LMFDB - Embedded newform 1120.2.h.a.111.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1120,2,Mod(111,1120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1120, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 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("1120.111");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1120 = 2^{5} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1120.h (of order $2$, degree $1$, not 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:	$8.94324502638$
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} - x^{15} - 2x^{12} + 6x^{11} - 12x^{9} + 8x^{8} - 24x^{7} + 48x^{5} - 32x^{4} - 128x + 256 $ x^16 - x^15 - 2x^12 + 6x^11 - 12x^9 + 8x^8 - 24x^7 + 48x^5 - 32x^4 - 128x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			111.6
Root			$-1.38133 - 0.303194i$ of defining polynomial
Character	$\chi$	$=$	1120.111
Dual form			1120.2.h.a.111.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.34113i q^{3} -1.00000 q^{5} +(-1.28003 + 2.31550i) q^{7} +1.20136 q^{9} +O(q^{10})$	$q-1.34113i q^{3} -1.00000 q^{5} +(-1.28003 + 2.31550i) q^{7} +1.20136 q^{9} -2.44809 q^{11} +1.57090 q^{13} +1.34113i q^{15} -1.11987i q^{17} -8.44773i q^{19} +(3.10539 + 1.71669i) q^{21} -2.62959i q^{23} +1.00000 q^{25} -5.63459i q^{27} -3.43282i q^{29} -9.70304 q^{31} +3.28321i q^{33} +(1.28003 - 2.31550i) q^{35} -6.22712i q^{37} -2.10679i q^{39} -3.13128i q^{41} -7.45492 q^{43} -1.20136 q^{45} +9.40956 q^{47} +(-3.72304 - 5.92781i) q^{49} -1.50190 q^{51} +11.6067i q^{53} +2.44809 q^{55} -11.3295 q^{57} +6.16041i q^{59} -9.44231 q^{61} +(-1.53778 + 2.78175i) q^{63} -1.57090 q^{65} +2.15461 q^{67} -3.52663 q^{69} -7.87185i q^{71} -12.6145i q^{73} -1.34113i q^{75} +(3.13362 - 5.66853i) q^{77} +7.70558i q^{79} -3.95264 q^{81} +0.813234i q^{83} +1.11987i q^{85} -4.60387 q^{87} -5.12287i q^{89} +(-2.01081 + 3.63742i) q^{91} +13.0131i q^{93} +8.44773i q^{95} +0.833955i q^{97} -2.94104 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{5} - 16 q^{9}+O(q^{10}) $ 16 * q - 16 * q^5 - 16 * q^9	$ 16 q - 16 q^{5} - 16 q^{9} + 4 q^{11} + 4 q^{21} + 16 q^{25} - 16 q^{31} + 4 q^{43} + 16 q^{45} - 8 q^{49} + 40 q^{51} - 4 q^{55} - 16 q^{57} + 8 q^{61} + 28 q^{63} - 20 q^{67} + 40 q^{69} + 4 q^{77} + 24 q^{81} + 72 q^{87} + 32 q^{91} - 20 q^{99}+O(q^{100}) $ 16 * q - 16 * q^5 - 16 * q^9 + 4 * q^11 + 4 * q^21 + 16 * q^25 - 16 * q^31 + 4 * q^43 + 16 * q^45 - 8 * q^49 + 40 * q^51 - 4 * q^55 - 16 * q^57 + 8 * q^61 + 28 * q^63 - 20 * q^67 + 40 * q^69 + 4 * q^77 + 24 * q^81 + 72 * q^87 + 32 * q^91 - 20 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times$.

$n$	$351$	$421$	$801$	$897$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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
$3$		−	1.34113i		−	0.774303i	−0.922016	−	0.387152i	$-0.873459\pi$
$3$		−	1.34113i		−	0.774303i	0.922016	−	0.387152i	$-0.126541\pi$
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$	−1.28003	+	2.31550i	−0.483806	+	0.875175i
$7$	−1.28003	+	2.31550i	−0.483806	+	0.875175i
$8$		0			0
$9$	1.20136			0.400454
$10$		0			0
$11$	−2.44809			−0.738125			−0.369063	−	0.929404i	$-0.620321\pi$
$11$	−2.44809			−0.738125			−0.369063	+	0.929404i	$0.620321\pi$
$12$		0			0
$13$	1.57090			0.435690			0.217845	−	0.975983i	$-0.430097\pi$
$13$	1.57090			0.435690			0.217845	+	0.975983i	$0.430097\pi$
$14$		0			0
$15$			1.34113i			0.346279i
$16$		0			0
$17$		−	1.11987i		−	0.271609i	−0.990736	−	0.135804i	$-0.956638\pi$
$17$		−	1.11987i		−	0.271609i	0.990736	−	0.135804i	$-0.0433619\pi$
$18$		0			0
$19$		−	8.44773i		−	1.93804i	−0.246978	−	0.969021i	$-0.579437\pi$
$19$		−	8.44773i		−	1.93804i	0.246978	−	0.969021i	$-0.420563\pi$
$20$		0			0
$21$	3.10539	+	1.71669i	0.677651	+	0.374613i
$22$		0			0
$23$		−	2.62959i		−	0.548307i	−0.961686	−	0.274154i	$-0.911602\pi$
$23$		−	2.62959i		−	0.548307i	0.961686	−	0.274154i	$-0.0883977\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		−	5.63459i		−	1.08438i
$28$		0			0
$29$		−	3.43282i		−	0.637458i	−0.947846	−	0.318729i	$-0.896744\pi$
$29$		−	3.43282i		−	0.637458i	0.947846	−	0.318729i	$-0.103256\pi$
$30$		0			0
$31$	−9.70304			−1.74272			−0.871359	−	0.490647i	$-0.836761\pi$
$31$	−9.70304			−1.74272			−0.871359	+	0.490647i	$0.836761\pi$
$32$		0			0
$33$			3.28321i			0.571533i
$34$		0			0
$35$	1.28003	−	2.31550i	0.216365	−	0.391390i
$36$		0			0
$37$		−	6.22712i		−	1.02373i	−0.859065	−	0.511866i	$-0.828954\pi$
$37$		−	6.22712i		−	1.02373i	0.859065	−	0.511866i	$-0.171046\pi$
$38$		0			0
$39$		−	2.10679i		−	0.337357i
$40$		0			0
$41$		−	3.13128i		−	0.489024i	−0.969646	−	0.244512i	$-0.921372\pi$
$41$		−	3.13128i		−	0.489024i	0.969646	−	0.244512i	$-0.0786277\pi$
$42$		0			0
$43$	−7.45492			−1.13687			−0.568433	−	0.822730i	$-0.692450\pi$
$43$	−7.45492			−1.13687			−0.568433	+	0.822730i	$0.692450\pi$
$44$		0			0
$45$	−1.20136			−0.179089
$46$		0			0
$47$	9.40956			1.37253			0.686263	−	0.727354i	$-0.259250\pi$
$47$	9.40956			1.37253			0.686263	+	0.727354i	$0.259250\pi$
$48$		0			0
$49$	−3.72304	−	5.92781i	−0.531863	−	0.846830i
$50$		0			0
$51$	−1.50190			−0.210308
$52$		0			0
$53$			11.6067i			1.59430i	0.603778	+	0.797152i	$0.293661\pi$
$53$			11.6067i			1.59430i	−0.603778	+	0.797152i	$0.706339\pi$
$54$		0			0
$55$	2.44809			0.330100
$56$		0			0
$57$	−11.3295			−1.50063
$58$		0			0
$59$			6.16041i			0.802017i	0.916074	+	0.401009i	$0.131340\pi$
$59$			6.16041i			0.802017i	−0.916074	+	0.401009i	$0.868660\pi$
$60$		0			0
$61$	−9.44231			−1.20896			−0.604482	−	0.796619i	$-0.706620\pi$
$61$	−9.44231			−1.20896			−0.604482	+	0.796619i	$0.706620\pi$
$62$		0			0
$63$	−1.53778	+	2.78175i	−0.193742	+	0.350468i
$64$		0			0
$65$	−1.57090			−0.194847
$66$		0			0
$67$	2.15461			0.263228			0.131614	−	0.991301i	$-0.457984\pi$
$67$	2.15461			0.263228			0.131614	+	0.991301i	$0.457984\pi$
$68$		0			0
$69$	−3.52663			−0.424556
$70$		0			0
$71$		−	7.87185i		−	0.934217i	−0.884200	−	0.467108i	$-0.845296\pi$
$71$		−	7.87185i		−	0.934217i	0.884200	−	0.467108i	$-0.154704\pi$
$72$		0			0
$73$		−	12.6145i		−	1.47642i	−0.674571	−	0.738210i	$-0.735671\pi$
$73$		−	12.6145i		−	1.47642i	0.674571	−	0.738210i	$-0.264329\pi$
$74$		0			0
$75$		−	1.34113i		−	0.154861i
$76$		0			0
$77$	3.13362	−	5.66853i	0.357110	−	0.645989i
$78$		0			0
$79$			7.70558i			0.866946i	0.901167	+	0.433473i	$0.142712\pi$
$79$			7.70558i			0.866946i	−0.901167	+	0.433473i	$0.857288\pi$
$80$		0			0
$81$	−3.95264			−0.439182
$82$		0			0
$83$			0.813234i			0.0892640i	0.999003	+	0.0446320i	$0.0142115\pi$
$83$			0.813234i			0.0892640i	−0.999003	+	0.0446320i	$0.985788\pi$
$84$		0			0
$85$			1.11987i			0.121467i
$86$		0			0
$87$	−4.60387			−0.493586
$88$		0			0
$89$		−	5.12287i		−	0.543023i	−0.962435	−	0.271512i	$-0.912476\pi$
$89$		−	5.12287i		−	0.543023i	0.962435	−	0.271512i	$-0.0875235\pi$
$90$		0			0
$91$	−2.01081	+	3.63742i	−0.210790	+	0.381305i
$92$		0			0
$93$			13.0131i			1.34939i
$94$		0			0
$95$			8.44773i			0.866719i
$96$		0			0
$97$			0.833955i			0.0846753i	0.999103	+	0.0423377i	$0.0134805\pi$
$97$			0.833955i			0.0846753i	−0.999103	+	0.0423377i	$0.986519\pi$
$98$		0			0
$99$	−2.94104			−0.295585
$100$		0			0
$101$	−5.52401			−0.549660			−0.274830	−	0.961493i	$-0.588622\pi$
$101$	−5.52401			−0.549660			−0.274830	+	0.961493i	$0.588622\pi$
$102$		0			0
$103$	11.7675			1.15949			0.579745	−	0.814798i	$-0.303152\pi$
$103$	11.7675			1.15949			0.579745	+	0.814798i	$0.303152\pi$
$104$		0			0
$105$	−3.10539	−	1.71669i	−0.303055	−	0.167532i
$106$		0			0
$107$	1.38992			0.134369			0.0671844	−	0.997741i	$-0.478598\pi$
$107$	1.38992			0.134369			0.0671844	+	0.997741i	$0.478598\pi$
$108$		0			0
$109$		−	1.07524i		−	0.102990i	−0.998673	−	0.0514949i	$-0.983601\pi$
$109$		−	1.07524i		−	0.102990i	0.998673	−	0.0514949i	$-0.0163986\pi$
$110$		0			0
$111$	−8.35140			−0.792680
$112$		0			0
$113$	11.1967			1.05329			0.526647	−	0.850084i	$-0.323449\pi$
$113$	11.1967			1.05329			0.526647	+	0.850084i	$0.323449\pi$
$114$		0			0
$115$			2.62959i			0.245210i
$116$		0			0
$117$	1.88722			0.174474
$118$		0			0
$119$	2.59306	+	1.43347i	0.237705	+	0.131406i
$120$		0			0
$121$	−5.00688			−0.455171
$122$		0			0
$123$	−4.19946			−0.378653
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$		−	16.4186i		−	1.45692i	−0.685089	−	0.728459i	$-0.740237\pi$
$127$		−	16.4186i		−	1.45692i	0.685089	−	0.728459i	$-0.259763\pi$
$128$		0			0
$129$			9.99804i			0.880279i
$130$		0			0
$131$			2.00348i			0.175045i	0.996163	+	0.0875223i	$0.0278949\pi$
$131$			2.00348i			0.175045i	−0.996163	+	0.0875223i	$0.972105\pi$
$132$		0			0
$133$	19.5607	+	10.8134i	1.69613	+	0.937637i
$134$		0			0
$135$			5.63459i			0.484948i
$136$		0			0
$137$	4.20941			0.359634			0.179817	−	0.983700i	$-0.442449\pi$
$137$	4.20941			0.359634			0.179817	+	0.983700i	$0.442449\pi$
$138$		0			0
$139$			4.20658i			0.356798i	0.983958	+	0.178399i	$0.0570917\pi$
$139$			4.20658i			0.356798i	−0.983958	+	0.178399i	$0.942908\pi$
$140$		0			0
$141$		−	12.6195i		−	1.06275i
$142$		0			0
$143$	−3.84571			−0.321594
$144$		0			0
$145$			3.43282i			0.285080i
$146$		0			0
$147$	−7.94998	+	4.99310i	−0.655704	+	0.411824i
$148$		0			0
$149$		−	19.1438i		−	1.56832i	−0.620560	−	0.784159i	$-0.713094\pi$
$149$		−	19.1438i		−	1.56832i	0.620560	−	0.784159i	$-0.286906\pi$
$150$		0			0
$151$		−	2.73232i		−	0.222353i	−0.993801	−	0.111177i	$-0.964538\pi$
$151$		−	2.73232i		−	0.222353i	0.993801	−	0.111177i	$-0.0354619\pi$
$152$		0			0
$153$		−	1.34537i		−	0.108767i
$154$		0			0
$155$	9.70304			0.779367
$156$		0			0
$157$	−13.5441			−1.08094			−0.540468	−	0.841365i	$-0.681753\pi$
$157$	−13.5441			−1.08094			−0.540468	+	0.841365i	$0.681753\pi$
$158$		0			0
$159$	15.5661			1.23448
$160$		0			0
$161$	6.08880	+	3.36595i	0.479865	+	0.265274i
$162$		0			0
$163$	18.5317			1.45151			0.725756	−	0.687953i	$-0.241490\pi$
$163$	18.5317			1.45151			0.725756	+	0.687953i	$0.241490\pi$
$164$		0			0
$165$		−	3.28321i		−	0.255597i
$166$		0			0
$167$	−7.74592			−0.599397			−0.299699	−	0.954034i	$-0.596886\pi$
$167$	−7.74592			−0.599397			−0.299699	+	0.954034i	$0.596886\pi$
$168$		0			0
$169$	−10.5323			−0.810174
$170$		0			0
$171$		−	10.1488i		−	0.776097i
$172$		0			0
$173$	13.6112			1.03484			0.517421	−	0.855731i	$-0.326892\pi$
$173$	13.6112			1.03484			0.517421	+	0.855731i	$0.326892\pi$
$174$		0			0
$175$	−1.28003	+	2.31550i	−0.0967612	+	0.175035i
$176$		0			0
$177$	8.26193			0.621005
$178$		0			0
$179$	17.3370			1.29583			0.647914	−	0.761714i	$-0.275642\pi$
$179$	17.3370			1.29583			0.647914	+	0.761714i	$0.275642\pi$
$180$		0			0
$181$	−11.8268			−0.879076			−0.439538	−	0.898224i	$-0.644858\pi$
$181$	−11.8268			−0.879076			−0.439538	+	0.898224i	$0.644858\pi$
$182$		0			0
$183$			12.6634i			0.936105i
$184$		0			0
$185$			6.22712i			0.457827i
$186$		0			0
$187$			2.74154i			0.200481i
$188$		0			0
$189$	13.0469	+	7.21244i	0.949019	+	0.524628i
$190$		0			0
$191$		−	20.5823i		−	1.48929i	−0.667463	−	0.744643i	$-0.732620\pi$
$191$		−	20.5823i		−	1.48929i	0.667463	−	0.744643i	$-0.267380\pi$
$192$		0			0
$193$	10.0690			0.724781			0.362391	−	0.932026i	$-0.381961\pi$
$193$	10.0690			0.724781			0.362391	+	0.932026i	$0.381961\pi$
$194$		0			0
$195$			2.10679i			0.150870i
$196$		0			0
$197$			4.59538i			0.327407i	0.986510	+	0.163704i	$0.0523441\pi$
$197$			4.59538i			0.327407i	−0.986510	+	0.163704i	$0.947656\pi$
$198$		0			0
$199$	−2.15447			−0.152727			−0.0763633	−	0.997080i	$-0.524331\pi$
$199$	−2.15447			−0.152727			−0.0763633	+	0.997080i	$0.524331\pi$
$200$		0			0
$201$		−	2.88962i		−	0.203818i
$202$		0			0
$203$	7.94868	+	4.39411i	0.557888	+	0.308406i
$204$		0			0
$205$			3.13128i			0.218698i
$206$		0			0
$207$		−	3.15909i		−	0.219572i
$208$		0			0
$209$			20.6808i			1.43052i
$210$		0			0
$211$	11.7444			0.808515			0.404258	−	0.914645i	$-0.367530\pi$
$211$	11.7444			0.808515			0.404258	+	0.914645i	$0.367530\pi$
$212$		0			0
$213$	−10.5572			−0.723367
$214$		0			0
$215$	7.45492			0.508422
$216$		0			0
$217$	12.4202	−	22.4673i	0.843137	−	1.52518i
$218$		0			0
$219$	−16.9178			−1.14320
$220$		0			0
$221$		−	1.75921i		−	0.118337i
$222$		0			0
$223$	−6.92127			−0.463482			−0.231741	−	0.972777i	$-0.574442\pi$
$223$	−6.92127			−0.463482			−0.231741	+	0.972777i	$0.574442\pi$
$224$		0			0
$225$	1.20136			0.0800908
$226$		0			0
$227$			12.2022i			0.809890i	0.914341	+	0.404945i	$0.132709\pi$
$227$			12.2022i			0.809890i	−0.914341	+	0.404945i	$0.867291\pi$
$228$		0			0
$229$	−12.2581			−0.810041			−0.405021	−	0.914308i	$-0.632736\pi$
$229$	−12.2581			−0.810041			−0.405021	+	0.914308i	$0.632736\pi$
$230$		0			0
$231$	−7.60226	−	4.20261i	−0.500192	−	0.276511i
$232$		0			0
$233$	−10.1521			−0.665083			−0.332542	−	0.943089i	$-0.607906\pi$
$233$	−10.1521			−0.665083			−0.332542	+	0.943089i	$0.607906\pi$
$234$		0			0
$235$	−9.40956			−0.613812
$236$		0			0
$237$	10.3342			0.671279
$238$		0			0
$239$			5.57510i			0.360624i	0.983610	+	0.180312i	$0.0577107\pi$
$239$			5.57510i			0.360624i	−0.983610	+	0.180312i	$0.942289\pi$
$240$		0			0
$241$			20.9813i			1.35152i	0.737120	+	0.675762i	$0.236185\pi$
$241$			20.9813i			1.35152i	−0.737120	+	0.675762i	$0.763815\pi$
$242$		0			0
$243$		−	11.6027i		−	0.744316i
$244$		0			0
$245$	3.72304	+	5.92781i	0.237857	+	0.378714i
$246$		0			0
$247$		−	13.2706i		−	0.844386i
$248$		0			0
$249$	1.09065			0.0691174
$250$		0			0
$251$			22.8742i			1.44381i	0.691995	+	0.721903i	$0.256732\pi$
$251$			22.8742i			1.44381i	−0.691995	+	0.721903i	$0.743268\pi$
$252$		0			0
$253$			6.43746i			0.404720i
$254$		0			0
$255$	1.50190			0.0940525
$256$		0			0
$257$			28.8767i			1.80128i	0.434566	+	0.900640i	$0.356902\pi$
$257$			28.8767i			1.80128i	−0.434566	+	0.900640i	$0.643098\pi$
$258$		0			0
$259$	14.4189	+	7.97091i	0.895946	+	0.495288i
$260$		0			0
$261$		−	4.12406i		−	0.255273i
$262$		0			0
$263$			19.6017i			1.20869i	0.796722	+	0.604347i	$0.206566\pi$
$263$			19.6017i			1.20869i	−0.796722	+	0.604347i	$0.793434\pi$
$264$		0			0
$265$		−	11.6067i		−	0.712995i
$266$		0			0
$267$	−6.87045			−0.420465
$268$		0			0
$269$	−27.2431			−1.66104			−0.830520	−	0.556988i	$-0.811957\pi$
$269$	−27.2431			−1.66104			−0.830520	+	0.556988i	$0.811957\pi$
$270$		0			0
$271$	26.0339			1.58144			0.790722	−	0.612175i	$-0.209705\pi$
$271$	26.0339			1.58144			0.790722	+	0.612175i	$0.209705\pi$
$272$		0			0
$273$	4.87827	+	2.69676i	0.295246	+	0.163215i
$274$		0			0
$275$	−2.44809			−0.147625
$276$		0			0
$277$		−	5.64829i		−	0.339373i	−0.985498	−	0.169687i	$-0.945724\pi$
$277$		−	5.64829i		−	0.339373i	0.985498	−	0.169687i	$-0.0542755\pi$
$278$		0			0
$279$	−11.6569			−0.697878
$280$		0			0
$281$	25.9379			1.54733			0.773663	−	0.633598i	$-0.218422\pi$
$281$	25.9379			1.54733			0.773663	+	0.633598i	$0.218422\pi$
$282$		0			0
$283$			12.6062i			0.749358i	0.927155	+	0.374679i	$0.122247\pi$
$283$			12.6062i			0.749358i	−0.927155	+	0.374679i	$0.877753\pi$
$284$		0			0
$285$	11.3295			0.671103
$286$		0			0
$287$	7.25047	+	4.00813i	0.427982	+	0.236593i
$288$		0			0
$289$	15.7459			0.926229
$290$		0			0
$291$	1.11844			0.0655644
$292$		0			0
$293$	16.7982			0.981362			0.490681	−	0.871339i	$-0.336748\pi$
$293$	16.7982			0.981362			0.490681	+	0.871339i	$0.336748\pi$
$294$		0			0
$295$		−	6.16041i		−	0.358673i
$296$		0			0
$297$			13.7939i			0.800406i
$298$		0			0
$299$		−	4.13083i		−	0.238892i
$300$		0			0
$301$	9.54253	−	17.2618i	0.550022	−	0.994956i
$302$		0			0
$303$			7.40844i			0.425604i
$304$		0			0
$305$	9.44231			0.540665
$306$		0			0
$307$		−	5.08609i		−	0.290279i	−0.989411	−	0.145139i	$-0.953637\pi$
$307$		−	5.08609i		−	0.290279i	0.989411	−	0.145139i	$-0.0463630\pi$
$308$		0			0
$309$		−	15.7818i		−	0.897797i
$310$		0			0
$311$	−16.9297			−0.959994			−0.479997	−	0.877270i	$-0.659362\pi$
$311$	−16.9297			−0.959994			−0.479997	+	0.877270i	$0.659362\pi$
$312$		0			0
$313$		−	8.30791i		−	0.469591i	−0.972045	−	0.234795i	$-0.924558\pi$
$313$		−	8.30791i		−	0.469591i	0.972045	−	0.234795i	$-0.0754420\pi$
$314$		0			0
$315$	1.53778	−	2.78175i	0.0866441	−	0.156734i
$316$		0			0
$317$			10.7428i			0.603376i	0.953407	+	0.301688i	$0.0975501\pi$
$317$			10.7428i			0.603376i	−0.953407	+	0.301688i	$0.902450\pi$
$318$		0			0
$319$			8.40383i			0.470524i
$320$		0			0
$321$		−	1.86407i		−	0.104042i
$322$		0			0
$323$	−9.46038			−0.526390
$324$		0			0
$325$	1.57090			0.0871381
$326$		0			0
$327$	−1.44205			−0.0797453
$328$		0			0
$329$	−12.0445	+	21.7878i	−0.664036	+	1.20120i
$330$		0			0
$331$	−12.1739			−0.669140			−0.334570	−	0.942371i	$-0.608591\pi$
$331$	−12.1739			−0.669140			−0.334570	+	0.942371i	$0.608591\pi$
$332$		0			0
$333$		−	7.48103i		−	0.409958i
$334$		0			0
$335$	−2.15461			−0.117719
$336$		0			0
$337$	−11.5686			−0.630179			−0.315090	−	0.949062i	$-0.602035\pi$
$337$	−11.5686			−0.630179			−0.315090	+	0.949062i	$0.602035\pi$
$338$		0			0
$339$		−	15.0162i		−	0.815569i
$340$		0			0
$341$	23.7539			1.28634
$342$		0			0
$343$	18.4914	−	1.03291i	0.998444	−	0.0557721i
$344$		0			0
$345$	3.52663			0.189867
$346$		0			0
$347$	10.6711			0.572854			0.286427	−	0.958102i	$-0.407532\pi$
$347$	10.6711			0.572854			0.286427	+	0.958102i	$0.407532\pi$
$348$		0			0
$349$	14.5060			0.776486			0.388243	−	0.921557i	$-0.373082\pi$
$349$	14.5060			0.776486			0.388243	+	0.921557i	$0.373082\pi$
$350$		0			0
$351$		−	8.85139i		−	0.472452i
$352$		0			0
$353$			33.2031i			1.76722i	0.468221	+	0.883611i	$0.344895\pi$
$353$			33.2031i			1.76722i	−0.468221	+	0.883611i	$0.655105\pi$
$354$		0			0
$355$			7.87185i			0.417794i
$356$		0			0
$357$	1.92247	−	3.47764i	0.101748	−	0.184056i
$358$		0			0
$359$			13.3218i			0.703097i	0.936170	+	0.351549i	$0.114345\pi$
$359$			13.3218i			0.703097i	−0.936170	+	0.351549i	$0.885655\pi$
$360$		0			0
$361$	−52.3642			−2.75601
$362$		0			0
$363$			6.71489i			0.352440i
$364$		0			0
$365$			12.6145i			0.660275i
$366$		0			0
$367$	−25.9412			−1.35412			−0.677059	−	0.735929i	$-0.736746\pi$
$367$	−25.9412			−1.35412			−0.677059	+	0.735929i	$0.736746\pi$
$368$		0			0
$369$		−	3.76180i		−	0.195832i
$370$		0			0
$371$	−26.8753	−	14.8569i	−1.39530	−	0.771334i
$372$		0			0
$373$			30.5923i			1.58401i	0.610515	+	0.792005i	$0.290963\pi$
$373$			30.5923i			1.58401i	−0.610515	+	0.792005i	$0.709037\pi$
$374$		0			0
$375$			1.34113i			0.0692558i
$376$		0			0
$377$		−	5.39263i		−	0.277734i
$378$		0			0
$379$	10.6930			0.549262			0.274631	−	0.961550i	$-0.411444\pi$
$379$	10.6930			0.549262			0.274631	+	0.961550i	$0.411444\pi$
$380$		0			0
$381$	−22.0196			−1.12810
$382$		0			0
$383$	9.36792			0.478679			0.239339	−	0.970936i	$-0.423069\pi$
$383$	9.36792			0.478679			0.239339	+	0.970936i	$0.423069\pi$
$384$		0			0
$385$	−3.13362	+	5.66853i	−0.159704	+	0.288895i
$386$		0			0
$387$	−8.95606			−0.455262
$388$		0			0
$389$		−	2.30373i		−	0.116804i	−0.998293	−	0.0584019i	$-0.981399\pi$
$389$		−	2.30373i		−	0.116804i	0.998293	−	0.0584019i	$-0.0186005\pi$
$390$		0			0
$391$	−2.94480			−0.148925
$392$		0			0
$393$	2.68693			0.135538
$394$		0			0
$395$		−	7.70558i		−	0.387710i
$396$		0			0
$397$	−1.77718			−0.0891942			−0.0445971	−	0.999005i	$-0.514200\pi$
$397$	−1.77718			−0.0891942			−0.0445971	+	0.999005i	$0.514200\pi$
$398$		0			0
$399$	14.5021	−	26.2335i	0.726015	−	1.31332i
$400$		0			0
$401$	17.8159			0.889682			0.444841	−	0.895609i	$-0.353260\pi$
$401$	17.8159			0.889682			0.444841	+	0.895609i	$0.353260\pi$
$402$		0			0
$403$	−15.2425			−0.759285
$404$		0			0
$405$	3.95264			0.196408
$406$		0			0
$407$			15.2445i			0.755643i
$408$		0			0
$409$		−	37.8691i		−	1.87250i	−0.351329	−	0.936252i	$-0.614270\pi$
$409$		−	37.8691i		−	1.87250i	0.351329	−	0.936252i	$-0.385730\pi$
$410$		0			0
$411$		−	5.64538i		−	0.278466i
$412$		0			0
$413$	−14.2644	−	7.88551i	−0.701906	−	0.388021i
$414$		0			0
$415$		−	0.813234i		−	0.0399201i
$416$		0			0
$417$	5.64159			0.276270
$418$		0			0
$419$			3.11076i			0.151971i	0.997109	+	0.0759854i	$0.0242102\pi$
$419$			3.11076i			0.151971i	−0.997109	+	0.0759854i	$0.975790\pi$
$420$		0			0
$421$		−	18.8952i		−	0.920893i	−0.887687	−	0.460447i	$-0.847689\pi$
$421$		−	18.8952i		−	0.920893i	0.887687	−	0.460447i	$-0.152311\pi$
$422$		0			0
$423$	11.3043			0.549634
$424$		0			0
$425$		−	1.11987i		−	0.0543218i
$426$		0			0
$427$	12.0864	−	21.8636i	0.584904	−	1.05805i
$428$		0			0
$429$			5.15760i			0.249011i
$430$		0			0
$431$		−	38.2730i		−	1.84355i	−0.387730	−	0.921773i	$-0.626741\pi$
$431$		−	38.2730i		−	1.84355i	0.387730	−	0.921773i	$-0.373259\pi$
$432$		0			0
$433$			1.50589i			0.0723684i	0.999345	+	0.0361842i	$0.0115203\pi$
$433$			1.50589i			0.0723684i	−0.999345	+	0.0361842i	$0.988480\pi$
$434$		0			0
$435$	4.60387			0.220738
$436$		0			0
$437$	−22.2141			−1.06264
$438$		0			0
$439$	−5.00779			−0.239009			−0.119504	−	0.992834i	$-0.538131\pi$
$439$	−5.00779			−0.239009			−0.119504	+	0.992834i	$0.538131\pi$
$440$		0			0
$441$	−4.47272	−	7.12145i	−0.212987	−	0.339117i
$442$		0			0
$443$	24.9714			1.18643			0.593213	−	0.805046i	$-0.297859\pi$
$443$	24.9714			1.18643			0.593213	+	0.805046i	$0.297859\pi$
$444$		0			0
$445$			5.12287i			0.242847i
$446$		0			0
$447$	−25.6743			−1.21435
$448$		0			0
$449$	−4.85721			−0.229226			−0.114613	−	0.993410i	$-0.536563\pi$
$449$	−4.85721			−0.229226			−0.114613	+	0.993410i	$0.536563\pi$
$450$		0			0
$451$			7.66564i			0.360961i
$452$		0			0
$453$	−3.66441			−0.172169
$454$		0			0
$455$	2.01081	−	3.63742i	0.0942680	−	0.170525i
$456$		0			0
$457$	−21.1002			−0.987025			−0.493513	−	0.869739i	$-0.664287\pi$
$457$	−21.1002			−0.987025			−0.493513	+	0.869739i	$0.664287\pi$
$458$		0			0
$459$	−6.31002			−0.294526
$460$		0			0
$461$	11.2426			0.523620			0.261810	−	0.965119i	$-0.415681\pi$
$461$	11.2426			0.523620			0.261810	+	0.965119i	$0.415681\pi$
$462$		0			0
$463$			39.4199i			1.83200i	0.401179	+	0.916000i	$0.368601\pi$
$463$			39.4199i			1.83200i	−0.401179	+	0.916000i	$0.631399\pi$
$464$		0			0
$465$		−	13.0131i		−	0.603466i
$466$		0			0
$467$			26.4941i			1.22600i	0.790083	+	0.613000i	$0.210037\pi$
$467$			26.4941i			1.22600i	−0.790083	+	0.613000i	$0.789963\pi$
$468$		0			0
$469$	−2.75797	+	4.98899i	−0.127351	+	0.230370i
$470$		0			0
$471$			18.1644i			0.836973i
$472$		0			0
$473$	18.2503			0.839149
$474$		0			0
$475$		−	8.44773i		−	0.387608i
$476$		0			0
$477$			13.9439i			0.638446i
$478$		0			0
$479$	−35.7827			−1.63495			−0.817477	−	0.575961i	$-0.804628\pi$
$479$	−35.7827			−1.63495			−0.817477	+	0.575961i	$0.804628\pi$
$480$		0			0
$481$		−	9.78221i		−	0.446031i
$482$		0			0
$483$	4.51419	−	8.16590i	0.205403	−	0.371561i
$484$		0			0
$485$		−	0.833955i		−	0.0378680i
$486$		0			0
$487$		−	12.4491i		−	0.564125i	−0.959396	−	0.282062i	$-0.908982\pi$
$487$		−	12.4491i		−	0.564125i	0.959396	−	0.282062i	$-0.0910185\pi$
$488$		0			0
$489$		−	24.8534i		−	1.12391i
$490$		0			0
$491$	14.4396			0.651649			0.325824	−	0.945430i	$-0.394358\pi$
$491$	14.4396			0.651649			0.325824	+	0.945430i	$0.394358\pi$
$492$		0			0
$493$	−3.84432			−0.173139
$494$		0			0
$495$	2.94104			0.132190
$496$		0			0
$497$	18.2272	+	10.0762i	0.817603	+	0.451980i
$498$		0			0
$499$	−22.0969			−0.989195			−0.494597	−	0.869122i	$-0.664685\pi$
$499$	−22.0969			−0.989195			−0.494597	+	0.869122i	$0.664685\pi$
$500$		0			0
$501$			10.3883i			0.464116i
$502$		0			0
$503$	21.8311			0.973401			0.486701	−	0.873569i	$-0.338200\pi$
$503$	21.8311			0.973401			0.486701	+	0.873569i	$0.338200\pi$
$504$		0			0
$505$	5.52401			0.245815
$506$		0			0
$507$			14.1252i			0.627320i
$508$		0			0
$509$	30.6514			1.35860			0.679300	−	0.733861i	$-0.262284\pi$
$509$	30.6514			1.35860			0.679300	+	0.733861i	$0.262284\pi$
$510$		0			0
$511$	29.2089	+	16.1470i	1.29213	+	0.714301i
$512$		0			0
$513$	−47.5995			−2.10157
$514$		0			0
$515$	−11.7675			−0.518539
$516$		0			0
$517$	−23.0354			−1.01310
$518$		0			0
$519$		−	18.2545i		−	0.801282i
$520$		0			0
$521$		−	7.13741i		−	0.312696i	−0.987702	−	0.156348i	$-0.950028\pi$
$521$		−	7.13741i		−	0.312696i	0.987702	−	0.156348i	$-0.0499721\pi$
$522$		0			0
$523$		−	31.6160i		−	1.38247i	−0.722630	−	0.691235i	$-0.757067\pi$
$523$		−	31.6160i		−	1.38247i	0.722630	−	0.691235i	$-0.242933\pi$
$524$		0			0
$525$	3.10539	+	1.71669i	0.135530	+	0.0749225i
$526$		0			0
$527$			10.8662i			0.473338i
$528$		0			0
$529$	16.0853			0.699359
$530$		0			0
$531$			7.40088i			0.321171i
$532$		0			0
$533$		−	4.91894i		−	0.213063i
$534$		0			0
$535$	−1.38992			−0.0600916
$536$		0			0
$537$		−	23.2512i		−	1.00336i
$538$		0			0
$539$	9.11433	+	14.5118i	0.392582	+	0.625067i
$540$		0			0
$541$		−	39.5347i		−	1.69973i	−0.526999	−	0.849866i	$-0.676683\pi$
$541$		−	39.5347i		−	1.69973i	0.526999	−	0.849866i	$-0.323317\pi$
$542$		0			0
$543$			15.8613i			0.680672i
$544$		0			0
$545$			1.07524i			0.0460584i
$546$		0			0
$547$	−27.6549			−1.18244			−0.591218	−	0.806512i	$-0.701353\pi$
$547$	−27.6549			−1.18244			−0.591218	+	0.806512i	$0.701353\pi$
$548$		0			0
$549$	−11.3436			−0.484134
$550$		0			0
$551$	−28.9995			−1.23542
$552$		0			0
$553$	−17.8422	−	9.86338i	−0.758730	−	0.419434i
$554$		0			0
$555$	8.35140			0.354497
$556$		0			0
$557$		−	29.2830i		−	1.24076i	−0.784301	−	0.620380i	$-0.786978\pi$
$557$		−	29.2830i		−	1.24076i	0.784301	−	0.620380i	$-0.213022\pi$
$558$		0			0
$559$	−11.7110			−0.495321
$560$		0			0
$561$	3.67677			0.155234
$562$		0			0
$563$		−	5.22412i		−	0.220170i	−0.993922	−	0.110085i	$-0.964888\pi$
$563$		−	5.22412i		−	0.220170i	0.993922	−	0.110085i	$-0.0351124\pi$
$564$		0			0
$565$	−11.1967			−0.471047
$566$		0			0
$567$	5.05950	−	9.15233i	0.212479	−	0.384362i
$568$		0			0
$569$	11.8297			0.495925			0.247963	−	0.968770i	$-0.420239\pi$
$569$	11.8297			0.495925			0.247963	+	0.968770i	$0.420239\pi$
$570$		0			0
$571$	22.3223			0.934158			0.467079	−	0.884216i	$-0.345306\pi$
$571$	22.3223			0.934158			0.467079	+	0.884216i	$0.345306\pi$
$572$		0			0
$573$	−27.6037			−1.15316
$574$		0			0
$575$		−	2.62959i		−	0.109661i
$576$		0			0
$577$		−	25.4830i		−	1.06087i	−0.847725	−	0.530436i	$-0.822028\pi$
$577$		−	25.4830i		−	1.06087i	0.847725	−	0.530436i	$-0.177972\pi$
$578$		0			0
$579$		−	13.5038i		−	0.561201i
$580$		0			0
$581$	−1.88304	−	1.04096i	−0.0781216	−	0.0431865i
$582$		0			0
$583$		−	28.4142i		−	1.17680i
$584$		0			0
$585$	−1.88722			−0.0780271
$586$		0			0
$587$		−	27.1365i		−	1.12004i	−0.828478	−	0.560021i	$-0.810793\pi$
$587$		−	27.1365i		−	1.12004i	0.828478	−	0.560021i	$-0.189207\pi$
$588$		0			0
$589$			81.9687i			3.37746i
$590$		0			0
$591$	6.16301			0.253512
$592$		0			0
$593$		−	33.7929i		−	1.38771i	−0.720116	−	0.693853i	$-0.755912\pi$
$593$		−	33.7929i		−	1.38771i	0.720116	−	0.693853i	$-0.244088\pi$
$594$		0			0
$595$	−2.59306	−	1.43347i	−0.106305	−	0.0587666i
$596$		0			0
$597$			2.88943i			0.118257i
$598$		0			0
$599$		−	26.0094i		−	1.06271i	−0.847148	−	0.531357i	$-0.821682\pi$
$599$		−	26.0094i		−	1.06271i	0.847148	−	0.531357i	$-0.178318\pi$
$600$		0			0
$601$		−	25.5296i		−	1.04137i	−0.853747	−	0.520687i	$-0.825676\pi$
$601$		−	25.5296i		−	1.04137i	0.853747	−	0.520687i	$-0.174324\pi$
$602$		0			0
$603$	2.58847			0.105411
$604$		0			0
$605$	5.00688			0.203559
$606$		0			0
$607$	21.2033			0.860615			0.430307	−	0.902682i	$-0.358405\pi$
$607$	21.2033			0.860615			0.430307	+	0.902682i	$0.358405\pi$
$608$		0			0
$609$	5.89309	−	10.6602i	0.238800	−	0.431974i
$610$		0			0
$611$	14.7815			0.597996
$612$		0			0
$613$		−	0.381509i		−	0.0154090i	−0.999970	−	0.00770450i	$-0.997548\pi$
$613$		−	0.381509i		−	0.0154090i	0.999970	−	0.00770450i	$-0.00245244\pi$
$614$		0			0
$615$	4.19946			0.169339
$616$		0			0
$617$	−23.7394			−0.955714			−0.477857	−	0.878438i	$-0.658586\pi$
$617$	−23.7394			−0.955714			−0.477857	+	0.878438i	$0.658586\pi$
$618$		0			0
$619$		−	3.66231i		−	0.147201i	−0.997288	−	0.0736004i	$-0.976551\pi$
$619$		−	3.66231i		−	0.147201i	0.997288	−	0.0736004i	$-0.0234489\pi$
$620$		0			0
$621$	−14.8166			−0.594571
$622$		0			0
$623$	11.8620	+	6.55743i	0.475240	+	0.262718i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	27.7357			1.10766
$628$		0			0
$629$	−6.97358			−0.278055
$630$		0			0
$631$			10.6277i			0.423080i	0.977369	+	0.211540i	$0.0678479\pi$
$631$			10.6277i			0.423080i	−0.977369	+	0.211540i	$0.932152\pi$
$632$		0			0
$633$		−	15.7508i		−	0.626036i
$634$		0			0
$635$			16.4186i			0.651554i
$636$		0			0
$637$	−5.84854	−	9.31202i	−0.231728	−	0.368956i
$638$		0			0
$639$		−	9.45694i		−	0.374111i
$640$		0			0
$641$	−31.4300			−1.24141			−0.620705	−	0.784044i	$-0.713154\pi$
$641$	−31.4300			−1.24141			−0.620705	+	0.784044i	$0.713154\pi$
$642$		0			0
$643$			7.82074i			0.308420i	0.988038	+	0.154210i	$0.0492832\pi$
$643$			7.82074i			0.308420i	−0.988038	+	0.154210i	$0.950717\pi$
$644$		0			0
$645$		−	9.99804i		−	0.393673i
$646$		0			0
$647$	27.3346			1.07463			0.537317	−	0.843381i	$-0.319438\pi$
$647$	27.3346			1.07463			0.537317	+	0.843381i	$0.319438\pi$
$648$		0			0
$649$		−	15.0812i		−	0.591989i
$650$		0			0
$651$	−30.1317	−	16.6571i	−1.18095	−	0.652844i
$652$		0			0
$653$		−	19.6566i		−	0.769221i	−0.923079	−	0.384611i	$-0.874336\pi$
$653$		−	19.6566i		−	0.769221i	0.923079	−	0.384611i	$-0.125664\pi$
$654$		0			0
$655$		−	2.00348i		−	0.0782823i
$656$		0			0
$657$		−	15.1546i		−	0.591239i
$658$		0			0
$659$	−14.1522			−0.551293			−0.275647	−	0.961259i	$-0.588892\pi$
$659$	−14.1522			−0.551293			−0.275647	+	0.961259i	$0.588892\pi$
$660$		0			0
$661$	7.32100			0.284754			0.142377	−	0.989812i	$-0.454525\pi$
$661$	7.32100			0.284754			0.142377	+	0.989812i	$0.454525\pi$
$662$		0			0
$663$	−2.35934			−0.0916291
$664$		0			0
$665$	−19.5607	−	10.8134i	−0.758531	−	0.419324i
$666$		0			0
$667$	−9.02690			−0.349523
$668$		0			0
$669$			9.28234i			0.358876i
$670$		0			0
$671$	23.1156			0.892367
$672$		0			0
$673$	21.5006			0.828786			0.414393	−	0.910098i	$-0.363994\pi$
$673$	21.5006			0.828786			0.414393	+	0.910098i	$0.363994\pi$
$674$		0			0
$675$		−	5.63459i		−	0.216875i
$676$		0			0
$677$	41.7438			1.60434			0.802172	−	0.597093i	$-0.203678\pi$
$677$	41.7438			1.60434			0.802172	+	0.597093i	$0.203678\pi$
$678$		0			0
$679$	−1.93102	−	1.06749i	−0.0741057	−	0.0409664i
$680$		0			0
$681$	16.3648			0.627101
$682$		0			0
$683$	−26.9066			−1.02955			−0.514777	−	0.857324i	$-0.672125\pi$
$683$	−26.9066			−1.02955			−0.514777	+	0.857324i	$0.672125\pi$
$684$		0			0
$685$	−4.20941			−0.160833
$686$		0			0
$687$			16.4398i			0.627218i
$688$		0			0
$689$			18.2330i			0.694623i
$690$		0			0
$691$		−	17.3610i		−	0.660443i	−0.943904	−	0.330221i	$-0.892877\pi$
$691$		−	17.3610i		−	0.660443i	0.943904	−	0.330221i	$-0.107123\pi$
$692$		0			0
$693$	3.76462	−	6.80996i	0.143006	−	0.258689i
$694$		0			0
$695$		−	4.20658i		−	0.159565i
$696$		0			0
$697$	−3.50663			−0.132823
$698$		0			0
$699$			13.6153i			0.514976i
$700$		0			0
$701$			31.9368i			1.20624i	0.797651	+	0.603119i	$0.206076\pi$
$701$			31.9368i			1.20624i	−0.797651	+	0.603119i	$0.793924\pi$
$702$		0			0
$703$	−52.6051			−1.98404
$704$		0			0
$705$			12.6195i			0.475277i
$706$		0			0
$707$	7.07091	−	12.7908i	0.265929	−	0.481049i
$708$		0			0
$709$			1.65295i			0.0620777i	0.999518	+	0.0310389i	$0.00988157\pi$
$709$			1.65295i			0.0620777i	−0.999518	+	0.0310389i	$0.990118\pi$
$710$		0			0
$711$			9.25720i			0.347172i
$712$		0			0
$713$			25.5150i			0.955544i
$714$		0			0
$715$	3.84571			0.143821
$716$		0			0
$717$	7.47696			0.279232
$718$		0			0
$719$	−16.0007			−0.596725			−0.298363	−	0.954453i	$-0.596440\pi$
$719$	−16.0007			−0.596725			−0.298363	+	0.954453i	$0.596440\pi$
$720$		0			0
$721$	−15.0628	+	27.2477i	−0.560968	+	1.01476i
$722$		0			0
$723$	28.1387			1.04649
$724$		0			0
$725$		−	3.43282i		−	0.127492i
$726$		0			0
$727$	−7.28793			−0.270294			−0.135147	−	0.990826i	$-0.543151\pi$
$727$	−7.28793			−0.270294			−0.135147	+	0.990826i	$0.543151\pi$
$728$		0			0
$729$	−27.4187			−1.01551
$730$		0			0
$731$			8.34856i			0.308783i
$732$		0			0
$733$	4.79246			0.177013			0.0885067	−	0.996076i	$-0.471791\pi$
$733$	4.79246			0.177013			0.0885067	+	0.996076i	$0.471791\pi$
$734$		0			0
$735$	7.94998	−	4.99310i	0.293240	−	0.184173i
$736$		0			0
$737$	−5.27467			−0.194295
$738$		0			0
$739$	−37.3432			−1.37369			−0.686845	−	0.726804i	$-0.741005\pi$
$739$	−37.3432			−1.37369			−0.686845	+	0.726804i	$0.741005\pi$
$740$		0			0
$741$	−17.7976			−0.653811
$742$		0			0
$743$		−	14.9317i		−	0.547789i	−0.961760	−	0.273895i	$-0.911688\pi$
$743$		−	14.9317i		−	0.547789i	0.961760	−	0.273895i	$-0.0883119\pi$
$744$		0			0
$745$			19.1438i			0.701373i
$746$		0			0
$747$			0.976988i			0.0357461i
$748$		0			0
$749$	−1.77914	+	3.21836i	−0.0650085	+	0.117596i
$750$		0			0
$751$			32.2962i			1.17851i	0.807948	+	0.589253i	$0.200578\pi$
$751$			32.2962i			1.17851i	−0.807948	+	0.589253i	$0.799422\pi$
$752$		0			0
$753$	30.6773			1.11794
$754$		0			0
$755$			2.73232i			0.0994394i
$756$		0			0
$757$		−	21.1662i		−	0.769299i	−0.923063	−	0.384650i	$-0.874322\pi$
$757$		−	21.1662i		−	0.769299i	0.923063	−	0.384650i	$-0.125678\pi$
$758$		0			0
$759$	8.63349			0.313376
$760$		0			0
$761$			8.98517i			0.325712i	0.986650	+	0.162856i	$0.0520706\pi$
$761$			8.98517i			0.325712i	−0.986650	+	0.162856i	$0.947929\pi$
$762$		0			0
$763$	2.48972	+	1.37635i	0.0901341	+	0.0498271i
$764$		0			0
$765$			1.34537i			0.0486420i
$766$		0			0
$767$			9.67741i			0.349431i
$768$		0			0
$769$			38.3718i			1.38372i	0.722031	+	0.691861i	$0.243209\pi$
$769$			38.3718i			1.38372i	−0.722031	+	0.691861i	$0.756791\pi$
$770$		0			0
$771$	38.7275			1.39474
$772$		0			0
$773$	−5.61383			−0.201916			−0.100958	−	0.994891i	$-0.532191\pi$
$773$	−5.61383			−0.201916			−0.100958	+	0.994891i	$0.532191\pi$
$774$		0			0
$775$	−9.70304			−0.348543
$776$		0			0
$777$	10.6900	−	19.3376i	0.383503	−	0.693734i
$778$		0			0
$779$	−26.4522			−0.947749
$780$		0			0
$781$			19.2710i			0.689569i
$782$		0			0
$783$	−19.3425			−0.691245
$784$		0			0
$785$	13.5441			0.483409
$786$		0			0
$787$		−	13.1424i		−	0.468474i	−0.972180	−	0.234237i	$-0.924741\pi$
$787$		−	13.1424i		−	0.468474i	0.972180	−	0.234237i	$-0.0752592\pi$
$788$		0			0
$789$	26.2885			0.935895
$790$		0			0
$791$	−14.3321	+	25.9258i	−0.509590	+	0.921817i
$792$		0			0
$793$	−14.8330			−0.526734
$794$		0			0
$795$	−15.5661			−0.552074
$796$		0			0
$797$	−26.8837			−0.952269			−0.476134	−	0.879373i	$-0.657962\pi$
$797$	−26.8837			−0.952269			−0.476134	+	0.879373i	$0.657962\pi$
$798$		0			0
$799$		−	10.5375i		−	0.372790i
$800$		0			0
$801$		−	6.15442i		−	0.217456i
$802$		0			0
$803$			30.8815i			1.08978i
$804$		0			0
$805$	−6.08880	−	3.36595i	−0.214602	−	0.118634i
$806$		0			0
$807$			36.5366i			1.28615i
$808$		0			0
$809$	9.28389			0.326404			0.163202	−	0.986593i	$-0.447818\pi$
$809$	9.28389			0.326404			0.163202	+	0.986593i	$0.447818\pi$
$810$		0			0
$811$		−	47.2011i		−	1.65745i	−0.559653	−	0.828727i	$-0.689065\pi$
$811$		−	47.2011i		−	1.65745i	0.559653	−	0.828727i	$-0.310935\pi$
$812$		0			0
$813$		−	34.9149i		−	1.22452i
$814$		0			0
$815$	−18.5317			−0.649136
$816$		0			0
$817$			62.9772i			2.20329i
$818$		0			0
$819$	−2.41571	+	4.36986i	−0.0844116	+	0.152695i
$820$		0			0
$821$			28.6713i			1.00063i	0.865842	+	0.500317i	$0.166783\pi$
$821$			28.6713i			1.00063i	−0.865842	+	0.500317i	$0.833217\pi$
$822$		0			0
$823$		−	15.8578i		−	0.552769i	−0.961047	−	0.276385i	$-0.910864\pi$
$823$		−	15.8578i		−	0.552769i	0.961047	−	0.276385i	$-0.0891364\pi$
$824$		0			0
$825$			3.28321i			0.114307i
$826$		0			0
$827$	−9.33173			−0.324496			−0.162248	−	0.986750i	$-0.551874\pi$
$827$	−9.33173			−0.324496			−0.162248	+	0.986750i	$0.551874\pi$
$828$		0			0
$829$	50.2966			1.74687			0.873436	−	0.486939i	$-0.161887\pi$
$829$	50.2966			1.74687			0.873436	+	0.486939i	$0.161887\pi$
$830$		0			0
$831$	−7.57511			−0.262778
$832$		0			0
$833$	−6.63839	+	4.16933i	−0.230007	+	0.144459i
$834$		0			0
$835$	7.74592			0.268059
$836$		0			0
$837$			54.6726i			1.88976i
$838$		0			0
$839$	39.0155			1.34697			0.673483	−	0.739203i	$-0.264797\pi$
$839$	39.0155			1.34697			0.673483	+	0.739203i	$0.264797\pi$
$840$		0			0
$841$	17.2158			0.593647
$842$		0			0
$843$		−	34.7862i		−	1.19810i
$844$		0			0
$845$	10.5323			0.362321
$846$		0			0
$847$	6.40896	−	11.5934i	0.220214	−	0.398354i
$848$		0			0
$849$	16.9065			0.580231
$850$		0			0
$851$	−16.3748			−0.561320
$852$		0			0
$853$	11.1812			0.382836			0.191418	−	0.981509i	$-0.438691\pi$
$853$	11.1812			0.382836			0.191418	+	0.981509i	$0.438691\pi$
$854$		0			0
$855$			10.1488i			0.347081i
$856$		0			0
$857$			36.5484i			1.24847i	0.781236	+	0.624235i	$0.214589\pi$
$857$			36.5484i			1.24847i	−0.781236	+	0.624235i	$0.785411\pi$
$858$		0			0
$859$		−	6.59801i		−	0.225121i	−0.993645	−	0.112561i	$-0.964095\pi$
$859$		−	6.59801i		−	0.225121i	0.993645	−	0.112561i	$-0.0359052\pi$
$860$		0			0
$861$	5.37544	−	9.72384i	0.183195	−	0.331388i
$862$		0			0
$863$		−	6.48314i		−	0.220689i	−0.993893	−	0.110344i	$-0.964805\pi$
$863$		−	6.48314i		−	0.220689i	0.993893	−	0.110344i	$-0.0351954\pi$
$864$		0			0
$865$	−13.6112			−0.462795
$866$		0			0
$867$		−	21.1173i		−	0.717182i
$868$		0			0
$869$		−	18.8639i		−	0.639915i
$870$		0			0
$871$	3.38469			0.114686
$872$		0			0
$873$			1.00188i			0.0339086i
$874$		0			0
$875$	1.28003	−	2.31550i	0.0432729	−	0.0782781i
$876$		0			0
$877$		−	23.0472i		−	0.778248i	−0.921185	−	0.389124i	$-0.872778\pi$
$877$		−	23.0472i		−	0.778248i	0.921185	−	0.389124i	$-0.127222\pi$
$878$		0			0
$879$		−	22.5286i		−	0.759872i
$880$		0			0
$881$			35.8703i			1.20850i	0.796794	+	0.604251i	$0.206528\pi$
$881$			35.8703i			1.20850i	−0.796794	+	0.604251i	$0.793472\pi$
$882$		0			0
$883$	−50.9029			−1.71302			−0.856509	−	0.516131i	$-0.827372\pi$
$883$	−50.9029			−1.71302			−0.856509	+	0.516131i	$0.827372\pi$
$884$		0			0
$885$	−8.26193			−0.277722
$886$		0			0
$887$	−26.6009			−0.893172			−0.446586	−	0.894741i	$-0.647360\pi$
$887$	−26.6009			−0.893172			−0.446586	+	0.894741i	$0.647360\pi$
$888$		0			0
$889$	38.0173	+	21.0164i	1.27506	+	0.704866i
$890$		0			0
$891$	9.67640			0.324172
$892$		0			0
$893$		−	79.4895i		−	2.66001i
$894$		0			0
$895$	−17.3370			−0.579511
$896$		0			0
$897$	−5.54000			−0.184975
$898$		0			0
$899$			33.3088i			1.11091i
$900$		0			0
$901$	12.9980			0.433027
$902$		0			0
$903$	−23.1504	−	12.7978i	−0.770398	−	0.425884i
$904$		0			0
$905$	11.8268			0.393135
$906$		0			0
$907$	27.7572			0.921662			0.460831	−	0.887488i	$-0.347551\pi$
$907$	27.7572			0.921662			0.460831	+	0.887488i	$0.347551\pi$
$908$		0			0
$909$	−6.63634			−0.220114
$910$		0			0
$911$		−	2.35676i		−	0.0780829i	−0.999238	−	0.0390414i	$-0.987570\pi$
$911$		−	2.35676i		−	0.0780829i	0.999238	−	0.0390414i	$-0.0124304\pi$
$912$		0			0
$913$		−	1.99087i		−	0.0658880i
$914$		0			0
$915$		−	12.6634i		−	0.418639i
$916$		0			0
$917$	−4.63904	−	2.56451i	−0.153195	−	0.0846876i
$918$		0			0
$919$			5.11186i			0.168625i	0.996439	+	0.0843124i	$0.0268694\pi$
$919$			5.11186i			0.168625i	−0.996439	+	0.0843124i	$0.973131\pi$
$920$		0			0
$921$	−6.82113			−0.224764
$922$		0			0
$923$		−	12.3659i		−	0.407029i
$924$		0			0
$925$		−	6.22712i		−	0.204747i
$926$		0			0
$927$	14.1371			0.464322
$928$		0			0
$929$		−	49.8134i		−	1.63432i	−0.576408	−	0.817162i	$-0.695546\pi$
$929$		−	49.8134i		−	1.63432i	0.576408	−	0.817162i	$-0.304454\pi$
$930$		0			0
$931$	−50.0766	+	31.4513i	−1.64119	+	1.03077i
$932$		0			0
$933$			22.7049i			0.743326i
$934$		0			0
$935$		−	2.74154i		−	0.0896580i
$936$		0			0
$937$			15.7887i			0.515794i	0.966172	+	0.257897i	$0.0830296\pi$
$937$			15.7887i			0.515794i	−0.966172	+	0.257897i	$0.916970\pi$
$938$		0			0
$939$	−11.1420			−0.363606
$940$		0			0
$941$	−2.58455			−0.0842538			−0.0421269	−	0.999112i	$-0.513413\pi$
$941$	−2.58455			−0.0842538			−0.0421269	+	0.999112i	$0.513413\pi$
$942$		0			0
$943$	−8.23398			−0.268135
$944$		0			0
$945$	−13.0469	−	7.21244i	−0.424414	−	0.234621i
$946$		0			0
$947$	−17.5390			−0.569940			−0.284970	−	0.958536i	$-0.591984\pi$
$947$	−17.5390			−0.569940			−0.284970	+	0.958536i	$0.591984\pi$
$948$		0			0
$949$		−	19.8162i		−	0.643262i
$950$		0			0
$951$	14.4075			0.467196
$952$		0			0
$953$	−25.0362			−0.811001			−0.405501	−	0.914095i	$-0.632903\pi$
$953$	−25.0362			−0.811001			−0.405501	+	0.914095i	$0.632903\pi$
$954$		0			0
$955$			20.5823i			0.666029i
$956$		0			0
$957$	11.2707			0.364329
$958$		0			0
$959$	−5.38817	+	9.74687i	−0.173993	+	0.314743i
$960$		0			0
$961$	63.1490			2.03706
$962$		0			0
$963$	1.66980			0.0538086
$964$		0			0
$965$	−10.0690			−0.324132
$966$		0			0
$967$			18.7066i			0.601564i	0.953693	+	0.300782i	$0.0972477\pi$
$967$			18.7066i			0.601564i	−0.953693	+	0.300782i	$0.902752\pi$
$968$		0			0
$969$			12.6876i			0.407585i
$970$		0			0
$971$			32.0975i			1.03006i	0.857173	+	0.515029i	$0.172219\pi$
$971$			32.0975i			1.03006i	−0.857173	+	0.515029i	$0.827781\pi$
$972$		0			0
$973$	−9.74033	−	5.38455i	−0.312261	−	0.172621i
$974$		0			0
$975$		−	2.10679i		−	0.0674713i
$976$		0			0
$977$	5.17448			0.165546			0.0827731	−	0.996568i	$-0.473622\pi$
$977$	5.17448			0.165546			0.0827731	+	0.996568i	$0.473622\pi$
$978$		0			0
$979$			12.5412i			0.400819i
$980$		0			0
$981$		−	1.29176i		−	0.0412427i
$982$		0			0
$983$	25.3909			0.809843			0.404921	−	0.914352i	$-0.367299\pi$
$983$	25.3909			0.809843			0.404921	+	0.914352i	$0.367299\pi$
$984$		0			0
$985$		−	4.59538i		−	0.146421i
$986$		0			0
$987$	29.2203	+	16.1533i	0.930094	+	0.514166i
$988$		0			0
$989$			19.6034i			0.623351i
$990$		0			0
$991$			37.1976i			1.18162i	0.806810	+	0.590810i	$0.201192\pi$
$991$			37.1976i			1.18162i	−0.806810	+	0.590810i	$0.798808\pi$
$992$		0			0
$993$			16.3269i			0.518117i
$994$		0			0
$995$	2.15447			0.0683014
$996$		0			0
$997$	45.0979			1.42827			0.714133	−	0.700011i	$-0.246821\pi$
$997$	45.0979			1.42827			0.714133	+	0.700011i	$0.246821\pi$
$998$		0			0
$999$	−35.0873			−1.11011

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1120.2.h.a.111.6		16
4.3	odd	2		280.2.h.a.251.1	✓	16
7.6	odd	2		1120.2.h.b.111.11		16
8.3	odd	2		1120.2.h.b.111.6		16
8.5	even	2		280.2.h.b.251.2	yes	16
28.27	even	2		280.2.h.b.251.1	yes	16
56.13	odd	2		280.2.h.a.251.2	yes	16
56.27	even	2	inner	1120.2.h.a.111.11		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.h.a.251.1	✓	16	4.3	odd	2
280.2.h.a.251.2	yes	16	56.13	odd	2
280.2.h.b.251.1	yes	16	28.27	even	2
280.2.h.b.251.2	yes	16	8.5	even	2
1120.2.h.a.111.6		16	1.1	even	1	trivial
1120.2.h.a.111.11		16	56.27	even	2	inner
1120.2.h.b.111.6		16	8.3	odd	2
1120.2.h.b.111.11		16	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1120.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.34113i q^{3} -1.00000 q^{5} +(-1.28003 + 2.31550i) q^{7} +1.20136 q^{9} +O(q^{10})\)	\(q-1.34113i q^{3} -1.00000 q^{5} +(-1.28003 + 2.31550i) q^{7} +1.20136 q^{9} -2.44809 q^{11} +1.57090 q^{13} +1.34113i q^{15} -1.11987i q^{17} -8.44773i q^{19} +(3.10539 + 1.71669i) q^{21} -2.62959i q^{23} +1.00000 q^{25} -5.63459i q^{27} -3.43282i q^{29} -9.70304 q^{31} +3.28321i q^{33} +(1.28003 - 2.31550i) q^{35} -6.22712i q^{37} -2.10679i q^{39} -3.13128i q^{41} -7.45492 q^{43} -1.20136 q^{45} +9.40956 q^{47} +(-3.72304 - 5.92781i) q^{49} -1.50190 q^{51} +11.6067i q^{53} +2.44809 q^{55} -11.3295 q^{57} +6.16041i q^{59} -9.44231 q^{61} +(-1.53778 + 2.78175i) q^{63} -1.57090 q^{65} +2.15461 q^{67} -3.52663 q^{69} -7.87185i q^{71} -12.6145i q^{73} -1.34113i q^{75} +(3.13362 - 5.66853i) q^{77} +7.70558i q^{79} -3.95264 q^{81} +0.813234i q^{83} +1.11987i q^{85} -4.60387 q^{87} -5.12287i q^{89} +(-2.01081 + 3.63742i) q^{91} +13.0131i q^{93} +8.44773i q^{95} +0.833955i q^{97} -2.94104 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{5} - 16 q^{9}+O(q^{10}) \) 16 * q - 16 * q^5 - 16 * q^9	\( 16 q - 16 q^{5} - 16 q^{9} + 4 q^{11} + 4 q^{21} + 16 q^{25} - 16 q^{31} + 4 q^{43} + 16 q^{45} - 8 q^{49} + 40 q^{51} - 4 q^{55} - 16 q^{57} + 8 q^{61} + 28 q^{63} - 20 q^{67} + 40 q^{69} + 4 q^{77} + 24 q^{81} + 72 q^{87} + 32 q^{91} - 20 q^{99}+O(q^{100}) \) 16 * q - 16 * q^5 - 16 * q^9 + 4 * q^11 + 4 * q^21 + 16 * q^25 - 16 * q^31 + 4 * q^43 + 16 * q^45 - 8 * q^49 + 40 * q^51 - 4 * q^55 - 16 * q^57 + 8 * q^61 + 28 * q^63 - 20 * q^67 + 40 * q^69 + 4 * q^77 + 24 * q^81 + 72 * q^87 + 32 * q^91 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more