LMFDB - Embedded newform 1920.2.h.j.1151.4

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			1151.4
Root			$1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1920.1151
Dual form			1920.2.h.j.1151.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 + 1.22474i) q^{3} +1.00000i q^{5} +0.449490i q^{7} +3.00000i q^{9} +2.00000 q^{11} -2.89898 q^{13} +(-1.22474 + 1.22474i) q^{15} +2.00000i q^{17} +2.00000i q^{19} +(-0.550510 + 0.550510i) q^{21} -4.44949 q^{23} -1.00000 q^{25} +(-3.67423 + 3.67423i) q^{27} -0.898979i q^{29} +8.00000i q^{31} +(2.44949 + 2.44949i) q^{33} -0.449490 q^{35} +6.00000 q^{37} +(-3.55051 - 3.55051i) q^{39} +7.79796i q^{41} -6.44949i q^{43} -3.00000 q^{45} -8.44949 q^{47} +6.79796 q^{49} +(-2.44949 + 2.44949i) q^{51} -6.89898i q^{53} +2.00000i q^{55} +(-2.44949 + 2.44949i) q^{57} +1.10102 q^{59} -8.89898 q^{61} -1.34847 q^{63} -2.89898i q^{65} +11.3485i q^{67} +(-5.44949 - 5.44949i) q^{69} -0.898979 q^{71} +10.8990 q^{73} +(-1.22474 - 1.22474i) q^{75} +0.898979i q^{77} +4.89898i q^{79} -9.00000 q^{81} -12.2474 q^{83} -2.00000 q^{85} +(1.10102 - 1.10102i) q^{87} +9.79796i q^{89} -1.30306i q^{91} +(-9.79796 + 9.79796i) q^{93} -2.00000 q^{95} +18.8990 q^{97} +6.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$			0.449490i			0.169891i	0.996386	+	0.0849456i	$0.0270716\pi$
$7$			0.449490i			0.169891i	−0.996386	+	0.0849456i	$0.972928\pi$
$8$		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
$13$	−2.89898			−0.804032			−0.402016	−	0.915633i	$-0.631690\pi$
$13$	−2.89898			−0.804032			−0.402016	+	0.915633i	$0.631690\pi$
$14$		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
$21$	−0.550510	+	0.550510i	−0.120131	+	0.120131i
$22$		0			0
$23$	−4.44949			−0.927783			−0.463891	−	0.885892i	$-0.653547\pi$
$23$	−4.44949			−0.927783			−0.463891	+	0.885892i	$0.653547\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	−3.67423	+	3.67423i	−0.707107	+	0.707107i
$28$		0			0
$29$		−	0.898979i		−	0.166936i	−0.996510	−	0.0834681i	$-0.973400\pi$
$29$		−	0.898979i		−	0.166936i	0.996510	−	0.0834681i	$-0.0265997\pi$
$30$		0			0
$31$			8.00000i			1.43684i	0.695608	+	0.718421i	$0.255135\pi$
$31$			8.00000i			1.43684i	−0.695608	+	0.718421i	$0.744865\pi$
$32$		0			0
$33$	2.44949	+	2.44949i	0.426401	+	0.426401i
$34$		0			0
$35$	−0.449490			−0.0759776
$36$		0			0
$37$	6.00000			0.986394			0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000			0.986394			0.493197	+	0.869918i	$0.335828\pi$
$38$		0			0
$39$	−3.55051	−	3.55051i	−0.568537	−	0.568537i
$40$		0			0
$41$			7.79796i			1.21784i	0.793233	+	0.608918i	$0.208396\pi$
$41$			7.79796i			1.21784i	−0.793233	+	0.608918i	$0.791604\pi$
$42$		0			0
$43$		−	6.44949i		−	0.983538i	−0.870726	−	0.491769i	$-0.836350\pi$
$43$		−	6.44949i		−	0.983538i	0.870726	−	0.491769i	$-0.163650\pi$
$44$		0			0
$45$	−3.00000			−0.447214
$46$		0			0
$47$	−8.44949			−1.23248			−0.616242	−	0.787557i	$-0.711346\pi$
$47$	−8.44949			−1.23248			−0.616242	+	0.787557i	$0.711346\pi$
$48$		0			0
$49$	6.79796			0.971137
$50$		0			0
$51$	−2.44949	+	2.44949i	−0.342997	+	0.342997i
$52$		0			0
$53$		−	6.89898i		−	0.947648i	−0.880620	−	0.473824i	$-0.842873\pi$
$53$		−	6.89898i		−	0.947648i	0.880620	−	0.473824i	$-0.157127\pi$
$54$		0			0
$55$			2.00000i			0.269680i
$56$		0			0
$57$	−2.44949	+	2.44949i	−0.324443	+	0.324443i
$58$		0			0
$59$	1.10102			0.143341			0.0716703	−	0.997428i	$-0.477167\pi$
$59$	1.10102			0.143341			0.0716703	+	0.997428i	$0.477167\pi$
$60$		0			0
$61$	−8.89898			−1.13940			−0.569699	−	0.821854i	$-0.692940\pi$
$61$	−8.89898			−1.13940			−0.569699	+	0.821854i	$0.692940\pi$
$62$		0			0
$63$	−1.34847			−0.169891
$64$		0			0
$65$		−	2.89898i		−	0.359574i
$66$		0			0
$67$			11.3485i			1.38644i	0.720728	+	0.693218i	$0.243808\pi$
$67$			11.3485i			1.38644i	−0.720728	+	0.693218i	$0.756192\pi$
$68$		0			0
$69$	−5.44949	−	5.44949i	−0.656041	−	0.656041i
$70$		0			0
$71$	−0.898979			−0.106689			−0.0533446	−	0.998576i	$-0.516988\pi$
$71$	−0.898979			−0.106689			−0.0533446	+	0.998576i	$0.516988\pi$
$72$		0			0
$73$	10.8990			1.27563			0.637815	−	0.770190i	$-0.279839\pi$
$73$	10.8990			1.27563			0.637815	+	0.770190i	$0.279839\pi$
$74$		0			0
$75$	−1.22474	−	1.22474i	−0.141421	−	0.141421i
$76$		0			0
$77$			0.898979i			0.102448i
$78$		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	$-0.734636\pi$
$83$	−12.2474			−1.34433			−0.672166	+	0.740400i	$0.734636\pi$
$84$		0			0
$85$	−2.00000			−0.216930
$86$		0			0
$87$	1.10102	−	1.10102i	0.118042	−	0.118042i
$88$		0			0
$89$			9.79796i			1.03858i	0.854598	+	0.519291i	$0.173804\pi$
$89$			9.79796i			1.03858i	−0.854598	+	0.519291i	$0.826196\pi$
$90$		0			0
$91$		−	1.30306i		−	0.136598i
$92$		0			0
$93$	−9.79796	+	9.79796i	−1.01600	+	1.01600i
$94$		0			0
$95$	−2.00000			−0.205196
$96$		0			0
$97$	18.8990			1.91890			0.959450	−	0.281878i	$-0.0909573\pi$
$97$	18.8990			1.91890			0.959450	+	0.281878i	$0.0909573\pi$
$98$		0			0
$99$			6.00000i			0.603023i
$100$		0			0
$101$		−	12.8990i		−	1.28350i	−0.766915	−	0.641748i	$-0.778209\pi$
$101$		−	12.8990i		−	1.28350i	0.766915	−	0.641748i	$-0.221791\pi$
$102$		0			0
$103$		−	1.34847i		−	0.132869i	−0.997791	−	0.0664343i	$-0.978838\pi$
$103$		−	1.34847i		−	0.132869i	0.997791	−	0.0664343i	$-0.0211623\pi$
$104$		0			0
$105$	−0.550510	−	0.550510i	−0.0537243	−	0.0537243i
$106$		0			0
$107$	−5.55051			−0.536588			−0.268294	−	0.963337i	$-0.586460\pi$
$107$	−5.55051			−0.536588			−0.268294	+	0.963337i	$0.586460\pi$
$108$		0			0
$109$	3.10102			0.297024			0.148512	−	0.988911i	$-0.452552\pi$
$109$	3.10102			0.297024			0.148512	+	0.988911i	$0.452552\pi$
$110$		0			0
$111$	7.34847	+	7.34847i	0.697486	+	0.697486i
$112$		0			0
$113$		−	10.8990i		−	1.02529i	−0.858601	−	0.512645i	$-0.828666\pi$
$113$		−	10.8990i		−	1.02529i	0.858601	−	0.512645i	$-0.171334\pi$
$114$		0			0
$115$		−	4.44949i		−	0.414917i
$116$		0			0
$117$		−	8.69694i		−	0.804032i
$118$		0			0
$119$	−0.898979			−0.0824093
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$	−9.55051	+	9.55051i	−0.861141	+	0.861141i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$			5.34847i			0.474600i	0.971436	+	0.237300i	$0.0762624\pi$
$127$			5.34847i			0.474600i	−0.971436	+	0.237300i	$0.923738\pi$
$128$		0			0
$129$	7.89898	−	7.89898i	0.695466	−	0.695466i
$130$		0			0
$131$	−0.202041			−0.0176524			−0.00882620	−	0.999961i	$-0.502810\pi$
$131$	−0.202041			−0.0176524			−0.00882620	+	0.999961i	$0.502810\pi$
$132$		0			0
$133$	−0.898979			−0.0779514
$134$		0			0
$135$	−3.67423	−	3.67423i	−0.316228	−	0.316228i
$136$		0			0
$137$		−	1.10102i		−	0.0940665i	−0.998893	−	0.0470333i	$-0.985023\pi$
$137$		−	1.10102i		−	0.0940665i	0.998893	−	0.0470333i	$-0.0149767\pi$
$138$		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
$141$	−10.3485	−	10.3485i	−0.871498	−	0.871498i
$142$		0			0
$143$	−5.79796			−0.484850
$144$		0			0
$145$	0.898979			0.0746562
$146$		0			0
$147$	8.32577	+	8.32577i	0.686698	+	0.686698i
$148$		0			0
$149$		−	3.79796i		−	0.311141i	−0.987825	−	0.155570i	$-0.950278\pi$
$149$		−	3.79796i		−	0.311141i	0.987825	−	0.155570i	$-0.0497216\pi$
$150$		0			0
$151$			23.5959i			1.92021i	0.279642	+	0.960104i	$0.409784\pi$
$151$			23.5959i			1.92021i	−0.279642	+	0.960104i	$0.590216\pi$
$152$		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	$-0.369342\pi$
$157$	10.0000			0.798087			0.399043	+	0.916932i	$0.369342\pi$
$158$		0			0
$159$	8.44949	−	8.44949i	0.670088	−	0.670088i
$160$		0			0
$161$		−	2.00000i		−	0.157622i
$162$		0			0
$163$		−	5.55051i		−	0.434750i	−0.976088	−	0.217375i	$-0.930251\pi$
$163$		−	5.55051i		−	0.434750i	0.976088	−	0.217375i	$-0.0697494\pi$
$164$		0			0
$165$	−2.44949	+	2.44949i	−0.190693	+	0.190693i
$166$		0			0
$167$	21.3485			1.65199			0.825997	−	0.563674i	$-0.190613\pi$
$167$	21.3485			1.65199			0.825997	+	0.563674i	$0.190613\pi$
$168$		0			0
$169$	−4.59592			−0.353532
$170$		0			0
$171$	−6.00000			−0.458831
$172$		0			0
$173$			7.79796i			0.592868i	0.955053	+	0.296434i	$0.0957975\pi$
$173$			7.79796i			0.592868i	−0.955053	+	0.296434i	$0.904203\pi$
$174$		0			0
$175$		−	0.449490i		−	0.0339782i
$176$		0			0
$177$	1.34847	+	1.34847i	0.101357	+	0.101357i
$178$		0			0
$179$	−22.8990			−1.71155			−0.855775	−	0.517348i	$-0.826919\pi$
$179$	−22.8990			−1.71155			−0.855775	+	0.517348i	$0.826919\pi$
$180$		0			0
$181$	17.5959			1.30789			0.653947	−	0.756540i	$-0.273112\pi$
$181$	17.5959			1.30789			0.653947	+	0.756540i	$0.273112\pi$
$182$		0			0
$183$	−10.8990	−	10.8990i	−0.805676	−	0.805676i
$184$		0			0
$185$			6.00000i			0.441129i
$186$		0			0
$187$			4.00000i			0.292509i
$188$		0			0
$189$	−1.65153	−	1.65153i	−0.120131	−	0.120131i
$190$		0			0
$191$	−3.10102			−0.224382			−0.112191	−	0.993687i	$-0.535787\pi$
$191$	−3.10102			−0.224382			−0.112191	+	0.993687i	$0.535787\pi$
$192$		0			0
$193$	24.6969			1.77772			0.888862	−	0.458175i	$-0.151497\pi$
$193$	24.6969			1.77772			0.888862	+	0.458175i	$0.151497\pi$
$194$		0			0
$195$	3.55051	−	3.55051i	0.254257	−	0.254257i
$196$		0			0
$197$		−	20.6969i		−	1.47460i	−0.675568	−	0.737298i	$-0.736101\pi$
$197$		−	20.6969i		−	1.47460i	0.675568	−	0.737298i	$-0.263899\pi$
$198$		0			0
$199$			12.8990i			0.914384i	0.889368	+	0.457192i	$0.151145\pi$
$199$			12.8990i			0.914384i	−0.889368	+	0.457192i	$0.848855\pi$
$200$		0			0
$201$	−13.8990	+	13.8990i	−0.980358	+	0.980358i
$202$		0			0
$203$	0.404082			0.0283610
$204$		0			0
$205$	−7.79796			−0.544633
$206$		0			0
$207$		−	13.3485i		−	0.927783i
$208$		0			0
$209$			4.00000i			0.276686i
$210$		0			0
$211$		−	18.8990i		−	1.30106i	−0.759481	−	0.650530i	$-0.774547\pi$
$211$		−	18.8990i		−	1.30106i	0.759481	−	0.650530i	$-0.225453\pi$
$212$		0			0
$213$	−1.10102	−	1.10102i	−0.0754407	−	0.0754407i
$214$		0			0
$215$	6.44949			0.439852
$216$		0			0
$217$	−3.59592			−0.244107
$218$		0			0
$219$	13.3485	+	13.3485i	0.902006	+	0.902006i
$220$		0			0
$221$		−	5.79796i		−	0.390013i
$222$		0			0
$223$		−	12.4495i		−	0.833679i	−0.908980	−	0.416840i	$-0.863138\pi$
$223$		−	12.4495i		−	0.833679i	0.908980	−	0.416840i	$-0.136862\pi$
$224$		0			0
$225$		−	3.00000i		−	0.200000i
$226$		0			0
$227$	21.1464			1.40354			0.701769	−	0.712405i	$-0.252394\pi$
$227$	21.1464			1.40354			0.701769	+	0.712405i	$0.252394\pi$
$228$		0			0
$229$	19.7980			1.30829			0.654143	−	0.756371i	$-0.273029\pi$
$229$	19.7980			1.30829			0.654143	+	0.756371i	$0.273029\pi$
$230$		0			0
$231$	−1.10102	+	1.10102i	−0.0724418	+	0.0724418i
$232$		0			0
$233$		−	11.7980i		−	0.772910i	−0.922308	−	0.386455i	$-0.873699\pi$
$233$		−	11.7980i		−	0.772910i	0.922308	−	0.386455i	$-0.126301\pi$
$234$		0			0
$235$		−	8.44949i		−	0.551184i
$236$		0			0
$237$	−6.00000	+	6.00000i	−0.389742	+	0.389742i
$238$		0			0
$239$	6.20204			0.401177			0.200588	−	0.979676i	$-0.435715\pi$
$239$	6.20204			0.401177			0.200588	+	0.979676i	$0.435715\pi$
$240$		0			0
$241$	−23.5959			−1.51995			−0.759973	−	0.649954i	$-0.774788\pi$
$241$	−23.5959			−1.51995			−0.759973	+	0.649954i	$0.774788\pi$
$242$		0			0
$243$	−11.0227	−	11.0227i	−0.707107	−	0.707107i
$244$		0			0
$245$			6.79796i			0.434306i
$246$		0			0
$247$		−	5.79796i		−	0.368915i
$248$		0			0
$249$	−15.0000	−	15.0000i	−0.950586	−	0.950586i
$250$		0			0
$251$	27.7980			1.75459			0.877296	−	0.479950i	$-0.159345\pi$
$251$	27.7980			1.75459			0.877296	+	0.479950i	$0.159345\pi$
$252$		0			0
$253$	−8.89898			−0.559474
$254$		0			0
$255$	−2.44949	−	2.44949i	−0.153393	−	0.153393i
$256$		0			0
$257$			10.8990i			0.679860i	0.940451	+	0.339930i	$0.110403\pi$
$257$			10.8990i			0.679860i	−0.940451	+	0.339930i	$0.889597\pi$
$258$		0			0
$259$			2.69694i			0.167580i
$260$		0			0
$261$	2.69694			0.166936
$262$		0			0
$263$	−12.4495			−0.767668			−0.383834	−	0.923402i	$-0.625397\pi$
$263$	−12.4495			−0.767668			−0.383834	+	0.923402i	$0.625397\pi$
$264$		0			0
$265$	6.89898			0.423801
$266$		0			0
$267$	−12.0000	+	12.0000i	−0.734388	+	0.734388i
$268$		0			0
$269$			21.5959i			1.31673i	0.752700	+	0.658363i	$0.228751\pi$
$269$			21.5959i			1.31673i	−0.752700	+	0.658363i	$0.771249\pi$
$270$		0			0
$271$			12.0000i			0.728948i	0.931214	+	0.364474i	$0.118751\pi$
$271$			12.0000i			0.728948i	−0.931214	+	0.364474i	$0.881249\pi$
$272$		0			0
$273$	1.59592	−	1.59592i	0.0965893	−	0.0965893i
$274$		0			0
$275$	−2.00000			−0.120605
$276$		0			0
$277$	2.00000			0.120168			0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000			0.120168			0.0600842	+	0.998193i	$0.480863\pi$
$278$		0			0
$279$	−24.0000			−1.43684
$280$		0			0
$281$			17.5959i			1.04968i	0.851200	+	0.524842i	$0.175876\pi$
$281$			17.5959i			1.04968i	−0.851200	+	0.524842i	$0.824124\pi$
$282$		0			0
$283$		−	13.1464i		−	0.781474i	−0.920502	−	0.390737i	$-0.872220\pi$
$283$		−	13.1464i		−	0.781474i	0.920502	−	0.390737i	$-0.127780\pi$
$284$		0			0
$285$	−2.44949	−	2.44949i	−0.145095	−	0.145095i
$286$		0			0
$287$	−3.50510			−0.206900
$288$		0			0
$289$	13.0000			0.764706
$290$		0			0
$291$	23.1464	+	23.1464i	1.35687	+	1.35687i
$292$		0			0
$293$			27.7980i			1.62397i	0.583675	+	0.811987i	$0.301614\pi$
$293$			27.7980i			1.62397i	−0.583675	+	0.811987i	$0.698386\pi$
$294$		0			0
$295$			1.10102i			0.0641039i
$296$		0			0
$297$	−7.34847	+	7.34847i	−0.426401	+	0.426401i
$298$		0			0
$299$	12.8990			0.745967
$300$		0			0
$301$	2.89898			0.167094
$302$		0			0
$303$	15.7980	−	15.7980i	0.907569	−	0.907569i
$304$		0			0
$305$		−	8.89898i		−	0.509554i
$306$		0			0
$307$		−	27.3485i		−	1.56086i	−0.625243	−	0.780430i	$-0.715000\pi$
$307$		−	27.3485i		−	1.56086i	0.625243	−	0.780430i	$-0.285000\pi$
$308$		0			0
$309$	1.65153	−	1.65153i	0.0939523	−	0.0939523i
$310$		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
$313$	−0.202041			−0.0114200			−0.00571002	−	0.999984i	$-0.501818\pi$
$313$	−0.202041			−0.0114200			−0.00571002	+	0.999984i	$0.501818\pi$
$314$		0			0
$315$		−	1.34847i		−	0.0759776i
$316$		0			0
$317$			20.6969i			1.16246i	0.813741	+	0.581228i	$0.197428\pi$
$317$			20.6969i			1.16246i	−0.813741	+	0.581228i	$0.802572\pi$
$318$		0			0
$319$		−	1.79796i		−	0.100666i
$320$		0			0
$321$	−6.79796	−	6.79796i	−0.379425	−	0.379425i
$322$		0			0
$323$	−4.00000			−0.222566
$324$		0			0
$325$	2.89898			0.160806
$326$		0			0
$327$	3.79796	+	3.79796i	0.210028	+	0.210028i
$328$		0			0
$329$		−	3.79796i		−	0.209388i
$330$		0			0
$331$			13.1010i			0.720097i	0.932934	+	0.360049i	$0.117240\pi$
$331$			13.1010i			0.720097i	−0.932934	+	0.360049i	$0.882760\pi$
$332$		0			0
$333$			18.0000i			0.986394i
$334$		0			0
$335$	−11.3485			−0.620033
$336$		0			0
$337$	−4.20204			−0.228900			−0.114450	−	0.993429i	$-0.536511\pi$
$337$	−4.20204			−0.228900			−0.114450	+	0.993429i	$0.536511\pi$
$338$		0			0
$339$	13.3485	−	13.3485i	0.724989	−	0.724989i
$340$		0			0
$341$			16.0000i			0.866449i
$342$		0			0
$343$			6.20204i			0.334879i
$344$		0			0
$345$	5.44949	−	5.44949i	0.293391	−	0.293391i
$346$		0			0
$347$	17.1464			0.920468			0.460234	−	0.887798i	$-0.347765\pi$
$347$	17.1464			0.920468			0.460234	+	0.887798i	$0.347765\pi$
$348$		0			0
$349$	14.0000			0.749403			0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000			0.749403			0.374701	+	0.927146i	$0.377745\pi$
$350$		0			0
$351$	10.6515	−	10.6515i	0.568537	−	0.568537i
$352$		0			0
$353$		−	26.4949i		−	1.41018i	−0.709117	−	0.705091i	$-0.750906\pi$
$353$		−	26.4949i		−	1.41018i	0.709117	−	0.705091i	$-0.249094\pi$
$354$		0			0
$355$		−	0.898979i		−	0.0477129i
$356$		0			0
$357$	−1.10102	−	1.10102i	−0.0582722	−	0.0582722i
$358$		0			0
$359$	8.00000			0.422224			0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000			0.422224			0.211112	+	0.977462i	$0.432292\pi$
$360$		0			0
$361$	15.0000			0.789474
$362$		0			0
$363$	−8.57321	−	8.57321i	−0.449977	−	0.449977i
$364$		0			0
$365$			10.8990i			0.570479i
$366$		0			0
$367$			4.44949i			0.232261i	0.993234	+	0.116131i	$0.0370492\pi$
$367$			4.44949i			0.232261i	−0.993234	+	0.116131i	$0.962951\pi$
$368$		0			0
$369$	−23.3939			−1.21784
$370$		0			0
$371$	3.10102			0.160997
$372$		0			0
$373$	−5.59592			−0.289746			−0.144873	−	0.989450i	$-0.546277\pi$
$373$	−5.59592			−0.289746			−0.144873	+	0.989450i	$0.546277\pi$
$374$		0			0
$375$	1.22474	−	1.22474i	0.0632456	−	0.0632456i
$376$		0			0
$377$			2.60612i			0.134222i
$378$		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
$381$	−6.55051	+	6.55051i	−0.335593	+	0.335593i
$382$		0			0
$383$	−35.1464			−1.79590			−0.897949	−	0.440099i	$-0.854943\pi$
$383$	−35.1464			−1.79590			−0.897949	+	0.440099i	$0.854943\pi$
$384$		0			0
$385$	−0.898979			−0.0458162
$386$		0			0
$387$	19.3485			0.983538
$388$		0			0
$389$			0.202041i			0.0102439i	0.999987	+	0.00512194i	$0.00163037\pi$
$389$			0.202041i			0.0102439i	−0.999987	+	0.00512194i	$0.998370\pi$
$390$		0			0
$391$		−	8.89898i		−	0.450041i
$392$		0			0
$393$	−0.247449	−	0.247449i	−0.0124821	−	0.0124821i
$394$		0			0
$395$	−4.89898			−0.246494
$396$		0			0
$397$	38.4949			1.93200			0.966002	−	0.258535i	$-0.0832399\pi$
$397$	38.4949			1.93200			0.966002	+	0.258535i	$0.0832399\pi$
$398$		0			0
$399$	−1.10102	−	1.10102i	−0.0551200	−	0.0551200i
$400$		0			0
$401$		−	15.5959i		−	0.778823i	−0.921064	−	0.389411i	$-0.872678\pi$
$401$		−	15.5959i		−	0.778823i	0.921064	−	0.389411i	$-0.127322\pi$
$402$		0			0
$403$		−	23.1918i		−	1.15527i
$404$		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
$411$	1.34847	−	1.34847i	0.0665151	−	0.0665151i
$412$		0			0
$413$			0.494897i			0.0243523i
$414$		0			0
$415$		−	12.2474i		−	0.601204i
$416$		0			0
$417$	−2.44949	+	2.44949i	−0.119952	+	0.119952i
$418$		0			0
$419$	−7.30306			−0.356778			−0.178389	−	0.983960i	$-0.557089\pi$
$419$	−7.30306			−0.356778			−0.178389	+	0.983960i	$0.557089\pi$
$420$		0			0
$421$	36.4949			1.77865			0.889326	−	0.457273i	$-0.151174\pi$
$421$	36.4949			1.77865			0.889326	+	0.457273i	$0.151174\pi$
$422$		0			0
$423$		−	25.3485i		−	1.23248i
$424$		0			0
$425$		−	2.00000i		−	0.0970143i
$426$		0			0
$427$		−	4.00000i		−	0.193574i
$428$		0			0
$429$	−7.10102	−	7.10102i	−0.342841	−	0.342841i
$430$		0			0
$431$	16.4949			0.794531			0.397266	−	0.917704i	$-0.369959\pi$
$431$	16.4949			0.794531			0.397266	+	0.917704i	$0.369959\pi$
$432$		0			0
$433$	9.10102			0.437367			0.218684	−	0.975796i	$-0.429824\pi$
$433$	9.10102			0.437367			0.218684	+	0.975796i	$0.429824\pi$
$434$		0			0
$435$	1.10102	+	1.10102i	0.0527899	+	0.0527899i
$436$		0			0
$437$		−	8.89898i		−	0.425696i
$438$		0			0
$439$		−	28.4949i		−	1.35999i	−0.733218	−	0.679994i	$-0.761983\pi$
$439$		−	28.4949i		−	1.35999i	0.733218	−	0.679994i	$-0.238017\pi$
$440$		0			0
$441$			20.3939i			0.971137i
$442$		0			0
$443$	24.2474			1.15203			0.576016	−	0.817439i	$-0.304607\pi$
$443$	24.2474			1.15203			0.576016	+	0.817439i	$0.304607\pi$
$444$		0			0
$445$	−9.79796			−0.464468
$446$		0			0
$447$	4.65153	−	4.65153i	0.220010	−	0.220010i
$448$		0			0
$449$			25.5959i			1.20795i	0.797005	+	0.603973i	$0.206417\pi$
$449$			25.5959i			1.20795i	−0.797005	+	0.603973i	$0.793583\pi$
$450$		0			0
$451$			15.5959i			0.734383i
$452$		0			0
$453$	−28.8990	+	28.8990i	−1.35779	+	1.35779i
$454$		0			0
$455$	1.30306			0.0610885
$456$		0			0
$457$	−27.7980			−1.30033			−0.650167	−	0.759791i	$-0.725301\pi$
$457$	−27.7980			−1.30033			−0.650167	+	0.759791i	$0.725301\pi$
$458$		0			0
$459$	−7.34847	−	7.34847i	−0.342997	−	0.342997i
$460$		0			0
$461$		−	42.2929i		−	1.96977i	−0.173196	−	0.984887i	$-0.555409\pi$
$461$		−	42.2929i		−	1.96977i	0.173196	−	0.984887i	$-0.444591\pi$
$462$		0			0
$463$		−	25.8434i		−	1.20104i	−0.799609	−	0.600522i	$-0.794960\pi$
$463$		−	25.8434i		−	1.20104i	0.799609	−	0.600522i	$-0.205040\pi$
$464$		0			0
$465$	−9.79796	−	9.79796i	−0.454369	−	0.454369i
$466$		0			0
$467$	−13.1464			−0.608344			−0.304172	−	0.952617i	$-0.598380\pi$
$467$	−13.1464			−0.608344			−0.304172	+	0.952617i	$0.598380\pi$
$468$		0			0
$469$	−5.10102			−0.235543
$470$		0			0
$471$	12.2474	+	12.2474i	0.564333	+	0.564333i
$472$		0			0
$473$		−	12.8990i		−	0.593096i
$474$		0			0
$475$		−	2.00000i		−	0.0917663i
$476$		0			0
$477$	20.6969			0.947648
$478$		0			0
$479$	41.3939			1.89133			0.945667	−	0.325136i	$-0.105410\pi$
$479$	41.3939			1.89133			0.945667	+	0.325136i	$0.105410\pi$
$480$		0			0
$481$	−17.3939			−0.793093
$482$		0			0
$483$	2.44949	−	2.44949i	0.111456	−	0.111456i
$484$		0			0
$485$			18.8990i			0.858158i
$486$		0			0
$487$			17.3485i			0.786134i	0.919510	+	0.393067i	$0.128586\pi$
$487$			17.3485i			0.786134i	−0.919510	+	0.393067i	$0.871414\pi$
$488$		0			0
$489$	6.79796	−	6.79796i	0.307414	−	0.307414i
$490$		0			0
$491$	3.79796			0.171399			0.0856997	−	0.996321i	$-0.472687\pi$
$491$	3.79796			0.171399			0.0856997	+	0.996321i	$0.472687\pi$
$492$		0			0
$493$	1.79796			0.0809760
$494$		0			0
$495$	−6.00000			−0.269680
$496$		0			0
$497$		−	0.404082i		−	0.0181256i
$498$		0			0
$499$		−	27.3939i		−	1.22632i	−0.789959	−	0.613159i	$-0.789898\pi$
$499$		−	27.3939i		−	1.22632i	0.789959	−	0.613159i	$-0.210102\pi$
$500$		0			0
$501$	26.1464	+	26.1464i	1.16814	+	1.16814i
$502$		0			0
$503$	−16.9444			−0.755513			−0.377756	−	0.925905i	$-0.623304\pi$
$503$	−16.9444			−0.755513			−0.377756	+	0.925905i	$0.623304\pi$
$504$		0			0
$505$	12.8990			0.573997
$506$		0			0
$507$	−5.62883	−	5.62883i	−0.249985	−	0.249985i
$508$		0			0
$509$			16.8990i			0.749034i	0.927220	+	0.374517i	$0.122191\pi$
$509$			16.8990i			0.749034i	−0.927220	+	0.374517i	$0.877809\pi$
$510$		0			0
$511$			4.89898i			0.216718i
$512$		0			0
$513$	−7.34847	−	7.34847i	−0.324443	−	0.324443i
$514$		0			0
$515$	1.34847			0.0594207
$516$		0			0
$517$	−16.8990			−0.743216
$518$		0			0
$519$	−9.55051	+	9.55051i	−0.419221	+	0.419221i
$520$		0			0
$521$		−	6.20204i		−	0.271716i	−0.990728	−	0.135858i	$-0.956621\pi$
$521$		−	6.20204i		−	0.271716i	0.990728	−	0.135858i	$-0.0433791\pi$
$522$		0			0
$523$		−	40.2474i		−	1.75990i	−0.475068	−	0.879949i	$-0.657577\pi$
$523$		−	40.2474i		−	1.75990i	0.475068	−	0.879949i	$-0.342423\pi$
$524$		0			0
$525$	0.550510	−	0.550510i	0.0240262	−	0.0240262i
$526$		0			0
$527$	−16.0000			−0.696971
$528$		0			0
$529$	−3.20204			−0.139219
$530$		0			0
$531$			3.30306i			0.143341i
$532$		0			0
$533$		−	22.6061i		−	0.979180i
$534$		0			0
$535$		−	5.55051i		−	0.239969i
$536$		0			0
$537$	−28.0454	−	28.0454i	−1.21025	−	1.21025i
$538$		0			0
$539$	13.5959			0.585618
$540$		0			0
$541$	26.0000			1.11783			0.558914	−	0.829226i	$-0.311218\pi$
$541$	26.0000			1.11783			0.558914	+	0.829226i	$0.311218\pi$
$542$		0			0
$543$	21.5505	+	21.5505i	0.924821	+	0.924821i
$544$		0			0
$545$			3.10102i			0.132833i
$546$		0			0
$547$			18.0454i			0.771566i	0.922590	+	0.385783i	$0.126069\pi$
$547$			18.0454i			0.771566i	−0.922590	+	0.385783i	$0.873931\pi$
$548$		0			0
$549$		−	26.6969i		−	1.13940i
$550$		0			0
$551$	1.79796			0.0765956
$552$		0			0
$553$	−2.20204			−0.0936403
$554$		0			0
$555$	−7.34847	+	7.34847i	−0.311925	+	0.311925i
$556$		0			0
$557$		−	6.00000i		−	0.254228i	−0.991888	−	0.127114i	$-0.959429\pi$
$557$		−	6.00000i		−	0.254228i	0.991888	−	0.127114i	$-0.0405714\pi$
$558$		0			0
$559$			18.6969i			0.790796i
$560$		0			0
$561$	−4.89898	+	4.89898i	−0.206835	+	0.206835i
$562$		0			0
$563$	−18.4495			−0.777553			−0.388777	−	0.921332i	$-0.627102\pi$
$563$	−18.4495			−0.777553			−0.388777	+	0.921332i	$0.627102\pi$
$564$		0			0
$565$	10.8990			0.458524
$566$		0			0
$567$		−	4.04541i		−	0.169891i
$568$		0			0
$569$		−	9.59592i		−	0.402282i	−0.979562	−	0.201141i	$-0.935535\pi$
$569$		−	9.59592i		−	0.402282i	0.979562	−	0.201141i	$-0.0644649\pi$
$570$		0			0
$571$			28.6969i			1.20093i	0.799651	+	0.600465i	$0.205018\pi$
$571$			28.6969i			1.20093i	−0.799651	+	0.600465i	$0.794982\pi$
$572$		0			0
$573$	−3.79796	−	3.79796i	−0.158662	−	0.158662i
$574$		0			0
$575$	4.44949			0.185557
$576$		0			0
$577$	−11.7980			−0.491155			−0.245578	−	0.969377i	$-0.578978\pi$
$577$	−11.7980			−0.491155			−0.245578	+	0.969377i	$0.578978\pi$
$578$		0			0
$579$	30.2474	+	30.2474i	1.25704	+	1.25704i
$580$		0			0
$581$		−	5.50510i		−	0.228390i
$582$		0			0
$583$		−	13.7980i		−	0.571453i
$584$		0			0
$585$	8.69694			0.359574
$586$		0			0
$587$	−4.24745			−0.175311			−0.0876555	−	0.996151i	$-0.527937\pi$
$587$	−4.24745			−0.175311			−0.0876555	+	0.996151i	$0.527937\pi$
$588$		0			0
$589$	−16.0000			−0.659269
$590$		0			0
$591$	25.3485	−	25.3485i	1.04270	−	1.04270i
$592$		0			0
$593$		−	11.3031i		−	0.464161i	−0.972697	−	0.232081i	$-0.925447\pi$
$593$		−	11.3031i		−	0.464161i	0.972697	−	0.232081i	$-0.0745533\pi$
$594$		0			0
$595$		−	0.898979i		−	0.0368546i
$596$		0			0
$597$	−15.7980	+	15.7980i	−0.646567	+	0.646567i
$598$		0			0
$599$	41.3939			1.69131			0.845654	−	0.533732i	$-0.179211\pi$
$599$	41.3939			1.69131			0.845654	+	0.533732i	$0.179211\pi$
$600$		0			0
$601$	13.7980			0.562830			0.281415	−	0.959586i	$-0.409196\pi$
$601$	13.7980			0.562830			0.281415	+	0.959586i	$0.409196\pi$
$602$		0			0
$603$	−34.0454			−1.38644
$604$		0			0
$605$		−	7.00000i		−	0.284590i
$606$		0			0
$607$			37.3485i			1.51593i	0.652297	+	0.757964i	$0.273806\pi$
$607$			37.3485i			1.51593i	−0.652297	+	0.757964i	$0.726194\pi$
$608$		0			0
$609$	0.494897	+	0.494897i	0.0200543	+	0.0200543i
$610$		0			0
$611$	24.4949			0.990957
$612$		0			0
$613$	−6.49490			−0.262326			−0.131163	−	0.991361i	$-0.541871\pi$
$613$	−6.49490			−0.262326			−0.131163	+	0.991361i	$0.541871\pi$
$614$		0			0
$615$	−9.55051	−	9.55051i	−0.385114	−	0.385114i
$616$		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
$619$		−	11.7980i		−	0.474200i	−0.971485	−	0.237100i	$-0.923803\pi$
$619$		−	11.7980i		−	0.474200i	0.971485	−	0.237100i	$-0.0761969\pi$
$620$		0			0
$621$	16.3485	−	16.3485i	0.656041	−	0.656041i
$622$		0			0
$623$	−4.40408			−0.176446
$624$		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	$-0.896829\pi$
$631$		−	16.0000i		−	0.636950i	0.947931	−	0.318475i	$-0.103171\pi$
$632$		0			0
$633$	23.1464	−	23.1464i	0.919988	−	0.919988i
$634$		0			0
$635$	−5.34847			−0.212248
$636$		0			0
$637$	−19.7071			−0.780825
$638$		0			0
$639$		−	2.69694i		−	0.106689i
$640$		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
$643$			23.3485i			0.920774i	0.887718	+	0.460387i	$0.152289\pi$
$643$			23.3485i			0.920774i	−0.887718	+	0.460387i	$0.847711\pi$
$644$		0			0
$645$	7.89898	+	7.89898i	0.311022	+	0.311022i
$646$		0			0
$647$	30.2474			1.18915			0.594575	−	0.804040i	$-0.297320\pi$
$647$	30.2474			1.18915			0.594575	+	0.804040i	$0.297320\pi$
$648$		0			0
$649$	2.20204			0.0864377
$650$		0			0
$651$	−4.40408	−	4.40408i	−0.172610	−	0.172610i
$652$		0			0
$653$			1.10102i			0.0430863i	0.999768	+	0.0215431i	$0.00685792\pi$
$653$			1.10102i			0.0430863i	−0.999768	+	0.0215431i	$0.993142\pi$
$654$		0			0
$655$		−	0.202041i		−	0.00789440i
$656$		0			0
$657$			32.6969i			1.27563i
$658$		0			0
$659$	−34.4949			−1.34373			−0.671865	−	0.740673i	$-0.734507\pi$
$659$	−34.4949			−1.34373			−0.671865	+	0.740673i	$0.734507\pi$
$660$		0			0
$661$	7.10102			0.276198			0.138099	−	0.990418i	$-0.455901\pi$
$661$	7.10102			0.276198			0.138099	+	0.990418i	$0.455901\pi$
$662$		0			0
$663$	7.10102	−	7.10102i	0.275781	−	0.275781i
$664$		0			0
$665$		−	0.898979i		−	0.0348609i
$666$		0			0
$667$			4.00000i			0.154881i
$668$		0			0
$669$	15.2474	−	15.2474i	0.589500	−	0.589500i
$670$		0			0
$671$	−17.7980			−0.687083
$672$		0			0
$673$	−3.30306			−0.127324			−0.0636618	−	0.997972i	$-0.520278\pi$
$673$	−3.30306			−0.127324			−0.0636618	+	0.997972i	$0.520278\pi$
$674$		0			0
$675$	3.67423	−	3.67423i	0.141421	−	0.141421i
$676$		0			0
$677$		−	12.6969i		−	0.487983i	−0.969777	−	0.243991i	$-0.921543\pi$
$677$		−	12.6969i		−	0.487983i	0.969777	−	0.243991i	$-0.0784569\pi$
$678$		0			0
$679$			8.49490i			0.326004i
$680$		0			0
$681$	25.8990	+	25.8990i	0.992451	+	0.992451i
$682$		0			0
$683$	4.65153			0.177986			0.0889929	−	0.996032i	$-0.471635\pi$
$683$	4.65153			0.177986			0.0889929	+	0.996032i	$0.471635\pi$
$684$		0			0
$685$	1.10102			0.0420678
$686$		0			0
$687$	24.2474	+	24.2474i	0.925098	+	0.925098i
$688$		0			0
$689$			20.0000i			0.761939i
$690$		0			0
$691$			20.6969i			0.787349i	0.919250	+	0.393674i	$0.128796\pi$
$691$			20.6969i			0.787349i	−0.919250	+	0.393674i	$0.871204\pi$
$692$		0			0
$693$	−2.69694			−0.102448
$694$		0			0
$695$	−2.00000			−0.0758643
$696$		0			0
$697$	−15.5959			−0.590738
$698$		0			0
$699$	14.4495	−	14.4495i	0.546530	−	0.546530i
$700$		0			0
$701$		−	43.3939i		−	1.63896i	−0.573105	−	0.819482i	$-0.694261\pi$
$701$		−	43.3939i		−	1.63896i	0.573105	−	0.819482i	$-0.305739\pi$
$702$		0			0
$703$			12.0000i			0.452589i
$704$		0			0
$705$	10.3485	−	10.3485i	0.389746	−	0.389746i
$706$		0			0
$707$	5.79796			0.218055
$708$		0			0
$709$	−43.7980			−1.64487			−0.822433	−	0.568861i	$-0.807384\pi$
$709$	−43.7980			−1.64487			−0.822433	+	0.568861i	$0.807384\pi$
$710$		0			0
$711$	−14.6969			−0.551178
$712$		0			0
$713$		−	35.5959i		−	1.33308i
$714$		0			0
$715$		−	5.79796i		−	0.216831i
$716$		0			0
$717$	7.59592	+	7.59592i	0.283675	+	0.283675i
$718$		0			0
$719$	−33.3939			−1.24538			−0.622691	−	0.782468i	$-0.713961\pi$
$719$	−33.3939			−1.24538			−0.622691	+	0.782468i	$0.713961\pi$
$720$		0			0
$721$	0.606123			0.0225732
$722$		0			0
$723$	−28.8990	−	28.8990i	−1.07476	−	1.07476i
$724$		0			0
$725$			0.898979i			0.0333873i
$726$		0			0
$727$			3.14643i			0.116695i	0.998296	+	0.0583473i	$0.0185831\pi$
$727$			3.14643i			0.116695i	−0.998296	+	0.0583473i	$0.981417\pi$
$728$		0			0
$729$		−	27.0000i		−	1.00000i
$730$		0			0
$731$	12.8990			0.477086
$732$		0			0
$733$	14.0000			0.517102			0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000			0.517102			0.258551	+	0.965998i	$0.416755\pi$
$734$		0			0
$735$	−8.32577	+	8.32577i	−0.307100	+	0.307100i
$736$		0			0
$737$			22.6969i			0.836052i
$738$		0			0
$739$			3.79796i			0.139710i	0.997557	+	0.0698551i	$0.0222537\pi$
$739$			3.79796i			0.139710i	−0.997557	+	0.0698551i	$0.977746\pi$
$740$		0			0
$741$	7.10102	−	7.10102i	0.260863	−	0.260863i
$742$		0			0
$743$	0.853572			0.0313145			0.0156573	−	0.999877i	$-0.495016\pi$
$743$	0.853572			0.0313145			0.0156573	+	0.999877i	$0.495016\pi$
$744$		0			0
$745$	3.79796			0.139146
$746$		0			0
$747$		−	36.7423i		−	1.34433i
$748$		0			0
$749$		−	2.49490i		−	0.0911616i
$750$		0			0
$751$			9.39388i			0.342787i	0.985203	+	0.171394i	$0.0548270\pi$
$751$			9.39388i			0.342787i	−0.985203	+	0.171394i	$0.945173\pi$
$752$		0			0
$753$	34.0454	+	34.0454i	1.24068	+	1.24068i
$754$		0			0
$755$	−23.5959			−0.858743
$756$		0			0
$757$	−19.3939			−0.704882			−0.352441	−	0.935834i	$-0.614648\pi$
$757$	−19.3939			−0.704882			−0.352441	+	0.935834i	$0.614648\pi$
$758$		0			0
$759$	−10.8990	−	10.8990i	−0.395608	−	0.395608i
$760$		0			0
$761$			33.7980i			1.22518i	0.790403	+	0.612588i	$0.209871\pi$
$761$			33.7980i			1.22518i	−0.790403	+	0.612588i	$0.790129\pi$
$762$		0			0
$763$			1.39388i			0.0504617i
$764$		0			0
$765$		−	6.00000i		−	0.216930i
$766$		0			0
$767$	−3.19184			−0.115251
$768$		0			0
$769$	−31.7980			−1.14666			−0.573332	−	0.819323i	$-0.694349\pi$
$769$	−31.7980			−1.14666			−0.573332	+	0.819323i	$0.694349\pi$
$770$		0			0
$771$	−13.3485	+	13.3485i	−0.480733	+	0.480733i
$772$		0			0
$773$			24.6969i			0.888287i	0.895956	+	0.444144i	$0.146492\pi$
$773$			24.6969i			0.888287i	−0.895956	+	0.444144i	$0.853508\pi$
$774$		0			0
$775$		−	8.00000i		−	0.287368i
$776$		0			0
$777$	−3.30306	+	3.30306i	−0.118497	+	0.118497i
$778$		0			0
$779$	−15.5959			−0.558782
$780$		0			0
$781$	−1.79796			−0.0643360
$782$		0			0
$783$	3.30306	+	3.30306i	0.118042	+	0.118042i
$784$		0			0
$785$			10.0000i			0.356915i
$786$		0			0
$787$		−	13.1464i		−	0.468620i	−0.972162	−	0.234310i	$-0.924717\pi$
$787$		−	13.1464i		−	0.468620i	0.972162	−	0.234310i	$-0.0752830\pi$
$788$		0			0
$789$	−15.2474	−	15.2474i	−0.542824	−	0.542824i
$790$		0			0
$791$	4.89898			0.174188
$792$		0			0
$793$	25.7980			0.916112
$794$		0			0
$795$	8.44949	+	8.44949i	0.299673	+	0.299673i
$796$		0			0
$797$			1.10102i			0.0390001i	0.999810	+	0.0195001i	$0.00620746\pi$
$797$			1.10102i			0.0390001i	−0.999810	+	0.0195001i	$0.993793\pi$
$798$		0			0
$799$		−	16.8990i		−	0.597843i
$800$		0			0
$801$	−29.3939			−1.03858
$802$		0			0
$803$	21.7980			0.769233
$804$		0			0
$805$	2.00000			0.0704907
$806$		0			0
$807$	−26.4495	+	26.4495i	−0.931066	+	0.931066i
$808$		0			0
$809$			8.40408i			0.295472i	0.989027	+	0.147736i	$0.0471986\pi$
$809$			8.40408i			0.295472i	−0.989027	+	0.147736i	$0.952801\pi$
$810$		0			0
$811$			1.50510i			0.0528513i	0.999651	+	0.0264256i	$0.00841252\pi$
$811$			1.50510i			0.0528513i	−0.999651	+	0.0264256i	$0.991587\pi$
$812$		0			0
$813$	−14.6969	+	14.6969i	−0.515444	+	0.515444i
$814$		0			0
$815$	5.55051			0.194426
$816$		0			0
$817$	12.8990			0.451278
$818$		0			0
$819$	3.90918			0.136598
$820$		0			0
$821$			37.5959i			1.31211i	0.754715	+	0.656053i	$0.227775\pi$
$821$			37.5959i			1.31211i	−0.754715	+	0.656053i	$0.772225\pi$
$822$		0			0
$823$		−	7.55051i		−	0.263194i	−0.991303	−	0.131597i	$-0.957989\pi$
$823$		−	7.55051i		−	0.263194i	0.991303	−	0.131597i	$-0.0420105\pi$
$824$		0			0
$825$	−2.44949	−	2.44949i	−0.0852803	−	0.0852803i
$826$		0			0
$827$	−1.14643			−0.0398652			−0.0199326	−	0.999801i	$-0.506345\pi$
$827$	−1.14643			−0.0398652			−0.0199326	+	0.999801i	$0.506345\pi$
$828$		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
$833$			13.5959i			0.471071i
$834$		0			0
$835$			21.3485i			0.738794i
$836$		0			0
$837$	−29.3939	−	29.3939i	−1.01600	−	1.01600i
$838$		0			0
$839$	−17.7980			−0.614454			−0.307227	−	0.951636i	$-0.599401\pi$
$839$	−17.7980			−0.614454			−0.307227	+	0.951636i	$0.599401\pi$
$840$		0			0
$841$	28.1918			0.972132
$842$		0			0
$843$	−21.5505	+	21.5505i	−0.742239	+	0.742239i
$844$		0			0
$845$		−	4.59592i		−	0.158104i
$846$		0			0
$847$		−	3.14643i		−	0.108113i
$848$		0			0
$849$	16.1010	−	16.1010i	0.552586	−	0.552586i
$850$		0			0
$851$	−26.6969			−0.915159
$852$		0			0
$853$	−50.8990			−1.74275			−0.871374	−	0.490619i	$-0.836771\pi$
$853$	−50.8990			−1.74275			−0.871374	+	0.490619i	$0.836771\pi$
$854$		0			0
$855$		−	6.00000i		−	0.205196i
$856$		0			0
$857$			8.69694i			0.297082i	0.988906	+	0.148541i	$0.0474577\pi$
$857$			8.69694i			0.297082i	−0.988906	+	0.148541i	$0.952542\pi$
$858$		0			0
$859$			37.5959i			1.28276i	0.767225	+	0.641378i	$0.221637\pi$
$859$			37.5959i			1.28276i	−0.767225	+	0.641378i	$0.778363\pi$
$860$		0			0
$861$	−4.29286	−	4.29286i	−0.146300	−	0.146300i
$862$		0			0
$863$	−44.9444			−1.52992			−0.764962	−	0.644075i	$-0.777242\pi$
$863$	−44.9444			−1.52992			−0.764962	+	0.644075i	$0.777242\pi$
$864$		0			0
$865$	−7.79796			−0.265139
$866$		0			0
$867$	15.9217	+	15.9217i	0.540729	+	0.540729i
$868$		0			0
$869$			9.79796i			0.332373i
$870$		0			0
$871$		−	32.8990i		−	1.11474i
$872$		0			0
$873$			56.6969i			1.91890i
$874$		0			0
$875$	0.449490			0.0151955
$876$		0			0
$877$	2.00000			0.0675352			0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000			0.0675352			0.0337676	+	0.999430i	$0.489249\pi$
$878$		0			0
$879$	−34.0454	+	34.0454i	−1.14832	+	1.14832i
$880$		0			0
$881$		−	8.20204i		−	0.276334i	−0.990409	−	0.138167i	$-0.955879\pi$
$881$		−	8.20204i		−	0.276334i	0.990409	−	0.138167i	$-0.0441210\pi$
$882$		0			0
$883$			18.4495i			0.620875i	0.950594	+	0.310437i	$0.100476\pi$
$883$			18.4495i			0.620875i	−0.950594	+	0.310437i	$0.899524\pi$
$884$		0			0
$885$	−1.34847	+	1.34847i	−0.0453283	+	0.0453283i
$886$		0			0
$887$	−47.1464			−1.58302			−0.791511	−	0.611155i	$-0.790705\pi$
$887$	−47.1464			−1.58302			−0.791511	+	0.611155i	$0.790705\pi$
$888$		0			0
$889$	−2.40408			−0.0806303
$890$		0			0
$891$	−18.0000			−0.603023
$892$		0			0
$893$		−	16.8990i		−	0.565503i
$894$		0			0
$895$		−	22.8990i		−	0.765428i
$896$		0			0
$897$	15.7980	+	15.7980i	0.527478	+	0.527478i
$898$		0			0
$899$	7.19184			0.239861
$900$		0			0
$901$	13.7980			0.459677
$902$		0			0
$903$	3.55051	+	3.55051i	0.118154	+	0.118154i
$904$		0			0
$905$			17.5959i			0.584908i
$906$		0			0
$907$			2.94439i			0.0977668i	0.998804	+	0.0488834i	$0.0155663\pi$
$907$			2.94439i			0.0977668i	−0.998804	+	0.0488834i	$0.984434\pi$
$908$		0			0
$909$	38.6969			1.28350
$910$		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
$915$	10.8990	−	10.8990i	0.360309	−	0.360309i
$916$		0			0
$917$		−	0.0908154i		−	0.00299899i
$918$		0			0
$919$			44.4949i			1.46775i	0.679284	+	0.733876i	$0.262291\pi$
$919$			44.4949i			1.46775i	−0.679284	+	0.733876i	$0.737709\pi$
$920$		0			0
$921$	33.4949	−	33.4949i	1.10369	−	1.10369i
$922$		0			0
$923$	2.60612			0.0857816
$924$		0			0
$925$	−6.00000			−0.197279
$926$		0			0
$927$	4.04541			0.132869
$928$		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
$931$			13.5959i			0.445588i
$932$		0			0
$933$	30.0000	+	30.0000i	0.982156	+	0.982156i
$934$		0			0
$935$	−4.00000			−0.130814
$936$		0			0
$937$	2.49490			0.0815047			0.0407524	−	0.999169i	$-0.487025\pi$
$937$	2.49490			0.0815047			0.0407524	+	0.999169i	$0.487025\pi$
$938$		0			0
$939$	−0.247449	−	0.247449i	−0.00807518	−	0.00807518i
$940$		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
$943$		−	34.6969i		−	1.12989i
$944$		0			0
$945$	1.65153	−	1.65153i	0.0537243	−	0.0537243i
$946$		0			0
$947$	−6.44949			−0.209580			−0.104790	−	0.994494i	$-0.533417\pi$
$947$	−6.44949			−0.209580			−0.104790	+	0.994494i	$0.533417\pi$
$948$		0			0
$949$	−31.5959			−1.02565
$950$		0			0
$951$	−25.3485	+	25.3485i	−0.821980	+	0.821980i
$952$		0			0
$953$		−	10.8990i		−	0.353053i	−0.984296	−	0.176526i	$-0.943514\pi$
$953$		−	10.8990i		−	0.353053i	0.984296	−	0.176526i	$-0.0564861\pi$
$954$		0			0
$955$		−	3.10102i		−	0.100347i
$956$		0			0
$957$	2.20204	−	2.20204i	0.0711819	−	0.0711819i
$958$		0			0
$959$	0.494897			0.0159811
$960$		0			0
$961$	−33.0000			−1.06452
$962$		0			0
$963$		−	16.6515i		−	0.536588i
$964$		0			0
$965$			24.6969i			0.795023i
$966$		0			0
$967$		−	5.75255i		−	0.184990i	−0.995713	−	0.0924948i	$-0.970516\pi$
$967$		−	5.75255i		−	0.184990i	0.995713	−	0.0924948i	$-0.0294841\pi$
$968$		0			0
$969$	−4.89898	−	4.89898i	−0.157378	−	0.157378i
$970$		0			0
$971$	20.2020			0.648314			0.324157	−	0.946003i	$-0.394919\pi$
$971$	20.2020			0.648314			0.324157	+	0.946003i	$0.394919\pi$
$972$		0			0
$973$	−0.898979			−0.0288200
$974$		0			0
$975$	3.55051	+	3.55051i	0.113707	+	0.113707i
$976$		0			0
$977$			45.1918i			1.44581i	0.690945	+	0.722907i	$0.257195\pi$
$977$			45.1918i			1.44581i	−0.690945	+	0.722907i	$0.742805\pi$
$978$		0			0
$979$			19.5959i			0.626288i
$980$		0			0
$981$			9.30306i			0.297024i
$982$		0			0
$983$	−7.95459			−0.253712			−0.126856	−	0.991921i	$-0.540489\pi$
$983$	−7.95459			−0.253712			−0.126856	+	0.991921i	$0.540489\pi$
$984$		0			0
$985$	20.6969			0.659459
$986$		0			0
$987$	4.65153	−	4.65153i	0.148060	−	0.148060i
$988$		0			0
$989$			28.6969i			0.912510i
$990$		0			0
$991$			47.1918i			1.49910i	0.661949	+	0.749549i	$0.269730\pi$
$991$			47.1918i			1.49910i	−0.661949	+	0.749549i	$0.730270\pi$
$992$		0			0
$993$	−16.0454	+	16.0454i	−0.509186	+	0.509186i
$994$		0			0
$995$	−12.8990			−0.408925
$996$		0			0
$997$	−42.0908			−1.33303			−0.666515	−	0.745492i	$-0.732215\pi$
$997$	−42.0908			−1.33303			−0.666515	+	0.745492i	$0.732215\pi$
$998$		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.j.1151.4	yes	4
3.2	odd	2		1920.2.h.h.1151.2	yes	4
4.3	odd	2		1920.2.h.h.1151.1	yes	4
8.3	odd	2		1920.2.h.i.1151.4	yes	4
8.5	even	2		1920.2.h.g.1151.1	✓	4
12.11	even	2	inner	1920.2.h.j.1151.3	yes	4
24.5	odd	2		1920.2.h.i.1151.3	yes	4
24.11	even	2		1920.2.h.g.1151.2	yes	4

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(1.22474 + 1.22474i) q^{3} +1.00000i q^{5} +0.449490i q^{7} +3.00000i q^{9} +2.00000 q^{11} -2.89898 q^{13} +(-1.22474 + 1.22474i) q^{15} +2.00000i q^{17} +2.00000i q^{19} +(-0.550510 + 0.550510i) q^{21} -4.44949 q^{23} -1.00000 q^{25} +(-3.67423 + 3.67423i) q^{27} -0.898979i q^{29} +8.00000i q^{31} +(2.44949 + 2.44949i) q^{33} -0.449490 q^{35} +6.00000 q^{37} +(-3.55051 - 3.55051i) q^{39} +7.79796i q^{41} -6.44949i q^{43} -3.00000 q^{45} -8.44949 q^{47} +6.79796 q^{49} +(-2.44949 + 2.44949i) q^{51} -6.89898i q^{53} +2.00000i q^{55} +(-2.44949 + 2.44949i) q^{57} +1.10102 q^{59} -8.89898 q^{61} -1.34847 q^{63} -2.89898i q^{65} +11.3485i q^{67} +(-5.44949 - 5.44949i) q^{69} -0.898979 q^{71} +10.8990 q^{73} +(-1.22474 - 1.22474i) q^{75} +0.898979i q^{77} +4.89898i q^{79} -9.00000 q^{81} -12.2474 q^{83} -2.00000 q^{85} +(1.10102 - 1.10102i) q^{87} +9.79796i q^{89} -1.30306i q^{91} +(-9.79796 + 9.79796i) q^{93} -2.00000 q^{95} +18.8990 q^{97} +6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{11} + 8 q^{13} - 12 q^{21} - 8 q^{23} - 4 q^{25} + 8 q^{35} + 24 q^{37} - 24 q^{39} - 12 q^{45} - 24 q^{47} - 12 q^{49} + 24 q^{59} - 16 q^{61} + 24 q^{63} - 12 q^{69} + 16 q^{71} + 24 q^{73}+ \cdots + 56 q^{97}+O(q^{100}) \) 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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