LMFDB - Embedded newform 1920.2.h.i.1151.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1920,2,Mod(1151,1920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1920, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("1920.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1920 = 2^{7} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1920.h (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:	$15.3312771881$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.1
Root			$-1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1920.1151
Dual form			1920.2.h.i.1151.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 - 1.22474i) q^{3} -1.00000i q^{5} +4.44949i q^{7} +3.00000i q^{9} +2.00000 q^{11} -6.89898 q^{13} +(-1.22474 + 1.22474i) q^{15} +2.00000i q^{17} +2.00000i q^{19} +(5.44949 - 5.44949i) q^{21} -0.449490 q^{23} -1.00000 q^{25} +(3.67423 - 3.67423i) q^{27} -8.89898i q^{29} -8.00000i q^{31} +(-2.44949 - 2.44949i) q^{33} +4.44949 q^{35} -6.00000 q^{37} +(8.44949 + 8.44949i) q^{39} -11.7980i q^{41} -1.55051i q^{43} +3.00000 q^{45} +3.55051 q^{47} -12.7980 q^{49} +(2.44949 - 2.44949i) q^{51} -2.89898i q^{53} -2.00000i q^{55} +(2.44949 - 2.44949i) q^{57} +10.8990 q^{59} -0.898979 q^{61} -13.3485 q^{63} +6.89898i q^{65} -3.34847i q^{67} +(0.550510 + 0.550510i) q^{69} -8.89898 q^{71} +1.10102 q^{73} +(1.22474 + 1.22474i) q^{75} +8.89898i q^{77} +4.89898i q^{79} -9.00000 q^{81} +12.2474 q^{83} +2.00000 q^{85} +(-10.8990 + 10.8990i) q^{87} -9.79796i q^{89} -30.6969i q^{91} +(-9.79796 + 9.79796i) q^{93} +2.00000 q^{95} +9.10102 q^{97} +6.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{11} - 8 q^{13} + 12 q^{21} + 8 q^{23} - 4 q^{25} + 8 q^{35} - 24 q^{37} + 24 q^{39} + 12 q^{45} + 24 q^{47} - 12 q^{49} + 24 q^{59} + 16 q^{61} - 24 q^{63} + 12 q^{69} - 16 q^{71} + 24 q^{73}+ \cdots + 56 q^{97}+O(q^{100}) $ 4 * q + 8 * q^11 - 8 * q^13 + 12 * q^21 + 8 * q^23 - 4 * q^25 + 8 * q^35 - 24 * q^37 + 24 * q^39 + 12 * q^45 + 24 * q^47 - 12 * q^49 + 24 * q^59 + 16 * q^61 - 24 * q^63 + 12 * q^69 - 16 * q^71 + 24 * q^73 - 36 * q^81 + 8 * q^85 - 24 * q^87 + 8 * q^95 + 56 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1920\mathbb{Z}\right)^\times$.

$n$	$511$	$641$	$901$	$1537$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$	−1.22474	−	1.22474i	−0.707107	−	0.707107i
$3$	−1.22474	−	1.22474i	−0.707107	−	0.707107i
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$			4.44949i			1.68175i	0.541230	+	0.840875i	$0.317959\pi$
$7$			4.44949i			1.68175i	−0.541230	+	0.840875i	$0.682041\pi$
$8$		0			0
$9$			3.00000i			1.00000i
$10$		0			0
$11$	2.00000			0.603023			0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000			0.603023			0.301511	+	0.953463i	$0.402509\pi$
$12$		0			0
$13$	−6.89898			−1.91343			−0.956716	−	0.291022i	$-0.906005\pi$
$13$	−6.89898			−1.91343			−0.956716	+	0.291022i	$0.906005\pi$
$14$		0			0
$15$	−1.22474	+	1.22474i	−0.316228	+	0.316228i
$16$		0			0
$17$			2.00000i			0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$			2.00000i			0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$		0			0
$19$			2.00000i			0.458831i	0.973329	+	0.229416i	$0.0736815\pi$
$19$			2.00000i			0.458831i	−0.973329	+	0.229416i	$0.926318\pi$
$20$		0			0
$21$	5.44949	−	5.44949i	1.18918	−	1.18918i
$22$		0			0
$23$	−0.449490			−0.0937251			−0.0468625	−	0.998901i	$-0.514922\pi$
$23$	−0.449490			−0.0937251			−0.0468625	+	0.998901i	$0.514922\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	3.67423	−	3.67423i	0.707107	−	0.707107i
$28$		0			0
$29$		−	8.89898i		−	1.65250i	−0.563304	−	0.826250i	$-0.690470\pi$
$29$		−	8.89898i		−	1.65250i	0.563304	−	0.826250i	$-0.309530\pi$
$30$		0			0
$31$		−	8.00000i		−	1.43684i	−0.695608	−	0.718421i	$-0.744865\pi$
$31$		−	8.00000i		−	1.43684i	0.695608	−	0.718421i	$-0.255135\pi$
$32$		0			0
$33$	−2.44949	−	2.44949i	−0.426401	−	0.426401i
$34$		0			0
$35$	4.44949			0.752101
$36$		0			0
$37$	−6.00000			−0.986394			−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000			−0.986394			−0.493197	+	0.869918i	$0.664172\pi$
$38$		0			0
$39$	8.44949	+	8.44949i	1.35300	+	1.35300i
$40$		0			0
$41$		−	11.7980i		−	1.84253i	−0.388934	−	0.921266i	$-0.627156\pi$
$41$		−	11.7980i		−	1.84253i	0.388934	−	0.921266i	$-0.372844\pi$
$42$		0			0
$43$		−	1.55051i		−	0.236451i	−0.992987	−	0.118225i	$-0.962279\pi$
$43$		−	1.55051i		−	0.236451i	0.992987	−	0.118225i	$-0.0377205\pi$
$44$		0			0
$45$	3.00000			0.447214
$46$		0			0
$47$	3.55051			0.517895			0.258948	−	0.965891i	$-0.416624\pi$
$47$	3.55051			0.517895			0.258948	+	0.965891i	$0.416624\pi$
$48$		0			0
$49$	−12.7980			−1.82828
$50$		0			0
$51$	2.44949	−	2.44949i	0.342997	−	0.342997i
$52$		0			0
$53$		−	2.89898i		−	0.398205i	−0.979979	−	0.199103i	$-0.936197\pi$
$53$		−	2.89898i		−	0.398205i	0.979979	−	0.199103i	$-0.0638027\pi$
$54$		0			0
$55$		−	2.00000i		−	0.269680i
$56$		0			0
$57$	2.44949	−	2.44949i	0.324443	−	0.324443i
$58$		0			0
$59$	10.8990			1.41893			0.709463	−	0.704743i	$-0.248937\pi$
$59$	10.8990			1.41893			0.709463	+	0.704743i	$0.248937\pi$
$60$		0			0
$61$	−0.898979			−0.115103			−0.0575513	−	0.998343i	$-0.518329\pi$
$61$	−0.898979			−0.115103			−0.0575513	+	0.998343i	$0.518329\pi$
$62$		0			0
$63$	−13.3485			−1.68175
$64$		0			0
$65$			6.89898i			0.855713i
$66$		0			0
$67$		−	3.34847i		−	0.409081i	−0.978858	−	0.204540i	$-0.934430\pi$
$67$		−	3.34847i		−	0.409081i	0.978858	−	0.204540i	$-0.0655699\pi$
$68$		0			0
$69$	0.550510	+	0.550510i	0.0662736	+	0.0662736i
$70$		0			0
$71$	−8.89898			−1.05611			−0.528057	−	0.849209i	$-0.677079\pi$
$71$	−8.89898			−1.05611			−0.528057	+	0.849209i	$0.677079\pi$
$72$		0			0
$73$	1.10102			0.128865			0.0644324	−	0.997922i	$-0.479476\pi$
$73$	1.10102			0.128865			0.0644324	+	0.997922i	$0.479476\pi$
$74$		0			0
$75$	1.22474	+	1.22474i	0.141421	+	0.141421i
$76$		0			0
$77$			8.89898i			1.01413i
$78$		0			0
$79$			4.89898i			0.551178i	0.961276	+	0.275589i	$0.0888729\pi$
$79$			4.89898i			0.551178i	−0.961276	+	0.275589i	$0.911127\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$	12.2474			1.34433			0.672166	−	0.740400i	$-0.265364\pi$
$83$	12.2474			1.34433			0.672166	+	0.740400i	$0.265364\pi$
$84$		0			0
$85$	2.00000			0.216930
$86$		0			0
$87$	−10.8990	+	10.8990i	−1.16849	+	1.16849i
$88$		0			0
$89$		−	9.79796i		−	1.03858i	−0.854598	−	0.519291i	$-0.826196\pi$
$89$		−	9.79796i		−	1.03858i	0.854598	−	0.519291i	$-0.173804\pi$
$90$		0			0
$91$		−	30.6969i		−	3.21791i
$92$		0			0
$93$	−9.79796	+	9.79796i	−1.01600	+	1.01600i
$94$		0			0
$95$	2.00000			0.205196
$96$		0			0
$97$	9.10102			0.924069			0.462034	−	0.886862i	$-0.347120\pi$
$97$	9.10102			0.924069			0.462034	+	0.886862i	$0.347120\pi$
$98$		0			0
$99$			6.00000i			0.603023i
$100$		0			0
$101$			3.10102i			0.308563i	0.988027	+	0.154282i	$0.0493063\pi$
$101$			3.10102i			0.308563i	−0.988027	+	0.154282i	$0.950694\pi$
$102$		0			0
$103$		−	13.3485i		−	1.31526i	−0.753339	−	0.657632i	$-0.771558\pi$
$103$		−	13.3485i		−	1.31526i	0.753339	−	0.657632i	$-0.228442\pi$
$104$		0			0
$105$	−5.44949	−	5.44949i	−0.531816	−	0.531816i
$106$		0			0
$107$	−10.4495			−1.01019			−0.505095	−	0.863064i	$-0.668543\pi$
$107$	−10.4495			−1.01019			−0.505095	+	0.863064i	$0.668543\pi$
$108$		0			0
$109$	−12.8990			−1.23550			−0.617749	−	0.786375i	$-0.711955\pi$
$109$	−12.8990			−1.23550			−0.617749	+	0.786375i	$0.711955\pi$
$110$		0			0
$111$	7.34847	+	7.34847i	0.697486	+	0.697486i
$112$		0			0
$113$		−	1.10102i		−	0.103575i	−0.998658	−	0.0517876i	$-0.983508\pi$
$113$		−	1.10102i		−	0.103575i	0.998658	−	0.0517876i	$-0.0164919\pi$
$114$		0			0
$115$			0.449490i			0.0419151i
$116$		0			0
$117$		−	20.6969i		−	1.91343i
$118$		0			0
$119$	−8.89898			−0.815768
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$	−14.4495	+	14.4495i	−1.30287	+	1.30287i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$			9.34847i			0.829543i	0.909926	+	0.414771i	$0.136138\pi$
$127$			9.34847i			0.829543i	−0.909926	+	0.414771i	$0.863862\pi$
$128$		0			0
$129$	−1.89898	+	1.89898i	−0.167196	+	0.167196i
$130$		0			0
$131$	−19.7980			−1.72976			−0.864878	−	0.501982i	$-0.832604\pi$
$131$	−19.7980			−1.72976			−0.864878	+	0.501982i	$0.832604\pi$
$132$		0			0
$133$	−8.89898			−0.771639
$134$		0			0
$135$	−3.67423	−	3.67423i	−0.316228	−	0.316228i
$136$		0			0
$137$		−	10.8990i		−	0.931163i	−0.885005	−	0.465581i	$-0.845845\pi$
$137$		−	10.8990i		−	0.931163i	0.885005	−	0.465581i	$-0.154155\pi$
$138$		0			0
$139$			2.00000i			0.169638i	0.996396	+	0.0848189i	$0.0270312\pi$
$139$			2.00000i			0.169638i	−0.996396	+	0.0848189i	$0.972969\pi$
$140$		0			0
$141$	−4.34847	−	4.34847i	−0.366207	−	0.366207i
$142$		0			0
$143$	−13.7980			−1.15384
$144$		0			0
$145$	−8.89898			−0.739020
$146$		0			0
$147$	15.6742	+	15.6742i	1.29279	+	1.29279i
$148$		0			0
$149$		−	15.7980i		−	1.29422i	−0.762397	−	0.647110i	$-0.775978\pi$
$149$		−	15.7980i		−	1.29422i	0.762397	−	0.647110i	$-0.224022\pi$
$150$		0			0
$151$			15.5959i			1.26918i	0.772850	+	0.634589i	$0.218831\pi$
$151$			15.5959i			1.26918i	−0.772850	+	0.634589i	$0.781169\pi$
$152$		0			0
$153$	−6.00000			−0.485071
$154$		0			0
$155$	−8.00000			−0.642575
$156$		0			0
$157$	−10.0000			−0.798087			−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000			−0.798087			−0.399043	+	0.916932i	$0.630658\pi$
$158$		0			0
$159$	−3.55051	+	3.55051i	−0.281574	+	0.281574i
$160$		0			0
$161$		−	2.00000i		−	0.157622i
$162$		0			0
$163$		−	10.4495i		−	0.818467i	−0.912430	−	0.409234i	$-0.865796\pi$
$163$		−	10.4495i		−	0.818467i	0.912430	−	0.409234i	$-0.134204\pi$
$164$		0			0
$165$	−2.44949	+	2.44949i	−0.190693	+	0.190693i
$166$		0			0
$167$	−6.65153			−0.514711			−0.257355	−	0.966317i	$-0.582851\pi$
$167$	−6.65153			−0.514711			−0.257355	+	0.966317i	$0.582851\pi$
$168$		0			0
$169$	34.5959			2.66122
$170$		0			0
$171$	−6.00000			−0.458831
$172$		0			0
$173$			11.7980i			0.896982i	0.893787	+	0.448491i	$0.148038\pi$
$173$			11.7980i			0.896982i	−0.893787	+	0.448491i	$0.851962\pi$
$174$		0			0
$175$		−	4.44949i		−	0.336350i
$176$		0			0
$177$	−13.3485	−	13.3485i	−1.00333	−	1.00333i
$178$		0			0
$179$	−13.1010			−0.979216			−0.489608	−	0.871943i	$-0.662860\pi$
$179$	−13.1010			−0.979216			−0.489608	+	0.871943i	$0.662860\pi$
$180$		0			0
$181$	21.5959			1.60521			0.802606	−	0.596510i	$-0.203446\pi$
$181$	21.5959			1.60521			0.802606	+	0.596510i	$0.203446\pi$
$182$		0			0
$183$	1.10102	+	1.10102i	0.0813898	+	0.0813898i
$184$		0			0
$185$			6.00000i			0.441129i
$186$		0			0
$187$			4.00000i			0.292509i
$188$		0			0
$189$	16.3485	+	16.3485i	1.18918	+	1.18918i
$190$		0			0
$191$	12.8990			0.933337			0.466669	−	0.884432i	$-0.345454\pi$
$191$	12.8990			0.933337			0.466669	+	0.884432i	$0.345454\pi$
$192$		0			0
$193$	−4.69694			−0.338093			−0.169047	−	0.985608i	$-0.554069\pi$
$193$	−4.69694			−0.338093			−0.169047	+	0.985608i	$0.554069\pi$
$194$		0			0
$195$	8.44949	−	8.44949i	0.605081	−	0.605081i
$196$		0			0
$197$		−	8.69694i		−	0.619631i	−0.950797	−	0.309816i	$-0.899733\pi$
$197$		−	8.69694i		−	0.619631i	0.950797	−	0.309816i	$-0.100267\pi$
$198$		0			0
$199$		−	3.10102i		−	0.219826i	−0.993941	−	0.109913i	$-0.964943\pi$
$199$		−	3.10102i		−	0.219826i	0.993941	−	0.109913i	$-0.0350571\pi$
$200$		0			0
$201$	−4.10102	+	4.10102i	−0.289264	+	0.289264i
$202$		0			0
$203$	39.5959			2.77909
$204$		0			0
$205$	−11.7980			−0.824005
$206$		0			0
$207$		−	1.34847i		−	0.0937251i
$208$		0			0
$209$			4.00000i			0.276686i
$210$		0			0
$211$		−	9.10102i		−	0.626540i	−0.949664	−	0.313270i	$-0.898576\pi$
$211$		−	9.10102i		−	0.626540i	0.949664	−	0.313270i	$-0.101424\pi$
$212$		0			0
$213$	10.8990	+	10.8990i	0.746786	+	0.746786i
$214$		0			0
$215$	−1.55051			−0.105744
$216$		0			0
$217$	35.5959			2.41641
$218$		0			0
$219$	−1.34847	−	1.34847i	−0.0911211	−	0.0911211i
$220$		0			0
$221$		−	13.7980i		−	0.928151i
$222$		0			0
$223$			7.55051i			0.505620i	0.967516	+	0.252810i	$0.0813547\pi$
$223$			7.55051i			0.505620i	−0.967516	+	0.252810i	$0.918645\pi$
$224$		0			0
$225$		−	3.00000i		−	0.200000i
$226$		0			0
$227$	−13.1464			−0.872559			−0.436280	−	0.899811i	$-0.643704\pi$
$227$	−13.1464			−0.872559			−0.436280	+	0.899811i	$0.643704\pi$
$228$		0			0
$229$	−0.202041			−0.0133512			−0.00667562	−	0.999978i	$-0.502125\pi$
$229$	−0.202041			−0.0133512			−0.00667562	+	0.999978i	$0.502125\pi$
$230$		0			0
$231$	10.8990	−	10.8990i	0.717100	−	0.717100i
$232$		0			0
$233$			7.79796i			0.510861i	0.966827	+	0.255431i	$0.0822172\pi$
$233$			7.79796i			0.510861i	−0.966827	+	0.255431i	$0.917783\pi$
$234$		0			0
$235$		−	3.55051i		−	0.231610i
$236$		0			0
$237$	6.00000	−	6.00000i	0.389742	−	0.389742i
$238$		0			0
$239$	−25.7980			−1.66873			−0.834366	−	0.551211i	$-0.814166\pi$
$239$	−25.7980			−1.66873			−0.834366	+	0.551211i	$0.814166\pi$
$240$		0			0
$241$	15.5959			1.00462			0.502311	−	0.864687i	$-0.332483\pi$
$241$	15.5959			1.00462			0.502311	+	0.864687i	$0.332483\pi$
$242$		0			0
$243$	11.0227	+	11.0227i	0.707107	+	0.707107i
$244$		0			0
$245$			12.7980i			0.817632i
$246$		0			0
$247$		−	13.7980i		−	0.877943i
$248$		0			0
$249$	−15.0000	−	15.0000i	−0.950586	−	0.950586i
$250$		0			0
$251$	8.20204			0.517708			0.258854	−	0.965916i	$-0.416655\pi$
$251$	8.20204			0.517708			0.258854	+	0.965916i	$0.416655\pi$
$252$		0			0
$253$	−0.898979			−0.0565184
$254$		0			0
$255$	−2.44949	−	2.44949i	−0.153393	−	0.153393i
$256$		0			0
$257$			1.10102i			0.0686798i	0.999410	+	0.0343399i	$0.0109329\pi$
$257$			1.10102i			0.0686798i	−0.999410	+	0.0343399i	$0.989067\pi$
$258$		0			0
$259$		−	26.6969i		−	1.65887i
$260$		0			0
$261$	26.6969			1.65250
$262$		0			0
$263$	7.55051			0.465584			0.232792	−	0.972526i	$-0.425214\pi$
$263$	7.55051			0.465584			0.232792	+	0.972526i	$0.425214\pi$
$264$		0			0
$265$	−2.89898			−0.178083
$266$		0			0
$267$	−12.0000	+	12.0000i	−0.734388	+	0.734388i
$268$		0			0
$269$			17.5959i			1.07284i	0.843951	+	0.536421i	$0.180224\pi$
$269$			17.5959i			1.07284i	−0.843951	+	0.536421i	$0.819776\pi$
$270$		0			0
$271$		−	12.0000i		−	0.728948i	−0.931214	−	0.364474i	$-0.881249\pi$
$271$		−	12.0000i		−	0.728948i	0.931214	−	0.364474i	$-0.118751\pi$
$272$		0			0
$273$	−37.5959	+	37.5959i	−2.27541	+	2.27541i
$274$		0			0
$275$	−2.00000			−0.120605
$276$		0			0
$277$	−2.00000			−0.120168			−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000			−0.120168			−0.0600842	+	0.998193i	$0.519137\pi$
$278$		0			0
$279$	24.0000			1.43684
$280$		0			0
$281$		−	21.5959i		−	1.28830i	−0.764897	−	0.644152i	$-0.777210\pi$
$281$		−	21.5959i		−	1.28830i	0.764897	−	0.644152i	$-0.222790\pi$
$282$		0			0
$283$			21.1464i			1.25702i	0.777800	+	0.628512i	$0.216336\pi$
$283$			21.1464i			1.25702i	−0.777800	+	0.628512i	$0.783664\pi$
$284$		0			0
$285$	−2.44949	−	2.44949i	−0.145095	−	0.145095i
$286$		0			0
$287$	52.4949			3.09868
$288$		0			0
$289$	13.0000			0.764706
$290$		0			0
$291$	−11.1464	−	11.1464i	−0.653415	−	0.653415i
$292$		0			0
$293$		−	8.20204i		−	0.479168i	−0.970876	−	0.239584i	$-0.922989\pi$
$293$		−	8.20204i		−	0.479168i	0.970876	−	0.239584i	$-0.0770111\pi$
$294$		0			0
$295$		−	10.8990i		−	0.634563i
$296$		0			0
$297$	7.34847	−	7.34847i	0.426401	−	0.426401i
$298$		0			0
$299$	3.10102			0.179337
$300$		0			0
$301$	6.89898			0.397651
$302$		0			0
$303$	3.79796	−	3.79796i	0.218187	−	0.218187i
$304$		0			0
$305$			0.898979i			0.0514754i
$306$		0			0
$307$		−	12.6515i		−	0.722061i	−0.932554	−	0.361030i	$-0.882425\pi$
$307$		−	12.6515i		−	0.722061i	0.932554	−	0.361030i	$-0.117575\pi$
$308$		0			0
$309$	−16.3485	+	16.3485i	−0.930032	+	0.930032i
$310$		0			0
$311$	24.4949			1.38898			0.694489	−	0.719503i	$-0.255630\pi$
$311$	24.4949			1.38898			0.694489	+	0.719503i	$0.255630\pi$
$312$		0			0
$313$	−19.7980			−1.11905			−0.559523	−	0.828815i	$-0.689016\pi$
$313$	−19.7980			−1.11905			−0.559523	+	0.828815i	$0.689016\pi$
$314$		0			0
$315$			13.3485i			0.752101i
$316$		0			0
$317$			8.69694i			0.488469i	0.969716	+	0.244234i	$0.0785366\pi$
$317$			8.69694i			0.488469i	−0.969716	+	0.244234i	$0.921463\pi$
$318$		0			0
$319$		−	17.7980i		−	0.996494i
$320$		0			0
$321$	12.7980	+	12.7980i	0.714312	+	0.714312i
$322$		0			0
$323$	−4.00000			−0.222566
$324$		0			0
$325$	6.89898			0.382687
$326$		0			0
$327$	15.7980	+	15.7980i	0.873629	+	0.873629i
$328$		0			0
$329$			15.7980i			0.870970i
$330$		0			0
$331$			22.8990i			1.25864i	0.777146	+	0.629321i	$0.216667\pi$
$331$			22.8990i			1.25864i	−0.777146	+	0.629321i	$0.783333\pi$
$332$		0			0
$333$		−	18.0000i		−	0.986394i
$334$		0			0
$335$	−3.34847			−0.182946
$336$		0			0
$337$	−23.7980			−1.29636			−0.648179	−	0.761488i	$-0.724469\pi$
$337$	−23.7980			−1.29636			−0.648179	+	0.761488i	$0.724469\pi$
$338$		0			0
$339$	−1.34847	+	1.34847i	−0.0732388	+	0.0732388i
$340$		0			0
$341$		−	16.0000i		−	0.866449i
$342$		0			0
$343$		−	25.7980i		−	1.39296i
$344$		0			0
$345$	0.550510	−	0.550510i	0.0296385	−	0.0296385i
$346$		0			0
$347$	−17.1464			−0.920468			−0.460234	−	0.887798i	$-0.652235\pi$
$347$	−17.1464			−0.920468			−0.460234	+	0.887798i	$0.652235\pi$
$348$		0			0
$349$	−14.0000			−0.749403			−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000			−0.749403			−0.374701	+	0.927146i	$0.622255\pi$
$350$		0			0
$351$	−25.3485	+	25.3485i	−1.35300	+	1.35300i
$352$		0			0
$353$			22.4949i			1.19728i	0.801017	+	0.598641i	$0.204292\pi$
$353$			22.4949i			1.19728i	−0.801017	+	0.598641i	$0.795708\pi$
$354$		0			0
$355$			8.89898i			0.472309i
$356$		0			0
$357$	10.8990	+	10.8990i	0.576835	+	0.576835i
$358$		0			0
$359$	−8.00000			−0.422224			−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000			−0.422224			−0.211112	+	0.977462i	$0.567708\pi$
$360$		0			0
$361$	15.0000			0.789474
$362$		0			0
$363$	8.57321	+	8.57321i	0.449977	+	0.449977i
$364$		0			0
$365$		−	1.10102i		−	0.0576300i
$366$		0			0
$367$			0.449490i			0.0234632i	0.999931	+	0.0117316i	$0.00373437\pi$
$367$			0.449490i			0.0234632i	−0.999931	+	0.0117316i	$0.996266\pi$
$368$		0			0
$369$	35.3939			1.84253
$370$		0			0
$371$	12.8990			0.669682
$372$		0			0
$373$	−33.5959			−1.73953			−0.869765	−	0.493466i	$-0.835730\pi$
$373$	−33.5959			−1.73953			−0.869765	+	0.493466i	$0.835730\pi$
$374$		0			0
$375$	1.22474	−	1.22474i	0.0632456	−	0.0632456i
$376$		0			0
$377$			61.3939i			3.16195i
$378$		0			0
$379$		−	10.0000i		−	0.513665i	−0.966456	−	0.256833i	$-0.917321\pi$
$379$		−	10.0000i		−	0.513665i	0.966456	−	0.256833i	$-0.0826790\pi$
$380$		0			0
$381$	11.4495	−	11.4495i	0.586575	−	0.586575i
$382$		0			0
$383$	0.853572			0.0436155			0.0218077	−	0.999762i	$-0.493058\pi$
$383$	0.853572			0.0436155			0.0218077	+	0.999762i	$0.493058\pi$
$384$		0			0
$385$	8.89898			0.453534
$386$		0			0
$387$	4.65153			0.236451
$388$		0			0
$389$		−	19.7980i		−	1.00380i	−0.864927	−	0.501898i	$-0.832635\pi$
$389$		−	19.7980i		−	1.00380i	0.864927	−	0.501898i	$-0.167365\pi$
$390$		0			0
$391$		−	0.898979i		−	0.0454633i
$392$		0			0
$393$	24.2474	+	24.2474i	1.22312	+	1.22312i
$394$		0			0
$395$	4.89898			0.246494
$396$		0			0
$397$	10.4949			0.526724			0.263362	−	0.964697i	$-0.415169\pi$
$397$	10.4949			0.526724			0.263362	+	0.964697i	$0.415169\pi$
$398$		0			0
$399$	10.8990	+	10.8990i	0.545631	+	0.545631i
$400$		0			0
$401$			23.5959i			1.17832i	0.808015	+	0.589162i	$0.200542\pi$
$401$			23.5959i			1.17832i	−0.808015	+	0.589162i	$0.799458\pi$
$402$		0			0
$403$			55.1918i			2.74930i
$404$		0			0
$405$			9.00000i			0.447214i
$406$		0			0
$407$	−12.0000			−0.594818
$408$		0			0
$409$	−4.00000			−0.197787			−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000			−0.197787			−0.0988936	+	0.995098i	$0.531530\pi$
$410$		0			0
$411$	−13.3485	+	13.3485i	−0.658431	+	0.658431i
$412$		0			0
$413$			48.4949i			2.38628i
$414$		0			0
$415$		−	12.2474i		−	0.601204i
$416$		0			0
$417$	2.44949	−	2.44949i	0.119952	−	0.119952i
$418$		0			0
$419$	−36.6969			−1.79276			−0.896381	−	0.443284i	$-0.853813\pi$
$419$	−36.6969			−1.79276			−0.896381	+	0.443284i	$0.853813\pi$
$420$		0			0
$421$	12.4949			0.608964			0.304482	−	0.952518i	$-0.401517\pi$
$421$	12.4949			0.608964			0.304482	+	0.952518i	$0.401517\pi$
$422$		0			0
$423$			10.6515i			0.517895i
$424$		0			0
$425$		−	2.00000i		−	0.0970143i
$426$		0			0
$427$		−	4.00000i		−	0.193574i
$428$		0			0
$429$	16.8990	+	16.8990i	0.815890	+	0.815890i
$430$		0			0
$431$	32.4949			1.56522			0.782612	−	0.622510i	$-0.213887\pi$
$431$	32.4949			1.56522			0.782612	+	0.622510i	$0.213887\pi$
$432$		0			0
$433$	18.8990			0.908227			0.454113	−	0.890944i	$-0.349956\pi$
$433$	18.8990			0.908227			0.454113	+	0.890944i	$0.349956\pi$
$434$		0			0
$435$	10.8990	+	10.8990i	0.522566	+	0.522566i
$436$		0			0
$437$		−	0.898979i		−	0.0430040i
$438$		0			0
$439$		−	20.4949i		−	0.978168i	−0.872237	−	0.489084i	$-0.837331\pi$
$439$		−	20.4949i		−	0.978168i	0.872237	−	0.489084i	$-0.162669\pi$
$440$		0			0
$441$		−	38.3939i		−	1.82828i
$442$		0			0
$443$	−0.247449			−0.0117566			−0.00587832	−	0.999983i	$-0.501871\pi$
$443$	−0.247449			−0.0117566			−0.00587832	+	0.999983i	$0.501871\pi$
$444$		0			0
$445$	−9.79796			−0.464468
$446$		0			0
$447$	−19.3485	+	19.3485i	−0.915151	+	0.915151i
$448$		0			0
$449$		−	13.5959i		−	0.641631i	−0.947142	−	0.320816i	$-0.896043\pi$
$449$		−	13.5959i		−	0.641631i	0.947142	−	0.320816i	$-0.103957\pi$
$450$		0			0
$451$		−	23.5959i		−	1.11109i
$452$		0			0
$453$	19.1010	−	19.1010i	0.897444	−	0.897444i
$454$		0			0
$455$	−30.6969			−1.43909
$456$		0			0
$457$	−8.20204			−0.383675			−0.191838	−	0.981427i	$-0.561445\pi$
$457$	−8.20204			−0.383675			−0.191838	+	0.981427i	$0.561445\pi$
$458$		0			0
$459$	7.34847	+	7.34847i	0.342997	+	0.342997i
$460$		0			0
$461$		−	26.2929i		−	1.22458i	−0.790633	−	0.612290i	$-0.790248\pi$
$461$		−	26.2929i		−	1.22458i	0.790633	−	0.612290i	$-0.209752\pi$
$462$		0			0
$463$		−	37.8434i		−	1.75873i	−0.476148	−	0.879365i	$-0.657967\pi$
$463$		−	37.8434i		−	1.75873i	0.476148	−	0.879365i	$-0.342033\pi$
$464$		0			0
$465$	9.79796	+	9.79796i	0.454369	+	0.454369i
$466$		0			0
$467$	21.1464			0.978540			0.489270	−	0.872132i	$-0.337263\pi$
$467$	21.1464			0.978540			0.489270	+	0.872132i	$0.337263\pi$
$468$		0			0
$469$	14.8990			0.687971
$470$		0			0
$471$	12.2474	+	12.2474i	0.564333	+	0.564333i
$472$		0			0
$473$		−	3.10102i		−	0.142585i
$474$		0			0
$475$		−	2.00000i		−	0.0917663i
$476$		0			0
$477$	8.69694			0.398205
$478$		0			0
$479$	17.3939			0.794747			0.397373	−	0.917657i	$-0.369922\pi$
$479$	17.3939			0.794747			0.397373	+	0.917657i	$0.369922\pi$
$480$		0			0
$481$	41.3939			1.88740
$482$		0			0
$483$	−2.44949	+	2.44949i	−0.111456	+	0.111456i
$484$		0			0
$485$		−	9.10102i		−	0.413256i
$486$		0			0
$487$		−	2.65153i		−	0.120152i	−0.998194	−	0.0600762i	$-0.980866\pi$
$487$		−	2.65153i		−	0.120152i	0.998194	−	0.0600762i	$-0.0191344\pi$
$488$		0			0
$489$	−12.7980	+	12.7980i	−0.578744	+	0.578744i
$490$		0			0
$491$	−15.7980			−0.712952			−0.356476	−	0.934304i	$-0.616022\pi$
$491$	−15.7980			−0.712952			−0.356476	+	0.934304i	$0.616022\pi$
$492$		0			0
$493$	17.7980			0.801580
$494$		0			0
$495$	6.00000			0.269680
$496$		0			0
$497$		−	39.5959i		−	1.77612i
$498$		0			0
$499$			31.3939i			1.40538i	0.711495	+	0.702691i	$0.248019\pi$
$499$			31.3939i			1.40538i	−0.711495	+	0.702691i	$0.751981\pi$
$500$		0			0
$501$	8.14643	+	8.14643i	0.363956	+	0.363956i
$502$		0			0
$503$	−36.9444			−1.64727			−0.823634	−	0.567121i	$-0.808057\pi$
$503$	−36.9444			−1.64727			−0.823634	+	0.567121i	$0.808057\pi$
$504$		0			0
$505$	3.10102			0.137994
$506$		0			0
$507$	−42.3712	−	42.3712i	−1.88177	−	1.88177i
$508$		0			0
$509$		−	7.10102i		−	0.314747i	−0.987539	−	0.157374i	$-0.949697\pi$
$509$		−	7.10102i		−	0.314747i	0.987539	−	0.157374i	$-0.0503027\pi$
$510$		0			0
$511$			4.89898i			0.216718i
$512$		0			0
$513$	7.34847	+	7.34847i	0.324443	+	0.324443i
$514$		0			0
$515$	−13.3485			−0.588204
$516$		0			0
$517$	7.10102			0.312303
$518$		0			0
$519$	14.4495	−	14.4495i	0.634262	−	0.634262i
$520$		0			0
$521$		−	25.7980i		−	1.13023i	−0.825012	−	0.565115i	$-0.808832\pi$
$521$		−	25.7980i		−	1.13023i	0.825012	−	0.565115i	$-0.191168\pi$
$522$		0			0
$523$		−	15.7526i		−	0.688811i	−0.938821	−	0.344405i	$-0.888081\pi$
$523$		−	15.7526i		−	0.688811i	0.938821	−	0.344405i	$-0.111919\pi$
$524$		0			0
$525$	−5.44949	+	5.44949i	−0.237835	+	0.237835i
$526$		0			0
$527$	16.0000			0.696971
$528$		0			0
$529$	−22.7980			−0.991216
$530$		0			0
$531$			32.6969i			1.41893i
$532$		0			0
$533$			81.3939i			3.52556i
$534$		0			0
$535$			10.4495i			0.451771i
$536$		0			0
$537$	16.0454	+	16.0454i	0.692410	+	0.692410i
$538$		0			0
$539$	−25.5959			−1.10249
$540$		0			0
$541$	−26.0000			−1.11783			−0.558914	−	0.829226i	$-0.688782\pi$
$541$	−26.0000			−1.11783			−0.558914	+	0.829226i	$0.688782\pi$
$542$		0			0
$543$	−26.4495	−	26.4495i	−1.13506	−	1.13506i
$544$		0			0
$545$			12.8990i			0.552532i
$546$		0			0
$547$		−	26.0454i		−	1.11362i	−0.830639	−	0.556811i	$-0.812025\pi$
$547$		−	26.0454i		−	1.11362i	0.830639	−	0.556811i	$-0.187975\pi$
$548$		0			0
$549$		−	2.69694i		−	0.115103i
$550$		0			0
$551$	17.7980			0.758219
$552$		0			0
$553$	−21.7980			−0.926944
$554$		0			0
$555$	7.34847	−	7.34847i	0.311925	−	0.311925i
$556$		0			0
$557$			6.00000i			0.254228i	0.991888	+	0.127114i	$0.0405714\pi$
$557$			6.00000i			0.254228i	−0.991888	+	0.127114i	$0.959429\pi$
$558$		0			0
$559$			10.6969i			0.452432i
$560$		0			0
$561$	4.89898	−	4.89898i	0.206835	−	0.206835i
$562$		0			0
$563$	−13.5505			−0.571086			−0.285543	−	0.958366i	$-0.592174\pi$
$563$	−13.5505			−0.571086			−0.285543	+	0.958366i	$0.592174\pi$
$564$		0			0
$565$	−1.10102			−0.0463203
$566$		0			0
$567$		−	40.0454i		−	1.68175i
$568$		0			0
$569$			29.5959i			1.24073i	0.784315	+	0.620363i	$0.213015\pi$
$569$			29.5959i			1.24073i	−0.784315	+	0.620363i	$0.786985\pi$
$570$		0			0
$571$		−	0.696938i		−	0.0291660i	−0.999894	−	0.0145830i	$-0.995358\pi$
$571$		−	0.696938i		−	0.0291660i	0.999894	−	0.0145830i	$-0.00464207\pi$
$572$		0			0
$573$	−15.7980	−	15.7980i	−0.659969	−	0.659969i
$574$		0			0
$575$	0.449490			0.0187450
$576$		0			0
$577$	7.79796			0.324633			0.162317	−	0.986739i	$-0.448103\pi$
$577$	7.79796			0.324633			0.162317	+	0.986739i	$0.448103\pi$
$578$		0			0
$579$	5.75255	+	5.75255i	0.239068	+	0.239068i
$580$		0			0
$581$			54.4949i			2.26083i
$582$		0			0
$583$		−	5.79796i		−	0.240127i
$584$		0			0
$585$	−20.6969			−0.855713
$586$		0			0
$587$	20.2474			0.835702			0.417851	−	0.908516i	$-0.362783\pi$
$587$	20.2474			0.835702			0.417851	+	0.908516i	$0.362783\pi$
$588$		0			0
$589$	16.0000			0.659269
$590$		0			0
$591$	−10.6515	+	10.6515i	−0.438145	+	0.438145i
$592$		0			0
$593$		−	40.6969i		−	1.67122i	−0.549321	−	0.835611i	$-0.685114\pi$
$593$		−	40.6969i		−	1.67122i	0.549321	−	0.835611i	$-0.314886\pi$
$594$		0			0
$595$			8.89898i			0.364823i
$596$		0			0
$597$	−3.79796	+	3.79796i	−0.155440	+	0.155440i
$598$		0			0
$599$	17.3939			0.710695			0.355347	−	0.934734i	$-0.384363\pi$
$599$	17.3939			0.710695			0.355347	+	0.934734i	$0.384363\pi$
$600$		0			0
$601$	−5.79796			−0.236504			−0.118252	−	0.992984i	$-0.537729\pi$
$601$	−5.79796			−0.236504			−0.118252	+	0.992984i	$0.537729\pi$
$602$		0			0
$603$	10.0454			0.409081
$604$		0			0
$605$			7.00000i			0.284590i
$606$		0			0
$607$		−	22.6515i		−	0.919397i	−0.888075	−	0.459699i	$-0.847957\pi$
$607$		−	22.6515i		−	0.919397i	0.888075	−	0.459699i	$-0.152043\pi$
$608$		0			0
$609$	−48.4949	−	48.4949i	−1.96511	−	1.96511i
$610$		0			0
$611$	−24.4949			−0.990957
$612$		0			0
$613$	−42.4949			−1.71635			−0.858176	−	0.513355i	$-0.828402\pi$
$613$	−42.4949			−1.71635			−0.858176	+	0.513355i	$0.828402\pi$
$614$		0			0
$615$	14.4495	+	14.4495i	0.582660	+	0.582660i
$616$		0			0
$617$		−	46.0000i		−	1.85189i	−0.377658	−	0.925945i	$-0.623271\pi$
$617$		−	46.0000i		−	1.85189i	0.377658	−	0.925945i	$-0.376729\pi$
$618$		0			0
$619$			7.79796i			0.313426i	0.987644	+	0.156713i	$0.0500898\pi$
$619$			7.79796i			0.313426i	−0.987644	+	0.156713i	$0.949910\pi$
$620$		0			0
$621$	−1.65153	+	1.65153i	−0.0662736	+	0.0662736i
$622$		0			0
$623$	43.5959			1.74663
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	4.89898	−	4.89898i	0.195646	−	0.195646i
$628$		0			0
$629$		−	12.0000i		−	0.478471i
$630$		0			0
$631$			16.0000i			0.636950i	0.947931	+	0.318475i	$0.103171\pi$
$631$			16.0000i			0.636950i	−0.947931	+	0.318475i	$0.896829\pi$
$632$		0			0
$633$	−11.1464	+	11.1464i	−0.443031	+	0.443031i
$634$		0			0
$635$	9.34847			0.370983
$636$		0			0
$637$	88.2929			3.49829
$638$		0			0
$639$		−	26.6969i		−	1.05611i
$640$		0			0
$641$		−	6.00000i		−	0.236986i	−0.992955	−	0.118493i	$-0.962194\pi$
$641$		−	6.00000i		−	0.236986i	0.992955	−	0.118493i	$-0.0378063\pi$
$642$		0			0
$643$			8.65153i			0.341183i	0.985342	+	0.170592i	$0.0545679\pi$
$643$			8.65153i			0.341183i	−0.985342	+	0.170592i	$0.945432\pi$
$644$		0			0
$645$	1.89898	+	1.89898i	0.0747722	+	0.0747722i
$646$		0			0
$647$	−5.75255			−0.226156			−0.113078	−	0.993586i	$-0.536071\pi$
$647$	−5.75255			−0.226156			−0.113078	+	0.993586i	$0.536071\pi$
$648$		0			0
$649$	21.7980			0.855645
$650$		0			0
$651$	−43.5959	−	43.5959i	−1.70866	−	1.70866i
$652$		0			0
$653$		−	10.8990i		−	0.426510i	−0.976997	−	0.213255i	$-0.931593\pi$
$653$		−	10.8990i		−	0.426510i	0.976997	−	0.213255i	$-0.0684065\pi$
$654$		0			0
$655$			19.7980i			0.773570i
$656$		0			0
$657$			3.30306i			0.128865i
$658$		0			0
$659$	14.4949			0.564641			0.282321	−	0.959320i	$-0.408896\pi$
$659$	14.4949			0.564641			0.282321	+	0.959320i	$0.408896\pi$
$660$		0			0
$661$	−16.8990			−0.657294			−0.328647	−	0.944453i	$-0.606593\pi$
$661$	−16.8990			−0.657294			−0.328647	+	0.944453i	$0.606593\pi$
$662$		0			0
$663$	−16.8990	+	16.8990i	−0.656302	+	0.656302i
$664$		0			0
$665$			8.89898i			0.345088i
$666$		0			0
$667$			4.00000i			0.154881i
$668$		0			0
$669$	9.24745	−	9.24745i	0.357527	−	0.357527i
$670$		0			0
$671$	−1.79796			−0.0694094
$672$		0			0
$673$	−32.6969			−1.26037			−0.630187	−	0.776443i	$-0.717022\pi$
$673$	−32.6969			−1.26037			−0.630187	+	0.776443i	$0.717022\pi$
$674$		0			0
$675$	−3.67423	+	3.67423i	−0.141421	+	0.141421i
$676$		0			0
$677$		−	16.6969i		−	0.641715i	−0.947127	−	0.320858i	$-0.896029\pi$
$677$		−	16.6969i		−	0.641715i	0.947127	−	0.320858i	$-0.103971\pi$
$678$		0			0
$679$			40.4949i			1.55405i
$680$		0			0
$681$	16.1010	+	16.1010i	0.616992	+	0.616992i
$682$		0			0
$683$	19.3485			0.740349			0.370174	−	0.928962i	$-0.379298\pi$
$683$	19.3485			0.740349			0.370174	+	0.928962i	$0.379298\pi$
$684$		0			0
$685$	−10.8990			−0.416429
$686$		0			0
$687$	0.247449	+	0.247449i	0.00944076	+	0.00944076i
$688$		0			0
$689$			20.0000i			0.761939i
$690$		0			0
$691$		−	8.69694i		−	0.330847i	−0.986223	−	0.165424i	$-0.947101\pi$
$691$		−	8.69694i		−	0.330847i	0.986223	−	0.165424i	$-0.0528991\pi$
$692$		0			0
$693$	−26.6969			−1.01413
$694$		0			0
$695$	2.00000			0.0758643
$696$		0			0
$697$	23.5959			0.893759
$698$		0			0
$699$	9.55051	−	9.55051i	0.361233	−	0.361233i
$700$		0			0
$701$		−	15.3939i		−	0.581419i	−0.956811	−	0.290709i	$-0.906109\pi$
$701$		−	15.3939i		−	0.581419i	0.956811	−	0.290709i	$-0.0938913\pi$
$702$		0			0
$703$		−	12.0000i		−	0.452589i
$704$		0			0
$705$	−4.34847	+	4.34847i	−0.163773	+	0.163773i
$706$		0			0
$707$	−13.7980			−0.518926
$708$		0			0
$709$	24.2020			0.908927			0.454463	−	0.890765i	$-0.349831\pi$
$709$	24.2020			0.908927			0.454463	+	0.890765i	$0.349831\pi$
$710$		0			0
$711$	−14.6969			−0.551178
$712$		0			0
$713$			3.59592i			0.134668i
$714$		0			0
$715$			13.7980i			0.516014i
$716$		0			0
$717$	31.5959	+	31.5959i	1.17997	+	1.17997i
$718$		0			0
$719$	−25.3939			−0.947032			−0.473516	−	0.880785i	$-0.657015\pi$
$719$	−25.3939			−0.947032			−0.473516	+	0.880785i	$0.657015\pi$
$720$		0			0
$721$	59.3939			2.21194
$722$		0			0
$723$	−19.1010	−	19.1010i	−0.710375	−	0.710375i
$724$		0			0
$725$			8.89898i			0.330500i
$726$		0			0
$727$			31.1464i			1.15516i	0.816335	+	0.577579i	$0.196002\pi$
$727$			31.1464i			1.15516i	−0.816335	+	0.577579i	$0.803998\pi$
$728$		0			0
$729$		−	27.0000i		−	1.00000i
$730$		0			0
$731$	3.10102			0.114695
$732$		0			0
$733$	−14.0000			−0.517102			−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000			−0.517102			−0.258551	+	0.965998i	$0.583245\pi$
$734$		0			0
$735$	15.6742	−	15.6742i	0.578153	−	0.578153i
$736$		0			0
$737$		−	6.69694i		−	0.246685i
$738$		0			0
$739$		−	15.7980i		−	0.581137i	−0.956854	−	0.290569i	$-0.906156\pi$
$739$		−	15.7980i		−	0.581137i	0.956854	−	0.290569i	$-0.0938445\pi$
$740$		0			0
$741$	−16.8990	+	16.8990i	−0.620800	+	0.620800i
$742$		0			0
$743$	−35.1464			−1.28940			−0.644699	−	0.764437i	$-0.723017\pi$
$743$	−35.1464			−1.28940			−0.644699	+	0.764437i	$0.723017\pi$
$744$		0			0
$745$	−15.7980			−0.578792
$746$		0			0
$747$			36.7423i			1.34433i
$748$		0			0
$749$		−	46.4949i		−	1.69889i
$750$		0			0
$751$			49.3939i			1.80241i	0.433395	+	0.901204i	$0.357316\pi$
$751$			49.3939i			1.80241i	−0.433395	+	0.901204i	$0.642684\pi$
$752$		0			0
$753$	−10.0454	−	10.0454i	−0.366075	−	0.366075i
$754$		0			0
$755$	15.5959			0.567594
$756$		0			0
$757$	−39.3939			−1.43179			−0.715897	−	0.698205i	$-0.753982\pi$
$757$	−39.3939			−1.43179			−0.715897	+	0.698205i	$0.753982\pi$
$758$		0			0
$759$	1.10102	+	1.10102i	0.0399645	+	0.0399645i
$760$		0			0
$761$			14.2020i			0.514824i	0.966302	+	0.257412i	$0.0828697\pi$
$761$			14.2020i			0.514824i	−0.966302	+	0.257412i	$0.917130\pi$
$762$		0			0
$763$		−	57.3939i		−	2.07780i
$764$		0			0
$765$			6.00000i			0.216930i
$766$		0			0
$767$	−75.1918			−2.71502
$768$		0			0
$769$	−12.2020			−0.440017			−0.220008	−	0.975498i	$-0.570608\pi$
$769$	−12.2020			−0.440017			−0.220008	+	0.975498i	$0.570608\pi$
$770$		0			0
$771$	1.34847	−	1.34847i	0.0485639	−	0.0485639i
$772$		0			0
$773$			4.69694i			0.168937i	0.996426	+	0.0844686i	$0.0269193\pi$
$773$			4.69694i			0.168937i	−0.996426	+	0.0844686i	$0.973081\pi$
$774$		0			0
$775$			8.00000i			0.287368i
$776$		0			0
$777$	−32.6969	+	32.6969i	−1.17300	+	1.17300i
$778$		0			0
$779$	23.5959			0.845411
$780$		0			0
$781$	−17.7980			−0.636861
$782$		0			0
$783$	−32.6969	−	32.6969i	−1.16849	−	1.16849i
$784$		0			0
$785$			10.0000i			0.356915i
$786$		0			0
$787$			21.1464i			0.753789i	0.926256	+	0.376894i	$0.123008\pi$
$787$			21.1464i			0.753789i	−0.926256	+	0.376894i	$0.876992\pi$
$788$		0			0
$789$	−9.24745	−	9.24745i	−0.329218	−	0.329218i
$790$		0			0
$791$	4.89898			0.174188
$792$		0			0
$793$	6.20204			0.220241
$794$		0			0
$795$	3.55051	+	3.55051i	0.125924	+	0.125924i
$796$		0			0
$797$		−	10.8990i		−	0.386062i	−0.981193	−	0.193031i	$-0.938168\pi$
$797$		−	10.8990i		−	0.386062i	0.981193	−	0.193031i	$-0.0618317\pi$
$798$		0			0
$799$			7.10102i			0.251216i
$800$		0			0
$801$	29.3939			1.03858
$802$		0			0
$803$	2.20204			0.0777083
$804$		0			0
$805$	−2.00000			−0.0704907
$806$		0			0
$807$	21.5505	−	21.5505i	0.758614	−	0.758614i
$808$		0			0
$809$			47.5959i			1.67338i	0.547674	+	0.836692i	$0.315513\pi$
$809$			47.5959i			1.67338i	−0.547674	+	0.836692i	$0.684487\pi$
$810$		0			0
$811$			50.4949i			1.77312i	0.462618	+	0.886558i	$0.346910\pi$
$811$			50.4949i			1.77312i	−0.462618	+	0.886558i	$0.653090\pi$
$812$		0			0
$813$	−14.6969	+	14.6969i	−0.515444	+	0.515444i
$814$		0			0
$815$	−10.4495			−0.366030
$816$		0			0
$817$	3.10102			0.108491
$818$		0			0
$819$	92.0908			3.21791
$820$		0			0
$821$			1.59592i			0.0556979i	0.999612	+	0.0278490i	$0.00886575\pi$
$821$			1.59592i			0.0556979i	−0.999612	+	0.0278490i	$0.991134\pi$
$822$		0			0
$823$			12.4495i			0.433962i	0.976176	+	0.216981i	$0.0696210\pi$
$823$			12.4495i			0.433962i	−0.976176	+	0.216981i	$0.930379\pi$
$824$		0			0
$825$	2.44949	+	2.44949i	0.0852803	+	0.0852803i
$826$		0			0
$827$	33.1464			1.15261			0.576307	−	0.817233i	$-0.304493\pi$
$827$	33.1464			1.15261			0.576307	+	0.817233i	$0.304493\pi$
$828$		0			0
$829$	24.4949			0.850743			0.425371	−	0.905019i	$-0.360143\pi$
$829$	24.4949			0.850743			0.425371	+	0.905019i	$0.360143\pi$
$830$		0			0
$831$	2.44949	+	2.44949i	0.0849719	+	0.0849719i
$832$		0			0
$833$		−	25.5959i		−	0.886846i
$834$		0			0
$835$			6.65153i			0.230186i
$836$		0			0
$837$	−29.3939	−	29.3939i	−1.01600	−	1.01600i
$838$		0			0
$839$	−1.79796			−0.0620724			−0.0310362	−	0.999518i	$-0.509881\pi$
$839$	−1.79796			−0.0620724			−0.0310362	+	0.999518i	$0.509881\pi$
$840$		0			0
$841$	−50.1918			−1.73075
$842$		0			0
$843$	−26.4495	+	26.4495i	−0.910969	+	0.910969i
$844$		0			0
$845$		−	34.5959i		−	1.19014i
$846$		0			0
$847$		−	31.1464i		−	1.07020i
$848$		0			0
$849$	25.8990	−	25.8990i	0.888851	−	0.888851i
$850$		0			0
$851$	2.69694			0.0924499
$852$		0			0
$853$	41.1010			1.40727			0.703636	−	0.710561i	$-0.251559\pi$
$853$	41.1010			1.40727			0.703636	+	0.710561i	$0.251559\pi$
$854$		0			0
$855$			6.00000i			0.205196i
$856$		0			0
$857$		−	20.6969i		−	0.706994i	−0.935436	−	0.353497i	$-0.884992\pi$
$857$		−	20.6969i		−	0.706994i	0.935436	−	0.353497i	$-0.115008\pi$
$858$		0			0
$859$		−	1.59592i		−	0.0544520i	−0.999629	−	0.0272260i	$-0.991333\pi$
$859$		−	1.59592i		−	0.0544520i	0.999629	−	0.0272260i	$-0.00866738\pi$
$860$		0			0
$861$	−64.2929	−	64.2929i	−2.19109	−	2.19109i
$862$		0			0
$863$	−8.94439			−0.304470			−0.152235	−	0.988344i	$-0.548647\pi$
$863$	−8.94439			−0.304470			−0.152235	+	0.988344i	$0.548647\pi$
$864$		0			0
$865$	11.7980			0.401143
$866$		0			0
$867$	−15.9217	−	15.9217i	−0.540729	−	0.540729i
$868$		0			0
$869$			9.79796i			0.332373i
$870$		0			0
$871$			23.1010i			0.782748i
$872$		0			0
$873$			27.3031i			0.924069i
$874$		0			0
$875$	−4.44949			−0.150420
$876$		0			0
$877$	−2.00000			−0.0675352			−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000			−0.0675352			−0.0337676	+	0.999430i	$0.510751\pi$
$878$		0			0
$879$	−10.0454	+	10.0454i	−0.338823	+	0.338823i
$880$		0			0
$881$		−	27.7980i		−	0.936537i	−0.883586	−	0.468269i	$-0.844878\pi$
$881$		−	27.7980i		−	0.936537i	0.883586	−	0.468269i	$-0.155122\pi$
$882$		0			0
$883$			13.5505i			0.456011i	0.973660	+	0.228005i	$0.0732204\pi$
$883$			13.5505i			0.456011i	−0.973660	+	0.228005i	$0.926780\pi$
$884$		0			0
$885$	−13.3485	+	13.3485i	−0.448704	+	0.448704i
$886$		0			0
$887$	12.8536			0.431581			0.215790	−	0.976440i	$-0.430767\pi$
$887$	12.8536			0.431581			0.215790	+	0.976440i	$0.430767\pi$
$888$		0			0
$889$	−41.5959			−1.39508
$890$		0			0
$891$	−18.0000			−0.603023
$892$		0			0
$893$			7.10102i			0.237627i
$894$		0			0
$895$			13.1010i			0.437919i
$896$		0			0
$897$	−3.79796	−	3.79796i	−0.126810	−	0.126810i
$898$		0			0
$899$	−71.1918			−2.37438
$900$		0			0
$901$	5.79796			0.193158
$902$		0			0
$903$	−8.44949	−	8.44949i	−0.281181	−	0.281181i
$904$		0			0
$905$		−	21.5959i		−	0.717873i
$906$		0			0
$907$		−	50.9444i		−	1.69158i	−0.533515	−	0.845790i	$-0.679129\pi$
$907$		−	50.9444i		−	1.69158i	0.533515	−	0.845790i	$-0.320871\pi$
$908$		0			0
$909$	−9.30306			−0.308563
$910$		0			0
$911$	−14.6969			−0.486931			−0.243466	−	0.969910i	$-0.578284\pi$
$911$	−14.6969			−0.486931			−0.243466	+	0.969910i	$0.578284\pi$
$912$		0			0
$913$	24.4949			0.810663
$914$		0			0
$915$	1.10102	−	1.10102i	0.0363986	−	0.0363986i
$916$		0			0
$917$		−	88.0908i		−	2.90902i
$918$		0			0
$919$			4.49490i			0.148273i	0.997248	+	0.0741365i	$0.0236200\pi$
$919$			4.49490i			0.148273i	−0.997248	+	0.0741365i	$0.976380\pi$
$920$		0			0
$921$	−15.4949	+	15.4949i	−0.510574	+	0.510574i
$922$		0			0
$923$	61.3939			2.02080
$924$		0			0
$925$	6.00000			0.197279
$926$		0			0
$927$	40.0454			1.31526
$928$		0			0
$929$			34.0000i			1.11550i	0.830008	+	0.557752i	$0.188336\pi$
$929$			34.0000i			1.11550i	−0.830008	+	0.557752i	$0.811664\pi$
$930$		0			0
$931$		−	25.5959i		−	0.838872i
$932$		0			0
$933$	−30.0000	−	30.0000i	−0.982156	−	0.982156i
$934$		0			0
$935$	4.00000			0.130814
$936$		0			0
$937$	−46.4949			−1.51892			−0.759461	−	0.650553i	$-0.774537\pi$
$937$	−46.4949			−1.51892			−0.759461	+	0.650553i	$0.774537\pi$
$938$		0			0
$939$	24.2474	+	24.2474i	0.791286	+	0.791286i
$940$		0			0
$941$		−	24.4949i		−	0.798511i	−0.916840	−	0.399255i	$-0.869269\pi$
$941$		−	24.4949i		−	0.798511i	0.916840	−	0.399255i	$-0.130731\pi$
$942$		0			0
$943$			5.30306i			0.172691i
$944$		0			0
$945$	16.3485	−	16.3485i	0.531816	−	0.531816i
$946$		0			0
$947$	−1.55051			−0.0503848			−0.0251924	−	0.999683i	$-0.508020\pi$
$947$	−1.55051			−0.0503848			−0.0251924	+	0.999683i	$0.508020\pi$
$948$		0			0
$949$	−7.59592			−0.246574
$950$		0			0
$951$	10.6515	−	10.6515i	0.345400	−	0.345400i
$952$		0			0
$953$		−	1.10102i		−	0.0356656i	−0.999841	−	0.0178328i	$-0.994323\pi$
$953$		−	1.10102i		−	0.0356656i	0.999841	−	0.0178328i	$-0.00567665\pi$
$954$		0			0
$955$		−	12.8990i		−	0.417401i
$956$		0			0
$957$	−21.7980	+	21.7980i	−0.704628	+	0.704628i
$958$		0			0
$959$	48.4949			1.56598
$960$		0			0
$961$	−33.0000			−1.06452
$962$		0			0
$963$		−	31.3485i		−	1.01019i
$964$		0			0
$965$			4.69694i			0.151200i
$966$		0			0
$967$			30.2474i			0.972692i	0.873766	+	0.486346i	$0.161671\pi$
$967$			30.2474i			0.972692i	−0.873766	+	0.486346i	$0.838329\pi$
$968$		0			0
$969$	4.89898	+	4.89898i	0.157378	+	0.157378i
$970$		0			0
$971$	39.7980			1.27718			0.638589	−	0.769548i	$-0.279519\pi$
$971$	39.7980			1.27718			0.638589	+	0.769548i	$0.279519\pi$
$972$		0			0
$973$	−8.89898			−0.285288
$974$		0			0
$975$	−8.44949	−	8.44949i	−0.270600	−	0.270600i
$976$		0			0
$977$		−	33.1918i		−	1.06190i	−0.847403	−	0.530950i	$-0.821835\pi$
$977$		−	33.1918i		−	1.06190i	0.847403	−	0.530950i	$-0.178165\pi$
$978$		0			0
$979$		−	19.5959i		−	0.626288i
$980$		0			0
$981$		−	38.6969i		−	1.23550i
$982$		0			0
$983$	52.0454			1.65999			0.829995	−	0.557770i	$-0.188343\pi$
$983$	52.0454			1.65999			0.829995	+	0.557770i	$0.188343\pi$
$984$		0			0
$985$	−8.69694			−0.277108
$986$		0			0
$987$	19.3485	−	19.3485i	0.615869	−	0.615869i
$988$		0			0
$989$			0.696938i			0.0221614i
$990$		0			0
$991$			31.1918i			0.990841i	0.868653	+	0.495421i	$0.164986\pi$
$991$			31.1918i			0.990841i	−0.868653	+	0.495421i	$0.835014\pi$
$992$		0			0
$993$	28.0454	−	28.0454i	0.889994	−	0.889994i
$994$		0			0
$995$	−3.10102			−0.0983090
$996$		0			0
$997$	−46.0908			−1.45971			−0.729855	−	0.683602i	$-0.760413\pi$
$997$	−46.0908			−1.45971			−0.729855	+	0.683602i	$0.760413\pi$
$998$		0			0
$999$	−22.0454	+	22.0454i	−0.697486	+	0.697486i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1920.2.h.i.1151.1	yes	4
3.2	odd	2		1920.2.h.g.1151.3	✓	4
4.3	odd	2		1920.2.h.g.1151.4	yes	4
8.3	odd	2		1920.2.h.j.1151.1	yes	4
8.5	even	2		1920.2.h.h.1151.4	yes	4
12.11	even	2	inner	1920.2.h.i.1151.2	yes	4
24.5	odd	2		1920.2.h.j.1151.2	yes	4
24.11	even	2		1920.2.h.h.1151.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.h.g.1151.3	✓	4	3.2	odd	2
1920.2.h.g.1151.4	yes	4	4.3	odd	2
1920.2.h.h.1151.3	yes	4	24.11	even	2
1920.2.h.h.1151.4	yes	4	8.5	even	2
1920.2.h.i.1151.1	yes	4	1.1	even	1	trivial
1920.2.h.i.1151.2	yes	4	12.11	even	2	inner
1920.2.h.j.1151.1	yes	4	8.3	odd	2
1920.2.h.j.1151.2	yes	4	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1920 = 2^{7} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1920.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.22474 - 1.22474i) q^{3} -1.00000i q^{5} +4.44949i q^{7} +3.00000i q^{9} +2.00000 q^{11} -6.89898 q^{13} +(-1.22474 + 1.22474i) q^{15} +2.00000i q^{17} +2.00000i q^{19} +(5.44949 - 5.44949i) q^{21} -0.449490 q^{23} -1.00000 q^{25} +(3.67423 - 3.67423i) q^{27} -8.89898i q^{29} -8.00000i q^{31} +(-2.44949 - 2.44949i) q^{33} +4.44949 q^{35} -6.00000 q^{37} +(8.44949 + 8.44949i) q^{39} -11.7980i q^{41} -1.55051i q^{43} +3.00000 q^{45} +3.55051 q^{47} -12.7980 q^{49} +(2.44949 - 2.44949i) q^{51} -2.89898i q^{53} -2.00000i q^{55} +(2.44949 - 2.44949i) q^{57} +10.8990 q^{59} -0.898979 q^{61} -13.3485 q^{63} +6.89898i q^{65} -3.34847i q^{67} +(0.550510 + 0.550510i) q^{69} -8.89898 q^{71} +1.10102 q^{73} +(1.22474 + 1.22474i) q^{75} +8.89898i q^{77} +4.89898i q^{79} -9.00000 q^{81} +12.2474 q^{83} +2.00000 q^{85} +(-10.8990 + 10.8990i) q^{87} -9.79796i q^{89} -30.6969i q^{91} +(-9.79796 + 9.79796i) q^{93} +2.00000 q^{95} +9.10102 q^{97} +6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{11} - 8 q^{13} + 12 q^{21} + 8 q^{23} - 4 q^{25} + 8 q^{35} - 24 q^{37} + 24 q^{39} + 12 q^{45} + 24 q^{47} - 12 q^{49} + 24 q^{59} + 16 q^{61} - 24 q^{63} + 12 q^{69} - 16 q^{71} + 24 q^{73}+ \cdots + 56 q^{97}+O(q^{100}) \) 4 * q + 8 * q^11 - 8 * q^13 + 12 * q^21 + 8 * q^23 - 4 * q^25 + 8 * q^35 - 24 * q^37 + 24 * q^39 + 12 * q^45 + 24 * q^47 - 12 * q^49 + 24 * q^59 + 16 * q^61 - 24 * q^63 + 12 * q^69 - 16 * q^71 + 24 * q^73 - 36 * q^81 + 8 * q^85 - 24 * q^87 + 8 * q^95 + 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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