LMFDB - Embedded newform 7920.2.a.bx.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:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3960)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	$q-1.00000 q^{5} +3.41421 q^{7} +O(q^{10})$	$q-1.00000 q^{5} +3.41421 q^{7} +1.00000 q^{11} -1.41421 q^{13} -4.24264 q^{17} +6.82843 q^{19} +2.82843 q^{23} +1.00000 q^{25} -3.65685 q^{29} -0.828427 q^{31} -3.41421 q^{35} +7.65685 q^{37} +7.65685 q^{41} -5.07107 q^{43} -12.4853 q^{47} +4.65685 q^{49} +10.4853 q^{53} -1.00000 q^{55} -3.17157 q^{59} +3.17157 q^{61} +1.41421 q^{65} +8.48528 q^{67} +6.48528 q^{71} -7.07107 q^{73} +3.41421 q^{77} +16.4853 q^{79} -1.75736 q^{83} +4.24264 q^{85} +2.00000 q^{89} -4.82843 q^{91} -6.82843 q^{95} +3.17157 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 4 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 + 4 * q^7	$ 2 q - 2 q^{5} + 4 q^{7} + 2 q^{11} + 8 q^{19} + 2 q^{25} + 4 q^{29} + 4 q^{31} - 4 q^{35} + 4 q^{37} + 4 q^{41} + 4 q^{43} - 8 q^{47} - 2 q^{49} + 4 q^{53} - 2 q^{55} - 12 q^{59} + 12 q^{61} - 4 q^{71} + 4 q^{77} + 16 q^{79} - 12 q^{83} + 4 q^{89} - 4 q^{91} - 8 q^{95} + 12 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 4 * q^7 + 2 * q^11 + 8 * q^19 + 2 * q^25 + 4 * q^29 + 4 * q^31 - 4 * q^35 + 4 * q^37 + 4 * q^41 + 4 * q^43 - 8 * q^47 - 2 * q^49 + 4 * q^53 - 2 * q^55 - 12 * q^59 + 12 * q^61 - 4 * q^71 + 4 * q^77 + 16 * q^79 - 12 * q^83 + 4 * q^89 - 4 * q^91 - 8 * q^95 + 12 * 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
$7$	3.41421	1.29045	0.645226	−	0.763992i	$-0.276763\pi$
$7$	3.41421	1.29045	0.645226	+	0.763992i	$0.276763\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.41421	−0.392232	−0.196116	−	0.980581i	$-0.562833\pi$
$13$	−1.41421	−0.392232	−0.196116	+	0.980581i	$0.562833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.24264	−1.02899	−0.514496	−	0.857493i	$-0.672021\pi$
$17$	−4.24264	−1.02899	−0.514496	+	0.857493i	$0.672021\pi$
$18$	0	0
$19$	6.82843	1.56655	0.783274	−	0.621676i	$-0.213548\pi$
$19$	6.82843	1.56655	0.783274	+	0.621676i	$0.213548\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.82843	0.589768	0.294884	−	0.955533i	$-0.404719\pi$
$23$	2.82843	0.589768	0.294884	+	0.955533i	$0.404719\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.65685	−0.679061	−0.339530	−	0.940595i	$-0.610268\pi$
$29$	−3.65685	−0.679061	−0.339530	+	0.940595i	$0.610268\pi$
$30$	0	0
$31$	−0.828427	−0.148790	−0.0743950	−	0.997229i	$-0.523703\pi$
$31$	−0.828427	−0.148790	−0.0743950	+	0.997229i	$0.523703\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.41421	−0.577107
$36$	0	0
$37$	7.65685	1.25878	0.629390	−	0.777090i	$-0.283305\pi$
$37$	7.65685	1.25878	0.629390	+	0.777090i	$0.283305\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.65685	1.19580	0.597900	−	0.801571i	$-0.296002\pi$
$41$	7.65685	1.19580	0.597900	+	0.801571i	$0.296002\pi$
$42$	0	0
$43$	−5.07107	−0.773331	−0.386665	−	0.922220i	$-0.626373\pi$
$43$	−5.07107	−0.773331	−0.386665	+	0.922220i	$0.626373\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−12.4853	−1.82117	−0.910583	−	0.413327i	$-0.864367\pi$
$47$	−12.4853	−1.82117	−0.910583	+	0.413327i	$0.864367\pi$
$48$	0	0
$49$	4.65685	0.665265
$50$	0	0
$51$	0	0
$52$	0	0
$53$	10.4853	1.44026	0.720132	−	0.693837i	$-0.244081\pi$
$53$	10.4853	1.44026	0.720132	+	0.693837i	$0.244081\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3.17157	−0.412904	−0.206452	−	0.978457i	$-0.566192\pi$
$59$	−3.17157	−0.412904	−0.206452	+	0.978457i	$0.566192\pi$
$60$	0	0
$61$	3.17157	0.406078	0.203039	−	0.979171i	$-0.434918\pi$
$61$	3.17157	0.406078	0.203039	+	0.979171i	$0.434918\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.41421	0.175412
$66$	0	0
$67$	8.48528	1.03664	0.518321	−	0.855186i	$-0.326557\pi$
$67$	8.48528	1.03664	0.518321	+	0.855186i	$0.326557\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.48528	0.769661	0.384831	−	0.922987i	$-0.374260\pi$
$71$	6.48528	0.769661	0.384831	+	0.922987i	$0.374260\pi$
$72$	0	0
$73$	−7.07107	−0.827606	−0.413803	−	0.910366i	$-0.635800\pi$
$73$	−7.07107	−0.827606	−0.413803	+	0.910366i	$0.635800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.41421	0.389086
$78$	0	0
$79$	16.4853	1.85474	0.927370	−	0.374147i	$-0.122064\pi$
$79$	16.4853	1.85474	0.927370	+	0.374147i	$0.122064\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1.75736	−0.192895	−0.0964476	−	0.995338i	$-0.530748\pi$
$83$	−1.75736	−0.192895	−0.0964476	+	0.995338i	$0.530748\pi$
$84$	0	0
$85$	4.24264	0.460179
$86$	0	0
$87$	0	0
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	−4.82843	−0.506157
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.82843	−0.700582
$96$	0	0
$97$	3.17157	0.322024	0.161012	−	0.986952i	$-0.448524\pi$
$97$	3.17157	0.322024	0.161012	+	0.986952i	$0.448524\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.51472	0.150720	0.0753601	−	0.997156i	$-0.475989\pi$
$101$	1.51472	0.150720	0.0753601	+	0.997156i	$0.475989\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.41421	−0.716759	−0.358380	−	0.933576i	$-0.616671\pi$
$107$	−7.41421	−0.716759	−0.358380	+	0.933576i	$0.616671\pi$
$108$	0	0
$109$	−13.3137	−1.27522	−0.637611	−	0.770358i	$-0.720077\pi$
$109$	−13.3137	−1.27522	−0.637611	+	0.770358i	$0.720077\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−3.65685	−0.344008	−0.172004	−	0.985096i	$-0.555024\pi$
$113$	−3.65685	−0.344008	−0.172004	+	0.985096i	$0.555024\pi$
$114$	0	0
$115$	−2.82843	−0.263752
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−14.4853	−1.32786
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−21.5563	−1.91282	−0.956408	−	0.292033i	$-0.905668\pi$
$127$	−21.5563	−1.91282	−0.956408	+	0.292033i	$0.905668\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−6.82843	−0.596602	−0.298301	−	0.954472i	$-0.596420\pi$
$131$	−6.82843	−0.596602	−0.298301	+	0.954472i	$0.596420\pi$
$132$	0	0
$133$	23.3137	2.02155
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17.3137	1.47921	0.739605	−	0.673041i	$-0.235012\pi$
$137$	17.3137	1.47921	0.739605	+	0.673041i	$0.235012\pi$
$138$	0	0
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.41421	−0.118262
$144$	0	0
$145$	3.65685	0.303685
$146$	0	0
$147$	0	0
$148$	0	0
$149$	14.4853	1.18668	0.593340	−	0.804952i	$-0.297809\pi$
$149$	14.4853	1.18668	0.593340	+	0.804952i	$0.297809\pi$
$150$	0	0
$151$	18.1421	1.47639	0.738193	−	0.674590i	$-0.235679\pi$
$151$	18.1421	1.47639	0.738193	+	0.674590i	$0.235679\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.828427	0.0665409
$156$	0	0
$157$	22.4853	1.79452	0.897260	−	0.441502i	$-0.145554\pi$
$157$	22.4853	1.79452	0.897260	+	0.441502i	$0.145554\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	9.65685	0.761067
$162$	0	0
$163$	11.3137	0.886158	0.443079	−	0.896483i	$-0.353886\pi$
$163$	11.3137	0.886158	0.443079	+	0.896483i	$0.353886\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.72792	0.520622	0.260311	−	0.965525i	$-0.416175\pi$
$167$	6.72792	0.520622	0.260311	+	0.965525i	$0.416175\pi$
$168$	0	0
$169$	−11.0000	−0.846154
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.10051	0.159698	0.0798492	−	0.996807i	$-0.474556\pi$
$173$	2.10051	0.159698	0.0798492	+	0.996807i	$0.474556\pi$
$174$	0	0
$175$	3.41421	0.258090
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−6.34315	−0.474109	−0.237054	−	0.971496i	$-0.576182\pi$
$179$	−6.34315	−0.474109	−0.237054	+	0.971496i	$0.576182\pi$
$180$	0	0
$181$	0.686292	0.0510116	0.0255058	−	0.999675i	$-0.491880\pi$
$181$	0.686292	0.0510116	0.0255058	+	0.999675i	$0.491880\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−7.65685	−0.562943
$186$	0	0
$187$	−4.24264	−0.310253
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−19.3137	−1.39749	−0.698745	−	0.715370i	$-0.746258\pi$
$191$	−19.3137	−1.39749	−0.698745	+	0.715370i	$0.746258\pi$
$192$	0	0
$193$	−1.89949	−0.136729	−0.0683643	−	0.997660i	$-0.521778\pi$
$193$	−1.89949	−0.136729	−0.0683643	+	0.997660i	$0.521778\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−0.242641	−0.0172874	−0.00864372	−	0.999963i	$-0.502751\pi$
$197$	−0.242641	−0.0172874	−0.00864372	+	0.999963i	$0.502751\pi$
$198$	0	0
$199$	−16.9706	−1.20301	−0.601506	−	0.798869i	$-0.705432\pi$
$199$	−16.9706	−1.20301	−0.601506	+	0.798869i	$0.705432\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−12.4853	−0.876295
$204$	0	0
$205$	−7.65685	−0.534778
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6.82843	0.472332
$210$	0	0
$211$	3.51472	0.241963	0.120982	−	0.992655i	$-0.461396\pi$
$211$	3.51472	0.241963	0.120982	+	0.992655i	$0.461396\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.07107	0.345844
$216$	0	0
$217$	−2.82843	−0.192006
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.00000	0.403604
$222$	0	0
$223$	−9.17157	−0.614174	−0.307087	−	0.951681i	$-0.599354\pi$
$223$	−9.17157	−0.614174	−0.307087	+	0.951681i	$0.599354\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−13.0711	−0.867557	−0.433779	−	0.901019i	$-0.642820\pi$
$227$	−13.0711	−0.867557	−0.433779	+	0.901019i	$0.642820\pi$
$228$	0	0
$229$	13.3137	0.879795	0.439897	−	0.898048i	$-0.355015\pi$
$229$	13.3137	0.879795	0.439897	+	0.898048i	$0.355015\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.7279	1.09588	0.547941	−	0.836517i	$-0.315412\pi$
$233$	16.7279	1.09588	0.547941	+	0.836517i	$0.315412\pi$
$234$	0	0
$235$	12.4853	0.814450
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−10.1421	−0.656040	−0.328020	−	0.944671i	$-0.606381\pi$
$239$	−10.1421	−0.656040	−0.328020	+	0.944671i	$0.606381\pi$
$240$	0	0
$241$	16.1421	1.03981	0.519903	−	0.854225i	$-0.325968\pi$
$241$	16.1421	1.03981	0.519903	+	0.854225i	$0.325968\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−4.65685	−0.297516
$246$	0	0
$247$	−9.65685	−0.614451
$248$	0	0
$249$	0	0
$250$	0	0
$251$	26.4853	1.67174	0.835868	−	0.548930i	$-0.184965\pi$
$251$	26.4853	1.67174	0.835868	+	0.548930i	$0.184965\pi$
$252$	0	0
$253$	2.82843	0.177822
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−29.7990	−1.85881	−0.929405	−	0.369062i	$-0.879679\pi$
$257$	−29.7990	−1.85881	−0.929405	+	0.369062i	$0.879679\pi$
$258$	0	0
$259$	26.1421	1.62439
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6.72792	0.414861	0.207431	−	0.978250i	$-0.433490\pi$
$263$	6.72792	0.414861	0.207431	+	0.978250i	$0.433490\pi$
$264$	0	0
$265$	−10.4853	−0.644106
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−9.65685	−0.588789	−0.294394	−	0.955684i	$-0.595118\pi$
$269$	−9.65685	−0.588789	−0.294394	+	0.955684i	$0.595118\pi$
$270$	0	0
$271$	−4.48528	−0.272461	−0.136231	−	0.990677i	$-0.543499\pi$
$271$	−4.48528	−0.272461	−0.136231	+	0.990677i	$0.543499\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	17.4142	1.04632	0.523159	−	0.852235i	$-0.324753\pi$
$277$	17.4142	1.04632	0.523159	+	0.852235i	$0.324753\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	29.3137	1.74871	0.874355	−	0.485288i	$-0.161285\pi$
$281$	29.3137	1.74871	0.874355	+	0.485288i	$0.161285\pi$
$282$	0	0
$283$	1.75736	0.104464	0.0522321	−	0.998635i	$-0.483366\pi$
$283$	1.75736	0.104464	0.0522321	+	0.998635i	$0.483366\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	26.1421	1.54312
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−6.58579	−0.384746	−0.192373	−	0.981322i	$-0.561618\pi$
$293$	−6.58579	−0.384746	−0.192373	+	0.981322i	$0.561618\pi$
$294$	0	0
$295$	3.17157	0.184656
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.00000	−0.231326
$300$	0	0
$301$	−17.3137	−0.997946
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3.17157	−0.181604
$306$	0	0
$307$	1.75736	0.100298	0.0501489	−	0.998742i	$-0.484030\pi$
$307$	1.75736	0.100298	0.0501489	+	0.998742i	$0.484030\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4.82843	−0.273795	−0.136897	−	0.990585i	$-0.543713\pi$
$311$	−4.82843	−0.273795	−0.136897	+	0.990585i	$0.543713\pi$
$312$	0	0
$313$	18.0000	1.01742	0.508710	−	0.860938i	$-0.330123\pi$
$313$	18.0000	1.01742	0.508710	+	0.860938i	$0.330123\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	26.0000	1.46031	0.730153	−	0.683284i	$-0.239449\pi$
$317$	26.0000	1.46031	0.730153	+	0.683284i	$0.239449\pi$
$318$	0	0
$319$	−3.65685	−0.204745
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−28.9706	−1.61197
$324$	0	0
$325$	−1.41421	−0.0784465
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−42.6274	−2.35013
$330$	0	0
$331$	2.48528	0.136603	0.0683017	−	0.997665i	$-0.478242\pi$
$331$	2.48528	0.136603	0.0683017	+	0.997665i	$0.478242\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−8.48528	−0.463600
$336$	0	0
$337$	−34.8701	−1.89949	−0.949747	−	0.313020i	$-0.898659\pi$
$337$	−34.8701	−1.89949	−0.949747	+	0.313020i	$0.898659\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−0.828427	−0.0448618
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9.75736	0.523802	0.261901	−	0.965095i	$-0.415651\pi$
$347$	9.75736	0.523802	0.261901	+	0.965095i	$0.415651\pi$
$348$	0	0
$349$	−16.8284	−0.900805	−0.450403	−	0.892826i	$-0.648720\pi$
$349$	−16.8284	−0.900805	−0.450403	+	0.892826i	$0.648720\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	6.48528	0.345177	0.172588	−	0.984994i	$-0.444787\pi$
$353$	6.48528	0.345177	0.172588	+	0.984994i	$0.444787\pi$
$354$	0	0
$355$	−6.48528	−0.344203
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8.48528	0.447836	0.223918	−	0.974608i	$-0.428115\pi$
$359$	8.48528	0.447836	0.223918	+	0.974608i	$0.428115\pi$
$360$	0	0
$361$	27.6274	1.45407
$362$	0	0
$363$	0	0
$364$	0	0
$365$	7.07107	0.370117
$366$	0	0
$367$	26.8284	1.40043	0.700216	−	0.713931i	$-0.253087\pi$
$367$	26.8284	1.40043	0.700216	+	0.713931i	$0.253087\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	35.7990	1.85859
$372$	0	0
$373$	24.2426	1.25524	0.627618	−	0.778521i	$-0.284030\pi$
$373$	24.2426	1.25524	0.627618	+	0.778521i	$0.284030\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.17157	0.266350
$378$	0	0
$379$	15.3137	0.786612	0.393306	−	0.919408i	$-0.371331\pi$
$379$	15.3137	0.786612	0.393306	+	0.919408i	$0.371331\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	32.9706	1.68472	0.842359	−	0.538918i	$-0.181167\pi$
$383$	32.9706	1.68472	0.842359	+	0.538918i	$0.181167\pi$
$384$	0	0
$385$	−3.41421	−0.174004
$386$	0	0
$387$	0	0
$388$	0	0
$389$	19.3137	0.979244	0.489622	−	0.871935i	$-0.337135\pi$
$389$	19.3137	0.979244	0.489622	+	0.871935i	$0.337135\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−16.4853	−0.829465
$396$	0	0
$397$	1.51472	0.0760215	0.0380108	−	0.999277i	$-0.487898\pi$
$397$	1.51472	0.0760215	0.0380108	+	0.999277i	$0.487898\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−13.3137	−0.664855	−0.332427	−	0.943129i	$-0.607868\pi$
$401$	−13.3137	−0.664855	−0.332427	+	0.943129i	$0.607868\pi$
$402$	0	0
$403$	1.17157	0.0583602
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.65685	0.379536
$408$	0	0
$409$	−26.9706	−1.33361	−0.666804	−	0.745233i	$-0.732338\pi$
$409$	−26.9706	−1.33361	−0.666804	+	0.745233i	$0.732338\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−10.8284	−0.532832
$414$	0	0
$415$	1.75736	0.0862654
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−18.6274	−0.910009	−0.455004	−	0.890489i	$-0.650362\pi$
$419$	−18.6274	−0.910009	−0.455004	+	0.890489i	$0.650362\pi$
$420$	0	0
$421$	22.3431	1.08894	0.544469	−	0.838781i	$-0.316731\pi$
$421$	22.3431	1.08894	0.544469	+	0.838781i	$0.316731\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−4.24264	−0.205798
$426$	0	0
$427$	10.8284	0.524024
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−24.9706	−1.20279	−0.601395	−	0.798952i	$-0.705388\pi$
$431$	−24.9706	−1.20279	−0.601395	+	0.798952i	$0.705388\pi$
$432$	0	0
$433$	−20.8284	−1.00095	−0.500475	−	0.865751i	$-0.666841\pi$
$433$	−20.8284	−1.00095	−0.500475	+	0.865751i	$0.666841\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	19.3137	0.923900
$438$	0	0
$439$	16.6863	0.796393	0.398197	−	0.917300i	$-0.369636\pi$
$439$	16.6863	0.796393	0.398197	+	0.917300i	$0.369636\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	0	0
$445$	−2.00000	−0.0948091
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4.34315	0.204966	0.102483	−	0.994735i	$-0.467321\pi$
$449$	4.34315	0.204966	0.102483	+	0.994735i	$0.467321\pi$
$450$	0	0
$451$	7.65685	0.360547
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4.82843	0.226360
$456$	0	0
$457$	25.6985	1.20212	0.601062	−	0.799202i	$-0.294744\pi$
$457$	25.6985	1.20212	0.601062	+	0.799202i	$0.294744\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	39.4558	1.83764	0.918821	−	0.394675i	$-0.129143\pi$
$461$	39.4558	1.83764	0.918821	+	0.394675i	$0.129143\pi$
$462$	0	0
$463$	−20.4853	−0.952032	−0.476016	−	0.879437i	$-0.657920\pi$
$463$	−20.4853	−0.952032	−0.476016	+	0.879437i	$0.657920\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.82843	0.130884	0.0654420	−	0.997856i	$-0.479154\pi$
$467$	2.82843	0.130884	0.0654420	+	0.997856i	$0.479154\pi$
$468$	0	0
$469$	28.9706	1.33774
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−5.07107	−0.233168
$474$	0	0
$475$	6.82843	0.313310
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2.14214	0.0978767	0.0489383	−	0.998802i	$-0.484416\pi$
$479$	2.14214	0.0978767	0.0489383	+	0.998802i	$0.484416\pi$
$480$	0	0
$481$	−10.8284	−0.493734
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.17157	−0.144014
$486$	0	0
$487$	11.7990	0.534663	0.267332	−	0.963605i	$-0.413858\pi$
$487$	11.7990	0.534663	0.267332	+	0.963605i	$0.413858\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	40.9706	1.84898	0.924488	−	0.381212i	$-0.124493\pi$
$491$	40.9706	1.84898	0.924488	+	0.381212i	$0.124493\pi$
$492$	0	0
$493$	15.5147	0.698748
$494$	0	0
$495$	0	0
$496$	0	0
$497$	22.1421	0.993211
$498$	0	0
$499$	36.0000	1.61158	0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000	1.61158	0.805791	+	0.592200i	$0.201741\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−17.0711	−0.761161	−0.380581	−	0.924748i	$-0.624276\pi$
$503$	−17.0711	−0.761161	−0.380581	+	0.924748i	$0.624276\pi$
$504$	0	0
$505$	−1.51472	−0.0674041
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−12.6863	−0.562310	−0.281155	−	0.959662i	$-0.590718\pi$
$509$	−12.6863	−0.562310	−0.281155	+	0.959662i	$0.590718\pi$
$510$	0	0
$511$	−24.1421	−1.06799
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−12.4853	−0.549102
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−24.6274	−1.07895	−0.539473	−	0.842003i	$-0.681377\pi$
$521$	−24.6274	−1.07895	−0.539473	+	0.842003i	$0.681377\pi$
$522$	0	0
$523$	−4.10051	−0.179303	−0.0896513	−	0.995973i	$-0.528575\pi$
$523$	−4.10051	−0.179303	−0.0896513	+	0.995973i	$0.528575\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.51472	0.153104
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−10.8284	−0.469031
$534$	0	0
$535$	7.41421	0.320544
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.65685	0.200585
$540$	0	0
$541$	31.6569	1.36103	0.680517	−	0.732732i	$-0.261755\pi$
$541$	31.6569	1.36103	0.680517	+	0.732732i	$0.261755\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	13.3137	0.570297
$546$	0	0
$547$	13.2721	0.567473	0.283737	−	0.958902i	$-0.408426\pi$
$547$	13.2721	0.567473	0.283737	+	0.958902i	$0.408426\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−24.9706	−1.06378
$552$	0	0
$553$	56.2843	2.39345
$554$	0	0
$555$	0	0
$556$	0	0
$557$	41.2132	1.74626	0.873130	−	0.487488i	$-0.162087\pi$
$557$	41.2132	1.74626	0.873130	+	0.487488i	$0.162087\pi$
$558$	0	0
$559$	7.17157	0.303325
$560$	0	0
$561$	0	0
$562$	0	0
$563$	5.07107	0.213720	0.106860	−	0.994274i	$-0.465920\pi$
$563$	5.07107	0.213720	0.106860	+	0.994274i	$0.465920\pi$
$564$	0	0
$565$	3.65685	0.153845
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−4.14214	−0.173647	−0.0868237	−	0.996224i	$-0.527672\pi$
$569$	−4.14214	−0.173647	−0.0868237	+	0.996224i	$0.527672\pi$
$570$	0	0
$571$	17.6569	0.738916	0.369458	−	0.929247i	$-0.379543\pi$
$571$	17.6569	0.738916	0.369458	+	0.929247i	$0.379543\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.82843	0.117954
$576$	0	0
$577$	8.82843	0.367532	0.183766	−	0.982970i	$-0.441171\pi$
$577$	8.82843	0.367532	0.183766	+	0.982970i	$0.441171\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−6.00000	−0.248922
$582$	0	0
$583$	10.4853	0.434256
$584$	0	0
$585$	0	0
$586$	0	0
$587$	24.4853	1.01062	0.505308	−	0.862939i	$-0.331379\pi$
$587$	24.4853	1.01062	0.505308	+	0.862939i	$0.331379\pi$
$588$	0	0
$589$	−5.65685	−0.233087
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−33.4142	−1.37216	−0.686079	−	0.727527i	$-0.740669\pi$
$593$	−33.4142	−1.37216	−0.686079	+	0.727527i	$0.740669\pi$
$594$	0	0
$595$	14.4853	0.593839
$596$	0	0
$597$	0	0
$598$	0	0
$599$	2.34315	0.0957383	0.0478692	−	0.998854i	$-0.484757\pi$
$599$	2.34315	0.0957383	0.0478692	+	0.998854i	$0.484757\pi$
$600$	0	0
$601$	20.1421	0.821615	0.410807	−	0.911722i	$-0.365247\pi$
$601$	20.1421	0.821615	0.410807	+	0.911722i	$0.365247\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	36.3848	1.47681	0.738406	−	0.674356i	$-0.235579\pi$
$607$	36.3848	1.47681	0.738406	+	0.674356i	$0.235579\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	17.6569	0.714320
$612$	0	0
$613$	4.44365	0.179477	0.0897387	−	0.995965i	$-0.471397\pi$
$613$	4.44365	0.179477	0.0897387	+	0.995965i	$0.471397\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−28.3431	−1.14105	−0.570526	−	0.821280i	$-0.693261\pi$
$617$	−28.3431	−1.14105	−0.570526	+	0.821280i	$0.693261\pi$
$618$	0	0
$619$	−15.4558	−0.621223	−0.310611	−	0.950537i	$-0.600534\pi$
$619$	−15.4558	−0.621223	−0.310611	+	0.950537i	$0.600534\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6.82843	0.273575
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−32.4853	−1.29527
$630$	0	0
$631$	22.6274	0.900783	0.450392	−	0.892831i	$-0.351284\pi$
$631$	22.6274	0.900783	0.450392	+	0.892831i	$0.351284\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	21.5563	0.855438
$636$	0	0
$637$	−6.58579	−0.260938
$638$	0	0
$639$	0	0
$640$	0	0
$641$	25.3137	0.999831	0.499916	−	0.866074i	$-0.333364\pi$
$641$	25.3137	0.999831	0.499916	+	0.866074i	$0.333364\pi$
$642$	0	0
$643$	−18.3431	−0.723383	−0.361692	−	0.932298i	$-0.617801\pi$
$643$	−18.3431	−0.723383	−0.361692	+	0.932298i	$0.617801\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6.62742	−0.260551	−0.130275	−	0.991478i	$-0.541586\pi$
$647$	−6.62742	−0.260551	−0.130275	+	0.991478i	$0.541586\pi$
$648$	0	0
$649$	−3.17157	−0.124495
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5.79899	0.226932	0.113466	−	0.993542i	$-0.463805\pi$
$653$	5.79899	0.226932	0.113466	+	0.993542i	$0.463805\pi$
$654$	0	0
$655$	6.82843	0.266809
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−22.6274	−0.881439	−0.440720	−	0.897645i	$-0.645277\pi$
$659$	−22.6274	−0.881439	−0.440720	+	0.897645i	$0.645277\pi$
$660$	0	0
$661$	39.6569	1.54247	0.771236	−	0.636549i	$-0.219639\pi$
$661$	39.6569	1.54247	0.771236	+	0.636549i	$0.219639\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−23.3137	−0.904067
$666$	0	0
$667$	−10.3431	−0.400488
$668$	0	0
$669$	0	0
$670$	0	0
$671$	3.17157	0.122437
$672$	0	0
$673$	25.8995	0.998352	0.499176	−	0.866501i	$-0.333636\pi$
$673$	25.8995	0.998352	0.499176	+	0.866501i	$0.333636\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−25.8995	−0.995398	−0.497699	−	0.867350i	$-0.665822\pi$
$677$	−25.8995	−0.995398	−0.497699	+	0.867350i	$0.665822\pi$
$678$	0	0
$679$	10.8284	0.415557
$680$	0	0
$681$	0	0
$682$	0	0
$683$	3.79899	0.145364	0.0726822	−	0.997355i	$-0.476844\pi$
$683$	3.79899	0.145364	0.0726822	+	0.997355i	$0.476844\pi$
$684$	0	0
$685$	−17.3137	−0.661523
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−14.8284	−0.564918
$690$	0	0
$691$	−6.34315	−0.241305	−0.120652	−	0.992695i	$-0.538499\pi$
$691$	−6.34315	−0.241305	−0.120652	+	0.992695i	$0.538499\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−16.0000	−0.606915
$696$	0	0
$697$	−32.4853	−1.23047
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−30.2843	−1.14382	−0.571911	−	0.820316i	$-0.693798\pi$
$701$	−30.2843	−1.14382	−0.571911	+	0.820316i	$0.693798\pi$
$702$	0	0
$703$	52.2843	1.97194
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.17157	0.194497
$708$	0	0
$709$	12.6863	0.476444	0.238222	−	0.971211i	$-0.423435\pi$
$709$	12.6863	0.476444	0.238222	+	0.971211i	$0.423435\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2.34315	−0.0877515
$714$	0	0
$715$	1.41421	0.0528886
$716$	0	0
$717$	0	0
$718$	0	0
$719$	5.65685	0.210965	0.105483	−	0.994421i	$-0.466361\pi$
$719$	5.65685	0.210965	0.105483	+	0.994421i	$0.466361\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−3.65685	−0.135812
$726$	0	0
$727$	28.2843	1.04901	0.524503	−	0.851409i	$-0.324251\pi$
$727$	28.2843	1.04901	0.524503	+	0.851409i	$0.324251\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	21.5147	0.795751
$732$	0	0
$733$	35.5563	1.31330	0.656652	−	0.754194i	$-0.271972\pi$
$733$	35.5563	1.31330	0.656652	+	0.754194i	$0.271972\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.48528	0.312559
$738$	0	0
$739$	−36.2843	−1.33474	−0.667369	−	0.744727i	$-0.732580\pi$
$739$	−36.2843	−1.33474	−0.667369	+	0.744727i	$0.732580\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−22.5269	−0.826432	−0.413216	−	0.910633i	$-0.635595\pi$
$743$	−22.5269	−0.826432	−0.413216	+	0.910633i	$0.635595\pi$
$744$	0	0
$745$	−14.4853	−0.530700
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−25.3137	−0.924943
$750$	0	0
$751$	−13.6569	−0.498346	−0.249173	−	0.968459i	$-0.580159\pi$
$751$	−13.6569	−0.498346	−0.249173	+	0.968459i	$0.580159\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−18.1421	−0.660260
$756$	0	0
$757$	22.4853	0.817241	0.408621	−	0.912704i	$-0.366010\pi$
$757$	22.4853	0.817241	0.408621	+	0.912704i	$0.366010\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−4.82843	−0.175030	−0.0875152	−	0.996163i	$-0.527893\pi$
$761$	−4.82843	−0.175030	−0.0875152	+	0.996163i	$0.527893\pi$
$762$	0	0
$763$	−45.4558	−1.64561
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.48528	0.161954
$768$	0	0
$769$	−32.8284	−1.18382	−0.591912	−	0.806003i	$-0.701627\pi$
$769$	−32.8284	−1.18382	−0.591912	+	0.806003i	$0.701627\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−40.1421	−1.44381	−0.721906	−	0.691991i	$-0.756734\pi$
$773$	−40.1421	−1.44381	−0.721906	+	0.691991i	$0.756734\pi$
$774$	0	0
$775$	−0.828427	−0.0297580
$776$	0	0
$777$	0	0
$778$	0	0
$779$	52.2843	1.87328
$780$	0	0
$781$	6.48528	0.232062
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−22.4853	−0.802534
$786$	0	0
$787$	−4.10051	−0.146167	−0.0730836	−	0.997326i	$-0.523284\pi$
$787$	−4.10051	−0.146167	−0.0730836	+	0.997326i	$0.523284\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−12.4853	−0.443925
$792$	0	0
$793$	−4.48528	−0.159277
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−28.6274	−1.01404	−0.507018	−	0.861936i	$-0.669252\pi$
$797$	−28.6274	−1.01404	−0.507018	+	0.861936i	$0.669252\pi$
$798$	0	0
$799$	52.9706	1.87396
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−7.07107	−0.249533
$804$	0	0
$805$	−9.65685	−0.340359
$806$	0	0
$807$	0	0
$808$	0	0
$809$	54.9706	1.93266	0.966331	−	0.257302i	$-0.0828335\pi$
$809$	54.9706	1.93266	0.966331	+	0.257302i	$0.0828335\pi$
$810$	0	0
$811$	−30.6274	−1.07547	−0.537737	−	0.843113i	$-0.680721\pi$
$811$	−30.6274	−1.07547	−0.537737	+	0.843113i	$0.680721\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−11.3137	−0.396302
$816$	0	0
$817$	−34.6274	−1.21146
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.31371	−0.0458487	−0.0229244	−	0.999737i	$-0.507298\pi$
$821$	−1.31371	−0.0458487	−0.0229244	+	0.999737i	$0.507298\pi$
$822$	0	0
$823$	−42.6274	−1.48590	−0.742949	−	0.669348i	$-0.766574\pi$
$823$	−42.6274	−1.48590	−0.742949	+	0.669348i	$0.766574\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−36.8701	−1.28210	−0.641049	−	0.767500i	$-0.721500\pi$
$827$	−36.8701	−1.28210	−0.641049	+	0.767500i	$0.721500\pi$
$828$	0	0
$829$	6.62742	0.230180	0.115090	−	0.993355i	$-0.463284\pi$
$829$	6.62742	0.230180	0.115090	+	0.993355i	$0.463284\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−19.7574	−0.684552
$834$	0	0
$835$	−6.72792	−0.232829
$836$	0	0
$837$	0	0
$838$	0	0
$839$	44.1421	1.52396	0.761978	−	0.647603i	$-0.224228\pi$
$839$	44.1421	1.52396	0.761978	+	0.647603i	$0.224228\pi$
$840$	0	0
$841$	−15.6274	−0.538876
$842$	0	0
$843$	0	0
$844$	0	0
$845$	11.0000	0.378412
$846$	0	0
$847$	3.41421	0.117314
$848$	0	0
$849$	0	0
$850$	0	0
$851$	21.6569	0.742387
$852$	0	0
$853$	34.5858	1.18419	0.592097	−	0.805866i	$-0.298300\pi$
$853$	34.5858	1.18419	0.592097	+	0.805866i	$0.298300\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−22.1005	−0.754939	−0.377469	−	0.926022i	$-0.623206\pi$
$857$	−22.1005	−0.754939	−0.377469	+	0.926022i	$0.623206\pi$
$858$	0	0
$859$	−36.1421	−1.23315	−0.616577	−	0.787295i	$-0.711481\pi$
$859$	−36.1421	−1.23315	−0.616577	+	0.787295i	$0.711481\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−4.68629	−0.159523	−0.0797616	−	0.996814i	$-0.525416\pi$
$863$	−4.68629	−0.159523	−0.0797616	+	0.996814i	$0.525416\pi$
$864$	0	0
$865$	−2.10051	−0.0714193
$866$	0	0
$867$	0	0
$868$	0	0
$869$	16.4853	0.559225
$870$	0	0
$871$	−12.0000	−0.406604
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.41421	−0.115421
$876$	0	0
$877$	22.3848	0.755880	0.377940	−	0.925830i	$-0.376633\pi$
$877$	22.3848	0.755880	0.377940	+	0.925830i	$0.376633\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−13.6569	−0.460111	−0.230056	−	0.973177i	$-0.573891\pi$
$881$	−13.6569	−0.460111	−0.230056	+	0.973177i	$0.573891\pi$
$882$	0	0
$883$	−40.2843	−1.35567	−0.677837	−	0.735212i	$-0.737082\pi$
$883$	−40.2843	−1.35567	−0.677837	+	0.735212i	$0.737082\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	23.8995	0.802467	0.401233	−	0.915976i	$-0.368582\pi$
$887$	23.8995	0.802467	0.401233	+	0.915976i	$0.368582\pi$
$888$	0	0
$889$	−73.5980	−2.46840
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−85.2548	−2.85294
$894$	0	0
$895$	6.34315	0.212028
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3.02944	0.101037
$900$	0	0
$901$	−44.4853	−1.48202
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−0.686292	−0.0228131
$906$	0	0
$907$	−56.4853	−1.87556	−0.937781	−	0.347226i	$-0.887124\pi$
$907$	−56.4853	−1.87556	−0.937781	+	0.347226i	$0.887124\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	59.3137	1.96515	0.982575	−	0.185864i	$-0.0595085\pi$
$911$	59.3137	1.96515	0.982575	+	0.185864i	$0.0595085\pi$
$912$	0	0
$913$	−1.75736	−0.0581601
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−23.3137	−0.769886
$918$	0	0
$919$	−35.3137	−1.16489	−0.582446	−	0.812869i	$-0.697904\pi$
$919$	−35.3137	−1.16489	−0.582446	+	0.812869i	$0.697904\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−9.17157	−0.301886
$924$	0	0
$925$	7.65685	0.251756
$926$	0	0
$927$	0	0
$928$	0	0
$929$	17.6569	0.579303	0.289651	−	0.957132i	$-0.406461\pi$
$929$	17.6569	0.579303	0.289651	+	0.957132i	$0.406461\pi$
$930$	0	0
$931$	31.7990	1.04217
$932$	0	0
$933$	0	0
$934$	0	0
$935$	4.24264	0.138749
$936$	0	0
$937$	−3.07107	−0.100327	−0.0501637	−	0.998741i	$-0.515974\pi$
$937$	−3.07107	−0.100327	−0.0501637	+	0.998741i	$0.515974\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−17.5147	−0.570964	−0.285482	−	0.958384i	$-0.592154\pi$
$941$	−17.5147	−0.570964	−0.285482	+	0.958384i	$0.592154\pi$
$942$	0	0
$943$	21.6569	0.705244
$944$	0	0
$945$	0	0
$946$	0	0
$947$	13.4558	0.437256	0.218628	−	0.975808i	$-0.429842\pi$
$947$	13.4558	0.437256	0.218628	+	0.975808i	$0.429842\pi$
$948$	0	0
$949$	10.0000	0.324614
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−2.10051	−0.0680420	−0.0340210	−	0.999421i	$-0.510831\pi$
$953$	−2.10051	−0.0680420	−0.0340210	+	0.999421i	$0.510831\pi$
$954$	0	0
$955$	19.3137	0.624977
$956$	0	0
$957$	0	0
$958$	0	0
$959$	59.1127	1.90885
$960$	0	0
$961$	−30.3137	−0.977862
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.89949	0.0611469
$966$	0	0
$967$	−8.87006	−0.285242	−0.142621	−	0.989777i	$-0.545553\pi$
$967$	−8.87006	−0.285242	−0.142621	+	0.989777i	$0.545553\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−29.9411	−0.960856	−0.480428	−	0.877034i	$-0.659519\pi$
$971$	−29.9411	−0.960856	−0.480428	+	0.877034i	$0.659519\pi$
$972$	0	0
$973$	54.6274	1.75127
$974$	0	0
$975$	0	0
$976$	0	0
$977$	50.2843	1.60874	0.804368	−	0.594131i	$-0.202504\pi$
$977$	50.2843	1.60874	0.804368	+	0.594131i	$0.202504\pi$
$978$	0	0
$979$	2.00000	0.0639203
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−23.0294	−0.734525	−0.367262	−	0.930117i	$-0.619705\pi$
$983$	−23.0294	−0.734525	−0.367262	+	0.930117i	$0.619705\pi$
$984$	0	0
$985$	0.242641	0.00773118
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−14.3431	−0.456086
$990$	0	0
$991$	−35.4558	−1.12629	−0.563146	−	0.826357i	$-0.690409\pi$
$991$	−35.4558	−1.12629	−0.563146	+	0.826357i	$0.690409\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	16.9706	0.538003
$996$	0	0
$997$	−8.24264	−0.261047	−0.130524	−	0.991445i	$-0.541666\pi$
$997$	−8.24264	−0.261047	−0.130524	+	0.991445i	$0.541666\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7920.2.a.bx.1.2		2
3.2	odd	2		7920.2.a.cf.1.2		2
4.3	odd	2		3960.2.a.u.1.1	✓	2
12.11	even	2		3960.2.a.bb.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3960.2.a.u.1.1	✓	2	4.3	odd	2
3960.2.a.bb.1.1	yes	2	12.11	even	2
7920.2.a.bx.1.2		2	1.1	even	1	trivial
7920.2.a.cf.1.2		2	3.2	odd	2

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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +3.41421 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +3.41421 q^{7} +1.00000 q^{11} -1.41421 q^{13} -4.24264 q^{17} +6.82843 q^{19} +2.82843 q^{23} +1.00000 q^{25} -3.65685 q^{29} -0.828427 q^{31} -3.41421 q^{35} +7.65685 q^{37} +7.65685 q^{41} -5.07107 q^{43} -12.4853 q^{47} +4.65685 q^{49} +10.4853 q^{53} -1.00000 q^{55} -3.17157 q^{59} +3.17157 q^{61} +1.41421 q^{65} +8.48528 q^{67} +6.48528 q^{71} -7.07107 q^{73} +3.41421 q^{77} +16.4853 q^{79} -1.75736 q^{83} +4.24264 q^{85} +2.00000 q^{89} -4.82843 q^{91} -6.82843 q^{95} +3.17157 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 4 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 + 4 * q^7	\( 2 q - 2 q^{5} + 4 q^{7} + 2 q^{11} + 8 q^{19} + 2 q^{25} + 4 q^{29} + 4 q^{31} - 4 q^{35} + 4 q^{37} + 4 q^{41} + 4 q^{43} - 8 q^{47} - 2 q^{49} + 4 q^{53} - 2 q^{55} - 12 q^{59} + 12 q^{61} - 4 q^{71} + 4 q^{77} + 16 q^{79} - 12 q^{83} + 4 q^{89} - 4 q^{91} - 8 q^{95} + 12 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 4 * q^7 + 2 * q^11 + 8 * q^19 + 2 * q^25 + 4 * q^29 + 4 * q^31 - 4 * q^35 + 4 * q^37 + 4 * q^41 + 4 * q^43 - 8 * q^47 - 2 * q^49 + 4 * q^53 - 2 * q^55 - 12 * q^59 + 12 * q^61 - 4 * q^71 + 4 * q^77 + 16 * q^79 - 12 * q^83 + 4 * q^89 - 4 * q^91 - 8 * q^95 + 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more