11.7825i		−	0.669201i
$311$	11.1667			0.633206			0.316603	−	0.948558i	$-0.397458\pi$
$311$	11.1667			0.633206			0.316603	+	0.948558i	$0.397458\pi$
$312$		0			0
$313$	17.8028			1.00627			0.503136	−	0.864208i	$-0.332180\pi$
$313$	17.8028			1.00627			0.503136	+	0.864208i	$0.332180\pi$
$314$		−	29.1272i		−	1.64374i
$315$		0			0
$316$	−0.329308			−0.0185250
$317$		−	30.8623i		−	1.73340i	−0.498829	−	0.866700i	$-0.666236\pi$
$317$		−	30.8623i		−	1.73340i	0.498829	−	0.866700i	$-0.333764\pi$
$318$		0			0
$319$			3.37306i			0.188855i
$320$			4.12862i			0.230797i
$321$		0			0
$322$	24.6417			1.37323
$323$		−	5.32634i		−	0.296365i
$324$		0			0
$325$	−13.5662	−	1.80528i	−0.752518	−	0.100139i
$326$	27.3691			1.51584
$327$		0			0
$328$	−11.6924			−0.645607
$329$	−32.9843			−1.81849
$330$		0			0
$331$			17.4048i			0.956657i	0.878181	+	0.478328i	$0.158757\pi$
$331$			17.4048i			0.956657i	−0.878181	+	0.478328i	$0.841243\pi$
$332$		−	10.5156i		−	0.577120i
$333$		0			0
$334$	8.72989			0.477678
$335$	9.06558			0.495305
$336$		0			0
$337$	8.75830			0.477095			0.238548	−	0.971131i	$-0.423329\pi$
$337$	8.75830			0.477095			0.238548	+	0.971131i	$0.423329\pi$
$338$	−5.56325	+	20.5330i	−0.302601	+	1.11685i
$339$		0			0
$340$			1.09612i			0.0594456i
$341$	−11.5379			−0.624811
$342$		0			0
$343$		−	11.8741i		−	0.641141i
$344$		−	3.84750i		−	0.207443i
$345$		0			0
$346$		−	20.1386i		−	1.08266i
$347$	−7.27756			−0.390680			−0.195340	−	0.980736i	$-0.562581\pi$
$347$	−7.27756			−0.390680			−0.195340	+	0.980736i	$0.562581\pi$
$348$		0			0
$349$		−	12.5256i		−	0.670479i	−0.942133	−	0.335240i	$-0.891183\pi$
$349$		−	12.5256i		−	0.670479i	0.942133	−	0.335240i	$-0.108817\pi$
$350$	19.9349			1.06556
$351$		0			0
$352$	−6.47999			−0.345385
$353$			1.64092i			0.0873371i	0.999046	+	0.0436686i	$0.0139046\pi$
$353$			1.64092i			0.0873371i	−0.999046	+	0.0436686i	$0.986095\pi$
$354$		0			0
$355$	−6.41813			−0.340639
$356$		−	9.09962i		−	0.482279i
$357$		0			0
$358$		−	30.8363i		−	1.62975i
$359$			7.00788i			0.369862i	0.982752	+	0.184931i	$0.0592061\pi$
$359$			7.00788i			0.369862i	−0.982752	+	0.184931i	$0.940794\pi$
$360$		0			0
$361$	5.93526			0.312382
$362$			17.7990i			0.935493i
$363$		0			0
$364$	1.03465	−	7.77512i	0.0542304	−	0.407527i
$365$	−1.36836			−0.0716233
$366$		0			0
$367$	−1.19268			−0.0622574			−0.0311287	−	0.999515i	$-0.509910\pi$
$367$	−1.19268			−0.0622574			−0.0311287	+	0.999515i	$0.509910\pi$
$368$	−22.9729			−1.19754
$369$		0			0
$370$		−	20.8735i		−	1.08516i
$371$		−	37.7868i		−	1.96179i
$372$		0			0
$373$	17.3040			0.895968			0.447984	−	0.894042i	$-0.352142\pi$
$373$	17.3040			0.895968			0.447984	+	0.894042i	$0.352142\pi$
$374$	4.24040			0.219266
$375$		0			0
$376$	22.2363			1.14675
$377$	6.85563	+	0.912292i	0.353083	+	0.0469854i
$378$		0			0
$379$		−	1.54294i		−	0.0792554i	−0.999215	−	0.0396277i	$-0.987383\pi$
$379$		−	1.54294i		−	0.0792554i	0.999215	−	0.0396277i	$-0.0126172\pi$
$380$	2.68863			0.137924
$381$		0			0
$382$			30.4324i			1.55706i
$383$			23.6612i			1.20903i	0.796593	+	0.604515i	$0.206633\pi$
$383$			23.6612i			1.20903i	−0.796593	+	0.604515i	$0.793367\pi$
$384$		0			0
$385$			6.19322i			0.315636i
$386$	21.2036			1.07923
$387$		0			0
$388$			4.03681i			0.204938i
$389$	−5.16671			−0.261963			−0.130981	−	0.991385i	$-0.541813\pi$
$389$	−5.16671			−0.261963			−0.130981	+	0.991385i	$0.541813\pi$
$390$		0			0
$391$	6.91407			0.349659
$392$		−	7.14033i		−	0.360641i
$393$		0			0
$394$	−23.0674			−1.16212
$395$		−	0.533132i		−	0.0268248i
$396$		0			0
$397$		−	14.0455i		−	0.704926i	−0.935826	−	0.352463i	$-0.885344\pi$
$397$		−	14.0455i		−	0.704926i	0.935826	−	0.352463i	$-0.114656\pi$
$398$			27.3138i			1.36912i
$399$		0			0
$400$	−18.5848			−0.929242
$401$			14.9882i			0.748474i	0.927333	+	0.374237i	$0.122095\pi$
$401$			14.9882i			0.748474i	−0.927333	+	0.374237i	$0.877905\pi$
$402$		0			0
$403$	−3.12058	+	23.4503i	−0.155447	+	1.16814i
$404$	10.7680			0.535726
$405$		0			0
$406$	−10.0740			−0.499964
$407$	−20.4401			−1.01318
$408$		0			0
$409$			0.475605i			0.0235172i	0.999931	+	0.0117586i	$0.00374296\pi$
$409$			0.475605i			0.0235172i	−0.999931	+	0.0117586i	$0.996257\pi$
$410$			9.70456i			0.479274i
$411$		0			0
$412$	4.23775			0.208779
$413$	−17.7358			−0.872722
$414$		0			0
$415$	17.0242			0.835687
$416$	−1.75260	+	13.1704i	−0.0859285	+	0.645730i
$417$		0			0
$418$		−	10.4011i		−	0.508734i
$419$	12.9182			0.631093			0.315547	−	0.948910i	$-0.397812\pi$
$419$	12.9182			0.631093			0.315547	+	0.948910i	$0.397812\pi$
$420$		0			0
$421$			31.6159i			1.54087i	0.637521	+	0.770433i	$0.279960\pi$
$421$			31.6159i			1.54087i	−0.637521	+	0.770433i	$0.720040\pi$
$422$			8.96971i			0.436639i
$423$		0			0
$424$			25.4738i			1.23712i
$425$	5.59342			0.271321
$426$		0			0
$427$		−	6.32346i		−	0.306014i
$428$	7.20804			0.348414
$429$		0			0
$430$	−3.19337			−0.153998
$431$		−	12.5560i		−	0.604799i	−0.953181	−	0.302400i	$-0.902212\pi$
$431$		−	12.5560i		−	0.604799i	0.953181	−	0.302400i	$-0.0977877\pi$
$432$		0			0
$433$	−27.9766			−1.34447			−0.672235	−	0.740338i	$-0.734665\pi$
$433$	−27.9766			−1.34447			−0.672235	+	0.740338i	$0.734665\pi$
$434$		−	34.4591i		−	1.65409i
$435$		0			0
$436$		−	10.8591i		−	0.520058i
$437$		−	16.9592i		−	0.811269i
$438$		0			0
$439$	−11.5356			−0.550564			−0.275282	−	0.961364i	$-0.588771\pi$
$439$	−11.5356			−0.550564			−0.275282	+	0.961364i	$0.588771\pi$
$440$		−	4.17514i		−	0.199042i
$441$		0			0
$442$	1.14688	−	8.61847i	0.0545514	−	0.409939i
$443$	26.2645			1.24786			0.623932	−	0.781479i	$-0.285534\pi$
$443$	26.2645			1.24786			0.623932	+	0.781479i	$0.285534\pi$
$444$		0			0
$445$	14.7318			0.698354
$446$	15.0145			0.710958
$447$		0			0
$448$			12.0746i			0.570469i
$449$			29.9575i			1.41378i	0.707323	+	0.706891i	$0.249903\pi$
$449$			29.9575i			1.41378i	−0.707323	+	0.706891i	$0.750097\pi$
$450$		0			0
$451$	9.50308			0.447483
$452$	2.01816			0.0949262
$453$		0			0
$454$	−5.63732			−0.264573
$455$	12.5875	+	1.67504i	0.590111	+	0.0785272i
$456$		0			0
$457$			11.4601i			0.536082i	0.963408	+	0.268041i	$0.0863762\pi$
$457$			11.4601i			0.536082i	−0.963408	+	0.268041i	$0.913624\pi$
$458$	−20.4253			−0.954410
$459$		0			0
$460$			3.49008i			0.162726i
$461$			28.5125i			1.32796i	0.747750	+	0.663981i	$0.231134\pi$
$461$			28.5125i			1.32796i	−0.747750	+	0.663981i	$0.768866\pi$
$462$		0			0
$463$		−	1.69588i		−	0.0788144i	−0.999223	−	0.0394072i	$-0.987453\pi$
$463$		−	1.69588i		−	0.0788144i	0.999223	−	0.0394072i	$-0.0125470\pi$
$464$	9.39177			0.436002
$465$		0			0
$466$		−	8.32639i		−	0.385713i
$467$	14.1217			0.653476			0.326738	−	0.945115i	$-0.394051\pi$
$467$	14.1217			0.653476			0.326738	+	0.945115i	$0.394051\pi$
$468$		0			0
$469$	26.5132			1.22427
$470$		−	18.4558i		−	0.851304i
$471$		0			0
$472$	11.9565			0.550344
$473$			3.12707i			0.143783i
$474$		0			0
$475$		−	13.7198i		−	0.629510i
$476$			3.20572i			0.146934i
$477$		0			0
$478$	2.48408			0.113619
$479$			27.7634i			1.26854i	0.773110	+	0.634272i	$0.218700\pi$
$479$			27.7634i			1.26854i	−0.773110	+	0.634272i	$0.781300\pi$
$480$		0			0
$481$	−5.52831	+	41.5438i	−0.252069	+	1.89423i
$482$	31.2600			1.42385
$483$		0			0
$484$	−5.36016			−0.243643
$485$	−6.53537			−0.296756
$486$		0			0
$487$			27.5898i			1.25021i	0.780539	+	0.625107i	$0.214945\pi$
$487$			27.5898i			1.25021i	−0.780539	+	0.625107i	$0.785055\pi$
$488$			4.26294i			0.192974i
$489$		0			0
$490$	−5.92638			−0.267726
$491$	40.7366			1.83842			0.919210	−	0.393769i	$-0.128829\pi$
$491$	40.7366			1.83842			0.919210	+	0.393769i	$0.128829\pi$
$492$		0			0
$493$	−2.82661			−0.127304
$494$	−21.1399	−	2.81312i	−0.951128	−	0.126568i
$495$		0			0
$496$			32.1254i			1.44248i
$497$	−18.7705			−0.841970
$498$		0			0
$499$			15.6223i			0.699349i	0.936871	+	0.349674i	$0.113708\pi$
$499$			15.6223i			0.699349i	−0.936871	+	0.349674i	$0.886292\pi$
$500$			6.54266i			0.292596i
$501$		0			0
$502$		−	13.0346i		−	0.581761i
$503$	30.2181			1.34736			0.673679	−	0.739024i	$-0.264713\pi$
$503$	30.2181			1.34736			0.673679	+	0.739024i	$0.264713\pi$
$504$		0			0
$505$			17.4327i			0.775746i
$506$	13.5016			0.600218
$507$		0			0
$508$	−4.15477			−0.184338
$509$		−	15.0873i		−	0.668734i	−0.942443	−	0.334367i	$-0.891478\pi$
$509$		−	15.0873i		−	0.668734i	0.942443	−	0.334367i	$-0.108522\pi$
$510$		0			0
$511$	−4.00191			−0.177034
$512$		−	3.14438i		−	0.138963i
$513$		0			0
$514$		−	34.4719i		−	1.52049i
$515$			6.86068i			0.302318i
$516$		0			0
$517$	−18.0727			−0.794834
$518$		−	61.0465i		−	2.68223i
$519$		0			0
$520$	−8.48582	−	1.12923i	−0.372128	−	0.0495198i
$521$	26.8690			1.17715			0.588576	−	0.808442i	$-0.299689\pi$
$521$	26.8690			1.17715			0.588576	+	0.808442i	$0.299689\pi$
$522$		0			0
$523$	24.9855			1.09254			0.546270	−	0.837609i	$-0.316047\pi$
$523$	24.9855			1.09254			0.546270	+	0.837609i	$0.316047\pi$
$524$	−5.71095			−0.249484
$525$		0			0
$526$			39.4455i			1.71991i
$527$		−	9.66869i		−	0.421175i
$528$		0			0
$529$	−0.985414			−0.0428441
$530$	21.1429			0.918390
$531$		0			0
$532$	7.86317			0.340911
$533$	2.57024	−	19.3147i	0.111330	−	0.836612i
$534$		0			0
$535$			11.6694i			0.504513i
$536$	−17.8738			−0.772030
$537$		0			0
$538$		−	35.6377i		−	1.53645i
$539$			5.80334i			0.249967i
$540$		0			0
$541$			33.1735i			1.42624i	0.701042	+	0.713120i	$0.252719\pi$
$541$			33.1735i			1.42624i	−0.701042	+	0.713120i	$0.747281\pi$
$542$	8.34488			0.358443
$543$		0			0
$544$		−	5.43020i		−	0.232818i
$545$	17.5803			0.753059
$546$		0			0
$547$	18.7240			0.800582			0.400291	−	0.916388i	$-0.368909\pi$
$547$	18.7240			0.800582			0.400291	+	0.916388i	$0.368909\pi$
$548$		−	5.24913i		−	0.224232i
$549$		0			0
$550$	10.9226			0.465743
$551$			6.93326i			0.295367i
$552$		0			0
$553$		−	1.55920i		−	0.0663038i
$554$			20.0911i			0.853590i
$555$		0			0
$556$	−8.53796			−0.362090
$557$		−	27.4258i		−	1.16207i	−0.813879	−	0.581035i	$-0.802648\pi$
$557$		−	27.4258i		−	1.16207i	0.813879	−	0.581035i	$-0.197352\pi$
$558$		0			0
$559$	6.35567	+	0.845762i	0.268816	+	0.0357719i
$560$	17.2441			0.728694
$561$		0			0
$562$	−11.9657			−0.504744
$563$	−19.7146			−0.830872			−0.415436	−	0.909622i	$-0.636371\pi$
$563$	−19.7146			−0.830872			−0.415436	+	0.909622i	$0.636371\pi$
$564$		0			0
$565$			3.26729i			0.137456i
$566$			20.9422i			0.880267i
$567$		0			0
$568$	12.6540			0.530952
$569$	19.4174			0.814021			0.407011	−	0.913423i	$-0.366571\pi$
$569$	19.4174			0.814021			0.407011	+	0.913423i	$0.366571\pi$
$570$		0			0
$571$	9.99073			0.418099			0.209050	−	0.977905i	$-0.432963\pi$
$571$	9.99073			0.418099			0.209050	+	0.977905i	$0.432963\pi$
$572$	0.566902	−	4.26011i	0.0237034	−	0.178124i
$573$		0			0
$574$			28.3820i			1.18464i
$575$	17.8096			0.742711
$576$		0			0
$577$		−	17.2976i		−	0.720107i	−0.932932	−	0.360054i	$-0.882758\pi$
$577$		−	17.2976i		−	0.720107i	0.932932	−	0.360054i	$-0.117242\pi$
$578$		−	24.2655i		−	1.00931i
$579$		0			0
$580$		−	1.42682i		−	0.0592453i
$581$	49.7891			2.06560
$582$		0			0
$583$		−	20.7040i		−	0.857471i
$584$	2.69787			0.111639
$585$		0			0
$586$	10.7582			0.444415
$587$			21.9939i			0.907786i	0.891056	+	0.453893i	$0.149965\pi$
$587$			21.9939i			0.907786i	−0.891056	+	0.453893i	$0.850035\pi$
$588$		0			0
$589$	−23.7159			−0.977196
$590$		−	9.92376i		−	0.408555i
$591$		0			0
$592$			56.9123i			2.33908i
$593$		−	29.8711i		−	1.22666i	−0.789826	−	0.613330i	$-0.789829\pi$
$593$		−	29.8711i		−	1.22666i	0.789826	−	0.613330i	$-0.210171\pi$
$594$		0			0
$595$	−5.18988			−0.212765
$596$		−	5.29691i		−	0.216970i
$597$		0			0
$598$	3.65169	−	27.4415i	0.149329	−	1.12216i
$599$	−0.0136575			−0.000558032			−0.000279016	−	1.00000i	$-0.500089\pi$
$599$	−0.0136575			−0.000558032			−0.000279016		1.00000i	$0.500089\pi$
$600$		0			0
$601$	6.51599			0.265793			0.132896	−	0.991130i	$-0.457572\pi$
$601$	6.51599			0.265793			0.132896	+	0.991130i	$0.457572\pi$
$602$	−9.33934			−0.380643
$603$		0			0
$604$		−	11.9816i		−	0.487526i
$605$		−	8.67780i		−	0.352803i
$606$		0			0
$607$	−35.5842			−1.44432			−0.722158	−	0.691728i	$-0.756850\pi$
$607$	−35.5842			−1.44432			−0.722158	+	0.691728i	$0.756850\pi$
$608$	−13.3195			−0.540177
$609$		0			0
$610$	3.53818			0.143257
$611$	−4.88801	+	36.7320i	−0.197748	+	1.48602i
$612$		0			0
$613$			2.03420i			0.0821607i	0.999156	+	0.0410803i	$0.0130800\pi$
$613$			2.03420i			0.0821607i	−0.999156	+	0.0410803i	$0.986920\pi$
$614$	−39.8151			−1.60681
$615$		0			0
$616$		−	12.2106i		−	0.491979i
$617$			3.06833i			0.123526i	0.998091	+	0.0617632i	$0.0196724\pi$
$617$			3.06833i			0.123526i	−0.998091	+	0.0617632i	$0.980328\pi$
$618$		0			0
$619$			3.96169i			0.159234i	0.996826	+	0.0796169i	$0.0253697\pi$
$619$			3.96169i			0.159234i	−0.996826	+	0.0796169i	$0.974630\pi$
$620$	4.88056			0.196008
$621$		0			0
$622$			18.2733i			0.732693i
$623$	43.0846			1.72615
$624$		0			0
$625$	8.38660			0.335464
$626$			29.1326i			1.16437i
$627$		0			0
$628$	12.0651			0.481451
$629$		−	17.1287i		−	0.682966i
$630$		0			0
$631$		−	8.56943i		−	0.341144i	−0.985345	−	0.170572i	$-0.945438\pi$
$631$		−	8.56943i		−	0.341144i	0.985345	−	0.170572i	$-0.0545615\pi$
$632$			1.05113i			0.0418116i
$633$		0			0
$634$	50.5034			2.00575
$635$		−	6.72635i		−	0.266927i
$636$		0			0
$637$	11.7951	+	1.56959i	0.467338	+	0.0621896i
$638$	−5.51971			−0.218527
$639$		0			0
$640$	−14.8438			−0.586752
$641$	−10.9151			−0.431121			−0.215561	−	0.976490i	$-0.569158\pi$
$641$	−10.9151			−0.431121			−0.215561	+	0.976490i	$0.569158\pi$
$642$		0			0
$643$		−	29.3102i		−	1.15588i	−0.816078	−	0.577941i	$-0.803856\pi$
$643$		−	29.3102i		−	1.15588i	0.816078	−	0.577941i	$-0.196144\pi$
$644$			10.2071i			0.402216i
$645$		0			0
$646$	8.71607			0.342929
$647$	−5.27064			−0.207210			−0.103605	−	0.994619i	$-0.533038\pi$
$647$	−5.27064			−0.207210			−0.103605	+	0.994619i	$0.533038\pi$
$648$		0			0
$649$	−9.71773			−0.381454
$650$	2.95418	−	22.1999i	0.115873	−	0.870751i
$651$		0			0
$652$			11.3369i			0.443986i
$653$	21.0634			0.824274			0.412137	−	0.911122i	$-0.364783\pi$
$653$	21.0634			0.824274			0.412137	+	0.911122i	$0.364783\pi$
$654$		0			0
$655$		−	9.24571i		−	0.361260i
$656$		−	26.4599i		−	1.03308i
$657$		0			0
$658$		−	53.9759i		−	2.10420i
$659$	9.20827			0.358703			0.179352	−	0.983785i	$-0.442600\pi$
$659$	9.20827			0.358703			0.179352	+	0.983785i	$0.442600\pi$
$660$		0			0
$661$		−	9.69679i		−	0.377162i	−0.982058	−	0.188581i	$-0.939611\pi$
$661$		−	9.69679i		−	0.377162i	0.982058	−	0.188581i	$-0.0603887\pi$
$662$	−28.4814			−1.10696
$663$		0			0
$664$	−33.5652			−1.30258
$665$			12.7300i			0.493650i
$666$		0			0
$667$	−9.00000			−0.348481
$668$			3.61611i			0.139911i
$669$		0			0
$670$			14.8350i			0.573126i
$671$		−	3.46472i		−	0.133754i
$672$		0			0
$673$	−27.6272			−1.06495			−0.532475	−	0.846445i	$-0.678738\pi$
$673$	−27.6272			−1.06495			−0.532475	+	0.846445i	$0.678738\pi$
$674$			14.3322i			0.552054i
$675$		0			0
$676$	−8.50521	−	2.30442i	−0.327123	−	0.0886314i
$677$	−22.9882			−0.883509			−0.441755	−	0.897136i	$-0.645644\pi$
$677$	−22.9882			−0.883509			−0.441755	+	0.897136i	$0.645644\pi$
$678$		0			0
$679$	−19.1134			−0.733503
$680$	3.49874			0.134171
$681$		0			0
$682$		−	18.8807i		−	0.722979i
$683$		−	19.2616i		−	0.737023i	−0.929623	−	0.368512i	$-0.879867\pi$
$683$		−	19.2616i		−	0.737023i	0.929623	−	0.368512i	$-0.120133\pi$
$684$		0			0
$685$	8.49806			0.324694
$686$	19.4309			0.741875
$687$		0			0
$688$	8.70686			0.331946
$689$	−42.0801	−	5.59968i	−1.60312	−	0.213331i
$690$		0			0
$691$			0.957657i			0.0364310i	0.999834	+	0.0182155i	$0.00579849\pi$
$691$			0.957657i			0.0364310i	−0.999834	+	0.0182155i	$0.994202\pi$
$692$	8.34185			0.317109
$693$		0			0
$694$		−	11.9091i		−	0.452062i
$695$		−	13.8225i		−	0.524317i
$696$		0			0
$697$			7.96353i			0.301640i
$698$	20.4970			0.775822
$699$		0			0
$700$			8.25745i			0.312102i
$701$	−5.78079			−0.218337			−0.109169	−	0.994023i	$-0.534819\pi$
$701$	−5.78079			−0.218337			−0.109169	+	0.994023i	$0.534819\pi$
$702$		0			0
$703$	−42.0142			−1.58460
$704$			6.61584i			0.249344i
$705$		0			0
$706$	−2.68521			−0.101059
$707$			50.9838i			1.91744i
$708$		0			0
$709$		−	5.59628i		−	0.210173i	−0.994463	−	0.105086i	$-0.966488\pi$
$709$		−	5.59628i		−	0.210173i	0.994463	−	0.105086i	$-0.0335119\pi$
$710$		−	10.5027i		−	0.394159i
$711$		0			0
$712$	−29.0453			−1.08852
$713$		−	30.7854i		−	1.15292i
$714$		0			0
$715$	6.89689	+	0.917783i	0.257929	+	0.0343231i
$716$	12.7730			0.477351
$717$		0			0
$718$	−11.4678			−0.427973
$719$	17.4320			0.650102			0.325051	−	0.945696i	$-0.394618\pi$
$719$	17.4320			0.650102			0.325051	+	0.945696i	$0.394618\pi$
$720$		0			0
$721$			20.0647i			0.747250i
$722$			9.71252i			0.361463i
$723$		0			0
$724$	−7.37271			−0.274004
$725$	−7.28092			−0.270406
$726$		0			0
$727$	24.0426			0.891689			0.445845	−	0.895110i	$-0.352903\pi$
$727$	24.0426			0.891689			0.445845	+	0.895110i	$0.352903\pi$
$728$	−24.8176	−	3.30253i	−0.919803	−	0.122400i
$729$		0			0
$730$		−	2.23920i		−	0.0828764i
$731$	−2.62047			−0.0969217
$732$		0			0
$733$		−	13.7324i		−	0.507217i	−0.967307	−	0.253608i	$-0.918383\pi$
$733$		−	13.7324i		−	0.507217i	0.967307	−	0.253608i	$-0.0816174\pi$
$734$		−	1.95171i		−	0.0720390i
$735$		0			0
$736$		−	17.2899i		−	0.637314i
$737$	14.5270			0.535109
$738$		0			0
$739$			13.1229i			0.482732i	0.970434	+	0.241366i	$0.0775954\pi$
$739$			13.1229i			0.482732i	−0.970434	+	0.241366i	$0.922405\pi$
$740$	8.64622			0.317842
$741$		0			0
$742$	61.8346			2.27002
$743$		−	35.5269i		−	1.30335i	−0.758497	−	0.651677i	$-0.774066\pi$
$743$		−	35.5269i		−	1.30335i	0.758497	−	0.651677i	$-0.225934\pi$
$744$		0			0
$745$	8.57541			0.314179
$746$			28.3164i			1.03674i
$747$		0			0
$748$			1.75646i			0.0642227i
$749$			34.1284i			1.24702i
$750$		0			0
$751$	23.4238			0.854747			0.427374	−	0.904075i	$-0.359439\pi$
$751$	23.4238			0.854747			0.427374	+	0.904075i	$0.359439\pi$
$752$			50.3205i			1.83500i
$753$		0			0
$754$	−1.49288	+	11.2186i	−0.0543676	+	0.408558i
$755$	19.3976			0.705951
$756$		0			0
$757$	18.7868			0.682817			0.341409	−	0.939915i	$-0.389096\pi$
$757$	18.7868			0.682817			0.341409	+	0.939915i	$0.389096\pi$
$758$	2.52488			0.0917077
$759$		0			0
$760$		−	8.58192i		−	0.311299i
$761$			16.9411i			0.614113i	0.951691	+	0.307057i	$0.0993441\pi$
$761$			16.9411i			0.614113i	−0.951691	+	0.307057i	$0.900656\pi$
$762$		0			0
$763$	51.4155			1.86136
$764$	−12.6057			−0.456060
$765$		0			0
$766$	−38.7194			−1.39899
$767$	−2.62830	+	19.7510i	−0.0949024	+	0.713166i
$768$		0			0
$769$		−	38.4177i		−	1.38538i	−0.721237	−	0.692688i	$-0.756426\pi$
$769$		−	38.4177i		−	1.38538i	0.721237	−	0.692688i	$-0.243574\pi$
$770$	−10.1346			−0.365227
$771$		0			0
$772$			8.78296i			0.316106i
$773$			21.1236i			0.759761i	0.925036	+	0.379881i	$0.124035\pi$
$773$			21.1236i			0.759761i	−0.925036	+	0.379881i	$0.875965\pi$
$774$		0			0
$775$		−	24.9051i		−	0.894617i
$776$	12.8852			0.462552
$777$		0			0
$778$		−	8.45485i		−	0.303121i
$779$	19.5334			0.699857
$780$		0			0
$781$	−10.2846			−0.368013
$782$			11.3142i			0.404597i
$783$		0			0
$784$	16.1585			0.577089
$785$			19.5328i			0.697154i
$786$		0			0
$787$		−	2.97375i		−	0.106003i	−0.998594	−	0.0530014i	$-0.983121\pi$
$787$		−	2.97375i		−	0.106003i	0.998594	−	0.0530014i	$-0.0168788\pi$
$788$		−	9.55500i		−	0.340383i
$789$		0			0
$790$	0.872422			0.0310394
$791$			9.55551i			0.339755i
$792$		0			0
$793$	−7.04193	−	0.937084i	−0.250066	−	0.0332768i
$794$	22.9843			0.815681
$795$		0			0
$796$	−11.3139			−0.401012
$797$	−29.7027			−1.05212			−0.526061	−	0.850447i	$-0.676332\pi$
$797$	−29.7027			−1.05212			−0.526061	+	0.850447i	$0.676332\pi$
$798$		0			0
$799$		−	15.1448i		−	0.535784i
$800$		−	13.9874i		−	0.494528i
$801$		0			0
$802$	−24.5268			−0.866071
$803$	−2.19271			−0.0773790
$804$		0			0
$805$	−16.5247			−0.582421
$806$	−38.3744	−	5.10655i	−1.35168	−	0.179871i
$807$		0			0
$808$		−	34.3705i		−	1.20915i
$809$	1.78891			0.0628946			0.0314473	−	0.999505i	$-0.489988\pi$
$809$	1.78891			0.0628946			0.0314473	+	0.999505i	$0.489988\pi$
$810$		0			0
$811$		−	17.3429i		−	0.608991i	−0.952514	−	0.304496i	$-0.901512\pi$
$811$		−	17.3429i		−	0.608991i	0.952514	−	0.304496i	$-0.0984879\pi$
$812$		−	4.17287i		−	0.146439i
$813$		0			0
$814$		−	33.4484i		−	1.17236i
$815$	−18.3538			−0.642905
$816$		0			0
$817$			6.42764i			0.224875i
$818$	−0.778285			−0.0272121
$819$		0			0
$820$	−4.01983			−0.140379
$821$		−	34.7618i		−	1.21320i	−0.795008	−	0.606598i	$-0.792534\pi$
$821$		−	34.7618i		−	1.21320i	0.795008	−	0.606598i	$-0.207466\pi$
$822$		0			0
$823$	−20.0338			−0.698335			−0.349167	−	0.937060i	$-0.613536\pi$
$823$	−20.0338			−0.698335			−0.349167	+	0.937060i	$0.613536\pi$
$824$		−	13.5266i		−	0.471221i
$825$		0			0
$826$		−	29.0230i		−	1.00984i
$827$		−	11.7034i		−	0.406966i	−0.979078	−	0.203483i	$-0.934774\pi$
$827$		−	11.7034i		−	0.406966i	0.979078	−	0.203483i	$-0.0652262\pi$
$828$		0			0
$829$	25.2016			0.875287			0.437644	−	0.899149i	$-0.355813\pi$
$829$	25.2016			0.875287			0.437644	+	0.899149i	$0.355813\pi$
$830$			27.8586i			0.966987i
$831$		0			0
$832$	13.4465	+	1.78935i	0.466172	+	0.0620345i
$833$	−4.86317			−0.168499
$834$		0			0
$835$	−5.85428			−0.202596
$836$	4.30836			0.149008
$837$		0			0
$838$			21.1394i			0.730248i
$839$			12.6922i			0.438183i	0.975704	+	0.219091i	$0.0703093\pi$
$839$			12.6922i			0.438183i	−0.975704	+	0.219091i	$0.929691\pi$
$840$		0			0
$841$	−25.3206			−0.873125
$842$	−51.7366			−1.78296
$843$		0			0
$844$	−3.71544			−0.127891
$845$	3.73072	−	13.7695i	0.128341	−	0.473684i
$846$		0			0
$847$		−	25.3791i		−	0.872036i
$848$	−57.6470			−1.97961
$849$		0			0
$850$			9.15312i			0.313949i
$851$		−	54.5382i		−	1.86955i
$852$		0			0
$853$			6.47301i			0.221632i	0.993841	+	0.110816i	$0.0353463\pi$
$853$			6.47301i			0.221632i	−0.993841	+	0.110816i	$0.964654\pi$
$854$	10.3478			0.354093
$855$		0			0
$856$		−	23.0076i		−	0.786382i
$857$	26.4305			0.902849			0.451425	−	0.892309i	$-0.350916\pi$
$857$	26.4305			0.902849			0.451425	+	0.892309i	$0.350916\pi$
$858$		0			0
$859$	−2.98193			−0.101742			−0.0508710	−	0.998705i	$-0.516200\pi$
$859$	−2.98193			−0.101742			−0.0508710	+	0.998705i	$0.516200\pi$
$860$		−	1.32276i		−	0.0451058i
$861$		0			0
$862$	20.5467			0.699823
$863$		−	3.40827i		−	0.116019i	−0.998316	−	0.0580095i	$-0.981525\pi$
$863$		−	3.40827i		−	0.116019i	0.998316	−	0.0580095i	$-0.0184754\pi$
$864$		0			0
$865$			13.5050i			0.459184i
$866$		−	45.7812i		−	1.55571i
$867$		0			0
$868$	14.2737			0.484481
$869$		−	0.854310i		−	0.0289805i
$870$		0			0
$871$	3.92903	−	29.5256i	0.133130	−	1.00044i
$872$	−34.6616			−1.17379
$873$		0			0
$874$	27.7522			0.938732
$875$	−30.9780			−1.04725
$876$		0			0
$877$			16.0227i			0.541049i	0.962713	+	0.270524i	$0.0871971\pi$
$877$			16.0227i			0.541049i	−0.962713	+	0.270524i	$0.912803\pi$
$878$		−	18.8769i		−	0.637066i
$879$		0			0
$880$	9.44829			0.318502
$881$	−26.8852			−0.905787			−0.452894	−	0.891565i	$-0.649608\pi$
$881$	−26.8852			−0.905787			−0.452894	+	0.891565i	$0.649608\pi$
$882$		0			0
$883$	−16.3368			−0.549778			−0.274889	−	0.961476i	$-0.588641\pi$
$883$	−16.3368			−0.549778			−0.274889	+	0.961476i	$0.588641\pi$
$884$	3.56995	+	0.475061i	0.120071	+	0.0159780i
$885$		0			0
$886$			42.9795i			1.44392i
$887$	24.8199			0.833371			0.416686	−	0.909051i	$-0.363192\pi$
$887$	24.8199			0.833371			0.416686	+	0.909051i	$0.363192\pi$
$888$		0			0
$889$		−	19.6719i		−	0.659774i
$890$			24.1072i			0.808076i
$891$		0			0
$892$			6.21933i			0.208239i
$893$	−37.1480			−1.24311
$894$		0			0
$895$			20.6789i			0.691218i
$896$	−43.4121			−1.45030
$897$		0			0
$898$	−49.0227			−1.63591
$899$			12.5857i			0.419756i
$900$		0			0
$901$	17.3498			0.578006
$902$			15.5509i			0.517789i
$903$		0			0
$904$		−	6.44182i		−	0.214252i
$905$		−	11.9360i		−	0.396766i
$906$		0			0
$907$	−28.8675			−0.958529			−0.479265	−	0.877670i	$-0.659097\pi$
$907$	−28.8675			−0.958529			−0.479265	+	0.877670i	$0.659097\pi$
$908$		−	2.33510i		−	0.0774930i
$909$		0			0
$910$	−2.74106	+	20.5983i	−0.0908651	+	0.682827i
$911$	13.2819			0.440048			0.220024	−	0.975494i	$-0.429386\pi$
$911$	13.2819			0.440048			0.220024	+	0.975494i	$0.429386\pi$
$912$		0			0
$913$	27.2802			0.902844
$914$	−18.7534			−0.620309
$915$		0			0
$916$		−	8.46058i		−	0.279545i
$917$		−	27.0400i		−	0.892940i
$918$		0			0
$919$	−3.98083			−0.131316			−0.0656578	−	0.997842i	$-0.520915\pi$
$919$	−3.98083			−0.131316			−0.0656578	+	0.997842i	$0.520915\pi$
$920$	11.1401			0.367278
$921$		0			0
$922$	−46.6582			−1.53661
$923$	−2.78163	+	20.9032i	−0.0915583	+	0.688036i
$924$		0			0
$925$		−	44.1209i		−	1.45069i
$926$	2.77516			0.0911975
$927$		0			0
$928$			7.06846i			0.232033i
$929$			7.35842i			0.241422i	0.992688	+	0.120711i	$0.0385174\pi$
$929$			7.35842i			0.241422i	−0.992688	+	0.120711i	$0.961483\pi$
$930$		0			0
$931$			11.9286i			0.390946i
$932$	3.44897			0.112975
$933$		0			0
$934$			23.1089i			0.756147i
$935$	−2.84362			−0.0929964
$936$		0			0
$937$	8.02666			0.262220			0.131110	−	0.991368i	$-0.458146\pi$
$937$	8.02666			0.262220			0.131110	+	0.991368i	$0.458146\pi$
$938$			43.3864i			1.41662i
$939$		0			0
$940$	7.64479			0.249346
$941$		−	14.2624i		−	0.464941i	−0.972603	−	0.232470i	$-0.925319\pi$
$941$		−	14.2624i		−	0.464941i	0.972603	−	0.232470i	$-0.0746809\pi$
$942$		0			0
$943$			25.3561i			0.825709i
$944$			27.0575i			0.880648i
$945$		0			0
$946$	−5.11717			−0.166374
$947$		−	8.22076i		−	0.267139i	−0.991039	−	0.133569i	$-0.957356\pi$
$947$		−	8.22076i		−	0.267139i	0.991039	−	0.133569i	$-0.0426439\pi$
$948$		0			0
$949$	−0.593049	+	4.45661i	−0.0192512	+	0.144668i
$950$	22.4513			0.728416
$951$		0			0
$952$	10.2324			0.331635
$953$	−34.3915			−1.11405			−0.557025	−	0.830496i	$-0.688057\pi$
$953$	−34.3915			−1.11405			−0.557025	+	0.830496i	$0.688057\pi$
$954$		0			0
$955$		−	20.4080i		−	0.660388i
$956$			1.02896i			0.0332789i
$957$		0			0
$958$	−45.4323			−1.46785
$959$	24.8534			0.802559
$960$		0			0
$961$	−12.0505			−0.388726
$962$	−67.9826	−	9.04658i	−2.19185	−	0.291673i
$963$		0			0
$964$			12.9486i			0.417045i
$965$	−14.2191			−0.457730
$966$		0			0
$967$		−	49.8119i		−	1.60184i	−0.598770	−	0.800921i	$-0.704344\pi$
$967$		−	49.8119i		−	1.60184i	0.598770	−	0.800921i	$-0.295656\pi$
$968$			17.1092i			0.549912i
$969$		0			0
$970$		−	10.6945i		−	0.343381i
$971$	49.8192			1.59878			0.799388	−	0.600815i	$-0.205157\pi$
$971$	49.8192			1.59878			0.799388	+	0.600815i	$0.205157\pi$
$972$		0			0
$973$		−	40.4253i		−	1.29597i
$974$	−45.1482			−1.44664
$975$		0			0
$976$	−9.64699			−0.308793
$977$			43.6262i			1.39573i	0.716231	+	0.697863i	$0.245866\pi$
$977$			43.6262i			1.39573i	−0.716231	+	0.697863i	$0.754134\pi$
$978$		0			0
$979$	23.6067			0.754474
$980$		−	2.45483i		−	0.0784167i
$981$		0			0
$982$			66.6618i			2.12726i
$983$		−	57.4316i		−	1.83178i	−0.401427	−	0.915891i	$-0.631486\pi$
$983$		−	57.4316i		−	1.83178i	0.401427	−	0.915891i	$-0.368514\pi$
$984$		0			0
$985$	15.4690			0.492884
$986$		−	4.62549i		−	0.147306i
$987$		0			0
$988$	1.16526	−	8.75658i	0.0370717	−	0.278584i
$989$	−8.34366			−0.265313
$990$		0			0
$991$	7.92798			0.251841			0.125920	−	0.992040i	$-0.459812\pi$
$991$	7.92798			0.251841			0.125920	+	0.992040i	$0.459812\pi$
$992$	−24.1783			−0.767663
$993$		0			0
$994$		−	30.7162i		−	0.974257i
$995$		−	18.3167i		−	0.580677i
$996$		0			0
$997$	−46.1993			−1.46315			−0.731573	−	0.681763i	$-0.761214\pi$
$997$	−46.1993			−1.46315			−0.731573	+	0.681763i	$0.761214\pi$
$998$	−25.5644			−0.809228
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1053.2.b.j.649.8		10
3.2	odd	2		1053.2.b.i.649.3		10
9.2	odd	6		351.2.t.c.64.3		20
9.4	even	3		117.2.t.c.25.3	✓	20
9.5	odd	6		351.2.t.c.181.8		20
9.7	even	3		117.2.t.c.103.8	yes	20
13.12	even	2	inner	1053.2.b.j.649.3		10
39.38	odd	2		1053.2.b.i.649.8		10
117.25	even	6		117.2.t.c.103.3	yes	20
117.38	odd	6		351.2.t.c.64.8		20
117.77	odd	6		351.2.t.c.181.3		20
117.103	even	6		117.2.t.c.25.8	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.2.t.c.25.3	✓	20	9.4	even	3
117.2.t.c.25.8	yes	20	117.103	even	6
117.2.t.c.103.3	yes	20	117.25	even	6
117.2.t.c.103.8	yes	20	9.7	even	3
351.2.t.c.64.3		20	9.2	odd	6
351.2.t.c.64.8		20	117.38	odd	6
351.2.t.c.181.3		20	117.77	odd	6
351.2.t.c.181.8		20	9.5	odd	6
1053.2.b.i.649.3		10	3.2	odd	2
1053.2.b.i.649.8		10	39.38	odd	2
1053.2.b.j.649.3		10	13.12	even	2	inner
1053.2.b.j.649.8		10	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:	\( N \)	\(=\)	\( 1053 = 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1053.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.63641i q^{2} -0.677835 q^{4} -1.09738i q^{5} -3.20939i q^{7} +2.16360i q^{8} +1.79576 q^{10} -1.75848i q^{11} +(-3.57405 - 0.475605i) q^{13} +5.25188 q^{14} -4.89621 q^{16} +1.47360 q^{17} -3.61452i q^{19} +0.743842i q^{20} +2.87759 q^{22} +4.69197 q^{23} +3.79576 q^{25} +(0.778285 - 5.84860i) q^{26} +2.17544i q^{28} -1.91817 q^{29} -6.56129i q^{31} -3.68500i q^{32} +2.41141i q^{34} -3.52192 q^{35} -11.6237i q^{37} +5.91483 q^{38} +2.37429 q^{40} +5.40415i q^{41} -1.77828 q^{43} +1.19196i q^{44} +7.67798i q^{46} -10.2774i q^{47} -3.30020 q^{49} +6.21142i q^{50} +(2.42261 + 0.322382i) q^{52} +11.7738 q^{53} -1.92972 q^{55} +6.94385 q^{56} -3.13891i q^{58} -5.52622i q^{59} +1.97030 q^{61} +10.7370 q^{62} -3.76225 q^{64} +(-0.521919 + 3.92208i) q^{65} +8.26112i q^{67} -0.998855 q^{68} -5.76330i q^{70} -5.84860i q^{71} -1.24694i q^{73} +19.0212 q^{74} +2.45005i q^{76} -5.64365 q^{77} +0.485823 q^{79} +5.37300i q^{80} -8.84340 q^{82} +15.5135i q^{83} -1.61709i q^{85} -2.91000i q^{86} +3.80465 q^{88} +13.4245i q^{89} +(-1.52640 + 11.4705i) q^{91} -3.18038 q^{92} +16.8181 q^{94} -3.96649 q^{95} -5.95544i q^{97} -5.40048i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 12 q^{4} - 8 q^{10} + 4 q^{13} + 18 q^{14} - 4 q^{16} - 6 q^{17} + 10 q^{22} - 24 q^{23} + 12 q^{25} - 6 q^{26} - 12 q^{29} - 6 q^{35} - 12 q^{38} + 8 q^{40} - 4 q^{43} + 10 q^{49} + 54 q^{53} + 10 q^{55}+ \cdots - 24 q^{95}+O(q^{100}) \) 10 * q - 12 * q^4 - 8 * q^10 + 4 * q^13 + 18 * q^14 - 4 * q^16 - 6 * q^17 + 10 * q^22 - 24 * q^23 + 12 * q^25 - 6 * q^26 - 12 * q^29 - 6 * q^35 - 12 * q^38 + 8 * q^40 - 4 * q^43 + 10 * q^49 + 54 * q^53 + 10 * q^55 - 36 * q^56 + 2 * q^61 - 36 * q^62 + 4 * q^64 + 24 * q^65 - 24 * q^68 + 42 * q^74 + 6 * q^77 + 14 * q^79 - 2 * q^82 - 22 * q^88 - 36 * q^91 + 84 * q^92 - 20 * q^94 - 24 * q^95

Introduction

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