LMFDB - Embedded newform 4950.2.a.bw.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4950,2,Mod(1,4950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4950, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4950.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4950.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:	$39.5259490005$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 110)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.37228$ of defining polynomial
Character	$\chi$	$=$	4950.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -3.37228 q^{7} -1.00000 q^{8} +1.00000 q^{11} -2.00000 q^{13} +3.37228 q^{14} +1.00000 q^{16} +1.37228 q^{17} +0.627719 q^{19} -1.00000 q^{22} +2.74456 q^{23} +2.00000 q^{26} -3.37228 q^{28} -1.37228 q^{29} +3.37228 q^{31} -1.00000 q^{32} -1.37228 q^{34} -9.37228 q^{37} -0.627719 q^{38} +11.4891 q^{41} +4.00000 q^{43} +1.00000 q^{44} -2.74456 q^{46} +2.74456 q^{47} +4.37228 q^{49} -2.00000 q^{52} -4.11684 q^{53} +3.37228 q^{56} +1.37228 q^{58} +2.74456 q^{59} -5.37228 q^{61} -3.37228 q^{62} +1.00000 q^{64} -8.00000 q^{67} +1.37228 q^{68} -10.1168 q^{71} +15.4891 q^{73} +9.37228 q^{74} +0.627719 q^{76} -3.37228 q^{77} -1.25544 q^{79} -11.4891 q^{82} -2.74456 q^{83} -4.00000 q^{86} -1.00000 q^{88} +1.37228 q^{89} +6.74456 q^{91} +2.74456 q^{92} -2.74456 q^{94} +12.7446 q^{97} -4.37228 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - q^{7} - 2 q^{8} + 2 q^{11} - 4 q^{13} + q^{14} + 2 q^{16} - 3 q^{17} + 7 q^{19} - 2 q^{22} - 6 q^{23} + 4 q^{26} - q^{28} + 3 q^{29} + q^{31} - 2 q^{32} + 3 q^{34} - 13 q^{37}+ \cdots - 3 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - q^7 - 2 * q^8 + 2 * q^11 - 4 * q^13 + q^14 + 2 * q^16 - 3 * q^17 + 7 * q^19 - 2 * q^22 - 6 * q^23 + 4 * q^26 - q^28 + 3 * q^29 + q^31 - 2 * q^32 + 3 * q^34 - 13 * q^37 - 7 * q^38 + 8 * q^43 + 2 * q^44 + 6 * q^46 - 6 * q^47 + 3 * q^49 - 4 * q^52 + 9 * q^53 + q^56 - 3 * q^58 - 6 * q^59 - 5 * q^61 - q^62 + 2 * q^64 - 16 * q^67 - 3 * q^68 - 3 * q^71 + 8 * q^73 + 13 * q^74 + 7 * q^76 - q^77 - 14 * q^79 + 6 * q^83 - 8 * q^86 - 2 * q^88 - 3 * q^89 + 2 * q^91 - 6 * q^92 + 6 * q^94 + 14 * q^97 - 3 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.37228	−1.27460	−0.637301	−	0.770615i	$-0.719949\pi$
$7$	−3.37228	−1.27460	−0.637301	+	0.770615i	$0.719949\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	3.37228	0.901280
$15$	0	0
$16$	1.00000	0.250000
$17$	1.37228	0.332827	0.166414	−	0.986056i	$-0.446781\pi$
$17$	1.37228	0.332827	0.166414	+	0.986056i	$0.446781\pi$
$18$	0	0
$19$	0.627719	0.144009	0.0720043	−	0.997404i	$-0.477060\pi$
$19$	0.627719	0.144009	0.0720043	+	0.997404i	$0.477060\pi$
$20$	0	0
$21$	0	0
$22$	−1.00000	−0.213201
$23$	2.74456	0.572281	0.286140	−	0.958188i	$-0.407628\pi$
$23$	2.74456	0.572281	0.286140	+	0.958188i	$0.407628\pi$
$24$	0	0
$25$	0	0
$26$	2.00000	0.392232
$27$	0	0
$28$	−3.37228	−0.637301
$29$	−1.37228	−0.254826	−0.127413	−	0.991850i	$-0.540667\pi$
$29$	−1.37228	−0.254826	−0.127413	+	0.991850i	$0.540667\pi$
$30$	0	0
$31$	3.37228	0.605680	0.302840	−	0.953041i	$-0.402065\pi$
$31$	3.37228	0.605680	0.302840	+	0.953041i	$0.402065\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−1.37228	−0.235344
$35$	0	0
$36$	0	0
$37$	−9.37228	−1.54079	−0.770397	−	0.637565i	$-0.779942\pi$
$37$	−9.37228	−1.54079	−0.770397	+	0.637565i	$0.779942\pi$
$38$	−0.627719	−0.101829
$39$	0	0
$40$	0	0
$41$	11.4891	1.79430	0.897150	−	0.441726i	$-0.145634\pi$
$41$	11.4891	1.79430	0.897150	+	0.441726i	$0.145634\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−2.74456	−0.404664
$47$	2.74456	0.400336	0.200168	−	0.979762i	$-0.435851\pi$
$47$	2.74456	0.400336	0.200168	+	0.979762i	$0.435851\pi$
$48$	0	0
$49$	4.37228	0.624612
$50$	0	0
$51$	0	0
$52$	−2.00000	−0.277350
$53$	−4.11684	−0.565492	−0.282746	−	0.959195i	$-0.591245\pi$
$53$	−4.11684	−0.565492	−0.282746	+	0.959195i	$0.591245\pi$
$54$	0	0
$55$	0	0
$56$	3.37228	0.450640
$57$	0	0
$58$	1.37228	0.180189
$59$	2.74456	0.357312	0.178656	−	0.983912i	$-0.442825\pi$
$59$	2.74456	0.357312	0.178656	+	0.983912i	$0.442825\pi$
$60$	0	0
$61$	−5.37228	−0.687850	−0.343925	−	0.938997i	$-0.611757\pi$
$61$	−5.37228	−0.687850	−0.343925	+	0.938997i	$0.611757\pi$
$62$	−3.37228	−0.428280
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	1.37228	0.166414
$69$	0	0
$70$	0	0
$71$	−10.1168	−1.20065	−0.600324	−	0.799757i	$-0.704962\pi$
$71$	−10.1168	−1.20065	−0.600324	+	0.799757i	$0.704962\pi$
$72$	0	0
$73$	15.4891	1.81286	0.906432	−	0.422351i	$-0.138795\pi$
$73$	15.4891	1.81286	0.906432	+	0.422351i	$0.138795\pi$
$74$	9.37228	1.08951
$75$	0	0
$76$	0.627719	0.0720043
$77$	−3.37228	−0.384307
$78$	0	0
$79$	−1.25544	−0.141248	−0.0706239	−	0.997503i	$-0.522499\pi$
$79$	−1.25544	−0.141248	−0.0706239	+	0.997503i	$0.522499\pi$
$80$	0	0
$81$	0	0
$82$	−11.4891	−1.26876
$83$	−2.74456	−0.301255	−0.150627	−	0.988591i	$-0.548129\pi$
$83$	−2.74456	−0.301255	−0.150627	+	0.988591i	$0.548129\pi$
$84$	0	0
$85$	0	0
$86$	−4.00000	−0.431331
$87$	0	0
$88$	−1.00000	−0.106600
$89$	1.37228	0.145462	0.0727308	−	0.997352i	$-0.476829\pi$
$89$	1.37228	0.145462	0.0727308	+	0.997352i	$0.476829\pi$
$90$	0	0
$91$	6.74456	0.707022
$92$	2.74456	0.286140
$93$	0	0
$94$	−2.74456	−0.283080
$95$	0	0
$96$	0	0
$97$	12.7446	1.29401	0.647007	−	0.762484i	$-0.276020\pi$
$97$	12.7446	1.29401	0.647007	+	0.762484i	$0.276020\pi$
$98$	−4.37228	−0.441667
$99$	0	0
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	9.48913	0.934991	0.467496	−	0.883995i	$-0.345156\pi$
$103$	9.48913	0.934991	0.467496	+	0.883995i	$0.345156\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	4.11684	0.399863
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	−15.4891	−1.48359	−0.741795	−	0.670627i	$-0.766025\pi$
$109$	−15.4891	−1.48359	−0.741795	+	0.670627i	$0.766025\pi$
$110$	0	0
$111$	0	0
$112$	−3.37228	−0.318651
$113$	3.25544	0.306246	0.153123	−	0.988207i	$-0.451067\pi$
$113$	3.25544	0.306246	0.153123	+	0.988207i	$0.451067\pi$
$114$	0	0
$115$	0	0
$116$	−1.37228	−0.127413
$117$	0	0
$118$	−2.74456	−0.252657
$119$	−4.62772	−0.424222
$120$	0	0
$121$	1.00000	0.0909091
$122$	5.37228	0.486383
$123$	0	0
$124$	3.37228	0.302840
$125$	0	0
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−22.1168	−1.93236	−0.966179	−	0.257873i	$-0.916978\pi$
$131$	−22.1168	−1.93236	−0.966179	+	0.257873i	$0.916978\pi$
$132$	0	0
$133$	−2.11684	−0.183554
$134$	8.00000	0.691095
$135$	0	0
$136$	−1.37228	−0.117672
$137$	8.74456	0.747098	0.373549	−	0.927610i	$-0.378141\pi$
$137$	8.74456	0.747098	0.373549	+	0.927610i	$0.378141\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	10.1168	0.848987
$143$	−2.00000	−0.167248
$144$	0	0
$145$	0	0
$146$	−15.4891	−1.28189
$147$	0	0
$148$	−9.37228	−0.770397
$149$	−21.6060	−1.77003	−0.885015	−	0.465563i	$-0.845852\pi$
$149$	−21.6060	−1.77003	−0.885015	+	0.465563i	$0.845852\pi$
$150$	0	0
$151$	−12.2337	−0.995563	−0.497782	−	0.867302i	$-0.665852\pi$
$151$	−12.2337	−0.995563	−0.497782	+	0.867302i	$0.665852\pi$
$152$	−0.627719	−0.0509147
$153$	0	0
$154$	3.37228	0.271746
$155$	0	0
$156$	0	0
$157$	−9.37228	−0.747989	−0.373995	−	0.927431i	$-0.622012\pi$
$157$	−9.37228	−0.747989	−0.373995	+	0.927431i	$0.622012\pi$
$158$	1.25544	0.0998772
$159$	0	0
$160$	0	0
$161$	−9.25544	−0.729431
$162$	0	0
$163$	5.88316	0.460804	0.230402	−	0.973095i	$-0.425996\pi$
$163$	5.88316	0.460804	0.230402	+	0.973095i	$0.425996\pi$
$164$	11.4891	0.897150
$165$	0	0
$166$	2.74456	0.213019
$167$	−4.62772	−0.358104	−0.179052	−	0.983840i	$-0.557303\pi$
$167$	−4.62772	−0.358104	−0.179052	+	0.983840i	$0.557303\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	4.00000	0.304997
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	0	0
$176$	1.00000	0.0753778
$177$	0	0
$178$	−1.37228	−0.102857
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	−6.74456	−0.499940
$183$	0	0
$184$	−2.74456	−0.202332
$185$	0	0
$186$	0	0
$187$	1.37228	0.100351
$188$	2.74456	0.200168
$189$	0	0
$190$	0	0
$191$	5.48913	0.397179	0.198590	−	0.980083i	$-0.436364\pi$
$191$	5.48913	0.397179	0.198590	+	0.980083i	$0.436364\pi$
$192$	0	0
$193$	−14.8614	−1.06975	−0.534874	−	0.844932i	$-0.679641\pi$
$193$	−14.8614	−1.06975	−0.534874	+	0.844932i	$0.679641\pi$
$194$	−12.7446	−0.915006
$195$	0	0
$196$	4.37228	0.312306
$197$	−20.7446	−1.47799	−0.738994	−	0.673712i	$-0.764699\pi$
$197$	−20.7446	−1.47799	−0.738994	+	0.673712i	$0.764699\pi$
$198$	0	0
$199$	18.1168	1.28427	0.642135	−	0.766592i	$-0.278049\pi$
$199$	18.1168	1.28427	0.642135	+	0.766592i	$0.278049\pi$
$200$	0	0
$201$	0	0
$202$	6.00000	0.422159
$203$	4.62772	0.324802
$204$	0	0
$205$	0	0
$206$	−9.48913	−0.661139
$207$	0	0
$208$	−2.00000	−0.138675
$209$	0.627719	0.0434202
$210$	0	0
$211$	6.11684	0.421101	0.210550	−	0.977583i	$-0.432474\pi$
$211$	6.11684	0.421101	0.210550	+	0.977583i	$0.432474\pi$
$212$	−4.11684	−0.282746
$213$	0	0
$214$	12.0000	0.820303
$215$	0	0
$216$	0	0
$217$	−11.3723	−0.772001
$218$	15.4891	1.04906
$219$	0	0
$220$	0	0
$221$	−2.74456	−0.184619
$222$	0	0
$223$	18.7446	1.25523	0.627614	−	0.778524i	$-0.284031\pi$
$223$	18.7446	1.25523	0.627614	+	0.778524i	$0.284031\pi$
$224$	3.37228	0.225320
$225$	0	0
$226$	−3.25544	−0.216548
$227$	−2.74456	−0.182163	−0.0910815	−	0.995843i	$-0.529032\pi$
$227$	−2.74456	−0.182163	−0.0910815	+	0.995843i	$0.529032\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	0	0
$232$	1.37228	0.0900947
$233$	1.37228	0.0899011	0.0449506	−	0.998989i	$-0.485687\pi$
$233$	1.37228	0.0899011	0.0449506	+	0.998989i	$0.485687\pi$
$234$	0	0
$235$	0	0
$236$	2.74456	0.178656
$237$	0	0
$238$	4.62772	0.299970
$239$	14.7446	0.953746	0.476873	−	0.878972i	$-0.341770\pi$
$239$	14.7446	0.953746	0.476873	+	0.878972i	$0.341770\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	−5.37228	−0.343925
$245$	0	0
$246$	0	0
$247$	−1.25544	−0.0798816
$248$	−3.37228	−0.214140
$249$	0	0
$250$	0	0
$251$	−2.74456	−0.173235	−0.0866176	−	0.996242i	$-0.527606\pi$
$251$	−2.74456	−0.173235	−0.0866176	+	0.996242i	$0.527606\pi$
$252$	0	0
$253$	2.74456	0.172549
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	31.6060	1.96390
$260$	0	0
$261$	0	0
$262$	22.1168	1.36638
$263$	−24.8614	−1.53302	−0.766510	−	0.642232i	$-0.778008\pi$
$263$	−24.8614	−1.53302	−0.766510	+	0.642232i	$0.778008\pi$
$264$	0	0
$265$	0	0
$266$	2.11684	0.129792
$267$	0	0
$268$	−8.00000	−0.488678
$269$	8.74456	0.533165	0.266583	−	0.963812i	$-0.414105\pi$
$269$	8.74456	0.533165	0.266583	+	0.963812i	$0.414105\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	1.37228	0.0832068
$273$	0	0
$274$	−8.74456	−0.528278
$275$	0	0
$276$	0	0
$277$	12.7446	0.765747	0.382873	−	0.923801i	$-0.374935\pi$
$277$	12.7446	0.765747	0.382873	+	0.923801i	$0.374935\pi$
$278$	4.00000	0.239904
$279$	0	0
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	−5.25544	−0.312403	−0.156202	−	0.987725i	$-0.549925\pi$
$283$	−5.25544	−0.312403	−0.156202	+	0.987725i	$0.549925\pi$
$284$	−10.1168	−0.600324
$285$	0	0
$286$	2.00000	0.118262
$287$	−38.7446	−2.28702
$288$	0	0
$289$	−15.1168	−0.889226
$290$	0	0
$291$	0	0
$292$	15.4891	0.906432
$293$	23.4891	1.37225	0.686125	−	0.727484i	$-0.259310\pi$
$293$	23.4891	1.37225	0.686125	+	0.727484i	$0.259310\pi$
$294$	0	0
$295$	0	0
$296$	9.37228	0.544753
$297$	0	0
$298$	21.6060	1.25160
$299$	−5.48913	−0.317444
$300$	0	0
$301$	−13.4891	−0.777500
$302$	12.2337	0.703970
$303$	0	0
$304$	0.627719	0.0360021
$305$	0	0
$306$	0	0
$307$	−5.25544	−0.299944	−0.149972	−	0.988690i	$-0.547918\pi$
$307$	−5.25544	−0.299944	−0.149972	+	0.988690i	$0.547918\pi$
$308$	−3.37228	−0.192154
$309$	0	0
$310$	0	0
$311$	−19.3723	−1.09850	−0.549251	−	0.835658i	$-0.685087\pi$
$311$	−19.3723	−1.09850	−0.549251	+	0.835658i	$0.685087\pi$
$312$	0	0
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	9.37228	0.528908
$315$	0	0
$316$	−1.25544	−0.0706239
$317$	−24.3505	−1.36766	−0.683831	−	0.729640i	$-0.739687\pi$
$317$	−24.3505	−1.36766	−0.683831	+	0.729640i	$0.739687\pi$
$318$	0	0
$319$	−1.37228	−0.0768330
$320$	0	0
$321$	0	0
$322$	9.25544	0.515785
$323$	0.861407	0.0479299
$324$	0	0
$325$	0	0
$326$	−5.88316	−0.325838
$327$	0	0
$328$	−11.4891	−0.634381
$329$	−9.25544	−0.510269
$330$	0	0
$331$	30.9783	1.70272	0.851359	−	0.524583i	$-0.175779\pi$
$331$	30.9783	1.70272	0.851359	+	0.524583i	$0.175779\pi$
$332$	−2.74456	−0.150627
$333$	0	0
$334$	4.62772	0.253217
$335$	0	0
$336$	0	0
$337$	−24.1168	−1.31373	−0.656864	−	0.754009i	$-0.728117\pi$
$337$	−24.1168	−1.31373	−0.656864	+	0.754009i	$0.728117\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	0	0
$341$	3.37228	0.182619
$342$	0	0
$343$	8.86141	0.478471
$344$	−4.00000	−0.215666
$345$	0	0
$346$	6.00000	0.322562
$347$	−32.2337	−1.73040	−0.865198	−	0.501431i	$-0.832807\pi$
$347$	−32.2337	−1.73040	−0.865198	+	0.501431i	$0.832807\pi$
$348$	0	0
$349$	19.4891	1.04323	0.521614	−	0.853181i	$-0.325330\pi$
$349$	19.4891	1.04323	0.521614	+	0.853181i	$0.325330\pi$
$350$	0	0
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	−0.510875	−0.0271911	−0.0135956	−	0.999908i	$-0.504328\pi$
$353$	−0.510875	−0.0271911	−0.0135956	+	0.999908i	$0.504328\pi$
$354$	0	0
$355$	0	0
$356$	1.37228	0.0727308
$357$	0	0
$358$	−12.0000	−0.634220
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−18.6060	−0.979262
$362$	10.0000	0.525588
$363$	0	0
$364$	6.74456	0.353511
$365$	0	0
$366$	0	0
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	2.74456	0.143070
$369$	0	0
$370$	0	0
$371$	13.8832	0.720778
$372$	0	0
$373$	−31.4891	−1.63045	−0.815223	−	0.579148i	$-0.803385\pi$
$373$	−31.4891	−1.63045	−0.815223	+	0.579148i	$0.803385\pi$
$374$	−1.37228	−0.0709590
$375$	0	0
$376$	−2.74456	−0.141540
$377$	2.74456	0.141352
$378$	0	0
$379$	−0.233688	−0.0120037	−0.00600187	−	0.999982i	$-0.501910\pi$
$379$	−0.233688	−0.0120037	−0.00600187	+	0.999982i	$0.501910\pi$
$380$	0	0
$381$	0	0
$382$	−5.48913	−0.280848
$383$	32.2337	1.64706	0.823532	−	0.567269i	$-0.192000\pi$
$383$	32.2337	1.64706	0.823532	+	0.567269i	$0.192000\pi$
$384$	0	0
$385$	0	0
$386$	14.8614	0.756426
$387$	0	0
$388$	12.7446	0.647007
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	3.76631	0.190471
$392$	−4.37228	−0.220834
$393$	0	0
$394$	20.7446	1.04510
$395$	0	0
$396$	0	0
$397$	−24.9783	−1.25362	−0.626811	−	0.779171i	$-0.715640\pi$
$397$	−24.9783	−1.25362	−0.626811	+	0.779171i	$0.715640\pi$
$398$	−18.1168	−0.908115
$399$	0	0
$400$	0	0
$401$	−13.3723	−0.667780	−0.333890	−	0.942612i	$-0.608361\pi$
$401$	−13.3723	−0.667780	−0.333890	+	0.942612i	$0.608361\pi$
$402$	0	0
$403$	−6.74456	−0.335971
$404$	−6.00000	−0.298511
$405$	0	0
$406$	−4.62772	−0.229670
$407$	−9.37228	−0.464567
$408$	0	0
$409$	−1.76631	−0.0873385	−0.0436693	−	0.999046i	$-0.513905\pi$
$409$	−1.76631	−0.0873385	−0.0436693	+	0.999046i	$0.513905\pi$
$410$	0	0
$411$	0	0
$412$	9.48913	0.467496
$413$	−9.25544	−0.455430
$414$	0	0
$415$	0	0
$416$	2.00000	0.0980581
$417$	0	0
$418$	−0.627719	−0.0307027
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	10.2337	0.498759	0.249380	−	0.968406i	$-0.419773\pi$
$421$	10.2337	0.498759	0.249380	+	0.968406i	$0.419773\pi$
$422$	−6.11684	−0.297763
$423$	0	0
$424$	4.11684	0.199932
$425$	0	0
$426$	0	0
$427$	18.1168	0.876736
$428$	−12.0000	−0.580042
$429$	0	0
$430$	0	0
$431$	34.9783	1.68484	0.842422	−	0.538819i	$-0.181129\pi$
$431$	34.9783	1.68484	0.842422	+	0.538819i	$0.181129\pi$
$432$	0	0
$433$	−27.7228	−1.33227	−0.666137	−	0.745830i	$-0.732053\pi$
$433$	−27.7228	−1.33227	−0.666137	+	0.745830i	$0.732053\pi$
$434$	11.3723	0.545887
$435$	0	0
$436$	−15.4891	−0.741795
$437$	1.72281	0.0824133
$438$	0	0
$439$	18.9783	0.905782	0.452891	−	0.891566i	$-0.350393\pi$
$439$	18.9783	0.905782	0.452891	+	0.891566i	$0.350393\pi$
$440$	0	0
$441$	0	0
$442$	2.74456	0.130546
$443$	29.4891	1.40107	0.700535	−	0.713618i	$-0.252945\pi$
$443$	29.4891	1.40107	0.700535	+	0.713618i	$0.252945\pi$
$444$	0	0
$445$	0	0
$446$	−18.7446	−0.887581
$447$	0	0
$448$	−3.37228	−0.159325
$449$	−28.9783	−1.36757	−0.683784	−	0.729684i	$-0.739667\pi$
$449$	−28.9783	−1.36757	−0.683784	+	0.729684i	$0.739667\pi$
$450$	0	0
$451$	11.4891	0.541002
$452$	3.25544	0.153123
$453$	0	0
$454$	2.74456	0.128809
$455$	0	0
$456$	0	0
$457$	16.3505	0.764846	0.382423	−	0.923987i	$-0.375090\pi$
$457$	16.3505	0.764846	0.382423	+	0.923987i	$0.375090\pi$
$458$	10.0000	0.467269
$459$	0	0
$460$	0	0
$461$	−16.1168	−0.750636	−0.375318	−	0.926896i	$-0.622467\pi$
$461$	−16.1168	−0.750636	−0.375318	+	0.926896i	$0.622467\pi$
$462$	0	0
$463$	0.233688	0.0108604	0.00543020	−	0.999985i	$-0.498272\pi$
$463$	0.233688	0.0108604	0.00543020	+	0.999985i	$0.498272\pi$
$464$	−1.37228	−0.0637066
$465$	0	0
$466$	−1.37228	−0.0635697
$467$	−19.3723	−0.896442	−0.448221	−	0.893923i	$-0.647942\pi$
$467$	−19.3723	−0.896442	−0.448221	+	0.893923i	$0.647942\pi$
$468$	0	0
$469$	26.9783	1.24574
$470$	0	0
$471$	0	0
$472$	−2.74456	−0.126329
$473$	4.00000	0.183920
$474$	0	0
$475$	0	0
$476$	−4.62772	−0.212111
$477$	0	0
$478$	−14.7446	−0.674401
$479$	−5.48913	−0.250805	−0.125402	−	0.992106i	$-0.540022\pi$
$479$	−5.48913	−0.250805	−0.125402	+	0.992106i	$0.540022\pi$
$480$	0	0
$481$	18.7446	0.854678
$482$	22.0000	1.00207
$483$	0	0
$484$	1.00000	0.0454545
$485$	0	0
$486$	0	0
$487$	−20.0000	−0.906287	−0.453143	−	0.891438i	$-0.649697\pi$
$487$	−20.0000	−0.906287	−0.453143	+	0.891438i	$0.649697\pi$
$488$	5.37228	0.243192
$489$	0	0
$490$	0	0
$491$	7.37228	0.332706	0.166353	−	0.986066i	$-0.446801\pi$
$491$	7.37228	0.332706	0.166353	+	0.986066i	$0.446801\pi$
$492$	0	0
$493$	−1.88316	−0.0848131
$494$	1.25544	0.0564848
$495$	0	0
$496$	3.37228	0.151420
$497$	34.1168	1.53035
$498$	0	0
$499$	−33.4891	−1.49918	−0.749590	−	0.661903i	$-0.769749\pi$
$499$	−33.4891	−1.49918	−0.749590	+	0.661903i	$0.769749\pi$
$500$	0	0
$501$	0	0
$502$	2.74456	0.122496
$503$	−34.9783	−1.55960	−0.779802	−	0.626027i	$-0.784680\pi$
$503$	−34.9783	−1.55960	−0.779802	+	0.626027i	$0.784680\pi$
$504$	0	0
$505$	0	0
$506$	−2.74456	−0.122011
$507$	0	0
$508$	−8.00000	−0.354943
$509$	−9.76631	−0.432884	−0.216442	−	0.976295i	$-0.569445\pi$
$509$	−9.76631	−0.432884	−0.216442	+	0.976295i	$0.569445\pi$
$510$	0	0
$511$	−52.2337	−2.31068
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−18.0000	−0.793946
$515$	0	0
$516$	0	0
$517$	2.74456	0.120706
$518$	−31.6060	−1.38869
$519$	0	0
$520$	0	0
$521$	−12.5109	−0.548111	−0.274056	−	0.961714i	$-0.588365\pi$
$521$	−12.5109	−0.548111	−0.274056	+	0.961714i	$0.588365\pi$
$522$	0	0
$523$	−30.9783	−1.35458	−0.677292	−	0.735714i	$-0.736847\pi$
$523$	−30.9783	−1.35458	−0.677292	+	0.735714i	$0.736847\pi$
$524$	−22.1168	−0.966179
$525$	0	0
$526$	24.8614	1.08401
$527$	4.62772	0.201587
$528$	0	0
$529$	−15.4674	−0.672495
$530$	0	0
$531$	0	0
$532$	−2.11684	−0.0917768
$533$	−22.9783	−0.995299
$534$	0	0
$535$	0	0
$536$	8.00000	0.345547
$537$	0	0
$538$	−8.74456	−0.377005
$539$	4.37228	0.188327
$540$	0	0
$541$	−20.1168	−0.864891	−0.432445	−	0.901660i	$-0.642349\pi$
$541$	−20.1168	−0.864891	−0.432445	+	0.901660i	$0.642349\pi$
$542$	16.0000	0.687259
$543$	0	0
$544$	−1.37228	−0.0588361
$545$	0	0
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	8.74456	0.373549
$549$	0	0
$550$	0	0
$551$	−0.861407	−0.0366972
$552$	0	0
$553$	4.23369	0.180035
$554$	−12.7446	−0.541465
$555$	0	0
$556$	−4.00000	−0.169638
$557$	4.97825	0.210935	0.105468	−	0.994423i	$-0.466366\pi$
$557$	4.97825	0.210935	0.105468	+	0.994423i	$0.466366\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	0	0
$562$	18.0000	0.759284
$563$	−8.23369	−0.347009	−0.173504	−	0.984833i	$-0.555509\pi$
$563$	−8.23369	−0.347009	−0.173504	+	0.984833i	$0.555509\pi$
$564$	0	0
$565$	0	0
$566$	5.25544	0.220903
$567$	0	0
$568$	10.1168	0.424493
$569$	15.2554	0.639541	0.319771	−	0.947495i	$-0.396394\pi$
$569$	15.2554	0.639541	0.319771	+	0.947495i	$0.396394\pi$
$570$	0	0
$571$	15.3723	0.643310	0.321655	−	0.946857i	$-0.395761\pi$
$571$	15.3723	0.643310	0.321655	+	0.946857i	$0.395761\pi$
$572$	−2.00000	−0.0836242
$573$	0	0
$574$	38.7446	1.61717
$575$	0	0
$576$	0	0
$577$	−36.9783	−1.53942	−0.769712	−	0.638391i	$-0.779600\pi$
$577$	−36.9783	−1.53942	−0.769712	+	0.638391i	$0.779600\pi$
$578$	15.1168	0.628778
$579$	0	0
$580$	0	0
$581$	9.25544	0.383980
$582$	0	0
$583$	−4.11684	−0.170502
$584$	−15.4891	−0.640945
$585$	0	0
$586$	−23.4891	−0.970327
$587$	24.8614	1.02614	0.513070	−	0.858347i	$-0.328508\pi$
$587$	24.8614	1.02614	0.513070	+	0.858347i	$0.328508\pi$
$588$	0	0
$589$	2.11684	0.0872230
$590$	0	0
$591$	0	0
$592$	−9.37228	−0.385198
$593$	−12.5109	−0.513760	−0.256880	−	0.966443i	$-0.582695\pi$
$593$	−12.5109	−0.513760	−0.256880	+	0.966443i	$0.582695\pi$
$594$	0	0
$595$	0	0
$596$	−21.6060	−0.885015
$597$	0	0
$598$	5.48913	0.224467
$599$	39.6060	1.61826	0.809128	−	0.587632i	$-0.199940\pi$
$599$	39.6060	1.61826	0.809128	+	0.587632i	$0.199940\pi$
$600$	0	0
$601$	−16.5109	−0.673493	−0.336746	−	0.941595i	$-0.609326\pi$
$601$	−16.5109	−0.673493	−0.336746	+	0.941595i	$0.609326\pi$
$602$	13.4891	0.549776
$603$	0	0
$604$	−12.2337	−0.497782
$605$	0	0
$606$	0	0
$607$	5.88316	0.238790	0.119395	−	0.992847i	$-0.461905\pi$
$607$	5.88316	0.238790	0.119395	+	0.992847i	$0.461905\pi$
$608$	−0.627719	−0.0254574
$609$	0	0
$610$	0	0
$611$	−5.48913	−0.222066
$612$	0	0
$613$	−20.5109	−0.828426	−0.414213	−	0.910180i	$-0.635943\pi$
$613$	−20.5109	−0.828426	−0.414213	+	0.910180i	$0.635943\pi$
$614$	5.25544	0.212092
$615$	0	0
$616$	3.37228	0.135873
$617$	−2.23369	−0.0899249	−0.0449624	−	0.998989i	$-0.514317\pi$
$617$	−2.23369	−0.0899249	−0.0449624	+	0.998989i	$0.514317\pi$
$618$	0	0
$619$	−44.4674	−1.78729	−0.893647	−	0.448770i	$-0.851862\pi$
$619$	−44.4674	−1.78729	−0.893647	+	0.448770i	$0.851862\pi$
$620$	0	0
$621$	0	0
$622$	19.3723	0.776758
$623$	−4.62772	−0.185406
$624$	0	0
$625$	0	0
$626$	−22.0000	−0.879297
$627$	0	0
$628$	−9.37228	−0.373995
$629$	−12.8614	−0.512818
$630$	0	0
$631$	42.1168	1.67665	0.838323	−	0.545175i	$-0.183537\pi$
$631$	42.1168	1.67665	0.838323	+	0.545175i	$0.183537\pi$
$632$	1.25544	0.0499386
$633$	0	0
$634$	24.3505	0.967083
$635$	0	0
$636$	0	0
$637$	−8.74456	−0.346472
$638$	1.37228	0.0543291
$639$	0	0
$640$	0	0
$641$	27.0951	1.07019	0.535096	−	0.844791i	$-0.320275\pi$
$641$	27.0951	1.07019	0.535096	+	0.844791i	$0.320275\pi$
$642$	0	0
$643$	5.88316	0.232009	0.116005	−	0.993249i	$-0.462991\pi$
$643$	5.88316	0.232009	0.116005	+	0.993249i	$0.462991\pi$
$644$	−9.25544	−0.364715
$645$	0	0
$646$	−0.861407	−0.0338916
$647$	−37.7228	−1.48304	−0.741518	−	0.670933i	$-0.765894\pi$
$647$	−37.7228	−1.48304	−0.741518	+	0.670933i	$0.765894\pi$
$648$	0	0
$649$	2.74456	0.107734
$650$	0	0
$651$	0	0
$652$	5.88316	0.230402
$653$	10.6277	0.415895	0.207947	−	0.978140i	$-0.433322\pi$
$653$	10.6277	0.415895	0.207947	+	0.978140i	$0.433322\pi$
$654$	0	0
$655$	0	0
$656$	11.4891	0.448575
$657$	0	0
$658$	9.25544	0.360815
$659$	−12.8614	−0.501009	−0.250505	−	0.968115i	$-0.580597\pi$
$659$	−12.8614	−0.501009	−0.250505	+	0.968115i	$0.580597\pi$
$660$	0	0
$661$	8.51087	0.331035	0.165517	−	0.986207i	$-0.447071\pi$
$661$	8.51087	0.331035	0.165517	+	0.986207i	$0.447071\pi$
$662$	−30.9783	−1.20400
$663$	0	0
$664$	2.74456	0.106510
$665$	0	0
$666$	0	0
$667$	−3.76631	−0.145832
$668$	−4.62772	−0.179052
$669$	0	0
$670$	0	0
$671$	−5.37228	−0.207395
$672$	0	0
$673$	−14.8614	−0.572865	−0.286433	−	0.958100i	$-0.592469\pi$
$673$	−14.8614	−0.572865	−0.286433	+	0.958100i	$0.592469\pi$
$674$	24.1168	0.928946
$675$	0	0
$676$	−9.00000	−0.346154
$677$	3.25544	0.125117	0.0625583	−	0.998041i	$-0.480074\pi$
$677$	3.25544	0.125117	0.0625583	+	0.998041i	$0.480074\pi$
$678$	0	0
$679$	−42.9783	−1.64935
$680$	0	0
$681$	0	0
$682$	−3.37228	−0.129131
$683$	−28.6277	−1.09541	−0.547705	−	0.836672i	$-0.684498\pi$
$683$	−28.6277	−1.09541	−0.547705	+	0.836672i	$0.684498\pi$
$684$	0	0
$685$	0	0
$686$	−8.86141	−0.338330
$687$	0	0
$688$	4.00000	0.152499
$689$	8.23369	0.313679
$690$	0	0
$691$	40.2337	1.53056	0.765281	−	0.643697i	$-0.222600\pi$
$691$	40.2337	1.53056	0.765281	+	0.643697i	$0.222600\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	32.2337	1.22357
$695$	0	0
$696$	0	0
$697$	15.7663	0.597192
$698$	−19.4891	−0.737674
$699$	0	0
$700$	0	0
$701$	37.3723	1.41153	0.705766	−	0.708445i	$-0.250603\pi$
$701$	37.3723	1.41153	0.705766	+	0.708445i	$0.250603\pi$
$702$	0	0
$703$	−5.88316	−0.221887
$704$	1.00000	0.0376889
$705$	0	0
$706$	0.510875	0.0192270
$707$	20.2337	0.760966
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	0	0
$712$	−1.37228	−0.0514284
$713$	9.25544	0.346619
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	0	0
$718$	0	0
$719$	−13.8832	−0.517754	−0.258877	−	0.965910i	$-0.583352\pi$
$719$	−13.8832	−0.517754	−0.258877	+	0.965910i	$0.583352\pi$
$720$	0	0
$721$	−32.0000	−1.19174
$722$	18.6060	0.692442
$723$	0	0
$724$	−10.0000	−0.371647
$725$	0	0
$726$	0	0
$727$	24.2337	0.898778	0.449389	−	0.893336i	$-0.351642\pi$
$727$	24.2337	0.898778	0.449389	+	0.893336i	$0.351642\pi$
$728$	−6.74456	−0.249970
$729$	0	0
$730$	0	0
$731$	5.48913	0.203023
$732$	0	0
$733$	−46.2337	−1.70768	−0.853840	−	0.520535i	$-0.825732\pi$
$733$	−46.2337	−1.70768	−0.853840	+	0.520535i	$0.825732\pi$
$734$	−4.00000	−0.147643
$735$	0	0
$736$	−2.74456	−0.101166
$737$	−8.00000	−0.294684
$738$	0	0
$739$	−20.4674	−0.752905	−0.376452	−	0.926436i	$-0.622856\pi$
$739$	−20.4674	−0.752905	−0.376452	+	0.926436i	$0.622856\pi$
$740$	0	0
$741$	0	0
$742$	−13.8832	−0.509667
$743$	−4.62772	−0.169775	−0.0848873	−	0.996391i	$-0.527053\pi$
$743$	−4.62772	−0.169775	−0.0848873	+	0.996391i	$0.527053\pi$
$744$	0	0
$745$	0	0
$746$	31.4891	1.15290
$747$	0	0
$748$	1.37228	0.0501756
$749$	40.4674	1.47865
$750$	0	0
$751$	8.86141	0.323357	0.161679	−	0.986843i	$-0.448309\pi$
$751$	8.86141	0.323357	0.161679	+	0.986843i	$0.448309\pi$
$752$	2.74456	0.100084
$753$	0	0
$754$	−2.74456	−0.0999511
$755$	0	0
$756$	0	0
$757$	20.9783	0.762467	0.381234	−	0.924479i	$-0.375499\pi$
$757$	20.9783	0.762467	0.381234	+	0.924479i	$0.375499\pi$
$758$	0.233688	0.00848793
$759$	0	0
$760$	0	0
$761$	−4.97825	−0.180461	−0.0902307	−	0.995921i	$-0.528760\pi$
$761$	−4.97825	−0.180461	−0.0902307	+	0.995921i	$0.528760\pi$
$762$	0	0
$763$	52.2337	1.89099
$764$	5.48913	0.198590
$765$	0	0
$766$	−32.2337	−1.16465
$767$	−5.48913	−0.198201
$768$	0	0
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	0	0
$772$	−14.8614	−0.534874
$773$	−33.6060	−1.20872	−0.604361	−	0.796710i	$-0.706572\pi$
$773$	−33.6060	−1.20872	−0.604361	+	0.796710i	$0.706572\pi$
$774$	0	0
$775$	0	0
$776$	−12.7446	−0.457503
$777$	0	0
$778$	6.00000	0.215110
$779$	7.21194	0.258395
$780$	0	0
$781$	−10.1168	−0.362009
$782$	−3.76631	−0.134683
$783$	0	0
$784$	4.37228	0.156153
$785$	0	0
$786$	0	0
$787$	44.4674	1.58509	0.792545	−	0.609813i	$-0.208755\pi$
$787$	44.4674	1.58509	0.792545	+	0.609813i	$0.208755\pi$
$788$	−20.7446	−0.738994
$789$	0	0
$790$	0	0
$791$	−10.9783	−0.390342
$792$	0	0
$793$	10.7446	0.381551
$794$	24.9783	0.886445
$795$	0	0
$796$	18.1168	0.642135
$797$	11.4891	0.406966	0.203483	−	0.979079i	$-0.434774\pi$
$797$	11.4891	0.406966	0.203483	+	0.979079i	$0.434774\pi$
$798$	0	0
$799$	3.76631	0.133243
$800$	0	0
$801$	0	0
$802$	13.3723	0.472192
$803$	15.4891	0.546599
$804$	0	0
$805$	0	0
$806$	6.74456	0.237567
$807$	0	0
$808$	6.00000	0.211079
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	44.8614	1.57530	0.787649	−	0.616125i	$-0.211298\pi$
$811$	44.8614	1.57530	0.787649	+	0.616125i	$0.211298\pi$
$812$	4.62772	0.162401
$813$	0	0
$814$	9.37228	0.328498
$815$	0	0
$816$	0	0
$817$	2.51087	0.0878444
$818$	1.76631	0.0617577
$819$	0	0
$820$	0	0
$821$	−11.4891	−0.400973	−0.200487	−	0.979696i	$-0.564252\pi$
$821$	−11.4891	−0.400973	−0.200487	+	0.979696i	$0.564252\pi$
$822$	0	0
$823$	28.0000	0.976019	0.488009	−	0.872838i	$-0.337723\pi$
$823$	28.0000	0.976019	0.488009	+	0.872838i	$0.337723\pi$
$824$	−9.48913	−0.330569
$825$	0	0
$826$	9.25544	0.322038
$827$	46.9783	1.63359	0.816797	−	0.576925i	$-0.195748\pi$
$827$	46.9783	1.63359	0.816797	+	0.576925i	$0.195748\pi$
$828$	0	0
$829$	−24.7446	−0.859414	−0.429707	−	0.902968i	$-0.641383\pi$
$829$	−24.7446	−0.859414	−0.429707	+	0.902968i	$0.641383\pi$
$830$	0	0
$831$	0	0
$832$	−2.00000	−0.0693375
$833$	6.00000	0.207888
$834$	0	0
$835$	0	0
$836$	0.627719	0.0217101
$837$	0	0
$838$	−12.0000	−0.414533
$839$	10.9783	0.379011	0.189506	−	0.981880i	$-0.439311\pi$
$839$	10.9783	0.379011	0.189506	+	0.981880i	$0.439311\pi$
$840$	0	0
$841$	−27.1168	−0.935064
$842$	−10.2337	−0.352676
$843$	0	0
$844$	6.11684	0.210550
$845$	0	0
$846$	0	0
$847$	−3.37228	−0.115873
$848$	−4.11684	−0.141373
$849$	0	0
$850$	0	0
$851$	−25.7228	−0.881767
$852$	0	0
$853$	38.4674	1.31710	0.658549	−	0.752538i	$-0.271171\pi$
$853$	38.4674	1.31710	0.658549	+	0.752538i	$0.271171\pi$
$854$	−18.1168	−0.619946
$855$	0	0
$856$	12.0000	0.410152
$857$	36.3505	1.24171	0.620855	−	0.783925i	$-0.286785\pi$
$857$	36.3505	1.24171	0.620855	+	0.783925i	$0.286785\pi$
$858$	0	0
$859$	−42.7446	−1.45843	−0.729213	−	0.684287i	$-0.760114\pi$
$859$	−42.7446	−1.45843	−0.729213	+	0.684287i	$0.760114\pi$
$860$	0	0
$861$	0	0
$862$	−34.9783	−1.19136
$863$	21.2554	0.723544	0.361772	−	0.932267i	$-0.382172\pi$
$863$	21.2554	0.723544	0.361772	+	0.932267i	$0.382172\pi$
$864$	0	0
$865$	0	0
$866$	27.7228	0.942060
$867$	0	0
$868$	−11.3723	−0.386000
$869$	−1.25544	−0.0425878
$870$	0	0
$871$	16.0000	0.542139
$872$	15.4891	0.524528
$873$	0	0
$874$	−1.72281	−0.0582750
$875$	0	0
$876$	0	0
$877$	−36.9783	−1.24867	−0.624333	−	0.781158i	$-0.714629\pi$
$877$	−36.9783	−1.24867	−0.624333	+	0.781158i	$0.714629\pi$
$878$	−18.9783	−0.640485
$879$	0	0
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	−3.37228	−0.113486	−0.0567432	−	0.998389i	$-0.518072\pi$
$883$	−3.37228	−0.113486	−0.0567432	+	0.998389i	$0.518072\pi$
$884$	−2.74456	−0.0923096
$885$	0	0
$886$	−29.4891	−0.990707
$887$	−10.9783	−0.368614	−0.184307	−	0.982869i	$-0.559004\pi$
$887$	−10.9783	−0.368614	−0.184307	+	0.982869i	$0.559004\pi$
$888$	0	0
$889$	26.9783	0.904821
$890$	0	0
$891$	0	0
$892$	18.7446	0.627614
$893$	1.72281	0.0576517
$894$	0	0
$895$	0	0
$896$	3.37228	0.112660
$897$	0	0
$898$	28.9783	0.967017
$899$	−4.62772	−0.154343
$900$	0	0
$901$	−5.64947	−0.188211
$902$	−11.4891	−0.382546
$903$	0	0
$904$	−3.25544	−0.108274
$905$	0	0
$906$	0	0
$907$	0.394031	0.0130836	0.00654179	−	0.999979i	$-0.497918\pi$
$907$	0.394031	0.0130836	0.00654179	+	0.999979i	$0.497918\pi$
$908$	−2.74456	−0.0910815
$909$	0	0
$910$	0	0
$911$	8.39403	0.278107	0.139053	−	0.990285i	$-0.455594\pi$
$911$	8.39403	0.278107	0.139053	+	0.990285i	$0.455594\pi$
$912$	0	0
$913$	−2.74456	−0.0908318
$914$	−16.3505	−0.540828
$915$	0	0
$916$	−10.0000	−0.330409
$917$	74.5842	2.46299
$918$	0	0
$919$	18.9783	0.626035	0.313017	−	0.949747i	$-0.398660\pi$
$919$	18.9783	0.626035	0.313017	+	0.949747i	$0.398660\pi$
$920$	0	0
$921$	0	0
$922$	16.1168	0.530780
$923$	20.2337	0.666000
$924$	0	0
$925$	0	0
$926$	−0.233688	−0.00767946
$927$	0	0
$928$	1.37228	0.0450473
$929$	−24.3505	−0.798915	−0.399458	−	0.916752i	$-0.630801\pi$
$929$	−24.3505	−0.798915	−0.399458	+	0.916752i	$0.630801\pi$
$930$	0	0
$931$	2.74456	0.0899494
$932$	1.37228	0.0449506
$933$	0	0
$934$	19.3723	0.633880
$935$	0	0
$936$	0	0
$937$	28.5109	0.931410	0.465705	−	0.884940i	$-0.345801\pi$
$937$	28.5109	0.931410	0.465705	+	0.884940i	$0.345801\pi$
$938$	−26.9783	−0.880871
$939$	0	0
$940$	0	0
$941$	15.0951	0.492086	0.246043	−	0.969259i	$-0.420870\pi$
$941$	15.0951	0.492086	0.246043	+	0.969259i	$0.420870\pi$
$942$	0	0
$943$	31.5326	1.02684
$944$	2.74456	0.0893279
$945$	0	0
$946$	−4.00000	−0.130051
$947$	48.8614	1.58778	0.793891	−	0.608060i	$-0.208052\pi$
$947$	48.8614	1.58778	0.793891	+	0.608060i	$0.208052\pi$
$948$	0	0
$949$	−30.9783	−1.00560
$950$	0	0
$951$	0	0
$952$	4.62772	0.149985
$953$	40.1168	1.29951	0.649756	−	0.760143i	$-0.274871\pi$
$953$	40.1168	1.29951	0.649756	+	0.760143i	$0.274871\pi$
$954$	0	0
$955$	0	0
$956$	14.7446	0.476873
$957$	0	0
$958$	5.48913	0.177346
$959$	−29.4891	−0.952254
$960$	0	0
$961$	−19.6277	−0.633152
$962$	−18.7446	−0.604349
$963$	0	0
$964$	−22.0000	−0.708572
$965$	0	0
$966$	0	0
$967$	−47.6060	−1.53090	−0.765452	−	0.643493i	$-0.777485\pi$
$967$	−47.6060	−1.53090	−0.765452	+	0.643493i	$0.777485\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	0	0
$971$	1.02175	0.0327895	0.0163947	−	0.999866i	$-0.494781\pi$
$971$	1.02175	0.0327895	0.0163947	+	0.999866i	$0.494781\pi$
$972$	0	0
$973$	13.4891	0.432442
$974$	20.0000	0.640841
$975$	0	0
$976$	−5.37228	−0.171963
$977$	14.2337	0.455376	0.227688	−	0.973734i	$-0.426883\pi$
$977$	14.2337	0.455376	0.227688	+	0.973734i	$0.426883\pi$
$978$	0	0
$979$	1.37228	0.0438583
$980$	0	0
$981$	0	0
$982$	−7.37228	−0.235259
$983$	−13.7228	−0.437690	−0.218845	−	0.975760i	$-0.570229\pi$
$983$	−13.7228	−0.437690	−0.218845	+	0.975760i	$0.570229\pi$
$984$	0	0
$985$	0	0
$986$	1.88316	0.0599719
$987$	0	0
$988$	−1.25544	−0.0399408
$989$	10.9783	0.349088
$990$	0	0
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	−3.37228	−0.107070
$993$	0	0
$994$	−34.1168	−1.08212
$995$	0	0
$996$	0	0
$997$	−22.2337	−0.704148	−0.352074	−	0.935972i	$-0.614523\pi$
$997$	−22.2337	−0.704148	−0.352074	+	0.935972i	$0.614523\pi$
$998$	33.4891	1.06008
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4950.2.a.bw.1.1		2
3.2	odd	2		550.2.a.n.1.2		2
5.2	odd	4		4950.2.c.bc.199.1		4
5.3	odd	4		4950.2.c.bc.199.4		4
5.4	even	2		990.2.a.m.1.2		2
12.11	even	2		4400.2.a.bl.1.1		2
15.2	even	4		550.2.b.f.199.3		4
15.8	even	4		550.2.b.f.199.2		4
15.14	odd	2		110.2.a.d.1.1	✓	2
20.19	odd	2		7920.2.a.bq.1.1		2
33.32	even	2		6050.2.a.cb.1.2		2
60.23	odd	4		4400.2.b.p.4049.1		4
60.47	odd	4		4400.2.b.p.4049.4		4
60.59	even	2		880.2.a.n.1.2		2
105.104	even	2		5390.2.a.bp.1.2		2
120.29	odd	2		3520.2.a.bq.1.2		2
120.59	even	2		3520.2.a.bj.1.1		2
165.164	even	2		1210.2.a.r.1.1		2
660.659	odd	2		9680.2.a.bt.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.2.a.d.1.1	✓	2	15.14	odd	2
550.2.a.n.1.2		2	3.2	odd	2
550.2.b.f.199.2		4	15.8	even	4
550.2.b.f.199.3		4	15.2	even	4
880.2.a.n.1.2		2	60.59	even	2
990.2.a.m.1.2		2	5.4	even	2
1210.2.a.r.1.1		2	165.164	even	2
3520.2.a.bj.1.1		2	120.59	even	2
3520.2.a.bq.1.2		2	120.29	odd	2
4400.2.a.bl.1.1		2	12.11	even	2
4400.2.b.p.4049.1		4	60.23	odd	4
4400.2.b.p.4049.4		4	60.47	odd	4
4950.2.a.bw.1.1		2	1.1	even	1	trivial
4950.2.c.bc.199.1		4	5.2	odd	4
4950.2.c.bc.199.4		4	5.3	odd	4
5390.2.a.bp.1.2		2	105.104	even	2
6050.2.a.cb.1.2		2	33.32	even	2
7920.2.a.bq.1.1		2	20.19	odd	2
9680.2.a.bt.1.2		2	660.659	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4950.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.37228 q^{7} -1.00000 q^{8} +1.00000 q^{11} -2.00000 q^{13} +3.37228 q^{14} +1.00000 q^{16} +1.37228 q^{17} +0.627719 q^{19} -1.00000 q^{22} +2.74456 q^{23} +2.00000 q^{26} -3.37228 q^{28} -1.37228 q^{29} +3.37228 q^{31} -1.00000 q^{32} -1.37228 q^{34} -9.37228 q^{37} -0.627719 q^{38} +11.4891 q^{41} +4.00000 q^{43} +1.00000 q^{44} -2.74456 q^{46} +2.74456 q^{47} +4.37228 q^{49} -2.00000 q^{52} -4.11684 q^{53} +3.37228 q^{56} +1.37228 q^{58} +2.74456 q^{59} -5.37228 q^{61} -3.37228 q^{62} +1.00000 q^{64} -8.00000 q^{67} +1.37228 q^{68} -10.1168 q^{71} +15.4891 q^{73} +9.37228 q^{74} +0.627719 q^{76} -3.37228 q^{77} -1.25544 q^{79} -11.4891 q^{82} -2.74456 q^{83} -4.00000 q^{86} -1.00000 q^{88} +1.37228 q^{89} +6.74456 q^{91} +2.74456 q^{92} -2.74456 q^{94} +12.7446 q^{97} -4.37228 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - q^{7} - 2 q^{8} + 2 q^{11} - 4 q^{13} + q^{14} + 2 q^{16} - 3 q^{17} + 7 q^{19} - 2 q^{22} - 6 q^{23} + 4 q^{26} - q^{28} + 3 q^{29} + q^{31} - 2 q^{32} + 3 q^{34} - 13 q^{37}+ \cdots - 3 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - q^7 - 2 * q^8 + 2 * q^11 - 4 * q^13 + q^14 + 2 * q^16 - 3 * q^17 + 7 * q^19 - 2 * q^22 - 6 * q^23 + 4 * q^26 - q^28 + 3 * q^29 + q^31 - 2 * q^32 + 3 * q^34 - 13 * q^37 - 7 * q^38 + 8 * q^43 + 2 * q^44 + 6 * q^46 - 6 * q^47 + 3 * q^49 - 4 * q^52 + 9 * q^53 + q^56 - 3 * q^58 - 6 * q^59 - 5 * q^61 - q^62 + 2 * q^64 - 16 * q^67 - 3 * q^68 - 3 * q^71 + 8 * q^73 + 13 * q^74 + 7 * q^76 - q^77 - 14 * q^79 + 6 * q^83 - 8 * q^86 - 2 * q^88 - 3 * q^89 + 2 * q^91 - 6 * q^92 + 6 * q^94 + 14 * q^97 - 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more