LMFDB - Embedded newform 1148.2.d.a.1065.13

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			1065.13
Root			$-0.379831i$ of defining polynomial
Character	$\chi$	$=$	1148.1065
Dual form			1148.2.d.a.1065.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.03684i q^{3} +2.44552 q^{5} +1.00000i q^{7} +1.92496 q^{9} +O(q^{10})$	$q+1.03684i q^{3} +2.44552 q^{5} +1.00000i q^{7} +1.92496 q^{9} +5.06947i q^{11} -2.54571i q^{13} +2.53562i q^{15} +5.30064i q^{17} -1.14984i q^{19} -1.03684 q^{21} -5.57153 q^{23} +0.980584 q^{25} +5.10640i q^{27} -8.69224i q^{29} +8.34429 q^{31} -5.25624 q^{33} +2.44552i q^{35} +11.0486 q^{37} +2.63950 q^{39} +(-6.22148 - 1.51433i) q^{41} -5.96616 q^{43} +4.70754 q^{45} +4.18168i q^{47} -1.00000 q^{49} -5.49592 q^{51} +6.93136i q^{53} +12.3975i q^{55} +1.19220 q^{57} +7.10317 q^{59} +1.10590 q^{61} +1.92496i q^{63} -6.22560i q^{65} +2.43504i q^{67} -5.77679i q^{69} -6.81857i q^{71} -5.66952 q^{73} +1.01671i q^{75} -5.06947 q^{77} +4.13643i q^{79} +0.480357 q^{81} -12.5457 q^{83} +12.9628i q^{85} +9.01247 q^{87} +2.55515i q^{89} +2.54571 q^{91} +8.65171i q^{93} -2.81195i q^{95} -6.85832i q^{97} +9.75854i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$			1.03684i			0.598620i	0.954156	+	0.299310i	$0.0967565\pi$
$3$			1.03684i			0.598620i	−0.954156	+	0.299310i	$0.903243\pi$
$4$		0			0
$5$	2.44552			1.09367			0.546836	−	0.837240i	$-0.315832\pi$
$5$	2.44552			1.09367			0.546836	+	0.837240i	$0.315832\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	1.92496			0.641654
$10$		0			0
$11$			5.06947i			1.52850i	0.644918	+	0.764252i	$0.276892\pi$
$11$			5.06947i			1.52850i	−0.644918	+	0.764252i	$0.723108\pi$
$12$		0			0
$13$		−	2.54571i		−	0.706053i	−0.935613	−	0.353027i	$-0.885152\pi$
$13$		−	2.54571i		−	0.706053i	0.935613	−	0.353027i	$-0.114848\pi$
$14$		0			0
$15$			2.53562i			0.654694i
$16$		0			0
$17$			5.30064i			1.28559i	0.766037	+	0.642797i	$0.222226\pi$
$17$			5.30064i			1.28559i	−0.766037	+	0.642797i	$0.777774\pi$
$18$		0			0
$19$		−	1.14984i		−	0.263791i	−0.991264	−	0.131895i	$-0.957894\pi$
$19$		−	1.14984i		−	0.263791i	0.991264	−	0.131895i	$-0.0421063\pi$
$20$		0			0
$21$	−1.03684			−0.226257
$22$		0			0
$23$	−5.57153			−1.16174			−0.580872	−	0.813995i	$-0.697288\pi$
$23$	−5.57153			−1.16174			−0.580872	+	0.813995i	$0.697288\pi$
$24$		0			0
$25$	0.980584			0.196117
$26$		0			0
$27$			5.10640i			0.982727i
$28$		0			0
$29$		−	8.69224i		−	1.61411i	−0.590477	−	0.807054i	$-0.701061\pi$
$29$		−	8.69224i		−	1.61411i	0.590477	−	0.807054i	$-0.298939\pi$
$30$		0			0
$31$	8.34429			1.49868			0.749340	−	0.662186i	$-0.230371\pi$
$31$	8.34429			1.49868			0.749340	+	0.662186i	$0.230371\pi$
$32$		0			0
$33$	−5.25624			−0.914994
$34$		0			0
$35$			2.44552i			0.413369i
$36$		0			0
$37$	11.0486			1.81637			0.908187	−	0.418564i	$-0.137466\pi$
$37$	11.0486			1.81637			0.908187	+	0.418564i	$0.137466\pi$
$38$		0			0
$39$	2.63950			0.422658
$40$		0			0
$41$	−6.22148	−	1.51433i	−0.971632	−	0.236498i
$41$	−6.22148	−	1.51433i	−0.971632	−	0.236498i
$42$		0			0
$43$	−5.96616			−0.909832			−0.454916	−	0.890534i	$-0.650331\pi$
$43$	−5.96616			−0.909832			−0.454916	+	0.890534i	$0.650331\pi$
$44$		0			0
$45$	4.70754			0.701758
$46$		0			0
$47$			4.18168i			0.609961i	0.952359	+	0.304980i	$0.0986499\pi$
$47$			4.18168i			0.609961i	−0.952359	+	0.304980i	$0.901350\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−5.49592			−0.769582
$52$		0			0
$53$			6.93136i			0.952096i	0.879419	+	0.476048i	$0.157931\pi$
$53$			6.93136i			0.952096i	−0.879419	+	0.476048i	$0.842069\pi$
$54$		0			0
$55$			12.3975i			1.67168i
$56$		0			0
$57$	1.19220			0.157910
$58$		0			0
$59$	7.10317			0.924754			0.462377	−	0.886684i	$-0.346997\pi$
$59$	7.10317			0.924754			0.462377	+	0.886684i	$0.346997\pi$
$60$		0			0
$61$	1.10590			0.141597			0.0707983	−	0.997491i	$-0.477445\pi$
$61$	1.10590			0.141597			0.0707983	+	0.997491i	$0.477445\pi$
$62$		0			0
$63$			1.92496i			0.242522i
$64$		0			0
$65$		−	6.22560i		−	0.772190i
$66$		0			0
$67$			2.43504i			0.297487i	0.988876	+	0.148744i	$0.0475229\pi$
$67$			2.43504i			0.297487i	−0.988876	+	0.148744i	$0.952477\pi$
$68$		0			0
$69$		−	5.77679i		−	0.695444i
$70$		0			0
$71$		−	6.81857i		−	0.809215i	−0.914490	−	0.404608i	$-0.867408\pi$
$71$		−	6.81857i		−	0.809215i	0.914490	−	0.404608i	$-0.132592\pi$
$72$		0			0
$73$	−5.66952			−0.663567			−0.331784	−	0.943356i	$-0.607650\pi$
$73$	−5.66952			−0.663567			−0.331784	+	0.943356i	$0.607650\pi$
$74$		0			0
$75$			1.01671i			0.117400i
$76$		0			0
$77$	−5.06947			−0.577720
$78$		0			0
$79$			4.13643i			0.465385i	0.972550	+	0.232693i	$0.0747536\pi$
$79$			4.13643i			0.465385i	−0.972550	+	0.232693i	$0.925246\pi$
$80$		0			0
$81$	0.480357			0.0533730
$82$		0			0
$83$	−12.5457			−1.37707			−0.688536	−	0.725203i	$-0.741746\pi$
$83$	−12.5457			−1.37707			−0.688536	+	0.725203i	$0.741746\pi$
$84$		0			0
$85$			12.9628i			1.40602i
$86$		0			0
$87$	9.01247			0.966238
$88$		0			0
$89$			2.55515i			0.270846i	0.990788	+	0.135423i	$0.0432393\pi$
$89$			2.55515i			0.270846i	−0.990788	+	0.135423i	$0.956761\pi$
$90$		0			0
$91$	2.54571			0.266863
$92$		0			0
$93$			8.65171i			0.897140i
$94$		0			0
$95$		−	2.81195i		−	0.288500i
$96$		0			0
$97$		−	6.85832i		−	0.696356i	−0.937428	−	0.348178i	$-0.886800\pi$
$97$		−	6.85832i		−	0.696356i	0.937428	−	0.348178i	$-0.113200\pi$
$98$		0			0
$99$			9.75854i			0.980770i
$100$		0			0
$101$		−	0.0728344i		−	0.00724729i	−0.999993	−	0.00362365i	$-0.998847\pi$
$101$		−	0.0728344i		−	0.00724729i	0.999993	−	0.00362365i	$-0.00115345\pi$
$102$		0			0
$103$	1.65441			0.163014			0.0815069	−	0.996673i	$-0.474027\pi$
$103$	1.65441			0.163014			0.0815069	+	0.996673i	$0.474027\pi$
$104$		0			0
$105$	−2.53562			−0.247451
$106$		0			0
$107$	13.3535			1.29093			0.645465	−	0.763790i	$-0.276664\pi$
$107$	13.3535			1.29093			0.645465	+	0.763790i	$0.276664\pi$
$108$		0			0
$109$			2.30709i			0.220979i	0.993877	+	0.110489i	$0.0352418\pi$
$109$			2.30709i			0.220979i	−0.993877	+	0.110489i	$0.964758\pi$
$110$		0			0
$111$			11.4556i			1.08732i
$112$		0			0
$113$	12.5497			1.18057			0.590287	−	0.807194i	$-0.299015\pi$
$113$	12.5497			1.18057			0.590287	+	0.807194i	$0.299015\pi$
$114$		0			0
$115$	−13.6253			−1.27057
$116$		0			0
$117$		−	4.90040i		−	0.453042i
$118$		0			0
$119$	−5.30064			−0.485909
$120$		0			0
$121$	−14.6996			−1.33632
$122$		0			0
$123$	1.57012	−	6.45069i	0.141573	−	0.581639i
$124$		0			0
$125$	−9.82957			−0.879184
$126$		0			0
$127$	5.73087			0.508532			0.254266	−	0.967134i	$-0.418166\pi$
$127$	5.73087			0.508532			0.254266	+	0.967134i	$0.418166\pi$
$128$		0			0
$129$		−	6.18596i		−	0.544644i
$130$		0			0
$131$	−4.51320			−0.394320			−0.197160	−	0.980371i	$-0.563172\pi$
$131$	−4.51320			−0.394320			−0.197160	+	0.980371i	$0.563172\pi$
$132$		0			0
$133$	1.14984			0.0997035
$134$		0			0
$135$			12.4878i			1.07478i
$136$		0			0
$137$			3.22565i			0.275585i	0.990461	+	0.137793i	$0.0440008\pi$
$137$			3.22565i			0.275585i	−0.990461	+	0.137793i	$0.955999\pi$
$138$		0			0
$139$	4.48715			0.380595			0.190298	−	0.981726i	$-0.439055\pi$
$139$	4.48715			0.380595			0.190298	+	0.981726i	$0.439055\pi$
$140$		0			0
$141$	−4.33574			−0.365135
$142$		0			0
$143$	12.9054			1.07921
$144$		0			0
$145$		−	21.2571i		−	1.76530i
$146$		0			0
$147$		−	1.03684i		−	0.0855172i
$148$		0			0
$149$			1.78471i			0.146209i	0.997324	+	0.0731044i	$0.0232906\pi$
$149$			1.78471i			0.146209i	−0.997324	+	0.0731044i	$0.976709\pi$
$150$		0			0
$151$		−	8.90375i		−	0.724577i	−0.932066	−	0.362288i	$-0.881996\pi$
$151$		−	8.90375i		−	0.724577i	0.932066	−	0.362288i	$-0.118004\pi$
$152$		0			0
$153$			10.2035i			0.824906i
$154$		0			0
$155$	20.4062			1.63906
$156$		0			0
$157$			0.205961i			0.0164375i	0.999966	+	0.00821873i	$0.00261613\pi$
$157$			0.205961i			0.0164375i	−0.999966	+	0.00821873i	$0.997384\pi$
$158$		0			0
$159$	−7.18672			−0.569944
$160$		0			0
$161$		−	5.57153i		−	0.439098i
$162$		0			0
$163$	−1.00934			−0.0790572			−0.0395286	−	0.999218i	$-0.512586\pi$
$163$	−1.00934			−0.0790572			−0.0395286	+	0.999218i	$0.512586\pi$
$164$		0			0
$165$	−12.8543			−1.00070
$166$		0			0
$167$		−	22.7851i		−	1.76316i	−0.472034	−	0.881580i	$-0.656480\pi$
$167$		−	22.7851i		−	1.76316i	0.472034	−	0.881580i	$-0.343520\pi$
$168$		0			0
$169$	6.51935			0.501489
$170$		0			0
$171$		−	2.21339i		−	0.169262i
$172$		0			0
$173$	−2.64615			−0.201183			−0.100591	−	0.994928i	$-0.532073\pi$
$173$	−2.64615			−0.201183			−0.100591	+	0.994928i	$0.532073\pi$
$174$		0			0
$175$			0.980584i			0.0741252i
$176$		0			0
$177$			7.36485i			0.553576i
$178$		0			0
$179$			12.1448i			0.907747i	0.891066	+	0.453873i	$0.149958\pi$
$179$			12.1448i			0.907747i	−0.891066	+	0.453873i	$0.850042\pi$
$180$		0			0
$181$		−	20.4684i		−	1.52141i	−0.649100	−	0.760703i	$-0.724854\pi$
$181$		−	20.4684i		−	1.52141i	0.649100	−	0.760703i	$-0.275146\pi$
$182$		0			0
$183$			1.14665i			0.0847626i
$184$		0			0
$185$	27.0195			1.98652
$186$		0			0
$187$	−26.8714			−1.96503
$188$		0			0
$189$	−5.10640			−0.371436
$190$		0			0
$191$			13.5091i			0.977487i	0.872428	+	0.488744i	$0.162545\pi$
$191$			13.5091i			0.977487i	−0.872428	+	0.488744i	$0.837455\pi$
$192$		0			0
$193$		−	14.1767i		−	1.02046i	−0.860037	−	0.510232i	$-0.829560\pi$
$193$		−	14.1767i		−	1.02046i	0.860037	−	0.510232i	$-0.170440\pi$
$194$		0			0
$195$	6.45495			0.462249
$196$		0			0
$197$	22.0306			1.56961			0.784807	−	0.619740i	$-0.212762\pi$
$197$	22.0306			1.56961			0.784807	+	0.619740i	$0.212762\pi$
$198$		0			0
$199$		−	14.9190i		−	1.05758i	−0.848752	−	0.528791i	$-0.822646\pi$
$199$		−	14.9190i		−	1.05758i	0.848752	−	0.528791i	$-0.177354\pi$
$200$		0			0
$201$	−2.52475			−0.178082
$202$		0			0
$203$	8.69224			0.610076
$204$		0			0
$205$	−15.2148	−	3.70332i	−1.06265	−	0.258651i
$206$		0			0
$207$	−10.7250			−0.745437
$208$		0			0
$209$	5.82907			0.403205
$210$		0			0
$211$		−	24.6004i		−	1.69356i	−0.531940	−	0.846782i	$-0.678537\pi$
$211$		−	24.6004i		−	1.69356i	0.531940	−	0.846782i	$-0.321463\pi$
$212$		0			0
$213$	7.06977			0.484413
$214$		0			0
$215$	−14.5904			−0.995057
$216$		0			0
$217$			8.34429i			0.566448i
$218$		0			0
$219$		−	5.87839i		−	0.397225i
$220$		0			0
$221$	13.4939			0.907698
$222$		0			0
$223$	2.63720			0.176600			0.0883001	−	0.996094i	$-0.471857\pi$
$223$	2.63720			0.176600			0.0883001	+	0.996094i	$0.471857\pi$
$224$		0			0
$225$	1.88759			0.125839
$226$		0			0
$227$		−	13.8673i		−	0.920403i	−0.887815	−	0.460201i	$-0.847777\pi$
$227$		−	13.8673i		−	0.920403i	0.887815	−	0.460201i	$-0.152223\pi$
$228$		0			0
$229$		−	18.3429i		−	1.21214i	−0.795413	−	0.606068i	$-0.792746\pi$
$229$		−	18.3429i		−	1.21214i	0.795413	−	0.606068i	$-0.207254\pi$
$230$		0			0
$231$		−	5.25624i		−	0.345835i
$232$		0			0
$233$			4.30305i			0.281902i	0.990017	+	0.140951i	$0.0450160\pi$
$233$			4.30305i			0.281902i	−0.990017	+	0.140951i	$0.954984\pi$
$234$		0			0
$235$			10.2264i			0.667097i
$236$		0			0
$237$	−4.28882			−0.278589
$238$		0			0
$239$		−	18.6883i		−	1.20885i	−0.796664	−	0.604423i	$-0.793404\pi$
$239$		−	18.6883i		−	1.20885i	0.796664	−	0.604423i	$-0.206596\pi$
$240$		0			0
$241$	12.9190			0.832188			0.416094	−	0.909322i	$-0.363399\pi$
$241$	12.9190			0.832188			0.416094	+	0.909322i	$0.363399\pi$
$242$		0			0
$243$			15.8173i			1.01468i
$244$		0			0
$245$	−2.44552			−0.156239
$246$		0			0
$247$	−2.92715			−0.186250
$248$		0			0
$249$		−	13.0079i		−	0.824343i
$250$		0			0
$251$	−24.3743			−1.53849			−0.769245	−	0.638954i	$-0.779367\pi$
$251$	−24.3743			−1.53849			−0.769245	+	0.638954i	$0.779367\pi$
$252$		0			0
$253$		−	28.2447i		−	1.77573i
$254$		0			0
$255$	−13.4404			−0.841670
$256$		0			0
$257$		−	14.1856i		−	0.884872i	−0.896800	−	0.442436i	$-0.854114\pi$
$257$		−	14.1856i		−	0.884872i	0.896800	−	0.442436i	$-0.145886\pi$
$258$		0			0
$259$			11.0486i			0.686525i
$260$		0			0
$261$		−	16.7322i		−	1.03570i
$262$		0			0
$263$		−	7.24943i		−	0.447019i	−0.974702	−	0.223510i	$-0.928249\pi$
$263$		−	7.24943i		−	0.447019i	0.974702	−	0.223510i	$-0.0717514\pi$
$264$		0			0
$265$			16.9508i			1.04128i
$266$		0			0
$267$	−2.64929			−0.162134
$268$		0			0
$269$	−29.5342			−1.80073			−0.900366	−	0.435133i	$-0.856701\pi$
$269$	−29.5342			−1.80073			−0.900366	+	0.435133i	$0.856701\pi$
$270$		0			0
$271$	−9.81750			−0.596371			−0.298185	−	0.954508i	$-0.596381\pi$
$271$	−9.81750			−0.596371			−0.298185	+	0.954508i	$0.596381\pi$
$272$		0			0
$273$			2.63950i			0.159750i
$274$		0			0
$275$			4.97105i			0.299765i
$276$		0			0
$277$	−9.02683			−0.542370			−0.271185	−	0.962527i	$-0.587415\pi$
$277$	−9.02683			−0.542370			−0.271185	+	0.962527i	$0.587415\pi$
$278$		0			0
$279$	16.0624			0.961633
$280$		0			0
$281$		−	20.2704i		−	1.20923i	−0.796518	−	0.604615i	$-0.793327\pi$
$281$		−	20.2704i		−	1.20923i	0.796518	−	0.604615i	$-0.206673\pi$
$282$		0			0
$283$	21.2024			1.26035			0.630176	−	0.776452i	$-0.282983\pi$
$283$	21.2024			1.26035			0.630176	+	0.776452i	$0.282983\pi$
$284$		0			0
$285$	2.91555			0.172702
$286$		0			0
$287$	1.51433	−	6.22148i	0.0893878	−	0.367242i
$288$		0			0
$289$	−11.0968			−0.652750
$290$		0			0
$291$	7.11098			0.416853
$292$		0			0
$293$			19.6438i			1.14761i	0.818994	+	0.573803i	$0.194532\pi$
$293$			19.6438i			1.14761i	−0.818994	+	0.573803i	$0.805468\pi$
$294$		0			0
$295$	17.3710			1.01138
$296$		0			0
$297$	−25.8868			−1.50210
$298$		0			0
$299$			14.1835i			0.820253i
$300$		0			0
$301$		−	5.96616i		−	0.343884i
$302$		0			0
$303$	0.0755177			0.00433838
$304$		0			0
$305$	2.70452			0.154860
$306$		0			0
$307$	−24.9629			−1.42471			−0.712353	−	0.701822i	$-0.752370\pi$
$307$	−24.9629			−1.42471			−0.712353	+	0.701822i	$0.752370\pi$
$308$		0			0
$309$			1.71536i			0.0975833i
$310$		0			0
$311$			19.5023i			1.10587i	0.833223	+	0.552937i	$0.186493\pi$
$311$			19.5023i			1.10587i	−0.833223	+	0.552937i	$0.813507\pi$
$312$		0			0
$313$		−	23.5846i		−	1.33308i	−0.745470	−	0.666539i	$-0.767775\pi$
$313$		−	23.5846i		−	1.33308i	0.745470	−	0.666539i	$-0.232225\pi$
$314$		0			0
$315$			4.70754i			0.265240i
$316$		0			0
$317$		−	23.2861i		−	1.30788i	−0.756547	−	0.653940i	$-0.773115\pi$
$317$		−	23.2861i		−	1.30788i	0.756547	−	0.653940i	$-0.226885\pi$
$318$		0			0
$319$	44.0651			2.46717
$320$		0			0
$321$			13.8454i			0.772777i
$322$		0			0
$323$	6.09487			0.339127
$324$		0			0
$325$		−	2.49629i		−	0.138469i
$326$		0			0
$327$	−2.39208			−0.132282
$328$		0			0
$329$	−4.18168			−0.230543
$330$		0			0
$331$		−	6.30854i		−	0.346749i	−0.984856	−	0.173374i	$-0.944533\pi$
$331$		−	6.30854i		−	0.346749i	0.984856	−	0.173374i	$-0.0554671\pi$
$332$		0			0
$333$	21.2681			1.16548
$334$		0			0
$335$			5.95494i			0.325353i
$336$		0			0
$337$	21.0885			1.14876			0.574381	−	0.818588i	$-0.305243\pi$
$337$	21.0885			1.14876			0.574381	+	0.818588i	$0.305243\pi$
$338$		0			0
$339$			13.0120i			0.706715i
$340$		0			0
$341$			42.3012i			2.29074i
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$		−	14.1273i		−	0.760587i
$346$		0			0
$347$			24.8209i			1.33245i	0.745749	+	0.666227i	$0.232092\pi$
$347$			24.8209i			1.33245i	−0.745749	+	0.666227i	$0.767908\pi$
$348$		0			0
$349$	−8.51924			−0.456025			−0.228012	−	0.973658i	$-0.573223\pi$
$349$	−8.51924			−0.456025			−0.228012	+	0.973658i	$0.573223\pi$
$350$		0			0
$351$	12.9994			0.693858
$352$		0			0
$353$	−26.3620			−1.40311			−0.701554	−	0.712617i	$-0.747510\pi$
$353$	−26.3620			−1.40311			−0.701554	+	0.712617i	$0.747510\pi$
$354$		0			0
$355$		−	16.6750i		−	0.885015i
$356$		0			0
$357$		−	5.49592i		−	0.290875i
$358$		0			0
$359$	10.2119			0.538963			0.269481	−	0.963006i	$-0.413148\pi$
$359$	10.2119			0.538963			0.269481	+	0.963006i	$0.413148\pi$
$360$		0			0
$361$	17.6779			0.930415
$362$		0			0
$363$		−	15.2411i		−	0.799951i
$364$		0			0
$365$	−13.8649			−0.725724
$366$		0			0
$367$	15.9741			0.833842			0.416921	−	0.908943i	$-0.363109\pi$
$367$	15.9741			0.833842			0.416921	+	0.908943i	$0.363109\pi$
$368$		0			0
$369$	−11.9761	−	2.91502i	−0.623451	−	0.151750i
$370$		0			0
$371$	−6.93136			−0.359858
$372$		0			0
$373$	−29.2093			−1.51240			−0.756200	−	0.654340i	$-0.772946\pi$
$373$	−29.2093			−1.51240			−0.756200	+	0.654340i	$0.772946\pi$
$374$		0			0
$375$		−	10.1917i		−	0.526297i
$376$		0			0
$377$	−22.1279			−1.13965
$378$		0			0
$379$	20.6013			1.05822			0.529108	−	0.848554i	$-0.322526\pi$
$379$	20.6013			1.05822			0.529108	+	0.848554i	$0.322526\pi$
$380$		0			0
$381$			5.94200i			0.304418i
$382$		0			0
$383$			9.48638i			0.484731i	0.970185	+	0.242366i	$0.0779233\pi$
$383$			9.48638i			0.484731i	−0.970185	+	0.242366i	$0.922077\pi$
$384$		0			0
$385$	−12.3975			−0.631836
$386$		0			0
$387$	−11.4846			−0.583797
$388$		0			0
$389$	9.88002			0.500937			0.250468	−	0.968125i	$-0.419415\pi$
$389$	9.88002			0.500937			0.250468	+	0.968125i	$0.419415\pi$
$390$		0			0
$391$		−	29.5326i		−	1.49353i
$392$		0			0
$393$		−	4.67947i		−	0.236048i
$394$		0			0
$395$			10.1157i			0.508978i
$396$		0			0
$397$			35.1575i			1.76450i	0.470778	+	0.882251i	$0.343973\pi$
$397$			35.1575i			1.76450i	−0.470778	+	0.882251i	$0.656027\pi$
$398$		0			0
$399$			1.19220i			0.0596845i
$400$		0			0
$401$	−29.9291			−1.49459			−0.747295	−	0.664493i	$-0.768648\pi$
$401$	−29.9291			−1.49459			−0.747295	+	0.664493i	$0.768648\pi$
$402$		0			0
$403$		−	21.2422i		−	1.05815i
$404$		0			0
$405$	1.17472			0.0583725
$406$		0			0
$407$			56.0105i			2.77634i
$408$		0			0
$409$	−24.6460			−1.21866			−0.609332	−	0.792915i	$-0.708562\pi$
$409$	−24.6460			−1.21866			−0.609332	+	0.792915i	$0.708562\pi$
$410$		0			0
$411$	−3.34448			−0.164971
$412$		0			0
$413$			7.10317i			0.349524i
$414$		0			0
$415$	−30.6808			−1.50606
$416$		0			0
$417$			4.65246i			0.227832i
$418$		0			0
$419$	1.57574			0.0769801			0.0384900	−	0.999259i	$-0.487745\pi$
$419$	1.57574			0.0769801			0.0384900	+	0.999259i	$0.487745\pi$
$420$		0			0
$421$		−	3.89300i		−	0.189733i	−0.995490	−	0.0948665i	$-0.969758\pi$
$421$		−	3.89300i		−	0.189733i	0.995490	−	0.0948665i	$-0.0302424\pi$
$422$		0			0
$423$			8.04957i			0.391383i
$424$		0			0
$425$			5.19772i			0.252127i
$426$		0			0
$427$			1.10590i			0.0535185i
$428$		0			0
$429$			13.3809i			0.646034i
$430$		0			0
$431$	−36.5130			−1.75877			−0.879385	−	0.476112i	$-0.842046\pi$
$431$	−36.5130			−1.75877			−0.879385	+	0.476112i	$0.842046\pi$
$432$		0			0
$433$	4.87501			0.234278			0.117139	−	0.993116i	$-0.462628\pi$
$433$	4.87501			0.234278			0.117139	+	0.993116i	$0.462628\pi$
$434$		0			0
$435$	22.0402			1.05675
$436$		0			0
$437$			6.40635i			0.306457i
$438$		0			0
$439$			26.1080i			1.24607i	0.782195	+	0.623033i	$0.214100\pi$
$439$			26.1080i			1.24607i	−0.782195	+	0.623033i	$0.785900\pi$
$440$		0			0
$441$	−1.92496			−0.0916648
$442$		0			0
$443$	14.6787			0.697405			0.348702	−	0.937233i	$-0.386622\pi$
$443$	14.6787			0.697405			0.348702	+	0.937233i	$0.386622\pi$
$444$		0			0
$445$			6.24868i			0.296216i
$446$		0			0
$447$	−1.85046			−0.0875236
$448$		0			0
$449$	35.7567			1.68746			0.843732	−	0.536764i	$-0.180354\pi$
$449$	35.7567			1.68746			0.843732	+	0.536764i	$0.180354\pi$
$450$		0			0
$451$	7.67684	−	31.5396i	0.361488	−	1.48514i
$452$		0			0
$453$	9.23177			0.433746
$454$		0			0
$455$	6.22560			0.291861
$456$		0			0
$457$		−	19.2988i		−	0.902761i	−0.892332	−	0.451380i	$-0.850932\pi$
$457$		−	19.2988i		−	0.902761i	0.892332	−	0.451380i	$-0.149068\pi$
$458$		0			0
$459$	−27.0672			−1.26339
$460$		0			0
$461$	13.1664			0.613219			0.306609	−	0.951835i	$-0.400805\pi$
$461$	13.1664			0.613219			0.306609	+	0.951835i	$0.400805\pi$
$462$		0			0
$463$		−	5.48571i		−	0.254942i	−0.991842	−	0.127471i	$-0.959314\pi$
$463$		−	5.48571i		−	0.254942i	0.991842	−	0.127471i	$-0.0406860\pi$
$464$		0			0
$465$			21.1579i			0.981176i
$466$		0			0
$467$	0.0926038			0.00428519			0.00214260	−	0.999998i	$-0.499318\pi$
$467$	0.0926038			0.00428519			0.00214260	+	0.999998i	$0.499318\pi$
$468$		0			0
$469$	−2.43504			−0.112440
$470$		0			0
$471$	−0.213549			−0.00983980
$472$		0			0
$473$		−	30.2453i		−	1.39068i
$474$		0			0
$475$		−	1.12751i		−	0.0517338i
$476$		0			0
$477$			13.3426i			0.610916i
$478$		0			0
$479$		−	18.5595i		−	0.848007i	−0.905661	−	0.424003i	$-0.860624\pi$
$479$		−	18.5595i		−	0.848007i	0.905661	−	0.424003i	$-0.139376\pi$
$480$		0			0
$481$		−	28.1265i		−	1.28246i
$482$		0			0
$483$	5.77679			0.262853
$484$		0			0
$485$		−	16.7722i		−	0.761585i
$486$		0			0
$487$	−11.0124			−0.499021			−0.249510	−	0.968372i	$-0.580270\pi$
$487$	−11.0124			−0.499021			−0.249510	+	0.968372i	$0.580270\pi$
$488$		0			0
$489$		−	1.04652i		−	0.0473253i
$490$		0			0
$491$	27.9332			1.26061			0.630304	−	0.776349i	$-0.282930\pi$
$491$	27.9332			1.26061			0.630304	+	0.776349i	$0.282930\pi$
$492$		0			0
$493$	46.0744			2.07509
$494$		0			0
$495$			23.8647i			1.07264i
$496$		0			0
$497$	6.81857			0.305855
$498$		0			0
$499$		−	35.9775i		−	1.61057i	−0.592885	−	0.805287i	$-0.702011\pi$
$499$		−	35.9775i		−	1.61057i	0.592885	−	0.805287i	$-0.297989\pi$
$500$		0			0
$501$	23.6245			1.05546
$502$		0			0
$503$		−	16.3821i		−	0.730442i	−0.930921	−	0.365221i	$-0.880993\pi$
$503$		−	16.3821i		−	0.730442i	0.930921	−	0.365221i	$-0.119007\pi$
$504$		0			0
$505$		−	0.178118i		−	0.00792616i
$506$		0			0
$507$			6.75953i			0.300201i
$508$		0			0
$509$			29.7528i			1.31877i	0.751805	+	0.659385i	$0.229183\pi$
$509$			29.7528i			1.31877i	−0.751805	+	0.659385i	$0.770817\pi$
$510$		0			0
$511$		−	5.66952i		−	0.250805i
$512$		0			0
$513$	5.87153			0.259234
$514$		0			0
$515$	4.04590			0.178283
$516$		0			0
$517$	−21.1989			−0.932327
$518$		0			0
$519$		−	2.74363i		−	0.120432i
$520$		0			0
$521$		−	42.2378i		−	1.85047i	−0.379394	−	0.925235i	$-0.623867\pi$
$521$		−	42.2378i		−	1.85047i	0.379394	−	0.925235i	$-0.376133\pi$
$522$		0			0
$523$	−43.9610			−1.92228			−0.961140	−	0.276062i	$-0.910971\pi$
$523$	−43.9610			−1.92228			−0.961140	+	0.276062i	$0.910971\pi$
$524$		0			0
$525$	−1.01671			−0.0443729
$526$		0			0
$527$			44.2301i			1.92669i
$528$		0			0
$529$	8.04193			0.349649
$530$		0			0
$531$	13.6733			0.593371
$532$		0			0
$533$	−3.85504	+	15.8381i	−0.166980	+	0.686024i
$534$		0			0
$535$	32.6562			1.41185
$536$		0			0
$537$	−12.5923			−0.543396
$538$		0			0
$539$		−	5.06947i		−	0.218358i
$540$		0			0
$541$	34.0851			1.46543			0.732717	−	0.680534i	$-0.238252\pi$
$541$	34.0851			1.46543			0.732717	+	0.680534i	$0.238252\pi$
$542$		0			0
$543$	21.2225			0.910745
$544$		0			0
$545$			5.64203i			0.241678i
$546$		0			0
$547$			37.2593i			1.59309i	0.604578	+	0.796546i	$0.293342\pi$
$547$			37.2593i			1.59309i	−0.604578	+	0.796546i	$0.706658\pi$
$548$		0			0
$549$	2.12882			0.0908559
$550$		0			0
$551$	−9.99465			−0.425787
$552$		0			0
$553$	−4.13643			−0.175899
$554$		0			0
$555$			28.0150i			1.18917i
$556$		0			0
$557$			41.4925i			1.75810i	0.476734	+	0.879048i	$0.341821\pi$
$557$			41.4925i			1.75810i	−0.476734	+	0.879048i	$0.658179\pi$
$558$		0			0
$559$			15.1881i			0.642390i
$560$		0			0
$561$		−	27.8614i		−	1.17631i
$562$		0			0
$563$			18.5817i			0.783127i	0.920151	+	0.391563i	$0.128066\pi$
$563$			18.5817i			0.783127i	−0.920151	+	0.391563i	$0.871934\pi$
$564$		0			0
$565$	30.6905			1.29116
$566$		0			0
$567$			0.480357i			0.0201731i
$568$		0			0
$569$	44.1911			1.85259			0.926293	−	0.376804i	$-0.122977\pi$
$569$	44.1911			1.85259			0.926293	+	0.376804i	$0.122977\pi$
$570$		0			0
$571$		−	35.8113i		−	1.49866i	−0.662199	−	0.749328i	$-0.730377\pi$
$571$		−	35.8113i		−	1.49866i	0.662199	−	0.749328i	$-0.269623\pi$
$572$		0			0
$573$	−14.0068			−0.585144
$574$		0			0
$575$	−5.46335			−0.227838
$576$		0			0
$577$		−	10.7811i		−	0.448821i	−0.974495	−	0.224411i	$-0.927954\pi$
$577$		−	10.7811i		−	0.448821i	0.974495	−	0.224411i	$-0.0720457\pi$
$578$		0			0
$579$	14.6990			0.610870
$580$		0			0
$581$		−	12.5457i		−	0.520484i
$582$		0			0
$583$	−35.1384			−1.45528
$584$		0			0
$585$		−	11.9840i		−	0.495479i
$586$		0			0
$587$			20.6643i			0.852909i	0.904509	+	0.426454i	$0.140238\pi$
$587$			20.6643i			0.852909i	−0.904509	+	0.426454i	$0.859762\pi$
$588$		0			0
$589$		−	9.59457i		−	0.395338i
$590$		0			0
$591$			22.8422i			0.939603i
$592$		0			0
$593$			5.15963i			0.211881i	0.994372	+	0.105940i	$0.0337852\pi$
$593$			5.15963i			0.211881i	−0.994372	+	0.105940i	$0.966215\pi$
$594$		0			0
$595$	−12.9628			−0.531424
$596$		0			0
$597$	15.4686			0.633090
$598$		0			0
$599$	24.0629			0.983183			0.491591	−	0.870826i	$-0.336415\pi$
$599$	24.0629			0.983183			0.491591	+	0.870826i	$0.336415\pi$
$600$		0			0
$601$			30.4468i			1.24195i	0.783830	+	0.620976i	$0.213263\pi$
$601$			30.4468i			1.24195i	−0.783830	+	0.620976i	$0.786737\pi$
$602$		0			0
$603$			4.68735i			0.190884i
$604$		0			0
$605$	−35.9481			−1.46150
$606$		0			0
$607$	−1.12398			−0.0456210			−0.0228105	−	0.999740i	$-0.507261\pi$
$607$	−1.12398			−0.0456210			−0.0228105	+	0.999740i	$0.507261\pi$
$608$		0			0
$609$			9.01247i			0.365204i
$610$		0			0
$611$	10.6454			0.430665
$612$		0			0
$613$	−18.0651			−0.729643			−0.364821	−	0.931077i	$-0.618870\pi$
$613$	−18.0651			−0.729643			−0.364821	+	0.931077i	$0.618870\pi$
$614$		0			0
$615$	3.83975	−	15.7753i	0.154834	−	0.636122i
$616$		0			0
$617$	0.700580			0.0282043			0.0141021	−	0.999901i	$-0.495511\pi$
$617$	0.700580			0.0282043			0.0141021	+	0.999901i	$0.495511\pi$
$618$		0			0
$619$	−3.83466			−0.154128			−0.0770640	−	0.997026i	$-0.524555\pi$
$619$	−3.83466			−0.154128			−0.0770640	+	0.997026i	$0.524555\pi$
$620$		0			0
$621$		−	28.4505i		−	1.14168i
$622$		0			0
$623$	−2.55515			−0.102370
$624$		0			0
$625$	−28.9414			−1.15766
$626$		0			0
$627$			6.04381i			0.241367i
$628$		0			0
$629$			58.5645i			2.33512i
$630$		0			0
$631$	−43.8159			−1.74429			−0.872143	−	0.489252i	$-0.837270\pi$
$631$	−43.8159			−1.74429			−0.872143	+	0.489252i	$0.837270\pi$
$632$		0			0
$633$	25.5067			1.01380
$634$		0			0
$635$	14.0150			0.556167
$636$		0			0
$637$			2.54571i			0.100865i
$638$		0			0
$639$		−	13.1255i		−	0.519236i
$640$		0			0
$641$			26.6857i			1.05402i	0.849858	+	0.527012i	$0.176688\pi$
$641$			26.6857i			1.05402i	−0.849858	+	0.527012i	$0.823312\pi$
$642$		0			0
$643$			10.3703i			0.408964i	0.978870	+	0.204482i	$0.0655510\pi$
$643$			10.3703i			0.408964i	−0.978870	+	0.204482i	$0.934449\pi$
$644$		0			0
$645$		−	15.1279i		−	0.595661i
$646$		0			0
$647$	−14.2927			−0.561903			−0.280951	−	0.959722i	$-0.590650\pi$
$647$	−14.2927			−0.561903			−0.280951	+	0.959722i	$0.590650\pi$
$648$		0			0
$649$			36.0093i			1.41349i
$650$		0			0
$651$	−8.65171			−0.339087
$652$		0			0
$653$			39.3548i			1.54007i	0.638000	+	0.770037i	$0.279762\pi$
$653$			39.3548i			1.54007i	−0.638000	+	0.770037i	$0.720238\pi$
$654$		0			0
$655$	−11.0371			−0.431257
$656$		0			0
$657$	−10.9136			−0.425780
$658$		0			0
$659$		−	15.6534i		−	0.609768i	−0.952389	−	0.304884i	$-0.901382\pi$
$659$		−	15.6534i		−	0.609768i	0.952389	−	0.304884i	$-0.0986178\pi$
$660$		0			0
$661$	−7.42227			−0.288693			−0.144347	−	0.989527i	$-0.546108\pi$
$661$	−7.42227			−0.288693			−0.144347	+	0.989527i	$0.546108\pi$
$662$		0			0
$663$			13.9910i			0.543366i
$664$		0			0
$665$	2.81195			0.109043
$666$		0			0
$667$			48.4291i			1.87518i
$668$		0			0
$669$			2.73436i			0.105716i
$670$		0			0
$671$			5.60635i			0.216431i
$672$		0			0
$673$			38.0277i			1.46586i	0.680304	+	0.732930i	$0.261848\pi$
$673$			38.0277i			1.46586i	−0.680304	+	0.732930i	$0.738152\pi$
$674$		0			0
$675$			5.00726i			0.192729i
$676$		0			0
$677$	−16.2475			−0.624444			−0.312222	−	0.950009i	$-0.601073\pi$
$677$	−16.2475			−0.624444			−0.312222	+	0.950009i	$0.601073\pi$
$678$		0			0
$679$	6.85832			0.263198
$680$		0			0
$681$	14.3781			0.550972
$682$		0			0
$683$			21.4141i			0.819387i	0.912223	+	0.409694i	$0.134364\pi$
$683$			21.4141i			0.819387i	−0.912223	+	0.409694i	$0.865636\pi$
$684$		0			0
$685$			7.88839i			0.301400i
$686$		0			0
$687$	19.0187			0.725609
$688$		0			0
$689$	17.6452			0.672230
$690$		0			0
$691$			7.95535i			0.302636i	0.988485	+	0.151318i	$0.0483517\pi$
$691$			7.95535i			0.302636i	−0.988485	+	0.151318i	$0.951648\pi$
$692$		0			0
$693$	−9.75854			−0.370696
$694$		0			0
$695$	10.9734			0.416246
$696$		0			0
$697$	8.02689	−	32.9778i	0.304040	−	1.24912i
$698$		0			0
$699$	−4.46158			−0.168752
$700$		0			0
$701$	31.1962			1.17826			0.589132	−	0.808037i	$-0.299470\pi$
$701$	31.1962			1.17826			0.589132	+	0.808037i	$0.299470\pi$
$702$		0			0
$703$		−	12.7041i		−	0.479143i
$704$		0			0
$705$	−10.6031			−0.399338
$706$		0			0
$707$	0.0728344			0.00273922
$708$		0			0
$709$			49.0697i			1.84285i	0.388554	+	0.921426i	$0.372975\pi$
$709$			49.0697i			1.84285i	−0.388554	+	0.921426i	$0.627025\pi$
$710$		0			0
$711$			7.96247i			0.298616i
$712$		0			0
$713$	−46.4905			−1.74108
$714$		0			0
$715$	31.5605			1.18030
$716$		0			0
$717$	19.3768			0.723640
$718$		0			0
$719$		−	33.7020i		−	1.25687i	−0.777861	−	0.628436i	$-0.783695\pi$
$719$		−	33.7020i		−	1.25687i	0.777861	−	0.628436i	$-0.216305\pi$
$720$		0			0
$721$			1.65441i			0.0616134i
$722$		0			0
$723$			13.3950i			0.498165i
$724$		0			0
$725$		−	8.52347i		−	0.316554i
$726$		0			0
$727$		−	39.1952i		−	1.45367i	−0.686812	−	0.726835i	$-0.740990\pi$
$727$		−	39.1952i		−	1.45367i	0.686812	−	0.726835i	$-0.259010\pi$
$728$		0			0
$729$	−14.9589			−0.554034
$730$		0			0
$731$		−	31.6245i		−	1.16967i
$732$		0			0
$733$	37.2355			1.37533			0.687663	−	0.726030i	$-0.258637\pi$
$733$	37.2355			1.37533			0.687663	+	0.726030i	$0.258637\pi$
$734$		0			0
$735$		−	2.53562i		−	0.0935277i
$736$		0			0
$737$	−12.3444			−0.454710
$738$		0			0
$739$	−27.9374			−1.02769			−0.513846	−	0.857882i	$-0.671780\pi$
$739$	−27.9374			−1.02769			−0.513846	+	0.857882i	$0.671780\pi$
$740$		0			0
$741$		−	3.03499i		−	0.111493i
$742$		0			0
$743$	−11.8458			−0.434582			−0.217291	−	0.976107i	$-0.569722\pi$
$743$	−11.8458			−0.434582			−0.217291	+	0.976107i	$0.569722\pi$
$744$		0			0
$745$			4.36454i			0.159904i
$746$		0			0
$747$	−24.1500			−0.883603
$748$		0			0
$749$			13.3535i			0.487925i
$750$		0			0
$751$		−	44.7271i		−	1.63211i	−0.577971	−	0.816057i	$-0.696155\pi$
$751$		−	44.7271i		−	1.63211i	0.577971	−	0.816057i	$-0.303845\pi$
$752$		0			0
$753$		−	25.2722i		−	0.920972i
$754$		0			0
$755$		−	21.7743i		−	0.792449i
$756$		0			0
$757$		−	1.18991i		−	0.0432481i	−0.999766	−	0.0216241i	$-0.993116\pi$
$757$		−	1.18991i		−	0.0432481i	0.999766	−	0.0216241i	$-0.00688369\pi$
$758$		0			0
$759$	29.2853			1.06299
$760$		0			0
$761$	14.8342			0.537741			0.268870	−	0.963176i	$-0.413350\pi$
$761$	14.8342			0.537741			0.268870	+	0.963176i	$0.413350\pi$
$762$		0			0
$763$	−2.30709			−0.0835221
$764$		0			0
$765$			24.9529i			0.902176i
$766$		0			0
$767$		−	18.0826i		−	0.652925i
$768$		0			0
$769$	9.17776			0.330959			0.165479	−	0.986213i	$-0.447083\pi$
$769$	9.17776			0.330959			0.165479	+	0.986213i	$0.447083\pi$
$770$		0			0
$771$	14.7082			0.529702
$772$		0			0
$773$			13.7536i			0.494681i	0.968929	+	0.247341i	$0.0795567\pi$
$773$			13.7536i			0.494681i	−0.968929	+	0.247341i	$0.920443\pi$
$774$		0			0
$775$	8.18229			0.293916
$776$		0			0
$777$	−11.4556			−0.410968
$778$		0			0
$779$	−1.74123	+	7.15369i	−0.0623859	+	0.256307i
$780$		0			0
$781$	34.5666			1.23689
$782$		0			0
$783$	44.3861			1.58623
$784$		0			0
$785$			0.503682i			0.0179772i
$786$		0			0
$787$	−2.33979			−0.0834045			−0.0417022	−	0.999130i	$-0.513278\pi$
$787$	−2.33979			−0.0834045			−0.0417022	+	0.999130i	$0.513278\pi$
$788$		0			0
$789$	7.51651			0.267595
$790$		0			0
$791$			12.5497i			0.446215i
$792$		0			0
$793$		−	2.81531i		−	0.0999747i
$794$		0			0
$795$	−17.5753			−0.623331
$796$		0			0
$797$	37.4073			1.32504			0.662518	−	0.749046i	$-0.269488\pi$
$797$	37.4073			1.32504			0.662518	+	0.749046i	$0.269488\pi$
$798$		0			0
$799$	−22.1656			−0.784161
$800$		0			0
$801$			4.91857i			0.173789i
$802$		0			0
$803$		−	28.7415i		−	1.01426i
$804$		0			0
$805$		−	13.6253i		−	0.480229i
$806$		0			0
$807$		−	30.6223i		−	1.07796i
$808$		0			0
$809$			14.7688i			0.519245i	0.965710	+	0.259622i	$0.0835981\pi$
$809$			14.7688i			0.519245i	−0.965710	+	0.259622i	$0.916402\pi$
$810$		0			0
$811$	−11.6730			−0.409895			−0.204947	−	0.978773i	$-0.565702\pi$
$811$	−11.6730			−0.409895			−0.204947	+	0.978773i	$0.565702\pi$
$812$		0			0
$813$		−	10.1792i		−	0.357000i
$814$		0			0
$815$	−2.46835			−0.0864626
$816$		0			0
$817$			6.86011i			0.240005i
$818$		0			0
$819$	4.90040			0.171234
$820$		0			0
$821$	−0.676309			−0.0236033			−0.0118017	−	0.999930i	$-0.503757\pi$
$821$	−0.676309			−0.0236033			−0.0118017	+	0.999930i	$0.503757\pi$
$822$		0			0
$823$			16.2848i			0.567651i	0.958876	+	0.283825i	$0.0916036\pi$
$823$			16.2848i			0.567651i	−0.958876	+	0.283825i	$0.908396\pi$
$824$		0			0
$825$	−5.15419			−0.179446
$826$		0			0
$827$			50.7661i			1.76531i	0.470022	+	0.882655i	$0.344246\pi$
$827$			50.7661i			1.76531i	−0.470022	+	0.882655i	$0.655754\pi$
$828$		0			0
$829$	3.18187			0.110511			0.0552554	−	0.998472i	$-0.482403\pi$
$829$	3.18187			0.110511			0.0552554	+	0.998472i	$0.482403\pi$
$830$		0			0
$831$		−	9.35939i		−	0.324674i
$832$		0			0
$833$		−	5.30064i		−	0.183656i
$834$		0			0
$835$		−	55.7214i		−	1.92832i
$836$		0			0
$837$			42.6093i			1.47279i
$838$		0			0
$839$			25.2071i			0.870247i	0.900371	+	0.435123i	$0.143295\pi$
$839$			25.2071i			0.870247i	−0.900371	+	0.435123i	$0.856705\pi$
$840$		0			0
$841$	−46.5550			−1.60535
$842$		0			0
$843$	21.0172			0.723870
$844$		0			0
$845$	15.9432			0.548464
$846$		0			0
$847$		−	14.6996i		−	0.505083i
$848$		0			0
$849$			21.9835i			0.754472i
$850$		0			0
$851$	−61.5574			−2.11016
$852$		0			0
$853$	−18.2407			−0.624550			−0.312275	−	0.949992i	$-0.601091\pi$
$853$	−18.2407			−0.624550			−0.312275	+	0.949992i	$0.601091\pi$
$854$		0			0
$855$		−	5.41290i		−	0.185117i
$856$		0			0
$857$	10.6525			0.363884			0.181942	−	0.983309i	$-0.441762\pi$
$857$	10.6525			0.363884			0.181942	+	0.983309i	$0.441762\pi$
$858$		0			0
$859$	−50.9602			−1.73874			−0.869370	−	0.494162i	$-0.835475\pi$
$859$	−50.9602			−1.73874			−0.869370	+	0.494162i	$0.835475\pi$
$860$		0			0
$861$	6.45069	+	1.57012i	0.219839	+	0.0535094i
$862$		0			0
$863$	15.8880			0.540835			0.270417	−	0.962743i	$-0.412838\pi$
$863$	15.8880			0.540835			0.270417	+	0.962743i	$0.412838\pi$
$864$		0			0
$865$	−6.47122			−0.220028
$866$		0			0
$867$		−	11.5056i		−	0.390750i
$868$		0			0
$869$	−20.9695			−0.711343
$870$		0			0
$871$	6.19891			0.210042
$872$		0			0
$873$		−	13.2020i		−	0.446820i
$874$		0			0
$875$		−	9.82957i		−	0.332300i
$876$		0			0
$877$	−43.6344			−1.47343			−0.736714	−	0.676205i	$-0.763623\pi$
$877$	−43.6344			−1.47343			−0.736714	+	0.676205i	$0.763623\pi$
$878$		0			0
$879$	−20.3675			−0.686980
$880$		0			0
$881$	−0.143418			−0.00483188			−0.00241594	−	0.999997i	$-0.500769\pi$
$881$	−0.143418			−0.00483188			−0.00241594	+	0.999997i	$0.500769\pi$
$882$		0			0
$883$		−	20.3642i		−	0.685309i	−0.939461	−	0.342655i	$-0.888674\pi$
$883$		−	20.3642i		−	0.685309i	0.939461	−	0.342655i	$-0.111326\pi$
$884$		0			0
$885$			18.0109i			0.605431i
$886$		0			0
$887$		−	16.0145i		−	0.537715i	−0.963180	−	0.268858i	$-0.913354\pi$
$887$		−	16.0145i		−	0.537715i	0.963180	−	0.268858i	$-0.0866461\pi$
$888$		0			0
$889$			5.73087i			0.192207i
$890$		0			0
$891$			2.43516i			0.0815808i
$892$		0			0
$893$	4.80825			0.160902
$894$		0			0
$895$			29.7005i			0.992777i
$896$		0			0
$897$	−14.7060			−0.491020
$898$		0			0
$899$		−	72.5306i		−	2.41903i
$900$		0			0
$901$	−36.7406			−1.22401
$902$		0			0
$903$	6.18596			0.205856
$904$		0			0
$905$		−	50.0560i		−	1.66392i
$906$		0			0
$907$	−26.1467			−0.868186			−0.434093	−	0.900868i	$-0.642931\pi$
$907$	−26.1467			−0.868186			−0.434093	+	0.900868i	$0.642931\pi$
$908$		0			0
$909$		−	0.140203i		−	0.00465025i
$910$		0			0
$911$	−0.0933489			−0.00309279			−0.00154639	−	0.999999i	$-0.500492\pi$
$911$	−0.0933489			−0.00309279			−0.00154639	+	0.999999i	$0.500492\pi$
$912$		0			0
$913$		−	63.6002i		−	2.10486i
$914$		0			0
$915$			2.80415i			0.0927024i
$916$		0			0
$917$		−	4.51320i		−	0.149039i
$918$		0			0
$919$		−	20.4905i		−	0.675918i	−0.941161	−	0.337959i	$-0.890264\pi$
$919$		−	20.4905i		−	0.675918i	0.941161	−	0.337959i	$-0.109736\pi$
$920$		0			0
$921$		−	25.8825i		−	0.852858i
$922$		0			0
$923$	−17.3581			−0.571349
$924$		0			0
$925$	10.8341			0.356222
$926$		0			0
$927$	3.18467			0.104598
$928$		0			0
$929$		−	22.0654i		−	0.723941i	−0.932189	−	0.361971i	$-0.882104\pi$
$929$		−	22.0654i		−	0.723941i	0.932189	−	0.361971i	$-0.117896\pi$
$930$		0			0
$931$			1.14984i			0.0376844i
$932$		0			0
$933$	−20.2208			−0.661999
$934$		0			0
$935$	−65.7147			−2.14910
$936$		0			0
$937$		−	18.0375i		−	0.589258i	−0.955612	−	0.294629i	$-0.904804\pi$
$937$		−	18.0375i		−	0.589258i	0.955612	−	0.294629i	$-0.0951961\pi$
$938$		0			0
$939$	24.4534			0.798008
$940$		0			0
$941$	8.25835			0.269215			0.134607	−	0.990899i	$-0.457023\pi$
$941$	8.25835			0.269215			0.134607	+	0.990899i	$0.457023\pi$
$942$		0			0
$943$	34.6632	+	8.43711i	1.12879	+	0.274750i
$944$		0			0
$945$	−12.4878			−0.406229
$946$		0			0
$947$	12.1134			0.393634			0.196817	−	0.980440i	$-0.436939\pi$
$947$	12.1134			0.393634			0.196817	+	0.980440i	$0.436939\pi$
$948$		0			0
$949$			14.4330i			0.468514i
$950$		0			0
$951$	24.1440			0.782923
$952$		0			0
$953$	−22.1122			−0.716286			−0.358143	−	0.933667i	$-0.616590\pi$
$953$	−22.1122			−0.716286			−0.358143	+	0.933667i	$0.616590\pi$
$954$		0			0
$955$			33.0369i			1.06905i
$956$		0			0
$957$			45.6885i			1.47690i
$958$		0			0
$959$	−3.22565			−0.104162
$960$		0			0
$961$	38.6272			1.24604
$962$		0			0
$963$	25.7049			0.828330
$964$		0			0
$965$		−	34.6695i		−	1.11605i
$966$		0			0
$967$		−	38.4287i		−	1.23578i	−0.786263	−	0.617892i	$-0.787987\pi$
$967$		−	38.4287i		−	1.23578i	0.786263	−	0.617892i	$-0.212013\pi$
$968$		0			0
$969$			6.31941i			0.203009i
$970$		0			0
$971$		−	41.7036i		−	1.33833i	−0.743113	−	0.669166i	$-0.766651\pi$
$971$		−	41.7036i		−	1.33833i	0.743113	−	0.669166i	$-0.233349\pi$
$972$		0			0
$973$			4.48715i			0.143851i
$974$		0			0
$975$	2.58825			0.0828904
$976$		0			0
$977$		−	22.6514i		−	0.724683i	−0.932045	−	0.362342i	$-0.881977\pi$
$977$		−	22.6514i		−	0.724683i	0.932045	−	0.362342i	$-0.118023\pi$
$978$		0			0
$979$	−12.9533			−0.413989
$980$		0			0
$981$			4.44105i			0.141792i
$982$		0			0
$983$	−22.5106			−0.717975			−0.358988	−	0.933342i	$-0.616878\pi$
$983$	−22.5106			−0.717975			−0.358988	+	0.933342i	$0.616878\pi$
$984$		0			0
$985$	53.8763			1.71664
$986$		0			0
$987$		−	4.33574i		−	0.138008i
$988$		0			0
$989$	33.2407			1.05699
$990$		0			0
$991$		−	5.96932i		−	0.189622i	−0.995495	−	0.0948109i	$-0.969775\pi$
$991$		−	5.96932i		−	0.189622i	0.995495	−	0.0948109i	$-0.0302247\pi$
$992$		0			0
$993$	6.54095			0.207571
$994$		0			0
$995$		−	36.4848i		−	1.15665i
$996$		0			0
$997$			54.7157i			1.73286i	0.499296	+	0.866432i	$0.333592\pi$
$997$			54.7157i			1.73286i	−0.499296	+	0.866432i	$0.666408\pi$
$998$		0			0
$999$			56.4185i			1.78500i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1148.2.d.a.1065.13	yes	20
41.40	even	2	inner	1148.2.d.a.1065.8	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.d.a.1065.8	✓	20	41.40	even	2	inner
1148.2.d.a.1065.13	yes	20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.03684i q^{3} +2.44552 q^{5} +1.00000i q^{7} +1.92496 q^{9} +O(q^{10})\)	\(q+1.03684i q^{3} +2.44552 q^{5} +1.00000i q^{7} +1.92496 q^{9} +5.06947i q^{11} -2.54571i q^{13} +2.53562i q^{15} +5.30064i q^{17} -1.14984i q^{19} -1.03684 q^{21} -5.57153 q^{23} +0.980584 q^{25} +5.10640i q^{27} -8.69224i q^{29} +8.34429 q^{31} -5.25624 q^{33} +2.44552i q^{35} +11.0486 q^{37} +2.63950 q^{39} +(-6.22148 - 1.51433i) q^{41} -5.96616 q^{43} +4.70754 q^{45} +4.18168i q^{47} -1.00000 q^{49} -5.49592 q^{51} +6.93136i q^{53} +12.3975i q^{55} +1.19220 q^{57} +7.10317 q^{59} +1.10590 q^{61} +1.92496i q^{63} -6.22560i q^{65} +2.43504i q^{67} -5.77679i q^{69} -6.81857i q^{71} -5.66952 q^{73} +1.01671i q^{75} -5.06947 q^{77} +4.13643i q^{79} +0.480357 q^{81} -12.5457 q^{83} +12.9628i q^{85} +9.01247 q^{87} +2.55515i q^{89} +2.54571 q^{91} +8.65171i q^{93} -2.81195i q^{95} -6.85832i q^{97} +9.75854i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{5} - 20 q^{9}+O(q^{10}) \) 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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