LMFDB - Embedded newform 2880.2.h.d.1151.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(1151,2880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2880, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2880.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.h (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:	$22.9969157821$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1440)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.4
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2880.1151
Dual form			2880.2.h.d.1151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{5} -0.585786i q^{7} +O(q^{10})$	$q+1.00000i q^{5} -0.585786i q^{7} +5.41421 q^{11} +0.585786 q^{13} +6.82843i q^{17} +0.828427 q^{23} -1.00000 q^{25} -6.00000i q^{29} +10.4853i q^{31} +0.585786 q^{35} -5.07107 q^{37} -3.07107i q^{41} -1.17157i q^{43} -5.65685 q^{47} +6.65685 q^{49} -6.82843i q^{53} +5.41421i q^{55} +9.41421 q^{59} -7.17157 q^{61} +0.585786i q^{65} +8.00000i q^{67} +5.65685 q^{71} +6.48528 q^{73} -3.17157i q^{77} +2.48528i q^{79} +14.8284 q^{83} -6.82843 q^{85} +4.24264i q^{89} -0.343146i q^{91} -14.4853 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 16 q^{11} + 8 q^{13} - 8 q^{23} - 4 q^{25} + 8 q^{35} + 8 q^{37} + 4 q^{49} + 32 q^{59} - 40 q^{61} - 8 q^{73} + 48 q^{83} - 16 q^{85} - 24 q^{97}+O(q^{100}) $ 4 * q + 16 * q^11 + 8 * q^13 - 8 * q^23 - 4 * q^25 + 8 * q^35 + 8 * q^37 + 4 * q^49 + 32 * q^59 - 40 * q^61 - 8 * q^73 + 48 * q^83 - 16 * q^85 - 24 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	− 0.585786i	− 0.221406i	−0.993854	−	0.110703i	$-0.964690\pi$
$7$	− 0.585786i	− 0.221406i	0.993854	−	0.110703i	$-0.0353103\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.41421	1.63245	0.816223	−	0.577736i	$-0.196064\pi$
$11$	5.41421	1.63245	0.816223	+	0.577736i	$0.196064\pi$
$12$	0	0
$13$	0.585786	0.162468	0.0812340	−	0.996695i	$-0.474114\pi$
$13$	0.585786	0.162468	0.0812340	+	0.996695i	$0.474114\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.82843i	1.65614i	0.560627	+	0.828068i	$0.310560\pi$
$17$	6.82843i	1.65614i	−0.560627	+	0.828068i	$0.689440\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.828427	0.172739	0.0863695	−	0.996263i	$-0.472473\pi$
$23$	0.828427	0.172739	0.0863695	+	0.996263i	$0.472473\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 6.00000i	− 1.11417i	−0.830455	−	0.557086i	$-0.811919\pi$
$29$	− 6.00000i	− 1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$	0	0
$31$	10.4853i	1.88321i	0.336717	+	0.941606i	$0.390684\pi$
$31$	10.4853i	1.88321i	−0.336717	+	0.941606i	$0.609316\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.585786	0.0990160
$36$	0	0
$37$	−5.07107	−0.833678	−0.416839	−	0.908980i	$-0.636862\pi$
$37$	−5.07107	−0.833678	−0.416839	+	0.908980i	$0.636862\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 3.07107i	− 0.479620i	−0.970820	−	0.239810i	$-0.922915\pi$
$41$	− 3.07107i	− 0.479620i	0.970820	−	0.239810i	$-0.0770852\pi$
$42$	0	0
$43$	− 1.17157i	− 0.178663i	−0.996002	−	0.0893316i	$-0.971527\pi$
$43$	− 1.17157i	− 0.178663i	0.996002	−	0.0893316i	$-0.0284731\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.65685	−0.825137	−0.412568	−	0.910927i	$-0.635368\pi$
$47$	−5.65685	−0.825137	−0.412568	+	0.910927i	$0.635368\pi$
$48$	0	0
$49$	6.65685	0.950979
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 6.82843i	− 0.937957i	−0.883210	−	0.468978i	$-0.844622\pi$
$53$	− 6.82843i	− 0.937957i	0.883210	−	0.468978i	$-0.155378\pi$
$54$	0	0
$55$	5.41421i	0.730052i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.41421	1.22563	0.612813	−	0.790228i	$-0.290038\pi$
$59$	9.41421	1.22563	0.612813	+	0.790228i	$0.290038\pi$
$60$	0	0
$61$	−7.17157	−0.918226	−0.459113	−	0.888378i	$-0.651833\pi$
$61$	−7.17157	−0.918226	−0.459113	+	0.888378i	$0.651833\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.585786i	0.0726579i
$66$	0	0
$67$	8.00000i	0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$	8.00000i	0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.65685	0.671345	0.335673	−	0.941979i	$-0.391036\pi$
$71$	5.65685	0.671345	0.335673	+	0.941979i	$0.391036\pi$
$72$	0	0
$73$	6.48528	0.759045	0.379522	−	0.925183i	$-0.376088\pi$
$73$	6.48528	0.759045	0.379522	+	0.925183i	$0.376088\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 3.17157i	− 0.361434i
$78$	0	0
$79$	2.48528i	0.279616i	0.990179	+	0.139808i	$0.0446485\pi$
$79$	2.48528i	0.279616i	−0.990179	+	0.139808i	$0.955351\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	14.8284	1.62763	0.813816	−	0.581123i	$-0.197387\pi$
$83$	14.8284	1.62763	0.813816	+	0.581123i	$0.197387\pi$
$84$	0	0
$85$	−6.82843	−0.740647
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.24264i	0.449719i	0.974391	+	0.224860i	$0.0721923\pi$
$89$	4.24264i	0.449719i	−0.974391	+	0.224860i	$0.927808\pi$
$90$	0	0
$91$	− 0.343146i	− 0.0359714i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−14.4853	−1.47076	−0.735379	−	0.677656i	$-0.762996\pi$
$97$	−14.4853	−1.47076	−0.735379	+	0.677656i	$0.762996\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0.828427i	0.0824316i	0.999150	+	0.0412158i	$0.0131231\pi$
$101$	0.828427i	0.0824316i	−0.999150	+	0.0412158i	$0.986877\pi$
$102$	0	0
$103$	13.5563i	1.33575i	0.744275	+	0.667873i	$0.232795\pi$
$103$	13.5563i	1.33575i	−0.744275	+	0.667873i	$0.767205\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.17157	0.886649	0.443325	−	0.896361i	$-0.353799\pi$
$107$	9.17157	0.886649	0.443325	+	0.896361i	$0.353799\pi$
$108$	0	0
$109$	12.8284	1.22874	0.614370	−	0.789018i	$-0.289410\pi$
$109$	12.8284	1.22874	0.614370	+	0.789018i	$0.289410\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	9.65685i	0.908440i	0.890889	+	0.454220i	$0.150082\pi$
$113$	9.65685i	0.908440i	−0.890889	+	0.454220i	$0.849918\pi$
$114$	0	0
$115$	0.828427i	0.0772512i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	18.3137	1.66488
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	14.7279i	1.30689i	0.756973	+	0.653446i	$0.226677\pi$
$127$	14.7279i	1.30689i	−0.756973	+	0.653446i	$0.773323\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−12.7279	−1.11204	−0.556022	−	0.831168i	$-0.687673\pi$
$131$	−12.7279	−1.11204	−0.556022	+	0.831168i	$0.687673\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.31371i	0.283109i	0.989930	+	0.141555i	$0.0452101\pi$
$137$	3.31371i	0.283109i	−0.989930	+	0.141555i	$0.954790\pi$
$138$	0	0
$139$	7.65685i	0.649446i	0.945809	+	0.324723i	$0.105271\pi$
$139$	7.65685i	0.649446i	−0.945809	+	0.324723i	$0.894729\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.17157	0.265220
$144$	0	0
$145$	6.00000	0.498273
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.17157i	0.259825i	0.991525	+	0.129913i	$0.0414697\pi$
$149$	3.17157i	0.259825i	−0.991525	+	0.129913i	$0.958530\pi$
$150$	0	0
$151$	4.34315i	0.353440i	0.984261	+	0.176720i	$0.0565487\pi$
$151$	4.34315i	0.353440i	−0.984261	+	0.176720i	$0.943451\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−10.4853	−0.842198
$156$	0	0
$157$	18.2426	1.45592	0.727961	−	0.685619i	$-0.240468\pi$
$157$	18.2426	1.45592	0.727961	+	0.685619i	$0.240468\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 0.485281i	− 0.0382455i
$162$	0	0
$163$	16.4853i	1.29123i	0.763664	+	0.645613i	$0.223398\pi$
$163$	16.4853i	1.29123i	−0.763664	+	0.645613i	$0.776602\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−17.7990	−1.37733	−0.688664	−	0.725081i	$-0.741802\pi$
$167$	−17.7990	−1.37733	−0.688664	+	0.725081i	$0.741802\pi$
$168$	0	0
$169$	−12.6569	−0.973604
$170$	0	0
$171$	0	0
$172$	0	0
$173$	8.48528i	0.645124i	0.946548	+	0.322562i	$0.104544\pi$
$173$	8.48528i	0.645124i	−0.946548	+	0.322562i	$0.895456\pi$
$174$	0	0
$175$	0.585786i	0.0442813i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	24.0416	1.79696	0.898478	−	0.439019i	$-0.144674\pi$
$179$	24.0416	1.79696	0.898478	+	0.439019i	$0.144674\pi$
$180$	0	0
$181$	−18.4853	−1.37400	−0.687000	−	0.726657i	$-0.741073\pi$
$181$	−18.4853	−1.37400	−0.687000	+	0.726657i	$0.741073\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 5.07107i	− 0.372832i
$186$	0	0
$187$	36.9706i	2.70356i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.4853	1.48226	0.741131	−	0.671360i	$-0.234290\pi$
$191$	20.4853	1.48226	0.741131	+	0.671360i	$0.234290\pi$
$192$	0	0
$193$	−14.9706	−1.07760	−0.538802	−	0.842432i	$-0.681123\pi$
$193$	−14.9706	−1.07760	−0.538802	+	0.842432i	$0.681123\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 1.31371i	− 0.0935979i	−0.998904	−	0.0467989i	$-0.985098\pi$
$197$	− 1.31371i	− 0.0935979i	0.998904	−	0.0467989i	$-0.0149020\pi$
$198$	0	0
$199$	− 4.34315i	− 0.307877i	−0.988080	−	0.153939i	$-0.950804\pi$
$199$	− 4.34315i	− 0.307877i	0.988080	−	0.153939i	$-0.0491958\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−3.51472	−0.246685
$204$	0	0
$205$	3.07107	0.214493
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 26.0000i	− 1.78991i	−0.446153	−	0.894957i	$-0.647206\pi$
$211$	− 26.0000i	− 1.78991i	0.446153	−	0.894957i	$-0.352794\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.17157	0.0799006
$216$	0	0
$217$	6.14214	0.416955
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.00000i	0.269069i
$222$	0	0
$223$	− 22.7279i	− 1.52197i	−0.648767	−	0.760987i	$-0.724715\pi$
$223$	− 22.7279i	− 1.52197i	0.648767	−	0.760987i	$-0.275285\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5.65685	−0.375459	−0.187729	−	0.982221i	$-0.560113\pi$
$227$	−5.65685	−0.375459	−0.187729	+	0.982221i	$0.560113\pi$
$228$	0	0
$229$	−18.9706	−1.25361	−0.626805	−	0.779176i	$-0.715638\pi$
$229$	−18.9706	−1.25361	−0.626805	+	0.779176i	$0.715638\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 26.1421i	− 1.71263i	−0.516455	−	0.856314i	$-0.672749\pi$
$233$	− 26.1421i	− 1.71263i	0.516455	−	0.856314i	$-0.327251\pi$
$234$	0	0
$235$	− 5.65685i	− 0.369012i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	25.4558	1.64660	0.823301	−	0.567605i	$-0.192130\pi$
$239$	25.4558	1.64660	0.823301	+	0.567605i	$0.192130\pi$
$240$	0	0
$241$	−4.97056	−0.320182	−0.160091	−	0.987102i	$-0.551179\pi$
$241$	−4.97056	−0.320182	−0.160091	+	0.987102i	$0.551179\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.65685i	0.425291i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−5.89949	−0.372373	−0.186186	−	0.982514i	$-0.559613\pi$
$251$	−5.89949	−0.372373	−0.186186	+	0.982514i	$0.559613\pi$
$252$	0	0
$253$	4.48528	0.281987
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	2.97056i	0.184582i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	26.4853	1.63315	0.816576	−	0.577238i	$-0.195869\pi$
$263$	26.4853	1.63315	0.816576	+	0.577238i	$0.195869\pi$
$264$	0	0
$265$	6.82843	0.419467
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 3.65685i	− 0.222962i	−0.993767	−	0.111481i	$-0.964441\pi$
$269$	− 3.65685i	− 0.222962i	0.993767	−	0.111481i	$-0.0355595\pi$
$270$	0	0
$271$	22.0000i	1.33640i	0.743980	+	0.668202i	$0.232936\pi$
$271$	22.0000i	1.33640i	−0.743980	+	0.668202i	$0.767064\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.41421	−0.326489
$276$	0	0
$277$	30.7279	1.84626	0.923131	−	0.384486i	$-0.125621\pi$
$277$	30.7279	1.84626	0.923131	+	0.384486i	$0.125621\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 25.4142i	− 1.51608i	−0.652205	−	0.758042i	$-0.726156\pi$
$281$	− 25.4142i	− 1.51608i	0.652205	−	0.758042i	$-0.273844\pi$
$282$	0	0
$283$	− 15.3137i	− 0.910305i	−0.890413	−	0.455153i	$-0.849585\pi$
$283$	− 15.3137i	− 0.910305i	0.890413	−	0.455153i	$-0.150415\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.79899	−0.106191
$288$	0	0
$289$	−29.6274	−1.74279
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6.68629i	0.390617i	0.980742	+	0.195309i	$0.0625709\pi$
$293$	6.68629i	0.390617i	−0.980742	+	0.195309i	$0.937429\pi$
$294$	0	0
$295$	9.41421i	0.548117i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.485281	0.0280645
$300$	0	0
$301$	−0.686292	−0.0395572
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 7.17157i	− 0.410643i
$306$	0	0
$307$	24.0000i	1.36975i	0.728659	+	0.684876i	$0.240144\pi$
$307$	24.0000i	1.36975i	−0.728659	+	0.684876i	$0.759856\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2.14214	0.121469	0.0607347	−	0.998154i	$-0.480656\pi$
$311$	2.14214	0.121469	0.0607347	+	0.998154i	$0.480656\pi$
$312$	0	0
$313$	−3.65685	−0.206698	−0.103349	−	0.994645i	$-0.532956\pi$
$313$	−3.65685	−0.206698	−0.103349	+	0.994645i	$0.532956\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 28.6274i	− 1.60788i	−0.594713	−	0.803938i	$-0.702734\pi$
$317$	− 28.6274i	− 1.60788i	0.594713	−	0.803938i	$-0.297266\pi$
$318$	0	0
$319$	− 32.4853i	− 1.81883i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−0.585786	−0.0324936
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.31371i	0.182691i
$330$	0	0
$331$	4.97056i	0.273207i	0.990626	+	0.136603i	$0.0436186\pi$
$331$	4.97056i	0.273207i	−0.990626	+	0.136603i	$0.956381\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	−9.51472	−0.518300	−0.259150	−	0.965837i	$-0.583442\pi$
$337$	−9.51472	−0.518300	−0.259150	+	0.965837i	$0.583442\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	56.7696i	3.07424i
$342$	0	0
$343$	− 8.00000i	− 0.431959i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−33.6569	−1.80679	−0.903397	−	0.428805i	$-0.858935\pi$
$347$	−33.6569	−1.80679	−0.903397	+	0.428805i	$0.858935\pi$
$348$	0	0
$349$	19.4558	1.04145	0.520724	−	0.853725i	$-0.325662\pi$
$349$	19.4558	1.04145	0.520724	+	0.853725i	$0.325662\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	30.1421i	1.60430i	0.597120	+	0.802152i	$0.296312\pi$
$353$	30.1421i	1.60430i	−0.597120	+	0.802152i	$0.703688\pi$
$354$	0	0
$355$	5.65685i	0.300235i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	13.1716	0.695169	0.347585	−	0.937649i	$-0.387002\pi$
$359$	13.1716	0.695169	0.347585	+	0.937649i	$0.387002\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6.48528i	0.339455i
$366$	0	0
$367$	− 28.8701i	− 1.50700i	−0.657445	−	0.753502i	$-0.728363\pi$
$367$	− 28.8701i	− 1.50700i	0.657445	−	0.753502i	$-0.271637\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−4.00000	−0.207670
$372$	0	0
$373$	−14.7279	−0.762583	−0.381291	−	0.924455i	$-0.624521\pi$
$373$	−14.7279	−0.762583	−0.381291	+	0.924455i	$0.624521\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 3.51472i	− 0.181017i
$378$	0	0
$379$	− 13.3137i	− 0.683879i	−0.939722	−	0.341940i	$-0.888916\pi$
$379$	− 13.3137i	− 0.683879i	0.939722	−	0.341940i	$-0.111084\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−21.6569	−1.10661	−0.553307	−	0.832978i	$-0.686634\pi$
$383$	−21.6569	−1.10661	−0.553307	+	0.832978i	$0.686634\pi$
$384$	0	0
$385$	3.17157	0.161638
$386$	0	0
$387$	0	0
$388$	0	0
$389$	23.6569i	1.19945i	0.800206	+	0.599725i	$0.204723\pi$
$389$	23.6569i	1.19945i	−0.800206	+	0.599725i	$0.795277\pi$
$390$	0	0
$391$	5.65685i	0.286079i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.48528	−0.125048
$396$	0	0
$397$	10.7279	0.538419	0.269209	−	0.963082i	$-0.413238\pi$
$397$	10.7279	0.538419	0.269209	+	0.963082i	$0.413238\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 24.7279i	− 1.23485i	−0.786628	−	0.617427i	$-0.788175\pi$
$401$	− 24.7279i	− 1.23485i	0.786628	−	0.617427i	$-0.211825\pi$
$402$	0	0
$403$	6.14214i	0.305962i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−27.4558	−1.36094
$408$	0	0
$409$	18.2843	0.904099	0.452050	−	0.891993i	$-0.350693\pi$
$409$	18.2843	0.904099	0.452050	+	0.891993i	$0.350693\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 5.51472i	− 0.271362i
$414$	0	0
$415$	14.8284i	0.727899i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.61522	−0.0789088	−0.0394544	−	0.999221i	$-0.512562\pi$
$419$	−1.61522	−0.0789088	−0.0394544	+	0.999221i	$0.512562\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 6.82843i	− 0.331227i
$426$	0	0
$427$	4.20101i	0.203301i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	26.1421	1.25922	0.629611	−	0.776910i	$-0.283214\pi$
$431$	26.1421	1.25922	0.629611	+	0.776910i	$0.283214\pi$
$432$	0	0
$433$	2.48528	0.119435	0.0597175	−	0.998215i	$-0.480980\pi$
$433$	2.48528	0.119435	0.0597175	+	0.998215i	$0.480980\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	− 38.2843i	− 1.82721i	−0.406604	−	0.913604i	$-0.633287\pi$
$439$	− 38.2843i	− 1.82721i	0.406604	−	0.913604i	$-0.366713\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	18.6274	0.885015	0.442508	−	0.896765i	$-0.354089\pi$
$443$	18.6274	0.885015	0.442508	+	0.896765i	$0.354089\pi$
$444$	0	0
$445$	−4.24264	−0.201120
$446$	0	0
$447$	0	0
$448$	0	0
$449$	21.4142i	1.01060i	0.862944	+	0.505300i	$0.168618\pi$
$449$	21.4142i	1.01060i	−0.862944	+	0.505300i	$0.831382\pi$
$450$	0	0
$451$	− 16.6274i	− 0.782954i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0.343146	0.0160869
$456$	0	0
$457$	−14.4853	−0.677593	−0.338796	−	0.940860i	$-0.610020\pi$
$457$	−14.4853	−0.677593	−0.338796	+	0.940860i	$0.610020\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 19.1716i	− 0.892909i	−0.894806	−	0.446455i	$-0.852686\pi$
$461$	− 19.1716i	− 0.892909i	0.894806	−	0.446455i	$-0.147314\pi$
$462$	0	0
$463$	2.92893i	0.136119i	0.997681	+	0.0680595i	$0.0216808\pi$
$463$	2.92893i	0.136119i	−0.997681	+	0.0680595i	$0.978319\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−31.1127	−1.43972	−0.719862	−	0.694117i	$-0.755795\pi$
$467$	−31.1127	−1.43972	−0.719862	+	0.694117i	$0.755795\pi$
$468$	0	0
$469$	4.68629	0.216393
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 6.34315i	− 0.291658i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−4.68629	−0.214122	−0.107061	−	0.994252i	$-0.534144\pi$
$479$	−4.68629	−0.214122	−0.107061	+	0.994252i	$0.534144\pi$
$480$	0	0
$481$	−2.97056	−0.135446
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 14.4853i	− 0.657743i
$486$	0	0
$487$	− 14.7279i	− 0.667386i	−0.942682	−	0.333693i	$-0.891705\pi$
$487$	− 14.7279i	− 0.667386i	0.942682	−	0.333693i	$-0.108295\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−16.9289	−0.763992	−0.381996	−	0.924164i	$-0.624763\pi$
$491$	−16.9289	−0.763992	−0.381996	+	0.924164i	$0.624763\pi$
$492$	0	0
$493$	40.9706	1.84522
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 3.31371i	− 0.148640i
$498$	0	0
$499$	− 26.6274i	− 1.19201i	−0.802982	−	0.596003i	$-0.796754\pi$
$499$	− 26.6274i	− 1.19201i	0.802982	−	0.596003i	$-0.203246\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	20.2843	0.904431	0.452215	−	0.891909i	$-0.350634\pi$
$503$	20.2843	0.904431	0.452215	+	0.891909i	$0.350634\pi$
$504$	0	0
$505$	−0.828427	−0.0368645
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 20.1421i	− 0.892784i	−0.894837	−	0.446392i	$-0.852709\pi$
$509$	− 20.1421i	− 0.892784i	0.894837	−	0.446392i	$-0.147291\pi$
$510$	0	0
$511$	− 3.79899i	− 0.168057i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−13.5563	−0.597364
$516$	0	0
$517$	−30.6274	−1.34699
$518$	0	0
$519$	0	0
$520$	0	0
$521$	24.7279i	1.08335i	0.840588	+	0.541675i	$0.182210\pi$
$521$	24.7279i	1.08335i	−0.840588	+	0.541675i	$0.817790\pi$
$522$	0	0
$523$	− 24.4853i	− 1.07067i	−0.844641	−	0.535333i	$-0.820186\pi$
$523$	− 24.4853i	− 1.07067i	0.844641	−	0.535333i	$-0.179814\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−71.5980	−3.11886
$528$	0	0
$529$	−22.3137	−0.970161
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 1.79899i	− 0.0779229i
$534$	0	0
$535$	9.17157i	0.396522i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	36.0416	1.55242
$540$	0	0
$541$	−23.1716	−0.996224	−0.498112	−	0.867113i	$-0.665973\pi$
$541$	−23.1716	−0.996224	−0.498112	+	0.867113i	$0.665973\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	12.8284i	0.549509i
$546$	0	0
$547$	− 0.485281i	− 0.0207491i	−0.999946	−	0.0103746i	$-0.996698\pi$
$547$	− 0.485281i	− 0.0207491i	0.999946	−	0.0103746i	$-0.00330239\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	1.45584	0.0619088
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3.51472i	0.148923i	0.997224	+	0.0744617i	$0.0237239\pi$
$557$	3.51472i	0.148923i	−0.997224	+	0.0744617i	$0.976276\pi$
$558$	0	0
$559$	− 0.686292i	− 0.0290270i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−17.6569	−0.744148	−0.372074	−	0.928203i	$-0.621353\pi$
$563$	−17.6569	−0.744148	−0.372074	+	0.928203i	$0.621353\pi$
$564$	0	0
$565$	−9.65685	−0.406267
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 33.2132i	− 1.39237i	−0.717862	−	0.696185i	$-0.754879\pi$
$569$	− 33.2132i	− 1.39237i	0.717862	−	0.696185i	$-0.245121\pi$
$570$	0	0
$571$	− 28.9706i	− 1.21238i	−0.795320	−	0.606190i	$-0.792697\pi$
$571$	− 28.9706i	− 1.21238i	0.795320	−	0.606190i	$-0.207303\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−0.828427	−0.0345478
$576$	0	0
$577$	−35.4558	−1.47605	−0.738023	−	0.674775i	$-0.764240\pi$
$577$	−35.4558	−1.47605	−0.738023	+	0.674775i	$0.764240\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 8.68629i	− 0.360368i
$582$	0	0
$583$	− 36.9706i	− 1.53116i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2.14214	−0.0884154	−0.0442077	−	0.999022i	$-0.514076\pi$
$587$	−2.14214	−0.0884154	−0.0442077	+	0.999022i	$0.514076\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 21.6569i	− 0.889340i	−0.895694	−	0.444670i	$-0.853321\pi$
$593$	− 21.6569i	− 0.889340i	0.895694	−	0.444670i	$-0.146679\pi$
$594$	0	0
$595$	4.00000i	0.163984i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−13.8579	−0.566217	−0.283108	−	0.959088i	$-0.591366\pi$
$599$	−13.8579	−0.566217	−0.283108	+	0.959088i	$0.591366\pi$
$600$	0	0
$601$	39.3137	1.60364	0.801820	−	0.597566i	$-0.203865\pi$
$601$	39.3137	1.60364	0.801820	+	0.597566i	$0.203865\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	18.3137i	0.744558i
$606$	0	0
$607$	39.2132i	1.59161i	0.605550	+	0.795807i	$0.292953\pi$
$607$	39.2132i	1.59161i	−0.605550	+	0.795807i	$0.707047\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−3.31371	−0.134058
$612$	0	0
$613$	−39.2132	−1.58381	−0.791903	−	0.610647i	$-0.790910\pi$
$613$	−39.2132	−1.58381	−0.791903	+	0.610647i	$0.790910\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	21.1716i	0.852335i	0.904644	+	0.426168i	$0.140137\pi$
$617$	21.1716i	0.852335i	−0.904644	+	0.426168i	$0.859863\pi$
$618$	0	0
$619$	− 10.9706i	− 0.440944i	−0.975393	−	0.220472i	$-0.929240\pi$
$619$	− 10.9706i	− 0.440944i	0.975393	−	0.220472i	$-0.0707598\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.48528	0.0995707
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 34.6274i	− 1.38069i
$630$	0	0
$631$	− 12.8284i	− 0.510692i	−0.966850	−	0.255346i	$-0.917811\pi$
$631$	− 12.8284i	− 0.510692i	0.966850	−	0.255346i	$-0.0821893\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−14.7279	−0.584460
$636$	0	0
$637$	3.89949	0.154504
$638$	0	0
$639$	0	0
$640$	0	0
$641$	12.4437i	0.491495i	0.969334	+	0.245747i	$0.0790333\pi$
$641$	12.4437i	0.491495i	−0.969334	+	0.245747i	$0.920967\pi$
$642$	0	0
$643$	− 27.1127i	− 1.06922i	−0.845099	−	0.534610i	$-0.820458\pi$
$643$	− 27.1127i	− 1.06922i	0.845099	−	0.534610i	$-0.179542\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−35.3137	−1.38833	−0.694163	−	0.719818i	$-0.744225\pi$
$647$	−35.3137	−1.38833	−0.694163	+	0.719818i	$0.744225\pi$
$648$	0	0
$649$	50.9706	2.00077
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 8.34315i	− 0.326493i	−0.986585	−	0.163246i	$-0.947803\pi$
$653$	− 8.34315i	− 0.326493i	0.986585	−	0.163246i	$-0.0521965\pi$
$654$	0	0
$655$	− 12.7279i	− 0.497321i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	9.41421	0.366726	0.183363	−	0.983045i	$-0.441302\pi$
$659$	9.41421	0.366726	0.183363	+	0.983045i	$0.441302\pi$
$660$	0	0
$661$	−44.4264	−1.72799	−0.863993	−	0.503503i	$-0.832044\pi$
$661$	−44.4264	−1.72799	−0.863993	+	0.503503i	$0.832044\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 4.97056i	− 0.192461i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−38.8284	−1.49895
$672$	0	0
$673$	0.544156	0.0209757	0.0104878	−	0.999945i	$-0.496662\pi$
$673$	0.544156	0.0209757	0.0104878	+	0.999945i	$0.496662\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	29.3137i	1.12662i	0.826247	+	0.563309i	$0.190472\pi$
$677$	29.3137i	1.12662i	−0.826247	+	0.563309i	$0.809528\pi$
$678$	0	0
$679$	8.48528i	0.325635i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	5.17157	0.197885	0.0989424	−	0.995093i	$-0.468454\pi$
$683$	5.17157	0.197885	0.0989424	+	0.995093i	$0.468454\pi$
$684$	0	0
$685$	−3.31371	−0.126610
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 4.00000i	− 0.152388i
$690$	0	0
$691$	8.00000i	0.304334i	0.988355	+	0.152167i	$0.0486252\pi$
$691$	8.00000i	0.304334i	−0.988355	+	0.152167i	$0.951375\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7.65685	−0.290441
$696$	0	0
$697$	20.9706	0.794317
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 26.4853i	− 1.00034i	−0.865929	−	0.500168i	$-0.833272\pi$
$701$	− 26.4853i	− 1.00034i	0.865929	−	0.500168i	$-0.166728\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.485281	0.0182509
$708$	0	0
$709$	26.9706	1.01290	0.506450	−	0.862269i	$-0.330957\pi$
$709$	26.9706	1.01290	0.506450	+	0.862269i	$0.330957\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.68629i	0.325304i
$714$	0	0
$715$	3.17157i	0.118610i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	5.17157	0.192867	0.0964336	−	0.995339i	$-0.469256\pi$
$719$	5.17157	0.192867	0.0964336	+	0.995339i	$0.469256\pi$
$720$	0	0
$721$	7.94113	0.295743
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6.00000i	0.222834i
$726$	0	0
$727$	41.0711i	1.52324i	0.648023	+	0.761621i	$0.275596\pi$
$727$	41.0711i	1.52324i	−0.648023	+	0.761621i	$0.724404\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	31.8995	1.17823	0.589117	−	0.808047i	$-0.299476\pi$
$733$	31.8995	1.17823	0.589117	+	0.808047i	$0.299476\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	43.3137i	1.59548i
$738$	0	0
$739$	− 33.6569i	− 1.23809i	−0.785357	−	0.619044i	$-0.787520\pi$
$739$	− 33.6569i	− 1.23809i	0.785357	−	0.619044i	$-0.212480\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−48.0000	−1.76095	−0.880475	−	0.474093i	$-0.842776\pi$
$743$	−48.0000	−1.76095	−0.880475	+	0.474093i	$0.842776\pi$
$744$	0	0
$745$	−3.17157	−0.116197
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 5.37258i	− 0.196310i
$750$	0	0
$751$	− 23.1716i	− 0.845543i	−0.906236	−	0.422771i	$-0.861057\pi$
$751$	− 23.1716i	− 0.845543i	0.906236	−	0.422771i	$-0.138943\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−4.34315	−0.158063
$756$	0	0
$757$	−0.870058	−0.0316228	−0.0158114	−	0.999875i	$-0.505033\pi$
$757$	−0.870058	−0.0316228	−0.0158114	+	0.999875i	$0.505033\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	22.8701i	0.829039i	0.910040	+	0.414519i	$0.136050\pi$
$761$	22.8701i	0.829039i	−0.910040	+	0.414519i	$0.863950\pi$
$762$	0	0
$763$	− 7.51472i	− 0.272051i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5.51472	0.199125
$768$	0	0
$769$	−13.6569	−0.492479	−0.246239	−	0.969209i	$-0.579195\pi$
$769$	−13.6569	−0.492479	−0.246239	+	0.969209i	$0.579195\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 18.6863i	− 0.672099i	−0.941844	−	0.336050i	$-0.890909\pi$
$773$	− 18.6863i	− 0.672099i	0.941844	−	0.336050i	$-0.109091\pi$
$774$	0	0
$775$	− 10.4853i	− 0.376642i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	30.6274	1.09594
$782$	0	0
$783$	0	0
$784$	0	0
$785$	18.2426i	0.651108i
$786$	0	0
$787$	13.8579i	0.493980i	0.969018	+	0.246990i	$0.0794414\pi$
$787$	13.8579i	0.493980i	−0.969018	+	0.246990i	$0.920559\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	5.65685	0.201135
$792$	0	0
$793$	−4.20101	−0.149182
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 13.1716i	− 0.466561i	−0.972409	−	0.233281i	$-0.925054\pi$
$797$	− 13.1716i	− 0.466561i	0.972409	−	0.233281i	$-0.0749460\pi$
$798$	0	0
$799$	− 38.6274i	− 1.36654i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	35.1127	1.23910
$804$	0	0
$805$	0.485281	0.0171039
$806$	0	0
$807$	0	0
$808$	0	0
$809$	12.0416i	0.423361i	0.977339	+	0.211681i	$0.0678937\pi$
$809$	12.0416i	0.423361i	−0.977339	+	0.211681i	$0.932106\pi$
$810$	0	0
$811$	− 48.6274i	− 1.70754i	−0.520651	−	0.853770i	$-0.674311\pi$
$811$	− 48.6274i	− 1.70754i	0.520651	−	0.853770i	$-0.325689\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−16.4853	−0.577454
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	6.68629i	0.233353i	0.993170	+	0.116677i	$0.0372241\pi$
$821$	6.68629i	0.233353i	−0.993170	+	0.116677i	$0.962776\pi$
$822$	0	0
$823$	− 2.24264i	− 0.0781735i	−0.999236	−	0.0390868i	$-0.987555\pi$
$823$	− 2.24264i	− 0.0781735i	0.999236	−	0.0390868i	$-0.0124449\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.62742	−0.0913642	−0.0456821	−	0.998956i	$-0.514546\pi$
$827$	−2.62742	−0.0913642	−0.0456821	+	0.998956i	$0.514546\pi$
$828$	0	0
$829$	−31.4558	−1.09251	−0.546253	−	0.837620i	$-0.683946\pi$
$829$	−31.4558	−1.09251	−0.546253	+	0.837620i	$0.683946\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	45.4558i	1.57495i
$834$	0	0
$835$	− 17.7990i	− 0.615959i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	42.4264	1.46472	0.732361	−	0.680916i	$-0.238418\pi$
$839$	42.4264	1.46472	0.732361	+	0.680916i	$0.238418\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 12.6569i	− 0.435409i
$846$	0	0
$847$	− 10.7279i	− 0.368616i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−4.20101	−0.144009
$852$	0	0
$853$	−30.2426	−1.03549	−0.517744	−	0.855536i	$-0.673228\pi$
$853$	−30.2426	−1.03549	−0.517744	+	0.855536i	$0.673228\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.79899i	0.129771i	0.997893	+	0.0648855i	$0.0206682\pi$
$857$	3.79899i	0.129771i	−0.997893	+	0.0648855i	$0.979332\pi$
$858$	0	0
$859$	− 26.0000i	− 0.887109i	−0.896248	−	0.443554i	$-0.853717\pi$
$859$	− 26.0000i	− 0.887109i	0.896248	−	0.443554i	$-0.146283\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	45.7990	1.55902	0.779508	−	0.626392i	$-0.215469\pi$
$863$	45.7990	1.55902	0.779508	+	0.626392i	$0.215469\pi$
$864$	0	0
$865$	−8.48528	−0.288508
$866$	0	0
$867$	0	0
$868$	0	0
$869$	13.4558i	0.456458i
$870$	0	0
$871$	4.68629i	0.158789i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−0.585786	−0.0198032
$876$	0	0
$877$	−29.5563	−0.998047	−0.499023	−	0.866588i	$-0.666308\pi$
$877$	−29.5563	−0.998047	−0.499023	+	0.866588i	$0.666308\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 35.3553i	− 1.19115i	−0.803299	−	0.595576i	$-0.796924\pi$
$881$	− 35.3553i	− 1.19115i	0.803299	−	0.595576i	$-0.203076\pi$
$882$	0	0
$883$	− 42.4264i	− 1.42776i	−0.700267	−	0.713881i	$-0.746936\pi$
$883$	− 42.4264i	− 1.42776i	0.700267	−	0.713881i	$-0.253064\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	7.17157	0.240798	0.120399	−	0.992726i	$-0.461583\pi$
$887$	7.17157	0.240798	0.120399	+	0.992726i	$0.461583\pi$
$888$	0	0
$889$	8.62742	0.289354
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	24.0416i	0.803623i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	62.9117	2.09822
$900$	0	0
$901$	46.6274	1.55338
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 18.4853i	− 0.614472i
$906$	0	0
$907$	30.1421i	1.00085i	0.865779	+	0.500427i	$0.166823\pi$
$907$	30.1421i	1.00085i	−0.865779	+	0.500427i	$0.833177\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−29.6569	−0.982575	−0.491288	−	0.870997i	$-0.663474\pi$
$911$	−29.6569	−0.982575	−0.491288	+	0.870997i	$0.663474\pi$
$912$	0	0
$913$	80.2843	2.65702
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7.45584i	0.246214i
$918$	0	0
$919$	− 7.17157i	− 0.236568i	−0.992980	−	0.118284i	$-0.962261\pi$
$919$	− 7.17157i	− 0.236568i	0.992980	−	0.118284i	$-0.0377394\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	3.31371	0.109072
$924$	0	0
$925$	5.07107	0.166736
$926$	0	0
$927$	0	0
$928$	0	0
$929$	26.1005i	0.856330i	0.903701	+	0.428165i	$0.140840\pi$
$929$	26.1005i	0.856330i	−0.903701	+	0.428165i	$0.859160\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−36.9706	−1.20907
$936$	0	0
$937$	18.9706	0.619741	0.309871	−	0.950779i	$-0.399714\pi$
$937$	18.9706	0.619741	0.309871	+	0.950779i	$0.399714\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 14.2843i	− 0.465654i	−0.972518	−	0.232827i	$-0.925202\pi$
$941$	− 14.2843i	− 0.465654i	0.972518	−	0.232827i	$-0.0747976\pi$
$942$	0	0
$943$	− 2.54416i	− 0.0828491i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−27.5980	−0.896814	−0.448407	−	0.893830i	$-0.648008\pi$
$947$	−27.5980	−0.896814	−0.448407	+	0.893830i	$0.648008\pi$
$948$	0	0
$949$	3.79899	0.123320
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 45.2548i	− 1.46595i	−0.680257	−	0.732974i	$-0.738132\pi$
$953$	− 45.2548i	− 1.46595i	0.680257	−	0.732974i	$-0.261868\pi$
$954$	0	0
$955$	20.4853i	0.662888i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.94113	0.0626822
$960$	0	0
$961$	−78.9411	−2.54649
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 14.9706i	− 0.481919i
$966$	0	0
$967$	1.27208i	0.0409073i	0.999791	+	0.0204536i	$0.00651105\pi$
$967$	1.27208i	0.0409073i	−0.999791	+	0.0204536i	$0.993489\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	12.9289	0.414909	0.207455	−	0.978245i	$-0.433482\pi$
$971$	12.9289	0.414909	0.207455	+	0.978245i	$0.433482\pi$
$972$	0	0
$973$	4.48528	0.143792
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 42.8284i	− 1.37020i	−0.728448	−	0.685101i	$-0.759758\pi$
$977$	− 42.8284i	− 1.37020i	0.728448	−	0.685101i	$-0.240242\pi$
$978$	0	0
$979$	22.9706i	0.734142i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−39.5980	−1.26298	−0.631490	−	0.775384i	$-0.717556\pi$
$983$	−39.5980	−1.26298	−0.631490	+	0.775384i	$0.717556\pi$
$984$	0	0
$985$	1.31371	0.0418582
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 0.970563i	− 0.0308621i
$990$	0	0
$991$	− 27.6569i	− 0.878549i	−0.898353	−	0.439274i	$-0.855236\pi$
$991$	− 27.6569i	− 0.878549i	0.898353	−	0.439274i	$-0.144764\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	4.34315	0.137687
$996$	0	0
$997$	−30.7279	−0.973163	−0.486582	−	0.873635i	$-0.661756\pi$
$997$	−30.7279	−0.973163	−0.486582	+	0.873635i	$0.661756\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.h.d.1151.4		4
3.2	odd	2		2880.2.h.a.1151.2		4
4.3	odd	2		2880.2.h.a.1151.3		4
8.3	odd	2		1440.2.h.d.1151.1	yes	4
8.5	even	2		1440.2.h.a.1151.2	✓	4
12.11	even	2	inner	2880.2.h.d.1151.1		4
24.5	odd	2		1440.2.h.d.1151.4	yes	4
24.11	even	2		1440.2.h.a.1151.3	yes	4
40.3	even	4		7200.2.o.m.7199.1		4
40.13	odd	4		7200.2.o.b.7199.3		4
40.19	odd	2		7200.2.h.i.1151.2		4
40.27	even	4		7200.2.o.e.7199.4		4
40.29	even	2		7200.2.h.c.1151.3		4
40.37	odd	4		7200.2.o.j.7199.2		4
120.29	odd	2		7200.2.h.i.1151.3		4
120.53	even	4		7200.2.o.e.7199.3		4
120.59	even	2		7200.2.h.c.1151.2		4
120.77	even	4		7200.2.o.m.7199.2		4
120.83	odd	4		7200.2.o.j.7199.1		4
120.107	odd	4		7200.2.o.b.7199.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1440.2.h.a.1151.2	✓	4	8.5	even	2
1440.2.h.a.1151.3	yes	4	24.11	even	2
1440.2.h.d.1151.1	yes	4	8.3	odd	2
1440.2.h.d.1151.4	yes	4	24.5	odd	2
2880.2.h.a.1151.2		4	3.2	odd	2
2880.2.h.a.1151.3		4	4.3	odd	2
2880.2.h.d.1151.1		4	12.11	even	2	inner
2880.2.h.d.1151.4		4	1.1	even	1	trivial
7200.2.h.c.1151.2		4	120.59	even	2
7200.2.h.c.1151.3		4	40.29	even	2
7200.2.h.i.1151.2		4	40.19	odd	2
7200.2.h.i.1151.3		4	120.29	odd	2
7200.2.o.b.7199.3		4	40.13	odd	4
7200.2.o.b.7199.4		4	120.107	odd	4
7200.2.o.e.7199.3		4	120.53	even	4
7200.2.o.e.7199.4		4	40.27	even	4
7200.2.o.j.7199.1		4	120.83	odd	4
7200.2.o.j.7199.2		4	40.37	odd	4
7200.2.o.m.7199.1		4	40.3	even	4
7200.2.o.m.7199.2		4	120.77	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:	\( N \)	\(=\)	\( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2880.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{5} -0.585786i q^{7} +O(q^{10})\)	\(q+1.00000i q^{5} -0.585786i q^{7} +5.41421 q^{11} +0.585786 q^{13} +6.82843i q^{17} +0.828427 q^{23} -1.00000 q^{25} -6.00000i q^{29} +10.4853i q^{31} +0.585786 q^{35} -5.07107 q^{37} -3.07107i q^{41} -1.17157i q^{43} -5.65685 q^{47} +6.65685 q^{49} -6.82843i q^{53} +5.41421i q^{55} +9.41421 q^{59} -7.17157 q^{61} +0.585786i q^{65} +8.00000i q^{67} +5.65685 q^{71} +6.48528 q^{73} -3.17157i q^{77} +2.48528i q^{79} +14.8284 q^{83} -6.82843 q^{85} +4.24264i q^{89} -0.343146i q^{91} -14.4853 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 16 q^{11} + 8 q^{13} - 8 q^{23} - 4 q^{25} + 8 q^{35} + 8 q^{37} + 4 q^{49} + 32 q^{59} - 40 q^{61} - 8 q^{73} + 48 q^{83} - 16 q^{85} - 24 q^{97}+O(q^{100}) \) 4 * q + 16 * q^11 + 8 * q^13 - 8 * q^23 - 4 * q^25 + 8 * q^35 + 8 * q^37 + 4 * q^49 + 32 * q^59 - 40 * q^61 - 8 * q^73 + 48 * q^83 - 16 * q^85 - 24 * q^97

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