LMFDB - Embedded newform 2304.3.g.q.1279.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,3,Mod(1279,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2304.1279");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2304.g (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:	$62.7794529086$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 192)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1279.4
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2304.1279
Dual form			2304.3.g.q.1279.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.46410 q^{5} +10.0000i q^{7} +O(q^{10})$	$q+3.46410 q^{5} +10.0000i q^{7} -13.8564i q^{11} +6.92820 q^{13} -30.0000 q^{17} -6.92820i q^{19} -12.0000i q^{23} -13.0000 q^{25} +51.9615 q^{29} -14.0000i q^{31} +34.6410i q^{35} -55.4256 q^{37} -6.00000 q^{41} -62.3538i q^{43} -84.0000i q^{47} -51.0000 q^{49} -17.3205 q^{53} -48.0000i q^{55} -62.3538i q^{59} -96.9948 q^{61} +24.0000 q^{65} +48.4974i q^{67} +60.0000i q^{71} +86.0000 q^{73} +138.564 q^{77} -38.0000i q^{79} -13.8564i q^{83} -103.923 q^{85} +78.0000 q^{89} +69.2820i q^{91} -24.0000i q^{95} +62.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 120 q^{17} - 52 q^{25} - 24 q^{41} - 204 q^{49} + 96 q^{65} + 344 q^{73} + 312 q^{89} + 248 q^{97}+O(q^{100}) $ 4 * q - 120 * q^17 - 52 * q^25 - 24 * q^41 - 204 * q^49 + 96 * q^65 + 344 * q^73 + 312 * q^89 + 248 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$.

$n$	$1279$	$1793$	$2053$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	3.46410	0.692820	0.346410	−	0.938083i	$-0.387401\pi$
$5$	3.46410	0.692820	0.346410	+	0.938083i	$0.387401\pi$
$6$	0	0
$7$	10.0000i	1.42857i	0.699854	+	0.714286i	$0.253248\pi$
$7$	10.0000i	1.42857i	−0.699854	+	0.714286i	$0.746752\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 13.8564i	− 1.25967i	−0.776728	−	0.629837i	$-0.783122\pi$
$11$	− 13.8564i	− 1.25967i	0.776728	−	0.629837i	$-0.216878\pi$
$12$	0	0
$13$	6.92820	0.532939	0.266469	−	0.963843i	$-0.414143\pi$
$13$	6.92820	0.532939	0.266469	+	0.963843i	$0.414143\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−30.0000	−1.76471	−0.882353	−	0.470588i	$-0.844042\pi$
$17$	−30.0000	−1.76471	−0.882353	+	0.470588i	$0.844042\pi$
$18$	0	0
$19$	− 6.92820i	− 0.364642i	−0.983239	−	0.182321i	$-0.941639\pi$
$19$	− 6.92820i	− 0.364642i	0.983239	−	0.182321i	$-0.0583610\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 12.0000i	− 0.521739i	−0.965374	−	0.260870i	$-0.915991\pi$
$23$	− 12.0000i	− 0.521739i	0.965374	−	0.260870i	$-0.0840093\pi$
$24$	0	0
$25$	−13.0000	−0.520000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	51.9615	1.79178	0.895888	−	0.444279i	$-0.146540\pi$
$29$	51.9615	1.79178	0.895888	+	0.444279i	$0.146540\pi$
$30$	0	0
$31$	− 14.0000i	− 0.451613i	−0.974172	−	0.225806i	$-0.927498\pi$
$31$	− 14.0000i	− 0.451613i	0.974172	−	0.225806i	$-0.0725017\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	34.6410i	0.989743i
$36$	0	0
$37$	−55.4256	−1.49799	−0.748995	−	0.662576i	$-0.769463\pi$
$37$	−55.4256	−1.49799	−0.748995	+	0.662576i	$0.769463\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.00000	−0.146341	−0.0731707	−	0.997319i	$-0.523312\pi$
$41$	−6.00000	−0.146341	−0.0731707	+	0.997319i	$0.523312\pi$
$42$	0	0
$43$	− 62.3538i	− 1.45009i	−0.688702	−	0.725045i	$-0.741819\pi$
$43$	− 62.3538i	− 1.45009i	0.688702	−	0.725045i	$-0.258181\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 84.0000i	− 1.78723i	−0.448830	−	0.893617i	$-0.648159\pi$
$47$	− 84.0000i	− 1.78723i	0.448830	−	0.893617i	$-0.351841\pi$
$48$	0	0
$49$	−51.0000	−1.04082
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−17.3205	−0.326802	−0.163401	−	0.986560i	$-0.552246\pi$
$53$	−17.3205	−0.326802	−0.163401	+	0.986560i	$0.552246\pi$
$54$	0	0
$55$	− 48.0000i	− 0.872727i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 62.3538i	− 1.05684i	−0.848982	−	0.528422i	$-0.822784\pi$
$59$	− 62.3538i	− 1.05684i	0.848982	−	0.528422i	$-0.177216\pi$
$60$	0	0
$61$	−96.9948	−1.59008	−0.795040	−	0.606557i	$-0.792550\pi$
$61$	−96.9948	−1.59008	−0.795040	+	0.606557i	$0.792550\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	24.0000	0.369231
$66$	0	0
$67$	48.4974i	0.723842i	0.932209	+	0.361921i	$0.117879\pi$
$67$	48.4974i	0.723842i	−0.932209	+	0.361921i	$0.882121\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	60.0000i	0.845070i	0.906347	+	0.422535i	$0.138860\pi$
$71$	60.0000i	0.845070i	−0.906347	+	0.422535i	$0.861140\pi$
$72$	0	0
$73$	86.0000	1.17808	0.589041	−	0.808103i	$-0.299506\pi$
$73$	86.0000	1.17808	0.589041	+	0.808103i	$0.299506\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	138.564	1.79953
$78$	0	0
$79$	− 38.0000i	− 0.481013i	−0.970648	−	0.240506i	$-0.922687\pi$
$79$	− 38.0000i	− 0.481013i	0.970648	−	0.240506i	$-0.0773135\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 13.8564i	− 0.166945i	−0.996510	−	0.0834723i	$-0.973399\pi$
$83$	− 13.8564i	− 0.166945i	0.996510	−	0.0834723i	$-0.0266010\pi$
$84$	0	0
$85$	−103.923	−1.22262
$86$	0	0
$87$	0	0
$88$	0	0
$89$	78.0000	0.876404	0.438202	−	0.898876i	$-0.355615\pi$
$89$	78.0000	0.876404	0.438202	+	0.898876i	$0.355615\pi$
$90$	0	0
$91$	69.2820i	0.761341i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 24.0000i	− 0.252632i
$96$	0	0
$97$	62.0000	0.639175	0.319588	−	0.947557i	$-0.396456\pi$
$97$	62.0000	0.639175	0.319588	+	0.947557i	$0.396456\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	65.8179	0.651663	0.325831	−	0.945428i	$-0.394356\pi$
$101$	65.8179	0.651663	0.325831	+	0.945428i	$0.394356\pi$
$102$	0	0
$103$	14.0000i	0.135922i	0.997688	+	0.0679612i	$0.0216494\pi$
$103$	14.0000i	0.135922i	−0.997688	+	0.0679612i	$0.978351\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	76.2102i	0.712245i	0.934439	+	0.356123i	$0.115901\pi$
$107$	76.2102i	0.712245i	−0.934439	+	0.356123i	$0.884099\pi$
$108$	0	0
$109$	20.7846	0.190684	0.0953422	−	0.995445i	$-0.469605\pi$
$109$	20.7846	0.190684	0.0953422	+	0.995445i	$0.469605\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.00000	0.0530973	0.0265487	−	0.999648i	$-0.491548\pi$
$113$	6.00000	0.0530973	0.0265487	+	0.999648i	$0.491548\pi$
$114$	0	0
$115$	− 41.5692i	− 0.361471i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 300.000i	− 2.52101i
$120$	0	0
$121$	−71.0000	−0.586777
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−131.636	−1.05309
$126$	0	0
$127$	− 82.0000i	− 0.645669i	−0.946455	−	0.322835i	$-0.895364\pi$
$127$	− 82.0000i	− 0.645669i	0.946455	−	0.322835i	$-0.104636\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	76.2102i	0.581758i	0.956760	+	0.290879i	$0.0939476\pi$
$131$	76.2102i	0.581758i	−0.956760	+	0.290879i	$0.906052\pi$
$132$	0	0
$133$	69.2820	0.520918
$134$	0	0
$135$	0	0
$136$	0	0
$137$	42.0000	0.306569	0.153285	−	0.988182i	$-0.451015\pi$
$137$	42.0000	0.306569	0.153285	+	0.988182i	$0.451015\pi$
$138$	0	0
$139$	− 48.4974i	− 0.348902i	−0.984666	−	0.174451i	$-0.944185\pi$
$139$	− 48.4974i	− 0.348902i	0.984666	−	0.174451i	$-0.0558151\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 96.0000i	− 0.671329i
$144$	0	0
$145$	180.000	1.24138
$146$	0	0
$147$	0	0
$148$	0	0
$149$	183.597	1.23220	0.616099	−	0.787669i	$-0.288712\pi$
$149$	183.597	1.23220	0.616099	+	0.787669i	$0.288712\pi$
$150$	0	0
$151$	− 134.000i	− 0.887417i	−0.896171	−	0.443709i	$-0.853663\pi$
$151$	− 134.000i	− 0.887417i	0.896171	−	0.443709i	$-0.146337\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 48.4974i	− 0.312887i
$156$	0	0
$157$	83.1384	0.529544	0.264772	−	0.964311i	$-0.414703\pi$
$157$	83.1384	0.529544	0.264772	+	0.964311i	$0.414703\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	120.000	0.745342
$162$	0	0
$163$	− 76.2102i	− 0.467547i	−0.972291	−	0.233774i	$-0.924893\pi$
$163$	− 76.2102i	− 0.467547i	0.972291	−	0.233774i	$-0.0751075\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 216.000i	− 1.29341i	−0.762739	−	0.646707i	$-0.776146\pi$
$167$	− 216.000i	− 1.29341i	0.762739	−	0.646707i	$-0.223854\pi$
$168$	0	0
$169$	−121.000	−0.715976
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−162.813	−0.941114	−0.470557	−	0.882370i	$-0.655947\pi$
$173$	−162.813	−0.941114	−0.470557	+	0.882370i	$0.655947\pi$
$174$	0	0
$175$	− 130.000i	− 0.742857i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 242.487i	− 1.35468i	−0.735672	−	0.677338i	$-0.763133\pi$
$179$	− 242.487i	− 1.35468i	0.735672	−	0.677338i	$-0.236867\pi$
$180$	0	0
$181$	297.913	1.64593	0.822963	−	0.568094i	$-0.192319\pi$
$181$	297.913	1.64593	0.822963	+	0.568094i	$0.192319\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−192.000	−1.03784
$186$	0	0
$187$	415.692i	2.22295i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 216.000i	− 1.13089i	−0.824786	−	0.565445i	$-0.808704\pi$
$191$	− 216.000i	− 1.13089i	0.824786	−	0.565445i	$-0.191296\pi$
$192$	0	0
$193$	−286.000	−1.48187	−0.740933	−	0.671579i	$-0.765616\pi$
$193$	−286.000	−1.48187	−0.740933	+	0.671579i	$0.765616\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−169.741	−0.861629	−0.430815	−	0.902440i	$-0.641774\pi$
$197$	−169.741	−0.861629	−0.430815	+	0.902440i	$0.641774\pi$
$198$	0	0
$199$	− 130.000i	− 0.653266i	−0.945151	−	0.326633i	$-0.894086\pi$
$199$	− 130.000i	− 0.653266i	0.945151	−	0.326633i	$-0.105914\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	519.615i	2.55968i
$204$	0	0
$205$	−20.7846	−0.101388
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−96.0000	−0.459330
$210$	0	0
$211$	325.626i	1.54325i	0.636078	+	0.771625i	$0.280556\pi$
$211$	325.626i	1.54325i	−0.636078	+	0.771625i	$0.719444\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 216.000i	− 1.00465i
$216$	0	0
$217$	140.000	0.645161
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−207.846	−0.940480
$222$	0	0
$223$	302.000i	1.35426i	0.735863	+	0.677130i	$0.236777\pi$
$223$	302.000i	1.35426i	−0.735863	+	0.677130i	$0.763223\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 166.277i	− 0.732497i	−0.930517	−	0.366249i	$-0.880642\pi$
$227$	− 166.277i	− 0.732497i	0.930517	−	0.366249i	$-0.119358\pi$
$228$	0	0
$229$	−20.7846	−0.0907625	−0.0453812	−	0.998970i	$-0.514450\pi$
$229$	−20.7846	−0.0907625	−0.0453812	+	0.998970i	$0.514450\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−282.000	−1.21030	−0.605150	−	0.796111i	$-0.706887\pi$
$233$	−282.000	−1.21030	−0.605150	+	0.796111i	$0.706887\pi$
$234$	0	0
$235$	− 290.985i	− 1.23823i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 192.000i	− 0.803347i	−0.915783	−	0.401674i	$-0.868429\pi$
$239$	− 192.000i	− 0.803347i	0.915783	−	0.401674i	$-0.131571\pi$
$240$	0	0
$241$	−182.000	−0.755187	−0.377593	−	0.925972i	$-0.623248\pi$
$241$	−182.000	−0.755187	−0.377593	+	0.925972i	$0.623248\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−176.669	−0.721099
$246$	0	0
$247$	− 48.0000i	− 0.194332i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 290.985i	− 1.15930i	−0.814865	−	0.579650i	$-0.803189\pi$
$251$	− 290.985i	− 1.15930i	0.814865	−	0.579650i	$-0.196811\pi$
$252$	0	0
$253$	−166.277	−0.657221
$254$	0	0
$255$	0	0
$256$	0	0
$257$	78.0000	0.303502	0.151751	−	0.988419i	$-0.451509\pi$
$257$	78.0000	0.303502	0.151751	+	0.988419i	$0.451509\pi$
$258$	0	0
$259$	− 554.256i	− 2.13999i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	48.0000i	0.182510i	0.995828	+	0.0912548i	$0.0290878\pi$
$263$	48.0000i	0.182510i	−0.995828	+	0.0912548i	$0.970912\pi$
$264$	0	0
$265$	−60.0000	−0.226415
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−280.592	−1.04309	−0.521547	−	0.853223i	$-0.674645\pi$
$269$	−280.592	−1.04309	−0.521547	+	0.853223i	$0.674645\pi$
$270$	0	0
$271$	− 130.000i	− 0.479705i	−0.970809	−	0.239852i	$-0.922901\pi$
$271$	− 130.000i	− 0.479705i	0.970809	−	0.239852i	$-0.0770990\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	180.133i	0.655030i
$276$	0	0
$277$	270.200	0.975451	0.487725	−	0.872997i	$-0.337827\pi$
$277$	270.200	0.975451	0.487725	+	0.872997i	$0.337827\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−234.000	−0.832740	−0.416370	−	0.909195i	$-0.636698\pi$
$281$	−234.000	−0.832740	−0.416370	+	0.909195i	$0.636698\pi$
$282$	0	0
$283$	− 311.769i	− 1.10166i	−0.834618	−	0.550829i	$-0.814312\pi$
$283$	− 311.769i	− 1.10166i	0.834618	−	0.550829i	$-0.185688\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 60.0000i	− 0.209059i
$288$	0	0
$289$	611.000	2.11419
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−155.885	−0.532029	−0.266015	−	0.963969i	$-0.585707\pi$
$293$	−155.885	−0.532029	−0.266015	+	0.963969i	$0.585707\pi$
$294$	0	0
$295$	− 216.000i	− 0.732203i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 83.1384i	− 0.278055i
$300$	0	0
$301$	623.538	2.07156
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−336.000	−1.10164
$306$	0	0
$307$	103.923i	0.338512i	0.985572	+	0.169256i	$0.0541364\pi$
$307$	103.923i	0.338512i	−0.985572	+	0.169256i	$0.945864\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	24.0000i	0.0771704i	0.999255	+	0.0385852i	$0.0122851\pi$
$311$	24.0000i	0.0771704i	−0.999255	+	0.0385852i	$0.987715\pi$
$312$	0	0
$313$	−190.000	−0.607029	−0.303514	−	0.952827i	$-0.598160\pi$
$313$	−190.000	−0.607029	−0.303514	+	0.952827i	$0.598160\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	107.387	0.338761	0.169380	−	0.985551i	$-0.445823\pi$
$317$	107.387	0.338761	0.169380	+	0.985551i	$0.445823\pi$
$318$	0	0
$319$	− 720.000i	− 2.25705i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	207.846i	0.643486i
$324$	0	0
$325$	−90.0666	−0.277128
$326$	0	0
$327$	0	0
$328$	0	0
$329$	840.000	2.55319
$330$	0	0
$331$	34.6410i	0.104656i	0.998630	+	0.0523278i	$0.0166641\pi$
$331$	34.6410i	0.104656i	−0.998630	+	0.0523278i	$0.983336\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	168.000i	0.501493i
$336$	0	0
$337$	−490.000	−1.45401	−0.727003	−	0.686634i	$-0.759087\pi$
$337$	−490.000	−1.45401	−0.727003	+	0.686634i	$0.759087\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−193.990	−0.568885
$342$	0	0
$343$	− 20.0000i	− 0.0583090i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 443.405i	− 1.27782i	−0.769280	−	0.638912i	$-0.779385\pi$
$347$	− 443.405i	− 1.27782i	0.769280	−	0.638912i	$-0.220615\pi$
$348$	0	0
$349$	96.9948	0.277922	0.138961	−	0.990298i	$-0.455624\pi$
$349$	96.9948	0.277922	0.138961	+	0.990298i	$0.455624\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	246.000	0.696884	0.348442	−	0.937330i	$-0.386711\pi$
$353$	246.000	0.696884	0.348442	+	0.937330i	$0.386711\pi$
$354$	0	0
$355$	207.846i	0.585482i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 708.000i	− 1.97214i	−0.166316	−	0.986072i	$-0.553187\pi$
$359$	− 708.000i	− 1.97214i	0.166316	−	0.986072i	$-0.446813\pi$
$360$	0	0
$361$	313.000	0.867036
$362$	0	0
$363$	0	0
$364$	0	0
$365$	297.913	0.816199
$366$	0	0
$367$	82.0000i	0.223433i	0.993740	+	0.111717i	$0.0356349\pi$
$367$	82.0000i	0.223433i	−0.993740	+	0.111717i	$0.964365\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 173.205i	− 0.466860i
$372$	0	0
$373$	−512.687	−1.37450	−0.687248	−	0.726423i	$-0.741181\pi$
$373$	−512.687	−1.37450	−0.687248	+	0.726423i	$0.741181\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	360.000	0.954907
$378$	0	0
$379$	533.472i	1.40758i	0.710410	+	0.703788i	$0.248510\pi$
$379$	533.472i	1.40758i	−0.710410	+	0.703788i	$0.751490\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 408.000i	− 1.06527i	−0.846344	−	0.532637i	$-0.821201\pi$
$383$	− 408.000i	− 1.06527i	0.846344	−	0.532637i	$-0.178799\pi$
$384$	0	0
$385$	480.000	1.24675
$386$	0	0
$387$	0	0
$388$	0	0
$389$	142.028	0.365111	0.182555	−	0.983196i	$-0.441563\pi$
$389$	142.028	0.365111	0.182555	+	0.983196i	$0.441563\pi$
$390$	0	0
$391$	360.000i	0.920716i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 131.636i	− 0.333255i
$396$	0	0
$397$	−41.5692	−0.104708	−0.0523542	−	0.998629i	$-0.516672\pi$
$397$	−41.5692	−0.104708	−0.0523542	+	0.998629i	$0.516672\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−270.000	−0.673317	−0.336658	−	0.941627i	$-0.609297\pi$
$401$	−270.000	−0.673317	−0.336658	+	0.941627i	$0.609297\pi$
$402$	0	0
$403$	− 96.9948i	− 0.240682i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	768.000i	1.88698i
$408$	0	0
$409$	110.000	0.268949	0.134474	−	0.990917i	$-0.457065\pi$
$409$	110.000	0.268949	0.134474	+	0.990917i	$0.457065\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	623.538	1.50978
$414$	0	0
$415$	− 48.0000i	− 0.115663i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	415.692i	0.992105i	0.868292	+	0.496053i	$0.165218\pi$
$419$	415.692i	0.992105i	−0.868292	+	0.496053i	$0.834782\pi$
$420$	0	0
$421$	575.041	1.36589	0.682946	−	0.730468i	$-0.260698\pi$
$421$	575.041	1.36589	0.682946	+	0.730468i	$0.260698\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	390.000	0.917647
$426$	0	0
$427$	− 969.948i	− 2.27154i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 396.000i	− 0.918794i	−0.888231	−	0.459397i	$-0.848066\pi$
$431$	− 396.000i	− 0.918794i	0.888231	−	0.459397i	$-0.151934\pi$
$432$	0	0
$433$	−386.000	−0.891455	−0.445727	−	0.895169i	$-0.647055\pi$
$433$	−386.000	−0.891455	−0.445727	+	0.895169i	$0.647055\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−83.1384	−0.190248
$438$	0	0
$439$	470.000i	1.07062i	0.844657	+	0.535308i	$0.179804\pi$
$439$	470.000i	1.07062i	−0.844657	+	0.535308i	$0.820196\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 332.554i	− 0.750686i	−0.926886	−	0.375343i	$-0.877525\pi$
$443$	− 332.554i	− 0.750686i	0.926886	−	0.375343i	$-0.122475\pi$
$444$	0	0
$445$	270.200	0.607191
$446$	0	0
$447$	0	0
$448$	0	0
$449$	786.000	1.75056	0.875278	−	0.483619i	$-0.160678\pi$
$449$	786.000	1.75056	0.875278	+	0.483619i	$0.160678\pi$
$450$	0	0
$451$	83.1384i	0.184342i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	240.000i	0.527473i
$456$	0	0
$457$	−34.0000	−0.0743982	−0.0371991	−	0.999308i	$-0.511844\pi$
$457$	−34.0000	−0.0743982	−0.0371991	+	0.999308i	$0.511844\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	349.874	0.758946	0.379473	−	0.925203i	$-0.376105\pi$
$461$	349.874	0.758946	0.379473	+	0.925203i	$0.376105\pi$
$462$	0	0
$463$	614.000i	1.32613i	0.748560	+	0.663067i	$0.230746\pi$
$463$	614.000i	1.32613i	−0.748560	+	0.663067i	$0.769254\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	443.405i	0.949475i	0.880127	+	0.474738i	$0.157457\pi$
$467$	443.405i	0.949475i	−0.880127	+	0.474738i	$0.842543\pi$
$468$	0	0
$469$	−484.974	−1.03406
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−864.000	−1.82664
$474$	0	0
$475$	90.0666i	0.189614i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	300.000i	0.626305i	0.949703	+	0.313152i	$0.101385\pi$
$479$	300.000i	0.626305i	−0.949703	+	0.313152i	$0.898615\pi$
$480$	0	0
$481$	−384.000	−0.798337
$482$	0	0
$483$	0	0
$484$	0	0
$485$	214.774	0.442834
$486$	0	0
$487$	− 638.000i	− 1.31006i	−0.755602	−	0.655031i	$-0.772656\pi$
$487$	− 638.000i	− 1.31006i	0.755602	−	0.655031i	$-0.227344\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	672.036i	1.36871i	0.729150	+	0.684354i	$0.239916\pi$
$491$	672.036i	1.36871i	−0.729150	+	0.684354i	$0.760084\pi$
$492$	0	0
$493$	−1558.85	−3.16196
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−600.000	−1.20724
$498$	0	0
$499$	561.184i	1.12462i	0.826927	+	0.562309i	$0.190087\pi$
$499$	561.184i	1.12462i	−0.826927	+	0.562309i	$0.809913\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	588.000i	1.16899i	0.811399	+	0.584493i	$0.198707\pi$
$503$	588.000i	1.16899i	−0.811399	+	0.584493i	$0.801293\pi$
$504$	0	0
$505$	228.000	0.451485
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−169.741	−0.333479	−0.166740	−	0.986001i	$-0.553324\pi$
$509$	−169.741	−0.333479	−0.166740	+	0.986001i	$0.553324\pi$
$510$	0	0
$511$	860.000i	1.68297i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	48.4974i	0.0941698i
$516$	0	0
$517$	−1163.94	−2.25133
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−366.000	−0.702495	−0.351248	−	0.936283i	$-0.614242\pi$
$521$	−366.000	−0.702495	−0.351248	+	0.936283i	$0.614242\pi$
$522$	0	0
$523$	367.195i	0.702093i	0.936358	+	0.351047i	$0.114174\pi$
$523$	367.195i	0.702093i	−0.936358	+	0.351047i	$0.885826\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	420.000i	0.796964i
$528$	0	0
$529$	385.000	0.727788
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−41.5692	−0.0779910
$534$	0	0
$535$	264.000i	0.493458i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	706.677i	1.31109i
$540$	0	0
$541$	48.4974	0.0896440	0.0448220	−	0.998995i	$-0.485728\pi$
$541$	48.4974	0.0896440	0.0448220	+	0.998995i	$0.485728\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	72.0000	0.132110
$546$	0	0
$547$	422.620i	0.772615i	0.922370	+	0.386307i	$0.126250\pi$
$547$	422.620i	0.772615i	−0.922370	+	0.386307i	$0.873750\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 360.000i	− 0.653358i
$552$	0	0
$553$	380.000	0.687161
$554$	0	0
$555$	0	0
$556$	0	0
$557$	557.720	1.00129	0.500647	−	0.865652i	$-0.333096\pi$
$557$	557.720	1.00129	0.500647	+	0.865652i	$0.333096\pi$
$558$	0	0
$559$	− 432.000i	− 0.772809i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 775.959i	− 1.37826i	−0.724639	−	0.689129i	$-0.757994\pi$
$563$	− 775.959i	− 1.37826i	0.724639	−	0.689129i	$-0.242006\pi$
$564$	0	0
$565$	20.7846	0.0367869
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−54.0000	−0.0949033	−0.0474517	−	0.998874i	$-0.515110\pi$
$569$	−54.0000	−0.0949033	−0.0474517	+	0.998874i	$0.515110\pi$
$570$	0	0
$571$	214.774i	0.376137i	0.982156	+	0.188069i	$0.0602227\pi$
$571$	214.774i	0.376137i	−0.982156	+	0.188069i	$0.939777\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	156.000i	0.271304i
$576$	0	0
$577$	46.0000	0.0797227	0.0398614	−	0.999205i	$-0.487308\pi$
$577$	46.0000	0.0797227	0.0398614	+	0.999205i	$0.487308\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	138.564	0.238492
$582$	0	0
$583$	240.000i	0.411664i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 630.466i	− 1.07405i	−0.843567	−	0.537024i	$-0.819548\pi$
$587$	− 630.466i	− 1.07405i	0.843567	−	0.537024i	$-0.180452\pi$
$588$	0	0
$589$	−96.9948	−0.164677
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−858.000	−1.44688	−0.723440	−	0.690387i	$-0.757440\pi$
$593$	−858.000	−1.44688	−0.723440	+	0.690387i	$0.757440\pi$
$594$	0	0
$595$	− 1039.23i	− 1.74661i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 228.000i	− 0.380634i	−0.981723	−	0.190317i	$-0.939048\pi$
$599$	− 228.000i	− 0.380634i	0.981723	−	0.190317i	$-0.0609516\pi$
$600$	0	0
$601$	−1010.00	−1.68053	−0.840266	−	0.542174i	$-0.817601\pi$
$601$	−1010.00	−1.68053	−0.840266	+	0.542174i	$0.817601\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−245.951	−0.406531
$606$	0	0
$607$	− 850.000i	− 1.40033i	−0.713981	−	0.700165i	$-0.753110\pi$
$607$	− 850.000i	− 1.40033i	0.713981	−	0.700165i	$-0.246890\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 581.969i	− 0.952486i
$612$	0	0
$613$	429.549	0.700732	0.350366	−	0.936613i	$-0.386057\pi$
$613$	429.549	0.700732	0.350366	+	0.936613i	$0.386057\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−234.000	−0.379254	−0.189627	−	0.981856i	$-0.560728\pi$
$617$	−234.000	−0.379254	−0.189627	+	0.981856i	$0.560728\pi$
$618$	0	0
$619$	1129.30i	1.82439i	0.409757	+	0.912195i	$0.365614\pi$
$619$	1129.30i	1.82439i	−0.409757	+	0.912195i	$0.634386\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	780.000i	1.25201i
$624$	0	0
$625$	−131.000	−0.209600
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1662.77	2.64351
$630$	0	0
$631$	− 446.000i	− 0.706815i	−0.935470	−	0.353407i	$-0.885023\pi$
$631$	− 446.000i	− 0.706815i	0.935470	−	0.353407i	$-0.114977\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 284.056i	− 0.447333i
$636$	0	0
$637$	−353.338	−0.554691
$638$	0	0
$639$	0	0
$640$	0	0
$641$	522.000	0.814353	0.407176	−	0.913350i	$-0.366513\pi$
$641$	522.000	0.814353	0.407176	+	0.913350i	$0.366513\pi$
$642$	0	0
$643$	145.492i	0.226271i	0.993580	+	0.113136i	$0.0360894\pi$
$643$	145.492i	0.226271i	−0.993580	+	0.113136i	$0.963911\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	996.000i	1.53941i	0.638398	+	0.769706i	$0.279597\pi$
$647$	996.000i	1.53941i	−0.638398	+	0.769706i	$0.720403\pi$
$648$	0	0
$649$	−864.000	−1.33128
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−994.197	−1.52251	−0.761254	−	0.648454i	$-0.775416\pi$
$653$	−994.197	−1.52251	−0.761254	+	0.648454i	$0.775416\pi$
$654$	0	0
$655$	264.000i	0.403053i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	48.4974i	0.0735924i	0.999323	+	0.0367962i	$0.0117152\pi$
$659$	48.4974i	0.0735924i	−0.999323	+	0.0367962i	$0.988285\pi$
$660$	0	0
$661$	138.564	0.209628	0.104814	−	0.994492i	$-0.466575\pi$
$661$	138.564	0.209628	0.104814	+	0.994492i	$0.466575\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	240.000	0.360902
$666$	0	0
$667$	− 623.538i	− 0.934840i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1344.00i	2.00298i
$672$	0	0
$673$	46.0000	0.0683507	0.0341753	−	0.999416i	$-0.489120\pi$
$673$	46.0000	0.0683507	0.0341753	+	0.999416i	$0.489120\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	696.284	1.02849	0.514243	−	0.857645i	$-0.328073\pi$
$677$	696.284	1.02849	0.514243	+	0.857645i	$0.328073\pi$
$678$	0	0
$679$	620.000i	0.913108i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	124.708i	0.182588i	0.995824	+	0.0912940i	$0.0291003\pi$
$683$	124.708i	0.182588i	−0.995824	+	0.0912940i	$0.970900\pi$
$684$	0	0
$685$	145.492	0.212397
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−120.000	−0.174165
$690$	0	0
$691$	− 602.754i	− 0.872292i	−0.899876	−	0.436146i	$-0.856343\pi$
$691$	− 602.754i	− 0.872292i	0.899876	−	0.436146i	$-0.143657\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 168.000i	− 0.241727i
$696$	0	0
$697$	180.000	0.258250
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−128.172	−0.182841	−0.0914207	−	0.995812i	$-0.529141\pi$
$701$	−128.172	−0.182841	−0.0914207	+	0.995812i	$0.529141\pi$
$702$	0	0
$703$	384.000i	0.546230i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	658.179i	0.930947i
$708$	0	0
$709$	533.472	0.752428	0.376214	−	0.926533i	$-0.377226\pi$
$709$	533.472	0.752428	0.376214	+	0.926533i	$0.377226\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−168.000	−0.235624
$714$	0	0
$715$	− 332.554i	− 0.465110i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1332.00i	1.85257i	0.376820	+	0.926287i	$0.377018\pi$
$719$	1332.00i	1.85257i	−0.376820	+	0.926287i	$0.622982\pi$
$720$	0	0
$721$	−140.000	−0.194175
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−675.500	−0.931724
$726$	0	0
$727$	754.000i	1.03714i	0.855036	+	0.518569i	$0.173535\pi$
$727$	754.000i	1.03714i	−0.855036	+	0.518569i	$0.826465\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1870.61i	2.55898i
$732$	0	0
$733$	727.461	0.992444	0.496222	−	0.868196i	$-0.334720\pi$
$733$	727.461	0.992444	0.496222	+	0.868196i	$0.334720\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	672.000	0.911805
$738$	0	0
$739$	145.492i	0.196877i	0.995143	+	0.0984386i	$0.0313848\pi$
$739$	145.492i	0.196877i	−0.995143	+	0.0984386i	$0.968615\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	816.000i	1.09825i	0.835740	+	0.549125i	$0.185039\pi$
$743$	816.000i	1.09825i	−0.835740	+	0.549125i	$0.814961\pi$
$744$	0	0
$745$	636.000	0.853691
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−762.102	−1.01749
$750$	0	0
$751$	466.000i	0.620506i	0.950654	+	0.310253i	$0.100414\pi$
$751$	466.000i	0.620506i	−0.950654	+	0.310253i	$0.899586\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 464.190i	− 0.614821i
$756$	0	0
$757$	450.333	0.594892	0.297446	−	0.954739i	$-0.403865\pi$
$757$	450.333	0.594892	0.297446	+	0.954739i	$0.403865\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	18.0000	0.0236531	0.0118265	−	0.999930i	$-0.496235\pi$
$761$	18.0000	0.0236531	0.0118265	+	0.999930i	$0.496235\pi$
$762$	0	0
$763$	207.846i	0.272406i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 432.000i	− 0.563233i
$768$	0	0
$769$	1298.00	1.68791	0.843953	−	0.536417i	$-0.180222\pi$
$769$	1298.00	1.68791	0.843953	+	0.536417i	$0.180222\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−197.454	−0.255438	−0.127719	−	0.991810i	$-0.540766\pi$
$773$	−197.454	−0.255438	−0.127719	+	0.991810i	$0.540766\pi$
$774$	0	0
$775$	182.000i	0.234839i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	41.5692i	0.0533623i
$780$	0	0
$781$	831.384	1.06451
$782$	0	0
$783$	0	0
$784$	0	0
$785$	288.000	0.366879
$786$	0	0
$787$	− 256.344i	− 0.325722i	−0.986649	−	0.162861i	$-0.947928\pi$
$787$	− 256.344i	− 0.325722i	0.986649	−	0.162861i	$-0.0520723\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	60.0000i	0.0758534i
$792$	0	0
$793$	−672.000	−0.847415
$794$	0	0
$795$	0	0
$796$	0	0
$797$	959.556	1.20396	0.601980	−	0.798511i	$-0.294379\pi$
$797$	959.556	1.20396	0.601980	+	0.798511i	$0.294379\pi$
$798$	0	0
$799$	2520.00i	3.15394i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 1191.65i	− 1.48400i
$804$	0	0
$805$	415.692	0.516388
$806$	0	0
$807$	0	0
$808$	0	0
$809$	378.000	0.467244	0.233622	−	0.972328i	$-0.424942\pi$
$809$	378.000	0.467244	0.233622	+	0.972328i	$0.424942\pi$
$810$	0	0
$811$	228.631i	0.281912i	0.990016	+	0.140956i	$0.0450176\pi$
$811$	228.631i	0.281912i	−0.990016	+	0.140956i	$0.954982\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 264.000i	− 0.323926i
$816$	0	0
$817$	−432.000	−0.528764
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1236.68	−1.50631	−0.753157	−	0.657840i	$-0.771470\pi$
$821$	−1236.68	−1.50631	−0.753157	+	0.657840i	$0.771470\pi$
$822$	0	0
$823$	− 758.000i	− 0.921021i	−0.887654	−	0.460510i	$-0.847666\pi$
$823$	− 758.000i	− 0.921021i	0.887654	−	0.460510i	$-0.152334\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 644.323i	− 0.779109i	−0.921003	−	0.389554i	$-0.872629\pi$
$827$	− 644.323i	− 0.779109i	0.921003	−	0.389554i	$-0.127371\pi$
$828$	0	0
$829$	−921.451	−1.11152	−0.555761	−	0.831342i	$-0.687573\pi$
$829$	−921.451	−1.11152	−0.555761	+	0.831342i	$0.687573\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1530.00	1.83673
$834$	0	0
$835$	− 748.246i	− 0.896103i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	612.000i	0.729440i	0.931117	+	0.364720i	$0.118835\pi$
$839$	612.000i	0.729440i	−0.931117	+	0.364720i	$0.881165\pi$
$840$	0	0
$841$	1859.00	2.21046
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−419.156	−0.496043
$846$	0	0
$847$	− 710.000i	− 0.838253i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	665.108i	0.781560i
$852$	0	0
$853$	−55.4256	−0.0649773	−0.0324886	−	0.999472i	$-0.510343\pi$
$853$	−55.4256	−0.0649773	−0.0324886	+	0.999472i	$0.510343\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−582.000	−0.679113	−0.339557	−	0.940586i	$-0.610277\pi$
$857$	−582.000	−0.679113	−0.339557	+	0.940586i	$0.610277\pi$
$858$	0	0
$859$	− 1364.86i	− 1.58889i	−0.607336	−	0.794445i	$-0.707762\pi$
$859$	− 1364.86i	− 1.58889i	0.607336	−	0.794445i	$-0.292238\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	456.000i	0.528389i	0.964469	+	0.264195i	$0.0851061\pi$
$863$	456.000i	0.528389i	−0.964469	+	0.264195i	$0.914894\pi$
$864$	0	0
$865$	−564.000	−0.652023
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−526.543	−0.605919
$870$	0	0
$871$	336.000i	0.385763i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 1316.36i	− 1.50441i
$876$	0	0
$877$	13.8564	0.0157998	0.00789989	−	0.999969i	$-0.497485\pi$
$877$	13.8564	0.0157998	0.00789989	+	0.999969i	$0.497485\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1362.00	−1.54597	−0.772985	−	0.634424i	$-0.781237\pi$
$881$	−1362.00	−1.54597	−0.772985	+	0.634424i	$0.781237\pi$
$882$	0	0
$883$	1586.56i	1.79678i	0.439197	+	0.898391i	$0.355263\pi$
$883$	1586.56i	1.79678i	−0.439197	+	0.898391i	$0.644737\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	240.000i	0.270575i	0.990806	+	0.135287i	$0.0431958\pi$
$887$	240.000i	0.270575i	−0.990806	+	0.135287i	$0.956804\pi$
$888$	0	0
$889$	820.000	0.922385
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−581.969	−0.651701
$894$	0	0
$895$	− 840.000i	− 0.938547i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 727.461i	− 0.809189i
$900$	0	0
$901$	519.615	0.576709
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1032.00	1.14033
$906$	0	0
$907$	159.349i	0.175688i	0.996134	+	0.0878438i	$0.0279976\pi$
$907$	159.349i	0.175688i	−0.996134	+	0.0878438i	$0.972002\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1392.00i	− 1.52799i	−0.645221	−	0.763996i	$-0.723235\pi$
$911$	− 1392.00i	− 1.52799i	0.645221	−	0.763996i	$-0.276765\pi$
$912$	0	0
$913$	−192.000	−0.210296
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−762.102	−0.831082
$918$	0	0
$919$	− 422.000i	− 0.459195i	−0.973286	−	0.229597i	$-0.926259\pi$
$919$	− 422.000i	− 0.459195i	0.973286	−	0.229597i	$-0.0737409\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	415.692i	0.450371i
$924$	0	0
$925$	720.533	0.778955
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1026.00	1.10441	0.552207	−	0.833707i	$-0.313786\pi$
$929$	1026.00	1.10441	0.552207	+	0.833707i	$0.313786\pi$
$930$	0	0
$931$	353.338i	0.379526i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1440.00i	1.54011i
$936$	0	0
$937$	−814.000	−0.868730	−0.434365	−	0.900737i	$-0.643027\pi$
$937$	−814.000	−0.868730	−0.434365	+	0.900737i	$0.643027\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−980.341	−1.04181	−0.520904	−	0.853615i	$-0.674405\pi$
$941$	−980.341	−1.04181	−0.520904	+	0.853615i	$0.674405\pi$
$942$	0	0
$943$	72.0000i	0.0763521i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1503.42i	1.58756i	0.608204	+	0.793780i	$0.291890\pi$
$947$	1503.42i	1.58756i	−0.608204	+	0.793780i	$0.708110\pi$
$948$	0	0
$949$	595.825	0.627846
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−366.000	−0.384050	−0.192025	−	0.981390i	$-0.561506\pi$
$953$	−366.000	−0.384050	−0.192025	+	0.981390i	$0.561506\pi$
$954$	0	0
$955$	− 748.246i	− 0.783504i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	420.000i	0.437956i
$960$	0	0
$961$	765.000	0.796046
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−990.733	−1.02667
$966$	0	0
$967$	686.000i	0.709411i	0.934978	+	0.354705i	$0.115419\pi$
$967$	686.000i	0.709411i	−0.934978	+	0.354705i	$0.884581\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1344.07i	− 1.38421i	−0.721795	−	0.692107i	$-0.756683\pi$
$971$	− 1344.07i	− 1.38421i	0.721795	−	0.692107i	$-0.243317\pi$
$972$	0	0
$973$	484.974	0.498432
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1470.00	−1.50461	−0.752303	−	0.658817i	$-0.771057\pi$
$977$	−1470.00	−1.50461	−0.752303	+	0.658817i	$0.771057\pi$
$978$	0	0
$979$	− 1080.80i	− 1.10398i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 720.000i	− 0.732452i	−0.930526	−	0.366226i	$-0.880650\pi$
$983$	− 720.000i	− 0.732452i	0.930526	−	0.366226i	$-0.119350\pi$
$984$	0	0
$985$	−588.000	−0.596954
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−748.246	−0.756568
$990$	0	0
$991$	− 1598.00i	− 1.61251i	−0.591566	−	0.806256i	$-0.701490\pi$
$991$	− 1598.00i	− 1.61251i	0.591566	−	0.806256i	$-0.298510\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 450.333i	− 0.452596i
$996$	0	0
$997$	−1454.92	−1.45930	−0.729650	−	0.683820i	$-0.760317\pi$
$997$	−1454.92	−1.45930	−0.729650	+	0.683820i	$0.760317\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.3.g.q.1279.4		4
3.2	odd	2		768.3.g.f.511.3		4
4.3	odd	2	inner	2304.3.g.q.1279.3		4
8.3	odd	2	inner	2304.3.g.q.1279.1		4
8.5	even	2	inner	2304.3.g.q.1279.2		4
12.11	even	2		768.3.g.f.511.1		4
16.3	odd	4		576.3.b.a.415.2		4
16.5	even	4		576.3.b.a.415.3		4
16.11	odd	4		576.3.b.a.415.4		4
16.13	even	4		576.3.b.a.415.1		4
24.5	odd	2		768.3.g.f.511.2		4
24.11	even	2		768.3.g.f.511.4		4
48.5	odd	4		192.3.b.b.31.3	yes	4
48.11	even	4		192.3.b.b.31.1	✓	4
48.29	odd	4		192.3.b.b.31.2	yes	4
48.35	even	4		192.3.b.b.31.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.3.b.b.31.1	✓	4	48.11	even	4
192.3.b.b.31.2	yes	4	48.29	odd	4
192.3.b.b.31.3	yes	4	48.5	odd	4
192.3.b.b.31.4	yes	4	48.35	even	4
576.3.b.a.415.1		4	16.13	even	4
576.3.b.a.415.2		4	16.3	odd	4
576.3.b.a.415.3		4	16.5	even	4
576.3.b.a.415.4		4	16.11	odd	4
768.3.g.f.511.1		4	12.11	even	2
768.3.g.f.511.2		4	24.5	odd	2
768.3.g.f.511.3		4	3.2	odd	2
768.3.g.f.511.4		4	24.11	even	2
2304.3.g.q.1279.1		4	8.3	odd	2	inner
2304.3.g.q.1279.2		4	8.5	even	2	inner
2304.3.g.q.1279.3		4	4.3	odd	2	inner
2304.3.g.q.1279.4		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:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2304.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.46410 q^{5} +10.0000i q^{7} +O(q^{10})\)	\(q+3.46410 q^{5} +10.0000i q^{7} -13.8564i q^{11} +6.92820 q^{13} -30.0000 q^{17} -6.92820i q^{19} -12.0000i q^{23} -13.0000 q^{25} +51.9615 q^{29} -14.0000i q^{31} +34.6410i q^{35} -55.4256 q^{37} -6.00000 q^{41} -62.3538i q^{43} -84.0000i q^{47} -51.0000 q^{49} -17.3205 q^{53} -48.0000i q^{55} -62.3538i q^{59} -96.9948 q^{61} +24.0000 q^{65} +48.4974i q^{67} +60.0000i q^{71} +86.0000 q^{73} +138.564 q^{77} -38.0000i q^{79} -13.8564i q^{83} -103.923 q^{85} +78.0000 q^{89} +69.2820i q^{91} -24.0000i q^{95} +62.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 120 q^{17} - 52 q^{25} - 24 q^{41} - 204 q^{49} + 96 q^{65} + 344 q^{73} + 312 q^{89} + 248 q^{97}+O(q^{100}) \) 4 * q - 120 * q^17 - 52 * q^25 - 24 * q^41 - 204 * q^49 + 96 * q^65 + 344 * q^73 + 312 * q^89 + 248 * q^97

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