LMFDB - Embedded newform 7920.2.a.cn.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7920,2,Mod(1,7920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7920, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7920.a (trivial)

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:	yes
Analytic conductor:	$63.2415184009$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.48704.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 6x^{2} + 4x + 6 $ x^4 - 2x^3 - 6x^2 + 4*x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 495)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.28632$ of defining polynomial
Character	$\chi$	$=$	7920.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -2.51962 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -2.51962 q^{7} +1.00000 q^{11} +6.05302 q^{13} +4.97417 q^{17} +7.02720 q^{19} +4.45455 q^{23} +1.00000 q^{25} -0.921150 q^{29} -3.03925 q^{31} -2.51962 q^{35} -3.49380 q^{37} +10.0664 q^{41} -1.48038 q^{43} -8.10605 q^{47} -0.651497 q^{49} +1.54545 q^{53} +1.00000 q^{55} -7.59984 q^{59} +10.1060 q^{61} +6.05302 q^{65} +8.69074 q^{67} +1.54545 q^{71} +6.05302 q^{73} -2.51962 q^{77} -11.0272 q^{79} -8.50757 q^{83} +4.97417 q^{85} -15.1453 q^{89} -15.2513 q^{91} +7.02720 q^{95} -11.5998 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} - 4 q^{7}+O(q^{10}) $ 4 * q + 4 * q^5 - 4 * q^7	$ 4 q + 4 q^{5} - 4 q^{7} + 4 q^{11} + 8 q^{13} + 4 q^{17} - 4 q^{19} + 8 q^{23} + 4 q^{25} - 4 q^{29} - 4 q^{35} + 8 q^{37} - 4 q^{41} - 12 q^{43} + 20 q^{49} + 16 q^{53} + 4 q^{55} + 24 q^{59} + 8 q^{61} + 8 q^{65} + 16 q^{71} + 8 q^{73} - 4 q^{77} - 12 q^{79} - 8 q^{83} + 4 q^{85} - 16 q^{89} + 16 q^{91} - 4 q^{95} + 8 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 - 4 * q^7 + 4 * q^11 + 8 * q^13 + 4 * q^17 - 4 * q^19 + 8 * q^23 + 4 * q^25 - 4 * q^29 - 4 * q^35 + 8 * q^37 - 4 * q^41 - 12 * q^43 + 20 * q^49 + 16 * q^53 + 4 * q^55 + 24 * q^59 + 8 * q^61 + 8 * q^65 + 16 * q^71 + 8 * q^73 - 4 * q^77 - 12 * q^79 - 8 * q^83 + 4 * q^85 - 16 * q^89 + 16 * q^91 - 4 * q^95 + 8 * q^97

Display 100 coefficients 10 coefficients

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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.51962	−0.952328	−0.476164	−	0.879356i	$-0.657973\pi$
$7$	−2.51962	−0.952328	−0.476164	+	0.879356i	$0.657973\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	6.05302	1.67881	0.839403	−	0.543509i	$-0.182905\pi$
$13$	6.05302	1.67881	0.839403	+	0.543509i	$0.182905\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.97417	1.20641	0.603207	−	0.797585i	$-0.293889\pi$
$17$	4.97417	1.20641	0.603207	+	0.797585i	$0.293889\pi$
$18$	0	0
$19$	7.02720	1.61215	0.806075	−	0.591814i	$-0.201588\pi$
$19$	7.02720	1.61215	0.806075	+	0.591814i	$0.201588\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.45455	0.928838	0.464419	−	0.885616i	$-0.346263\pi$
$23$	4.45455	0.928838	0.464419	+	0.885616i	$0.346263\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.921150	−0.171053	−0.0855266	−	0.996336i	$-0.527257\pi$
$29$	−0.921150	−0.171053	−0.0855266	+	0.996336i	$0.527257\pi$
$30$	0	0
$31$	−3.03925	−0.545865	−0.272932	−	0.962033i	$-0.587994\pi$
$31$	−3.03925	−0.545865	−0.272932	+	0.962033i	$0.587994\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.51962	−0.425894
$36$	0	0
$37$	−3.49380	−0.574377	−0.287188	−	0.957874i	$-0.592721\pi$
$37$	−3.49380	−0.574377	−0.287188	+	0.957874i	$0.592721\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	10.0664	1.57211	0.786057	−	0.618154i	$-0.212119\pi$
$41$	10.0664	1.57211	0.786057	+	0.618154i	$0.212119\pi$
$42$	0	0
$43$	−1.48038	−0.225755	−0.112878	−	0.993609i	$-0.536007\pi$
$43$	−1.48038	−0.225755	−0.112878	+	0.993609i	$0.536007\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.10605	−1.18239	−0.591194	−	0.806529i	$-0.701343\pi$
$47$	−8.10605	−1.18239	−0.591194	+	0.806529i	$0.701343\pi$
$48$	0	0
$49$	−0.651497	−0.0930710
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.54545	0.212284	0.106142	−	0.994351i	$-0.466150\pi$
$53$	1.54545	0.212284	0.106142	+	0.994351i	$0.466150\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−7.59984	−0.989415	−0.494708	−	0.869059i	$-0.664725\pi$
$59$	−7.59984	−0.989415	−0.494708	+	0.869059i	$0.664725\pi$
$60$	0	0
$61$	10.1060	1.29395	0.646973	−	0.762513i	$-0.276034\pi$
$61$	10.1060	1.29395	0.646973	+	0.762513i	$0.276034\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.05302	0.750785
$66$	0	0
$67$	8.69074	1.06174	0.530872	−	0.847452i	$-0.321865\pi$
$67$	8.69074	1.06174	0.530872	+	0.847452i	$0.321865\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.54545	0.183411	0.0917056	−	0.995786i	$-0.470768\pi$
$71$	1.54545	0.183411	0.0917056	+	0.995786i	$0.470768\pi$
$72$	0	0
$73$	6.05302	0.708453	0.354226	−	0.935160i	$-0.384744\pi$
$73$	6.05302	0.708453	0.354226	+	0.935160i	$0.384744\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.51962	−0.287138
$78$	0	0
$79$	−11.0272	−1.24066	−0.620328	−	0.784342i	$-0.713001\pi$
$79$	−11.0272	−1.24066	−0.620328	+	0.784342i	$0.713001\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−8.50757	−0.933827	−0.466914	−	0.884303i	$-0.654634\pi$
$83$	−8.50757	−0.933827	−0.466914	+	0.884303i	$0.654634\pi$
$84$	0	0
$85$	4.97417	0.539525
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−15.1453	−1.60540	−0.802699	−	0.596384i	$-0.796603\pi$
$89$	−15.1453	−1.60540	−0.802699	+	0.596384i	$0.796603\pi$
$90$	0	0
$91$	−15.2513	−1.59877
$92$	0	0
$93$	0	0
$94$	0	0
$95$	7.02720	0.720975
$96$	0	0
$97$	−11.5998	−1.17779	−0.588893	−	0.808211i	$-0.700436\pi$
$97$	−11.5998	−1.17779	−0.588893	+	0.808211i	$0.700436\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0.118097	0.0117511	0.00587556	−	0.999983i	$-0.498130\pi$
$101$	0.118097	0.0117511	0.00587556	+	0.999983i	$0.498130\pi$
$102$	0	0
$103$	−10.6391	−1.04830	−0.524150	−	0.851626i	$-0.675617\pi$
$103$	−10.6391	−1.04830	−0.524150	+	0.851626i	$0.675617\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.6921	1.80703	0.903517	−	0.428551i	$-0.140976\pi$
$107$	18.6921	1.80703	0.903517	+	0.428551i	$0.140976\pi$
$108$	0	0
$109$	−7.14529	−0.684395	−0.342198	−	0.939628i	$-0.611171\pi$
$109$	−7.14529	−0.684395	−0.342198	+	0.939628i	$0.611171\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.58470	0.243148	0.121574	−	0.992582i	$-0.461206\pi$
$113$	2.58470	0.243148	0.121574	+	0.992582i	$0.461206\pi$
$114$	0	0
$115$	4.45455	0.415389
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−12.5330	−1.14890
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−11.4287	−1.01414	−0.507068	−	0.861906i	$-0.669271\pi$
$127$	−11.4287	−1.01414	−0.507068	+	0.861906i	$0.669271\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	8.10605	0.708229	0.354114	−	0.935202i	$-0.384782\pi$
$131$	8.10605	0.708229	0.354114	+	0.935202i	$0.384782\pi$
$132$	0	0
$133$	−17.7059	−1.53530
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	20.0664	1.70201	0.851007	−	0.525155i	$-0.175993\pi$
$139$	20.0664	1.70201	0.851007	+	0.525155i	$0.175993\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	6.05302	0.506179
$144$	0	0
$145$	−0.921150	−0.0764973
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−21.2634	−1.74196	−0.870982	−	0.491314i	$-0.836517\pi$
$149$	−21.2634	−1.74196	−0.870982	+	0.491314i	$0.836517\pi$
$150$	0	0
$151$	7.02720	0.571865	0.285933	−	0.958250i	$-0.407697\pi$
$151$	7.02720	0.571865	0.285933	+	0.958250i	$0.407697\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3.03925	−0.244118
$156$	0	0
$157$	21.0936	1.68346	0.841728	−	0.539902i	$-0.181539\pi$
$157$	21.0936	1.68346	0.841728	+	0.539902i	$0.181539\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−11.2238	−0.884558
$162$	0	0
$163$	7.65150	0.599311	0.299656	−	0.954047i	$-0.403128\pi$
$163$	7.65150	0.599311	0.299656	+	0.954047i	$0.403128\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.49243	−0.270252	−0.135126	−	0.990828i	$-0.543144\pi$
$167$	−3.49243	−0.270252	−0.135126	+	0.990828i	$0.543144\pi$
$168$	0	0
$169$	23.6391	1.81839
$170$	0	0
$171$	0	0
$172$	0	0
$173$	16.9742	1.29052	0.645261	−	0.763962i	$-0.276749\pi$
$173$	16.9742	1.29052	0.645261	+	0.763962i	$0.276749\pi$
$174$	0	0
$175$	−2.51962	−0.190466
$176$	0	0
$177$	0	0
$178$	0	0
$179$	9.14529	0.683551	0.341776	−	0.939782i	$-0.388972\pi$
$179$	9.14529	0.683551	0.341776	+	0.939782i	$0.388972\pi$
$180$	0	0
$181$	22.6391	1.68275	0.841375	−	0.540452i	$-0.181747\pi$
$181$	22.6391	1.68275	0.841375	+	0.540452i	$0.181747\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−3.49380	−0.256869
$186$	0	0
$187$	4.97417	0.363748
$188$	0	0
$189$	0	0
$190$	0	0
$191$	12.2362	0.885380	0.442690	−	0.896675i	$-0.354024\pi$
$191$	12.2362	0.885380	0.442690	+	0.896675i	$0.354024\pi$
$192$	0	0
$193$	1.03788	0.0747081	0.0373540	−	0.999302i	$-0.488107\pi$
$193$	1.03788	0.0747081	0.0373540	+	0.999302i	$0.488107\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.06507	0.574613	0.287306	−	0.957839i	$-0.407240\pi$
$197$	8.06507	0.574613	0.287306	+	0.957839i	$0.407240\pi$
$198$	0	0
$199$	13.1453	0.931845	0.465923	−	0.884825i	$-0.345723\pi$
$199$	13.1453	0.931845	0.465923	+	0.884825i	$0.345723\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.32095	0.162899
$204$	0	0
$205$	10.0664	0.703071
$206$	0	0
$207$	0	0
$208$	0	0
$209$	7.02720	0.486081
$210$	0	0
$211$	3.05130	0.210060	0.105030	−	0.994469i	$-0.466506\pi$
$211$	3.05130	0.210060	0.105030	+	0.994469i	$0.466506\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.48038	−0.100961
$216$	0	0
$217$	7.65776	0.519843
$218$	0	0
$219$	0	0
$220$	0	0
$221$	30.1088	2.02534
$222$	0	0
$223$	13.9483	0.934050	0.467025	−	0.884244i	$-0.345326\pi$
$223$	13.9483	0.934050	0.467025	+	0.884244i	$0.345326\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−29.6529	−1.96813	−0.984065	−	0.177809i	$-0.943099\pi$
$227$	−29.6529	−1.96813	−0.984065	+	0.177809i	$0.943099\pi$
$228$	0	0
$229$	18.2121	1.20349	0.601744	−	0.798689i	$-0.294473\pi$
$229$	18.2121	1.20349	0.601744	+	0.798689i	$0.294473\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−13.0802	−0.856914	−0.428457	−	0.903562i	$-0.640943\pi$
$233$	−13.0802	−0.856914	−0.428457	+	0.903562i	$0.640943\pi$
$234$	0	0
$235$	−8.10605	−0.528780
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0.803053	0.0519452	0.0259726	−	0.999663i	$-0.491732\pi$
$239$	0.803053	0.0519452	0.0259726	+	0.999663i	$0.491732\pi$
$240$	0	0
$241$	3.03925	0.195775	0.0978876	−	0.995197i	$-0.468791\pi$
$241$	3.03925	0.195775	0.0978876	+	0.995197i	$0.468791\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−0.651497	−0.0416226
$246$	0	0
$247$	42.5358	2.70649
$248$	0	0
$249$	0	0
$250$	0	0
$251$	7.36365	0.464789	0.232395	−	0.972622i	$-0.425344\pi$
$251$	7.36365	0.464789	0.232395	+	0.972622i	$0.425344\pi$
$252$	0	0
$253$	4.45455	0.280055
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.94835	0.246291	0.123146	−	0.992389i	$-0.460702\pi$
$257$	3.94835	0.246291	0.123146	+	0.992389i	$0.460702\pi$
$258$	0	0
$259$	8.80305	0.546995
$260$	0	0
$261$	0	0
$262$	0	0
$263$	28.6405	1.76605	0.883023	−	0.469329i	$-0.155504\pi$
$263$	28.6405	1.76605	0.883023	+	0.469329i	$0.155504\pi$
$264$	0	0
$265$	1.54545	0.0949363
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−28.2121	−1.72012	−0.860061	−	0.510191i	$-0.829575\pi$
$269$	−28.2121	−1.72012	−0.860061	+	0.510191i	$0.829575\pi$
$270$	0	0
$271$	−11.2634	−0.684202	−0.342101	−	0.939663i	$-0.611139\pi$
$271$	−11.2634	−0.684202	−0.342101	+	0.939663i	$0.611139\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	−29.0165	−1.74343	−0.871717	−	0.490010i	$-0.836993\pi$
$277$	−29.0165	−1.74343	−0.871717	+	0.490010i	$0.836993\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−30.9755	−1.84785	−0.923923	−	0.382579i	$-0.875036\pi$
$281$	−30.9755	−1.84785	−0.923923	+	0.382579i	$0.875036\pi$
$282$	0	0
$283$	23.7951	1.41447	0.707235	−	0.706979i	$-0.249942\pi$
$283$	23.7951	1.41447	0.707235	+	0.706979i	$0.249942\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−25.3636	−1.49717
$288$	0	0
$289$	7.74240	0.455435
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0.998275	0.0583198	0.0291599	−	0.999575i	$-0.490717\pi$
$293$	0.998275	0.0583198	0.0291599	+	0.999575i	$0.490717\pi$
$294$	0	0
$295$	−7.59984	−0.442480
$296$	0	0
$297$	0	0
$298$	0	0
$299$	26.9635	1.55934
$300$	0	0
$301$	3.72999	0.214993
$302$	0	0
$303$	0	0
$304$	0	0
$305$	10.1060	0.578671
$306$	0	0
$307$	27.7710	1.58497	0.792486	−	0.609890i	$-0.208786\pi$
$307$	27.7710	1.58497	0.792486	+	0.609890i	$0.208786\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−17.5213	−0.993545	−0.496772	−	0.867881i	$-0.665482\pi$
$311$	−17.5213	−0.993545	−0.496772	+	0.867881i	$0.665482\pi$
$312$	0	0
$313$	5.41530	0.306091	0.153045	−	0.988219i	$-0.451092\pi$
$313$	5.41530	0.306091	0.153045	+	0.988219i	$0.451092\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6.79679	−0.381746	−0.190873	−	0.981615i	$-0.561132\pi$
$317$	−6.79679	−0.381746	−0.190873	+	0.981615i	$0.561132\pi$
$318$	0	0
$319$	−0.921150	−0.0515745
$320$	0	0
$321$	0	0
$322$	0	0
$323$	34.9545	1.94492
$324$	0	0
$325$	6.05302	0.335761
$326$	0	0
$327$	0	0
$328$	0	0
$329$	20.4242	1.12602
$330$	0	0
$331$	7.11845	0.391266	0.195633	−	0.980677i	$-0.437324\pi$
$331$	7.11845	0.391266	0.195633	+	0.980677i	$0.437324\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8.69074	0.474826
$336$	0	0
$337$	24.9105	1.35696	0.678480	−	0.734619i	$-0.262639\pi$
$337$	24.9105	1.35696	0.678480	+	0.734619i	$0.262639\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3.03925	−0.164584
$342$	0	0
$343$	19.2789	1.04096
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−23.6253	−1.26827	−0.634137	−	0.773221i	$-0.718644\pi$
$347$	−23.6253	−1.26827	−0.634137	+	0.773221i	$0.718644\pi$
$348$	0	0
$349$	−31.1721	−1.66861	−0.834303	−	0.551306i	$-0.814130\pi$
$349$	−31.1721	−1.66861	−0.834303	+	0.551306i	$0.814130\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0.533044	0.0283711	0.0141855	−	0.999899i	$-0.495484\pi$
$353$	0.533044	0.0283711	0.0141855	+	0.999899i	$0.495484\pi$
$354$	0	0
$355$	1.54545	0.0820240
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−12.0268	−0.634752	−0.317376	−	0.948300i	$-0.602802\pi$
$359$	−12.0268	−0.634752	−0.317376	+	0.948300i	$0.602802\pi$
$360$	0	0
$361$	30.3815	1.59903
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6.05302	0.316830
$366$	0	0
$367$	14.9608	0.780945	0.390472	−	0.920615i	$-0.372312\pi$
$367$	14.9608	0.780945	0.390472	+	0.920615i	$0.372312\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−3.89395	−0.202164
$372$	0	0
$373$	20.1315	1.04237	0.521185	−	0.853444i	$-0.325490\pi$
$373$	20.1315	1.04237	0.521185	+	0.853444i	$0.325490\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5.57574	−0.287165
$378$	0	0
$379$	−15.8967	−0.816558	−0.408279	−	0.912857i	$-0.633871\pi$
$379$	−15.8967	−0.816558	−0.408279	+	0.912857i	$0.633871\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2.63909	−0.134851	−0.0674256	−	0.997724i	$-0.521479\pi$
$383$	−2.63909	−0.134851	−0.0674256	+	0.997724i	$0.521479\pi$
$384$	0	0
$385$	−2.51962	−0.128412
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−13.8423	−0.701832	−0.350916	−	0.936407i	$-0.614130\pi$
$389$	−13.8423	−0.701832	−0.350916	+	0.936407i	$0.614130\pi$
$390$	0	0
$391$	22.1577	1.12056
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−11.0272	−0.554838
$396$	0	0
$397$	−25.6783	−1.28876	−0.644379	−	0.764706i	$-0.722884\pi$
$397$	−25.6783	−1.28876	−0.644379	+	0.764706i	$0.722884\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−31.1212	−1.55412	−0.777059	−	0.629428i	$-0.783289\pi$
$401$	−31.1212	−1.55412	−0.777059	+	0.629428i	$0.783289\pi$
$402$	0	0
$403$	−18.3966	−0.916402
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3.49380	−0.173181
$408$	0	0
$409$	12.7514	0.630516	0.315258	−	0.949006i	$-0.397909\pi$
$409$	12.7514	0.630516	0.315258	+	0.949006i	$0.397909\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	19.1487	0.942248
$414$	0	0
$415$	−8.50757	−0.417620
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0.236195	0.0115389	0.00576943	−	0.999983i	$-0.498164\pi$
$419$	0.236195	0.0115389	0.00576943	+	0.999983i	$0.498164\pi$
$420$	0	0
$421$	−21.7300	−1.05905	−0.529527	−	0.848293i	$-0.677631\pi$
$421$	−21.7300	−1.05905	−0.529527	+	0.848293i	$0.677631\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.97417	0.241283
$426$	0	0
$427$	−25.4634	−1.23226
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−16.2121	−0.780909	−0.390455	−	0.920622i	$-0.627682\pi$
$431$	−16.2121	−0.780909	−0.390455	+	0.920622i	$0.627682\pi$
$432$	0	0
$433$	13.5213	0.649795	0.324897	−	0.945749i	$-0.394670\pi$
$433$	13.5213	0.649795	0.324897	+	0.945749i	$0.394670\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	31.3030	1.49743
$438$	0	0
$439$	−0.0664435	−0.00317118	−0.00158559	−	0.999999i	$-0.500505\pi$
$439$	−0.0664435	−0.00317118	−0.00158559	+	0.999999i	$0.500505\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−0.560596	−0.0266347	−0.0133174	−	0.999911i	$-0.504239\pi$
$443$	−0.560596	−0.0266347	−0.0133174	+	0.999911i	$0.504239\pi$
$444$	0	0
$445$	−15.1453	−0.717956
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−9.92079	−0.468191	−0.234096	−	0.972214i	$-0.575213\pi$
$449$	−9.92079	−0.468191	−0.234096	+	0.972214i	$0.575213\pi$
$450$	0	0
$451$	10.0664	0.474010
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−15.2513	−0.714994
$456$	0	0
$457$	8.93457	0.417942	0.208971	−	0.977922i	$-0.432989\pi$
$457$	8.93457	0.417942	0.208971	+	0.977922i	$0.432989\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−34.1484	−1.59045	−0.795225	−	0.606315i	$-0.792647\pi$
$461$	−34.1484	−1.59045	−0.795225	+	0.606315i	$0.792647\pi$
$462$	0	0
$463$	24.9359	1.15887	0.579436	−	0.815018i	$-0.303273\pi$
$463$	24.9359	1.15887	0.579436	+	0.815018i	$0.303273\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−2.84844	−0.131810	−0.0659051	−	0.997826i	$-0.520993\pi$
$467$	−2.84844	−0.131810	−0.0659051	+	0.997826i	$0.520993\pi$
$468$	0	0
$469$	−21.8974	−1.01113
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.48038	−0.0680678
$474$	0	0
$475$	7.02720	0.322430
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−15.1177	−0.690747	−0.345374	−	0.938465i	$-0.612248\pi$
$479$	−15.1177	−0.690747	−0.345374	+	0.938465i	$0.612248\pi$
$480$	0	0
$481$	−21.1480	−0.964267
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−11.5998	−0.526722
$486$	0	0
$487$	4.53376	0.205444	0.102722	−	0.994710i	$-0.467245\pi$
$487$	4.53376	0.205444	0.102722	+	0.994710i	$0.467245\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−15.0909	−0.681043	−0.340521	−	0.940237i	$-0.610604\pi$
$491$	−15.0909	−0.681043	−0.340521	+	0.940237i	$0.610604\pi$
$492$	0	0
$493$	−4.58196	−0.206361
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.89395	−0.174668
$498$	0	0
$499$	8.93320	0.399905	0.199952	−	0.979806i	$-0.435921\pi$
$499$	8.93320	0.399905	0.199952	+	0.979806i	$0.435921\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	26.7162	1.19122	0.595609	−	0.803275i	$-0.296911\pi$
$503$	26.7162	1.19122	0.595609	+	0.803275i	$0.296911\pi$
$504$	0	0
$505$	0.118097	0.00525526
$506$	0	0
$507$	0	0
$508$	0	0
$509$	10.9876	0.487017	0.243508	−	0.969899i	$-0.421702\pi$
$509$	10.9876	0.487017	0.243508	+	0.969899i	$0.421702\pi$
$510$	0	0
$511$	−15.2513	−0.674680
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−10.6391	−0.468814
$516$	0	0
$517$	−8.10605	−0.356504
$518$	0	0
$519$	0	0
$520$	0	0
$521$	23.2782	1.01984	0.509918	−	0.860223i	$-0.329676\pi$
$521$	23.2782	1.01984	0.509918	+	0.860223i	$0.329676\pi$
$522$	0	0
$523$	18.4163	0.805289	0.402645	−	0.915356i	$-0.368091\pi$
$523$	18.4163	0.805289	0.402645	+	0.915356i	$0.368091\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−15.1177	−0.658539
$528$	0	0
$529$	−3.15699	−0.137260
$530$	0	0
$531$	0	0
$532$	0	0
$533$	60.9324	2.63928
$534$	0	0
$535$	18.6921	0.808131
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−0.651497	−0.0280620
$540$	0	0
$541$	−31.3815	−1.34920	−0.674598	−	0.738186i	$-0.735683\pi$
$541$	−31.3815	−1.34920	−0.674598	+	0.738186i	$0.735683\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−7.14529	−0.306071
$546$	0	0
$547$	3.74412	0.160087	0.0800436	−	0.996791i	$-0.474494\pi$
$547$	3.74412	0.160087	0.0800436	+	0.996791i	$0.474494\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6.47310	−0.275763
$552$	0	0
$553$	27.7844	1.18151
$554$	0	0
$555$	0	0
$556$	0	0
$557$	25.3164	1.07269	0.536345	−	0.843999i	$-0.319804\pi$
$557$	25.3164	1.07269	0.536345	+	0.843999i	$0.319804\pi$
$558$	0	0
$559$	−8.96075	−0.378999
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−24.7465	−1.04294	−0.521470	−	0.853269i	$-0.674616\pi$
$563$	−24.7465	−1.04294	−0.521470	+	0.853269i	$0.674616\pi$
$564$	0	0
$565$	2.58470	0.108739
$566$	0	0
$567$	0	0
$568$	0	0
$569$	44.0148	1.84520	0.922598	−	0.385763i	$-0.126062\pi$
$569$	44.0148	1.84520	0.922598	+	0.385763i	$0.126062\pi$
$570$	0	0
$571$	2.01205	0.0842017	0.0421009	−	0.999113i	$-0.486595\pi$
$571$	2.01205	0.0842017	0.0421009	+	0.999113i	$0.486595\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.45455	0.185768
$576$	0	0
$577$	26.3181	1.09564	0.547819	−	0.836597i	$-0.315458\pi$
$577$	26.3181	1.09564	0.547819	+	0.836597i	$0.315458\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	21.4359	0.889310
$582$	0	0
$583$	1.54545	0.0640060
$584$	0	0
$585$	0	0
$586$	0	0
$587$	35.8112	1.47809	0.739044	−	0.673657i	$-0.235278\pi$
$587$	35.8112	1.47809	0.739044	+	0.673657i	$0.235278\pi$
$588$	0	0
$589$	−21.3574	−0.880016
$590$	0	0
$591$	0	0
$592$	0	0
$593$	7.61953	0.312896	0.156448	−	0.987686i	$-0.449996\pi$
$593$	7.61953	0.312896	0.156448	+	0.987686i	$0.449996\pi$
$594$	0	0
$595$	−12.5330	−0.513805
$596$	0	0
$597$	0	0
$598$	0	0
$599$	40.2121	1.64302	0.821511	−	0.570193i	$-0.193132\pi$
$599$	40.2121	1.64302	0.821511	+	0.570193i	$0.193132\pi$
$600$	0	0
$601$	26.3181	1.07354	0.536770	−	0.843729i	$-0.319644\pi$
$601$	26.3181	1.07354	0.536770	+	0.843729i	$0.319644\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	−11.4287	−0.463878	−0.231939	−	0.972730i	$-0.574507\pi$
$607$	−11.4287	−0.463878	−0.231939	+	0.972730i	$0.574507\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−49.0661	−1.98500
$612$	0	0
$613$	−21.1198	−0.853022	−0.426511	−	0.904482i	$-0.640257\pi$
$613$	−21.1198	−0.853022	−0.426511	+	0.904482i	$0.640257\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−31.3905	−1.26373	−0.631867	−	0.775077i	$-0.717711\pi$
$617$	−31.3905	−1.26373	−0.631867	+	0.775077i	$0.717711\pi$
$618$	0	0
$619$	7.11845	0.286115	0.143057	−	0.989714i	$-0.454307\pi$
$619$	7.11845	0.286115	0.143057	+	0.989714i	$0.454307\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	38.1604	1.52887
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−17.3787	−0.692936
$630$	0	0
$631$	−21.3574	−0.850224	−0.425112	−	0.905141i	$-0.639765\pi$
$631$	−21.3574	−0.850224	−0.425112	+	0.905141i	$0.639765\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−11.4287	−0.453535
$636$	0	0
$637$	−3.94353	−0.156248
$638$	0	0
$639$	0	0
$640$	0	0
$641$	26.1329	1.03219	0.516093	−	0.856532i	$-0.327386\pi$
$641$	26.1329	1.03219	0.516093	+	0.856532i	$0.327386\pi$
$642$	0	0
$643$	−30.5358	−1.20421	−0.602107	−	0.798416i	$-0.705672\pi$
$643$	−30.5358	−1.20421	−0.602107	+	0.798416i	$0.705672\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	11.5482	0.454006	0.227003	−	0.973894i	$-0.427107\pi$
$647$	11.5482	0.454006	0.227003	+	0.973894i	$0.427107\pi$
$648$	0	0
$649$	−7.59984	−0.298320
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−22.2389	−0.870277	−0.435138	−	0.900364i	$-0.643301\pi$
$653$	−22.2389	−0.870277	−0.435138	+	0.900364i	$0.643301\pi$
$654$	0	0
$655$	8.10605	0.316729
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−39.4359	−1.53620	−0.768102	−	0.640328i	$-0.778799\pi$
$659$	−39.4359	−1.53620	−0.768102	+	0.640328i	$0.778799\pi$
$660$	0	0
$661$	21.8967	0.851683	0.425841	−	0.904798i	$-0.359978\pi$
$661$	21.8967	0.851683	0.425841	+	0.904798i	$0.359978\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−17.7059	−0.686605
$666$	0	0
$667$	−4.10331	−0.158881
$668$	0	0
$669$	0	0
$670$	0	0
$671$	10.1060	0.390140
$672$	0	0
$673$	−18.2920	−0.705103	−0.352552	−	0.935792i	$-0.614686\pi$
$673$	−18.2920	−0.705103	−0.352552	+	0.935792i	$0.614686\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0.817184	0.0314069	0.0157035	−	0.999877i	$-0.495001\pi$
$677$	0.817184	0.0314069	0.0157035	+	0.999877i	$0.495001\pi$
$678$	0	0
$679$	29.2272	1.12164
$680$	0	0
$681$	0	0
$682$	0	0
$683$	12.8031	0.489895	0.244948	−	0.969536i	$-0.421229\pi$
$683$	12.8031	0.489895	0.244948	+	0.969536i	$0.421229\pi$
$684$	0	0
$685$	−6.00000	−0.229248
$686$	0	0
$687$	0	0
$688$	0	0
$689$	9.35465	0.356384
$690$	0	0
$691$	22.9394	0.872656	0.436328	−	0.899788i	$-0.356279\pi$
$691$	22.9394	0.872656	0.436328	+	0.899788i	$0.356279\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	20.0664	0.761164
$696$	0	0
$697$	50.0722	1.89662
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−32.3329	−1.22120	−0.610599	−	0.791940i	$-0.709071\pi$
$701$	−32.3329	−1.22120	−0.610599	+	0.791940i	$0.709071\pi$
$702$	0	0
$703$	−24.5516	−0.925981
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−0.297561	−0.0111909
$708$	0	0
$709$	−9.25760	−0.347677	−0.173838	−	0.984774i	$-0.555617\pi$
$709$	−9.25760	−0.347677	−0.173838	+	0.984774i	$0.555617\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−13.5385	−0.507020
$714$	0	0
$715$	6.05302	0.226370
$716$	0	0
$717$	0	0
$718$	0	0
$719$	7.01170	0.261492	0.130746	−	0.991416i	$-0.458263\pi$
$719$	7.01170	0.261492	0.130746	+	0.991416i	$0.458263\pi$
$720$	0	0
$721$	26.8065	0.998326
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−0.921150	−0.0342106
$726$	0	0
$727$	−18.0303	−0.668706	−0.334353	−	0.942448i	$-0.608518\pi$
$727$	−18.0303	−0.668706	−0.334353	+	0.942448i	$0.608518\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−7.36365	−0.272354
$732$	0	0
$733$	−22.9890	−0.849117	−0.424558	−	0.905401i	$-0.639571\pi$
$733$	−22.9890	−0.849117	−0.424558	+	0.905401i	$0.639571\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.69074	0.320128
$738$	0	0
$739$	30.0967	1.10713	0.553563	−	0.832807i	$-0.313268\pi$
$739$	30.0967	1.10713	0.553563	+	0.832807i	$0.313268\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−32.7706	−1.20224	−0.601119	−	0.799160i	$-0.705278\pi$
$743$	−32.7706	−1.20224	−0.601119	+	0.799160i	$0.705278\pi$
$744$	0	0
$745$	−21.2634	−0.779030
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−47.0971	−1.72089
$750$	0	0
$751$	0.132887	0.00484912	0.00242456	−	0.999997i	$-0.499228\pi$
$751$	0.132887	0.00484912	0.00242456	+	0.999997i	$0.499228\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	7.02720	0.255746
$756$	0	0
$757$	−40.1363	−1.45878	−0.729390	−	0.684098i	$-0.760196\pi$
$757$	−40.1363	−1.45878	−0.729390	+	0.684098i	$0.760196\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−30.9487	−1.12189	−0.560945	−	0.827853i	$-0.689562\pi$
$761$	−30.9487	−1.12189	−0.560945	+	0.827853i	$0.689562\pi$
$762$	0	0
$763$	18.0035	0.651769
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−46.0020	−1.66104
$768$	0	0
$769$	−7.17213	−0.258634	−0.129317	−	0.991603i	$-0.541278\pi$
$769$	−7.17213	−0.258634	−0.129317	+	0.991603i	$0.541278\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	15.3546	0.552268	0.276134	−	0.961119i	$-0.410947\pi$
$773$	15.3546	0.552268	0.276134	+	0.961119i	$0.410947\pi$
$774$	0	0
$775$	−3.03925	−0.109173
$776$	0	0
$777$	0	0
$778$	0	0
$779$	70.7389	2.53448
$780$	0	0
$781$	1.54545	0.0553006
$782$	0	0
$783$	0	0
$784$	0	0
$785$	21.0936	0.752864
$786$	0	0
$787$	−10.8619	−0.387184	−0.193592	−	0.981082i	$-0.562014\pi$
$787$	−10.8619	−0.387184	−0.193592	+	0.981082i	$0.562014\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−6.51247	−0.231557
$792$	0	0
$793$	61.1721	2.17229
$794$	0	0
$795$	0	0
$796$	0	0
$797$	40.7720	1.44422	0.722109	−	0.691780i	$-0.243173\pi$
$797$	40.7720	1.44422	0.722109	+	0.691780i	$0.243173\pi$
$798$	0	0
$799$	−40.3209	−1.42645
$800$	0	0
$801$	0	0
$802$	0	0
$803$	6.05302	0.213607
$804$	0	0
$805$	−11.2238	−0.395587
$806$	0	0
$807$	0	0
$808$	0	0
$809$	14.7634	0.519055	0.259528	−	0.965736i	$-0.416433\pi$
$809$	14.7634	0.519055	0.259528	+	0.965736i	$0.416433\pi$
$810$	0	0
$811$	−3.93356	−0.138126	−0.0690629	−	0.997612i	$-0.522001\pi$
$811$	−3.93356	−0.138126	−0.0690629	+	0.997612i	$0.522001\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	7.65150	0.268020
$816$	0	0
$817$	−10.4029	−0.363951
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−20.4603	−0.714071	−0.357035	−	0.934091i	$-0.616212\pi$
$821$	−20.4603	−0.714071	−0.357035	+	0.934091i	$0.616212\pi$
$822$	0	0
$823$	−23.7603	−0.828231	−0.414116	−	0.910224i	$-0.635909\pi$
$823$	−23.7603	−0.828231	−0.414116	+	0.910224i	$0.635909\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	30.1253	1.04756	0.523779	−	0.851854i	$-0.324522\pi$
$827$	30.1253	1.04756	0.523779	+	0.851854i	$0.324522\pi$
$828$	0	0
$829$	37.8388	1.31420	0.657098	−	0.753806i	$-0.271784\pi$
$829$	37.8388	1.31420	0.657098	+	0.753806i	$0.271784\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.24066	−0.112282
$834$	0	0
$835$	−3.49243	−0.120860
$836$	0	0
$837$	0	0
$838$	0	0
$839$	45.7871	1.58075	0.790374	−	0.612625i	$-0.209886\pi$
$839$	45.7871	1.58075	0.790374	+	0.612625i	$0.209886\pi$
$840$	0	0
$841$	−28.1515	−0.970741
$842$	0	0
$843$	0	0
$844$	0	0
$845$	23.6391	0.813209
$846$	0	0
$847$	−2.51962	−0.0865753
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−15.5633	−0.533503
$852$	0	0
$853$	23.3312	0.798845	0.399423	−	0.916767i	$-0.369211\pi$
$853$	23.3312	0.798845	0.399423	+	0.916767i	$0.369211\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−16.0892	−0.549596	−0.274798	−	0.961502i	$-0.588611\pi$
$857$	−16.0892	−0.549596	−0.274798	+	0.961502i	$0.588611\pi$
$858$	0	0
$859$	−13.1969	−0.450274	−0.225137	−	0.974327i	$-0.572283\pi$
$859$	−13.1969	−0.450274	−0.225137	+	0.974327i	$0.572283\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	48.5606	1.65302	0.826511	−	0.562921i	$-0.190322\pi$
$863$	48.5606	1.65302	0.826511	+	0.562921i	$0.190322\pi$
$864$	0	0
$865$	16.9742	0.577139
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−11.0272	−0.374072
$870$	0	0
$871$	52.6053	1.78246
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2.51962	−0.0851788
$876$	0	0
$877$	17.4862	0.590466	0.295233	−	0.955425i	$-0.404603\pi$
$877$	17.4862	0.590466	0.295233	+	0.955425i	$0.404603\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−23.2238	−0.782429	−0.391215	−	0.920299i	$-0.627945\pi$
$881$	−23.2238	−0.782429	−0.391215	+	0.920299i	$0.627945\pi$
$882$	0	0
$883$	−39.8967	−1.34263	−0.671315	−	0.741172i	$-0.734270\pi$
$883$	−39.8967	−1.34263	−0.671315	+	0.741172i	$0.734270\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−13.6770	−0.459228	−0.229614	−	0.973282i	$-0.573746\pi$
$887$	−13.6770	−0.459228	−0.229614	+	0.973282i	$0.573746\pi$
$888$	0	0
$889$	28.7961	0.965789
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−56.9628	−1.90619
$894$	0	0
$895$	9.14529	0.305693
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.79960	0.0933720
$900$	0	0
$901$	7.68734	0.256102
$902$	0	0
$903$	0	0
$904$	0	0
$905$	22.6391	0.752549
$906$	0	0
$907$	−1.73625	−0.0576513	−0.0288257	−	0.999584i	$-0.509177\pi$
$907$	−1.73625	−0.0576513	−0.0288257	+	0.999584i	$0.509177\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−49.4662	−1.63889	−0.819444	−	0.573160i	$-0.805717\pi$
$911$	−49.4662	−1.63889	−0.819444	+	0.573160i	$0.805717\pi$
$912$	0	0
$913$	−8.50757	−0.281560
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−20.4242	−0.674466
$918$	0	0
$919$	−40.0423	−1.32087	−0.660437	−	0.750881i	$-0.729629\pi$
$919$	−40.0423	−1.32087	−0.660437	+	0.750881i	$0.729629\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	9.35465	0.307912
$924$	0	0
$925$	−3.49380	−0.114875
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−29.0420	−0.952837	−0.476418	−	0.879219i	$-0.658065\pi$
$929$	−29.0420	−0.952837	−0.476418	+	0.879219i	$0.658065\pi$
$930$	0	0
$931$	−4.57820	−0.150044
$932$	0	0
$933$	0	0
$934$	0	0
$935$	4.97417	0.162673
$936$	0	0
$937$	40.0282	1.30766	0.653832	−	0.756639i	$-0.273160\pi$
$937$	40.0282	1.30766	0.653832	+	0.756639i	$0.273160\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−13.9604	−0.455096	−0.227548	−	0.973767i	$-0.573071\pi$
$941$	−13.9604	−0.455096	−0.227548	+	0.973767i	$0.573071\pi$
$942$	0	0
$943$	44.8415	1.46024
$944$	0	0
$945$	0	0
$946$	0	0
$947$	17.4875	0.568269	0.284134	−	0.958785i	$-0.408294\pi$
$947$	17.4875	0.568269	0.284134	+	0.958785i	$0.408294\pi$
$948$	0	0
$949$	36.6391	1.18936
$950$	0	0
$951$	0	0
$952$	0	0
$953$	49.1890	1.59339	0.796694	−	0.604383i	$-0.206580\pi$
$953$	49.1890	1.59339	0.796694	+	0.604383i	$0.206580\pi$
$954$	0	0
$955$	12.2362	0.395954
$956$	0	0
$957$	0	0
$958$	0	0
$959$	15.1177	0.488177
$960$	0	0
$961$	−21.7630	−0.702032
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.03788	0.0334105
$966$	0	0
$967$	23.5589	0.757602	0.378801	−	0.925478i	$-0.376336\pi$
$967$	23.5589	0.757602	0.378801	+	0.925478i	$0.376336\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−5.16940	−0.165894	−0.0829469	−	0.996554i	$-0.526433\pi$
$971$	−5.16940	−0.165894	−0.0829469	+	0.996554i	$0.526433\pi$
$972$	0	0
$973$	−50.5599	−1.62088
$974$	0	0
$975$	0	0
$976$	0	0
$977$	9.73970	0.311601	0.155800	−	0.987789i	$-0.450204\pi$
$977$	9.73970	0.311601	0.155800	+	0.987789i	$0.450204\pi$
$978$	0	0
$979$	−15.1453	−0.484046
$980$	0	0
$981$	0	0
$982$	0	0
$983$	2.16670	0.0691070	0.0345535	−	0.999403i	$-0.488999\pi$
$983$	2.16670	0.0691070	0.0345535	+	0.999403i	$0.488999\pi$
$984$	0	0
$985$	8.06507	0.256975
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.59441	−0.209690
$990$	0	0
$991$	25.2265	0.801347	0.400674	−	0.916221i	$-0.368776\pi$
$991$	25.2265	0.801347	0.400674	+	0.916221i	$0.368776\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	13.1453	0.416734
$996$	0	0
$997$	29.5950	0.937281	0.468641	−	0.883389i	$-0.344744\pi$
$997$	29.5950	0.937281	0.468641	+	0.883389i	$0.344744\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7920.2.a.cn.1.2		4
3.2	odd	2		7920.2.a.cm.1.2		4
4.3	odd	2		495.2.a.g.1.1	yes	4
12.11	even	2		495.2.a.f.1.4	✓	4
20.3	even	4		2475.2.c.s.199.7		8
20.7	even	4		2475.2.c.s.199.2		8
20.19	odd	2		2475.2.a.bf.1.4		4
44.43	even	2		5445.2.a.bh.1.4		4
60.23	odd	4		2475.2.c.t.199.2		8
60.47	odd	4		2475.2.c.t.199.7		8
60.59	even	2		2475.2.a.bj.1.1		4
132.131	odd	2		5445.2.a.bs.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.2.a.f.1.4	✓	4	12.11	even	2
495.2.a.g.1.1	yes	4	4.3	odd	2
2475.2.a.bf.1.4		4	20.19	odd	2
2475.2.a.bj.1.1		4	60.59	even	2
2475.2.c.s.199.2		8	20.7	even	4
2475.2.c.s.199.7		8	20.3	even	4
2475.2.c.t.199.2		8	60.23	odd	4
2475.2.c.t.199.7		8	60.47	odd	4
5445.2.a.bh.1.4		4	44.43	even	2
5445.2.a.bs.1.1		4	132.131	odd	2
7920.2.a.cm.1.2		4	3.2	odd	2
7920.2.a.cn.1.2		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:	\( N \)	\(=\)	\( 7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7920.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -2.51962 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -2.51962 q^{7} +1.00000 q^{11} +6.05302 q^{13} +4.97417 q^{17} +7.02720 q^{19} +4.45455 q^{23} +1.00000 q^{25} -0.921150 q^{29} -3.03925 q^{31} -2.51962 q^{35} -3.49380 q^{37} +10.0664 q^{41} -1.48038 q^{43} -8.10605 q^{47} -0.651497 q^{49} +1.54545 q^{53} +1.00000 q^{55} -7.59984 q^{59} +10.1060 q^{61} +6.05302 q^{65} +8.69074 q^{67} +1.54545 q^{71} +6.05302 q^{73} -2.51962 q^{77} -11.0272 q^{79} -8.50757 q^{83} +4.97417 q^{85} -15.1453 q^{89} -15.2513 q^{91} +7.02720 q^{95} -11.5998 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 4 q^{7}+O(q^{10}) \) 4 * q + 4 * q^5 - 4 * q^7	\( 4 q + 4 q^{5} - 4 q^{7} + 4 q^{11} + 8 q^{13} + 4 q^{17} - 4 q^{19} + 8 q^{23} + 4 q^{25} - 4 q^{29} - 4 q^{35} + 8 q^{37} - 4 q^{41} - 12 q^{43} + 20 q^{49} + 16 q^{53} + 4 q^{55} + 24 q^{59} + 8 q^{61} + 8 q^{65} + 16 q^{71} + 8 q^{73} - 4 q^{77} - 12 q^{79} - 8 q^{83} + 4 q^{85} - 16 q^{89} + 16 q^{91} - 4 q^{95} + 8 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 - 4 * q^7 + 4 * q^11 + 8 * q^13 + 4 * q^17 - 4 * q^19 + 8 * q^23 + 4 * q^25 - 4 * q^29 - 4 * q^35 + 8 * q^37 - 4 * q^41 - 12 * q^43 + 20 * q^49 + 16 * q^53 + 4 * q^55 + 24 * q^59 + 8 * q^61 + 8 * q^65 + 16 * q^71 + 8 * q^73 - 4 * q^77 - 12 * q^79 - 8 * q^83 + 4 * q^85 - 16 * q^89 + 16 * q^91 - 4 * q^95 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more