LMFDB - Embedded newform 1148.2.d.a.1065.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1148,2,Mod(1065,1148)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1148, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1148.1065");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1148 = 2^{2} \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1148.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$9.16682615204$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 39 x^{18} + 531 x^{16} + 3488 x^{14} + 12661 x^{12} + 27027 x^{10} + 34540 x^{8} + 25909 x^{6} + \cdots + 144 $ x^20 + 39x^18 + 531x^16 + 3488x^14 + 12661x^12 + 27027x^10 + 34540x^8 + 25909x^6 + 10677x^4 + 2092*x^2 + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1065.3
Root			$-1.02366i$ of defining polynomial
Character	$\chi$	$=$	1148.1065
Dual form			1148.2.d.a.1065.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.57093i q^{3} -2.38500 q^{5} -1.00000i q^{7} -3.60969 q^{9} +O(q^{10})$	$q-2.57093i q^{3} -2.38500 q^{5} -1.00000i q^{7} -3.60969 q^{9} +2.88295i q^{11} +4.97483i q^{13} +6.13166i q^{15} +7.25525i q^{17} -4.84438i q^{19} -2.57093 q^{21} -1.50062 q^{23} +0.688204 q^{25} +1.56748i q^{27} +4.90169i q^{29} -5.66399 q^{31} +7.41187 q^{33} +2.38500i q^{35} +9.73403 q^{37} +12.7900 q^{39} +(4.46933 + 4.58531i) q^{41} -4.00048 q^{43} +8.60910 q^{45} -8.40651i q^{47} -1.00000 q^{49} +18.6528 q^{51} +5.39570i q^{53} -6.87582i q^{55} -12.4546 q^{57} +9.99346 q^{59} -10.3215 q^{61} +3.60969i q^{63} -11.8649i q^{65} -1.61869i q^{67} +3.85799i q^{69} +15.4780i q^{71} -1.66466 q^{73} -1.76933i q^{75} +2.88295 q^{77} +13.1796i q^{79} -6.79919 q^{81} -7.34592 q^{83} -17.3037i q^{85} +12.6019 q^{87} -6.68089i q^{89} +4.97483 q^{91} +14.5617i q^{93} +11.5538i q^{95} +3.42621i q^{97} -10.4066i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 4 q^{5} - 20 q^{9}+O(q^{10}) $ 20 * q + 4 * q^5 - 20 * q^9	$ 20 q + 4 q^{5} - 20 q^{9} + 4 q^{21} + 8 q^{31} + 20 q^{37} + 4 q^{39} - 16 q^{41} + 20 q^{43} - 4 q^{45} - 20 q^{49} + 52 q^{51} - 36 q^{57} + 20 q^{59} - 4 q^{61} - 12 q^{73} + 8 q^{77} + 20 q^{81} - 48 q^{83} + 44 q^{87} - 4 q^{91}+O(q^{100}) $ 20 * q + 4 * q^5 - 20 * q^9 + 4 * q^21 + 8 * q^31 + 20 * q^37 + 4 * q^39 - 16 * q^41 + 20 * q^43 - 4 * q^45 - 20 * q^49 + 52 * q^51 - 36 * q^57 + 20 * q^59 - 4 * q^61 - 12 * q^73 + 8 * q^77 + 20 * q^81 - 48 * q^83 + 44 * q^87 - 4 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1148\mathbb{Z}\right)^\times$.

$n$	$493$	$575$	$785$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		−	2.57093i		−	1.48433i	−0.670218	−	0.742164i	$-0.733799\pi$
$3$		−	2.57093i		−	1.48433i	0.670218	−	0.742164i	$-0.266201\pi$
$4$		0			0
$5$	−2.38500			−1.06660			−0.533301	−	0.845925i	$-0.679049\pi$
$5$	−2.38500			−1.06660			−0.533301	+	0.845925i	$0.679049\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	−3.60969			−1.20323
$10$		0			0
$11$			2.88295i			0.869242i	0.900614	+	0.434621i	$0.143118\pi$
$11$			2.88295i			0.869242i	−0.900614	+	0.434621i	$0.856882\pi$
$12$		0			0
$13$			4.97483i			1.37977i	0.723919	+	0.689885i	$0.242339\pi$
$13$			4.97483i			1.37977i	−0.723919	+	0.689885i	$0.757661\pi$
$14$		0			0
$15$			6.13166i			1.58319i
$16$		0			0
$17$			7.25525i			1.75966i	0.475291	+	0.879829i	$0.342343\pi$
$17$			7.25525i			1.75966i	−0.475291	+	0.879829i	$0.657657\pi$
$18$		0			0
$19$		−	4.84438i		−	1.11138i	−0.831390	−	0.555689i	$-0.812455\pi$
$19$		−	4.84438i		−	1.11138i	0.831390	−	0.555689i	$-0.187545\pi$
$20$		0			0
$21$	−2.57093			−0.561023
$22$		0			0
$23$	−1.50062			−0.312901			−0.156450	−	0.987686i	$-0.550005\pi$
$23$	−1.50062			−0.312901			−0.156450	+	0.987686i	$0.550005\pi$
$24$		0			0
$25$	0.688204			0.137641
$26$		0			0
$27$			1.56748i			0.301662i
$28$		0			0
$29$			4.90169i			0.910222i	0.890435	+	0.455111i	$0.150400\pi$
$29$			4.90169i			0.910222i	−0.890435	+	0.455111i	$0.849600\pi$
$30$		0			0
$31$	−5.66399			−1.01728			−0.508641	−	0.860978i	$-0.669852\pi$
$31$	−5.66399			−1.01728			−0.508641	+	0.860978i	$0.669852\pi$
$32$		0			0
$33$	7.41187			1.29024
$34$		0			0
$35$			2.38500i			0.403138i
$36$		0			0
$37$	9.73403			1.60026			0.800132	−	0.599823i	$-0.204763\pi$
$37$	9.73403			1.60026			0.800132	+	0.599823i	$0.204763\pi$
$38$		0			0
$39$	12.7900			2.04803
$40$		0			0
$41$	4.46933	+	4.58531i	0.697992	+	0.716105i
$41$	4.46933	+	4.58531i	0.697992	+	0.716105i
$42$		0			0
$43$	−4.00048			−0.610068			−0.305034	−	0.952341i	$-0.598668\pi$
$43$	−4.00048			−0.610068			−0.305034	+	0.952341i	$0.598668\pi$
$44$		0			0
$45$	8.60910			1.28337
$46$		0			0
$47$		−	8.40651i		−	1.22622i	−0.789999	−	0.613108i	$-0.789919\pi$
$47$		−	8.40651i		−	1.22622i	0.789999	−	0.613108i	$-0.210081\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	18.6528			2.61191
$52$		0			0
$53$			5.39570i			0.741157i	0.928801	+	0.370578i	$0.120840\pi$
$53$			5.39570i			0.741157i	−0.928801	+	0.370578i	$0.879160\pi$
$54$		0			0
$55$		−	6.87582i		−	0.927135i
$56$		0			0
$57$	−12.4546			−1.64965
$58$		0			0
$59$	9.99346			1.30104			0.650519	−	0.759490i	$-0.274552\pi$
$59$	9.99346			1.30104			0.650519	+	0.759490i	$0.274552\pi$
$60$		0			0
$61$	−10.3215			−1.32153			−0.660765	−	0.750593i	$-0.729768\pi$
$61$	−10.3215			−1.32153			−0.660765	+	0.750593i	$0.729768\pi$
$62$		0			0
$63$			3.60969i			0.454779i
$64$		0			0
$65$		−	11.8649i		−	1.47167i
$66$		0			0
$67$		−	1.61869i		−	0.197754i	−0.995100	−	0.0988771i	$-0.968475\pi$
$67$		−	1.61869i		−	0.197754i	0.995100	−	0.0988771i	$-0.0315251\pi$
$68$		0			0
$69$			3.85799i			0.464448i
$70$		0			0
$71$			15.4780i			1.83690i	0.395533	+	0.918452i	$0.370560\pi$
$71$			15.4780i			1.83690i	−0.395533	+	0.918452i	$0.629440\pi$
$72$		0			0
$73$	−1.66466			−0.194834			−0.0974169	−	0.995244i	$-0.531058\pi$
$73$	−1.66466			−0.194834			−0.0974169	+	0.995244i	$0.531058\pi$
$74$		0			0
$75$		−	1.76933i		−	0.204304i
$76$		0			0
$77$	2.88295			0.328542
$78$		0			0
$79$			13.1796i			1.48282i	0.671050	+	0.741412i	$0.265844\pi$
$79$			13.1796i			1.48282i	−0.671050	+	0.741412i	$0.734156\pi$
$80$		0			0
$81$	−6.79919			−0.755466
$82$		0			0
$83$	−7.34592			−0.806319			−0.403160	−	0.915130i	$-0.632088\pi$
$83$	−7.34592			−0.806319			−0.403160	+	0.915130i	$0.632088\pi$
$84$		0			0
$85$		−	17.3037i		−	1.87685i
$86$		0			0
$87$	12.6019			1.35107
$88$		0			0
$89$		−	6.68089i		−	0.708173i	−0.935213	−	0.354086i	$-0.884792\pi$
$89$		−	6.68089i		−	0.708173i	0.935213	−	0.354086i	$-0.115208\pi$
$90$		0			0
$91$	4.97483			0.521504
$92$		0			0
$93$			14.5617i			1.50998i
$94$		0			0
$95$			11.5538i			1.18540i
$96$		0			0
$97$			3.42621i			0.347879i	0.984756	+	0.173940i	$0.0556497\pi$
$97$			3.42621i			0.347879i	−0.984756	+	0.173940i	$0.944350\pi$
$98$		0			0
$99$		−	10.4066i		−	1.04590i
$100$		0			0
$101$		−	9.43120i		−	0.938439i	−0.883081	−	0.469220i	$-0.844535\pi$
$101$		−	9.43120i		−	0.938439i	0.883081	−	0.469220i	$-0.155465\pi$
$102$		0			0
$103$	−6.58092			−0.648438			−0.324219	−	0.945982i	$-0.605101\pi$
$103$	−6.58092			−0.648438			−0.324219	+	0.945982i	$0.605101\pi$
$104$		0			0
$105$	6.13166			0.598389
$106$		0			0
$107$	14.2759			1.38010			0.690050	−	0.723761i	$-0.257588\pi$
$107$	14.2759			1.38010			0.690050	+	0.723761i	$0.257588\pi$
$108$		0			0
$109$			12.0066i			1.15002i	0.818146	+	0.575011i	$0.195002\pi$
$109$			12.0066i			1.15002i	−0.818146	+	0.575011i	$0.804998\pi$
$110$		0			0
$111$		−	25.0255i		−	2.37532i
$112$		0			0
$113$	−12.2164			−1.14922			−0.574610	−	0.818428i	$-0.694846\pi$
$113$	−12.2164			−1.14922			−0.574610	+	0.818428i	$0.694846\pi$
$114$		0			0
$115$	3.57897			0.333741
$116$		0			0
$117$		−	17.9576i		−	1.66018i
$118$		0			0
$119$	7.25525			0.665088
$120$		0			0
$121$	2.68861			0.244419
$122$		0			0
$123$	11.7885	−	11.4904i	1.06294	−	1.03605i
$124$		0			0
$125$	10.2836			0.919794
$126$		0			0
$127$	−10.9331			−0.970158			−0.485079	−	0.874470i	$-0.661209\pi$
$127$	−10.9331			−0.970158			−0.485079	+	0.874470i	$0.661209\pi$
$128$		0			0
$129$			10.2850i			0.905541i
$130$		0			0
$131$	11.6438			1.01732			0.508661	−	0.860967i	$-0.330141\pi$
$131$	11.6438			1.01732			0.508661	+	0.860967i	$0.330141\pi$
$132$		0			0
$133$	−4.84438			−0.420061
$134$		0			0
$135$		−	3.73844i		−	0.321753i
$136$		0			0
$137$			9.83027i			0.839856i	0.907557	+	0.419928i	$0.137945\pi$
$137$			9.83027i			0.839856i	−0.907557	+	0.419928i	$0.862055\pi$
$138$		0			0
$139$	−6.62459			−0.561890			−0.280945	−	0.959724i	$-0.590648\pi$
$139$	−6.62459			−0.561890			−0.280945	+	0.959724i	$0.590648\pi$
$140$		0			0
$141$	−21.6126			−1.82011
$142$		0			0
$143$	−14.3422			−1.19935
$144$		0			0
$145$		−	11.6905i		−	0.970844i
$146$		0			0
$147$			2.57093i			0.212047i
$148$		0			0
$149$			10.3049i			0.844211i	0.906547	+	0.422105i	$0.138709\pi$
$149$			10.3049i			0.844211i	−0.906547	+	0.422105i	$0.861291\pi$
$150$		0			0
$151$			7.00670i			0.570198i	0.958498	+	0.285099i	$0.0920264\pi$
$151$			7.00670i			0.570198i	−0.958498	+	0.285099i	$0.907974\pi$
$152$		0			0
$153$		−	26.1892i		−	2.11727i
$154$		0			0
$155$	13.5086			1.08504
$156$		0			0
$157$			16.0449i			1.28052i	0.768157	+	0.640262i	$0.221174\pi$
$157$			16.0449i			1.28052i	−0.768157	+	0.640262i	$0.778826\pi$
$158$		0			0
$159$	13.8720			1.10012
$160$		0			0
$161$			1.50062i			0.118265i
$162$		0			0
$163$	−11.6231			−0.910395			−0.455198	−	0.890390i	$-0.650431\pi$
$163$	−11.6231			−0.910395			−0.455198	+	0.890390i	$0.650431\pi$
$164$		0			0
$165$	−17.6773			−1.37617
$166$		0			0
$167$			14.5888i			1.12892i	0.825462	+	0.564458i	$0.190915\pi$
$167$			14.5888i			1.12892i	−0.825462	+	0.564458i	$0.809085\pi$
$168$		0			0
$169$	−11.7489			−0.903764
$170$		0			0
$171$			17.4867i			1.33724i
$172$		0			0
$173$	−15.2827			−1.16192			−0.580961	−	0.813931i	$-0.697323\pi$
$173$	−15.2827			−1.16192			−0.580961	+	0.813931i	$0.697323\pi$
$174$		0			0
$175$		−	0.688204i		−	0.0520233i
$176$		0			0
$177$		−	25.6925i		−	1.93117i
$178$		0			0
$179$			9.23350i			0.690144i	0.938576	+	0.345072i	$0.112146\pi$
$179$			9.23350i			0.690144i	−0.938576	+	0.345072i	$0.887854\pi$
$180$		0			0
$181$		−	14.0782i		−	1.04643i	−0.852202	−	0.523213i	$-0.824733\pi$
$181$		−	14.0782i		−	1.04643i	0.852202	−	0.523213i	$-0.175267\pi$
$182$		0			0
$183$			26.5358i			1.96159i
$184$		0			0
$185$	−23.2156			−1.70685
$186$		0			0
$187$	−20.9165			−1.52957
$188$		0			0
$189$	1.56748			0.114018
$190$		0			0
$191$			5.26930i			0.381273i	0.981661	+	0.190636i	$0.0610552\pi$
$191$			5.26930i			0.381273i	−0.981661	+	0.190636i	$0.938945\pi$
$192$		0			0
$193$		−	12.3281i		−	0.887399i	−0.896176	−	0.443700i	$-0.853666\pi$
$193$		−	12.3281i		−	0.887399i	0.896176	−	0.443700i	$-0.146334\pi$
$194$		0			0
$195$	−30.5040			−2.18444
$196$		0			0
$197$	6.90635			0.492057			0.246029	−	0.969263i	$-0.420874\pi$
$197$	6.90635			0.492057			0.246029	+	0.969263i	$0.420874\pi$
$198$		0			0
$199$		−	14.7055i		−	1.04244i	−0.853422	−	0.521221i	$-0.825477\pi$
$199$		−	14.7055i		−	1.04244i	0.853422	−	0.521221i	$-0.174523\pi$
$200$		0			0
$201$	−4.16154			−0.293532
$202$		0			0
$203$	4.90169			0.344031
$204$		0			0
$205$	−10.6593	−	10.9359i	−0.744480	−	0.763799i
$206$		0			0
$207$	5.41678			0.376492
$208$		0			0
$209$	13.9661			0.966056
$210$		0			0
$211$			10.3928i			0.715469i	0.933823	+	0.357735i	$0.116451\pi$
$211$			10.3928i			0.715469i	−0.933823	+	0.357735i	$0.883549\pi$
$212$		0			0
$213$	39.7930			2.72657
$214$		0			0
$215$	9.54114			0.650700
$216$		0			0
$217$			5.66399i			0.384497i
$218$		0			0
$219$			4.27973i			0.289197i
$220$		0			0
$221$	−36.0937			−2.42792
$222$		0			0
$223$	3.33749			0.223495			0.111747	−	0.993737i	$-0.464355\pi$
$223$	3.33749			0.223495			0.111747	+	0.993737i	$0.464355\pi$
$224$		0			0
$225$	−2.48420			−0.165614
$226$		0			0
$227$		−	18.3494i		−	1.21789i	−0.793213	−	0.608945i	$-0.791593\pi$
$227$		−	18.3494i		−	1.21789i	0.793213	−	0.608945i	$-0.208407\pi$
$228$		0			0
$229$		−	5.76073i		−	0.380680i	−0.981718	−	0.190340i	$-0.939041\pi$
$229$		−	5.76073i		−	0.380680i	0.981718	−	0.190340i	$-0.0609590\pi$
$230$		0			0
$231$		−	7.41187i		−	0.487665i
$232$		0			0
$233$			3.97693i			0.260537i	0.991479	+	0.130269i	$0.0415840\pi$
$233$			3.97693i			0.260537i	−0.991479	+	0.130269i	$0.958416\pi$
$234$		0			0
$235$			20.0495i			1.30788i
$236$		0			0
$237$	33.8839			2.20100
$238$		0			0
$239$			15.3470i			0.992714i	0.868118	+	0.496357i	$0.165329\pi$
$239$			15.3470i			0.992714i	−0.868118	+	0.496357i	$0.834671\pi$
$240$		0			0
$241$	−20.3337			−1.30981			−0.654904	−	0.755712i	$-0.727291\pi$
$241$	−20.3337			−1.30981			−0.654904	+	0.755712i	$0.727291\pi$
$242$		0			0
$243$			22.1827i			1.42302i
$244$		0			0
$245$	2.38500			0.152372
$246$		0			0
$247$	24.1000			1.53345
$248$		0			0
$249$			18.8859i			1.19684i
$250$		0			0
$251$	−15.8791			−1.00228			−0.501141	−	0.865366i	$-0.667086\pi$
$251$	−15.8791			−1.00228			−0.501141	+	0.865366i	$0.667086\pi$
$252$		0			0
$253$		−	4.32621i		−	0.271986i
$254$		0			0
$255$	−44.4868			−2.78587
$256$		0			0
$257$			3.03747i			0.189472i	0.995502	+	0.0947362i	$0.0302008\pi$
$257$			3.03747i			0.189472i	−0.995502	+	0.0947362i	$0.969799\pi$
$258$		0			0
$259$		−	9.73403i		−	0.604843i
$260$		0			0
$261$		−	17.6936i		−	1.09521i
$262$		0			0
$263$		−	25.2239i		−	1.55537i	−0.628654	−	0.777685i	$-0.716394\pi$
$263$		−	25.2239i		−	1.55537i	0.628654	−	0.777685i	$-0.283606\pi$
$264$		0			0
$265$		−	12.8687i		−	0.790519i
$266$		0			0
$267$	−17.1761			−1.05116
$268$		0			0
$269$	−26.3116			−1.60425			−0.802125	−	0.597156i	$-0.796297\pi$
$269$	−26.3116			−1.60425			−0.802125	+	0.597156i	$0.796297\pi$
$270$		0			0
$271$	−31.2970			−1.90116			−0.950578	−	0.310487i	$-0.899508\pi$
$271$	−31.2970			−1.90116			−0.950578	+	0.310487i	$0.899508\pi$
$272$		0			0
$273$		−	12.7900i		−	0.774083i
$274$		0			0
$275$			1.98406i			0.119643i
$276$		0			0
$277$	24.6439			1.48071			0.740355	−	0.672217i	$-0.234658\pi$
$277$	24.6439			1.48071			0.740355	+	0.672217i	$0.234658\pi$
$278$		0			0
$279$	20.4453			1.22403
$280$		0			0
$281$		−	14.3107i		−	0.853706i	−0.904321	−	0.426853i	$-0.859622\pi$
$281$		−	14.3107i		−	0.853706i	0.904321	−	0.426853i	$-0.140378\pi$
$282$		0			0
$283$	33.2933			1.97908			0.989541	−	0.144252i	$-0.0460775\pi$
$283$	33.2933			1.97908			0.989541	+	0.144252i	$0.0460775\pi$
$284$		0			0
$285$	29.7041			1.75952
$286$		0			0
$287$	4.58531	−	4.46933i	0.270662	−	0.263816i
$288$		0			0
$289$	−35.6387			−2.09639
$290$		0			0
$291$	8.80856			0.516367
$292$		0			0
$293$			22.9000i			1.33783i	0.743339	+	0.668915i	$0.233241\pi$
$293$			22.9000i			1.33783i	−0.743339	+	0.668915i	$0.766759\pi$
$294$		0			0
$295$	−23.8344			−1.38769
$296$		0			0
$297$	−4.51897			−0.262217
$298$		0			0
$299$		−	7.46533i		−	0.431731i
$300$		0			0
$301$			4.00048i			0.230584i
$302$		0			0
$303$	−24.2470			−1.39295
$304$		0			0
$305$	24.6167			1.40955
$306$		0			0
$307$	9.86379			0.562956			0.281478	−	0.959568i	$-0.409175\pi$
$307$	9.86379			0.562956			0.281478	+	0.959568i	$0.409175\pi$
$308$		0			0
$309$			16.9191i			0.962494i
$310$		0			0
$311$			16.4799i			0.934490i	0.884128	+	0.467245i	$0.154753\pi$
$311$			16.4799i			0.934490i	−0.884128	+	0.467245i	$0.845247\pi$
$312$		0			0
$313$		−	26.4767i		−	1.49655i	−0.663388	−	0.748276i	$-0.730882\pi$
$313$		−	26.4767i		−	1.49655i	0.663388	−	0.748276i	$-0.269118\pi$
$314$		0			0
$315$		−	8.60910i		−	0.485068i
$316$		0			0
$317$		−	8.10018i		−	0.454951i	−0.973784	−	0.227476i	$-0.926953\pi$
$317$		−	8.10018i		−	0.454951i	0.973784	−	0.227476i	$-0.0730472\pi$
$318$		0			0
$319$	−14.1313			−0.791202
$320$		0			0
$321$		−	36.7023i		−	2.04852i
$322$		0			0
$323$	35.1472			1.95564
$324$		0			0
$325$			3.42370i			0.189912i
$326$		0			0
$327$	30.8681			1.70701
$328$		0			0
$329$	−8.40651			−0.463466
$330$		0			0
$331$		−	8.90873i		−	0.489668i	−0.969565	−	0.244834i	$-0.921266\pi$
$331$		−	8.90873i		−	0.489668i	0.969565	−	0.244834i	$-0.0787335\pi$
$332$		0			0
$333$	−35.1369			−1.92549
$334$		0			0
$335$			3.86057i			0.210925i
$336$		0			0
$337$	13.1099			0.714141			0.357071	−	0.934077i	$-0.383776\pi$
$337$	13.1099			0.714141			0.357071	+	0.934077i	$0.383776\pi$
$338$		0			0
$339$			31.4075i			1.70582i
$340$		0			0
$341$		−	16.3290i		−	0.884265i
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$		−	9.20129i		−	0.495381i
$346$		0			0
$347$		−	28.4940i		−	1.52964i	−0.644246	−	0.764818i	$-0.722829\pi$
$347$		−	28.4940i		−	1.52964i	0.644246	−	0.764818i	$-0.277171\pi$
$348$		0			0
$349$	21.2620			1.13813			0.569065	−	0.822292i	$-0.307305\pi$
$349$	21.2620			1.13813			0.569065	+	0.822292i	$0.307305\pi$
$350$		0			0
$351$	−7.79796			−0.416224
$352$		0			0
$353$	7.83587			0.417061			0.208531	−	0.978016i	$-0.433132\pi$
$353$	7.83587			0.417061			0.208531	+	0.978016i	$0.433132\pi$
$354$		0			0
$355$		−	36.9150i		−	1.95925i
$356$		0			0
$357$		−	18.6528i		−	0.987209i
$358$		0			0
$359$	−19.8342			−1.04681			−0.523406	−	0.852084i	$-0.675339\pi$
$359$	−19.8342			−1.04681			−0.523406	+	0.852084i	$0.675339\pi$
$360$		0			0
$361$	−4.46806			−0.235161
$362$		0			0
$363$		−	6.91223i		−	0.362798i
$364$		0			0
$365$	3.97021			0.207810
$366$		0			0
$367$	−21.6274			−1.12894			−0.564470	−	0.825453i	$-0.690920\pi$
$367$	−21.6274			−1.12894			−0.564470	+	0.825453i	$0.690920\pi$
$368$		0			0
$369$	−16.1329	−	16.5516i	−0.839846	−	0.861640i
$370$		0			0
$371$	5.39570			0.280131
$372$		0			0
$373$	36.3537			1.88232			0.941162	−	0.337955i	$-0.109735\pi$
$373$	36.3537			1.88232			0.941162	+	0.337955i	$0.109735\pi$
$374$		0			0
$375$		−	26.4385i		−	1.36528i
$376$		0			0
$377$	−24.3851			−1.25590
$378$		0			0
$379$	−26.2551			−1.34864			−0.674318	−	0.738441i	$-0.735562\pi$
$379$	−26.2551			−1.34864			−0.674318	+	0.738441i	$0.735562\pi$
$380$		0			0
$381$			28.1083i			1.44003i
$382$		0			0
$383$			16.9081i			0.863964i	0.901882	+	0.431982i	$0.142186\pi$
$383$			16.9081i			0.863964i	−0.901882	+	0.431982i	$0.857814\pi$
$384$		0			0
$385$	−6.87582			−0.350424
$386$		0			0
$387$	14.4405			0.734053
$388$		0			0
$389$	24.7825			1.25652			0.628261	−	0.778003i	$-0.283767\pi$
$389$	24.7825			1.25652			0.628261	+	0.778003i	$0.283767\pi$
$390$		0			0
$391$		−	10.8874i		−	0.550598i
$392$		0			0
$393$		−	29.9354i		−	1.51004i
$394$		0			0
$395$		−	31.4334i		−	1.58158i
$396$		0			0
$397$			6.05367i			0.303825i	0.988394	+	0.151913i	$0.0485432\pi$
$397$			6.05367i			0.303825i	−0.988394	+	0.151913i	$0.951457\pi$
$398$		0			0
$399$			12.4546i			0.623509i
$400$		0			0
$401$	6.92393			0.345764			0.172882	−	0.984943i	$-0.444692\pi$
$401$	6.92393			0.345764			0.172882	+	0.984943i	$0.444692\pi$
$402$		0			0
$403$		−	28.1774i		−	1.40362i
$404$		0			0
$405$	16.2160			0.805782
$406$		0			0
$407$			28.0627i			1.39102i
$408$		0			0
$409$	−18.5030			−0.914912			−0.457456	−	0.889232i	$-0.651239\pi$
$409$	−18.5030			−0.914912			−0.457456	+	0.889232i	$0.651239\pi$
$410$		0			0
$411$	25.2730			1.24662
$412$		0			0
$413$		−	9.99346i		−	0.491746i
$414$		0			0
$415$	17.5200			0.860022
$416$		0			0
$417$			17.0314i			0.834029i
$418$		0			0
$419$	25.2270			1.23242			0.616210	−	0.787582i	$-0.288667\pi$
$419$	25.2270			1.23242			0.616210	+	0.787582i	$0.288667\pi$
$420$		0			0
$421$		−	2.83208i		−	0.138027i	−0.997616	−	0.0690135i	$-0.978015\pi$
$421$		−	2.83208i		−	0.138027i	0.997616	−	0.0690135i	$-0.0219852\pi$
$422$		0			0
$423$			30.3449i			1.47542i
$424$		0			0
$425$			4.99309i			0.242201i
$426$		0			0
$427$			10.3215i			0.499491i
$428$		0			0
$429$			36.8728i			1.78023i
$430$		0			0
$431$	−15.1590			−0.730183			−0.365091	−	0.930972i	$-0.618962\pi$
$431$	−15.1590			−0.730183			−0.365091	+	0.930972i	$0.618962\pi$
$432$		0			0
$433$	−10.7792			−0.518014			−0.259007	−	0.965875i	$-0.583395\pi$
$433$	−10.7792			−0.518014			−0.259007	+	0.965875i	$0.583395\pi$
$434$		0			0
$435$	−30.0555			−1.44105
$436$		0			0
$437$			7.26958i			0.347751i
$438$		0			0
$439$		−	1.11594i		−	0.0532608i	−0.999645	−	0.0266304i	$-0.991522\pi$
$439$		−	1.11594i		−	0.0532608i	0.999645	−	0.0266304i	$-0.00847772\pi$
$440$		0			0
$441$	3.60969			0.171890
$442$		0			0
$443$	−13.8984			−0.660334			−0.330167	−	0.943923i	$-0.607105\pi$
$443$	−13.8984			−0.660334			−0.330167	+	0.943923i	$0.607105\pi$
$444$		0			0
$445$			15.9339i			0.755339i
$446$		0			0
$447$	26.4932			1.25309
$448$		0			0
$449$	0.294012			0.0138753			0.00693764	−	0.999976i	$-0.497792\pi$
$449$	0.294012			0.0138753			0.00693764	+	0.999976i	$0.497792\pi$
$450$		0			0
$451$	−13.2192	+	12.8849i	−0.622468	+	0.606724i
$452$		0			0
$453$	18.0138			0.846360
$454$		0			0
$455$	−11.8649			−0.556237
$456$		0			0
$457$			11.7879i			0.551416i	0.961241	+	0.275708i	$0.0889122\pi$
$457$			11.7879i			0.551416i	−0.961241	+	0.275708i	$0.911088\pi$
$458$		0			0
$459$	−11.3725			−0.530822
$460$		0			0
$461$	−11.2768			−0.525214			−0.262607	−	0.964903i	$-0.584582\pi$
$461$	−11.2768			−0.525214			−0.262607	+	0.964903i	$0.584582\pi$
$462$		0			0
$463$			28.6116i			1.32970i	0.746979	+	0.664848i	$0.231504\pi$
$463$			28.6116i			1.32970i	−0.746979	+	0.664848i	$0.768496\pi$
$464$		0			0
$465$		−	34.7297i		−	1.61055i
$466$		0			0
$467$	19.3804			0.896818			0.448409	−	0.893828i	$-0.351991\pi$
$467$	19.3804			0.896818			0.448409	+	0.893828i	$0.351991\pi$
$468$		0			0
$469$	−1.61869			−0.0747441
$470$		0			0
$471$	41.2504			1.90072
$472$		0			0
$473$		−	11.5332i		−	0.530297i
$474$		0			0
$475$		−	3.33392i		−	0.152971i
$476$		0			0
$477$		−	19.4768i		−	0.891783i
$478$		0			0
$479$			14.3718i			0.656663i	0.944563	+	0.328332i	$0.106486\pi$
$479$			14.3718i			0.656663i	−0.944563	+	0.328332i	$0.893514\pi$
$480$		0			0
$481$			48.4251i			2.20800i
$482$		0			0
$483$	3.85799			0.175545
$484$		0			0
$485$		−	8.17150i		−	0.371049i
$486$		0			0
$487$	28.6259			1.29716			0.648581	−	0.761146i	$-0.275363\pi$
$487$	28.6259			1.29716			0.648581	+	0.761146i	$0.275363\pi$
$488$		0			0
$489$			29.8823i			1.35133i
$490$		0			0
$491$	15.6508			0.706312			0.353156	−	0.935564i	$-0.385108\pi$
$491$	15.6508			0.706312			0.353156	+	0.935564i	$0.385108\pi$
$492$		0			0
$493$	−35.5630			−1.60168
$494$		0			0
$495$			24.8196i			1.11556i
$496$		0			0
$497$	15.4780			0.694284
$498$		0			0
$499$		−	24.6188i		−	1.10209i	−0.834477	−	0.551043i	$-0.814230\pi$
$499$		−	24.6188i		−	1.10209i	0.834477	−	0.551043i	$-0.185770\pi$
$500$		0			0
$501$	37.5069			1.67568
$502$		0			0
$503$			35.4258i			1.57956i	0.613392	+	0.789779i	$0.289805\pi$
$503$			35.4258i			1.57956i	−0.613392	+	0.789779i	$0.710195\pi$
$504$		0			0
$505$			22.4934i			1.00094i
$506$		0			0
$507$			30.2057i			1.34148i
$508$		0			0
$509$			16.0385i			0.710894i	0.934697	+	0.355447i	$0.115671\pi$
$509$			16.0385i			0.710894i	−0.934697	+	0.355447i	$0.884329\pi$
$510$		0			0
$511$			1.66466i			0.0736403i
$512$		0			0
$513$	7.59348			0.335261
$514$		0			0
$515$	15.6955			0.691625
$516$		0			0
$517$	24.2355			1.06588
$518$		0			0
$519$			39.2908i			1.72467i
$520$		0			0
$521$		−	29.4260i		−	1.28918i	−0.764529	−	0.644589i	$-0.777028\pi$
$521$		−	29.4260i		−	1.28918i	0.764529	−	0.644589i	$-0.222972\pi$
$522$		0			0
$523$	21.5468			0.942178			0.471089	−	0.882086i	$-0.343861\pi$
$523$	21.5468			0.942178			0.471089	+	0.882086i	$0.343861\pi$
$524$		0			0
$525$	−1.76933			−0.0772197
$526$		0			0
$527$		−	41.0937i		−	1.79007i
$528$		0			0
$529$	−20.7481			−0.902093
$530$		0			0
$531$	−36.0733			−1.56545
$532$		0			0
$533$	−22.8111	+	22.2342i	−0.988060	+	0.963069i
$534$		0			0
$535$	−34.0479			−1.47202
$536$		0			0
$537$	23.7387			1.02440
$538$		0			0
$539$		−	2.88295i		−	0.124177i
$540$		0			0
$541$	15.6475			0.672740			0.336370	−	0.941730i	$-0.390801\pi$
$541$	15.6475			0.672740			0.336370	+	0.941730i	$0.390801\pi$
$542$		0			0
$543$	−36.1942			−1.55324
$544$		0			0
$545$		−	28.6356i		−	1.22662i
$546$		0			0
$547$		−	33.4259i		−	1.42919i	−0.699538	−	0.714595i	$-0.746611\pi$
$547$		−	33.4259i		−	1.42919i	0.699538	−	0.714595i	$-0.253389\pi$
$548$		0			0
$549$	37.2574			1.59011
$550$		0			0
$551$	23.7457			1.01160
$552$		0			0
$553$	13.1796			0.560455
$554$		0			0
$555$			59.6858i			2.53352i
$556$		0			0
$557$		−	1.78645i		−	0.0756942i	−0.999284	−	0.0378471i	$-0.987950\pi$
$557$		−	1.78645i		−	0.0756942i	0.999284	−	0.0378471i	$-0.0120500\pi$
$558$		0			0
$559$		−	19.9017i		−	0.841753i
$560$		0			0
$561$			53.7750i			2.27038i
$562$		0			0
$563$			35.2787i			1.48682i	0.668836	+	0.743410i	$0.266793\pi$
$563$			35.2787i			1.48682i	−0.668836	+	0.743410i	$0.733207\pi$
$564$		0			0
$565$	29.1360			1.22576
$566$		0			0
$567$			6.79919i			0.285539i
$568$		0			0
$569$	−30.0684			−1.26053			−0.630267	−	0.776379i	$-0.717054\pi$
$569$	−30.0684			−1.26053			−0.630267	+	0.776379i	$0.717054\pi$
$570$		0			0
$571$		−	22.0288i		−	0.921876i	−0.887432	−	0.460938i	$-0.847513\pi$
$571$		−	22.0288i		−	0.921876i	0.887432	−	0.460938i	$-0.152487\pi$
$572$		0			0
$573$	13.5470			0.565934
$574$		0			0
$575$	−1.03273			−0.0430679
$576$		0			0
$577$		−	32.9259i		−	1.37072i	−0.728202	−	0.685362i	$-0.759644\pi$
$577$		−	32.9259i		−	1.37072i	0.728202	−	0.685362i	$-0.240356\pi$
$578$		0			0
$579$	−31.6948			−1.31719
$580$		0			0
$581$			7.34592i			0.304760i
$582$		0			0
$583$	−15.5555			−0.644244
$584$		0			0
$585$			42.8288i			1.77075i
$586$		0			0
$587$			4.53262i			0.187081i	0.995615	+	0.0935407i	$0.0298185\pi$
$587$			4.53262i			0.187081i	−0.995615	+	0.0935407i	$0.970181\pi$
$588$		0			0
$589$			27.4385i			1.13059i
$590$		0			0
$591$		−	17.7558i		−	0.730375i
$592$		0			0
$593$		−	18.3442i		−	0.753305i	−0.926355	−	0.376653i	$-0.877075\pi$
$593$		−	18.3442i		−	0.753305i	0.926355	−	0.376653i	$-0.122925\pi$
$594$		0			0
$595$	−17.3037			−0.709384
$596$		0			0
$597$	−37.8068			−1.54733
$598$		0			0
$599$	−35.8389			−1.46434			−0.732168	−	0.681124i	$-0.761491\pi$
$599$	−35.8389			−1.46434			−0.732168	+	0.681124i	$0.761491\pi$
$600$		0			0
$601$			36.9317i			1.50648i	0.657748	+	0.753238i	$0.271509\pi$
$601$			36.9317i			1.50648i	−0.657748	+	0.753238i	$0.728491\pi$
$602$		0			0
$603$			5.84297i			0.237944i
$604$		0			0
$605$	−6.41232			−0.260698
$606$		0			0
$607$	30.6081			1.24235			0.621173	−	0.783674i	$-0.286656\pi$
$607$	30.6081			1.24235			0.621173	+	0.783674i	$0.286656\pi$
$608$		0			0
$609$		−	12.6019i		−	0.510656i
$610$		0			0
$611$	41.8210			1.69189
$612$		0			0
$613$	6.21870			0.251171			0.125586	−	0.992083i	$-0.459919\pi$
$613$	6.21870			0.251171			0.125586	+	0.992083i	$0.459919\pi$
$614$		0			0
$615$	−28.1156	+	27.4044i	−1.13373	+	1.10505i
$616$		0			0
$617$	11.4734			0.461900			0.230950	−	0.972966i	$-0.425817\pi$
$617$	11.4734			0.461900			0.230950	+	0.972966i	$0.425817\pi$
$618$		0			0
$619$	4.62250			0.185794			0.0928970	−	0.995676i	$-0.470387\pi$
$619$	4.62250			0.185794			0.0928970	+	0.995676i	$0.470387\pi$
$620$		0			0
$621$		−	2.35219i		−	0.0943903i
$622$		0			0
$623$	−6.68089			−0.267664
$624$		0			0
$625$	−27.9674			−1.11870
$626$		0			0
$627$		−	35.9059i		−	1.43394i
$628$		0			0
$629$			70.6229i			2.81592i
$630$		0			0
$631$	−20.3436			−0.809866			−0.404933	−	0.914346i	$-0.632705\pi$
$631$	−20.3436			−0.809866			−0.404933	+	0.914346i	$0.632705\pi$
$632$		0			0
$633$	26.7192			1.06199
$634$		0			0
$635$	26.0754			1.03477
$636$		0			0
$637$		−	4.97483i		−	0.197110i
$638$		0			0
$639$		−	55.8709i		−	2.21022i
$640$		0			0
$641$		−	16.3439i		−	0.645545i	−0.946477	−	0.322772i	$-0.895385\pi$
$641$		−	16.3439i		−	0.645545i	0.946477	−	0.322772i	$-0.104615\pi$
$642$		0			0
$643$		−	25.1933i		−	0.993525i	−0.867887	−	0.496762i	$-0.834522\pi$
$643$		−	25.1933i		−	0.993525i	0.867887	−	0.496762i	$-0.165478\pi$
$644$		0			0
$645$		−	24.5296i		−	0.965853i
$646$		0			0
$647$	4.09131			0.160846			0.0804231	−	0.996761i	$-0.474373\pi$
$647$	4.09131			0.160846			0.0804231	+	0.996761i	$0.474373\pi$
$648$		0			0
$649$			28.8106i			1.13092i
$650$		0			0
$651$	14.5617			0.570720
$652$		0			0
$653$		−	8.55620i		−	0.334830i	−0.985887	−	0.167415i	$-0.946458\pi$
$653$		−	8.55620i		−	0.334830i	0.985887	−	0.167415i	$-0.0535419\pi$
$654$		0			0
$655$	−27.7704			−1.08508
$656$		0			0
$657$	6.00892			0.234430
$658$		0			0
$659$			12.6234i			0.491739i	0.969303	+	0.245869i	$0.0790734\pi$
$659$			12.6234i			0.491739i	−0.969303	+	0.245869i	$0.920927\pi$
$660$		0			0
$661$	−24.1049			−0.937572			−0.468786	−	0.883312i	$-0.655308\pi$
$661$	−24.1049			−0.937572			−0.468786	+	0.883312i	$0.655308\pi$
$662$		0			0
$663$			92.7943i			3.60383i
$664$		0			0
$665$	11.5538			0.448038
$666$		0			0
$667$		−	7.35558i		−	0.284809i
$668$		0			0
$669$		−	8.58045i		−	0.331739i
$670$		0			0
$671$		−	29.7563i		−	1.14873i
$672$		0			0
$673$			21.7532i			0.838524i	0.907865	+	0.419262i	$0.137711\pi$
$673$			21.7532i			0.838524i	−0.907865	+	0.419262i	$0.862289\pi$
$674$		0			0
$675$			1.07875i			0.0415210i
$676$		0			0
$677$	−3.31433			−0.127380			−0.0636900	−	0.997970i	$-0.520287\pi$
$677$	−3.31433			−0.127380			−0.0636900	+	0.997970i	$0.520287\pi$
$678$		0			0
$679$	3.42621			0.131486
$680$		0			0
$681$	−47.1749			−1.80775
$682$		0			0
$683$			9.68245i			0.370489i	0.982692	+	0.185244i	$0.0593077\pi$
$683$			9.68245i			0.370489i	−0.982692	+	0.185244i	$0.940692\pi$
$684$		0			0
$685$		−	23.4451i		−	0.895793i
$686$		0			0
$687$	−14.8105			−0.565054
$688$		0			0
$689$	−26.8427			−1.02263
$690$		0			0
$691$			11.4847i			0.436900i	0.975848	+	0.218450i	$0.0701000\pi$
$691$			11.4847i			0.436900i	−0.975848	+	0.218450i	$0.929900\pi$
$692$		0			0
$693$	−10.4066			−0.395313
$694$		0			0
$695$	15.7996			0.599313
$696$		0			0
$697$	−33.2676	+	32.4261i	−1.26010	+	1.22823i
$698$		0			0
$699$	10.2244			0.386723
$700$		0			0
$701$	9.74836			0.368191			0.184095	−	0.982908i	$-0.441064\pi$
$701$	9.74836			0.368191			0.184095	+	0.982908i	$0.441064\pi$
$702$		0			0
$703$		−	47.1554i		−	1.77850i
$704$		0			0
$705$	51.5459			1.94133
$706$		0			0
$707$	−9.43120			−0.354697
$708$		0			0
$709$		−	32.8261i		−	1.23281i	−0.787430	−	0.616404i	$-0.788589\pi$
$709$		−	32.8261i		−	1.23281i	0.787430	−	0.616404i	$-0.211411\pi$
$710$		0			0
$711$		−	47.5744i		−	1.78418i
$712$		0			0
$713$	8.49950			0.318309
$714$		0			0
$715$	34.2060			1.27923
$716$		0			0
$717$	39.4561			1.47351
$718$		0			0
$719$		−	15.4578i		−	0.576480i	−0.957558	−	0.288240i	$-0.906930\pi$
$719$		−	15.4578i		−	0.576480i	0.957558	−	0.288240i	$-0.0930701\pi$
$720$		0			0
$721$			6.58092i			0.245086i
$722$		0			0
$723$			52.2766i			1.94419i
$724$		0			0
$725$			3.37336i			0.125284i
$726$		0			0
$727$		−	37.8376i		−	1.40332i	−0.712512	−	0.701660i	$-0.752443\pi$
$727$		−	37.8376i		−	1.40332i	0.712512	−	0.701660i	$-0.247557\pi$
$728$		0			0
$729$	36.6327			1.35677
$730$		0			0
$731$		−	29.0245i		−	1.07351i
$732$		0			0
$733$	30.5423			1.12811			0.564053	−	0.825738i	$-0.309241\pi$
$733$	30.5423			1.12811			0.564053	+	0.825738i	$0.309241\pi$
$734$		0			0
$735$		−	6.13166i		−	0.226170i
$736$		0			0
$737$	4.66660			0.171896
$738$		0			0
$739$	20.7293			0.762539			0.381269	−	0.924464i	$-0.375487\pi$
$739$	20.7293			0.762539			0.381269	+	0.924464i	$0.375487\pi$
$740$		0			0
$741$		−	61.9594i		−	2.27614i
$742$		0			0
$743$	18.0275			0.661364			0.330682	−	0.943742i	$-0.392721\pi$
$743$	18.0275			0.661364			0.330682	+	0.943742i	$0.392721\pi$
$744$		0			0
$745$		−	24.5772i		−	0.900437i
$746$		0			0
$747$	26.5165			0.970189
$748$		0			0
$749$		−	14.2759i		−	0.521629i
$750$		0			0
$751$		−	51.8386i		−	1.89162i	−0.324726	−	0.945808i	$-0.605272\pi$
$751$		−	51.8386i		−	1.89162i	0.324726	−	0.945808i	$-0.394728\pi$
$752$		0			0
$753$			40.8242i			1.48771i
$754$		0			0
$755$		−	16.7110i		−	0.608174i
$756$		0			0
$757$			23.7894i			0.864641i	0.901720	+	0.432321i	$0.142305\pi$
$757$			23.7894i			0.864641i	−0.901720	+	0.432321i	$0.857695\pi$
$758$		0			0
$759$	−11.1224			−0.403717
$760$		0			0
$761$	−17.1518			−0.621751			−0.310875	−	0.950451i	$-0.600622\pi$
$761$	−17.1518			−0.621751			−0.310875	+	0.950451i	$0.600622\pi$
$762$		0			0
$763$	12.0066			0.434667
$764$		0			0
$765$			62.4612i			2.25829i
$766$		0			0
$767$			49.7158i			1.79513i
$768$		0			0
$769$	16.7004			0.602233			0.301117	−	0.953587i	$-0.402641\pi$
$769$	16.7004			0.602233			0.301117	+	0.953587i	$0.402641\pi$
$770$		0			0
$771$	7.80914			0.281239
$772$		0			0
$773$			35.9480i			1.29296i	0.762931	+	0.646480i	$0.223760\pi$
$773$			35.9480i			1.29296i	−0.762931	+	0.646480i	$0.776240\pi$
$774$		0			0
$775$	−3.89798			−0.140020
$776$		0			0
$777$	−25.0255			−0.897786
$778$		0			0
$779$	22.2130	−	21.6512i	0.795863	−	0.775733i
$780$		0			0
$781$	−44.6223			−1.59671
$782$		0			0
$783$	−7.68332			−0.274579
$784$		0			0
$785$		−	38.2670i		−	1.36581i
$786$		0			0
$787$	9.36484			0.333821			0.166910	−	0.985972i	$-0.446621\pi$
$787$	9.36484			0.333821			0.166910	+	0.985972i	$0.446621\pi$
$788$		0			0
$789$	−64.8488			−2.30868
$790$		0			0
$791$			12.2164i			0.434364i
$792$		0			0
$793$		−	51.3476i		−	1.82341i
$794$		0			0
$795$	−33.0846			−1.17339
$796$		0			0
$797$	28.4440			1.00754			0.503769	−	0.863839i	$-0.331946\pi$
$797$	28.4440			1.00754			0.503769	+	0.863839i	$0.331946\pi$
$798$		0			0
$799$	60.9914			2.15772
$800$		0			0
$801$			24.1160i			0.852096i
$802$		0			0
$803$		−	4.79913i		−	0.169358i
$804$		0			0
$805$		−	3.57897i		−	0.126142i
$806$		0			0
$807$			67.6455i			2.38123i
$808$		0			0
$809$			54.4730i			1.91517i	0.288150	+	0.957585i	$0.406960\pi$
$809$			54.4730i			1.91517i	−0.288150	+	0.957585i	$0.593040\pi$
$810$		0			0
$811$	−3.38589			−0.118895			−0.0594473	−	0.998231i	$-0.518934\pi$
$811$	−3.38589			−0.118895			−0.0594473	+	0.998231i	$0.518934\pi$
$812$		0			0
$813$			80.4624i			2.82194i
$814$		0			0
$815$	27.7212			0.971030
$816$		0			0
$817$			19.3799i			0.678016i
$818$		0			0
$819$	−17.9576			−0.627490
$820$		0			0
$821$	−13.9088			−0.485420			−0.242710	−	0.970099i	$-0.578036\pi$
$821$	−13.9088			−0.485420			−0.242710	+	0.970099i	$0.578036\pi$
$822$		0			0
$823$		−	17.9013i		−	0.623999i	−0.950082	−	0.311999i	$-0.899001\pi$
$823$		−	17.9013i		−	0.623999i	0.950082	−	0.311999i	$-0.100999\pi$
$824$		0			0
$825$	5.10087			0.177590
$826$		0			0
$827$		−	1.16642i		−	0.0405604i	−0.999794	−	0.0202802i	$-0.993544\pi$
$827$		−	1.16642i		−	0.0405604i	0.999794	−	0.0202802i	$-0.00645583\pi$
$828$		0			0
$829$	46.6958			1.62181			0.810906	−	0.585177i	$-0.198975\pi$
$829$	46.6958			1.62181			0.810906	+	0.585177i	$0.198975\pi$
$830$		0			0
$831$		−	63.3578i		−	2.19786i
$832$		0			0
$833$		−	7.25525i		−	0.251380i
$834$		0			0
$835$		−	34.7943i		−	1.20411i
$836$		0			0
$837$		−	8.87820i		−	0.306876i
$838$		0			0
$839$		−	5.49038i		−	0.189549i	−0.995499	−	0.0947744i	$-0.969787\pi$
$839$		−	5.49038i		−	0.189549i	0.995499	−	0.0947744i	$-0.0302130\pi$
$840$		0			0
$841$	4.97341			0.171497
$842$		0			0
$843$	−36.7919			−1.26718
$844$		0			0
$845$	28.0212			0.963957
$846$		0			0
$847$		−	2.68861i		−	0.0923817i
$848$		0			0
$849$		−	85.5949i		−	2.93761i
$850$		0			0
$851$	−14.6071			−0.500724
$852$		0			0
$853$	9.07359			0.310674			0.155337	−	0.987862i	$-0.450354\pi$
$853$	9.07359			0.310674			0.155337	+	0.987862i	$0.450354\pi$
$854$		0			0
$855$		−	41.7058i		−	1.42631i
$856$		0			0
$857$	54.4841			1.86114			0.930571	−	0.366112i	$-0.119311\pi$
$857$	54.4841			1.86114			0.930571	+	0.366112i	$0.119311\pi$
$858$		0			0
$859$	−22.7331			−0.775643			−0.387822	−	0.921734i	$-0.626772\pi$
$859$	−22.7331			−0.775643			−0.387822	+	0.921734i	$0.626772\pi$
$860$		0			0
$861$	−11.4904	−	11.7885i	−0.391590	−	0.401752i
$862$		0			0
$863$	−21.9695			−0.747851			−0.373926	−	0.927459i	$-0.621988\pi$
$863$	−21.9695			−0.747851			−0.373926	+	0.927459i	$0.621988\pi$
$864$		0			0
$865$	36.4492			1.23931
$866$		0			0
$867$			91.6247i			3.11174i
$868$		0			0
$869$	−37.9962			−1.28893
$870$		0			0
$871$	8.05270			0.272855
$872$		0			0
$873$		−	12.3676i		−	0.418579i
$874$		0			0
$875$		−	10.2836i		−	0.347650i
$876$		0			0
$877$	−18.9212			−0.638924			−0.319462	−	0.947599i	$-0.603502\pi$
$877$	−18.9212			−0.638924			−0.319462	+	0.947599i	$0.603502\pi$
$878$		0			0
$879$	58.8743			1.98578
$880$		0			0
$881$	47.5381			1.60160			0.800799	−	0.598933i	$-0.204408\pi$
$881$	47.5381			1.60160			0.800799	+	0.598933i	$0.204408\pi$
$882$		0			0
$883$		−	6.21651i		−	0.209202i	−0.994514	−	0.104601i	$-0.966643\pi$
$883$		−	6.21651i		−	0.209202i	0.994514	−	0.104601i	$-0.0333566\pi$
$884$		0			0
$885$			61.2765i			2.05979i
$886$		0			0
$887$			5.97504i			0.200622i	0.994956	+	0.100311i	$0.0319838\pi$
$887$			5.97504i			0.200622i	−0.994956	+	0.100311i	$0.968016\pi$
$888$		0			0
$889$			10.9331i			0.366685i
$890$		0			0
$891$		−	19.6017i		−	0.656682i
$892$		0			0
$893$	−40.7244			−1.36279
$894$		0			0
$895$		−	22.0219i		−	0.736110i
$896$		0			0
$897$	−19.1929			−0.640831
$898$		0			0
$899$		−	27.7631i		−	0.925953i
$900$		0			0
$901$	−39.1472			−1.30418
$902$		0			0
$903$	10.2850			0.342263
$904$		0			0
$905$			33.5765i			1.11612i
$906$		0			0
$907$	28.1178			0.933637			0.466819	−	0.884353i	$-0.345400\pi$
$907$	28.1178			0.933637			0.466819	+	0.884353i	$0.345400\pi$
$908$		0			0
$909$			34.0437i			1.12916i
$910$		0			0
$911$	−8.45762			−0.280213			−0.140107	−	0.990136i	$-0.544745\pi$
$911$	−8.45762			−0.280213			−0.140107	+	0.990136i	$0.544745\pi$
$912$		0			0
$913$		−	21.1779i		−	0.700886i
$914$		0			0
$915$		−	63.2878i		−	2.09223i
$916$		0			0
$917$		−	11.6438i		−	0.384512i
$918$		0			0
$919$		−	35.1801i		−	1.16049i	−0.814444	−	0.580243i	$-0.802958\pi$
$919$		−	35.1801i		−	1.16049i	0.814444	−	0.580243i	$-0.197042\pi$
$920$		0			0
$921$		−	25.3591i		−	0.835612i
$922$		0			0
$923$	−77.0005			−2.53450
$924$		0			0
$925$	6.69899			0.220262
$926$		0			0
$927$	23.7551			0.780220
$928$		0			0
$929$		−	23.9236i		−	0.784907i	−0.919772	−	0.392453i	$-0.871626\pi$
$929$		−	23.9236i		−	0.784907i	0.919772	−	0.392453i	$-0.128374\pi$
$930$		0			0
$931$			4.84438i			0.158768i
$932$		0			0
$933$	42.3688			1.38709
$934$		0			0
$935$	49.8858			1.63144
$936$		0			0
$937$		−	14.5809i		−	0.476338i	−0.971224	−	0.238169i	$-0.923453\pi$
$937$		−	14.5809i		−	0.476338i	0.971224	−	0.238169i	$-0.0765472\pi$
$938$		0			0
$939$	−68.0698			−2.22137
$940$		0			0
$941$	15.8926			0.518083			0.259042	−	0.965866i	$-0.416593\pi$
$941$	15.8926			0.518083			0.259042	+	0.965866i	$0.416593\pi$
$942$		0			0
$943$	−6.70677	−	6.88081i	−0.218402	−	0.224070i
$944$		0			0
$945$	−3.73844			−0.121611
$946$		0			0
$947$	−2.73424			−0.0888508			−0.0444254	−	0.999013i	$-0.514146\pi$
$947$	−2.73424			−0.0888508			−0.0444254	+	0.999013i	$0.514146\pi$
$948$		0			0
$949$		−	8.28140i		−	0.268826i
$950$		0			0
$951$	−20.8250			−0.675297
$952$		0			0
$953$	−27.6387			−0.895304			−0.447652	−	0.894208i	$-0.647740\pi$
$953$	−27.6387			−0.895304			−0.447652	+	0.894208i	$0.647740\pi$
$954$		0			0
$955$		−	12.5673i		−	0.406667i
$956$		0			0
$957$			36.3307i			1.17440i
$958$		0			0
$959$	9.83027			0.317436
$960$		0			0
$961$	1.08080			0.0348644
$962$		0			0
$963$	−51.5315			−1.66058
$964$		0			0
$965$			29.4026i			0.946502i
$966$		0			0
$967$		−	20.2019i		−	0.649648i	−0.945774	−	0.324824i	$-0.894695\pi$
$967$		−	20.2019i		−	0.649648i	0.945774	−	0.324824i	$-0.105305\pi$
$968$		0			0
$969$		−	90.3612i		−	2.90282i
$970$		0			0
$971$			16.3307i			0.524078i	0.965057	+	0.262039i	$0.0843950\pi$
$971$			16.3307i			0.524078i	−0.965057	+	0.262039i	$0.915605\pi$
$972$		0			0
$973$			6.62459i			0.212374i
$974$		0			0
$975$	8.80209			0.281893
$976$		0			0
$977$			0.0505900i			0.00161852i	1.00000		0.000809259i	$0.000257595\pi$
$977$			0.0505900i			0.00161852i	−1.00000		0.000809259i	$0.999742\pi$
$978$		0			0
$979$	19.2607			0.615573
$980$		0			0
$981$		−	43.3401i		−	1.38374i
$982$		0			0
$983$	45.6718			1.45670			0.728352	−	0.685203i	$-0.240287\pi$
$983$	45.6718			1.45670			0.728352	+	0.685203i	$0.240287\pi$
$984$		0			0
$985$	−16.4716			−0.524829
$986$		0			0
$987$			21.6126i			0.687936i
$988$		0			0
$989$	6.00320			0.190891
$990$		0			0
$991$			8.58782i			0.272801i	0.990654	+	0.136401i	$0.0435534\pi$
$991$			8.58782i			0.272801i	−0.990654	+	0.136401i	$0.956447\pi$
$992$		0			0
$993$	−22.9038			−0.726829
$994$		0			0
$995$			35.0725i			1.11187i
$996$		0			0
$997$		−	12.5734i		−	0.398205i	−0.979979	−	0.199103i	$-0.936197\pi$
$997$		−	12.5734i		−	0.398205i	0.979979	−	0.199103i	$-0.0638027\pi$
$998$		0			0
$999$			15.2579i			0.482739i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1148.2.d.a.1065.3	✓	20
41.40	even	2	inner	1148.2.d.a.1065.18	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.d.a.1065.3	✓	20	1.1	even	1	trivial
1148.2.d.a.1065.18	yes	20	41.40	even	2	inner

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:	\( N \)	\(=\)	\( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1148.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.57093i q^{3} -2.38500 q^{5} -1.00000i q^{7} -3.60969 q^{9} +O(q^{10})\)	\(q-2.57093i q^{3} -2.38500 q^{5} -1.00000i q^{7} -3.60969 q^{9} +2.88295i q^{11} +4.97483i q^{13} +6.13166i q^{15} +7.25525i q^{17} -4.84438i q^{19} -2.57093 q^{21} -1.50062 q^{23} +0.688204 q^{25} +1.56748i q^{27} +4.90169i q^{29} -5.66399 q^{31} +7.41187 q^{33} +2.38500i q^{35} +9.73403 q^{37} +12.7900 q^{39} +(4.46933 + 4.58531i) q^{41} -4.00048 q^{43} +8.60910 q^{45} -8.40651i q^{47} -1.00000 q^{49} +18.6528 q^{51} +5.39570i q^{53} -6.87582i q^{55} -12.4546 q^{57} +9.99346 q^{59} -10.3215 q^{61} +3.60969i q^{63} -11.8649i q^{65} -1.61869i q^{67} +3.85799i q^{69} +15.4780i q^{71} -1.66466 q^{73} -1.76933i q^{75} +2.88295 q^{77} +13.1796i q^{79} -6.79919 q^{81} -7.34592 q^{83} -17.3037i q^{85} +12.6019 q^{87} -6.68089i q^{89} +4.97483 q^{91} +14.5617i q^{93} +11.5538i q^{95} +3.42621i q^{97} -10.4066i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{5} - 20 q^{9}+O(q^{10}) \) 20 * q + 4 * q^5 - 20 * q^9	\( 20 q + 4 q^{5} - 20 q^{9} + 4 q^{21} + 8 q^{31} + 20 q^{37} + 4 q^{39} - 16 q^{41} + 20 q^{43} - 4 q^{45} - 20 q^{49} + 52 q^{51} - 36 q^{57} + 20 q^{59} - 4 q^{61} - 12 q^{73} + 8 q^{77} + 20 q^{81} - 48 q^{83} + 44 q^{87} - 4 q^{91}+O(q^{100}) \) 20 * q + 4 * q^5 - 20 * q^9 + 4 * q^21 + 8 * q^31 + 20 * q^37 + 4 * q^39 - 16 * q^41 + 20 * q^43 - 4 * q^45 - 20 * q^49 + 52 * q^51 - 36 * q^57 + 20 * q^59 - 4 * q^61 - 12 * q^73 + 8 * q^77 + 20 * q^81 - 48 * q^83 + 44 * q^87 - 4 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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