LMFDB - Embedded newform 920.2.e.c.369.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [920,2,Mod(369,920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(920, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("920.369");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 920 = 2^{3} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	920.e (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:	$7.34623698596$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 2 x^{14} + 6 x^{13} + 100 x^{12} - 196 x^{11} + 210 x^{10} + 702 x^{9} + 1572 x^{8} + \cdots + 1024 $ x^16 - 2x^15 + 2x^14 + 6x^13 + 100x^12 - 196x^11 + 210x^10 + 702x^9 + 1572x^8 - 1594x^7 + 2464x^6 + 9568x^5 + 15457x^4 + 4336x^3 + 128x^2 - 512*x + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			369.12
Root			$-2.15699 - 2.15699i$ of defining polynomial
Character	$\chi$	$=$	920.369
Dual form			920.2.e.c.369.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.69755i q^{3} +(0.589417 - 2.15699i) q^{5} +4.22860i q^{7} +0.118308 q^{9} +O(q^{10})$	$q+1.69755i q^{3} +(0.589417 - 2.15699i) q^{5} +4.22860i q^{7} +0.118308 q^{9} +4.59157 q^{11} +0.978254i q^{13} +(3.66160 + 1.00057i) q^{15} -3.04005i q^{17} -1.91463 q^{19} -7.17829 q^{21} -1.00000i q^{23} +(-4.30518 - 2.54273i) q^{25} +5.29350i q^{27} +0.737607 q^{29} +2.97939 q^{31} +7.79445i q^{33} +(9.12104 + 2.49241i) q^{35} +8.93637i q^{37} -1.66064 q^{39} +9.08240 q^{41} +6.97873i q^{43} +(0.0697326 - 0.255188i) q^{45} +2.58560i q^{47} -10.8811 q^{49} +5.16065 q^{51} +2.71400i q^{53} +(2.70635 - 9.90396i) q^{55} -3.25019i q^{57} -7.13514 q^{59} -0.731629 q^{61} +0.500277i q^{63} +(2.11008 + 0.576599i) q^{65} -7.16679i q^{67} +1.69755 q^{69} -6.08775 q^{71} +5.96311i q^{73} +(4.31642 - 7.30827i) q^{75} +19.4159i q^{77} +2.06100 q^{79} -8.63108 q^{81} -4.39408i q^{83} +(-6.55735 - 1.79186i) q^{85} +1.25213i q^{87} +7.25034 q^{89} -4.13665 q^{91} +5.05767i q^{93} +(-1.12852 + 4.12983i) q^{95} -7.04560i q^{97} +0.543219 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{5} - 22 q^{9}+O(q^{10}) $ 16 * q - 2 * q^5 - 22 * q^9	$ 16 q - 2 q^{5} - 22 q^{9} + 14 q^{11} + 6 q^{15} - 22 q^{19} + 12 q^{25} - 44 q^{29} + 18 q^{31} + 20 q^{35} + 14 q^{41} + 14 q^{45} - 78 q^{49} - 38 q^{51} + 30 q^{55} - 64 q^{59} + 34 q^{61} + 6 q^{65} + 6 q^{69} + 30 q^{71} + 56 q^{75} + 4 q^{79} + 48 q^{81} + 52 q^{85} - 92 q^{89} - 70 q^{91} + 38 q^{95} - 122 q^{99}+O(q^{100}) $ 16 * q - 2 * q^5 - 22 * q^9 + 14 * q^11 + 6 * q^15 - 22 * q^19 + 12 * q^25 - 44 * q^29 + 18 * q^31 + 20 * q^35 + 14 * q^41 + 14 * q^45 - 78 * q^49 - 38 * q^51 + 30 * q^55 - 64 * q^59 + 34 * q^61 + 6 * q^65 + 6 * q^69 + 30 * q^71 + 56 * q^75 + 4 * q^79 + 48 * q^81 + 52 * q^85 - 92 * q^89 - 70 * q^91 + 38 * q^95 - 122 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/920\mathbb{Z}\right)^\times$.

$n$	$231$	$281$	$461$	$737$
$\chi(n)$	$1$	$1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$			1.69755i			0.980084i	0.871699	+	0.490042i	$0.163019\pi$
$3$			1.69755i			0.980084i	−0.871699	+	0.490042i	$0.836981\pi$
$4$		0			0
$5$	0.589417	−	2.15699i	0.263595	−	0.964633i
$5$	0.589417	−	2.15699i	0.263595	−	0.964633i
$6$		0			0
$7$			4.22860i			1.59826i	0.601157	+	0.799131i	$0.294707\pi$
$7$			4.22860i			1.59826i	−0.601157	+	0.799131i	$0.705293\pi$
$8$		0			0
$9$	0.118308			0.0394359
$10$		0			0
$11$	4.59157			1.38441			0.692206	−	0.721700i	$-0.256639\pi$
$11$	4.59157			1.38441			0.692206	+	0.721700i	$0.256639\pi$
$12$		0			0
$13$			0.978254i			0.271319i	0.990756	+	0.135659i	$0.0433153\pi$
$13$			0.978254i			0.271319i	−0.990756	+	0.135659i	$0.956685\pi$
$14$		0			0
$15$	3.66160	+	1.00057i	0.945421	+	0.258345i
$16$		0			0
$17$		−	3.04005i		−	0.737321i	−0.929564	−	0.368660i	$-0.879817\pi$
$17$		−	3.04005i		−	0.737321i	0.929564	−	0.368660i	$-0.120183\pi$
$18$		0			0
$19$	−1.91463			−0.439247			−0.219623	−	0.975585i	$-0.570483\pi$
$19$	−1.91463			−0.439247			−0.219623	+	0.975585i	$0.570483\pi$
$20$		0			0
$21$	−7.17829			−1.56643
$22$		0			0
$23$		−	1.00000i		−	0.208514i
$23$		−	1.00000i		−	0.208514i
$24$		0			0
$25$	−4.30518	−	2.54273i	−0.861035	−	0.508545i
$26$		0			0
$27$			5.29350i			1.01873i
$28$		0			0
$29$	0.737607			0.136970			0.0684851	−	0.997652i	$-0.478183\pi$
$29$	0.737607			0.136970			0.0684851	+	0.997652i	$0.478183\pi$
$30$		0			0
$31$	2.97939			0.535114			0.267557	−	0.963542i	$-0.413784\pi$
$31$	2.97939			0.535114			0.267557	+	0.963542i	$0.413784\pi$
$32$		0			0
$33$			7.79445i			1.35684i
$34$		0			0
$35$	9.12104	+	2.49241i	1.54174	+	0.421294i
$36$		0			0
$37$			8.93637i			1.46913i	0.678539	+	0.734565i	$0.262614\pi$
$37$			8.93637i			1.46913i	−0.678539	+	0.734565i	$0.737386\pi$
$38$		0			0
$39$	−1.66064			−0.265915
$40$		0			0
$41$	9.08240			1.41843			0.709216	−	0.704991i	$-0.249049\pi$
$41$	9.08240			1.41843			0.709216	+	0.704991i	$0.249049\pi$
$42$		0			0
$43$			6.97873i			1.06425i	0.846667	+	0.532123i	$0.178606\pi$
$43$			6.97873i			1.06425i	−0.846667	+	0.532123i	$0.821394\pi$
$44$		0			0
$45$	0.0697326	−	0.255188i	0.0103951	−	0.0380412i
$46$		0			0
$47$			2.58560i			0.377148i	0.982059	+	0.188574i	$0.0603865\pi$
$47$			2.58560i			0.377148i	−0.982059	+	0.188574i	$0.939613\pi$
$48$		0			0
$49$	−10.8811			−1.55444
$50$		0			0
$51$	5.16065			0.722636
$52$		0			0
$53$			2.71400i			0.372796i	0.982474	+	0.186398i	$0.0596814\pi$
$53$			2.71400i			0.372796i	−0.982474	+	0.186398i	$0.940319\pi$
$54$		0			0
$55$	2.70635	−	9.90396i	0.364924	−	1.33545i
$56$		0			0
$57$		−	3.25019i		−	0.430499i
$58$		0			0
$59$	−7.13514			−0.928916			−0.464458	−	0.885595i	$-0.653751\pi$
$59$	−7.13514			−0.928916			−0.464458	+	0.885595i	$0.653751\pi$
$60$		0			0
$61$	−0.731629			−0.0936755			−0.0468377	−	0.998903i	$-0.514914\pi$
$61$	−0.731629			−0.0936755			−0.0468377	+	0.998903i	$0.514914\pi$
$62$		0			0
$63$			0.500277i			0.0630290i
$64$		0			0
$65$	2.11008	+	0.576599i	0.261723	+	0.0715183i
$66$		0			0
$67$		−	7.16679i		−	0.875563i	−0.899081	−	0.437781i	$-0.855764\pi$
$67$		−	7.16679i		−	0.875563i	0.899081	−	0.437781i	$-0.144236\pi$
$68$		0			0
$69$	1.69755			0.204362
$70$		0			0
$71$	−6.08775			−0.722483			−0.361241	−	0.932472i	$-0.617647\pi$
$71$	−6.08775			−0.722483			−0.361241	+	0.932472i	$0.617647\pi$
$72$		0			0
$73$			5.96311i			0.697930i	0.937136	+	0.348965i	$0.113467\pi$
$73$			5.96311i			0.697930i	−0.937136	+	0.348965i	$0.886533\pi$
$74$		0			0
$75$	4.31642	−	7.30827i	0.498417	−	0.843887i
$76$		0			0
$77$			19.4159i			2.21265i
$78$		0			0
$79$	2.06100			0.231880			0.115940	−	0.993256i	$-0.463012\pi$
$79$	2.06100			0.231880			0.115940	+	0.993256i	$0.463012\pi$
$80$		0			0
$81$	−8.63108			−0.959009
$82$		0			0
$83$		−	4.39408i		−	0.482312i	−0.970486	−	0.241156i	$-0.922473\pi$
$83$		−	4.39408i		−	0.482312i	0.970486	−	0.241156i	$-0.0775266\pi$
$84$		0			0
$85$	−6.55735	−	1.79186i	−0.711244	−	0.194354i
$86$		0			0
$87$			1.25213i			0.134242i
$88$		0			0
$89$	7.25034			0.768534			0.384267	−	0.923222i	$-0.374454\pi$
$89$	7.25034			0.768534			0.384267	+	0.923222i	$0.374454\pi$
$90$		0			0
$91$	−4.13665			−0.433638
$92$		0			0
$93$			5.05767i			0.524456i
$94$		0			0
$95$	−1.12852	+	4.12983i	−0.115783	+	0.423712i
$96$		0			0
$97$		−	7.04560i		−	0.715372i	−0.933842	−	0.357686i	$-0.883566\pi$
$97$		−	7.04560i		−	0.715372i	0.933842	−	0.357686i	$-0.116434\pi$
$98$		0			0
$99$	0.543219			0.0545956
$100$		0			0
$101$	7.63173			0.759385			0.379693	−	0.925113i	$-0.376030\pi$
$101$	7.63173			0.759385			0.379693	+	0.925113i	$0.376030\pi$
$102$		0			0
$103$			16.7855i			1.65393i	0.562254	+	0.826964i	$0.309934\pi$
$103$			16.7855i			1.65393i	−0.562254	+	0.826964i	$0.690066\pi$
$104$		0			0
$105$	−4.23100	+	15.4835i	−0.412903	+	1.51103i
$106$		0			0
$107$		−	15.3282i		−	1.48183i	−0.671599	−	0.740915i	$-0.734392\pi$
$107$		−	15.3282i		−	1.48183i	0.671599	−	0.740915i	$-0.265608\pi$
$108$		0			0
$109$	−14.9619			−1.43309			−0.716544	−	0.697542i	$-0.754277\pi$
$109$	−14.9619			−1.43309			−0.716544	+	0.697542i	$0.754277\pi$
$110$		0			0
$111$	−15.1700			−1.43987
$112$		0			0
$113$		−	19.5478i		−	1.83890i	−0.393207	−	0.919450i	$-0.628634\pi$
$113$		−	19.5478i		−	1.83890i	0.393207	−	0.919450i	$-0.371366\pi$
$114$		0			0
$115$	−2.15699	−	0.589417i	−0.201140	−	0.0549634i
$116$		0			0
$117$			0.115735i			0.0106997i
$118$		0			0
$119$	12.8552			1.17843
$120$		0			0
$121$	10.0826			0.916596
$122$		0			0
$123$			15.4179i			1.39018i
$124$		0			0
$125$	−8.02217	+	7.78748i	−0.717525	+	0.696533i
$126$		0			0
$127$		−	20.6469i		−	1.83212i	−0.401042	−	0.916060i	$-0.631352\pi$
$127$		−	20.6469i		−	1.83212i	0.401042	−	0.916060i	$-0.368648\pi$
$128$		0			0
$129$	−11.8468			−1.04305
$130$		0			0
$131$	10.1208			0.884255			0.442127	−	0.896952i	$-0.354224\pi$
$131$	10.1208			0.884255			0.442127	+	0.896952i	$0.354224\pi$
$132$		0			0
$133$		−	8.09622i		−	0.702031i
$134$		0			0
$135$	11.4180	+	3.12008i	0.982705	+	0.268533i
$136$		0			0
$137$		−	5.40734i		−	0.461980i	−0.972956	−	0.230990i	$-0.925804\pi$
$137$		−	5.40734i		−	0.461980i	0.972956	−	0.230990i	$-0.0741965\pi$
$138$		0			0
$139$	−6.41028			−0.543713			−0.271856	−	0.962338i	$-0.587638\pi$
$139$	−6.41028			−0.543713			−0.271856	+	0.962338i	$0.587638\pi$
$140$		0			0
$141$	−4.38919			−0.369637
$142$		0			0
$143$			4.49172i			0.375617i
$144$		0			0
$145$	0.434758	−	1.59101i	0.0361047	−	0.132126i
$146$		0			0
$147$		−	18.4712i		−	1.52348i
$148$		0			0
$149$	−13.3334			−1.09232			−0.546159	−	0.837682i	$-0.683911\pi$
$149$	−13.3334			−1.09232			−0.546159	+	0.837682i	$0.683911\pi$
$150$		0			0
$151$	19.4208			1.58044			0.790222	−	0.612821i	$-0.209965\pi$
$151$	19.4208			1.58044			0.790222	+	0.612821i	$0.209965\pi$
$152$		0			0
$153$		−	0.359662i		−	0.0290769i
$154$		0			0
$155$	1.75610	−	6.42650i	0.141053	−	0.516189i
$156$		0			0
$157$		−	15.2879i		−	1.22011i	−0.792361	−	0.610053i	$-0.791148\pi$
$157$		−	15.2879i		−	1.22011i	0.792361	−	0.610053i	$-0.208852\pi$
$158$		0			0
$159$	−4.60716			−0.365371
$160$		0			0
$161$	4.22860			0.333261
$162$		0			0
$163$		−	14.4308i		−	1.13031i	−0.824986	−	0.565153i	$-0.808817\pi$
$163$		−	14.4308i		−	1.13031i	0.824986	−	0.565153i	$-0.191183\pi$
$164$		0			0
$165$	16.8125	+	4.59418i	1.30885	+	0.357656i
$166$		0			0
$167$			23.7935i			1.84120i	0.390511	+	0.920598i	$0.372298\pi$
$167$			23.7935i			1.84120i	−0.390511	+	0.920598i	$0.627702\pi$
$168$		0			0
$169$	12.0430			0.926386
$170$		0			0
$171$	−0.226516			−0.0173221
$172$		0			0
$173$			1.54396i			0.117385i	0.998276	+	0.0586927i	$0.0186932\pi$
$173$			1.54396i			0.117385i	−0.998276	+	0.0586927i	$0.981307\pi$
$174$		0			0
$175$	10.7522	−	18.2049i	0.812789	−	1.37616i
$176$		0			0
$177$		−	12.1123i		−	0.910415i
$178$		0			0
$179$	−13.0390			−0.974583			−0.487292	−	0.873239i	$-0.662015\pi$
$179$	−13.0390			−0.974583			−0.487292	+	0.873239i	$0.662015\pi$
$180$		0			0
$181$	1.77458			0.131903			0.0659517	−	0.997823i	$-0.478992\pi$
$181$	1.77458			0.131903			0.0659517	+	0.997823i	$0.478992\pi$
$182$		0			0
$183$		−	1.24198i		−	0.0918098i
$184$		0			0
$185$	19.2756	+	5.26724i	1.41717	+	0.387255i
$186$		0			0
$187$		−	13.9586i		−	1.02076i
$188$		0			0
$189$	−22.3841			−1.62820
$190$		0			0
$191$	11.8332			0.856222			0.428111	−	0.903726i	$-0.359179\pi$
$191$	11.8332			0.856222			0.428111	+	0.903726i	$0.359179\pi$
$192$		0			0
$193$		−	20.9549i		−	1.50837i	−0.656663	−	0.754184i	$-0.728033\pi$
$193$		−	20.9549i		−	1.50837i	0.656663	−	0.754184i	$-0.271967\pi$
$194$		0			0
$195$	−0.978808	+	3.58197i	−0.0700939	+	0.256511i
$196$		0			0
$197$		−	14.1752i		−	1.00994i	−0.863137	−	0.504970i	$-0.831504\pi$
$197$		−	14.1752i		−	1.00994i	0.863137	−	0.504970i	$-0.168496\pi$
$198$		0			0
$199$	20.6813			1.46606			0.733030	−	0.680197i	$-0.238106\pi$
$199$	20.6813			1.46606			0.733030	+	0.680197i	$0.238106\pi$
$200$		0			0
$201$	12.1660			0.858125
$202$		0			0
$203$			3.11905i			0.218914i
$204$		0			0
$205$	5.35332	−	19.5906i	0.373892	−	1.36827i
$206$		0			0
$207$		−	0.118308i		−	0.00822296i
$208$		0			0
$209$	−8.79118			−0.608098
$210$		0			0
$211$	4.31377			0.296972			0.148486	−	0.988914i	$-0.452560\pi$
$211$	4.31377			0.296972			0.148486	+	0.988914i	$0.452560\pi$
$212$		0			0
$213$		−	10.3343i		−	0.708094i
$214$		0			0
$215$	15.0530	+	4.11338i	1.02661	+	0.280530i
$216$		0			0
$217$			12.5987i			0.855252i
$218$		0			0
$219$	−10.1227			−0.684030
$220$		0			0
$221$	2.97394			0.200049
$222$		0			0
$223$		−	3.78248i		−	0.253293i	−0.991948	−	0.126647i	$-0.959579\pi$
$223$		−	3.78248i		−	0.253293i	0.991948	−	0.126647i	$-0.0404214\pi$
$224$		0			0
$225$	−0.509336	−	0.300824i	−0.0339557	−	0.0200550i
$226$		0			0
$227$		−	14.4058i		−	0.956148i	−0.878320	−	0.478074i	$-0.841335\pi$
$227$		−	14.4058i		−	0.956148i	0.878320	−	0.478074i	$-0.158665\pi$
$228$		0			0
$229$	−22.6701			−1.49808			−0.749041	−	0.662524i	$-0.769485\pi$
$229$	−22.6701			−1.49808			−0.749041	+	0.662524i	$0.769485\pi$
$230$		0			0
$231$	−32.9596			−2.16858
$232$		0			0
$233$			0.433977i			0.0284308i	0.999899	+	0.0142154i	$0.00452505\pi$
$233$			0.433977i			0.0284308i	−0.999899	+	0.0142154i	$0.995475\pi$
$234$		0			0
$235$	5.57709	+	1.52399i	0.363810	+	0.0994144i
$236$		0			0
$237$			3.49866i			0.227262i
$238$		0			0
$239$	−27.3045			−1.76618			−0.883090	−	0.469204i	$-0.844541\pi$
$239$	−27.3045			−1.76618			−0.883090	+	0.469204i	$0.844541\pi$
$240$		0			0
$241$	23.9763			1.54445			0.772226	−	0.635348i	$-0.219143\pi$
$241$	23.9763			1.54445			0.772226	+	0.635348i	$0.219143\pi$
$242$		0			0
$243$			1.22876i			0.0788253i
$244$		0			0
$245$	−6.41350	+	23.4704i	−0.409743	+	1.49947i
$246$		0			0
$247$		−	1.87300i		−	0.119176i
$248$		0			0
$249$	7.45918			0.472707
$250$		0			0
$251$	−5.25761			−0.331857			−0.165929	−	0.986138i	$-0.553062\pi$
$251$	−5.25761			−0.331857			−0.165929	+	0.986138i	$0.553062\pi$
$252$		0			0
$253$		−	4.59157i		−	0.288670i
$254$		0			0
$255$	3.04178	−	11.1315i	0.190483	−	0.697079i
$256$		0			0
$257$			20.6630i			1.28892i	0.764637	+	0.644461i	$0.222918\pi$
$257$			20.6630i			1.28892i	−0.764637	+	0.644461i	$0.777082\pi$
$258$		0			0
$259$	−37.7884			−2.34805
$260$		0			0
$261$	0.0872647			0.00540155
$262$		0			0
$263$		−	4.88893i		−	0.301464i	−0.988575	−	0.150732i	$-0.951837\pi$
$263$		−	4.88893i		−	0.301464i	0.988575	−	0.150732i	$-0.0481631\pi$
$264$		0			0
$265$	5.85405	+	1.59968i	0.359612	+	0.0982673i
$266$		0			0
$267$			12.3078i			0.753228i
$268$		0			0
$269$	−2.12989			−0.129862			−0.0649309	−	0.997890i	$-0.520683\pi$
$269$	−2.12989			−0.129862			−0.0649309	+	0.997890i	$0.520683\pi$
$270$		0			0
$271$	−25.6680			−1.55922			−0.779608	−	0.626267i	$-0.784582\pi$
$271$	−25.6680			−1.55922			−0.779608	+	0.626267i	$0.784582\pi$
$272$		0			0
$273$		−	7.02218i		−	0.425002i
$274$		0			0
$275$	−19.7675	−	11.6751i	−1.19203	−	0.704036i
$276$		0			0
$277$			14.0110i			0.841837i	0.907099	+	0.420918i	$0.138292\pi$
$277$			14.0110i			0.841837i	−0.907099	+	0.420918i	$0.861708\pi$
$278$		0			0
$279$	0.352485			0.0211027
$280$		0			0
$281$	−12.4269			−0.741324			−0.370662	−	0.928768i	$-0.620869\pi$
$281$	−12.4269			−0.741324			−0.370662	+	0.928768i	$0.620869\pi$
$282$		0			0
$283$		−	8.94768i		−	0.531884i	−0.963989	−	0.265942i	$-0.914317\pi$
$283$		−	8.94768i		−	0.531884i	0.963989	−	0.265942i	$-0.0856830\pi$
$284$		0			0
$285$	−7.01062	−	1.91572i	−0.415273	−	0.113477i
$286$		0			0
$287$			38.4059i			2.26703i
$288$		0			0
$289$	7.75808			0.456358
$290$		0			0
$291$	11.9603			0.701125
$292$		0			0
$293$		−	30.7264i		−	1.79506i	−0.440956	−	0.897529i	$-0.645360\pi$
$293$		−	30.7264i		−	1.79506i	0.440956	−	0.897529i	$-0.354640\pi$
$294$		0			0
$295$	−4.20557	+	15.3904i	−0.244858	+	0.896063i
$296$		0			0
$297$			24.3055i			1.41035i
$298$		0			0
$299$	0.978254			0.0565739
$300$		0			0
$301$	−29.5103			−1.70094
$302$		0			0
$303$			12.9553i			0.744261i
$304$		0			0
$305$	−0.431234	+	1.57811i	−0.0246924	+	0.0903625i
$306$		0			0
$307$			13.6480i			0.778933i	0.921041	+	0.389467i	$0.127341\pi$
$307$			13.6480i			0.778933i	−0.921041	+	0.389467i	$0.872659\pi$
$308$		0			0
$309$	−28.4944			−1.62099
$310$		0			0
$311$	−28.7646			−1.63109			−0.815545	−	0.578693i	$-0.803563\pi$
$311$	−28.7646			−1.63109			−0.815545	+	0.578693i	$0.803563\pi$
$312$		0			0
$313$		−	11.4851i		−	0.649176i	−0.945856	−	0.324588i	$-0.894774\pi$
$313$		−	11.4851i		−	0.649176i	0.945856	−	0.324588i	$-0.105226\pi$
$314$		0			0
$315$	1.07909	+	0.294872i	0.0607998	+	0.0166141i
$316$		0			0
$317$		−	4.51108i		−	0.253368i	−0.991943	−	0.126684i	$-0.959567\pi$
$317$		−	4.51108i		−	0.253368i	0.991943	−	0.126684i	$-0.0404334\pi$
$318$		0			0
$319$	3.38678			0.189623
$320$		0			0
$321$	26.0204			1.45232
$322$		0			0
$323$			5.82058i			0.323866i
$324$		0			0
$325$	2.48743	−	4.21155i	0.137978	−	0.233615i
$326$		0			0
$327$		−	25.3986i		−	1.40455i
$328$		0			0
$329$	−10.9335			−0.602781
$330$		0			0
$331$	32.0738			1.76293			0.881467	−	0.472246i	$-0.156557\pi$
$331$	32.0738			1.76293			0.881467	+	0.472246i	$0.156557\pi$
$332$		0			0
$333$			1.05724i			0.0579365i
$334$		0			0
$335$	−15.4587	−	4.22423i	−0.844597	−	0.230794i
$336$		0			0
$337$		−	32.9334i		−	1.79399i	−0.442036	−	0.896997i	$-0.645744\pi$
$337$		−	32.9334i		−	1.79399i	0.442036	−	0.896997i	$-0.354256\pi$
$338$		0			0
$339$	33.1834			1.80228
$340$		0			0
$341$	13.6801			0.740818
$342$		0			0
$343$		−	16.4116i		−	0.886143i
$344$		0			0
$345$	1.00057	−	3.66160i	0.0538687	−	0.197134i
$346$		0			0
$347$			11.9194i			0.639867i	0.947440	+	0.319933i	$0.103661\pi$
$347$			11.9194i			0.639867i	−0.947440	+	0.319933i	$0.896339\pi$
$348$		0			0
$349$	4.63550			0.248132			0.124066	−	0.992274i	$-0.460406\pi$
$349$	4.63550			0.248132			0.124066	+	0.992274i	$0.460406\pi$
$350$		0			0
$351$	−5.17838			−0.276402
$352$		0			0
$353$		−	16.2153i		−	0.863055i	−0.902100	−	0.431528i	$-0.857975\pi$
$353$		−	16.2153i		−	0.863055i	0.902100	−	0.431528i	$-0.142025\pi$
$354$		0			0
$355$	−3.58822	+	13.1312i	−0.190443	+	0.696931i
$356$		0			0
$357$			21.8224i			1.15496i
$358$		0			0
$359$	33.5549			1.77096			0.885481	−	0.464676i	$-0.153829\pi$
$359$	33.5549			1.77096			0.885481	+	0.464676i	$0.153829\pi$
$360$		0			0
$361$	−15.3342			−0.807062
$362$		0			0
$363$			17.1157i			0.898341i
$364$		0			0
$365$	12.8624	+	3.51476i	0.673246	+	0.183971i
$366$		0			0
$367$		−	4.44019i		−	0.231776i	−0.993262	−	0.115888i	$-0.963029\pi$
$367$		−	4.44019i		−	0.231776i	0.993262	−	0.115888i	$-0.0369714\pi$
$368$		0			0
$369$	1.07452			0.0559372
$370$		0			0
$371$	−11.4764			−0.595826
$372$		0			0
$373$		−	29.3416i		−	1.51925i	−0.650362	−	0.759624i	$-0.725383\pi$
$373$		−	29.3416i		−	1.51925i	0.650362	−	0.759624i	$-0.274617\pi$
$374$		0			0
$375$	−13.2197	−	13.6181i	−0.682661	−	0.703234i
$376$		0			0
$377$			0.721567i			0.0371626i
$378$		0			0
$379$	−18.7557			−0.963415			−0.481708	−	0.876332i	$-0.659983\pi$
$379$	−18.7557			−0.963415			−0.481708	+	0.876332i	$0.659983\pi$
$380$		0			0
$381$	35.0493			1.79563
$382$		0			0
$383$			29.1684i			1.49043i	0.666822	+	0.745217i	$0.267654\pi$
$383$			29.1684i			1.49043i	−0.666822	+	0.745217i	$0.732346\pi$
$384$		0			0
$385$	41.8799	+	11.4441i	2.13440	+	0.583245i
$386$		0			0
$387$			0.825638i			0.0419695i
$388$		0			0
$389$	27.2943			1.38388			0.691939	−	0.721956i	$-0.256757\pi$
$389$	27.2943			1.38388			0.691939	+	0.721956i	$0.256757\pi$
$390$		0			0
$391$	−3.04005			−0.153742
$392$		0			0
$393$			17.1805i			0.866644i
$394$		0			0
$395$	1.21479	−	4.44554i	0.0611225	−	0.223680i
$396$		0			0
$397$			22.8548i			1.14705i	0.819188	+	0.573525i	$0.194424\pi$
$397$			22.8548i			1.14705i	−0.819188	+	0.573525i	$0.805576\pi$
$398$		0			0
$399$	13.7438			0.688049
$400$		0			0
$401$	19.6913			0.983337			0.491668	−	0.870783i	$-0.336387\pi$
$401$	19.6913			0.983337			0.491668	+	0.870783i	$0.336387\pi$
$402$		0			0
$403$			2.91460i			0.145186i
$404$		0			0
$405$	−5.08730	+	18.6171i	−0.252790	+	0.925092i
$406$		0			0
$407$			41.0320i			2.03388i
$408$		0			0
$409$	33.8863			1.67557			0.837785	−	0.546001i	$-0.183851\pi$
$409$	33.8863			1.67557			0.837785	+	0.546001i	$0.183851\pi$
$410$		0			0
$411$	9.17925			0.452779
$412$		0			0
$413$		−	30.1717i		−	1.48465i
$414$		0			0
$415$	−9.47796	−	2.58994i	−0.465255	−	0.127135i
$416$		0			0
$417$		−	10.8818i		−	0.532884i
$418$		0			0
$419$	−15.8605			−0.774837			−0.387419	−	0.921904i	$-0.626633\pi$
$419$	−15.8605			−0.774837			−0.387419	+	0.921904i	$0.626633\pi$
$420$		0			0
$421$	−13.9043			−0.677652			−0.338826	−	0.940849i	$-0.610030\pi$
$421$	−13.9043			−0.677652			−0.338826	+	0.940849i	$0.610030\pi$
$422$		0			0
$423$			0.305896i			0.0148732i
$424$		0			0
$425$	−7.73002	+	13.0880i	−0.374961	+	0.634859i
$426$		0			0
$427$		−	3.09377i		−	0.149718i
$428$		0			0
$429$	−7.62495			−0.368136
$430$		0			0
$431$	−0.736783			−0.0354896			−0.0177448	−	0.999843i	$-0.505649\pi$
$431$	−0.736783			−0.0354896			−0.0177448	+	0.999843i	$0.505649\pi$
$432$		0			0
$433$		−	3.79670i		−	0.182458i	−0.995830	−	0.0912288i	$-0.970921\pi$
$433$		−	3.79670i		−	0.182458i	0.995830	−	0.0912288i	$-0.0290794\pi$
$434$		0			0
$435$	2.70082	+	0.738025i	0.129495	+	0.0353856i
$436$		0			0
$437$			1.91463i			0.0915893i
$438$		0			0
$439$	5.74582			0.274233			0.137117	−	0.990555i	$-0.456217\pi$
$439$	5.74582			0.274233			0.137117	+	0.990555i	$0.456217\pi$
$440$		0			0
$441$	−1.28732			−0.0613009
$442$		0			0
$443$			6.53509i			0.310492i	0.987876	+	0.155246i	$0.0496170\pi$
$443$			6.53509i			0.310492i	−0.987876	+	0.155246i	$0.950383\pi$
$444$		0			0
$445$	4.27347	−	15.6389i	0.202582	−	0.741354i
$446$		0			0
$447$		−	22.6342i		−	1.07056i
$448$		0			0
$449$	−4.53781			−0.214152			−0.107076	−	0.994251i	$-0.534149\pi$
$449$	−4.53781			−0.214152			−0.107076	+	0.994251i	$0.534149\pi$
$450$		0			0
$451$	41.7025			1.96369
$452$		0			0
$453$			32.9679i			1.54897i
$454$		0			0
$455$	−2.43821	+	8.92269i	−0.114305	+	0.418302i
$456$		0			0
$457$			15.7911i			0.738676i	0.929295	+	0.369338i	$0.120416\pi$
$457$			15.7911i			0.738676i	−0.929295	+	0.369338i	$0.879584\pi$
$458$		0			0
$459$	16.0925			0.751134
$460$		0			0
$461$	15.6324			0.728074			0.364037	−	0.931384i	$-0.381398\pi$
$461$	15.6324			0.728074			0.364037	+	0.931384i	$0.381398\pi$
$462$		0			0
$463$			4.29485i			0.199598i	0.995008	+	0.0997992i	$0.0318201\pi$
$463$			4.29485i			0.199598i	−0.995008	+	0.0997992i	$0.968180\pi$
$464$		0			0
$465$	10.9093	+	2.98108i	0.505908	+	0.138244i
$466$		0			0
$467$			29.7617i			1.37720i	0.725139	+	0.688602i	$0.241775\pi$
$467$			29.7617i			1.37720i	−0.725139	+	0.688602i	$0.758225\pi$
$468$		0			0
$469$	30.3055			1.39938
$470$		0			0
$471$	25.9520			1.19581
$472$		0			0
$473$			32.0433i			1.47335i
$474$		0			0
$475$	8.24283	+	4.86839i	0.378207	+	0.223377i
$476$		0			0
$477$			0.321087i			0.0147016i
$478$		0			0
$479$	−36.8371			−1.68313			−0.841564	−	0.540157i	$-0.818365\pi$
$479$	−36.8371			−1.68313			−0.841564	+	0.540157i	$0.818365\pi$
$480$		0			0
$481$	−8.74203			−0.398602
$482$		0			0
$483$			7.17829i			0.326623i
$484$		0			0
$485$	−15.1973	−	4.15279i	−0.690072	−	0.188569i
$486$		0			0
$487$		−	26.0340i		−	1.17971i	−0.807508	−	0.589857i	$-0.799184\pi$
$487$		−	26.0340i		−	1.17971i	0.807508	−	0.589857i	$-0.200816\pi$
$488$		0			0
$489$	24.4970			1.10779
$490$		0			0
$491$	−12.2553			−0.553074			−0.276537	−	0.961003i	$-0.589187\pi$
$491$	−12.2553			−0.553074			−0.276537	+	0.961003i	$0.589187\pi$
$492$		0			0
$493$		−	2.24236i		−	0.100991i
$494$		0			0
$495$	0.320182	−	1.17172i	0.0143911	−	0.0526647i
$496$		0			0
$497$		−	25.7427i		−	1.15472i
$498$		0			0
$499$	−32.4229			−1.45145			−0.725724	−	0.687986i	$-0.758495\pi$
$499$	−32.4229			−1.45145			−0.725724	+	0.687986i	$0.758495\pi$
$500$		0			0
$501$	−40.3908			−1.80453
$502$		0			0
$503$		−	32.1726i		−	1.43451i	−0.696813	−	0.717253i	$-0.745399\pi$
$503$		−	32.1726i		−	1.43451i	0.696813	−	0.717253i	$-0.254601\pi$
$504$		0			0
$505$	4.49827	−	16.4615i	0.200170	−	0.732528i
$506$		0			0
$507$			20.4437i			0.907936i
$508$		0			0
$509$	−25.1343			−1.11406			−0.557030	−	0.830492i	$-0.688059\pi$
$509$	−25.1343			−1.11406			−0.557030	+	0.830492i	$0.688059\pi$
$510$		0			0
$511$	−25.2156			−1.11547
$512$		0			0
$513$		−	10.1351i		−	0.447476i
$514$		0			0
$515$	36.2062	+	9.89368i	1.59543	+	0.435968i
$516$		0			0
$517$			11.8720i			0.522128i
$518$		0			0
$519$	−2.62097			−0.115048
$520$		0			0
$521$	9.45730			0.414332			0.207166	−	0.978306i	$-0.433576\pi$
$521$	9.45730			0.414332			0.207166	+	0.978306i	$0.433576\pi$
$522$		0			0
$523$			39.9479i			1.74680i	0.487003	+	0.873400i	$0.338090\pi$
$523$			39.9479i			1.74680i	−0.487003	+	0.873400i	$0.661910\pi$
$524$		0			0
$525$	30.9038	+	18.2524i	1.34875	+	0.796601i
$526$		0			0
$527$		−	9.05749i		−	0.394551i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	−0.844143			−0.0366327
$532$		0			0
$533$			8.88489i			0.384847i
$534$		0			0
$535$	−33.0626	−	9.03467i	−1.42942	−	0.390603i
$536$		0			0
$537$		−	22.1345i		−	0.955173i
$538$		0			0
$539$	−49.9613			−2.15199
$540$		0			0
$541$	−40.1829			−1.72760			−0.863800	−	0.503835i	$-0.831922\pi$
$541$	−40.1829			−1.72760			−0.863800	+	0.503835i	$0.831922\pi$
$542$		0			0
$543$			3.01244i			0.129276i
$544$		0			0
$545$	−8.81878	+	32.2726i	−0.377755	+	1.38241i
$546$		0			0
$547$		−	2.86104i		−	0.122329i	−0.998128	−	0.0611647i	$-0.980519\pi$
$547$		−	2.86104i		−	0.122329i	0.998128	−	0.0611647i	$-0.0194815\pi$
$548$		0			0
$549$	−0.0865574			−0.00369418
$550$		0			0
$551$	−1.41225			−0.0601637
$552$		0			0
$553$			8.71514i			0.370606i
$554$		0			0
$555$	−8.94143	+	32.7214i	−0.379543	+	1.38895i
$556$		0			0
$557$			15.4417i			0.654286i	0.944975	+	0.327143i	$0.106086\pi$
$557$			15.4417i			0.654286i	−0.944975	+	0.327143i	$0.893914\pi$
$558$		0			0
$559$	−6.82696			−0.288750
$560$		0			0
$561$	23.6955			1.00043
$562$		0			0
$563$		−	17.5808i		−	0.740941i	−0.928844	−	0.370470i	$-0.879197\pi$
$563$		−	17.5808i		−	0.740941i	0.928844	−	0.370470i	$-0.120803\pi$
$564$		0			0
$565$	−42.1643	−	11.5218i	−1.77386	−	0.484725i
$566$		0			0
$567$		−	36.4974i		−	1.53275i
$568$		0			0
$569$	43.1463			1.80879			0.904394	−	0.426698i	$-0.140323\pi$
$569$	43.1463			1.80879			0.904394	+	0.426698i	$0.140323\pi$
$570$		0			0
$571$	−13.9876			−0.585365			−0.292682	−	0.956210i	$-0.594548\pi$
$571$	−13.9876			−0.585365			−0.292682	+	0.956210i	$0.594548\pi$
$572$		0			0
$573$			20.0875i			0.839169i
$574$		0			0
$575$	−2.54273	+	4.30518i	−0.106039	+	0.179538i
$576$		0			0
$577$			41.5627i			1.73028i	0.501533	+	0.865139i	$0.332770\pi$
$577$			41.5627i			1.73028i	−0.501533	+	0.865139i	$0.667230\pi$
$578$		0			0
$579$	35.5721			1.47833
$580$		0			0
$581$	18.5808			0.770862
$582$		0			0
$583$			12.4615i			0.516103i
$584$		0			0
$585$	0.249639	+	0.0682162i	0.0103213	+	0.00282039i
$586$		0			0
$587$			40.4457i			1.66937i	0.550724	+	0.834687i	$0.314352\pi$
$587$			40.4457i			1.66937i	−0.550724	+	0.834687i	$0.685648\pi$
$588$		0			0
$589$	−5.70443			−0.235047
$590$		0			0
$591$	24.0631			0.989825
$592$		0			0
$593$		−	28.0011i		−	1.14987i	−0.818200	−	0.574934i	$-0.805028\pi$
$593$		−	28.0011i		−	1.14987i	0.818200	−	0.574934i	$-0.194972\pi$
$594$		0			0
$595$	7.57705	−	27.7284i	0.310629	−	1.13675i
$596$		0			0
$597$			35.1077i			1.43686i
$598$		0			0
$599$	6.61659			0.270347			0.135173	−	0.990822i	$-0.456841\pi$
$599$	6.61659			0.270347			0.135173	+	0.990822i	$0.456841\pi$
$600$		0			0
$601$	−44.3974			−1.81101			−0.905504	−	0.424337i	$-0.860507\pi$
$601$	−44.3974			−1.81101			−0.905504	+	0.424337i	$0.860507\pi$
$602$		0			0
$603$		−	0.847887i		−	0.0345286i
$604$		0			0
$605$	5.94283	−	21.7479i	0.241610	−	0.884179i
$606$		0			0
$607$		−	23.0131i		−	0.934074i	−0.884238	−	0.467037i	$-0.845322\pi$
$607$		−	23.0131i		−	0.934074i	0.884238	−	0.467037i	$-0.154678\pi$
$608$		0			0
$609$	−5.29475			−0.214554
$610$		0			0
$611$	−2.52937			−0.102327
$612$		0			0
$613$			7.77554i			0.314051i	0.987595	+	0.157025i	$0.0501905\pi$
$613$			7.77554i			0.314051i	−0.987595	+	0.157025i	$0.949810\pi$
$614$		0			0
$615$	33.2561	+	9.08755i	1.34102	+	0.366445i
$616$		0			0
$617$		−	14.6868i		−	0.591268i	−0.955301	−	0.295634i	$-0.904469\pi$
$617$		−	14.6868i		−	0.591268i	0.955301	−	0.295634i	$-0.0955309\pi$
$618$		0			0
$619$	−18.1633			−0.730045			−0.365022	−	0.930999i	$-0.618939\pi$
$619$	−18.1633			−0.730045			−0.365022	+	0.930999i	$0.618939\pi$
$620$		0			0
$621$	5.29350			0.212421
$622$		0			0
$623$			30.6588i			1.22832i
$624$		0			0
$625$	12.0691	+	21.8938i	0.482763	+	0.875751i
$626$		0			0
$627$		−	14.9235i		−	0.595987i
$628$		0			0
$629$	27.1670			1.08322
$630$		0			0
$631$	11.0472			0.439780			0.219890	−	0.975525i	$-0.429430\pi$
$631$	11.0472			0.439780			0.219890	+	0.975525i	$0.429430\pi$
$632$		0			0
$633$			7.32286i			0.291058i
$634$		0			0
$635$	−44.5351	−	12.1696i	−1.76732	−	0.482938i
$636$		0			0
$637$		−	10.6445i		−	0.421749i
$638$		0			0
$639$	−0.720228			−0.0284918
$640$		0			0
$641$	44.7133			1.76607			0.883035	−	0.469306i	$-0.155496\pi$
$641$	44.7133			1.76607			0.883035	+	0.469306i	$0.155496\pi$
$642$		0			0
$643$			8.30391i			0.327474i	0.986504	+	0.163737i	$0.0523549\pi$
$643$			8.30391i			0.327474i	−0.986504	+	0.163737i	$0.947645\pi$
$644$		0			0
$645$	−6.98268	+	25.5533i	−0.274943	+	1.00616i
$646$		0			0
$647$		−	50.0603i		−	1.96807i	−0.177971	−	0.984036i	$-0.556953\pi$
$647$		−	50.0603i		−	1.96807i	0.177971	−	0.984036i	$-0.443047\pi$
$648$		0			0
$649$	−32.7615			−1.28600
$650$		0			0
$651$	−21.3869			−0.838219
$652$		0			0
$653$			22.4150i			0.877166i	0.898691	+	0.438583i	$0.144519\pi$
$653$			22.4150i			0.877166i	−0.898691	+	0.438583i	$0.855481\pi$
$654$		0			0
$655$	5.96534	−	21.8303i	0.233085	−	0.852982i
$656$		0			0
$657$			0.705483i			0.0275235i
$658$		0			0
$659$	32.3712			1.26100			0.630502	−	0.776187i	$-0.282849\pi$
$659$	32.3712			1.26100			0.630502	+	0.776187i	$0.282849\pi$
$660$		0			0
$661$	−23.2386			−0.903875			−0.451938	−	0.892050i	$-0.649267\pi$
$661$	−23.2386			−0.903875			−0.451938	+	0.892050i	$0.649267\pi$
$662$		0			0
$663$			5.04843i			0.196065i
$664$		0			0
$665$	−17.4634	−	4.77205i	−0.677203	−	0.185052i
$666$		0			0
$667$		−	0.737607i		−	0.0285602i
$668$		0			0
$669$	6.42096			0.248249
$670$		0			0
$671$	−3.35933			−0.129685
$672$		0			0
$673$		−	3.02445i		−	0.116584i	−0.998300	−	0.0582919i	$-0.981435\pi$
$673$		−	3.02445i		−	0.116584i	0.998300	−	0.0582919i	$-0.0185654\pi$
$674$		0			0
$675$	13.4599	−	22.7894i	0.518073	−	0.877166i
$676$		0			0
$677$		−	5.34663i		−	0.205488i	−0.994708	−	0.102744i	$-0.967238\pi$
$677$		−	5.34663i		−	0.205488i	0.994708	−	0.102744i	$-0.0327622\pi$
$678$		0			0
$679$	29.7931			1.14335
$680$		0			0
$681$	24.4547			0.937105
$682$		0			0
$683$			21.7778i			0.833305i	0.909066	+	0.416653i	$0.136797\pi$
$683$			21.7778i			0.833305i	−0.909066	+	0.416653i	$0.863203\pi$
$684$		0			0
$685$	−11.6636	−	3.18718i	−0.445641	−	0.121776i
$686$		0			0
$687$		−	38.4837i		−	1.46825i
$688$		0			0
$689$	−2.65498			−0.101147
$690$		0			0
$691$	17.1839			0.653705			0.326852	−	0.945075i	$-0.394012\pi$
$691$	17.1839			0.653705			0.326852	+	0.945075i	$0.394012\pi$
$692$		0			0
$693$			2.29706i			0.0872580i
$694$		0			0
$695$	−3.77833	+	13.8269i	−0.143320	+	0.524483i
$696$		0			0
$697$		−	27.6110i		−	1.04584i
$698$		0			0
$699$	−0.736700			−0.0278646
$700$		0			0
$701$	−10.9444			−0.413363			−0.206682	−	0.978408i	$-0.566266\pi$
$701$	−10.9444			−0.413363			−0.206682	+	0.978408i	$0.566266\pi$
$702$		0			0
$703$		−	17.1099i		−	0.645310i
$704$		0			0
$705$	−2.58706	+	9.46742i	−0.0974344	+	0.356564i
$706$		0			0
$707$			32.2715i			1.21370i
$708$		0			0
$709$	35.1463			1.31995			0.659973	−	0.751290i	$-0.270568\pi$
$709$	35.1463			1.31995			0.659973	+	0.751290i	$0.270568\pi$
$710$		0			0
$711$	0.243832			0.00914442
$712$		0			0
$713$		−	2.97939i		−	0.111579i
$714$		0			0
$715$	9.68859	+	2.64750i	0.362333	+	0.0990108i
$716$		0			0
$717$		−	46.3508i		−	1.73100i
$718$		0			0
$719$	−7.07135			−0.263717			−0.131858	−	0.991269i	$-0.542094\pi$
$719$	−7.07135			−0.263717			−0.131858	+	0.991269i	$0.542094\pi$
$720$		0			0
$721$	−70.9794			−2.64341
$722$		0			0
$723$			40.7012i			1.51369i
$724$		0			0
$725$	−3.17553	−	1.87553i	−0.117936	−	0.0696555i
$726$		0			0
$727$		−	1.75083i		−	0.0649345i	−0.999473	−	0.0324673i	$-0.989664\pi$
$727$		−	1.75083i		−	0.0649345i	0.999473	−	0.0324673i	$-0.0103365\pi$
$728$		0			0
$729$	−27.9791			−1.03626
$730$		0			0
$731$	21.2157			0.784691
$732$		0			0
$733$		−	29.8141i		−	1.10121i	−0.834766	−	0.550605i	$-0.814397\pi$
$733$		−	29.8141i		−	1.10121i	0.834766	−	0.550605i	$-0.185603\pi$
$734$		0			0
$735$	−39.8422	−	10.8873i	−1.46960	−	0.401583i
$736$		0			0
$737$		−	32.9069i		−	1.21214i
$738$		0			0
$739$	−14.2585			−0.524508			−0.262254	−	0.964999i	$-0.584466\pi$
$739$	−14.2585			−0.524508			−0.262254	+	0.964999i	$0.584466\pi$
$740$		0			0
$741$	3.17951			0.116802
$742$		0			0
$743$			1.02428i			0.0375772i	0.999823	+	0.0187886i	$0.00598095\pi$
$743$			1.02428i			0.0375772i	−0.999823	+	0.0187886i	$0.994019\pi$
$744$		0			0
$745$	−7.85895	+	28.7600i	−0.287930	+	1.05369i
$746$		0			0
$747$		−	0.519854i		−	0.0190204i
$748$		0			0
$749$	64.8167			2.36835
$750$		0			0
$751$	21.1173			0.770582			0.385291	−	0.922795i	$-0.374101\pi$
$751$	21.1173			0.770582			0.385291	+	0.922795i	$0.374101\pi$
$752$		0			0
$753$		−	8.92508i		−	0.325248i
$754$		0			0
$755$	11.4470	−	41.8904i	0.416597	−	1.52455i
$756$		0			0
$757$			20.6048i			0.748895i	0.927248	+	0.374447i	$0.122168\pi$
$757$			20.6048i			0.748895i	−0.927248	+	0.374447i	$0.877832\pi$
$758$		0			0
$759$	7.79445			0.282921
$760$		0			0
$761$	−42.4576			−1.53909			−0.769543	−	0.638595i	$-0.779516\pi$
$761$	−42.4576			−1.53909			−0.769543	+	0.638595i	$0.779516\pi$
$762$		0			0
$763$		−	63.2679i		−	2.29045i
$764$		0			0
$765$	−0.775786	−	0.211991i	−0.0280486	−	0.00766454i
$766$		0			0
$767$		−	6.97997i		−	0.252032i
$768$		0			0
$769$	−14.2501			−0.513872			−0.256936	−	0.966428i	$-0.582713\pi$
$769$	−14.2501			−0.513872			−0.256936	+	0.966428i	$0.582713\pi$
$770$		0			0
$771$	−35.0766			−1.26325
$772$		0			0
$773$			8.71017i			0.313283i	0.987656	+	0.156642i	$0.0500667\pi$
$773$			8.71017i			0.313283i	−0.987656	+	0.156642i	$0.949933\pi$
$774$		0			0
$775$	−12.8268	−	7.57577i	−0.460752	−	0.272130i
$776$		0			0
$777$		−	64.1478i		−	2.30129i
$778$		0			0
$779$	−17.3895			−0.623042
$780$		0			0
$781$	−27.9524			−1.00021
$782$		0			0
$783$			3.90452i			0.139536i
$784$		0			0
$785$	−32.9757	−	9.01093i	−1.17695	−	0.321614i
$786$		0			0
$787$		−	19.5185i		−	0.695760i	−0.937539	−	0.347880i	$-0.886902\pi$
$787$		−	19.5185i		−	0.695760i	0.937539	−	0.347880i	$-0.113098\pi$
$788$		0			0
$789$	8.29922			0.295460
$790$		0			0
$791$	82.6598			2.93904
$792$		0			0
$793$		−	0.715718i		−	0.0254159i
$794$		0			0
$795$	−2.71554	+	9.93758i	−0.0963101	+	0.352450i
$796$		0			0
$797$			26.8913i			0.952538i	0.879300	+	0.476269i	$0.158011\pi$
$797$			26.8913i			0.952538i	−0.879300	+	0.476269i	$0.841989\pi$
$798$		0			0
$799$	7.86035			0.278079
$800$		0			0
$801$	0.857772			0.0303079
$802$		0			0
$803$			27.3801i			0.966222i
$804$		0			0
$805$	2.49241	−	9.12104i	0.0878459	−	0.321474i
$806$		0			0
$807$		−	3.61561i		−	0.127275i
$808$		0			0
$809$	−16.5468			−0.581755			−0.290878	−	0.956760i	$-0.593947\pi$
$809$	−16.5468			−0.581755			−0.290878	+	0.956760i	$0.593947\pi$
$810$		0			0
$811$	−25.3327			−0.889551			−0.444775	−	0.895642i	$-0.646716\pi$
$811$	−25.3327			−0.889551			−0.444775	+	0.895642i	$0.646716\pi$
$812$		0			0
$813$		−	43.5728i		−	1.52816i
$814$		0			0
$815$	−31.1270	−	8.50574i	−1.09033	−	0.297943i
$816$		0			0
$817$		−	13.3617i		−	0.467466i
$818$		0			0
$819$	−0.489398			−0.0171009
$820$		0			0
$821$	41.5845			1.45131			0.725655	−	0.688059i	$-0.241537\pi$
$821$	41.5845			1.45131			0.725655	+	0.688059i	$0.241537\pi$
$822$		0			0
$823$			43.2049i			1.50603i	0.658003	+	0.753015i	$0.271401\pi$
$823$			43.2049i			1.50603i	−0.658003	+	0.753015i	$0.728599\pi$
$824$		0			0
$825$	19.8192	−	33.5565i	0.690014	−	1.16829i
$826$		0			0
$827$		−	25.6851i		−	0.893159i	−0.894744	−	0.446580i	$-0.852642\pi$
$827$		−	25.6851i		−	0.893159i	0.894744	−	0.446580i	$-0.147358\pi$
$828$		0			0
$829$	−39.3428			−1.36643			−0.683215	−	0.730217i	$-0.739419\pi$
$829$	−39.3428			−1.36643			−0.683215	+	0.730217i	$0.739419\pi$
$830$		0			0
$831$	−23.7844			−0.825070
$832$		0			0
$833$			33.0791i			1.14612i
$834$		0			0
$835$	51.3222	+	14.0243i	1.77608	+	0.485330i
$836$		0			0
$837$			15.7714i			0.545139i
$838$		0			0
$839$	−32.7583			−1.13094			−0.565471	−	0.824768i	$-0.691306\pi$
$839$	−32.7583			−1.13094			−0.565471	+	0.824768i	$0.691306\pi$
$840$		0			0
$841$	−28.4559			−0.981239
$842$		0			0
$843$		−	21.0953i		−	0.726560i
$844$		0			0
$845$	7.09836	−	25.9766i	0.244191	−	0.893623i
$846$		0			0
$847$			42.6351i			1.46496i
$848$		0			0
$849$	15.1892			0.521291
$850$		0			0
$851$	8.93637			0.306335
$852$		0			0
$853$			2.15477i			0.0737778i	0.999319	+	0.0368889i	$0.0117448\pi$
$853$			2.15477i			0.0737778i	−0.999319	+	0.0368889i	$0.988255\pi$
$854$		0			0
$855$	−0.133512	+	0.488592i	−0.00456602	+	0.0167095i
$856$		0			0
$857$			31.0312i			1.06000i	0.847996	+	0.530002i	$0.177809\pi$
$857$			31.0312i			1.06000i	−0.847996	+	0.530002i	$0.822191\pi$
$858$		0			0
$859$	23.0328			0.785868			0.392934	−	0.919567i	$-0.371460\pi$
$859$	23.0328			0.785868			0.392934	+	0.919567i	$0.371460\pi$
$860$		0			0
$861$	−65.1961			−2.22188
$862$		0			0
$863$			18.9638i			0.645536i	0.946478	+	0.322768i	$0.104613\pi$
$863$			18.9638i			0.645536i	−0.946478	+	0.322768i	$0.895387\pi$
$864$		0			0
$865$	3.33031	+	0.910039i	0.113234	+	0.0309422i
$866$		0			0
$867$			13.1698i			0.447269i
$868$		0			0
$869$	9.46322			0.321018
$870$		0			0
$871$	7.01094			0.237557
$872$		0			0
$873$		−	0.833550i		−	0.0282114i
$874$		0			0
$875$	−32.9302	−	33.9226i	−1.11324	−	1.14679i
$876$		0			0
$877$			26.9808i			0.911076i	0.890216	+	0.455538i	$0.150553\pi$
$877$			26.9808i			0.911076i	−0.890216	+	0.455538i	$0.849447\pi$
$878$		0			0
$879$	52.1598			1.75931
$880$		0			0
$881$	8.64313			0.291195			0.145597	−	0.989344i	$-0.453490\pi$
$881$	8.64313			0.291195			0.145597	+	0.989344i	$0.453490\pi$
$882$		0			0
$883$			19.2554i			0.647996i	0.946058	+	0.323998i	$0.105027\pi$
$883$			19.2554i			0.647996i	−0.946058	+	0.323998i	$0.894973\pi$
$884$		0			0
$885$	−26.1260	−	7.13918i	−0.878217	−	0.239981i
$886$		0			0
$887$			38.0027i			1.27600i	0.770034	+	0.638002i	$0.220239\pi$
$887$			38.0027i			1.27600i	−0.770034	+	0.638002i	$0.779761\pi$
$888$		0			0
$889$	87.3077			2.92821
$890$		0			0
$891$	−39.6302			−1.32766
$892$		0			0
$893$		−	4.95047i		−	0.165661i
$894$		0			0
$895$	−7.68542	+	28.1250i	−0.256895	+	0.940115i
$896$		0			0
$897$			1.66064i			0.0554471i
$898$		0			0
$899$	2.19762			0.0732946
$900$		0			0
$901$	8.25069			0.274870
$902$		0			0
$903$		−	50.0953i		−	1.66707i
$904$		0			0
$905$	1.04597	−	3.82774i	0.0347691	−	0.127238i
$906$		0			0
$907$		−	33.1430i		−	1.10050i	−0.835001	−	0.550248i	$-0.814533\pi$
$907$		−	33.1430i		−	1.10050i	0.835001	−	0.550248i	$-0.185467\pi$
$908$		0			0
$909$	0.902893			0.0299471
$910$		0			0
$911$	14.9983			0.496916			0.248458	−	0.968643i	$-0.420076\pi$
$911$	14.9983			0.496916			0.248458	+	0.968643i	$0.420076\pi$
$912$		0			0
$913$		−	20.1757i		−	0.667719i
$914$		0			0
$915$	−2.67893	−	0.732044i	−0.0885628	−	0.0242006i
$916$		0			0
$917$			42.7967i			1.41327i
$918$		0			0
$919$	21.9030			0.722513			0.361257	−	0.932466i	$-0.382348\pi$
$919$	21.9030			0.722513			0.361257	+	0.932466i	$0.382348\pi$
$920$		0			0
$921$	−23.1682			−0.763420
$922$		0			0
$923$		−	5.95536i		−	0.196023i
$924$		0			0
$925$	22.7227	−	38.4726i	0.747119	−	1.26497i
$926$		0			0
$927$			1.98586i			0.0652242i
$928$		0			0
$929$	13.5384			0.444181			0.222091	−	0.975026i	$-0.428712\pi$
$929$	13.5384			0.444181			0.222091	+	0.975026i	$0.428712\pi$
$930$		0			0
$931$	20.8333			0.682783
$932$		0			0
$933$		−	48.8295i		−	1.59861i
$934$		0			0
$935$	−30.1086	−	8.22745i	−0.984655	−	0.269066i
$936$		0			0
$937$		−	16.6360i		−	0.543474i	−0.962372	−	0.271737i	$-0.912402\pi$
$937$		−	16.6360i		−	0.543474i	0.962372	−	0.271737i	$-0.0875981\pi$
$938$		0			0
$939$	19.4966			0.636247
$940$		0			0
$941$	40.4273			1.31789			0.658945	−	0.752191i	$-0.271003\pi$
$941$	40.4273			1.31789			0.658945	+	0.752191i	$0.271003\pi$
$942$		0			0
$943$		−	9.08240i		−	0.295764i
$944$		0			0
$945$	−13.1936	+	48.2822i	−0.429187	+	1.57062i
$946$		0			0
$947$		−	48.1251i		−	1.56386i	−0.623369	−	0.781928i	$-0.714237\pi$
$947$		−	48.1251i		−	1.56386i	0.623369	−	0.781928i	$-0.285763\pi$
$948$		0			0
$949$	−5.83344			−0.189361
$950$		0			0
$951$	7.65781			0.248322
$952$		0			0
$953$		−	38.1482i		−	1.23574i	−0.786279	−	0.617871i	$-0.787995\pi$
$953$		−	38.1482i		−	1.23574i	0.786279	−	0.617871i	$-0.212005\pi$
$954$		0			0
$955$	6.97470	−	25.5241i	0.225696	−	0.825940i
$956$		0			0
$957$			5.74924i			0.185846i
$958$		0			0
$959$	22.8655			0.738365
$960$		0			0
$961$	−22.1232			−0.713653
$962$		0			0
$963$		−	1.81344i		−	0.0584373i
$964$		0			0
$965$	−45.1995	−	12.3512i	−1.45502	−	0.397598i
$966$		0			0
$967$			29.0361i			0.933737i	0.884327	+	0.466869i	$0.154618\pi$
$967$			29.0361i			0.933737i	−0.884327	+	0.466869i	$0.845382\pi$
$968$		0			0
$969$	−9.88076			−0.317416
$970$		0			0
$971$	27.8281			0.893046			0.446523	−	0.894772i	$-0.352662\pi$
$971$	27.8281			0.893046			0.446523	+	0.894772i	$0.352662\pi$
$972$		0			0
$973$		−	27.1065i		−	0.868995i
$974$		0			0
$975$	7.14934	+	4.22255i	0.228962	+	0.135230i
$976$		0			0
$977$		−	55.1406i		−	1.76411i	−0.471151	−	0.882053i	$-0.656161\pi$
$977$		−	55.1406i		−	1.76411i	0.471151	−	0.882053i	$-0.343839\pi$
$978$		0			0
$979$	33.2905			1.06397
$980$		0			0
$981$	−1.77011			−0.0565152
$982$		0			0
$983$		−	11.1171i		−	0.354581i	−0.984159	−	0.177290i	$-0.943267\pi$
$983$		−	11.1171i		−	0.354581i	0.984159	−	0.177290i	$-0.0567331\pi$
$984$		0			0
$985$	−30.5757	−	8.35509i	−0.974222	−	0.266215i
$986$		0			0
$987$		−	18.5602i		−	0.590776i
$988$		0			0
$989$	6.97873			0.221911
$990$		0			0
$991$	−40.3976			−1.28327			−0.641637	−	0.767009i	$-0.721744\pi$
$991$	−40.3976			−1.28327			−0.641637	+	0.767009i	$0.721744\pi$
$992$		0			0
$993$			54.4470i			1.72782i
$994$		0			0
$995$	12.1899	−	44.6093i	0.386446	−	1.41421i
$996$		0			0
$997$		−	8.01261i		−	0.253762i	−0.991918	−	0.126881i	$-0.959503\pi$
$997$		−	8.01261i		−	0.253762i	0.991918	−	0.126881i	$-0.0404966\pi$
$998$		0			0
$999$	−47.3046			−1.49665

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	920.2.e.c.369.12	yes	16
4.3	odd	2		1840.2.e.h.369.5		16
5.2	odd	4		4600.2.a.bk.1.6		8
5.3	odd	4		4600.2.a.bj.1.3		8
5.4	even	2	inner	920.2.e.c.369.5	✓	16
20.3	even	4		9200.2.a.de.1.6		8
20.7	even	4		9200.2.a.dd.1.3		8
20.19	odd	2		1840.2.e.h.369.12		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.e.c.369.5	✓	16	5.4	even	2	inner
920.2.e.c.369.12	yes	16	1.1	even	1	trivial
1840.2.e.h.369.5		16	4.3	odd	2
1840.2.e.h.369.12		16	20.19	odd	2
4600.2.a.bj.1.3		8	5.3	odd	4
4600.2.a.bk.1.6		8	5.2	odd	4
9200.2.a.dd.1.3		8	20.7	even	4
9200.2.a.de.1.6		8	20.3	even	4

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:	\( N \)	\(=\)	\( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	920.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.69755i q^{3} +(0.589417 - 2.15699i) q^{5} +4.22860i q^{7} +0.118308 q^{9} +O(q^{10})\)	\(q+1.69755i q^{3} +(0.589417 - 2.15699i) q^{5} +4.22860i q^{7} +0.118308 q^{9} +4.59157 q^{11} +0.978254i q^{13} +(3.66160 + 1.00057i) q^{15} -3.04005i q^{17} -1.91463 q^{19} -7.17829 q^{21} -1.00000i q^{23} +(-4.30518 - 2.54273i) q^{25} +5.29350i q^{27} +0.737607 q^{29} +2.97939 q^{31} +7.79445i q^{33} +(9.12104 + 2.49241i) q^{35} +8.93637i q^{37} -1.66064 q^{39} +9.08240 q^{41} +6.97873i q^{43} +(0.0697326 - 0.255188i) q^{45} +2.58560i q^{47} -10.8811 q^{49} +5.16065 q^{51} +2.71400i q^{53} +(2.70635 - 9.90396i) q^{55} -3.25019i q^{57} -7.13514 q^{59} -0.731629 q^{61} +0.500277i q^{63} +(2.11008 + 0.576599i) q^{65} -7.16679i q^{67} +1.69755 q^{69} -6.08775 q^{71} +5.96311i q^{73} +(4.31642 - 7.30827i) q^{75} +19.4159i q^{77} +2.06100 q^{79} -8.63108 q^{81} -4.39408i q^{83} +(-6.55735 - 1.79186i) q^{85} +1.25213i q^{87} +7.25034 q^{89} -4.13665 q^{91} +5.05767i q^{93} +(-1.12852 + 4.12983i) q^{95} -7.04560i q^{97} +0.543219 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{5} - 22 q^{9}+O(q^{10}) \) 16 * q - 2 * q^5 - 22 * q^9	\( 16 q - 2 q^{5} - 22 q^{9} + 14 q^{11} + 6 q^{15} - 22 q^{19} + 12 q^{25} - 44 q^{29} + 18 q^{31} + 20 q^{35} + 14 q^{41} + 14 q^{45} - 78 q^{49} - 38 q^{51} + 30 q^{55} - 64 q^{59} + 34 q^{61} + 6 q^{65} + 6 q^{69} + 30 q^{71} + 56 q^{75} + 4 q^{79} + 48 q^{81} + 52 q^{85} - 92 q^{89} - 70 q^{91} + 38 q^{95} - 122 q^{99}+O(q^{100}) \) 16 * q - 2 * q^5 - 22 * q^9 + 14 * q^11 + 6 * q^15 - 22 * q^19 + 12 * q^25 - 44 * q^29 + 18 * q^31 + 20 * q^35 + 14 * q^41 + 14 * q^45 - 78 * q^49 - 38 * q^51 + 30 * q^55 - 64 * q^59 + 34 * q^61 + 6 * q^65 + 6 * q^69 + 30 * q^71 + 56 * q^75 + 4 * q^79 + 48 * q^81 + 52 * q^85 - 92 * q^89 - 70 * q^91 + 38 * q^95 - 122 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more