LMFDB - Embedded newform 7350.2.a.dk.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7350,2,Mod(1,7350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7350, 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("7350.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7350.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:	$58.6900454856$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	7350.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +2.24264 q^{11} +1.00000 q^{12} -5.82843 q^{13} +1.00000 q^{16} -6.65685 q^{17} +1.00000 q^{18} -5.41421 q^{19} +2.24264 q^{22} +4.41421 q^{23} +1.00000 q^{24} -5.82843 q^{26} +1.00000 q^{27} +0.656854 q^{29} +1.24264 q^{31} +1.00000 q^{32} +2.24264 q^{33} -6.65685 q^{34} +1.00000 q^{36} -8.24264 q^{37} -5.41421 q^{38} -5.82843 q^{39} -7.00000 q^{41} -6.41421 q^{43} +2.24264 q^{44} +4.41421 q^{46} -8.00000 q^{47} +1.00000 q^{48} -6.65685 q^{51} -5.82843 q^{52} -8.65685 q^{53} +1.00000 q^{54} -5.41421 q^{57} +0.656854 q^{58} +4.41421 q^{59} +8.65685 q^{61} +1.24264 q^{62} +1.00000 q^{64} +2.24264 q^{66} -1.65685 q^{67} -6.65685 q^{68} +4.41421 q^{69} -2.00000 q^{71} +1.00000 q^{72} +0.242641 q^{73} -8.24264 q^{74} -5.41421 q^{76} -5.82843 q^{78} -2.24264 q^{79} +1.00000 q^{81} -7.00000 q^{82} +8.41421 q^{83} -6.41421 q^{86} +0.656854 q^{87} +2.24264 q^{88} +7.31371 q^{89} +4.41421 q^{92} +1.24264 q^{93} -8.00000 q^{94} +1.00000 q^{96} -14.0000 q^{97} +2.24264 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{8} + 2 q^{9} - 4 q^{11} + 2 q^{12} - 6 q^{13} + 2 q^{16} - 2 q^{17} + 2 q^{18} - 8 q^{19} - 4 q^{22} + 6 q^{23} + 2 q^{24} - 6 q^{26} + 2 q^{27} - 10 q^{29} - 6 q^{31} + 2 q^{32} - 4 q^{33} - 2 q^{34} + 2 q^{36} - 8 q^{37} - 8 q^{38} - 6 q^{39} - 14 q^{41} - 10 q^{43} - 4 q^{44} + 6 q^{46} - 16 q^{47} + 2 q^{48} - 2 q^{51} - 6 q^{52} - 6 q^{53} + 2 q^{54} - 8 q^{57} - 10 q^{58} + 6 q^{59} + 6 q^{61} - 6 q^{62} + 2 q^{64} - 4 q^{66} + 8 q^{67} - 2 q^{68} + 6 q^{69} - 4 q^{71} + 2 q^{72} - 8 q^{73} - 8 q^{74} - 8 q^{76} - 6 q^{78} + 4 q^{79} + 2 q^{81} - 14 q^{82} + 14 q^{83} - 10 q^{86} - 10 q^{87} - 4 q^{88} - 8 q^{89} + 6 q^{92} - 6 q^{93} - 16 q^{94} + 2 q^{96} - 28 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^8 + 2 * q^9 - 4 * q^11 + 2 * q^12 - 6 * q^13 + 2 * q^16 - 2 * q^17 + 2 * q^18 - 8 * q^19 - 4 * q^22 + 6 * q^23 + 2 * q^24 - 6 * q^26 + 2 * q^27 - 10 * q^29 - 6 * q^31 + 2 * q^32 - 4 * q^33 - 2 * q^34 + 2 * q^36 - 8 * q^37 - 8 * q^38 - 6 * q^39 - 14 * q^41 - 10 * q^43 - 4 * q^44 + 6 * q^46 - 16 * q^47 + 2 * q^48 - 2 * q^51 - 6 * q^52 - 6 * q^53 + 2 * q^54 - 8 * q^57 - 10 * q^58 + 6 * q^59 + 6 * q^61 - 6 * q^62 + 2 * q^64 - 4 * q^66 + 8 * q^67 - 2 * q^68 + 6 * q^69 - 4 * q^71 + 2 * q^72 - 8 * q^73 - 8 * q^74 - 8 * q^76 - 6 * q^78 + 4 * q^79 + 2 * q^81 - 14 * q^82 + 14 * q^83 - 10 * q^86 - 10 * q^87 - 4 * q^88 - 8 * q^89 + 6 * q^92 - 6 * q^93 - 16 * q^94 + 2 * q^96 - 28 * q^97 - 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	2.24264	0.676182	0.338091	−	0.941113i	$-0.390219\pi$
$11$	2.24264	0.676182	0.338091	+	0.941113i	$0.390219\pi$
$12$	1.00000	0.288675
$13$	−5.82843	−1.61651	−0.808257	−	0.588829i	$-0.799589\pi$
$13$	−5.82843	−1.61651	−0.808257	+	0.588829i	$0.799589\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.65685	−1.61452	−0.807262	−	0.590193i	$-0.799052\pi$
$17$	−6.65685	−1.61452	−0.807262	+	0.590193i	$0.799052\pi$
$18$	1.00000	0.235702
$19$	−5.41421	−1.24211	−0.621053	−	0.783769i	$-0.713295\pi$
$19$	−5.41421	−1.24211	−0.621053	+	0.783769i	$0.713295\pi$
$20$	0	0
$21$	0	0
$22$	2.24264	0.478133
$23$	4.41421	0.920427	0.460214	−	0.887808i	$-0.347773\pi$
$23$	4.41421	0.920427	0.460214	+	0.887808i	$0.347773\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−5.82843	−1.14305
$27$	1.00000	0.192450
$28$	0	0
$29$	0.656854	0.121975	0.0609874	−	0.998139i	$-0.480575\pi$
$29$	0.656854	0.121975	0.0609874	+	0.998139i	$0.480575\pi$
$30$	0	0
$31$	1.24264	0.223185	0.111592	−	0.993754i	$-0.464405\pi$
$31$	1.24264	0.223185	0.111592	+	0.993754i	$0.464405\pi$
$32$	1.00000	0.176777
$33$	2.24264	0.390394
$34$	−6.65685	−1.14164
$35$	0	0
$36$	1.00000	0.166667
$37$	−8.24264	−1.35508	−0.677541	−	0.735485i	$-0.736954\pi$
$37$	−8.24264	−1.35508	−0.677541	+	0.735485i	$0.736954\pi$
$38$	−5.41421	−0.878301
$39$	−5.82843	−0.933295
$40$	0	0
$41$	−7.00000	−1.09322	−0.546608	−	0.837389i	$-0.684081\pi$
$41$	−7.00000	−1.09322	−0.546608	+	0.837389i	$0.684081\pi$
$42$	0	0
$43$	−6.41421	−0.978158	−0.489079	−	0.872239i	$-0.662667\pi$
$43$	−6.41421	−0.978158	−0.489079	+	0.872239i	$0.662667\pi$
$44$	2.24264	0.338091
$45$	0	0
$46$	4.41421	0.650840
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	0	0
$51$	−6.65685	−0.932146
$52$	−5.82843	−0.808257
$53$	−8.65685	−1.18911	−0.594555	−	0.804055i	$-0.702672\pi$
$53$	−8.65685	−1.18911	−0.594555	+	0.804055i	$0.702672\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	0	0
$57$	−5.41421	−0.717130
$58$	0.656854	0.0862492
$59$	4.41421	0.574682	0.287341	−	0.957828i	$-0.407229\pi$
$59$	4.41421	0.574682	0.287341	+	0.957828i	$0.407229\pi$
$60$	0	0
$61$	8.65685	1.10840	0.554198	−	0.832385i	$-0.313025\pi$
$61$	8.65685	1.10840	0.554198	+	0.832385i	$0.313025\pi$
$62$	1.24264	0.157816
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	2.24264	0.276050
$67$	−1.65685	−0.202417	−0.101208	−	0.994865i	$-0.532271\pi$
$67$	−1.65685	−0.202417	−0.101208	+	0.994865i	$0.532271\pi$
$68$	−6.65685	−0.807262
$69$	4.41421	0.531409
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	1.00000	0.117851
$73$	0.242641	0.0283989	0.0141995	−	0.999899i	$-0.495480\pi$
$73$	0.242641	0.0283989	0.0141995	+	0.999899i	$0.495480\pi$
$74$	−8.24264	−0.958188
$75$	0	0
$76$	−5.41421	−0.621053
$77$	0	0
$78$	−5.82843	−0.659939
$79$	−2.24264	−0.252317	−0.126158	−	0.992010i	$-0.540265\pi$
$79$	−2.24264	−0.252317	−0.126158	+	0.992010i	$0.540265\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−7.00000	−0.773021
$83$	8.41421	0.923580	0.461790	−	0.886989i	$-0.347207\pi$
$83$	8.41421	0.923580	0.461790	+	0.886989i	$0.347207\pi$
$84$	0	0
$85$	0	0
$86$	−6.41421	−0.691662
$87$	0.656854	0.0704222
$88$	2.24264	0.239066
$89$	7.31371	0.775252	0.387626	−	0.921817i	$-0.373295\pi$
$89$	7.31371	0.775252	0.387626	+	0.921817i	$0.373295\pi$
$90$	0	0
$91$	0	0
$92$	4.41421	0.460214
$93$	1.24264	0.128856
$94$	−8.00000	−0.825137
$95$	0	0
$96$	1.00000	0.102062
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	2.24264	0.225394
$100$	0	0
$101$	6.34315	0.631167	0.315583	−	0.948898i	$-0.397800\pi$
$101$	6.34315	0.631167	0.315583	+	0.948898i	$0.397800\pi$
$102$	−6.65685	−0.659127
$103$	−4.41421	−0.434945	−0.217473	−	0.976066i	$-0.569781\pi$
$103$	−4.41421	−0.434945	−0.217473	+	0.976066i	$0.569781\pi$
$104$	−5.82843	−0.571524
$105$	0	0
$106$	−8.65685	−0.840828
$107$	16.1421	1.56052	0.780260	−	0.625456i	$-0.215087\pi$
$107$	16.1421	1.56052	0.780260	+	0.625456i	$0.215087\pi$
$108$	1.00000	0.0962250
$109$	−15.3137	−1.46679	−0.733394	−	0.679804i	$-0.762065\pi$
$109$	−15.3137	−1.46679	−0.733394	+	0.679804i	$0.762065\pi$
$110$	0	0
$111$	−8.24264	−0.782357
$112$	0	0
$113$	3.75736	0.353463	0.176731	−	0.984259i	$-0.443448\pi$
$113$	3.75736	0.353463	0.176731	+	0.984259i	$0.443448\pi$
$114$	−5.41421	−0.507088
$115$	0	0
$116$	0.656854	0.0609874
$117$	−5.82843	−0.538838
$118$	4.41421	0.406361
$119$	0	0
$120$	0	0
$121$	−5.97056	−0.542778
$122$	8.65685	0.783755
$123$	−7.00000	−0.631169
$124$	1.24264	0.111592
$125$	0	0
$126$	0	0
$127$	20.8284	1.84822	0.924112	−	0.382122i	$-0.124806\pi$
$127$	20.8284	1.84822	0.924112	+	0.382122i	$0.124806\pi$
$128$	1.00000	0.0883883
$129$	−6.41421	−0.564740
$130$	0	0
$131$	−2.82843	−0.247121	−0.123560	−	0.992337i	$-0.539431\pi$
$131$	−2.82843	−0.247121	−0.123560	+	0.992337i	$0.539431\pi$
$132$	2.24264	0.195197
$133$	0	0
$134$	−1.65685	−0.143130
$135$	0	0
$136$	−6.65685	−0.570821
$137$	−9.07107	−0.774994	−0.387497	−	0.921871i	$-0.626660\pi$
$137$	−9.07107	−0.774994	−0.387497	+	0.921871i	$0.626660\pi$
$138$	4.41421	0.375763
$139$	−9.41421	−0.798503	−0.399252	−	0.916841i	$-0.630730\pi$
$139$	−9.41421	−0.798503	−0.399252	+	0.916841i	$0.630730\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	−2.00000	−0.167836
$143$	−13.0711	−1.09306
$144$	1.00000	0.0833333
$145$	0	0
$146$	0.242641	0.0200811
$147$	0	0
$148$	−8.24264	−0.677541
$149$	−16.6569	−1.36458	−0.682291	−	0.731080i	$-0.739016\pi$
$149$	−16.6569	−1.36458	−0.682291	+	0.731080i	$0.739016\pi$
$150$	0	0
$151$	9.55635	0.777685	0.388842	−	0.921304i	$-0.372875\pi$
$151$	9.55635	0.777685	0.388842	+	0.921304i	$0.372875\pi$
$152$	−5.41421	−0.439151
$153$	−6.65685	−0.538175
$154$	0	0
$155$	0	0
$156$	−5.82843	−0.466648
$157$	−18.8284	−1.50267	−0.751336	−	0.659920i	$-0.770590\pi$
$157$	−18.8284	−1.50267	−0.751336	+	0.659920i	$0.770590\pi$
$158$	−2.24264	−0.178415
$159$	−8.65685	−0.686533
$160$	0	0
$161$	0	0
$162$	1.00000	0.0785674
$163$	10.8995	0.853714	0.426857	−	0.904319i	$-0.359621\pi$
$163$	10.8995	0.853714	0.426857	+	0.904319i	$0.359621\pi$
$164$	−7.00000	−0.546608
$165$	0	0
$166$	8.41421	0.653070
$167$	−1.51472	−0.117212	−0.0586062	−	0.998281i	$-0.518666\pi$
$167$	−1.51472	−0.117212	−0.0586062	+	0.998281i	$0.518666\pi$
$168$	0	0
$169$	20.9706	1.61312
$170$	0	0
$171$	−5.41421	−0.414035
$172$	−6.41421	−0.489079
$173$	21.5563	1.63890	0.819449	−	0.573151i	$-0.194279\pi$
$173$	21.5563	1.63890	0.819449	+	0.573151i	$0.194279\pi$
$174$	0.656854	0.0497960
$175$	0	0
$176$	2.24264	0.169045
$177$	4.41421	0.331793
$178$	7.31371	0.548186
$179$	−18.2426	−1.36352	−0.681759	−	0.731576i	$-0.738785\pi$
$179$	−18.2426	−1.36352	−0.681759	+	0.731576i	$0.738785\pi$
$180$	0	0
$181$	−12.9706	−0.964094	−0.482047	−	0.876145i	$-0.660107\pi$
$181$	−12.9706	−0.964094	−0.482047	+	0.876145i	$0.660107\pi$
$182$	0	0
$183$	8.65685	0.639933
$184$	4.41421	0.325420
$185$	0	0
$186$	1.24264	0.0911148
$187$	−14.9289	−1.09171
$188$	−8.00000	−0.583460
$189$	0	0
$190$	0	0
$191$	−17.7279	−1.28275	−0.641374	−	0.767229i	$-0.721635\pi$
$191$	−17.7279	−1.28275	−0.641374	+	0.767229i	$0.721635\pi$
$192$	1.00000	0.0721688
$193$	−23.7990	−1.71309	−0.856544	−	0.516073i	$-0.827393\pi$
$193$	−23.7990	−1.71309	−0.856544	+	0.516073i	$0.827393\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	0	0
$197$	17.1421	1.22133	0.610663	−	0.791890i	$-0.290903\pi$
$197$	17.1421	1.22133	0.610663	+	0.791890i	$0.290903\pi$
$198$	2.24264	0.159378
$199$	18.9706	1.34479	0.672394	−	0.740194i	$-0.265266\pi$
$199$	18.9706	1.34479	0.672394	+	0.740194i	$0.265266\pi$
$200$	0	0
$201$	−1.65685	−0.116865
$202$	6.34315	0.446302
$203$	0	0
$204$	−6.65685	−0.466073
$205$	0	0
$206$	−4.41421	−0.307553
$207$	4.41421	0.306809
$208$	−5.82843	−0.404129
$209$	−12.1421	−0.839889
$210$	0	0
$211$	28.0711	1.93249	0.966246	−	0.257621i	$-0.0829387\pi$
$211$	28.0711	1.93249	0.966246	+	0.257621i	$0.0829387\pi$
$212$	−8.65685	−0.594555
$213$	−2.00000	−0.137038
$214$	16.1421	1.10345
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	−15.3137	−1.03718
$219$	0.242641	0.0163961
$220$	0	0
$221$	38.7990	2.60990
$222$	−8.24264	−0.553210
$223$	3.92893	0.263101	0.131550	−	0.991309i	$-0.458004\pi$
$223$	3.92893	0.263101	0.131550	+	0.991309i	$0.458004\pi$
$224$	0	0
$225$	0	0
$226$	3.75736	0.249936
$227$	−20.7574	−1.37771	−0.688857	−	0.724897i	$-0.741887\pi$
$227$	−20.7574	−1.37771	−0.688857	+	0.724897i	$0.741887\pi$
$228$	−5.41421	−0.358565
$229$	−13.3137	−0.879795	−0.439897	−	0.898048i	$-0.644985\pi$
$229$	−13.3137	−0.879795	−0.439897	+	0.898048i	$0.644985\pi$
$230$	0	0
$231$	0	0
$232$	0.656854	0.0431246
$233$	10.7279	0.702810	0.351405	−	0.936224i	$-0.385704\pi$
$233$	10.7279	0.702810	0.351405	+	0.936224i	$0.385704\pi$
$234$	−5.82843	−0.381016
$235$	0	0
$236$	4.41421	0.287341
$237$	−2.24264	−0.145675
$238$	0	0
$239$	3.31371	0.214346	0.107173	−	0.994240i	$-0.465820\pi$
$239$	3.31371	0.214346	0.107173	+	0.994240i	$0.465820\pi$
$240$	0	0
$241$	8.34315	0.537429	0.268715	−	0.963220i	$-0.413401\pi$
$241$	8.34315	0.537429	0.268715	+	0.963220i	$0.413401\pi$
$242$	−5.97056	−0.383802
$243$	1.00000	0.0641500
$244$	8.65685	0.554198
$245$	0	0
$246$	−7.00000	−0.446304
$247$	31.5563	2.00788
$248$	1.24264	0.0789078
$249$	8.41421	0.533229
$250$	0	0
$251$	28.2132	1.78080	0.890401	−	0.455177i	$-0.150424\pi$
$251$	28.2132	1.78080	0.890401	+	0.455177i	$0.150424\pi$
$252$	0	0
$253$	9.89949	0.622376
$254$	20.8284	1.30689
$255$	0	0
$256$	1.00000	0.0625000
$257$	−27.8284	−1.73589	−0.867945	−	0.496661i	$-0.834559\pi$
$257$	−27.8284	−1.73589	−0.867945	+	0.496661i	$0.834559\pi$
$258$	−6.41421	−0.399331
$259$	0	0
$260$	0	0
$261$	0.656854	0.0406583
$262$	−2.82843	−0.174741
$263$	29.5858	1.82434	0.912169	−	0.409815i	$-0.134407\pi$
$263$	29.5858	1.82434	0.912169	+	0.409815i	$0.134407\pi$
$264$	2.24264	0.138025
$265$	0	0
$266$	0	0
$267$	7.31371	0.447592
$268$	−1.65685	−0.101208
$269$	1.41421	0.0862261	0.0431131	−	0.999070i	$-0.486272\pi$
$269$	1.41421	0.0862261	0.0431131	+	0.999070i	$0.486272\pi$
$270$	0	0
$271$	21.4558	1.30335	0.651675	−	0.758498i	$-0.274067\pi$
$271$	21.4558	1.30335	0.651675	+	0.758498i	$0.274067\pi$
$272$	−6.65685	−0.403631
$273$	0	0
$274$	−9.07107	−0.548003
$275$	0	0
$276$	4.41421	0.265704
$277$	−24.0416	−1.44452	−0.722261	−	0.691621i	$-0.756897\pi$
$277$	−24.0416	−1.44452	−0.722261	+	0.691621i	$0.756897\pi$
$278$	−9.41421	−0.564627
$279$	1.24264	0.0743950
$280$	0	0
$281$	−4.00000	−0.238620	−0.119310	−	0.992857i	$-0.538068\pi$
$281$	−4.00000	−0.238620	−0.119310	+	0.992857i	$0.538068\pi$
$282$	−8.00000	−0.476393
$283$	28.2426	1.67885	0.839425	−	0.543475i	$-0.182892\pi$
$283$	28.2426	1.67885	0.839425	+	0.543475i	$0.182892\pi$
$284$	−2.00000	−0.118678
$285$	0	0
$286$	−13.0711	−0.772908
$287$	0	0
$288$	1.00000	0.0589256
$289$	27.3137	1.60669
$290$	0	0
$291$	−14.0000	−0.820695
$292$	0.242641	0.0141995
$293$	19.3137	1.12832	0.564159	−	0.825666i	$-0.309200\pi$
$293$	19.3137	1.12832	0.564159	+	0.825666i	$0.309200\pi$
$294$	0	0
$295$	0	0
$296$	−8.24264	−0.479094
$297$	2.24264	0.130131
$298$	−16.6569	−0.964906
$299$	−25.7279	−1.48788
$300$	0	0
$301$	0	0
$302$	9.55635	0.549906
$303$	6.34315	0.364404
$304$	−5.41421	−0.310526
$305$	0	0
$306$	−6.65685	−0.380547
$307$	−8.58579	−0.490017	−0.245008	−	0.969521i	$-0.578791\pi$
$307$	−8.58579	−0.490017	−0.245008	+	0.969521i	$0.578791\pi$
$308$	0	0
$309$	−4.41421	−0.251116
$310$	0	0
$311$	13.5563	0.768710	0.384355	−	0.923185i	$-0.374424\pi$
$311$	13.5563	0.768710	0.384355	+	0.923185i	$0.374424\pi$
$312$	−5.82843	−0.329970
$313$	−25.2132	−1.42513	−0.712567	−	0.701604i	$-0.752468\pi$
$313$	−25.2132	−1.42513	−0.712567	+	0.701604i	$0.752468\pi$
$314$	−18.8284	−1.06255
$315$	0	0
$316$	−2.24264	−0.126158
$317$	−18.3137	−1.02860	−0.514300	−	0.857610i	$-0.671948\pi$
$317$	−18.3137	−1.02860	−0.514300	+	0.857610i	$0.671948\pi$
$318$	−8.65685	−0.485452
$319$	1.47309	0.0824771
$320$	0	0
$321$	16.1421	0.900966
$322$	0	0
$323$	36.0416	2.00541
$324$	1.00000	0.0555556
$325$	0	0
$326$	10.8995	0.603667
$327$	−15.3137	−0.846850
$328$	−7.00000	−0.386510
$329$	0	0
$330$	0	0
$331$	−10.4142	−0.572417	−0.286208	−	0.958167i	$-0.592395\pi$
$331$	−10.4142	−0.572417	−0.286208	+	0.958167i	$0.592395\pi$
$332$	8.41421	0.461790
$333$	−8.24264	−0.451694
$334$	−1.51472	−0.0828817
$335$	0	0
$336$	0	0
$337$	−25.8284	−1.40696	−0.703482	−	0.710713i	$-0.748372\pi$
$337$	−25.8284	−1.40696	−0.703482	+	0.710713i	$0.748372\pi$
$338$	20.9706	1.14065
$339$	3.75736	0.204072
$340$	0	0
$341$	2.78680	0.150913
$342$	−5.41421	−0.292767
$343$	0	0
$344$	−6.41421	−0.345831
$345$	0	0
$346$	21.5563	1.15888
$347$	−9.79899	−0.526037	−0.263019	−	0.964791i	$-0.584718\pi$
$347$	−9.79899	−0.526037	−0.263019	+	0.964791i	$0.584718\pi$
$348$	0.656854	0.0352111
$349$	4.51472	0.241667	0.120834	−	0.992673i	$-0.461443\pi$
$349$	4.51472	0.241667	0.120834	+	0.992673i	$0.461443\pi$
$350$	0	0
$351$	−5.82843	−0.311098
$352$	2.24264	0.119533
$353$	5.85786	0.311783	0.155891	−	0.987774i	$-0.450175\pi$
$353$	5.85786	0.311783	0.155891	+	0.987774i	$0.450175\pi$
$354$	4.41421	0.234613
$355$	0	0
$356$	7.31371	0.387626
$357$	0	0
$358$	−18.2426	−0.964154
$359$	1.72792	0.0911962	0.0455981	−	0.998960i	$-0.485481\pi$
$359$	1.72792	0.0911962	0.0455981	+	0.998960i	$0.485481\pi$
$360$	0	0
$361$	10.3137	0.542827
$362$	−12.9706	−0.681718
$363$	−5.97056	−0.313373
$364$	0	0
$365$	0	0
$366$	8.65685	0.452501
$367$	−16.5563	−0.864234	−0.432117	−	0.901817i	$-0.642233\pi$
$367$	−16.5563	−0.864234	−0.432117	+	0.901817i	$0.642233\pi$
$368$	4.41421	0.230107
$369$	−7.00000	−0.364405
$370$	0	0
$371$	0	0
$372$	1.24264	0.0644279
$373$	−1.65685	−0.0857887	−0.0428943	−	0.999080i	$-0.513658\pi$
$373$	−1.65685	−0.0857887	−0.0428943	+	0.999080i	$0.513658\pi$
$374$	−14.9289	−0.771957
$375$	0	0
$376$	−8.00000	−0.412568
$377$	−3.82843	−0.197174
$378$	0	0
$379$	31.7279	1.62975	0.814877	−	0.579634i	$-0.196804\pi$
$379$	31.7279	1.62975	0.814877	+	0.579634i	$0.196804\pi$
$380$	0	0
$381$	20.8284	1.06707
$382$	−17.7279	−0.907039
$383$	−19.0711	−0.974486	−0.487243	−	0.873266i	$-0.661997\pi$
$383$	−19.0711	−0.974486	−0.487243	+	0.873266i	$0.661997\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−23.7990	−1.21134
$387$	−6.41421	−0.326053
$388$	−14.0000	−0.710742
$389$	−36.2843	−1.83969	−0.919843	−	0.392287i	$-0.871684\pi$
$389$	−36.2843	−1.83969	−0.919843	+	0.392287i	$0.871684\pi$
$390$	0	0
$391$	−29.3848	−1.48605
$392$	0	0
$393$	−2.82843	−0.142675
$394$	17.1421	0.863608
$395$	0	0
$396$	2.24264	0.112697
$397$	28.1127	1.41094	0.705468	−	0.708742i	$-0.250737\pi$
$397$	28.1127	1.41094	0.705468	+	0.708742i	$0.250737\pi$
$398$	18.9706	0.950908
$399$	0	0
$400$	0	0
$401$	−23.6985	−1.18345	−0.591723	−	0.806141i	$-0.701552\pi$
$401$	−23.6985	−1.18345	−0.591723	+	0.806141i	$0.701552\pi$
$402$	−1.65685	−0.0826364
$403$	−7.24264	−0.360782
$404$	6.34315	0.315583
$405$	0	0
$406$	0	0
$407$	−18.4853	−0.916281
$408$	−6.65685	−0.329563
$409$	−9.21320	−0.455564	−0.227782	−	0.973712i	$-0.573147\pi$
$409$	−9.21320	−0.455564	−0.227782	+	0.973712i	$0.573147\pi$
$410$	0	0
$411$	−9.07107	−0.447443
$412$	−4.41421	−0.217473
$413$	0	0
$414$	4.41421	0.216947
$415$	0	0
$416$	−5.82843	−0.285762
$417$	−9.41421	−0.461016
$418$	−12.1421	−0.593891
$419$	13.7279	0.670653	0.335326	−	0.942102i	$-0.391153\pi$
$419$	13.7279	0.670653	0.335326	+	0.942102i	$0.391153\pi$
$420$	0	0
$421$	19.6569	0.958016	0.479008	−	0.877810i	$-0.340996\pi$
$421$	19.6569	0.958016	0.479008	+	0.877810i	$0.340996\pi$
$422$	28.0711	1.36648
$423$	−8.00000	−0.388973
$424$	−8.65685	−0.420414
$425$	0	0
$426$	−2.00000	−0.0969003
$427$	0	0
$428$	16.1421	0.780260
$429$	−13.0711	−0.631077
$430$	0	0
$431$	−26.6985	−1.28602	−0.643010	−	0.765857i	$-0.722315\pi$
$431$	−26.6985	−1.28602	−0.643010	+	0.765857i	$0.722315\pi$
$432$	1.00000	0.0481125
$433$	15.6569	0.752420	0.376210	−	0.926534i	$-0.377227\pi$
$433$	15.6569	0.752420	0.376210	+	0.926534i	$0.377227\pi$
$434$	0	0
$435$	0	0
$436$	−15.3137	−0.733394
$437$	−23.8995	−1.14327
$438$	0.242641	0.0115938
$439$	6.21320	0.296540	0.148270	−	0.988947i	$-0.452630\pi$
$439$	6.21320	0.296540	0.148270	+	0.988947i	$0.452630\pi$
$440$	0	0
$441$	0	0
$442$	38.7990	1.84548
$443$	−30.6274	−1.45515	−0.727576	−	0.686027i	$-0.759353\pi$
$443$	−30.6274	−1.45515	−0.727576	+	0.686027i	$0.759353\pi$
$444$	−8.24264	−0.391178
$445$	0	0
$446$	3.92893	0.186040
$447$	−16.6569	−0.787842
$448$	0	0
$449$	−1.27208	−0.0600331	−0.0300165	−	0.999549i	$-0.509556\pi$
$449$	−1.27208	−0.0600331	−0.0300165	+	0.999549i	$0.509556\pi$
$450$	0	0
$451$	−15.6985	−0.739213
$452$	3.75736	0.176731
$453$	9.55635	0.448996
$454$	−20.7574	−0.974191
$455$	0	0
$456$	−5.41421	−0.253544
$457$	40.4558	1.89244	0.946222	−	0.323517i	$-0.104865\pi$
$457$	40.4558	1.89244	0.946222	+	0.323517i	$0.104865\pi$
$458$	−13.3137	−0.622109
$459$	−6.65685	−0.310715
$460$	0	0
$461$	8.00000	0.372597	0.186299	−	0.982493i	$-0.440351\pi$
$461$	8.00000	0.372597	0.186299	+	0.982493i	$0.440351\pi$
$462$	0	0
$463$	−0.443651	−0.0206182	−0.0103091	−	0.999947i	$-0.503282\pi$
$463$	−0.443651	−0.0206182	−0.0103091	+	0.999947i	$0.503282\pi$
$464$	0.656854	0.0304937
$465$	0	0
$466$	10.7279	0.496961
$467$	−25.7279	−1.19055	−0.595273	−	0.803523i	$-0.702956\pi$
$467$	−25.7279	−1.19055	−0.595273	+	0.803523i	$0.702956\pi$
$468$	−5.82843	−0.269419
$469$	0	0
$470$	0	0
$471$	−18.8284	−0.867568
$472$	4.41421	0.203181
$473$	−14.3848	−0.661413
$474$	−2.24264	−0.103008
$475$	0	0
$476$	0	0
$477$	−8.65685	−0.396370
$478$	3.31371	0.151565
$479$	−24.7279	−1.12985	−0.564924	−	0.825143i	$-0.691094\pi$
$479$	−24.7279	−1.12985	−0.564924	+	0.825143i	$0.691094\pi$
$480$	0	0
$481$	48.0416	2.19051
$482$	8.34315	0.380020
$483$	0	0
$484$	−5.97056	−0.271389
$485$	0	0
$486$	1.00000	0.0453609
$487$	−28.1421	−1.27524	−0.637621	−	0.770350i	$-0.720081\pi$
$487$	−28.1421	−1.27524	−0.637621	+	0.770350i	$0.720081\pi$
$488$	8.65685	0.391877
$489$	10.8995	0.492892
$490$	0	0
$491$	4.82843	0.217904	0.108952	−	0.994047i	$-0.465251\pi$
$491$	4.82843	0.217904	0.108952	+	0.994047i	$0.465251\pi$
$492$	−7.00000	−0.315584
$493$	−4.37258	−0.196931
$494$	31.5563	1.41979
$495$	0	0
$496$	1.24264	0.0557962
$497$	0	0
$498$	8.41421	0.377050
$499$	5.92893	0.265415	0.132708	−	0.991155i	$-0.457633\pi$
$499$	5.92893	0.265415	0.132708	+	0.991155i	$0.457633\pi$
$500$	0	0
$501$	−1.51472	−0.0676726
$502$	28.2132	1.25922
$503$	0.384776	0.0171563	0.00857816	−	0.999963i	$-0.497269\pi$
$503$	0.384776	0.0171563	0.00857816	+	0.999963i	$0.497269\pi$
$504$	0	0
$505$	0	0
$506$	9.89949	0.440086
$507$	20.9706	0.931335
$508$	20.8284	0.924112
$509$	−6.72792	−0.298210	−0.149105	−	0.988821i	$-0.547639\pi$
$509$	−6.72792	−0.298210	−0.149105	+	0.988821i	$0.547639\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	−5.41421	−0.239043
$514$	−27.8284	−1.22746
$515$	0	0
$516$	−6.41421	−0.282370
$517$	−17.9411	−0.789050
$518$	0	0
$519$	21.5563	0.946219
$520$	0	0
$521$	−3.82843	−0.167726	−0.0838632	−	0.996477i	$-0.526726\pi$
$521$	−3.82843	−0.167726	−0.0838632	+	0.996477i	$0.526726\pi$
$522$	0.656854	0.0287497
$523$	19.3137	0.844530	0.422265	−	0.906473i	$-0.361235\pi$
$523$	19.3137	0.844530	0.422265	+	0.906473i	$0.361235\pi$
$524$	−2.82843	−0.123560
$525$	0	0
$526$	29.5858	1.29000
$527$	−8.27208	−0.360337
$528$	2.24264	0.0975984
$529$	−3.51472	−0.152814
$530$	0	0
$531$	4.41421	0.191561
$532$	0	0
$533$	40.7990	1.76720
$534$	7.31371	0.316495
$535$	0	0
$536$	−1.65685	−0.0715652
$537$	−18.2426	−0.787228
$538$	1.41421	0.0609711
$539$	0	0
$540$	0	0
$541$	33.0711	1.42184	0.710918	−	0.703275i	$-0.248280\pi$
$541$	33.0711	1.42184	0.710918	+	0.703275i	$0.248280\pi$
$542$	21.4558	0.921607
$543$	−12.9706	−0.556620
$544$	−6.65685	−0.285410
$545$	0	0
$546$	0	0
$547$	29.5858	1.26500	0.632498	−	0.774562i	$-0.282029\pi$
$547$	29.5858	1.26500	0.632498	+	0.774562i	$0.282029\pi$
$548$	−9.07107	−0.387497
$549$	8.65685	0.369466
$550$	0	0
$551$	−3.55635	−0.151506
$552$	4.41421	0.187881
$553$	0	0
$554$	−24.0416	−1.02143
$555$	0	0
$556$	−9.41421	−0.399252
$557$	−16.0000	−0.677942	−0.338971	−	0.940797i	$-0.610079\pi$
$557$	−16.0000	−0.677942	−0.338971	+	0.940797i	$0.610079\pi$
$558$	1.24264	0.0526052
$559$	37.3848	1.58121
$560$	0	0
$561$	−14.9289	−0.630300
$562$	−4.00000	−0.168730
$563$	−20.6985	−0.872337	−0.436169	−	0.899865i	$-0.643665\pi$
$563$	−20.6985	−0.872337	−0.436169	+	0.899865i	$0.643665\pi$
$564$	−8.00000	−0.336861
$565$	0	0
$566$	28.2426	1.18713
$567$	0	0
$568$	−2.00000	−0.0839181
$569$	−28.7279	−1.20434	−0.602169	−	0.798369i	$-0.705696\pi$
$569$	−28.7279	−1.20434	−0.602169	+	0.798369i	$0.705696\pi$
$570$	0	0
$571$	9.10051	0.380844	0.190422	−	0.981702i	$-0.439014\pi$
$571$	9.10051	0.380844	0.190422	+	0.981702i	$0.439014\pi$
$572$	−13.0711	−0.546529
$573$	−17.7279	−0.740595
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	16.6274	0.692208	0.346104	−	0.938196i	$-0.387504\pi$
$577$	16.6274	0.692208	0.346104	+	0.938196i	$0.387504\pi$
$578$	27.3137	1.13610
$579$	−23.7990	−0.989052
$580$	0	0
$581$	0	0
$582$	−14.0000	−0.580319
$583$	−19.4142	−0.804055
$584$	0.242641	0.0100405
$585$	0	0
$586$	19.3137	0.797842
$587$	31.5269	1.30125	0.650627	−	0.759397i	$-0.274506\pi$
$587$	31.5269	1.30125	0.650627	+	0.759397i	$0.274506\pi$
$588$	0	0
$589$	−6.72792	−0.277219
$590$	0	0
$591$	17.1421	0.705133
$592$	−8.24264	−0.338770
$593$	−34.0000	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$594$	2.24264	0.0920167
$595$	0	0
$596$	−16.6569	−0.682291
$597$	18.9706	0.776413
$598$	−25.7279	−1.05209
$599$	−20.2132	−0.825889	−0.412945	−	0.910756i	$-0.635500\pi$
$599$	−20.2132	−0.825889	−0.412945	+	0.910756i	$0.635500\pi$
$600$	0	0
$601$	25.7574	1.05066	0.525332	−	0.850897i	$-0.323941\pi$
$601$	25.7574	1.05066	0.525332	+	0.850897i	$0.323941\pi$
$602$	0	0
$603$	−1.65685	−0.0674723
$604$	9.55635	0.388842
$605$	0	0
$606$	6.34315	0.257673
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	−5.41421	−0.219575
$609$	0	0
$610$	0	0
$611$	46.6274	1.88634
$612$	−6.65685	−0.269087
$613$	−25.3137	−1.02241	−0.511206	−	0.859458i	$-0.670801\pi$
$613$	−25.3137	−1.02241	−0.511206	+	0.859458i	$0.670801\pi$
$614$	−8.58579	−0.346494
$615$	0	0
$616$	0	0
$617$	−12.7279	−0.512407	−0.256203	−	0.966623i	$-0.582472\pi$
$617$	−12.7279	−0.512407	−0.256203	+	0.966623i	$0.582472\pi$
$618$	−4.41421	−0.177566
$619$	−32.1421	−1.29190	−0.645951	−	0.763379i	$-0.723539\pi$
$619$	−32.1421	−1.29190	−0.645951	+	0.763379i	$0.723539\pi$
$620$	0	0
$621$	4.41421	0.177136
$622$	13.5563	0.543560
$623$	0	0
$624$	−5.82843	−0.233324
$625$	0	0
$626$	−25.2132	−1.00772
$627$	−12.1421	−0.484910
$628$	−18.8284	−0.751336
$629$	54.8701	2.18781
$630$	0	0
$631$	−8.82843	−0.351454	−0.175727	−	0.984439i	$-0.556228\pi$
$631$	−8.82843	−0.351454	−0.175727	+	0.984439i	$0.556228\pi$
$632$	−2.24264	−0.0892075
$633$	28.0711	1.11572
$634$	−18.3137	−0.727330
$635$	0	0
$636$	−8.65685	−0.343267
$637$	0	0
$638$	1.47309	0.0583201
$639$	−2.00000	−0.0791188
$640$	0	0
$641$	29.2132	1.15385	0.576926	−	0.816796i	$-0.304252\pi$
$641$	29.2132	1.15385	0.576926	+	0.816796i	$0.304252\pi$
$642$	16.1421	0.637079
$643$	−1.95837	−0.0772306	−0.0386153	−	0.999254i	$-0.512295\pi$
$643$	−1.95837	−0.0772306	−0.0386153	+	0.999254i	$0.512295\pi$
$644$	0	0
$645$	0	0
$646$	36.0416	1.41804
$647$	−25.0711	−0.985645	−0.492823	−	0.870130i	$-0.664035\pi$
$647$	−25.0711	−0.985645	−0.492823	+	0.870130i	$0.664035\pi$
$648$	1.00000	0.0392837
$649$	9.89949	0.388589
$650$	0	0
$651$	0	0
$652$	10.8995	0.426857
$653$	29.6569	1.16056	0.580281	−	0.814416i	$-0.302943\pi$
$653$	29.6569	1.16056	0.580281	+	0.814416i	$0.302943\pi$
$654$	−15.3137	−0.598813
$655$	0	0
$656$	−7.00000	−0.273304
$657$	0.242641	0.00946631
$658$	0	0
$659$	33.7990	1.31662	0.658311	−	0.752746i	$-0.271271\pi$
$659$	33.7990	1.31662	0.658311	+	0.752746i	$0.271271\pi$
$660$	0	0
$661$	−8.97056	−0.348914	−0.174457	−	0.984665i	$-0.555817\pi$
$661$	−8.97056	−0.348914	−0.174457	+	0.984665i	$0.555817\pi$
$662$	−10.4142	−0.404760
$663$	38.7990	1.50683
$664$	8.41421	0.326535
$665$	0	0
$666$	−8.24264	−0.319396
$667$	2.89949	0.112269
$668$	−1.51472	−0.0586062
$669$	3.92893	0.151901
$670$	0	0
$671$	19.4142	0.749477
$672$	0	0
$673$	−14.8579	−0.572729	−0.286364	−	0.958121i	$-0.592447\pi$
$673$	−14.8579	−0.572729	−0.286364	+	0.958121i	$0.592447\pi$
$674$	−25.8284	−0.994874
$675$	0	0
$676$	20.9706	0.806560
$677$	30.2426	1.16232	0.581160	−	0.813790i	$-0.302599\pi$
$677$	30.2426	1.16232	0.581160	+	0.813790i	$0.302599\pi$
$678$	3.75736	0.144301
$679$	0	0
$680$	0	0
$681$	−20.7574	−0.795424
$682$	2.78680	0.106712
$683$	−37.0711	−1.41848	−0.709242	−	0.704965i	$-0.750963\pi$
$683$	−37.0711	−1.41848	−0.709242	+	0.704965i	$0.750963\pi$
$684$	−5.41421	−0.207018
$685$	0	0
$686$	0	0
$687$	−13.3137	−0.507950
$688$	−6.41421	−0.244540
$689$	50.4558	1.92221
$690$	0	0
$691$	27.4558	1.04447	0.522235	−	0.852802i	$-0.325098\pi$
$691$	27.4558	1.04447	0.522235	+	0.852802i	$0.325098\pi$
$692$	21.5563	0.819449
$693$	0	0
$694$	−9.79899	−0.371965
$695$	0	0
$696$	0.656854	0.0248980
$697$	46.5980	1.76502
$698$	4.51472	0.170885
$699$	10.7279	0.405767
$700$	0	0
$701$	19.0000	0.717620	0.358810	−	0.933411i	$-0.383183\pi$
$701$	19.0000	0.717620	0.358810	+	0.933411i	$0.383183\pi$
$702$	−5.82843	−0.219980
$703$	44.6274	1.68315
$704$	2.24264	0.0845227
$705$	0	0
$706$	5.85786	0.220464
$707$	0	0
$708$	4.41421	0.165896
$709$	23.8995	0.897564	0.448782	−	0.893641i	$-0.351858\pi$
$709$	23.8995	0.897564	0.448782	+	0.893641i	$0.351858\pi$
$710$	0	0
$711$	−2.24264	−0.0841056
$712$	7.31371	0.274093
$713$	5.48528	0.205425
$714$	0	0
$715$	0	0
$716$	−18.2426	−0.681759
$717$	3.31371	0.123753
$718$	1.72792	0.0644855
$719$	28.1838	1.05108	0.525539	−	0.850770i	$-0.323864\pi$
$719$	28.1838	1.05108	0.525539	+	0.850770i	$0.323864\pi$
$720$	0	0
$721$	0	0
$722$	10.3137	0.383836
$723$	8.34315	0.310285
$724$	−12.9706	−0.482047
$725$	0	0
$726$	−5.97056	−0.221588
$727$	−27.8701	−1.03364	−0.516822	−	0.856093i	$-0.672885\pi$
$727$	−27.8701	−1.03364	−0.516822	+	0.856093i	$0.672885\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	42.6985	1.57926
$732$	8.65685	0.319967
$733$	−32.9411	−1.21671	−0.608354	−	0.793666i	$-0.708170\pi$
$733$	−32.9411	−1.21671	−0.608354	+	0.793666i	$0.708170\pi$
$734$	−16.5563	−0.611106
$735$	0	0
$736$	4.41421	0.162710
$737$	−3.71573	−0.136871
$738$	−7.00000	−0.257674
$739$	34.5563	1.27118	0.635588	−	0.772028i	$-0.280758\pi$
$739$	34.5563	1.27118	0.635588	+	0.772028i	$0.280758\pi$
$740$	0	0
$741$	31.5563	1.15925
$742$	0	0
$743$	−9.38478	−0.344294	−0.172147	−	0.985071i	$-0.555070\pi$
$743$	−9.38478	−0.344294	−0.172147	+	0.985071i	$0.555070\pi$
$744$	1.24264	0.0455574
$745$	0	0
$746$	−1.65685	−0.0606617
$747$	8.41421	0.307860
$748$	−14.9289	−0.545856
$749$	0	0
$750$	0	0
$751$	54.6274	1.99338	0.996691	−	0.0812791i	$-0.0259005\pi$
$751$	54.6274	1.99338	0.996691	+	0.0812791i	$0.0259005\pi$
$752$	−8.00000	−0.291730
$753$	28.2132	1.02815
$754$	−3.82843	−0.139423
$755$	0	0
$756$	0	0
$757$	16.4437	0.597655	0.298827	−	0.954307i	$-0.403405\pi$
$757$	16.4437	0.597655	0.298827	+	0.954307i	$0.403405\pi$
$758$	31.7279	1.15241
$759$	9.89949	0.359329
$760$	0	0
$761$	−49.9411	−1.81036	−0.905182	−	0.425024i	$-0.860266\pi$
$761$	−49.9411	−1.81036	−0.905182	+	0.425024i	$0.860266\pi$
$762$	20.8284	0.754534
$763$	0	0
$764$	−17.7279	−0.641374
$765$	0	0
$766$	−19.0711	−0.689066
$767$	−25.7279	−0.928981
$768$	1.00000	0.0360844
$769$	−32.0000	−1.15395	−0.576975	−	0.816762i	$-0.695767\pi$
$769$	−32.0000	−1.15395	−0.576975	+	0.816762i	$0.695767\pi$
$770$	0	0
$771$	−27.8284	−1.00222
$772$	−23.7990	−0.856544
$773$	−45.9411	−1.65239	−0.826194	−	0.563386i	$-0.809498\pi$
$773$	−45.9411	−1.65239	−0.826194	+	0.563386i	$0.809498\pi$
$774$	−6.41421	−0.230554
$775$	0	0
$776$	−14.0000	−0.502571
$777$	0	0
$778$	−36.2843	−1.30085
$779$	37.8995	1.35789
$780$	0	0
$781$	−4.48528	−0.160496
$782$	−29.3848	−1.05080
$783$	0.656854	0.0234741
$784$	0	0
$785$	0	0
$786$	−2.82843	−0.100887
$787$	−52.1421	−1.85867	−0.929333	−	0.369242i	$-0.879617\pi$
$787$	−52.1421	−1.85867	−0.929333	+	0.369242i	$0.879617\pi$
$788$	17.1421	0.610663
$789$	29.5858	1.05328
$790$	0	0
$791$	0	0
$792$	2.24264	0.0796888
$793$	−50.4558	−1.79174
$794$	28.1127	0.997682
$795$	0	0
$796$	18.9706	0.672394
$797$	−6.10051	−0.216091	−0.108045	−	0.994146i	$-0.534459\pi$
$797$	−6.10051	−0.216091	−0.108045	+	0.994146i	$0.534459\pi$
$798$	0	0
$799$	53.2548	1.88402
$800$	0	0
$801$	7.31371	0.258417
$802$	−23.6985	−0.836823
$803$	0.544156	0.0192028
$804$	−1.65685	−0.0584327
$805$	0	0
$806$	−7.24264	−0.255111
$807$	1.41421	0.0497827
$808$	6.34315	0.223151
$809$	−16.0000	−0.562530	−0.281265	−	0.959630i	$-0.590754\pi$
$809$	−16.0000	−0.562530	−0.281265	+	0.959630i	$0.590754\pi$
$810$	0	0
$811$	−26.0416	−0.914445	−0.457223	−	0.889352i	$-0.651156\pi$
$811$	−26.0416	−0.914445	−0.457223	+	0.889352i	$0.651156\pi$
$812$	0	0
$813$	21.4558	0.752489
$814$	−18.4853	−0.647909
$815$	0	0
$816$	−6.65685	−0.233037
$817$	34.7279	1.21498
$818$	−9.21320	−0.322132
$819$	0	0
$820$	0	0
$821$	−23.5147	−0.820669	−0.410335	−	0.911935i	$-0.634588\pi$
$821$	−23.5147	−0.820669	−0.410335	+	0.911935i	$0.634588\pi$
$822$	−9.07107	−0.316390
$823$	27.2132	0.948593	0.474296	−	0.880365i	$-0.342703\pi$
$823$	27.2132	0.948593	0.474296	+	0.880365i	$0.342703\pi$
$824$	−4.41421	−0.153776
$825$	0	0
$826$	0	0
$827$	−38.7279	−1.34670	−0.673351	−	0.739323i	$-0.735146\pi$
$827$	−38.7279	−1.34670	−0.673351	+	0.739323i	$0.735146\pi$
$828$	4.41421	0.153405
$829$	42.6569	1.48153	0.740767	−	0.671762i	$-0.234462\pi$
$829$	42.6569	1.48153	0.740767	+	0.671762i	$0.234462\pi$
$830$	0	0
$831$	−24.0416	−0.833995
$832$	−5.82843	−0.202064
$833$	0	0
$834$	−9.41421	−0.325988
$835$	0	0
$836$	−12.1421	−0.419945
$837$	1.24264	0.0429519
$838$	13.7279	0.474223
$839$	−21.2132	−0.732361	−0.366181	−	0.930544i	$-0.619335\pi$
$839$	−21.2132	−0.732361	−0.366181	+	0.930544i	$0.619335\pi$
$840$	0	0
$841$	−28.5685	−0.985122
$842$	19.6569	0.677420
$843$	−4.00000	−0.137767
$844$	28.0711	0.966246
$845$	0	0
$846$	−8.00000	−0.275046
$847$	0	0
$848$	−8.65685	−0.297278
$849$	28.2426	0.969285
$850$	0	0
$851$	−36.3848	−1.24725
$852$	−2.00000	−0.0685189
$853$	6.85786	0.234809	0.117404	−	0.993084i	$-0.462543\pi$
$853$	6.85786	0.234809	0.117404	+	0.993084i	$0.462543\pi$
$854$	0	0
$855$	0	0
$856$	16.1421	0.551727
$857$	27.5147	0.939885	0.469942	−	0.882697i	$-0.344275\pi$
$857$	27.5147	0.939885	0.469942	+	0.882697i	$0.344275\pi$
$858$	−13.0711	−0.446239
$859$	16.6863	0.569329	0.284664	−	0.958627i	$-0.408118\pi$
$859$	16.6863	0.569329	0.284664	+	0.958627i	$0.408118\pi$
$860$	0	0
$861$	0	0
$862$	−26.6985	−0.909354
$863$	−3.31371	−0.112800	−0.0564000	−	0.998408i	$-0.517962\pi$
$863$	−3.31371	−0.112800	−0.0564000	+	0.998408i	$0.517962\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	15.6569	0.532042
$867$	27.3137	0.927622
$868$	0	0
$869$	−5.02944	−0.170612
$870$	0	0
$871$	9.65685	0.327210
$872$	−15.3137	−0.518588
$873$	−14.0000	−0.473828
$874$	−23.8995	−0.808412
$875$	0	0
$876$	0.242641	0.00819807
$877$	7.02944	0.237367	0.118684	−	0.992932i	$-0.462133\pi$
$877$	7.02944	0.237367	0.118684	+	0.992932i	$0.462133\pi$
$878$	6.21320	0.209685
$879$	19.3137	0.651435
$880$	0	0
$881$	41.4264	1.39569	0.697846	−	0.716248i	$-0.254142\pi$
$881$	41.4264	1.39569	0.697846	+	0.716248i	$0.254142\pi$
$882$	0	0
$883$	−19.2426	−0.647566	−0.323783	−	0.946131i	$-0.604955\pi$
$883$	−19.2426	−0.647566	−0.323783	+	0.946131i	$0.604955\pi$
$884$	38.7990	1.30495
$885$	0	0
$886$	−30.6274	−1.02895
$887$	16.5858	0.556896	0.278448	−	0.960451i	$-0.410180\pi$
$887$	16.5858	0.556896	0.278448	+	0.960451i	$0.410180\pi$
$888$	−8.24264	−0.276605
$889$	0	0
$890$	0	0
$891$	2.24264	0.0751313
$892$	3.92893	0.131550
$893$	43.3137	1.44944
$894$	−16.6569	−0.557089
$895$	0	0
$896$	0	0
$897$	−25.7279	−0.859030
$898$	−1.27208	−0.0424498
$899$	0.816234	0.0272229
$900$	0	0
$901$	57.6274	1.91985
$902$	−15.6985	−0.522702
$903$	0	0
$904$	3.75736	0.124968
$905$	0	0
$906$	9.55635	0.317488
$907$	25.1005	0.833449	0.416724	−	0.909033i	$-0.363178\pi$
$907$	25.1005	0.833449	0.416724	+	0.909033i	$0.363178\pi$
$908$	−20.7574	−0.688857
$909$	6.34315	0.210389
$910$	0	0
$911$	−30.7574	−1.01904	−0.509518	−	0.860460i	$-0.670176\pi$
$911$	−30.7574	−1.01904	−0.509518	+	0.860460i	$0.670176\pi$
$912$	−5.41421	−0.179283
$913$	18.8701	0.624508
$914$	40.4558	1.33816
$915$	0	0
$916$	−13.3137	−0.439897
$917$	0	0
$918$	−6.65685	−0.219709
$919$	−38.8701	−1.28220	−0.641102	−	0.767455i	$-0.721523\pi$
$919$	−38.8701	−1.28220	−0.641102	+	0.767455i	$0.721523\pi$
$920$	0	0
$921$	−8.58579	−0.282911
$922$	8.00000	0.263466
$923$	11.6569	0.383690
$924$	0	0
$925$	0	0
$926$	−0.443651	−0.0145793
$927$	−4.41421	−0.144982
$928$	0.656854	0.0215623
$929$	2.85786	0.0937635	0.0468817	−	0.998900i	$-0.485072\pi$
$929$	2.85786	0.0937635	0.0468817	+	0.998900i	$0.485072\pi$
$930$	0	0
$931$	0	0
$932$	10.7279	0.351405
$933$	13.5563	0.443815
$934$	−25.7279	−0.841843
$935$	0	0
$936$	−5.82843	−0.190508
$937$	−14.8701	−0.485784	−0.242892	−	0.970053i	$-0.578096\pi$
$937$	−14.8701	−0.485784	−0.242892	+	0.970053i	$0.578096\pi$
$938$	0	0
$939$	−25.2132	−0.822802
$940$	0	0
$941$	35.3137	1.15119	0.575597	−	0.817734i	$-0.304770\pi$
$941$	35.3137	1.15119	0.575597	+	0.817734i	$0.304770\pi$
$942$	−18.8284	−0.613463
$943$	−30.8995	−1.00623
$944$	4.41421	0.143670
$945$	0	0
$946$	−14.3848	−0.467689
$947$	3.55635	0.115566	0.0577829	−	0.998329i	$-0.481597\pi$
$947$	3.55635	0.115566	0.0577829	+	0.998329i	$0.481597\pi$
$948$	−2.24264	−0.0728376
$949$	−1.41421	−0.0459073
$950$	0	0
$951$	−18.3137	−0.593863
$952$	0	0
$953$	23.5980	0.764414	0.382207	−	0.924077i	$-0.375164\pi$
$953$	23.5980	0.764414	0.382207	+	0.924077i	$0.375164\pi$
$954$	−8.65685	−0.280276
$955$	0	0
$956$	3.31371	0.107173
$957$	1.47309	0.0476182
$958$	−24.7279	−0.798923
$959$	0	0
$960$	0	0
$961$	−29.4558	−0.950189
$962$	48.0416	1.54892
$963$	16.1421	0.520173
$964$	8.34315	0.268715
$965$	0	0
$966$	0	0
$967$	−10.1005	−0.324810	−0.162405	−	0.986724i	$-0.551925\pi$
$967$	−10.1005	−0.324810	−0.162405	+	0.986724i	$0.551925\pi$
$968$	−5.97056	−0.191901
$969$	36.0416	1.15782
$970$	0	0
$971$	−45.9411	−1.47432	−0.737160	−	0.675718i	$-0.763834\pi$
$971$	−45.9411	−1.47432	−0.737160	+	0.675718i	$0.763834\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−28.1421	−0.901732
$975$	0	0
$976$	8.65685	0.277099
$977$	23.3553	0.747203	0.373602	−	0.927589i	$-0.378123\pi$
$977$	23.3553	0.747203	0.373602	+	0.927589i	$0.378123\pi$
$978$	10.8995	0.348527
$979$	16.4020	0.524211
$980$	0	0
$981$	−15.3137	−0.488929
$982$	4.82843	0.154081
$983$	14.6274	0.466542	0.233271	−	0.972412i	$-0.425057\pi$
$983$	14.6274	0.466542	0.233271	+	0.972412i	$0.425057\pi$
$984$	−7.00000	−0.223152
$985$	0	0
$986$	−4.37258	−0.139251
$987$	0	0
$988$	31.5563	1.00394
$989$	−28.3137	−0.900324
$990$	0	0
$991$	−57.2548	−1.81876	−0.909380	−	0.415967i	$-0.863443\pi$
$991$	−57.2548	−1.81876	−0.909380	+	0.415967i	$0.863443\pi$
$992$	1.24264	0.0394539
$993$	−10.4142	−0.330485
$994$	0	0
$995$	0	0
$996$	8.41421	0.266615
$997$	9.31371	0.294968	0.147484	−	0.989064i	$-0.452882\pi$
$997$	9.31371	0.294968	0.147484	+	0.989064i	$0.452882\pi$
$998$	5.92893	0.187677
$999$	−8.24264	−0.260786

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7350.2.a.dk.1.2	yes	2
5.4	even	2		7350.2.a.db.1.2	✓	2
7.6	odd	2		7350.2.a.di.1.2	yes	2
35.34	odd	2		7350.2.a.de.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7350.2.a.db.1.2	✓	2	5.4	even	2
7350.2.a.de.1.2	yes	2	35.34	odd	2
7350.2.a.di.1.2	yes	2	7.6	odd	2
7350.2.a.dk.1.2	yes	2	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:	\( N \)	\(=\)	\( 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7350.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +2.24264 q^{11} +1.00000 q^{12} -5.82843 q^{13} +1.00000 q^{16} -6.65685 q^{17} +1.00000 q^{18} -5.41421 q^{19} +2.24264 q^{22} +4.41421 q^{23} +1.00000 q^{24} -5.82843 q^{26} +1.00000 q^{27} +0.656854 q^{29} +1.24264 q^{31} +1.00000 q^{32} +2.24264 q^{33} -6.65685 q^{34} +1.00000 q^{36} -8.24264 q^{37} -5.41421 q^{38} -5.82843 q^{39} -7.00000 q^{41} -6.41421 q^{43} +2.24264 q^{44} +4.41421 q^{46} -8.00000 q^{47} +1.00000 q^{48} -6.65685 q^{51} -5.82843 q^{52} -8.65685 q^{53} +1.00000 q^{54} -5.41421 q^{57} +0.656854 q^{58} +4.41421 q^{59} +8.65685 q^{61} +1.24264 q^{62} +1.00000 q^{64} +2.24264 q^{66} -1.65685 q^{67} -6.65685 q^{68} +4.41421 q^{69} -2.00000 q^{71} +1.00000 q^{72} +0.242641 q^{73} -8.24264 q^{74} -5.41421 q^{76} -5.82843 q^{78} -2.24264 q^{79} +1.00000 q^{81} -7.00000 q^{82} +8.41421 q^{83} -6.41421 q^{86} +0.656854 q^{87} +2.24264 q^{88} +7.31371 q^{89} +4.41421 q^{92} +1.24264 q^{93} -8.00000 q^{94} +1.00000 q^{96} -14.0000 q^{97} +2.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{8} + 2 q^{9} - 4 q^{11} + 2 q^{12} - 6 q^{13} + 2 q^{16} - 2 q^{17} + 2 q^{18} - 8 q^{19} - 4 q^{22} + 6 q^{23} + 2 q^{24} - 6 q^{26} + 2 q^{27} - 10 q^{29} - 6 q^{31} + 2 q^{32} - 4 q^{33} - 2 q^{34} + 2 q^{36} - 8 q^{37} - 8 q^{38} - 6 q^{39} - 14 q^{41} - 10 q^{43} - 4 q^{44} + 6 q^{46} - 16 q^{47} + 2 q^{48} - 2 q^{51} - 6 q^{52} - 6 q^{53} + 2 q^{54} - 8 q^{57} - 10 q^{58} + 6 q^{59} + 6 q^{61} - 6 q^{62} + 2 q^{64} - 4 q^{66} + 8 q^{67} - 2 q^{68} + 6 q^{69} - 4 q^{71} + 2 q^{72} - 8 q^{73} - 8 q^{74} - 8 q^{76} - 6 q^{78} + 4 q^{79} + 2 q^{81} - 14 q^{82} + 14 q^{83} - 10 q^{86} - 10 q^{87} - 4 q^{88} - 8 q^{89} + 6 q^{92} - 6 q^{93} - 16 q^{94} + 2 q^{96} - 28 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^8 + 2 * q^9 - 4 * q^11 + 2 * q^12 - 6 * q^13 + 2 * q^16 - 2 * q^17 + 2 * q^18 - 8 * q^19 - 4 * q^22 + 6 * q^23 + 2 * q^24 - 6 * q^26 + 2 * q^27 - 10 * q^29 - 6 * q^31 + 2 * q^32 - 4 * q^33 - 2 * q^34 + 2 * q^36 - 8 * q^37 - 8 * q^38 - 6 * q^39 - 14 * q^41 - 10 * q^43 - 4 * q^44 + 6 * q^46 - 16 * q^47 + 2 * q^48 - 2 * q^51 - 6 * q^52 - 6 * q^53 + 2 * q^54 - 8 * q^57 - 10 * q^58 + 6 * q^59 + 6 * q^61 - 6 * q^62 + 2 * q^64 - 4 * q^66 + 8 * q^67 - 2 * q^68 + 6 * q^69 - 4 * q^71 + 2 * q^72 - 8 * q^73 - 8 * q^74 - 8 * q^76 - 6 * q^78 + 4 * q^79 + 2 * q^81 - 14 * q^82 + 14 * q^83 - 10 * q^86 - 10 * q^87 - 4 * q^88 - 8 * q^89 + 6 * q^92 - 6 * q^93 - 16 * q^94 + 2 * q^96 - 28 * q^97 - 4 * q^99

Introduction

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