LMFDB - Embedded newform 1200.2.o.f.1199.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,2,Mod(1199,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1200.1199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1200.o (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$9.58204824255$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 240)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1199.4
Root			$-1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1200.1199
Dual form			1200.2.o.f.1199.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 + 1.22474i) q^{3} -2.44949 q^{7} +3.00000i q^{9} +O(q^{10})$	$q+(1.22474 + 1.22474i) q^{3} -2.44949 q^{7} +3.00000i q^{9} +4.89898 q^{11} -2.00000i q^{13} +6.00000 q^{17} +4.89898i q^{19} +(-3.00000 - 3.00000i) q^{21} -2.44949i q^{23} +(-3.67423 + 3.67423i) q^{27} +9.79796i q^{31} +(6.00000 + 6.00000i) q^{33} -2.00000i q^{37} +(2.44949 - 2.44949i) q^{39} +6.00000i q^{41} -2.44949 q^{43} +12.2474i q^{47} -1.00000 q^{49} +(7.34847 + 7.34847i) q^{51} -6.00000 q^{53} +(-6.00000 + 6.00000i) q^{57} +9.79796 q^{59} +8.00000 q^{61} -7.34847i q^{63} -7.34847 q^{67} +(3.00000 - 3.00000i) q^{69} +4.89898 q^{71} -14.0000i q^{73} -12.0000 q^{77} -4.89898i q^{79} -9.00000 q^{81} +7.34847i q^{83} -12.0000i q^{89} +4.89898i q^{91} +(-12.0000 + 12.0000i) q^{93} -10.0000i q^{97} +14.6969i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 24 q^{17} - 12 q^{21} + 24 q^{33} - 4 q^{49} - 24 q^{53} - 24 q^{57} + 32 q^{61} + 12 q^{69} - 48 q^{77} - 36 q^{81} - 48 q^{93}+O(q^{100}) $ 4 * q + 24 * q^17 - 12 * q^21 + 24 * q^33 - 4 * q^49 - 24 * q^53 - 24 * q^57 + 32 * q^61 + 12 * q^69 - 48 * q^77 - 36 * q^81 - 48 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.44949			−0.925820			−0.462910	−	0.886405i	$-0.653195\pi$
$7$	−2.44949			−0.925820			−0.462910	+	0.886405i	$0.653195\pi$
$8$		0			0
$9$			3.00000i			1.00000i
$10$		0			0
$11$	4.89898			1.47710			0.738549	−	0.674200i	$-0.235511\pi$
$11$	4.89898			1.47710			0.738549	+	0.674200i	$0.235511\pi$
$12$		0			0
$13$		−	2.00000i		−	0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$		−	2.00000i		−	0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	6.00000			1.45521			0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000			1.45521			0.727607	+	0.685994i	$0.240633\pi$
$18$		0			0
$19$			4.89898i			1.12390i	0.827170	+	0.561951i	$0.189949\pi$
$19$			4.89898i			1.12390i	−0.827170	+	0.561951i	$0.810051\pi$
$20$		0			0
$21$	−3.00000	−	3.00000i	−0.654654	−	0.654654i
$22$		0			0
$23$		−	2.44949i		−	0.510754i	−0.966842	−	0.255377i	$-0.917800\pi$
$23$		−	2.44949i		−	0.510754i	0.966842	−	0.255377i	$-0.0821996\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−3.67423	+	3.67423i	−0.707107	+	0.707107i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$			9.79796i			1.75977i	0.475191	+	0.879883i	$0.342379\pi$
$31$			9.79796i			1.75977i	−0.475191	+	0.879883i	$0.657621\pi$
$32$		0			0
$33$	6.00000	+	6.00000i	1.04447	+	1.04447i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	2.00000i		−	0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$		−	2.00000i		−	0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$		0			0
$39$	2.44949	−	2.44949i	0.392232	−	0.392232i
$40$		0			0
$41$			6.00000i			0.937043i	0.883452	+	0.468521i	$0.155213\pi$
$41$			6.00000i			0.937043i	−0.883452	+	0.468521i	$0.844787\pi$
$42$		0			0
$43$	−2.44949			−0.373544			−0.186772	−	0.982403i	$-0.559803\pi$
$43$	−2.44949			−0.373544			−0.186772	+	0.982403i	$0.559803\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			12.2474i			1.78647i	0.449586	+	0.893237i	$0.351571\pi$
$47$			12.2474i			1.78647i	−0.449586	+	0.893237i	$0.648429\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	7.34847	+	7.34847i	1.02899	+	1.02899i
$52$		0			0
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−6.00000	+	6.00000i	−0.794719	+	0.794719i
$58$		0			0
$59$	9.79796			1.27559			0.637793	−	0.770208i	$-0.279848\pi$
$59$	9.79796			1.27559			0.637793	+	0.770208i	$0.279848\pi$
$60$		0			0
$61$	8.00000			1.02430			0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000			1.02430			0.512148	+	0.858898i	$0.328850\pi$
$62$		0			0
$63$		−	7.34847i		−	0.925820i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−7.34847			−0.897758			−0.448879	−	0.893592i	$-0.648177\pi$
$67$	−7.34847			−0.897758			−0.448879	+	0.893592i	$0.648177\pi$
$68$		0			0
$69$	3.00000	−	3.00000i	0.361158	−	0.361158i
$70$		0			0
$71$	4.89898			0.581402			0.290701	−	0.956814i	$-0.406112\pi$
$71$	4.89898			0.581402			0.290701	+	0.956814i	$0.406112\pi$
$72$		0			0
$73$		−	14.0000i		−	1.63858i	−0.573382	−	0.819288i	$-0.694369\pi$
$73$		−	14.0000i		−	1.63858i	0.573382	−	0.819288i	$-0.305631\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−12.0000			−1.36753
$78$		0			0
$79$		−	4.89898i		−	0.551178i	−0.961276	−	0.275589i	$-0.911127\pi$
$79$		−	4.89898i		−	0.551178i	0.961276	−	0.275589i	$-0.0888729\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$			7.34847i			0.806599i	0.915068	+	0.403300i	$0.132137\pi$
$83$			7.34847i			0.806599i	−0.915068	+	0.403300i	$0.867863\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	12.0000i		−	1.27200i	−0.771690	−	0.635999i	$-0.780588\pi$
$89$		−	12.0000i		−	1.27200i	0.771690	−	0.635999i	$-0.219412\pi$
$90$		0			0
$91$			4.89898i			0.513553i
$92$		0			0
$93$	−12.0000	+	12.0000i	−1.24434	+	1.24434i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	10.0000i		−	1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$		−	10.0000i		−	1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$		0			0
$99$			14.6969i			1.47710i
$100$		0			0
$101$		−	12.0000i		−	1.19404i	−0.802225	−	0.597022i	$-0.796350\pi$
$101$		−	12.0000i		−	1.19404i	0.802225	−	0.597022i	$-0.203650\pi$
$102$		0			0
$103$	12.2474			1.20678			0.603388	−	0.797447i	$-0.293817\pi$
$103$	12.2474			1.20678			0.603388	+	0.797447i	$0.293817\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	2.44949i		−	0.236801i	−0.992966	−	0.118401i	$-0.962223\pi$
$107$		−	2.44949i		−	0.236801i	0.992966	−	0.118401i	$-0.0377767\pi$
$108$		0			0
$109$	−4.00000			−0.383131			−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000			−0.383131			−0.191565	+	0.981480i	$0.561356\pi$
$110$		0			0
$111$	2.44949	−	2.44949i	0.232495	−	0.232495i
$112$		0			0
$113$	6.00000			0.564433			0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000			0.564433			0.282216	+	0.959351i	$0.408930\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	6.00000			0.554700
$118$		0			0
$119$	−14.6969			−1.34727
$120$		0			0
$121$	13.0000			1.18182
$122$		0			0
$123$	−7.34847	+	7.34847i	−0.662589	+	0.662589i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−7.34847			−0.652071			−0.326036	−	0.945357i	$-0.605713\pi$
$127$	−7.34847			−0.652071			−0.326036	+	0.945357i	$0.605713\pi$
$128$		0			0
$129$	−3.00000	−	3.00000i	−0.264135	−	0.264135i
$130$		0			0
$131$	4.89898			0.428026			0.214013	−	0.976831i	$-0.431347\pi$
$131$	4.89898			0.428026			0.214013	+	0.976831i	$0.431347\pi$
$132$		0			0
$133$		−	12.0000i		−	1.04053i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.00000			−0.512615			−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000			−0.512615			−0.256307	+	0.966595i	$0.582506\pi$
$138$		0			0
$139$			4.89898i			0.415526i	0.978179	+	0.207763i	$0.0666183\pi$
$139$			4.89898i			0.415526i	−0.978179	+	0.207763i	$0.933382\pi$
$140$		0			0
$141$	−15.0000	+	15.0000i	−1.26323	+	1.26323i
$142$		0			0
$143$		−	9.79796i		−	0.819346i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−1.22474	−	1.22474i	−0.101015	−	0.101015i
$148$		0			0
$149$			18.0000i			1.47462i	0.675556	+	0.737309i	$0.263904\pi$
$149$			18.0000i			1.47462i	−0.675556	+	0.737309i	$0.736096\pi$
$150$		0			0
$151$		−	19.5959i		−	1.59469i	−0.603522	−	0.797347i	$-0.706236\pi$
$151$		−	19.5959i		−	1.59469i	0.603522	−	0.797347i	$-0.293764\pi$
$152$		0			0
$153$			18.0000i			1.45521i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			10.0000i			0.798087i	0.916932	+	0.399043i	$0.130658\pi$
$157$			10.0000i			0.798087i	−0.916932	+	0.399043i	$0.869342\pi$
$158$		0			0
$159$	−7.34847	−	7.34847i	−0.582772	−	0.582772i
$160$		0			0
$161$			6.00000i			0.472866i
$162$		0			0
$163$	−7.34847			−0.575577			−0.287788	−	0.957694i	$-0.592920\pi$
$163$	−7.34847			−0.575577			−0.287788	+	0.957694i	$0.592920\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	17.1464i		−	1.32683i	−0.748251	−	0.663415i	$-0.769106\pi$
$167$		−	17.1464i		−	1.32683i	0.748251	−	0.663415i	$-0.230894\pi$
$168$		0			0
$169$	9.00000			0.692308
$170$		0			0
$171$	−14.6969			−1.12390
$172$		0			0
$173$	18.0000			1.36851			0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000			1.36851			0.684257	+	0.729241i	$0.260127\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	12.0000	+	12.0000i	0.901975	+	0.901975i
$178$		0			0
$179$	9.79796			0.732334			0.366167	−	0.930549i	$-0.380670\pi$
$179$	9.79796			0.732334			0.366167	+	0.930549i	$0.380670\pi$
$180$		0			0
$181$	10.0000			0.743294			0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000			0.743294			0.371647	+	0.928374i	$0.378793\pi$
$182$		0			0
$183$	9.79796	+	9.79796i	0.724286	+	0.724286i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	29.3939			2.14949
$188$		0			0
$189$	9.00000	−	9.00000i	0.654654	−	0.654654i
$190$		0			0
$191$	−14.6969			−1.06343			−0.531717	−	0.846922i	$-0.678453\pi$
$191$	−14.6969			−1.06343			−0.531717	+	0.846922i	$0.678453\pi$
$192$		0			0
$193$		−	10.0000i		−	0.719816i	−0.932988	−	0.359908i	$-0.882808\pi$
$193$		−	10.0000i		−	0.719816i	0.932988	−	0.359908i	$-0.117192\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	18.0000			1.28245			0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000			1.28245			0.641223	+	0.767354i	$0.278427\pi$
$198$		0			0
$199$		−	14.6969i		−	1.04184i	−0.853606	−	0.520919i	$-0.825589\pi$
$199$		−	14.6969i		−	1.04184i	0.853606	−	0.520919i	$-0.174411\pi$
$200$		0			0
$201$	−9.00000	−	9.00000i	−0.634811	−	0.634811i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	7.34847			0.510754
$208$		0			0
$209$			24.0000i			1.66011i
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$	6.00000	+	6.00000i	0.411113	+	0.411113i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	24.0000i		−	1.62923i
$218$		0			0
$219$	17.1464	−	17.1464i	1.15865	−	1.15865i
$220$		0			0
$221$		−	12.0000i		−	0.807207i
$222$		0			0
$223$	−2.44949			−0.164030			−0.0820150	−	0.996631i	$-0.526136\pi$
$223$	−2.44949			−0.164030			−0.0820150	+	0.996631i	$0.526136\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			2.44949i			0.162578i	0.996691	+	0.0812892i	$0.0259037\pi$
$227$			2.44949i			0.162578i	−0.996691	+	0.0812892i	$0.974096\pi$
$228$		0			0
$229$	−22.0000			−1.45380			−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000			−1.45380			−0.726900	+	0.686743i	$0.759040\pi$
$230$		0			0
$231$	−14.6969	−	14.6969i	−0.966988	−	0.966988i
$232$		0			0
$233$	−18.0000			−1.17922			−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000			−1.17922			−0.589610	+	0.807688i	$0.700718\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	6.00000	−	6.00000i	0.389742	−	0.389742i
$238$		0			0
$239$	−9.79796			−0.633777			−0.316889	−	0.948463i	$-0.602638\pi$
$239$	−9.79796			−0.633777			−0.316889	+	0.948463i	$0.602638\pi$
$240$		0			0
$241$	4.00000			0.257663			0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000			0.257663			0.128831	+	0.991667i	$0.458877\pi$
$242$		0			0
$243$	−11.0227	−	11.0227i	−0.707107	−	0.707107i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	9.79796			0.623429
$248$		0			0
$249$	−9.00000	+	9.00000i	−0.570352	+	0.570352i
$250$		0			0
$251$	−24.4949			−1.54610			−0.773052	−	0.634343i	$-0.781271\pi$
$251$	−24.4949			−1.54610			−0.773052	+	0.634343i	$0.781271\pi$
$252$		0			0
$253$		−	12.0000i		−	0.754434i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	6.00000			0.374270			0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000			0.374270			0.187135	+	0.982334i	$0.440080\pi$
$258$		0			0
$259$			4.89898i			0.304408i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	2.44949i		−	0.151042i	−0.997144	−	0.0755210i	$-0.975938\pi$
$263$		−	2.44949i		−	0.151042i	0.997144	−	0.0755210i	$-0.0240620\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	14.6969	−	14.6969i	0.899438	−	0.899438i
$268$		0			0
$269$		−	18.0000i		−	1.09748i	−0.835993	−	0.548740i	$-0.815108\pi$
$269$		−	18.0000i		−	1.09748i	0.835993	−	0.548740i	$-0.184892\pi$
$270$		0			0
$271$		−	9.79796i		−	0.595184i	−0.954693	−	0.297592i	$-0.903817\pi$
$271$		−	9.79796i		−	0.595184i	0.954693	−	0.297592i	$-0.0961834\pi$
$272$		0			0
$273$	−6.00000	+	6.00000i	−0.363137	+	0.363137i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			2.00000i			0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$			2.00000i			0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$		0			0
$279$	−29.3939			−1.75977
$280$		0			0
$281$		−	6.00000i		−	0.357930i	−0.983855	−	0.178965i	$-0.942725\pi$
$281$		−	6.00000i		−	0.357930i	0.983855	−	0.178965i	$-0.0572749\pi$
$282$		0			0
$283$	−26.9444			−1.60168			−0.800839	−	0.598880i	$-0.795613\pi$
$283$	−26.9444			−1.60168			−0.800839	+	0.598880i	$0.795613\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	14.6969i		−	0.867533i
$288$		0			0
$289$	19.0000			1.11765
$290$		0			0
$291$	12.2474	−	12.2474i	0.717958	−	0.717958i
$292$		0			0
$293$	6.00000			0.350524			0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000			0.350524			0.175262	+	0.984522i	$0.443923\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−18.0000	+	18.0000i	−1.04447	+	1.04447i
$298$		0			0
$299$	−4.89898			−0.283315
$300$		0			0
$301$	6.00000			0.345834
$302$		0			0
$303$	14.6969	−	14.6969i	0.844317	−	0.844317i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−2.44949			−0.139800			−0.0698999	−	0.997554i	$-0.522268\pi$
$307$	−2.44949			−0.139800			−0.0698999	+	0.997554i	$0.522268\pi$
$308$		0			0
$309$	15.0000	+	15.0000i	0.853320	+	0.853320i
$310$		0			0
$311$	−24.4949			−1.38898			−0.694489	−	0.719503i	$-0.744370\pi$
$311$	−24.4949			−1.38898			−0.694489	+	0.719503i	$0.744370\pi$
$312$		0			0
$313$		−	10.0000i		−	0.565233i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$		−	10.0000i		−	0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−18.0000			−1.01098			−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000			−1.01098			−0.505490	+	0.862832i	$0.668688\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	3.00000	−	3.00000i	0.167444	−	0.167444i
$322$		0			0
$323$			29.3939i			1.63552i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−4.89898	−	4.89898i	−0.270914	−	0.270914i
$328$		0			0
$329$		−	30.0000i		−	1.65395i
$330$		0			0
$331$		−	19.5959i		−	1.07709i	−0.842597	−	0.538545i	$-0.818974\pi$
$331$		−	19.5959i		−	1.07709i	0.842597	−	0.538545i	$-0.181026\pi$
$332$		0			0
$333$	6.00000			0.328798
$334$		0			0
$335$		0			0
$336$		0			0
$337$			14.0000i			0.762629i	0.924445	+	0.381314i	$0.124528\pi$
$337$			14.0000i			0.762629i	−0.924445	+	0.381314i	$0.875472\pi$
$338$		0			0
$339$	7.34847	+	7.34847i	0.399114	+	0.399114i
$340$		0			0
$341$			48.0000i			2.59935i
$342$		0			0
$343$	19.5959			1.05808
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	7.34847i		−	0.394486i	−0.980355	−	0.197243i	$-0.936801\pi$
$347$		−	7.34847i		−	0.394486i	0.980355	−	0.197243i	$-0.0631989\pi$
$348$		0			0
$349$	−2.00000			−0.107058			−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000			−0.107058			−0.0535288	+	0.998566i	$0.517047\pi$
$350$		0			0
$351$	7.34847	+	7.34847i	0.392232	+	0.392232i
$352$		0			0
$353$	−6.00000			−0.319348			−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000			−0.319348			−0.159674	+	0.987170i	$0.551044\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−18.0000	−	18.0000i	−0.952661	−	0.952661i
$358$		0			0
$359$	−9.79796			−0.517116			−0.258558	−	0.965996i	$-0.583247\pi$
$359$	−9.79796			−0.517116			−0.258558	+	0.965996i	$0.583247\pi$
$360$		0			0
$361$	−5.00000			−0.263158
$362$		0			0
$363$	15.9217	+	15.9217i	0.835672	+	0.835672i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	7.34847			0.383587			0.191793	−	0.981435i	$-0.438570\pi$
$367$	7.34847			0.383587			0.191793	+	0.981435i	$0.438570\pi$
$368$		0			0
$369$	−18.0000			−0.937043
$370$		0			0
$371$	14.6969			0.763027
$372$		0			0
$373$			26.0000i			1.34623i	0.739538	+	0.673114i	$0.235044\pi$
$373$			26.0000i			1.34623i	−0.739538	+	0.673114i	$0.764956\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			14.6969i			0.754931i	0.926024	+	0.377466i	$0.123204\pi$
$379$			14.6969i			0.754931i	−0.926024	+	0.377466i	$0.876796\pi$
$380$		0			0
$381$	−9.00000	−	9.00000i	−0.461084	−	0.461084i
$382$		0			0
$383$			2.44949i			0.125163i	0.998040	+	0.0625815i	$0.0199333\pi$
$383$			2.44949i			0.125163i	−0.998040	+	0.0625815i	$0.980067\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	7.34847i		−	0.373544i
$388$		0			0
$389$		−	18.0000i		−	0.912636i	−0.889817	−	0.456318i	$-0.849168\pi$
$389$		−	18.0000i		−	0.912636i	0.889817	−	0.456318i	$-0.150832\pi$
$390$		0			0
$391$		−	14.6969i		−	0.743256i
$392$		0			0
$393$	6.00000	+	6.00000i	0.302660	+	0.302660i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	2.00000i		−	0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$		−	2.00000i		−	0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$		0			0
$399$	14.6969	−	14.6969i	0.735767	−	0.735767i
$400$		0			0
$401$		−	24.0000i		−	1.19850i	−0.800561	−	0.599251i	$-0.795465\pi$
$401$		−	24.0000i		−	1.19850i	0.800561	−	0.599251i	$-0.204535\pi$
$402$		0			0
$403$	19.5959			0.976142
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	9.79796i		−	0.485667i
$408$		0			0
$409$	−16.0000			−0.791149			−0.395575	−	0.918434i	$-0.629455\pi$
$409$	−16.0000			−0.791149			−0.395575	+	0.918434i	$0.629455\pi$
$410$		0			0
$411$	−7.34847	−	7.34847i	−0.362473	−	0.362473i
$412$		0			0
$413$	−24.0000			−1.18096
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−6.00000	+	6.00000i	−0.293821	+	0.293821i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−8.00000			−0.389896			−0.194948	−	0.980814i	$-0.562454\pi$
$421$	−8.00000			−0.389896			−0.194948	+	0.980814i	$0.562454\pi$
$422$		0			0
$423$	−36.7423			−1.78647
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−19.5959			−0.948313
$428$		0			0
$429$	12.0000	−	12.0000i	0.579365	−	0.579365i
$430$		0			0
$431$	24.4949			1.17988			0.589939	−	0.807448i	$-0.299152\pi$
$431$	24.4949			1.17988			0.589939	+	0.807448i	$0.299152\pi$
$432$		0			0
$433$			2.00000i			0.0961139i	0.998845	+	0.0480569i	$0.0153029\pi$
$433$			2.00000i			0.0961139i	−0.998845	+	0.0480569i	$0.984697\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	12.0000			0.574038
$438$		0			0
$439$		−	24.4949i		−	1.16908i	−0.811366	−	0.584539i	$-0.801275\pi$
$439$		−	24.4949i		−	1.16908i	0.811366	−	0.584539i	$-0.198725\pi$
$440$		0			0
$441$		−	3.00000i		−	0.142857i
$442$		0			0
$443$		−	17.1464i		−	0.814651i	−0.913283	−	0.407326i	$-0.866461\pi$
$443$		−	17.1464i		−	0.814651i	0.913283	−	0.407326i	$-0.133539\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−22.0454	+	22.0454i	−1.04271	+	1.04271i
$448$		0			0
$449$			6.00000i			0.283158i	0.989927	+	0.141579i	$0.0452178\pi$
$449$			6.00000i			0.283158i	−0.989927	+	0.141579i	$0.954782\pi$
$450$		0			0
$451$			29.3939i			1.38410i
$452$		0			0
$453$	24.0000	−	24.0000i	1.12762	−	1.12762i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	22.0000i		−	1.02912i	−0.857455	−	0.514558i	$-0.827956\pi$
$457$		−	22.0000i		−	1.02912i	0.857455	−	0.514558i	$-0.172044\pi$
$458$		0			0
$459$	−22.0454	+	22.0454i	−1.02899	+	1.02899i
$460$		0			0
$461$			12.0000i			0.558896i	0.960161	+	0.279448i	$0.0901514\pi$
$461$			12.0000i			0.558896i	−0.960161	+	0.279448i	$0.909849\pi$
$462$		0			0
$463$	−2.44949			−0.113837			−0.0569187	−	0.998379i	$-0.518128\pi$
$463$	−2.44949			−0.113837			−0.0569187	+	0.998379i	$0.518128\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			17.1464i			0.793442i	0.917939	+	0.396721i	$0.129852\pi$
$467$			17.1464i			0.793442i	−0.917939	+	0.396721i	$0.870148\pi$
$468$		0			0
$469$	18.0000			0.831163
$470$		0			0
$471$	−12.2474	+	12.2474i	−0.564333	+	0.564333i
$472$		0			0
$473$	−12.0000			−0.551761
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	18.0000i		−	0.824163i
$478$		0			0
$479$	29.3939			1.34304			0.671520	−	0.740986i	$-0.265642\pi$
$479$	29.3939			1.34304			0.671520	+	0.740986i	$0.265642\pi$
$480$		0			0
$481$	−4.00000			−0.182384
$482$		0			0
$483$	−7.34847	+	7.34847i	−0.334367	+	0.334367i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	22.0454			0.998973			0.499486	−	0.866322i	$-0.333522\pi$
$487$	22.0454			0.998973			0.499486	+	0.866322i	$0.333522\pi$
$488$		0			0
$489$	−9.00000	−	9.00000i	−0.406994	−	0.406994i
$490$		0			0
$491$	14.6969			0.663264			0.331632	−	0.943409i	$-0.392401\pi$
$491$	14.6969			0.663264			0.331632	+	0.943409i	$0.392401\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−12.0000			−0.538274
$498$		0			0
$499$		−	4.89898i		−	0.219308i	−0.993970	−	0.109654i	$-0.965026\pi$
$499$		−	4.89898i		−	0.219308i	0.993970	−	0.109654i	$-0.0349744\pi$
$500$		0			0
$501$	21.0000	−	21.0000i	0.938211	−	0.938211i
$502$		0			0
$503$		−	36.7423i		−	1.63826i	−0.573608	−	0.819130i	$-0.694457\pi$
$503$		−	36.7423i		−	1.63826i	0.573608	−	0.819130i	$-0.305543\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	11.0227	+	11.0227i	0.489535	+	0.489535i
$508$		0			0
$509$			24.0000i			1.06378i	0.846813	+	0.531891i	$0.178518\pi$
$509$			24.0000i			1.06378i	−0.846813	+	0.531891i	$0.821482\pi$
$510$		0			0
$511$			34.2929i			1.51703i
$512$		0			0
$513$	−18.0000	−	18.0000i	−0.794719	−	0.794719i
$514$		0			0
$515$		0			0
$516$		0			0
$517$			60.0000i			2.63880i
$518$		0			0
$519$	22.0454	+	22.0454i	0.967686	+	0.967686i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	−2.44949			−0.107109			−0.0535544	−	0.998565i	$-0.517055\pi$
$523$	−2.44949			−0.107109			−0.0535544	+	0.998565i	$0.517055\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			58.7878i			2.56083i
$528$		0			0
$529$	17.0000			0.739130
$530$		0			0
$531$			29.3939i			1.27559i
$532$		0			0
$533$	12.0000			0.519778
$534$		0			0
$535$		0			0
$536$		0			0
$537$	12.0000	+	12.0000i	0.517838	+	0.517838i
$538$		0			0
$539$	−4.89898			−0.211014
$540$		0			0
$541$	−2.00000			−0.0859867			−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000			−0.0859867			−0.0429934	+	0.999075i	$0.513689\pi$
$542$		0			0
$543$	12.2474	+	12.2474i	0.525588	+	0.525588i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−41.6413			−1.78045			−0.890227	−	0.455517i	$-0.849455\pi$
$547$	−41.6413			−1.78045			−0.890227	+	0.455517i	$0.849455\pi$
$548$		0			0
$549$			24.0000i			1.02430i
$550$		0			0
$551$		0			0
$552$		0			0
$553$			12.0000i			0.510292i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	18.0000			0.762684			0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000			0.762684			0.381342	+	0.924434i	$0.375462\pi$
$558$		0			0
$559$			4.89898i			0.207205i
$560$		0			0
$561$	36.0000	+	36.0000i	1.51992	+	1.51992i
$562$		0			0
$563$		−	12.2474i		−	0.516168i	−0.966122	−	0.258084i	$-0.916909\pi$
$563$		−	12.2474i		−	0.516168i	0.966122	−	0.258084i	$-0.0830912\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	22.0454			0.925820
$568$		0			0
$569$			30.0000i			1.25767i	0.777541	+	0.628833i	$0.216467\pi$
$569$			30.0000i			1.25767i	−0.777541	+	0.628833i	$0.783533\pi$
$570$		0			0
$571$			9.79796i			0.410032i	0.978759	+	0.205016i	$0.0657246\pi$
$571$			9.79796i			0.410032i	−0.978759	+	0.205016i	$0.934275\pi$
$572$		0			0
$573$	−18.0000	−	18.0000i	−0.751961	−	0.751961i
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	38.0000i		−	1.58196i	−0.611842	−	0.790980i	$-0.709571\pi$
$577$		−	38.0000i		−	1.58196i	0.611842	−	0.790980i	$-0.290429\pi$
$578$		0			0
$579$	12.2474	−	12.2474i	0.508987	−	0.508987i
$580$		0			0
$581$		−	18.0000i		−	0.746766i
$582$		0			0
$583$	−29.3939			−1.21737
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	36.7423i		−	1.51652i	−0.651953	−	0.758259i	$-0.726050\pi$
$587$		−	36.7423i		−	1.51652i	0.651953	−	0.758259i	$-0.273950\pi$
$588$		0			0
$589$	−48.0000			−1.97781
$590$		0			0
$591$	22.0454	+	22.0454i	0.906827	+	0.906827i
$592$		0			0
$593$	42.0000			1.72473			0.862367	−	0.506284i	$-0.168981\pi$
$593$	42.0000			1.72473			0.862367	+	0.506284i	$0.168981\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	18.0000	−	18.0000i	0.736691	−	0.736691i
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−28.0000			−1.14214			−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000			−1.14214			−0.571072	+	0.820900i	$0.693472\pi$
$602$		0			0
$603$		−	22.0454i		−	0.897758i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−26.9444			−1.09364			−0.546819	−	0.837251i	$-0.684162\pi$
$607$	−26.9444			−1.09364			−0.546819	+	0.837251i	$0.684162\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	24.4949			0.990957
$612$		0			0
$613$			38.0000i			1.53481i	0.641165	+	0.767403i	$0.278451\pi$
$613$			38.0000i			1.53481i	−0.641165	+	0.767403i	$0.721549\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	6.00000			0.241551			0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000			0.241551			0.120775	+	0.992680i	$0.461462\pi$
$618$		0			0
$619$			4.89898i			0.196907i	0.995142	+	0.0984533i	$0.0313895\pi$
$619$			4.89898i			0.196907i	−0.995142	+	0.0984533i	$0.968610\pi$
$620$		0			0
$621$	9.00000	+	9.00000i	0.361158	+	0.361158i
$622$		0			0
$623$			29.3939i			1.17764i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−29.3939	+	29.3939i	−1.17388	+	1.17388i
$628$		0			0
$629$		−	12.0000i		−	0.478471i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$			2.00000i			0.0792429i
$638$		0			0
$639$			14.6969i			0.581402i
$640$		0			0
$641$		−	18.0000i		−	0.710957i	−0.934684	−	0.355479i	$-0.884318\pi$
$641$		−	18.0000i		−	0.710957i	0.934684	−	0.355479i	$-0.115682\pi$
$642$		0			0
$643$	36.7423			1.44898			0.724488	−	0.689287i	$-0.242076\pi$
$643$	36.7423			1.44898			0.724488	+	0.689287i	$0.242076\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			26.9444i			1.05929i	0.848218	+	0.529647i	$0.177675\pi$
$647$			26.9444i			1.05929i	−0.848218	+	0.529647i	$0.822325\pi$
$648$		0			0
$649$	48.0000			1.88416
$650$		0			0
$651$	29.3939	−	29.3939i	1.15204	−	1.15204i
$652$		0			0
$653$	18.0000			0.704394			0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000			0.704394			0.352197	+	0.935926i	$0.385435\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	42.0000			1.63858
$658$		0			0
$659$	29.3939			1.14502			0.572511	−	0.819897i	$-0.305969\pi$
$659$	29.3939			1.14502			0.572511	+	0.819897i	$0.305969\pi$
$660$		0			0
$661$	−40.0000			−1.55582			−0.777910	−	0.628376i	$-0.783720\pi$
$661$	−40.0000			−1.55582			−0.777910	+	0.628376i	$0.783720\pi$
$662$		0			0
$663$	14.6969	−	14.6969i	0.570782	−	0.570782i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−3.00000	−	3.00000i	−0.115987	−	0.115987i
$670$		0			0
$671$	39.1918			1.51298
$672$		0			0
$673$			26.0000i			1.00223i	0.865382	+	0.501113i	$0.167076\pi$
$673$			26.0000i			1.00223i	−0.865382	+	0.501113i	$0.832924\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	18.0000			0.691796			0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000			0.691796			0.345898	+	0.938272i	$0.387574\pi$
$678$		0			0
$679$			24.4949i			0.940028i
$680$		0			0
$681$	−3.00000	+	3.00000i	−0.114960	+	0.114960i
$682$		0			0
$683$			31.8434i			1.21845i	0.792996	+	0.609226i	$0.208520\pi$
$683$			31.8434i			1.21845i	−0.792996	+	0.609226i	$0.791480\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−26.9444	−	26.9444i	−1.02799	−	1.02799i
$688$		0			0
$689$			12.0000i			0.457164i
$690$		0			0
$691$			29.3939i			1.11820i	0.829102	+	0.559098i	$0.188852\pi$
$691$			29.3939i			1.11820i	−0.829102	+	0.559098i	$0.811148\pi$
$692$		0			0
$693$		−	36.0000i		−	1.36753i
$694$		0			0
$695$		0			0
$696$		0			0
$697$			36.0000i			1.36360i
$698$		0			0
$699$	−22.0454	−	22.0454i	−0.833834	−	0.833834i
$700$		0			0
$701$		−	30.0000i		−	1.13308i	−0.824033	−	0.566542i	$-0.808281\pi$
$701$		−	30.0000i		−	1.13308i	0.824033	−	0.566542i	$-0.191719\pi$
$702$		0			0
$703$	9.79796			0.369537
$704$		0			0
$705$		0			0
$706$		0			0
$707$			29.3939i			1.10547i
$708$		0			0
$709$	−10.0000			−0.375558			−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000			−0.375558			−0.187779	+	0.982211i	$0.560129\pi$
$710$		0			0
$711$	14.6969			0.551178
$712$		0			0
$713$	24.0000			0.898807
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−12.0000	−	12.0000i	−0.448148	−	0.448148i
$718$		0			0
$719$	−9.79796			−0.365402			−0.182701	−	0.983169i	$-0.558484\pi$
$719$	−9.79796			−0.365402			−0.182701	+	0.983169i	$0.558484\pi$
$720$		0			0
$721$	−30.0000			−1.11726
$722$		0			0
$723$	4.89898	+	4.89898i	0.182195	+	0.182195i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−26.9444			−0.999312			−0.499656	−	0.866224i	$-0.666540\pi$
$727$	−26.9444			−0.999312			−0.499656	+	0.866224i	$0.666540\pi$
$728$		0			0
$729$		−	27.0000i		−	1.00000i
$730$		0			0
$731$	−14.6969			−0.543586
$732$		0			0
$733$			34.0000i			1.25582i	0.778287	+	0.627909i	$0.216089\pi$
$733$			34.0000i			1.25582i	−0.778287	+	0.627909i	$0.783911\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−36.0000			−1.32608
$738$		0			0
$739$			14.6969i			0.540636i	0.962771	+	0.270318i	$0.0871288\pi$
$739$			14.6969i			0.540636i	−0.962771	+	0.270318i	$0.912871\pi$
$740$		0			0
$741$	12.0000	+	12.0000i	0.440831	+	0.440831i
$742$		0			0
$743$			41.6413i			1.52767i	0.645410	+	0.763836i	$0.276686\pi$
$743$			41.6413i			1.52767i	−0.645410	+	0.763836i	$0.723314\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−22.0454			−0.806599
$748$		0			0
$749$			6.00000i			0.219235i
$750$		0			0
$751$			19.5959i			0.715065i	0.933901	+	0.357533i	$0.116382\pi$
$751$			19.5959i			0.715065i	−0.933901	+	0.357533i	$0.883618\pi$
$752$		0			0
$753$	−30.0000	−	30.0000i	−1.09326	−	1.09326i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			2.00000i			0.0726912i	0.999339	+	0.0363456i	$0.0115717\pi$
$757$			2.00000i			0.0726912i	−0.999339	+	0.0363456i	$0.988428\pi$
$758$		0			0
$759$	14.6969	−	14.6969i	0.533465	−	0.533465i
$760$		0			0
$761$			48.0000i			1.74000i	0.493053	+	0.869999i	$0.335881\pi$
$761$			48.0000i			1.74000i	−0.493053	+	0.869999i	$0.664119\pi$
$762$		0			0
$763$	9.79796			0.354710
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	19.5959i		−	0.707568i
$768$		0			0
$769$	−14.0000			−0.504853			−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000			−0.504853			−0.252426	+	0.967616i	$0.581229\pi$
$770$		0			0
$771$	7.34847	+	7.34847i	0.264649	+	0.264649i
$772$		0			0
$773$	−42.0000			−1.51064			−0.755318	−	0.655359i	$-0.772517\pi$
$773$	−42.0000			−1.51064			−0.755318	+	0.655359i	$0.772517\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−6.00000	+	6.00000i	−0.215249	+	0.215249i
$778$		0			0
$779$	−29.3939			−1.05314
$780$		0			0
$781$	24.0000			0.858788
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	46.5403			1.65898			0.829491	−	0.558520i	$-0.188630\pi$
$787$	46.5403			1.65898			0.829491	+	0.558520i	$0.188630\pi$
$788$		0			0
$789$	3.00000	−	3.00000i	0.106803	−	0.106803i
$790$		0			0
$791$	−14.6969			−0.522563
$792$		0			0
$793$		−	16.0000i		−	0.568177i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−42.0000			−1.48772			−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000			−1.48772			−0.743858	+	0.668338i	$0.767006\pi$
$798$		0			0
$799$			73.4847i			2.59970i
$800$		0			0
$801$	36.0000			1.27200
$802$		0			0
$803$		−	68.5857i		−	2.42034i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	22.0454	−	22.0454i	0.776035	−	0.776035i
$808$		0			0
$809$			12.0000i			0.421898i	0.977497	+	0.210949i	$0.0676553\pi$
$809$			12.0000i			0.421898i	−0.977497	+	0.210949i	$0.932345\pi$
$810$		0			0
$811$		−	39.1918i		−	1.37621i	−0.725610	−	0.688106i	$-0.758443\pi$
$811$		−	39.1918i		−	1.37621i	0.725610	−	0.688106i	$-0.241557\pi$
$812$		0			0
$813$	12.0000	−	12.0000i	0.420858	−	0.420858i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	12.0000i		−	0.419827i
$818$		0			0
$819$	−14.6969			−0.513553
$820$		0			0
$821$		−	6.00000i		−	0.209401i	−0.994504	−	0.104701i	$-0.966612\pi$
$821$		−	6.00000i		−	0.209401i	0.994504	−	0.104701i	$-0.0333885\pi$
$822$		0			0
$823$	−36.7423			−1.28076			−0.640379	−	0.768059i	$-0.721223\pi$
$823$	−36.7423			−1.28076			−0.640379	+	0.768059i	$0.721223\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			17.1464i			0.596240i	0.954528	+	0.298120i	$0.0963595\pi$
$827$			17.1464i			0.596240i	−0.954528	+	0.298120i	$0.903640\pi$
$828$		0			0
$829$	20.0000			0.694629			0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000			0.694629			0.347314	+	0.937749i	$0.387094\pi$
$830$		0			0
$831$	−2.44949	+	2.44949i	−0.0849719	+	0.0849719i
$832$		0			0
$833$	−6.00000			−0.207888
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−36.0000	−	36.0000i	−1.24434	−	1.24434i
$838$		0			0
$839$	19.5959			0.676526			0.338263	−	0.941052i	$-0.390161\pi$
$839$	19.5959			0.676526			0.338263	+	0.941052i	$0.390161\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$	7.34847	−	7.34847i	0.253095	−	0.253095i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−31.8434			−1.09415
$848$		0			0
$849$	−33.0000	−	33.0000i	−1.13256	−	1.13256i
$850$		0			0
$851$	−4.89898			−0.167935
$852$		0			0
$853$			14.0000i			0.479351i	0.970853	+	0.239675i	$0.0770410\pi$
$853$			14.0000i			0.479351i	−0.970853	+	0.239675i	$0.922959\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−6.00000			−0.204956			−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000			−0.204956			−0.102478	+	0.994735i	$0.532677\pi$
$858$		0			0
$859$		−	53.8888i		−	1.83866i	−0.393486	−	0.919331i	$-0.628731\pi$
$859$		−	53.8888i		−	1.83866i	0.393486	−	0.919331i	$-0.371269\pi$
$860$		0			0
$861$	18.0000	−	18.0000i	0.613438	−	0.613438i
$862$		0			0
$863$		−	7.34847i		−	0.250145i	−0.992148	−	0.125072i	$-0.960084\pi$
$863$		−	7.34847i		−	0.250145i	0.992148	−	0.125072i	$-0.0399163\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	23.2702	+	23.2702i	0.790296	+	0.790296i
$868$		0			0
$869$		−	24.0000i		−	0.814144i
$870$		0			0
$871$			14.6969i			0.497987i
$872$		0			0
$873$	30.0000			1.01535
$874$		0			0
$875$		0			0
$876$		0			0
$877$			58.0000i			1.95852i	0.202606	+	0.979260i	$0.435059\pi$
$877$			58.0000i			1.95852i	−0.202606	+	0.979260i	$0.564941\pi$
$878$		0			0
$879$	7.34847	+	7.34847i	0.247858	+	0.247858i
$880$		0			0
$881$		−	18.0000i		−	0.606435i	−0.952921	−	0.303218i	$-0.901939\pi$
$881$		−	18.0000i		−	0.606435i	0.952921	−	0.303218i	$-0.0980609\pi$
$882$		0			0
$883$	−56.3383			−1.89593			−0.947967	−	0.318369i	$-0.896865\pi$
$883$	−56.3383			−1.89593			−0.947967	+	0.318369i	$0.896865\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	41.6413i		−	1.39818i	−0.715034	−	0.699089i	$-0.753589\pi$
$887$		−	41.6413i		−	1.39818i	0.715034	−	0.699089i	$-0.246411\pi$
$888$		0			0
$889$	18.0000			0.603701
$890$		0			0
$891$	−44.0908			−1.47710
$892$		0			0
$893$	−60.0000			−2.00782
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−6.00000	−	6.00000i	−0.200334	−	0.200334i
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−36.0000			−1.19933
$902$		0			0
$903$	7.34847	+	7.34847i	0.244542	+	0.244542i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	12.2474			0.406670			0.203335	−	0.979109i	$-0.434822\pi$
$907$	12.2474			0.406670			0.203335	+	0.979109i	$0.434822\pi$
$908$		0			0
$909$	36.0000			1.19404
$910$		0			0
$911$	24.4949			0.811552			0.405776	−	0.913973i	$-0.367001\pi$
$911$	24.4949			0.811552			0.405776	+	0.913973i	$0.367001\pi$
$912$		0			0
$913$			36.0000i			1.19143i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−12.0000			−0.396275
$918$		0			0
$919$		−	34.2929i		−	1.13122i	−0.824674	−	0.565608i	$-0.808641\pi$
$919$		−	34.2929i		−	1.13122i	0.824674	−	0.565608i	$-0.191359\pi$
$920$		0			0
$921$	−3.00000	−	3.00000i	−0.0988534	−	0.0988534i
$922$		0			0
$923$		−	9.79796i		−	0.322504i
$924$		0			0
$925$		0			0
$926$		0			0
$927$			36.7423i			1.20678i
$928$		0			0
$929$		−	54.0000i		−	1.77168i	−0.463988	−	0.885841i	$-0.653582\pi$
$929$		−	54.0000i		−	1.77168i	0.463988	−	0.885841i	$-0.346418\pi$
$930$		0			0
$931$		−	4.89898i		−	0.160558i
$932$		0			0
$933$	−30.0000	−	30.0000i	−0.982156	−	0.982156i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	38.0000i		−	1.24141i	−0.784046	−	0.620703i	$-0.786847\pi$
$937$		−	38.0000i		−	1.24141i	0.784046	−	0.620703i	$-0.213153\pi$
$938$		0			0
$939$	12.2474	−	12.2474i	0.399680	−	0.399680i
$940$		0			0
$941$			12.0000i			0.391189i	0.980685	+	0.195594i	$0.0626636\pi$
$941$			12.0000i			0.391189i	−0.980685	+	0.195594i	$0.937336\pi$
$942$		0			0
$943$	14.6969			0.478598
$944$		0			0
$945$		0			0
$946$		0			0
$947$			22.0454i			0.716379i	0.933649	+	0.358190i	$0.116606\pi$
$947$			22.0454i			0.716379i	−0.933649	+	0.358190i	$0.883394\pi$
$948$		0			0
$949$	−28.0000			−0.908918
$950$		0			0
$951$	−22.0454	−	22.0454i	−0.714871	−	0.714871i
$952$		0			0
$953$	30.0000			0.971795			0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000			0.971795			0.485898	+	0.874016i	$0.338493\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	14.6969			0.474589
$960$		0			0
$961$	−65.0000			−2.09677
$962$		0			0
$963$	7.34847			0.236801
$964$		0			0
$965$		0			0
$966$		0			0
$967$	56.3383			1.81172			0.905858	−	0.423581i	$-0.139227\pi$
$967$	56.3383			1.81172			0.905858	+	0.423581i	$0.139227\pi$
$968$		0			0
$969$	−36.0000	+	36.0000i	−1.15649	+	1.15649i
$970$		0			0
$971$	−53.8888			−1.72937			−0.864687	−	0.502312i	$-0.832483\pi$
$971$	−53.8888			−1.72937			−0.864687	+	0.502312i	$0.832483\pi$
$972$		0			0
$973$		−	12.0000i		−	0.384702i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	18.0000			0.575871			0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000			0.575871			0.287936	+	0.957650i	$0.407031\pi$
$978$		0			0
$979$		−	58.7878i		−	1.87886i
$980$		0			0
$981$		−	12.0000i		−	0.383131i
$982$		0			0
$983$			31.8434i			1.01565i	0.861462	+	0.507823i	$0.169550\pi$
$983$			31.8434i			1.01565i	−0.861462	+	0.507823i	$0.830450\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	36.7423	−	36.7423i	1.16952	−	1.16952i
$988$		0			0
$989$			6.00000i			0.190789i
$990$		0			0
$991$		−	9.79796i		−	0.311242i	−0.987817	−	0.155621i	$-0.950262\pi$
$991$		−	9.79796i		−	0.311242i	0.987817	−	0.155621i	$-0.0497379\pi$
$992$		0			0
$993$	24.0000	−	24.0000i	0.761617	−	0.761617i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	14.0000i		−	0.443384i	−0.975117	−	0.221692i	$-0.928842\pi$
$997$		−	14.0000i		−	0.443384i	0.975117	−	0.221692i	$-0.0711580\pi$
$998$		0			0
$999$	7.34847	+	7.34847i	0.232495	+	0.232495i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.2.o.f.1199.4		4
3.2	odd	2		1200.2.o.e.1199.3		4
4.3	odd	2	inner	1200.2.o.f.1199.1		4
5.2	odd	4		240.2.h.a.191.3	yes	4
5.3	odd	4		1200.2.h.k.1151.2		4
5.4	even	2		1200.2.o.e.1199.1		4
12.11	even	2		1200.2.o.e.1199.2		4
15.2	even	4		240.2.h.a.191.1	✓	4
15.8	even	4		1200.2.h.k.1151.4		4
15.14	odd	2	inner	1200.2.o.f.1199.2		4
20.3	even	4		1200.2.h.k.1151.3		4
20.7	even	4		240.2.h.a.191.2	yes	4
20.19	odd	2		1200.2.o.e.1199.4		4
40.27	even	4		960.2.h.c.191.3		4
40.37	odd	4		960.2.h.c.191.2		4
60.23	odd	4		1200.2.h.k.1151.1		4
60.47	odd	4		240.2.h.a.191.4	yes	4
60.59	even	2	inner	1200.2.o.f.1199.3		4
120.77	even	4		960.2.h.c.191.4		4
120.107	odd	4		960.2.h.c.191.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.2.h.a.191.1	✓	4	15.2	even	4
240.2.h.a.191.2	yes	4	20.7	even	4
240.2.h.a.191.3	yes	4	5.2	odd	4
240.2.h.a.191.4	yes	4	60.47	odd	4
960.2.h.c.191.1		4	120.107	odd	4
960.2.h.c.191.2		4	40.37	odd	4
960.2.h.c.191.3		4	40.27	even	4
960.2.h.c.191.4		4	120.77	even	4
1200.2.h.k.1151.1		4	60.23	odd	4
1200.2.h.k.1151.2		4	5.3	odd	4
1200.2.h.k.1151.3		4	20.3	even	4
1200.2.h.k.1151.4		4	15.8	even	4
1200.2.o.e.1199.1		4	5.4	even	2
1200.2.o.e.1199.2		4	12.11	even	2
1200.2.o.e.1199.3		4	3.2	odd	2
1200.2.o.e.1199.4		4	20.19	odd	2
1200.2.o.f.1199.1		4	4.3	odd	2	inner
1200.2.o.f.1199.2		4	15.14	odd	2	inner
1200.2.o.f.1199.3		4	60.59	even	2	inner
1200.2.o.f.1199.4		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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1200.o (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 + 1.22474i) q^{3} -2.44949 q^{7} +3.00000i q^{9} +O(q^{10})\)	\(q+(1.22474 + 1.22474i) q^{3} -2.44949 q^{7} +3.00000i q^{9} +4.89898 q^{11} -2.00000i q^{13} +6.00000 q^{17} +4.89898i q^{19} +(-3.00000 - 3.00000i) q^{21} -2.44949i q^{23} +(-3.67423 + 3.67423i) q^{27} +9.79796i q^{31} +(6.00000 + 6.00000i) q^{33} -2.00000i q^{37} +(2.44949 - 2.44949i) q^{39} +6.00000i q^{41} -2.44949 q^{43} +12.2474i q^{47} -1.00000 q^{49} +(7.34847 + 7.34847i) q^{51} -6.00000 q^{53} +(-6.00000 + 6.00000i) q^{57} +9.79796 q^{59} +8.00000 q^{61} -7.34847i q^{63} -7.34847 q^{67} +(3.00000 - 3.00000i) q^{69} +4.89898 q^{71} -14.0000i q^{73} -12.0000 q^{77} -4.89898i q^{79} -9.00000 q^{81} +7.34847i q^{83} -12.0000i q^{89} +4.89898i q^{91} +(-12.0000 + 12.0000i) q^{93} -10.0000i q^{97} +14.6969i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 24 q^{17} - 12 q^{21} + 24 q^{33} - 4 q^{49} - 24 q^{53} - 24 q^{57} + 32 q^{61} + 12 q^{69} - 48 q^{77} - 36 q^{81} - 48 q^{93}+O(q^{100}) \) 4 * q + 24 * q^17 - 12 * q^21 + 24 * q^33 - 4 * q^49 - 24 * q^53 - 24 * q^57 + 32 * q^61 + 12 * q^69 - 48 * q^77 - 36 * q^81 - 48 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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