LMFDB - Embedded newform 2400.2.d.g.49.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,2,Mod(49,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2400.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2400.d (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:	$19.1640964851$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.214798336.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 2x^{5} + 9x^{4} - 4x^{3} - 16x + 16 $ x^8 - 2x^7 - 2x^5 + 9x^4 - 4x^3 - 16*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 600)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.4
Root			$1.23291 + 0.692769i$ of defining polynomial
Character	$\chi$	$=$	2400.49
Dual form			2400.2.d.g.49.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -0.0802864i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.0802864i q^{7} +1.00000 q^{9} +2.41649i q^{11} +5.26785 q^{13} -0.255918i q^{17} +6.95864i q^{19} +0.0802864i q^{21} -1.64542i q^{23} -1.00000 q^{27} -4.51516i q^{29} -8.29484 q^{31} -2.41649i q^{33} -2.67241 q^{37} -5.26785 q^{39} -8.11921 q^{41} +4.08890 q^{43} -5.70272i q^{47} +6.99355 q^{49} +0.255918i q^{51} +11.5627 q^{53} -6.95864i q^{57} +12.6963i q^{59} +11.9403i q^{61} -0.0802864i q^{63} -7.27979 q^{67} +1.64542i q^{69} +11.3481 q^{71} +12.0779i q^{73} +0.194011 q^{77} +5.50539 q^{79} +1.00000 q^{81} +9.20811 q^{83} +4.51516i q^{87} -11.9173 q^{89} -0.422937i q^{91} +8.29484 q^{93} +8.50539i q^{97} +2.41649i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} + 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^3 + 8 * q^9	$ 8 q - 8 q^{3} + 8 q^{9} - 8 q^{27} - 8 q^{31} - 8 q^{43} - 8 q^{53} + 24 q^{67} + 40 q^{71} + 24 q^{77} - 16 q^{79} + 8 q^{81} - 32 q^{83} + 8 q^{93}+O(q^{100}) $ 8 * q - 8 * q^3 + 8 * q^9 - 8 * q^27 - 8 * q^31 - 8 * q^43 - 8 * q^53 + 24 * q^67 + 40 * q^71 + 24 * q^77 - 16 * q^79 + 8 * q^81 - 32 * q^83 + 8 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1601$	$1951$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 0.0802864i	− 0.0303454i	−0.999885	−	0.0151727i	$-0.995170\pi$
$7$	− 0.0802864i	− 0.0303454i	0.999885	−	0.0151727i	$-0.00482980\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.41649i	0.728599i	0.931282	+	0.364300i	$0.118692\pi$
$11$	2.41649i	0.728599i	−0.931282	+	0.364300i	$0.881308\pi$
$12$	0	0
$13$	5.26785	1.46104	0.730520	−	0.682892i	$-0.239278\pi$
$13$	5.26785	1.46104	0.730520	+	0.682892i	$0.239278\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.255918i	− 0.0620692i	−0.999518	−	0.0310346i	$-0.990120\pi$
$17$	− 0.255918i	− 0.0620692i	0.999518	−	0.0310346i	$-0.00988021\pi$
$18$	0	0
$19$	6.95864i	1.59642i	0.602378	+	0.798211i	$0.294220\pi$
$19$	6.95864i	1.59642i	−0.602378	+	0.798211i	$0.705780\pi$
$20$	0	0
$21$	0.0802864i	0.0175199i
$22$	0	0
$23$	− 1.64542i	− 0.343093i	−0.985176	−	0.171546i	$-0.945124\pi$
$23$	− 1.64542i	− 0.343093i	0.985176	−	0.171546i	$-0.0548764\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	− 4.51516i	− 0.838444i	−0.907884	−	0.419222i	$-0.862303\pi$
$29$	− 4.51516i	− 0.838444i	0.907884	−	0.419222i	$-0.137697\pi$
$30$	0	0
$31$	−8.29484	−1.48980	−0.744899	−	0.667177i	$-0.767502\pi$
$31$	−8.29484	−1.48980	−0.744899	+	0.667177i	$0.767502\pi$
$32$	0	0
$33$	− 2.41649i	− 0.420657i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.67241	−0.439341	−0.219671	−	0.975574i	$-0.570498\pi$
$37$	−2.67241	−0.439341	−0.219671	+	0.975574i	$0.570498\pi$
$38$	0	0
$39$	−5.26785	−0.843531
$40$	0	0
$41$	−8.11921	−1.26801	−0.634004	−	0.773330i	$-0.718590\pi$
$41$	−8.11921	−1.26801	−0.634004	+	0.773330i	$0.718590\pi$
$42$	0	0
$43$	4.08890	0.623551	0.311776	−	0.950156i	$-0.399076\pi$
$43$	4.08890	0.623551	0.311776	+	0.950156i	$0.399076\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 5.70272i	− 0.831827i	−0.909404	−	0.415914i	$-0.863462\pi$
$47$	− 5.70272i	− 0.831827i	0.909404	−	0.415914i	$-0.136538\pi$
$48$	0	0
$49$	6.99355	0.999079
$50$	0	0
$51$	0.255918i	0.0358357i
$52$	0	0
$53$	11.5627	1.58826	0.794129	−	0.607749i	$-0.207927\pi$
$53$	11.5627	1.58826	0.794129	+	0.607749i	$0.207927\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 6.95864i	− 0.921694i
$58$	0	0
$59$	12.6963i	1.65291i	0.563000	+	0.826457i	$0.309647\pi$
$59$	12.6963i	1.65291i	−0.563000	+	0.826457i	$0.690353\pi$
$60$	0	0
$61$	11.9403i	1.52879i	0.644746	+	0.764397i	$0.276963\pi$
$61$	11.9403i	1.52879i	−0.644746	+	0.764397i	$0.723037\pi$
$62$	0	0
$63$	− 0.0802864i	− 0.0101151i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−7.27979	−0.889367	−0.444684	−	0.895688i	$-0.646684\pi$
$67$	−7.27979	−0.889367	−0.444684	+	0.895688i	$0.646684\pi$
$68$	0	0
$69$	1.64542i	0.198085i
$70$	0	0
$71$	11.3481	1.34678	0.673388	−	0.739289i	$-0.264838\pi$
$71$	11.3481	1.34678	0.673388	+	0.739289i	$0.264838\pi$
$72$	0	0
$73$	12.0779i	1.41361i	0.707411	+	0.706803i	$0.249863\pi$
$73$	12.0779i	1.41361i	−0.707411	+	0.706803i	$0.750137\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.194011	0.0221096
$78$	0	0
$79$	5.50539	0.619405	0.309702	−	0.950834i	$-0.399771\pi$
$79$	5.50539	0.619405	0.309702	+	0.950834i	$0.399771\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.20811	1.01072	0.505361	−	0.862908i	$-0.331359\pi$
$83$	9.20811	1.01072	0.505361	+	0.862908i	$0.331359\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.51516i	0.484076i
$88$	0	0
$89$	−11.9173	−1.26323	−0.631615	−	0.775283i	$-0.717607\pi$
$89$	−11.9173	−1.26323	−0.631615	+	0.775283i	$0.717607\pi$
$90$	0	0
$91$	− 0.422937i	− 0.0443358i
$92$	0	0
$93$	8.29484	0.860135
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.50539i	0.863592i	0.901971	+	0.431796i	$0.142120\pi$
$97$	8.50539i	0.863592i	−0.901971	+	0.431796i	$0.857880\pi$
$98$	0	0
$99$	2.41649i	0.242866i
$100$	0	0
$101$	7.56270i	0.752516i	0.926515	+	0.376258i	$0.122789\pi$
$101$	7.56270i	0.752516i	−0.926515	+	0.376258i	$0.877211\pi$
$102$	0	0
$103$	− 1.78544i	− 0.175925i	−0.996124	−	0.0879624i	$-0.971964\pi$
$103$	− 1.78544i	− 0.175925i	0.996124	−	0.0879624i	$-0.0280355\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.4705	−1.01222	−0.506110	−	0.862469i	$-0.668917\pi$
$107$	−10.4705	−1.01222	−0.506110	+	0.862469i	$0.668917\pi$
$108$	0	0
$109$	3.64298i	0.348934i	0.984663	+	0.174467i	$0.0558203\pi$
$109$	3.64298i	0.348934i	−0.984663	+	0.174467i	$0.944180\pi$
$110$	0	0
$111$	2.67241	0.253654
$112$	0	0
$113$	− 8.83298i	− 0.830937i	−0.909608	−	0.415468i	$-0.863618\pi$
$113$	− 8.83298i	− 0.830937i	0.909608	−	0.415468i	$-0.136382\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	5.26785	0.487013
$118$	0	0
$119$	−0.0205467	−0.00188351
$120$	0	0
$121$	5.16057	0.469143
$122$	0	0
$123$	8.11921	0.732085
$124$	0	0
$125$	0	0
$126$	0	0
$127$	8.69628i	0.771670i	0.922568	+	0.385835i	$0.126087\pi$
$127$	8.69628i	0.771670i	−0.922568	+	0.385835i	$0.873913\pi$
$128$	0	0
$129$	−4.08890	−0.360008
$130$	0	0
$131$	10.7916i	0.942868i	0.881901	+	0.471434i	$0.156264\pi$
$131$	10.7916i	0.942868i	−0.881901	+	0.471434i	$0.843736\pi$
$132$	0	0
$133$	0.558684	0.0484440
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.5421i	0.986112i	0.869997	+	0.493056i	$0.164120\pi$
$137$	11.5421i	0.986112i	−0.869997	+	0.493056i	$0.835880\pi$
$138$	0	0
$139$	0.214558i	0.0181986i	0.999959	+	0.00909928i	$0.00289643\pi$
$139$	0.214558i	0.0181986i	−0.999959	+	0.00909928i	$0.997104\pi$
$140$	0	0
$141$	5.70272i	0.480256i
$142$	0	0
$143$	12.7297i	1.06451i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−6.99355	−0.576819
$148$	0	0
$149$	23.0475i	1.88813i	0.329762	+	0.944064i	$0.393031\pi$
$149$	23.0475i	1.88813i	−0.329762	+	0.944064i	$0.606969\pi$
$150$	0	0
$151$	−9.48573	−0.771938	−0.385969	−	0.922512i	$-0.626133\pi$
$151$	−9.48573	−0.771938	−0.385969	+	0.922512i	$0.626133\pi$
$152$	0	0
$153$	− 0.255918i	− 0.0206897i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	6.34413	0.506316	0.253158	−	0.967425i	$-0.418531\pi$
$157$	6.34413	0.506316	0.253158	+	0.967425i	$0.418531\pi$
$158$	0	0
$159$	−11.5627	−0.916981
$160$	0	0
$161$	−0.132104	−0.0104113
$162$	0	0
$163$	12.4100	0.972030	0.486015	−	0.873951i	$-0.338450\pi$
$163$	12.4100	0.972030	0.486015	+	0.873951i	$0.338450\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 23.2654i	− 1.80033i	−0.435547	−	0.900166i	$-0.643445\pi$
$167$	− 23.2654i	− 1.80033i	0.435547	−	0.900166i	$-0.356555\pi$
$168$	0	0
$169$	14.7503	1.13464
$170$	0	0
$171$	6.95864i	0.532140i
$172$	0	0
$173$	8.63897	0.656809	0.328404	−	0.944537i	$-0.393489\pi$
$173$	8.63897	0.656809	0.328404	+	0.944537i	$0.393489\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 12.6963i	− 0.954311i
$178$	0	0
$179$	− 9.40544i	− 0.702996i	−0.936189	−	0.351498i	$-0.885672\pi$
$179$	− 9.40544i	− 0.702996i	0.936189	−	0.351498i	$-0.114328\pi$
$180$	0	0
$181$	− 6.43487i	− 0.478300i	−0.970983	−	0.239150i	$-0.923131\pi$
$181$	− 6.43487i	− 0.478300i	0.970983	−	0.239150i	$-0.0768688\pi$
$182$	0	0
$183$	− 11.9403i	− 0.882649i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0.618423	0.0452236
$188$	0	0
$189$	0.0802864i	0.00583997i
$190$	0	0
$191$	5.56270	0.402503	0.201251	−	0.979540i	$-0.435499\pi$
$191$	5.56270	0.402503	0.201251	+	0.979540i	$0.435499\pi$
$192$	0	0
$193$	18.4227i	1.32609i	0.748578	+	0.663046i	$0.230737\pi$
$193$	18.4227i	1.32609i	−0.748578	+	0.663046i	$0.769263\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−18.0239	−1.28415	−0.642074	−	0.766643i	$-0.721926\pi$
$197$	−18.0239	−1.28415	−0.642074	+	0.766643i	$0.721926\pi$
$198$	0	0
$199$	20.1214	1.42637	0.713183	−	0.700977i	$-0.247253\pi$
$199$	20.1214	1.42637	0.713183	+	0.700977i	$0.247253\pi$
$200$	0	0
$201$	7.27979	0.513476
$202$	0	0
$203$	−0.362505	−0.0254429
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1.64542i	− 0.114364i
$208$	0	0
$209$	−16.8155	−1.16315
$210$	0	0
$211$	3.25592i	0.224147i	0.993700	+	0.112073i	$0.0357492\pi$
$211$	3.25592i	0.224147i	−0.993700	+	0.112073i	$0.964251\pi$
$212$	0	0
$213$	−11.3481	−0.777562
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0.665963i	0.0452085i
$218$	0	0
$219$	− 12.0779i	− 0.816146i
$220$	0	0
$221$	− 1.34814i	− 0.0906856i
$222$	0	0
$223$	26.9911i	1.80746i	0.428104	+	0.903730i	$0.359182\pi$
$223$	26.9911i	1.80746i	−0.428104	+	0.903730i	$0.640818\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	19.8219	1.31563	0.657814	−	0.753180i	$-0.271481\pi$
$227$	19.8219	1.31563	0.657814	+	0.753180i	$0.271481\pi$
$228$	0	0
$229$	21.6797i	1.43264i	0.697773	+	0.716319i	$0.254174\pi$
$229$	21.6797i	1.43264i	−0.697773	+	0.716319i	$0.745826\pi$
$230$	0	0
$231$	−0.194011	−0.0127650
$232$	0	0
$233$	− 17.2733i	− 1.13161i	−0.824538	−	0.565807i	$-0.808565\pi$
$233$	− 17.2733i	− 1.13161i	0.824538	−	0.565807i	$-0.191435\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−5.50539	−0.357614
$238$	0	0
$239$	−16.3718	−1.05900	−0.529502	−	0.848309i	$-0.677621\pi$
$239$	−16.3718	−1.05900	−0.529502	+	0.848309i	$0.677621\pi$
$240$	0	0
$241$	−6.82654	−0.439736	−0.219868	−	0.975530i	$-0.570563\pi$
$241$	−6.82654	−0.439736	−0.219868	+	0.975530i	$0.570563\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	36.6571i	2.33243i
$248$	0	0
$249$	−9.20811	−0.583540
$250$	0	0
$251$	− 2.96969i	− 0.187445i	−0.995598	−	0.0937225i	$-0.970123\pi$
$251$	− 2.96969i	− 0.187445i	0.995598	−	0.0937225i	$-0.0298766\pi$
$252$	0	0
$253$	3.97613	0.249977
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5.03031i	0.313782i	0.987616	+	0.156891i	$0.0501472\pi$
$257$	5.03031i	0.313782i	−0.987616	+	0.156891i	$0.949853\pi$
$258$	0	0
$259$	0.214558i	0.0133320i
$260$	0	0
$261$	− 4.51516i	− 0.279481i
$262$	0	0
$263$	− 2.70585i	− 0.166850i	−0.996514	−	0.0834248i	$-0.973414\pi$
$263$	− 2.70585i	− 0.166850i	0.996514	−	0.0834248i	$-0.0265859\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	11.9173	0.729326
$268$	0	0
$269$	− 22.3718i	− 1.36403i	−0.731337	−	0.682017i	$-0.761103\pi$
$269$	− 22.3718i	− 1.36403i	0.731337	−	0.682017i	$-0.238897\pi$
$270$	0	0
$271$	0.869741	0.0528330	0.0264165	−	0.999651i	$-0.491590\pi$
$271$	0.869741	0.0528330	0.0264165	+	0.999651i	$0.491590\pi$
$272$	0	0
$273$	0.422937i	0.0255973i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−28.6733	−1.72281	−0.861406	−	0.507918i	$-0.830415\pi$
$277$	−28.6733	−1.72281	−0.861406	+	0.507918i	$0.830415\pi$
$278$	0	0
$279$	−8.29484	−0.496599
$280$	0	0
$281$	15.1429	0.903349	0.451674	−	0.892183i	$-0.350827\pi$
$281$	15.1429	0.903349	0.451674	+	0.892183i	$0.350827\pi$
$282$	0	0
$283$	6.23225	0.370469	0.185234	−	0.982694i	$-0.440696\pi$
$283$	6.23225	0.370469	0.185234	+	0.982694i	$0.440696\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.651862i	0.0384782i
$288$	0	0
$289$	16.9345	0.996147
$290$	0	0
$291$	− 8.50539i	− 0.498595i
$292$	0	0
$293$	−21.5054	−1.25636	−0.628179	−	0.778069i	$-0.716200\pi$
$293$	−21.5054	−1.25636	−0.628179	+	0.778069i	$0.716200\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 2.41649i	− 0.140219i
$298$	0	0
$299$	− 8.66781i	− 0.501272i
$300$	0	0
$301$	− 0.328283i	− 0.0189219i
$302$	0	0
$303$	− 7.56270i	− 0.434466i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	3.57706	0.204154	0.102077	−	0.994777i	$-0.467451\pi$
$307$	3.57706	0.204154	0.102077	+	0.994777i	$0.467451\pi$
$308$	0	0
$309$	1.78544i	0.101570i
$310$	0	0
$311$	2.49461	0.141456	0.0707282	−	0.997496i	$-0.477468\pi$
$311$	2.49461	0.141456	0.0707282	+	0.997496i	$0.477468\pi$
$312$	0	0
$313$	− 9.57246i	− 0.541068i	−0.962710	−	0.270534i	$-0.912800\pi$
$313$	− 9.57246i	− 0.541068i	0.962710	−	0.270534i	$-0.0872002\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.16702	0.177877	0.0889387	−	0.996037i	$-0.471652\pi$
$317$	3.16702	0.177877	0.0889387	+	0.996037i	$0.471652\pi$
$318$	0	0
$319$	10.9108	0.610889
$320$	0	0
$321$	10.4705	0.584405
$322$	0	0
$323$	1.78084	0.0990887
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 3.64298i	− 0.201457i
$328$	0	0
$329$	−0.457851	−0.0252421
$330$	0	0
$331$	16.5118i	0.907573i	0.891111	+	0.453786i	$0.149927\pi$
$331$	16.5118i	0.907573i	−0.891111	+	0.453786i	$0.850073\pi$
$332$	0	0
$333$	−2.67241	−0.146447
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 11.8330i	− 0.644584i	−0.946640	−	0.322292i	$-0.895547\pi$
$337$	− 11.8330i	− 0.644584i	0.946640	−	0.322292i	$-0.104453\pi$
$338$	0	0
$339$	8.83298i	0.479742i
$340$	0	0
$341$	− 20.0444i	− 1.08547i
$342$	0	0
$343$	− 1.12349i	− 0.0606628i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	23.9713	1.28684	0.643422	−	0.765511i	$-0.277514\pi$
$347$	23.9713	1.28684	0.643422	+	0.765511i	$0.277514\pi$
$348$	0	0
$349$	− 8.91570i	− 0.477247i	−0.971112	−	0.238623i	$-0.923304\pi$
$349$	− 8.91570i	− 0.477247i	0.971112	−	0.238623i	$-0.0766961\pi$
$350$	0	0
$351$	−5.26785	−0.281177
$352$	0	0
$353$	7.35606i	0.391524i	0.980651	+	0.195762i	$0.0627179\pi$
$353$	7.35606i	0.391524i	−0.980651	+	0.195762i	$0.937282\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0.0205467	0.00108745
$358$	0	0
$359$	25.2114	1.33061	0.665304	−	0.746572i	$-0.268302\pi$
$359$	25.2114	1.33061	0.665304	+	0.746572i	$0.268302\pi$
$360$	0	0
$361$	−29.4227	−1.54856
$362$	0	0
$363$	−5.16057	−0.270860
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 5.86573i	− 0.306189i	−0.988212	−	0.153094i	$-0.951076\pi$
$367$	− 5.86573i	− 0.306189i	0.988212	−	0.153094i	$-0.0489238\pi$
$368$	0	0
$369$	−8.11921	−0.422669
$370$	0	0
$371$	− 0.928327i	− 0.0481963i
$372$	0	0
$373$	−27.5063	−1.42422	−0.712111	−	0.702067i	$-0.752261\pi$
$373$	−27.5063	−1.42422	−0.712111	+	0.702067i	$0.752261\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 23.7852i	− 1.22500i
$378$	0	0
$379$	− 11.7549i	− 0.603807i	−0.953339	−	0.301903i	$-0.902378\pi$
$379$	− 11.7549i	− 0.603807i	0.953339	−	0.301903i	$-0.0976220\pi$
$380$	0	0
$381$	− 8.69628i	− 0.445524i
$382$	0	0
$383$	− 34.3335i	− 1.75436i	−0.480162	−	0.877180i	$-0.659422\pi$
$383$	− 34.3335i	− 1.75436i	0.480162	−	0.877180i	$-0.340578\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.08890	0.207850
$388$	0	0
$389$	2.89515i	0.146790i	0.997303	+	0.0733951i	$0.0233834\pi$
$389$	2.89515i	0.146790i	−0.997303	+	0.0733951i	$0.976617\pi$
$390$	0	0
$391$	−0.421092	−0.0212955
$392$	0	0
$393$	− 10.7916i	− 0.544365i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	22.9099	1.14982	0.574909	−	0.818218i	$-0.305038\pi$
$397$	22.9099	1.14982	0.574909	+	0.818218i	$0.305038\pi$
$398$	0	0
$399$	−0.558684	−0.0279692
$400$	0	0
$401$	−12.4337	−0.620910	−0.310455	−	0.950588i	$-0.600481\pi$
$401$	−12.4337	−0.620910	−0.310455	+	0.950588i	$0.600481\pi$
$402$	0	0
$403$	−43.6960	−2.17665
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 6.45785i	− 0.320104i
$408$	0	0
$409$	32.0886	1.58668	0.793340	−	0.608778i	$-0.208340\pi$
$409$	32.0886	1.58668	0.793340	+	0.608778i	$0.208340\pi$
$410$	0	0
$411$	− 11.5421i	− 0.569332i
$412$	0	0
$413$	1.01934	0.0501583
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 0.214558i	− 0.0105069i
$418$	0	0
$419$	− 16.1364i	− 0.788317i	−0.919043	−	0.394158i	$-0.871036\pi$
$419$	− 16.1364i	− 0.788317i	0.919043	−	0.394158i	$-0.128964\pi$
$420$	0	0
$421$	− 28.7675i	− 1.40204i	−0.713141	−	0.701021i	$-0.752728\pi$
$421$	− 28.7675i	− 1.40204i	0.713141	−	0.701021i	$-0.247272\pi$
$422$	0	0
$423$	− 5.70272i	− 0.277276i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0.958640	0.0463918
$428$	0	0
$429$	− 12.7297i	− 0.614596i
$430$	0	0
$431$	−24.7297	−1.19119	−0.595594	−	0.803285i	$-0.703083\pi$
$431$	−24.7297	−1.19119	−0.595594	+	0.803285i	$0.703083\pi$
$432$	0	0
$433$	4.48816i	0.215687i	0.994168	+	0.107844i	$0.0343946\pi$
$433$	4.48816i	0.215687i	−0.994168	+	0.107844i	$0.965605\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	11.4499	0.547721
$438$	0	0
$439$	5.96081	0.284494	0.142247	−	0.989831i	$-0.454567\pi$
$439$	5.96081	0.284494	0.142247	+	0.989831i	$0.454567\pi$
$440$	0	0
$441$	6.99355	0.333026
$442$	0	0
$443$	−14.2924	−0.679053	−0.339526	−	0.940597i	$-0.610267\pi$
$443$	−14.2924	−0.679053	−0.339526	+	0.940597i	$0.610267\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 23.0475i	− 1.09011i
$448$	0	0
$449$	−24.5529	−1.15872	−0.579362	−	0.815070i	$-0.696698\pi$
$449$	−24.5529	−1.15872	−0.579362	+	0.815070i	$0.696698\pi$
$450$	0	0
$451$	− 19.6200i	− 0.923870i
$452$	0	0
$453$	9.48573	0.445678
$454$	0	0
$455$	0	0
$456$	0	0
$457$	28.9108i	1.35239i	0.736722	+	0.676196i	$0.236373\pi$
$457$	28.9108i	1.35239i	−0.736722	+	0.676196i	$0.763627\pi$
$458$	0	0
$459$	0.255918i	0.0119452i
$460$	0	0
$461$	4.35458i	0.202813i	0.994845	+	0.101407i	$0.0323343\pi$
$461$	4.35458i	0.202813i	−0.994845	+	0.101407i	$0.967666\pi$
$462$	0	0
$463$	11.1303i	0.517267i	0.965976	+	0.258634i	$0.0832722\pi$
$463$	11.1303i	0.517267i	−0.965976	+	0.258634i	$0.916728\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−19.1257	−0.885030	−0.442515	−	0.896761i	$-0.645914\pi$
$467$	−19.1257	−0.885030	−0.442515	+	0.896761i	$0.645914\pi$
$468$	0	0
$469$	0.584467i	0.0269882i
$470$	0	0
$471$	−6.34413	−0.292322
$472$	0	0
$473$	9.88079i	0.454319i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	11.5627	0.529419
$478$	0	0
$479$	−25.6358	−1.17133	−0.585666	−	0.810553i	$-0.699167\pi$
$479$	−25.6358	−1.17133	−0.585666	+	0.810553i	$0.699167\pi$
$480$	0	0
$481$	−14.0779	−0.641895
$482$	0	0
$483$	0.132104	0.00601096
$484$	0	0
$485$	0	0
$486$	0	0
$487$	12.8434i	0.581992i	0.956724	+	0.290996i	$0.0939866\pi$
$487$	12.8434i	0.581992i	−0.956724	+	0.290996i	$0.906013\pi$
$488$	0	0
$489$	−12.4100	−0.561202
$490$	0	0
$491$	− 16.9887i	− 0.766689i	−0.923605	−	0.383344i	$-0.874772\pi$
$491$	− 16.9887i	− 0.766689i	0.923605	−	0.383344i	$-0.125228\pi$
$492$	0	0
$493$	−1.15551	−0.0520415
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 0.911101i	− 0.0408684i
$498$	0	0
$499$	− 14.0521i	− 0.629060i	−0.949248	−	0.314530i	$-0.898153\pi$
$499$	− 14.0521i	− 0.629060i	0.949248	−	0.314530i	$-0.101847\pi$
$500$	0	0
$501$	23.2654i	1.03942i
$502$	0	0
$503$	− 9.53258i	− 0.425037i	−0.977157	−	0.212518i	$-0.931833\pi$
$503$	− 9.53258i	− 0.425037i	0.977157	−	0.212518i	$-0.0681665\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−14.7503	−0.655082
$508$	0	0
$509$	− 30.3450i	− 1.34502i	−0.740088	−	0.672510i	$-0.765216\pi$
$509$	− 30.3450i	− 1.34502i	0.740088	−	0.672510i	$-0.234784\pi$
$510$	0	0
$511$	0.969687	0.0428964
$512$	0	0
$513$	− 6.95864i	− 0.307231i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	13.7806	0.606069
$518$	0	0
$519$	−8.63897	−0.379209
$520$	0	0
$521$	14.4245	0.631949	0.315975	−	0.948768i	$-0.397669\pi$
$521$	14.4245	0.631949	0.315975	+	0.948768i	$0.397669\pi$
$522$	0	0
$523$	−28.2207	−1.23401	−0.617003	−	0.786961i	$-0.711654\pi$
$523$	−28.2207	−1.23401	−0.617003	+	0.786961i	$0.711654\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.12280i	0.0924706i
$528$	0	0
$529$	20.2926	0.882287
$530$	0	0
$531$	12.6963i	0.550971i
$532$	0	0
$533$	−42.7708	−1.85261
$534$	0	0
$535$	0	0
$536$	0	0
$537$	9.40544i	0.405875i
$538$	0	0
$539$	16.8999i	0.727928i
$540$	0	0
$541$	− 13.4695i	− 0.579100i	−0.957163	−	0.289550i	$-0.906494\pi$
$541$	− 13.4695i	− 0.579100i	0.957163	−	0.289550i	$-0.0935056\pi$
$542$	0	0
$543$	6.43487i	0.276147i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	4.42773	0.189316	0.0946581	−	0.995510i	$-0.469824\pi$
$547$	4.42773	0.189316	0.0946581	+	0.995510i	$0.469824\pi$
$548$	0	0
$549$	11.9403i	0.509598i
$550$	0	0
$551$	31.4193	1.33851
$552$	0	0
$553$	− 0.442008i	− 0.0187961i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	40.2017	1.70340	0.851700	−	0.524030i	$-0.175572\pi$
$557$	40.2017	1.70340	0.851700	+	0.524030i	$0.175572\pi$
$558$	0	0
$559$	21.5397	0.911033
$560$	0	0
$561$	−0.618423	−0.0261099
$562$	0	0
$563$	−13.1128	−0.552637	−0.276319	−	0.961066i	$-0.589115\pi$
$563$	−13.1128	−0.552637	−0.276319	+	0.961066i	$0.589115\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 0.0802864i	− 0.00337171i
$568$	0	0
$569$	11.0257	0.462222	0.231111	−	0.972927i	$-0.425764\pi$
$569$	11.0257	0.462222	0.231111	+	0.972927i	$0.425764\pi$
$570$	0	0
$571$	− 45.6960i	− 1.91232i	−0.292847	−	0.956159i	$-0.594603\pi$
$571$	− 45.6960i	− 1.91232i	0.292847	−	0.956159i	$-0.405397\pi$
$572$	0	0
$573$	−5.56270	−0.232385
$574$	0	0
$575$	0	0
$576$	0	0
$577$	17.2685i	0.718899i	0.933165	+	0.359449i	$0.117035\pi$
$577$	17.2685i	0.718899i	−0.933165	+	0.359449i	$0.882965\pi$
$578$	0	0
$579$	− 18.4227i	− 0.765620i
$580$	0	0
$581$	− 0.739286i	− 0.0306707i
$582$	0	0
$583$	27.9411i	1.15720i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−34.7155	−1.43286	−0.716432	−	0.697657i	$-0.754226\pi$
$587$	−34.7155	−1.43286	−0.716432	+	0.697657i	$0.754226\pi$
$588$	0	0
$589$	− 57.7208i	− 2.37835i
$590$	0	0
$591$	18.0239	0.741403
$592$	0	0
$593$	− 9.34022i	− 0.383557i	−0.981438	−	0.191778i	$-0.938575\pi$
$593$	− 9.34022i	− 0.383557i	0.981438	−	0.191778i	$-0.0614255\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−20.1214	−0.823513
$598$	0	0
$599$	−13.9110	−0.568389	−0.284195	−	0.958767i	$-0.591726\pi$
$599$	−13.9110	−0.568389	−0.284195	+	0.958767i	$0.591726\pi$
$600$	0	0
$601$	11.7330	0.478600	0.239300	−	0.970946i	$-0.423082\pi$
$601$	11.7330	0.478600	0.239300	+	0.970946i	$0.423082\pi$
$602$	0	0
$603$	−7.27979	−0.296456
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 10.8158i	− 0.438998i	−0.975613	−	0.219499i	$-0.929558\pi$
$607$	− 10.8158i	− 0.438998i	0.975613	−	0.219499i	$-0.0704423\pi$
$608$	0	0
$609$	0.362505	0.0146895
$610$	0	0
$611$	− 30.0411i	− 1.21533i
$612$	0	0
$613$	−17.9632	−0.725528	−0.362764	−	0.931881i	$-0.618167\pi$
$613$	−17.9632	−0.725528	−0.362764	+	0.931881i	$0.618167\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 12.3576i	− 0.497500i	−0.968568	−	0.248750i	$-0.919980\pi$
$617$	− 12.3576i	− 0.497500i	0.968568	−	0.248750i	$-0.0800197\pi$
$618$	0	0
$619$	4.99540i	0.200782i	0.994948	+	0.100391i	$0.0320094\pi$
$619$	4.99540i	0.200782i	−0.994948	+	0.100391i	$0.967991\pi$
$620$	0	0
$621$	1.64542i	0.0660283i
$622$	0	0
$623$	0.956795i	0.0383332i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	16.8155	0.671546
$628$	0	0
$629$	0.683917i	0.0272696i
$630$	0	0
$631$	17.9674	0.715273	0.357636	−	0.933861i	$-0.383583\pi$
$631$	17.9674	0.715273	0.357636	+	0.933861i	$0.383583\pi$
$632$	0	0
$633$	− 3.25592i	− 0.129411i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	36.8410	1.45969
$638$	0	0
$639$	11.3481	0.448925
$640$	0	0
$641$	27.3638	1.08081	0.540403	−	0.841406i	$-0.318272\pi$
$641$	27.3638	1.08081	0.540403	+	0.841406i	$0.318272\pi$
$642$	0	0
$643$	−2.27518	−0.0897245	−0.0448623	−	0.998993i	$-0.514285\pi$
$643$	−2.27518	−0.0897245	−0.0448623	+	0.998993i	$0.514285\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.4769i	0.490516i	0.969458	+	0.245258i	$0.0788726\pi$
$647$	12.4769i	0.490516i	−0.969458	+	0.245258i	$0.921127\pi$
$648$	0	0
$649$	−30.6804	−1.20431
$650$	0	0
$651$	− 0.665963i	− 0.0261011i
$652$	0	0
$653$	29.3055	1.14681	0.573406	−	0.819271i	$-0.305622\pi$
$653$	29.3055	1.14681	0.573406	+	0.819271i	$0.305622\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	12.0779i	0.471202i
$658$	0	0
$659$	18.6009i	0.724589i	0.932064	+	0.362295i	$0.118007\pi$
$659$	18.6009i	0.724589i	−0.932064	+	0.362295i	$0.881993\pi$
$660$	0	0
$661$	16.1318i	0.627456i	0.949513	+	0.313728i	$0.101578\pi$
$661$	16.1318i	0.627456i	−0.949513	+	0.313728i	$0.898422\pi$
$662$	0	0
$663$	1.34814i	0.0523573i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−7.42931	−0.287664
$668$	0	0
$669$	− 26.9911i	− 1.04354i
$670$	0	0
$671$	−28.8535	−1.11388
$672$	0	0
$673$	− 34.1385i	− 1.31594i	−0.753043	−	0.657971i	$-0.771415\pi$
$673$	− 34.1385i	− 1.31594i	0.753043	−	0.657971i	$-0.228585\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−12.1940	−0.468654	−0.234327	−	0.972158i	$-0.575289\pi$
$677$	−12.1940	−0.468654	−0.234327	+	0.972158i	$0.575289\pi$
$678$	0	0
$679$	0.682867	0.0262060
$680$	0	0
$681$	−19.8219	−0.759578
$682$	0	0
$683$	−21.8567	−0.836322	−0.418161	−	0.908373i	$-0.637325\pi$
$683$	−21.8567	−0.836322	−0.418161	+	0.908373i	$0.637325\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 21.6797i	− 0.827134i
$688$	0	0
$689$	60.9106	2.32051
$690$	0	0
$691$	− 6.17780i	− 0.235015i	−0.993072	−	0.117507i	$-0.962510\pi$
$691$	− 6.17780i	− 0.235015i	0.993072	−	0.117507i	$-0.0374903\pi$
$692$	0	0
$693$	0.194011	0.00736988
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.07785i	0.0787043i
$698$	0	0
$699$	17.2733i	0.653338i
$700$	0	0
$701$	33.9746i	1.28320i	0.767038	+	0.641601i	$0.221730\pi$
$701$	33.9746i	1.28320i	−0.767038	+	0.641601i	$0.778270\pi$
$702$	0	0
$703$	− 18.5963i	− 0.701374i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.607181	0.0228354
$708$	0	0
$709$	22.3441i	0.839151i	0.907720	+	0.419576i	$0.137821\pi$
$709$	22.3441i	0.839151i	−0.907720	+	0.419576i	$0.862179\pi$
$710$	0	0
$711$	5.50539	0.206468
$712$	0	0
$713$	13.6485i	0.511139i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	16.3718	0.611416
$718$	0	0
$719$	−17.8427	−0.665422	−0.332711	−	0.943029i	$-0.607963\pi$
$719$	−17.8427	−0.665422	−0.332711	+	0.943029i	$0.607963\pi$
$720$	0	0
$721$	−0.143347	−0.00533851
$722$	0	0
$723$	6.82654	0.253882
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 23.9148i	− 0.886953i	−0.896286	−	0.443476i	$-0.853745\pi$
$727$	− 23.9148i	− 0.886953i	0.896286	−	0.443476i	$-0.146255\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	− 1.04642i	− 0.0387034i
$732$	0	0
$733$	15.6789	0.579112	0.289556	−	0.957161i	$-0.406492\pi$
$733$	15.6789	0.579112	0.289556	+	0.957161i	$0.406492\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 17.5915i	− 0.647992i
$738$	0	0
$739$	22.3083i	0.820622i	0.911946	+	0.410311i	$0.134580\pi$
$739$	22.3083i	0.820622i	−0.911946	+	0.410311i	$0.865420\pi$
$740$	0	0
$741$	− 36.6571i	− 1.34663i
$742$	0	0
$743$	− 9.78057i	− 0.358814i	−0.983775	−	0.179407i	$-0.942582\pi$
$743$	− 9.78057i	− 0.358814i	0.983775	−	0.179407i	$-0.0574180\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	9.20811	0.336907
$748$	0	0
$749$	0.840636i	0.0307162i
$750$	0	0
$751$	8.05399	0.293894	0.146947	−	0.989144i	$-0.453055\pi$
$751$	8.05399	0.293894	0.146947	+	0.989144i	$0.453055\pi$
$752$	0	0
$753$	2.96969i	0.108221i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−28.4889	−1.03545	−0.517723	−	0.855549i	$-0.673220\pi$
$757$	−28.4889	−1.03545	−0.517723	+	0.855549i	$0.673220\pi$
$758$	0	0
$759$	−3.97613	−0.144324
$760$	0	0
$761$	−21.5005	−0.779393	−0.389697	−	0.920943i	$-0.627420\pi$
$761$	−21.5005	−0.779393	−0.389697	+	0.920943i	$0.627420\pi$
$762$	0	0
$763$	0.292482	0.0105886
$764$	0	0
$765$	0	0
$766$	0	0
$767$	66.8821i	2.41497i
$768$	0	0
$769$	−23.5596	−0.849580	−0.424790	−	0.905292i	$-0.639652\pi$
$769$	−23.5596	−0.849580	−0.424790	+	0.905292i	$0.639652\pi$
$770$	0	0
$771$	− 5.03031i	− 0.181162i
$772$	0	0
$773$	−31.1655	−1.12094	−0.560472	−	0.828173i	$-0.689380\pi$
$773$	−31.1655	−1.12094	−0.560472	+	0.828173i	$0.689380\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 0.214558i	− 0.00769722i
$778$	0	0
$779$	− 56.4987i	− 2.02428i
$780$	0	0
$781$	27.4227i	0.981260i
$782$	0	0
$783$	4.51516i	0.161359i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−5.07812	−0.181015	−0.0905077	−	0.995896i	$-0.528849\pi$
$787$	−5.07812	−0.181015	−0.0905077	+	0.995896i	$0.528849\pi$
$788$	0	0
$789$	2.70585i	0.0963307i
$790$	0	0
$791$	−0.709168	−0.0252151
$792$	0	0
$793$	62.8995i	2.23363i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.43418	0.0508012	0.0254006	−	0.999677i	$-0.491914\pi$
$797$	1.43418	0.0508012	0.0254006	+	0.999677i	$0.491914\pi$
$798$	0	0
$799$	−1.45943	−0.0516309
$800$	0	0
$801$	−11.9173	−0.421076
$802$	0	0
$803$	−29.1860	−1.02995
$804$	0	0
$805$	0	0
$806$	0	0
$807$	22.3718i	0.787525i
$808$	0	0
$809$	−23.7153	−0.833787	−0.416894	−	0.908955i	$-0.636881\pi$
$809$	−23.7153	−0.833787	−0.416894	+	0.908955i	$0.636881\pi$
$810$	0	0
$811$	47.8394i	1.67987i	0.542689	+	0.839933i	$0.317406\pi$
$811$	47.8394i	1.67987i	−0.542689	+	0.839933i	$0.682594\pi$
$812$	0	0
$813$	−0.869741	−0.0305031
$814$	0	0
$815$	0	0
$816$	0	0
$817$	28.4532i	0.995451i
$818$	0	0
$819$	− 0.422937i	− 0.0147786i
$820$	0	0
$821$	− 51.9216i	− 1.81208i	−0.423195	−	0.906038i	$-0.639092\pi$
$821$	− 51.9216i	− 1.81208i	0.423195	−	0.906038i	$-0.360908\pi$
$822$	0	0
$823$	− 27.8542i	− 0.970937i	−0.874254	−	0.485469i	$-0.838649\pi$
$823$	− 27.8542i	− 0.970937i	0.874254	−	0.485469i	$-0.161351\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	43.9365	1.52782	0.763912	−	0.645321i	$-0.223276\pi$
$827$	43.9365	1.52782	0.763912	+	0.645321i	$0.223276\pi$
$828$	0	0
$829$	− 41.6898i	− 1.44795i	−0.689827	−	0.723974i	$-0.742314\pi$
$829$	− 41.6898i	− 1.44795i	0.689827	−	0.723974i	$-0.257686\pi$
$830$	0	0
$831$	28.6733	0.994666
$832$	0	0
$833$	− 1.78978i	− 0.0620121i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	8.29484	0.286712
$838$	0	0
$839$	−25.4733	−0.879437	−0.439719	−	0.898136i	$-0.644922\pi$
$839$	−25.4733	−0.879437	−0.439719	+	0.898136i	$0.644922\pi$
$840$	0	0
$841$	8.61336	0.297012
$842$	0	0
$843$	−15.1429	−0.521549
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 0.414324i	− 0.0142363i
$848$	0	0
$849$	−6.23225	−0.213890
$850$	0	0
$851$	4.39722i	0.150735i
$852$	0	0
$853$	12.5366	0.429245	0.214622	−	0.976697i	$-0.431148\pi$
$853$	12.5366	0.429245	0.214622	+	0.976697i	$0.431148\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 2.61409i	− 0.0892956i	−0.999003	−	0.0446478i	$-0.985783\pi$
$857$	− 2.61409i	− 0.0892956i	0.999003	−	0.0446478i	$-0.0142166\pi$
$858$	0	0
$859$	31.8438i	1.08650i	0.839573	+	0.543248i	$0.182805\pi$
$859$	31.8438i	1.08650i	−0.839573	+	0.543248i	$0.817195\pi$
$860$	0	0
$861$	− 0.651862i	− 0.0222154i
$862$	0	0
$863$	22.3335i	0.760241i	0.924937	+	0.380121i	$0.124118\pi$
$863$	22.3335i	0.760241i	−0.924937	+	0.380121i	$0.875882\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−16.9345	−0.575126
$868$	0	0
$869$	13.3037i	0.451298i
$870$	0	0
$871$	−38.3488	−1.29940
$872$	0	0
$873$	8.50539i	0.287864i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−37.4408	−1.26429	−0.632143	−	0.774852i	$-0.717824\pi$
$877$	−37.4408	−1.26429	−0.632143	+	0.774852i	$0.717824\pi$
$878$	0	0
$879$	21.5054	0.725359
$880$	0	0
$881$	53.1952	1.79219	0.896096	−	0.443860i	$-0.146391\pi$
$881$	53.1952	1.79219	0.896096	+	0.443860i	$0.146391\pi$
$882$	0	0
$883$	36.0907	1.21455	0.607274	−	0.794493i	$-0.292263\pi$
$883$	36.0907	1.21455	0.607274	+	0.794493i	$0.292263\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 43.3018i	− 1.45393i	−0.686673	−	0.726966i	$-0.740930\pi$
$887$	− 43.3018i	− 1.45393i	0.686673	−	0.726966i	$-0.259070\pi$
$888$	0	0
$889$	0.698192	0.0234166
$890$	0	0
$891$	2.41649i	0.0809555i
$892$	0	0
$893$	39.6832	1.32795
$894$	0	0
$895$	0	0
$896$	0	0
$897$	8.66781i	0.289410i
$898$	0	0
$899$	37.4525i	1.24911i
$900$	0	0
$901$	− 2.95910i	− 0.0985820i
$902$	0	0
$903$	0.328283i	0.0109246i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	8.75026	0.290548	0.145274	−	0.989391i	$-0.453594\pi$
$907$	8.75026	0.290548	0.145274	+	0.989391i	$0.453594\pi$
$908$	0	0
$909$	7.56270i	0.250839i
$910$	0	0
$911$	13.7438	0.455353	0.227676	−	0.973737i	$-0.426887\pi$
$911$	13.7438	0.455353	0.227676	+	0.973737i	$0.426887\pi$
$912$	0	0
$913$	22.2513i	0.736411i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0.866420	0.0286117
$918$	0	0
$919$	−0.989347	−0.0326355	−0.0163178	−	0.999867i	$-0.505194\pi$
$919$	−0.989347	−0.0326355	−0.0163178	+	0.999867i	$0.505194\pi$
$920$	0	0
$921$	−3.57706	−0.117868
$922$	0	0
$923$	59.7803	1.96769
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 1.78544i	− 0.0586416i
$928$	0	0
$929$	8.49434	0.278690	0.139345	−	0.990244i	$-0.455500\pi$
$929$	8.49434	0.278690	0.139345	+	0.990244i	$0.455500\pi$
$930$	0	0
$931$	48.6656i	1.59495i
$932$	0	0
$933$	−2.49461	−0.0816699
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 18.6912i	− 0.610615i	−0.952254	−	0.305308i	$-0.901241\pi$
$937$	− 18.6912i	− 0.610615i	0.952254	−	0.305308i	$-0.0987593\pi$
$938$	0	0
$939$	9.57246i	0.312386i
$940$	0	0
$941$	− 3.03170i	− 0.0988305i	−0.998778	−	0.0494152i	$-0.984264\pi$
$941$	− 3.03170i	− 0.0988305i	0.998778	−	0.0494152i	$-0.0157358\pi$
$942$	0	0
$943$	13.3595i	0.435045i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−16.8327	−0.546990	−0.273495	−	0.961873i	$-0.588180\pi$
$947$	−16.8327	−0.546990	−0.273495	+	0.961873i	$0.588180\pi$
$948$	0	0
$949$	63.6243i	2.06533i
$950$	0	0
$951$	−3.16702	−0.102698
$952$	0	0
$953$	1.73948i	0.0563473i	0.999603	+	0.0281737i	$0.00896914\pi$
$953$	1.73948i	0.0563473i	−0.999603	+	0.0281737i	$0.991031\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−10.9108	−0.352697
$958$	0	0
$959$	0.926677	0.0299240
$960$	0	0
$961$	37.8044	1.21950
$962$	0	0
$963$	−10.4705	−0.337406
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 17.3399i	− 0.557615i	−0.960347	−	0.278808i	$-0.910061\pi$
$967$	− 17.3399i	− 0.557615i	0.960347	−	0.278808i	$-0.0899392\pi$
$968$	0	0
$969$	−1.78084	−0.0572089
$970$	0	0
$971$	45.4054i	1.45713i	0.684977	+	0.728565i	$0.259812\pi$
$971$	45.4054i	1.45713i	−0.684977	+	0.728565i	$0.740188\pi$
$972$	0	0
$973$	0.0172261	0.000552243 0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 43.4336i	− 1.38957i	−0.719220	−	0.694783i	$-0.755501\pi$
$977$	− 43.4336i	− 1.38957i	0.719220	−	0.694783i	$-0.244499\pi$
$978$	0	0
$979$	− 28.7980i	− 0.920388i
$980$	0	0
$981$	3.64298i	0.116311i
$982$	0	0
$983$	47.4465i	1.51331i	0.653815	+	0.756655i	$0.273168\pi$
$983$	47.4465i	1.51331i	−0.653815	+	0.756655i	$0.726832\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0.457851	0.0145735
$988$	0	0
$989$	− 6.72794i	− 0.213936i
$990$	0	0
$991$	28.8434	0.916242	0.458121	−	0.888890i	$-0.348523\pi$
$991$	28.8434	0.916242	0.458121	+	0.888890i	$0.348523\pi$
$992$	0	0
$993$	− 16.5118i	− 0.523987i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−33.1449	−1.04971	−0.524855	−	0.851192i	$-0.675881\pi$
$997$	−33.1449	−1.04971	−0.524855	+	0.851192i	$0.675881\pi$
$998$	0	0
$999$	2.67241	0.0845513

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.2.d.g.49.4		8
3.2	odd	2		7200.2.d.s.2449.4		8
4.3	odd	2		600.2.d.h.349.3		8
5.2	odd	4		2400.2.k.d.1201.7		8
5.3	odd	4		2400.2.k.e.1201.2		8
5.4	even	2		2400.2.d.h.49.5		8
8.3	odd	2		600.2.d.g.349.5		8
8.5	even	2		2400.2.d.h.49.4		8
12.11	even	2		1800.2.d.s.1549.6		8
15.2	even	4		7200.2.k.r.3601.5		8
15.8	even	4		7200.2.k.s.3601.3		8
15.14	odd	2		7200.2.d.t.2449.5		8
20.3	even	4		600.2.k.e.301.2	yes	8
20.7	even	4		600.2.k.d.301.7	✓	8
20.19	odd	2		600.2.d.g.349.6		8
24.5	odd	2		7200.2.d.t.2449.4		8
24.11	even	2		1800.2.d.t.1549.4		8
40.3	even	4		600.2.k.e.301.1	yes	8
40.13	odd	4		2400.2.k.e.1201.6		8
40.19	odd	2		600.2.d.h.349.4		8
40.27	even	4		600.2.k.d.301.8	yes	8
40.29	even	2	inner	2400.2.d.g.49.5		8
40.37	odd	4		2400.2.k.d.1201.3		8
60.23	odd	4		1800.2.k.q.901.7		8
60.47	odd	4		1800.2.k.t.901.2		8
60.59	even	2		1800.2.d.t.1549.3		8
120.29	odd	2		7200.2.d.s.2449.5		8
120.53	even	4		7200.2.k.s.3601.4		8
120.59	even	2		1800.2.d.s.1549.5		8
120.77	even	4		7200.2.k.r.3601.6		8
120.83	odd	4		1800.2.k.q.901.8		8
120.107	odd	4		1800.2.k.t.901.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.2.d.g.349.5		8	8.3	odd	2
600.2.d.g.349.6		8	20.19	odd	2
600.2.d.h.349.3		8	4.3	odd	2
600.2.d.h.349.4		8	40.19	odd	2
600.2.k.d.301.7	✓	8	20.7	even	4
600.2.k.d.301.8	yes	8	40.27	even	4
600.2.k.e.301.1	yes	8	40.3	even	4
600.2.k.e.301.2	yes	8	20.3	even	4
1800.2.d.s.1549.5		8	120.59	even	2
1800.2.d.s.1549.6		8	12.11	even	2
1800.2.d.t.1549.3		8	60.59	even	2
1800.2.d.t.1549.4		8	24.11	even	2
1800.2.k.q.901.7		8	60.23	odd	4
1800.2.k.q.901.8		8	120.83	odd	4
1800.2.k.t.901.1		8	120.107	odd	4
1800.2.k.t.901.2		8	60.47	odd	4
2400.2.d.g.49.4		8	1.1	even	1	trivial
2400.2.d.g.49.5		8	40.29	even	2	inner
2400.2.d.h.49.4		8	8.5	even	2
2400.2.d.h.49.5		8	5.4	even	2
2400.2.k.d.1201.3		8	40.37	odd	4
2400.2.k.d.1201.7		8	5.2	odd	4
2400.2.k.e.1201.2		8	5.3	odd	4
2400.2.k.e.1201.6		8	40.13	odd	4
7200.2.d.s.2449.4		8	3.2	odd	2
7200.2.d.s.2449.5		8	120.29	odd	2
7200.2.d.t.2449.4		8	24.5	odd	2
7200.2.d.t.2449.5		8	15.14	odd	2
7200.2.k.r.3601.5		8	15.2	even	4
7200.2.k.r.3601.6		8	120.77	even	4
7200.2.k.s.3601.3		8	15.8	even	4
7200.2.k.s.3601.4		8	120.53	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2400.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.0802864i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.0802864i q^{7} +1.00000 q^{9} +2.41649i q^{11} +5.26785 q^{13} -0.255918i q^{17} +6.95864i q^{19} +0.0802864i q^{21} -1.64542i q^{23} -1.00000 q^{27} -4.51516i q^{29} -8.29484 q^{31} -2.41649i q^{33} -2.67241 q^{37} -5.26785 q^{39} -8.11921 q^{41} +4.08890 q^{43} -5.70272i q^{47} +6.99355 q^{49} +0.255918i q^{51} +11.5627 q^{53} -6.95864i q^{57} +12.6963i q^{59} +11.9403i q^{61} -0.0802864i q^{63} -7.27979 q^{67} +1.64542i q^{69} +11.3481 q^{71} +12.0779i q^{73} +0.194011 q^{77} +5.50539 q^{79} +1.00000 q^{81} +9.20811 q^{83} +4.51516i q^{87} -11.9173 q^{89} -0.422937i q^{91} +8.29484 q^{93} +8.50539i q^{97} +2.41649i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} + 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^3 + 8 * q^9	\( 8 q - 8 q^{3} + 8 q^{9} - 8 q^{27} - 8 q^{31} - 8 q^{43} - 8 q^{53} + 24 q^{67} + 40 q^{71} + 24 q^{77} - 16 q^{79} + 8 q^{81} - 32 q^{83} + 8 q^{93}+O(q^{100}) \) 8 * q - 8 * q^3 + 8 * q^9 - 8 * q^27 - 8 * q^31 - 8 * q^43 - 8 * q^53 + 24 * q^67 + 40 * q^71 + 24 * q^77 - 16 * q^79 + 8 * q^81 - 32 * q^83 + 8 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more