LMFDB - Embedded newform 7350.2.a.dq.1.1

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:	$0$
Dimension:	$3$
Coefficient field:	3.3.2700.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 15x - 20 $ x^3 - 15*x - 20
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 210)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.61323$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -4.78426 q^{11} +1.00000 q^{12} -3.17103 q^{13} +1.00000 q^{16} +5.22646 q^{17} +1.00000 q^{18} -3.17103 q^{19} -4.78426 q^{22} +7.17103 q^{23} +1.00000 q^{24} -3.17103 q^{26} +1.00000 q^{27} +2.38677 q^{29} +4.17103 q^{31} +1.00000 q^{32} -4.78426 q^{33} +5.22646 q^{34} +1.00000 q^{36} +7.17103 q^{37} -3.17103 q^{38} -3.17103 q^{39} -2.05543 q^{41} -2.00000 q^{43} -4.78426 q^{44} +7.17103 q^{46} +11.5131 q^{47} +1.00000 q^{48} +5.22646 q^{51} -3.17103 q^{52} -8.11560 q^{53} +1.00000 q^{54} -3.17103 q^{57} +2.38677 q^{58} -10.7288 q^{59} +7.11560 q^{61} +4.17103 q^{62} +1.00000 q^{64} -4.78426 q^{66} +9.56852 q^{67} +5.22646 q^{68} +7.17103 q^{69} +6.00000 q^{71} +1.00000 q^{72} +4.00000 q^{73} +7.17103 q^{74} -3.17103 q^{76} -3.17103 q^{78} +12.1710 q^{79} +1.00000 q^{81} -2.05543 q^{82} -3.39749 q^{83} -2.00000 q^{86} +2.38677 q^{87} -4.78426 q^{88} +9.56852 q^{89} +7.17103 q^{92} +4.17103 q^{93} +11.5131 q^{94} +1.00000 q^{96} -1.27117 q^{97} -4.78426 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^6 + 3 * q^8 + 3 * q^9	$ 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{6} + 3 q^{8} + 3 q^{9} + 3 q^{11} + 3 q^{12} + 3 q^{13} + 3 q^{16} + 6 q^{17} + 3 q^{18} + 3 q^{19} + 3 q^{22} + 9 q^{23} + 3 q^{24} + 3 q^{26} + 3 q^{27} + 12 q^{29} + 3 q^{32} + 3 q^{33} + 6 q^{34} + 3 q^{36} + 9 q^{37} + 3 q^{38} + 3 q^{39} - 9 q^{41} - 6 q^{43} + 3 q^{44} + 9 q^{46} - 3 q^{47} + 3 q^{48} + 6 q^{51} + 3 q^{52} - 9 q^{53} + 3 q^{54} + 3 q^{57} + 12 q^{58} - 12 q^{59} + 6 q^{61} + 3 q^{64} + 3 q^{66} - 6 q^{67} + 6 q^{68} + 9 q^{69} + 18 q^{71} + 3 q^{72} + 12 q^{73} + 9 q^{74} + 3 q^{76} + 3 q^{78} + 24 q^{79} + 3 q^{81} - 9 q^{82} + 12 q^{83} - 6 q^{86} + 12 q^{87} + 3 q^{88} - 6 q^{89} + 9 q^{92} - 3 q^{94} + 3 q^{96} - 24 q^{97} + 3 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^6 + 3 * q^8 + 3 * q^9 + 3 * q^11 + 3 * q^12 + 3 * q^13 + 3 * q^16 + 6 * q^17 + 3 * q^18 + 3 * q^19 + 3 * q^22 + 9 * q^23 + 3 * q^24 + 3 * q^26 + 3 * q^27 + 12 * q^29 + 3 * q^32 + 3 * q^33 + 6 * q^34 + 3 * q^36 + 9 * q^37 + 3 * q^38 + 3 * q^39 - 9 * q^41 - 6 * q^43 + 3 * q^44 + 9 * q^46 - 3 * q^47 + 3 * q^48 + 6 * q^51 + 3 * q^52 - 9 * q^53 + 3 * q^54 + 3 * q^57 + 12 * q^58 - 12 * q^59 + 6 * q^61 + 3 * q^64 + 3 * q^66 - 6 * q^67 + 6 * q^68 + 9 * q^69 + 18 * q^71 + 3 * q^72 + 12 * q^73 + 9 * q^74 + 3 * q^76 + 3 * q^78 + 24 * q^79 + 3 * q^81 - 9 * q^82 + 12 * q^83 - 6 * q^86 + 12 * q^87 + 3 * q^88 - 6 * q^89 + 9 * q^92 - 3 * q^94 + 3 * q^96 - 24 * q^97 + 3 * 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
$11$	−4.78426	−1.44251	−0.721254	−	0.692670i	$-0.756434\pi$
$11$	−4.78426	−1.44251	−0.721254	+	0.692670i	$0.756434\pi$
$12$	1.00000	0.288675
$13$	−3.17103	−0.879485	−0.439743	−	0.898124i	$-0.644930\pi$
$13$	−3.17103	−0.879485	−0.439743	+	0.898124i	$0.644930\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	5.22646	1.26760	0.633801	−	0.773496i	$-0.281494\pi$
$17$	5.22646	1.26760	0.633801	+	0.773496i	$0.281494\pi$
$18$	1.00000	0.235702
$19$	−3.17103	−0.727484	−0.363742	−	0.931500i	$-0.618501\pi$
$19$	−3.17103	−0.727484	−0.363742	+	0.931500i	$0.618501\pi$
$20$	0	0
$21$	0	0
$22$	−4.78426	−1.02001
$23$	7.17103	1.49526	0.747632	−	0.664114i	$-0.231191\pi$
$23$	7.17103	1.49526	0.747632	+	0.664114i	$0.231191\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−3.17103	−0.621890
$27$	1.00000	0.192450
$28$	0	0
$29$	2.38677	0.443212	0.221606	−	0.975136i	$-0.428870\pi$
$29$	2.38677	0.443212	0.221606	+	0.975136i	$0.428870\pi$
$30$	0	0
$31$	4.17103	0.749139	0.374570	−	0.927199i	$-0.377791\pi$
$31$	4.17103	0.749139	0.374570	+	0.927199i	$0.377791\pi$
$32$	1.00000	0.176777
$33$	−4.78426	−0.832833
$34$	5.22646	0.896330
$35$	0	0
$36$	1.00000	0.166667
$37$	7.17103	1.17891	0.589455	−	0.807801i	$-0.299343\pi$
$37$	7.17103	1.17891	0.589455	+	0.807801i	$0.299343\pi$
$38$	−3.17103	−0.514409
$39$	−3.17103	−0.507771
$40$	0	0
$41$	−2.05543	−0.321004	−0.160502	−	0.987035i	$-0.551311\pi$
$41$	−2.05543	−0.321004	−0.160502	+	0.987035i	$0.551311\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	−4.78426	−0.721254
$45$	0	0
$46$	7.17103	1.05731
$47$	11.5131	1.67936	0.839678	−	0.543084i	$-0.182744\pi$
$47$	11.5131	1.67936	0.839678	+	0.543084i	$0.182744\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	0	0
$51$	5.22646	0.731851
$52$	−3.17103	−0.439743
$53$	−8.11560	−1.11476	−0.557382	−	0.830256i	$-0.688194\pi$
$53$	−8.11560	−1.11476	−0.557382	+	0.830256i	$0.688194\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	0	0
$57$	−3.17103	−0.420013
$58$	2.38677	0.313398
$59$	−10.7288	−1.39677	−0.698387	−	0.715720i	$-0.746099\pi$
$59$	−10.7288	−1.39677	−0.698387	+	0.715720i	$0.746099\pi$
$60$	0	0
$61$	7.11560	0.911059	0.455530	−	0.890221i	$-0.349450\pi$
$61$	7.11560	0.911059	0.455530	+	0.890221i	$0.349450\pi$
$62$	4.17103	0.529721
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	−4.78426	−0.588902
$67$	9.56852	1.16898	0.584490	−	0.811401i	$-0.301294\pi$
$67$	9.56852	1.16898	0.584490	+	0.811401i	$0.301294\pi$
$68$	5.22646	0.633801
$69$	7.17103	0.863291
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	1.00000	0.117851
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	7.17103	0.833615
$75$	0	0
$76$	−3.17103	−0.363742
$77$	0	0
$78$	−3.17103	−0.359048
$79$	12.1710	1.36935	0.684674	−	0.728850i	$-0.259945\pi$
$79$	12.1710	1.36935	0.684674	+	0.728850i	$0.259945\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−2.05543	−0.226984
$83$	−3.39749	−0.372923	−0.186461	−	0.982462i	$-0.559702\pi$
$83$	−3.39749	−0.372923	−0.186461	+	0.982462i	$0.559702\pi$
$84$	0	0
$85$	0	0
$86$	−2.00000	−0.215666
$87$	2.38677	0.255889
$88$	−4.78426	−0.510004
$89$	9.56852	1.01426	0.507131	−	0.861869i	$-0.330706\pi$
$89$	9.56852	1.01426	0.507131	+	0.861869i	$0.330706\pi$
$90$	0	0
$91$	0	0
$92$	7.17103	0.747632
$93$	4.17103	0.432516
$94$	11.5131	1.18748
$95$	0	0
$96$	1.00000	0.102062
$97$	−1.27117	−0.129068	−0.0645339	−	0.997916i	$-0.520556\pi$
$97$	−1.27117	−0.129068	−0.0645339	+	0.997916i	$0.520556\pi$
$98$	0	0
$99$	−4.78426	−0.480836
$100$	0	0
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	5.22646	0.517497
$103$	−11.2265	−1.10618	−0.553088	−	0.833123i	$-0.686551\pi$
$103$	−11.2265	−1.10618	−0.553088	+	0.833123i	$0.686551\pi$
$104$	−3.17103	−0.310945
$105$	0	0
$106$	−8.11560	−0.788257
$107$	−1.39749	−0.135100	−0.0675502	−	0.997716i	$-0.521518\pi$
$107$	−1.39749	−0.135100	−0.0675502	+	0.997716i	$0.521518\pi$
$108$	1.00000	0.0962250
$109$	12.4529	1.19277	0.596387	−	0.802697i	$-0.296603\pi$
$109$	12.4529	1.19277	0.596387	+	0.802697i	$0.296603\pi$
$110$	0	0
$111$	7.17103	0.680644
$112$	0	0
$113$	6.34206	0.596611	0.298305	−	0.954470i	$-0.403579\pi$
$113$	6.34206	0.596611	0.298305	+	0.954470i	$0.403579\pi$
$114$	−3.17103	−0.296994
$115$	0	0
$116$	2.38677	0.221606
$117$	−3.17103	−0.293162
$118$	−10.7288	−0.987669
$119$	0	0
$120$	0	0
$121$	11.8891	1.08083
$122$	7.11560	0.644216
$123$	−2.05543	−0.185332
$124$	4.17103	0.374570
$125$	0	0
$126$	0	0
$127$	18.0107	1.59819	0.799096	−	0.601203i	$-0.205312\pi$
$127$	18.0107	1.59819	0.799096	+	0.601203i	$0.205312\pi$
$128$	1.00000	0.0883883
$129$	−2.00000	−0.176090
$130$	0	0
$131$	−5.55780	−0.485587	−0.242794	−	0.970078i	$-0.578064\pi$
$131$	−5.55780	−0.485587	−0.242794	+	0.970078i	$0.578064\pi$
$132$	−4.78426	−0.416416
$133$	0	0
$134$	9.56852	0.826594
$135$	0	0
$136$	5.22646	0.448165
$137$	−14.7950	−1.26402	−0.632010	−	0.774960i	$-0.717770\pi$
$137$	−14.7950	−1.26402	−0.632010	+	0.774960i	$0.717770\pi$
$138$	7.17103	0.610439
$139$	3.65794	0.310262	0.155131	−	0.987894i	$-0.450420\pi$
$139$	3.65794	0.310262	0.155131	+	0.987894i	$0.450420\pi$
$140$	0	0
$141$	11.5131	0.969577
$142$	6.00000	0.503509
$143$	15.1710	1.26867
$144$	1.00000	0.0833333
$145$	0	0
$146$	4.00000	0.331042
$147$	0	0
$148$	7.17103	0.589455
$149$	22.3421	1.83033	0.915166	−	0.403076i	$-0.132059\pi$
$149$	22.3421	1.83033	0.915166	+	0.403076i	$0.132059\pi$
$150$	0	0
$151$	16.5131	1.34382	0.671908	−	0.740635i	$-0.265475\pi$
$151$	16.5131	1.34382	0.671908	+	0.740635i	$0.265475\pi$
$152$	−3.17103	−0.257204
$153$	5.22646	0.422534
$154$	0	0
$155$	0	0
$156$	−3.17103	−0.253886
$157$	9.94457	0.793663	0.396832	−	0.917891i	$-0.370110\pi$
$157$	9.94457	0.793663	0.396832	+	0.917891i	$0.370110\pi$
$158$	12.1710	0.968275
$159$	−8.11560	−0.643609
$160$	0	0
$161$	0	0
$162$	1.00000	0.0785674
$163$	2.88440	0.225924	0.112962	−	0.993599i	$-0.463966\pi$
$163$	2.88440	0.225924	0.112962	+	0.993599i	$0.463966\pi$
$164$	−2.05543	−0.160502
$165$	0	0
$166$	−3.39749	−0.263696
$167$	4.05543	0.313819	0.156909	−	0.987613i	$-0.449847\pi$
$167$	4.05543	0.313819	0.156909	+	0.987613i	$0.449847\pi$
$168$	0	0
$169$	−2.94457	−0.226505
$170$	0	0
$171$	−3.17103	−0.242495
$172$	−2.00000	−0.152499
$173$	6.39749	0.486392	0.243196	−	0.969977i	$-0.421804\pi$
$173$	6.39749	0.486392	0.243196	+	0.969977i	$0.421804\pi$
$174$	2.38677	0.180941
$175$	0	0
$176$	−4.78426	−0.360627
$177$	−10.7288	−0.806428
$178$	9.56852	0.717191
$179$	19.5131	1.45848	0.729238	−	0.684260i	$-0.239875\pi$
$179$	19.5131	1.45848	0.729238	+	0.684260i	$0.239875\pi$
$180$	0	0
$181$	−23.0262	−1.71152	−0.855761	−	0.517371i	$-0.826911\pi$
$181$	−23.0262	−1.71152	−0.855761	+	0.517371i	$0.826911\pi$
$182$	0	0
$183$	7.11560	0.526000
$184$	7.17103	0.528655
$185$	0	0
$186$	4.17103	0.305835
$187$	−25.0047	−1.82853
$188$	11.5131	0.839678
$189$	0	0
$190$	0	0
$191$	−10.2312	−0.740304	−0.370152	−	0.928971i	$-0.620694\pi$
$191$	−10.2312	−0.740304	−0.370152	+	0.928971i	$0.620694\pi$
$192$	1.00000	0.0721688
$193$	−11.6132	−0.835939	−0.417969	−	0.908461i	$-0.637258\pi$
$193$	−11.6132	−0.835939	−0.417969	+	0.908461i	$0.637258\pi$
$194$	−1.27117	−0.0912647
$195$	0	0
$196$	0	0
$197$	−2.39749	−0.170814	−0.0854070	−	0.996346i	$-0.527219\pi$
$197$	−2.39749	−0.170814	−0.0854070	+	0.996346i	$0.527219\pi$
$198$	−4.78426	−0.340003
$199$	−18.3421	−1.30023	−0.650117	−	0.759834i	$-0.725280\pi$
$199$	−18.3421	−1.30023	−0.650117	+	0.759834i	$0.725280\pi$
$200$	0	0
$201$	9.56852	0.674911
$202$	−8.00000	−0.562878
$203$	0	0
$204$	5.22646	0.365925
$205$	0	0
$206$	−11.2265	−0.782185
$207$	7.17103	0.498421
$208$	−3.17103	−0.219871
$209$	15.1710	1.04940
$210$	0	0
$211$	−2.82897	−0.194754	−0.0973772	−	0.995248i	$-0.531045\pi$
$211$	−2.82897	−0.194754	−0.0973772	+	0.995248i	$0.531045\pi$
$212$	−8.11560	−0.557382
$213$	6.00000	0.411113
$214$	−1.39749	−0.0955304
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	12.4529	0.843418
$219$	4.00000	0.270295
$220$	0	0
$221$	−16.5733	−1.11484
$222$	7.17103	0.481288
$223$	14.2973	0.957421	0.478711	−	0.877973i	$-0.341104\pi$
$223$	14.2973	0.957421	0.478711	+	0.877973i	$0.341104\pi$
$224$	0	0
$225$	0	0
$226$	6.34206	0.421868
$227$	11.0554	0.733775	0.366887	−	0.930265i	$-0.380423\pi$
$227$	11.0554	0.733775	0.366887	+	0.930265i	$0.380423\pi$
$228$	−3.17103	−0.210007
$229$	−18.4529	−1.21940	−0.609702	−	0.792631i	$-0.708711\pi$
$229$	−18.4529	−1.21940	−0.609702	+	0.792631i	$0.708711\pi$
$230$	0	0
$231$	0	0
$232$	2.38677	0.156699
$233$	−9.11560	−0.597183	−0.298591	−	0.954381i	$-0.596517\pi$
$233$	−9.11560	−0.597183	−0.298591	+	0.954381i	$0.596517\pi$
$234$	−3.17103	−0.207297
$235$	0	0
$236$	−10.7288	−0.698387
$237$	12.1710	0.790593
$238$	0	0
$239$	13.1156	0.848378	0.424189	−	0.905574i	$-0.360559\pi$
$239$	13.1156	0.848378	0.424189	+	0.905574i	$0.360559\pi$
$240$	0	0
$241$	−5.34206	−0.344112	−0.172056	−	0.985087i	$-0.555041\pi$
$241$	−5.34206	−0.344112	−0.172056	+	0.985087i	$0.555041\pi$
$242$	11.8891	0.764263
$243$	1.00000	0.0641500
$244$	7.11560	0.455530
$245$	0	0
$246$	−2.05543	−0.131049
$247$	10.0554	0.639812
$248$	4.17103	0.264861
$249$	−3.39749	−0.215307
$250$	0	0
$251$	27.9213	1.76238	0.881188	−	0.472765i	$-0.156744\pi$
$251$	27.9213	1.76238	0.881188	+	0.472765i	$0.156744\pi$
$252$	0	0
$253$	−34.3081	−2.15693
$254$	18.0107	1.13009
$255$	0	0
$256$	1.00000	0.0625000
$257$	−21.1370	−1.31849	−0.659246	−	0.751927i	$-0.729124\pi$
$257$	−21.1370	−1.31849	−0.659246	+	0.751927i	$0.729124\pi$
$258$	−2.00000	−0.124515
$259$	0	0
$260$	0	0
$261$	2.38677	0.147737
$262$	−5.55780	−0.343362
$263$	11.9106	0.734438	0.367219	−	0.930135i	$-0.380310\pi$
$263$	11.9106	0.734438	0.367219	+	0.930135i	$0.380310\pi$
$264$	−4.78426	−0.294451
$265$	0	0
$266$	0	0
$267$	9.56852	0.585584
$268$	9.56852	0.584490
$269$	−3.27117	−0.199447	−0.0997234	−	0.995015i	$-0.531796\pi$
$269$	−3.27117	−0.199447	−0.0997234	+	0.995015i	$0.531796\pi$
$270$	0	0
$271$	0.623949	0.0379022	0.0189511	−	0.999820i	$-0.493967\pi$
$271$	0.623949	0.0379022	0.0189511	+	0.999820i	$0.493967\pi$
$272$	5.22646	0.316901
$273$	0	0
$274$	−14.7950	−0.893797
$275$	0	0
$276$	7.17103	0.431645
$277$	−28.7950	−1.73012	−0.865061	−	0.501666i	$-0.832721\pi$
$277$	−28.7950	−1.73012	−0.865061	+	0.501666i	$0.832721\pi$
$278$	3.65794	0.219389
$279$	4.17103	0.249713
$280$	0	0
$281$	2.82897	0.168762	0.0843811	−	0.996434i	$-0.473109\pi$
$281$	2.82897	0.168762	0.0843811	+	0.996434i	$0.473109\pi$
$282$	11.5131	0.685594
$283$	−25.1370	−1.49424	−0.747121	−	0.664688i	$-0.768564\pi$
$283$	−25.1370	−1.49424	−0.747121	+	0.664688i	$0.768564\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	15.1710	0.897082
$287$	0	0
$288$	1.00000	0.0589256
$289$	10.3159	0.606817
$290$	0	0
$291$	−1.27117	−0.0745173
$292$	4.00000	0.234082
$293$	10.2265	0.597436	0.298718	−	0.954341i	$-0.403441\pi$
$293$	10.2265	0.597436	0.298718	+	0.954341i	$0.403441\pi$
$294$	0	0
$295$	0	0
$296$	7.17103	0.416808
$297$	−4.78426	−0.277611
$298$	22.3421	1.29424
$299$	−22.7395	−1.31506
$300$	0	0
$301$	0	0
$302$	16.5131	0.950222
$303$	−8.00000	−0.459588
$304$	−3.17103	−0.181871
$305$	0	0
$306$	5.22646	0.298777
$307$	−23.4577	−1.33880	−0.669400	−	0.742902i	$-0.733449\pi$
$307$	−23.4577	−1.33880	−0.669400	+	0.742902i	$0.733449\pi$
$308$	0	0
$309$	−11.2265	−0.638651
$310$	0	0
$311$	−5.65794	−0.320832	−0.160416	−	0.987049i	$-0.551284\pi$
$311$	−5.65794	−0.320832	−0.160416	+	0.987049i	$0.551284\pi$
$312$	−3.17103	−0.179524
$313$	−16.3868	−0.926235	−0.463118	−	0.886297i	$-0.653269\pi$
$313$	−16.3868	−0.926235	−0.463118	+	0.886297i	$0.653269\pi$
$314$	9.94457	0.561205
$315$	0	0
$316$	12.1710	0.684674
$317$	4.51309	0.253480	0.126740	−	0.991936i	$-0.459549\pi$
$317$	4.51309	0.253480	0.126740	+	0.991936i	$0.459549\pi$
$318$	−8.11560	−0.455100
$319$	−11.4189	−0.639337
$320$	0	0
$321$	−1.39749	−0.0780003
$322$	0	0
$323$	−16.5733	−0.922161
$324$	1.00000	0.0555556
$325$	0	0
$326$	2.88440	0.159752
$327$	12.4529	0.688648
$328$	−2.05543	−0.113492
$329$	0	0
$330$	0	0
$331$	−13.0816	−0.719030	−0.359515	−	0.933139i	$-0.617058\pi$
$331$	−13.0816	−0.719030	−0.359515	+	0.933139i	$0.617058\pi$
$332$	−3.39749	−0.186461
$333$	7.17103	0.392970
$334$	4.05543	0.221903
$335$	0	0
$336$	0	0
$337$	20.2973	1.10567	0.552834	−	0.833292i	$-0.313547\pi$
$337$	20.2973	1.10567	0.552834	+	0.833292i	$0.313547\pi$
$338$	−2.94457	−0.160163
$339$	6.34206	0.344453
$340$	0	0
$341$	−19.9553	−1.08064
$342$	−3.17103	−0.171470
$343$	0	0
$344$	−2.00000	−0.107833
$345$	0	0
$346$	6.39749	0.343931
$347$	14.4529	0.775873	0.387937	−	0.921686i	$-0.373188\pi$
$347$	14.4529	0.775873	0.387937	+	0.921686i	$0.373188\pi$
$348$	2.38677	0.127944
$349$	13.8891	0.743469	0.371734	−	0.928339i	$-0.378763\pi$
$349$	13.8891	0.743469	0.371734	+	0.928339i	$0.378763\pi$
$350$	0	0
$351$	−3.17103	−0.169257
$352$	−4.78426	−0.255002
$353$	−6.45292	−0.343454	−0.171727	−	0.985145i	$-0.554935\pi$
$353$	−6.45292	−0.343454	−0.171727	+	0.985145i	$0.554935\pi$
$354$	−10.7288	−0.570231
$355$	0	0
$356$	9.56852	0.507131
$357$	0	0
$358$	19.5131	1.03130
$359$	7.22646	0.381398	0.190699	−	0.981649i	$-0.438925\pi$
$359$	7.22646	0.381398	0.190699	+	0.981649i	$0.438925\pi$
$360$	0	0
$361$	−8.94457	−0.470767
$362$	−23.0262	−1.21023
$363$	11.8891	0.624018
$364$	0	0
$365$	0	0
$366$	7.11560	0.371938
$367$	−2.10014	−0.109626	−0.0548132	−	0.998497i	$-0.517456\pi$
$367$	−2.10014	−0.109626	−0.0548132	+	0.998497i	$0.517456\pi$
$368$	7.17103	0.373816
$369$	−2.05543	−0.107001
$370$	0	0
$371$	0	0
$372$	4.17103	0.216258
$373$	−23.2479	−1.20373	−0.601865	−	0.798598i	$-0.705576\pi$
$373$	−23.2479	−1.20373	−0.601865	+	0.798598i	$0.705576\pi$
$374$	−25.0047	−1.29296
$375$	0	0
$376$	11.5131	0.593742
$377$	−7.56852	−0.389799
$378$	0	0
$379$	2.16629	0.111275	0.0556374	−	0.998451i	$-0.482281\pi$
$379$	2.16629	0.111275	0.0556374	+	0.998451i	$0.482281\pi$
$380$	0	0
$381$	18.0107	0.922717
$382$	−10.2312	−0.523474
$383$	−8.05543	−0.411613	−0.205807	−	0.978593i	$-0.565982\pi$
$383$	−8.05543	−0.411613	−0.205807	+	0.978593i	$0.565982\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−11.6132	−0.591098
$387$	−2.00000	−0.101666
$388$	−1.27117	−0.0645339
$389$	11.8891	0.602803	0.301402	−	0.953497i	$-0.402545\pi$
$389$	11.8891	0.602803	0.301402	+	0.953497i	$0.402545\pi$
$390$	0	0
$391$	37.4791	1.89540
$392$	0	0
$393$	−5.55780	−0.280354
$394$	−2.39749	−0.120784
$395$	0	0
$396$	−4.78426	−0.240418
$397$	−16.1109	−0.808581	−0.404290	−	0.914631i	$-0.632481\pi$
$397$	−16.1109	−0.808581	−0.404290	+	0.914631i	$0.632481\pi$
$398$	−18.3421	−0.919404
$399$	0	0
$400$	0	0
$401$	24.9707	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$401$	24.9707	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$402$	9.56852	0.477234
$403$	−13.2265	−0.658857
$404$	−8.00000	−0.398015
$405$	0	0
$406$	0	0
$407$	−34.3081	−1.70059
$408$	5.22646	0.258748
$409$	3.05543	0.151081	0.0755406	−	0.997143i	$-0.475932\pi$
$409$	3.05543	0.151081	0.0755406	+	0.997143i	$0.475932\pi$
$410$	0	0
$411$	−14.7950	−0.729782
$412$	−11.2265	−0.553088
$413$	0	0
$414$	7.17103	0.352437
$415$	0	0
$416$	−3.17103	−0.155473
$417$	3.65794	0.179130
$418$	15.1710	0.742039
$419$	−20.1972	−0.986698	−0.493349	−	0.869831i	$-0.664227\pi$
$419$	−20.1972	−0.986698	−0.493349	+	0.869831i	$0.664227\pi$
$420$	0	0
$421$	−30.3635	−1.47983	−0.739913	−	0.672702i	$-0.765133\pi$
$421$	−30.3635	−1.47983	−0.739913	+	0.672702i	$0.765133\pi$
$422$	−2.82897	−0.137712
$423$	11.5131	0.559786
$424$	−8.11560	−0.394128
$425$	0	0
$426$	6.00000	0.290701
$427$	0	0
$428$	−1.39749	−0.0675502
$429$	15.1710	0.732464
$430$	0	0
$431$	24.5733	1.18365	0.591826	−	0.806066i	$-0.298407\pi$
$431$	24.5733	1.18365	0.591826	+	0.806066i	$0.298407\pi$
$432$	1.00000	0.0481125
$433$	4.68412	0.225104	0.112552	−	0.993646i	$-0.464097\pi$
$433$	4.68412	0.225104	0.112552	+	0.993646i	$0.464097\pi$
$434$	0	0
$435$	0	0
$436$	12.4529	0.596387
$437$	−22.7395	−1.08778
$438$	4.00000	0.191127
$439$	−30.5131	−1.45631	−0.728155	−	0.685412i	$-0.759622\pi$
$439$	−30.5131	−1.45631	−0.728155	+	0.685412i	$0.759622\pi$
$440$	0	0
$441$	0	0
$442$	−16.5733	−0.788310
$443$	−9.50835	−0.451755	−0.225878	−	0.974156i	$-0.572525\pi$
$443$	−9.50835	−0.451755	−0.225878	+	0.974156i	$0.572525\pi$
$444$	7.17103	0.340322
$445$	0	0
$446$	14.2973	0.676999
$447$	22.3421	1.05674
$448$	0	0
$449$	17.6239	0.831726	0.415863	−	0.909427i	$-0.363480\pi$
$449$	17.6239	0.831726	0.415863	+	0.909427i	$0.363480\pi$
$450$	0	0
$451$	9.83371	0.463051
$452$	6.34206	0.298305
$453$	16.5131	0.775853
$454$	11.0554	0.518857
$455$	0	0
$456$	−3.17103	−0.148497
$457$	0.386770	0.0180923	0.00904617	−	0.999959i	$-0.497120\pi$
$457$	0.386770	0.0180923	0.00904617	+	0.999959i	$0.497120\pi$
$458$	−18.4529	−0.862248
$459$	5.22646	0.243950
$460$	0	0
$461$	−37.3682	−1.74041	−0.870206	−	0.492688i	$-0.836014\pi$
$461$	−37.3682	−1.74041	−0.870206	+	0.492688i	$0.836014\pi$
$462$	0	0
$463$	24.3081	1.12969	0.564846	−	0.825196i	$-0.308936\pi$
$463$	24.3081	1.12969	0.564846	+	0.825196i	$0.308936\pi$
$464$	2.38677	0.110803
$465$	0	0
$466$	−9.11560	−0.422272
$467$	28.5733	1.32221	0.661106	−	0.750292i	$-0.270087\pi$
$467$	28.5733	1.32221	0.661106	+	0.750292i	$0.270087\pi$
$468$	−3.17103	−0.146581
$469$	0	0
$470$	0	0
$471$	9.94457	0.458222
$472$	−10.7288	−0.493834
$473$	9.56852	0.439961
$474$	12.1710	0.559034
$475$	0	0
$476$	0	0
$477$	−8.11560	−0.371588
$478$	13.1156	0.599894
$479$	−8.56378	−0.391289	−0.195645	−	0.980675i	$-0.562680\pi$
$479$	−8.56378	−0.391289	−0.195645	+	0.980675i	$0.562680\pi$
$480$	0	0
$481$	−22.7395	−1.03683
$482$	−5.34206	−0.243324
$483$	0	0
$484$	11.8891	0.540415
$485$	0	0
$486$	1.00000	0.0453609
$487$	−31.8444	−1.44301	−0.721504	−	0.692410i	$-0.756549\pi$
$487$	−31.8444	−1.44301	−0.721504	+	0.692410i	$0.756549\pi$
$488$	7.11560	0.322108
$489$	2.88440	0.130437
$490$	0	0
$491$	4.49763	0.202975	0.101488	−	0.994837i	$-0.467640\pi$
$491$	4.49763	0.202975	0.101488	+	0.994837i	$0.467640\pi$
$492$	−2.05543	−0.0926659
$493$	12.4744	0.561817
$494$	10.0554	0.452415
$495$	0	0
$496$	4.17103	0.187285
$497$	0	0
$498$	−3.39749	−0.152245
$499$	−0.342060	−0.0153127	−0.00765634	−	0.999971i	$-0.502437\pi$
$499$	−0.342060	−0.0153127	−0.00765634	+	0.999971i	$0.502437\pi$
$500$	0	0
$501$	4.05543	0.181183
$502$	27.9213	1.24619
$503$	−7.56852	−0.337464	−0.168732	−	0.985662i	$-0.553967\pi$
$503$	−7.56852	−0.337464	−0.168732	+	0.985662i	$0.553967\pi$
$504$	0	0
$505$	0	0
$506$	−34.3081	−1.52518
$507$	−2.94457	−0.130773
$508$	18.0107	0.799096
$509$	30.9600	1.37228	0.686140	−	0.727470i	$-0.259304\pi$
$509$	30.9600	1.37228	0.686140	+	0.727470i	$0.259304\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	−3.17103	−0.140004
$514$	−21.1370	−0.932315
$515$	0	0
$516$	−2.00000	−0.0880451
$517$	−55.0816	−2.42249
$518$	0	0
$519$	6.39749	0.280819
$520$	0	0
$521$	4.85041	0.212500	0.106250	−	0.994339i	$-0.466116\pi$
$521$	4.85041	0.212500	0.106250	+	0.994339i	$0.466116\pi$
$522$	2.38677	0.104466
$523$	4.25264	0.185955	0.0929774	−	0.995668i	$-0.470362\pi$
$523$	4.25264	0.185955	0.0929774	+	0.995668i	$0.470362\pi$
$524$	−5.55780	−0.242794
$525$	0	0
$526$	11.9106	0.519326
$527$	21.7997	0.949611
$528$	−4.78426	−0.208208
$529$	28.4237	1.23581
$530$	0	0
$531$	−10.7288	−0.465592
$532$	0	0
$533$	6.51783	0.282319
$534$	9.56852	0.414070
$535$	0	0
$536$	9.56852	0.413297
$537$	19.5131	0.842052
$538$	−3.27117	−0.141030
$539$	0	0
$540$	0	0
$541$	38.3635	1.64938	0.824688	−	0.565588i	$-0.191351\pi$
$541$	38.3635	1.64938	0.824688	+	0.565588i	$0.191351\pi$
$542$	0.623949	0.0268009
$543$	−23.0262	−0.988148
$544$	5.22646	0.224083
$545$	0	0
$546$	0	0
$547$	−1.44818	−0.0619197	−0.0309598	−	0.999521i	$-0.509856\pi$
$547$	−1.44818	−0.0619197	−0.0309598	+	0.999521i	$0.509856\pi$
$548$	−14.7950	−0.632010
$549$	7.11560	0.303686
$550$	0	0
$551$	−7.56852	−0.322430
$552$	7.17103	0.305219
$553$	0	0
$554$	−28.7950	−1.22338
$555$	0	0
$556$	3.65794	0.155131
$557$	13.0214	0.551736	0.275868	−	0.961196i	$-0.411035\pi$
$557$	13.0214	0.551736	0.275868	+	0.961196i	$0.411035\pi$
$558$	4.17103	0.176574
$559$	6.34206	0.268241
$560$	0	0
$561$	−25.0047	−1.05570
$562$	2.82897	0.119333
$563$	5.85041	0.246565	0.123283	−	0.992372i	$-0.460658\pi$
$563$	5.85041	0.246565	0.123283	+	0.992372i	$0.460658\pi$
$564$	11.5131	0.484789
$565$	0	0
$566$	−25.1370	−1.05659
$567$	0	0
$568$	6.00000	0.251754
$569$	−40.6501	−1.70414	−0.852071	−	0.523426i	$-0.824654\pi$
$569$	−40.6501	−1.70414	−0.852071	+	0.523426i	$0.824654\pi$
$570$	0	0
$571$	−3.22646	−0.135023	−0.0675116	−	0.997718i	$-0.521506\pi$
$571$	−3.22646	−0.135023	−0.0675116	+	0.997718i	$0.521506\pi$
$572$	15.1710	0.634333
$573$	−10.2312	−0.427415
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	11.6347	0.484358	0.242179	−	0.970232i	$-0.422138\pi$
$577$	11.6347	0.484358	0.242179	+	0.970232i	$0.422138\pi$
$578$	10.3159	0.429084
$579$	−11.6132	−0.482629
$580$	0	0
$581$	0	0
$582$	−1.27117	−0.0526917
$583$	38.8271	1.60806
$584$	4.00000	0.165521
$585$	0	0
$586$	10.2265	0.422451
$587$	28.9446	1.19467	0.597335	−	0.801992i	$-0.296226\pi$
$587$	28.9446	1.19467	0.597335	+	0.801992i	$0.296226\pi$
$588$	0	0
$589$	−13.2265	−0.544987
$590$	0	0
$591$	−2.39749	−0.0986195
$592$	7.17103	0.294728
$593$	−11.4577	−0.470510	−0.235255	−	0.971934i	$-0.575592\pi$
$593$	−11.4577	−0.470510	−0.235255	+	0.971934i	$0.575592\pi$
$594$	−4.78426	−0.196301
$595$	0	0
$596$	22.3421	0.915166
$597$	−18.3421	−0.750691
$598$	−22.7395	−0.929889
$599$	−3.00474	−0.122770	−0.0613852	−	0.998114i	$-0.519552\pi$
$599$	−3.00474	−0.122770	−0.0613852	+	0.998114i	$0.519552\pi$
$600$	0	0
$601$	40.0816	1.63496	0.817481	−	0.575955i	$-0.195370\pi$
$601$	40.0816	1.63496	0.817481	+	0.575955i	$0.195370\pi$
$602$	0	0
$603$	9.56852	0.389660
$604$	16.5131	0.671908
$605$	0	0
$606$	−8.00000	−0.324978
$607$	−26.1216	−1.06024	−0.530121	−	0.847922i	$-0.677854\pi$
$607$	−26.1216	−1.06024	−0.530121	+	0.847922i	$0.677854\pi$
$608$	−3.17103	−0.128602
$609$	0	0
$610$	0	0
$611$	−36.5083	−1.47697
$612$	5.22646	0.211267
$613$	−41.5131	−1.67670	−0.838349	−	0.545134i	$-0.816479\pi$
$613$	−41.5131	−1.67670	−0.838349	+	0.545134i	$0.816479\pi$
$614$	−23.4577	−0.946674
$615$	0	0
$616$	0	0
$617$	−31.1156	−1.25267	−0.626333	−	0.779555i	$-0.715445\pi$
$617$	−31.1156	−1.25267	−0.626333	+	0.779555i	$0.715445\pi$
$618$	−11.2265	−0.451594
$619$	32.9922	1.32607	0.663034	−	0.748589i	$-0.269268\pi$
$619$	32.9922	1.32607	0.663034	+	0.748589i	$0.269268\pi$
$620$	0	0
$621$	7.17103	0.287764
$622$	−5.65794	−0.226863
$623$	0	0
$624$	−3.17103	−0.126943
$625$	0	0
$626$	−16.3868	−0.654947
$627$	15.1710	0.605873
$628$	9.94457	0.396832
$629$	37.4791	1.49439
$630$	0	0
$631$	47.6501	1.89692	0.948461	−	0.316894i	$-0.102640\pi$
$631$	47.6501	1.89692	0.948461	+	0.316894i	$0.102640\pi$
$632$	12.1710	0.484138
$633$	−2.82897	−0.112441
$634$	4.51309	0.179238
$635$	0	0
$636$	−8.11560	−0.321804
$637$	0	0
$638$	−11.4189	−0.452080
$639$	6.00000	0.237356
$640$	0	0
$641$	10.2866	0.406297	0.203149	−	0.979148i	$-0.434883\pi$
$641$	10.2866	0.406297	0.203149	+	0.979148i	$0.434883\pi$
$642$	−1.39749	−0.0551545
$643$	−38.9368	−1.53552	−0.767758	−	0.640740i	$-0.778628\pi$
$643$	−38.9368	−1.53552	−0.767758	+	0.640740i	$0.778628\pi$
$644$	0	0
$645$	0	0
$646$	−16.5733	−0.652066
$647$	−33.8551	−1.33098	−0.665492	−	0.746405i	$-0.731778\pi$
$647$	−33.8551	−1.33098	−0.665492	+	0.746405i	$0.731778\pi$
$648$	1.00000	0.0392837
$649$	51.3295	2.01486
$650$	0	0
$651$	0	0
$652$	2.88440	0.112962
$653$	5.36350	0.209890	0.104945	−	0.994478i	$-0.466533\pi$
$653$	5.36350	0.209890	0.104945	+	0.994478i	$0.466533\pi$
$654$	12.4529	0.486948
$655$	0	0
$656$	−2.05543	−0.0802511
$657$	4.00000	0.156055
$658$	0	0
$659$	23.2265	0.904774	0.452387	−	0.891822i	$-0.350573\pi$
$659$	23.2265	0.904774	0.452387	+	0.891822i	$0.350573\pi$
$660$	0	0
$661$	−36.5947	−1.42337	−0.711684	−	0.702499i	$-0.752067\pi$
$661$	−36.5947	−1.42337	−0.711684	+	0.702499i	$0.752067\pi$
$662$	−13.0816	−0.508431
$663$	−16.5733	−0.643652
$664$	−3.39749	−0.131848
$665$	0	0
$666$	7.17103	0.277872
$667$	17.1156	0.662719
$668$	4.05543	0.156909
$669$	14.2973	0.552767
$670$	0	0
$671$	−34.0429	−1.31421
$672$	0	0
$673$	−17.7336	−0.683579	−0.341789	−	0.939777i	$-0.611033\pi$
$673$	−17.7336	−0.683579	−0.341789	+	0.939777i	$0.611033\pi$
$674$	20.2973	0.781825
$675$	0	0
$676$	−2.94457	−0.113253
$677$	26.2265	1.00796	0.503982	−	0.863714i	$-0.331868\pi$
$677$	26.2265	1.00796	0.503982	+	0.863714i	$0.331868\pi$
$678$	6.34206	0.243565
$679$	0	0
$680$	0	0
$681$	11.0554	0.423645
$682$	−19.9553	−0.764128
$683$	−49.2186	−1.88330	−0.941650	−	0.336595i	$-0.890725\pi$
$683$	−49.2186	−1.88330	−0.941650	+	0.336595i	$0.890725\pi$
$684$	−3.17103	−0.121247
$685$	0	0
$686$	0	0
$687$	−18.4529	−0.704023
$688$	−2.00000	−0.0762493
$689$	25.7348	0.980418
$690$	0	0
$691$	8.90584	0.338794	0.169397	−	0.985548i	$-0.445818\pi$
$691$	8.90584	0.338794	0.169397	+	0.985548i	$0.445818\pi$
$692$	6.39749	0.243196
$693$	0	0
$694$	14.4529	0.548625
$695$	0	0
$696$	2.38677	0.0904703
$697$	−10.7426	−0.406906
$698$	13.8891	0.525712
$699$	−9.11560	−0.344784
$700$	0	0
$701$	11.9553	0.451545	0.225773	−	0.974180i	$-0.427509\pi$
$701$	11.9553	0.451545	0.225773	+	0.974180i	$0.427509\pi$
$702$	−3.17103	−0.119683
$703$	−22.7395	−0.857638
$704$	−4.78426	−0.180314
$705$	0	0
$706$	−6.45292	−0.242859
$707$	0	0
$708$	−10.7288	−0.403214
$709$	52.1632	1.95903	0.979515	−	0.201369i	$-0.0645392\pi$
$709$	52.1632	1.95903	0.979515	+	0.201369i	$0.0645392\pi$
$710$	0	0
$711$	12.1710	0.456449
$712$	9.56852	0.358595
$713$	29.9106	1.12016
$714$	0	0
$715$	0	0
$716$	19.5131	0.729238
$717$	13.1156	0.489811
$718$	7.22646	0.269689
$719$	25.7902	0.961814	0.480907	−	0.876772i	$-0.340308\pi$
$719$	25.7902	0.961814	0.480907	+	0.876772i	$0.340308\pi$
$720$	0	0
$721$	0	0
$722$	−8.94457	−0.332882
$723$	−5.34206	−0.198673
$724$	−23.0262	−0.855761
$725$	0	0
$726$	11.8891	0.441247
$727$	34.6054	1.28344	0.641722	−	0.766937i	$-0.278220\pi$
$727$	34.6054	1.28344	0.641722	+	0.766937i	$0.278220\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−10.4529	−0.386615
$732$	7.11560	0.263000
$733$	−30.9707	−1.14393	−0.571965	−	0.820278i	$-0.693819\pi$
$733$	−30.9707	−1.14393	−0.571965	+	0.820278i	$0.693819\pi$
$734$	−2.10014	−0.0775176
$735$	0	0
$736$	7.17103	0.264328
$737$	−45.7783	−1.68626
$738$	−2.05543	−0.0756614
$739$	−2.17577	−0.0800370	−0.0400185	−	0.999199i	$-0.512742\pi$
$739$	−2.17577	−0.0800370	−0.0400185	+	0.999199i	$0.512742\pi$
$740$	0	0
$741$	10.0554	0.369395
$742$	0	0
$743$	−31.9446	−1.17193	−0.585966	−	0.810335i	$-0.699285\pi$
$743$	−31.9446	−1.17193	−0.585966	+	0.810335i	$0.699285\pi$
$744$	4.17103	0.152917
$745$	0	0
$746$	−23.2479	−0.851166
$747$	−3.39749	−0.124308
$748$	−25.0047	−0.914264
$749$	0	0
$750$	0	0
$751$	−5.60725	−0.204611	−0.102306	−	0.994753i	$-0.532622\pi$
$751$	−5.60725	−0.204611	−0.102306	+	0.994753i	$0.532622\pi$
$752$	11.5131	0.419839
$753$	27.9213	1.01751
$754$	−7.56852	−0.275629
$755$	0	0
$756$	0	0
$757$	−24.2312	−0.880698	−0.440349	−	0.897827i	$-0.645145\pi$
$757$	−24.2312	−0.880698	−0.440349	+	0.897827i	$0.645145\pi$
$758$	2.16629	0.0786832
$759$	−34.3081	−1.24530
$760$	0	0
$761$	39.8551	1.44475	0.722374	−	0.691503i	$-0.243051\pi$
$761$	39.8551	1.44475	0.722374	+	0.691503i	$0.243051\pi$
$762$	18.0107	0.652460
$763$	0	0
$764$	−10.2312	−0.370152
$765$	0	0
$766$	−8.05543	−0.291055
$767$	34.0214	1.22844
$768$	1.00000	0.0360844
$769$	29.7950	1.07443	0.537217	−	0.843444i	$-0.319476\pi$
$769$	29.7950	1.07443	0.537217	+	0.843444i	$0.319476\pi$
$770$	0	0
$771$	−21.1370	−0.761232
$772$	−11.6132	−0.417969
$773$	0.166290	0.00598102	0.00299051	−	0.999996i	$-0.499048\pi$
$773$	0.166290	0.00598102	0.00299051	+	0.999996i	$0.499048\pi$
$774$	−2.00000	−0.0718885
$775$	0	0
$776$	−1.27117	−0.0456324
$777$	0	0
$778$	11.8891	0.426246
$779$	6.51783	0.233525
$780$	0	0
$781$	−28.7056	−1.02717
$782$	37.4791	1.34025
$783$	2.38677	0.0852962
$784$	0	0
$785$	0	0
$786$	−5.55780	−0.198240
$787$	−28.4529	−1.01424	−0.507119	−	0.861876i	$-0.669289\pi$
$787$	−28.4529	−1.01424	−0.507119	+	0.861876i	$0.669289\pi$
$788$	−2.39749	−0.0854070
$789$	11.9106	0.424028
$790$	0	0
$791$	0	0
$792$	−4.78426	−0.170001
$793$	−22.5638	−0.801263
$794$	−16.1109	−0.571753
$795$	0	0
$796$	−18.3421	−0.650117
$797$	36.6334	1.29762	0.648811	−	0.760949i	$-0.275266\pi$
$797$	36.6334	1.29762	0.648811	+	0.760949i	$0.275266\pi$
$798$	0	0
$799$	60.1727	2.12876
$800$	0	0
$801$	9.56852	0.338087
$802$	24.9707	0.881748
$803$	−19.1370	−0.675331
$804$	9.56852	0.337456
$805$	0	0
$806$	−13.2265	−0.465882
$807$	−3.27117	−0.115151
$808$	−8.00000	−0.281439
$809$	−20.2866	−0.713240	−0.356620	−	0.934250i	$-0.616071\pi$
$809$	−20.2866	−0.713240	−0.356620	+	0.934250i	$0.616071\pi$
$810$	0	0
$811$	1.38079	0.0484861	0.0242431	−	0.999706i	$-0.492282\pi$
$811$	1.38079	0.0484861	0.0242431	+	0.999706i	$0.492282\pi$
$812$	0	0
$813$	0.623949	0.0218829
$814$	−34.3081	−1.20250
$815$	0	0
$816$	5.22646	0.182963
$817$	6.34206	0.221881
$818$	3.05543	0.106831
$819$	0	0
$820$	0	0
$821$	−17.4129	−0.607716	−0.303858	−	0.952717i	$-0.598275\pi$
$821$	−17.4129	−0.607716	−0.303858	+	0.952717i	$0.598275\pi$
$822$	−14.7950	−0.516034
$823$	20.5947	0.717886	0.358943	−	0.933359i	$-0.383137\pi$
$823$	20.5947	0.717886	0.358943	+	0.933359i	$0.383137\pi$
$824$	−11.2265	−0.391092
$825$	0	0
$826$	0	0
$827$	13.8504	0.481626	0.240813	−	0.970572i	$-0.422586\pi$
$827$	13.8504	0.481626	0.240813	+	0.970572i	$0.422586\pi$
$828$	7.17103	0.249211
$829$	−1.22646	−0.0425967	−0.0212984	−	0.999773i	$-0.506780\pi$
$829$	−1.22646	−0.0425967	−0.0212984	+	0.999773i	$0.506780\pi$
$830$	0	0
$831$	−28.7950	−0.998887
$832$	−3.17103	−0.109936
$833$	0	0
$834$	3.65794	0.126664
$835$	0	0
$836$	15.1710	0.524701
$837$	4.17103	0.144172
$838$	−20.1972	−0.697701
$839$	29.9106	1.03263	0.516314	−	0.856399i	$-0.327304\pi$
$839$	29.9106	1.03263	0.516314	+	0.856399i	$0.327304\pi$
$840$	0	0
$841$	−23.3033	−0.803563
$842$	−30.3635	−1.04640
$843$	2.82897	0.0974349
$844$	−2.82897	−0.0973772
$845$	0	0
$846$	11.5131	0.395828
$847$	0	0
$848$	−8.11560	−0.278691
$849$	−25.1370	−0.862701
$850$	0	0
$851$	51.4237	1.76278
$852$	6.00000	0.205557
$853$	−10.7181	−0.366981	−0.183491	−	0.983021i	$-0.558740\pi$
$853$	−10.7181	−0.366981	−0.183491	+	0.983021i	$0.558740\pi$
$854$	0	0
$855$	0	0
$856$	−1.39749	−0.0477652
$857$	39.6889	1.35575	0.677873	−	0.735179i	$-0.262902\pi$
$857$	39.6889	1.35575	0.677873	+	0.735179i	$0.262902\pi$
$858$	15.1710	0.517930
$859$	11.0047	0.375477	0.187738	−	0.982219i	$-0.439884\pi$
$859$	11.0047	0.375477	0.187738	+	0.982219i	$0.439884\pi$
$860$	0	0
$861$	0	0
$862$	24.5733	0.836969
$863$	8.62869	0.293724	0.146862	−	0.989157i	$-0.453083\pi$
$863$	8.62869	0.293724	0.146862	+	0.989157i	$0.453083\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	4.68412	0.159173
$867$	10.3159	0.350346
$868$	0	0
$869$	−58.2294	−1.97530
$870$	0	0
$871$	−30.3421	−1.02810
$872$	12.4529	0.421709
$873$	−1.27117	−0.0430226
$874$	−22.7395	−0.769177
$875$	0	0
$876$	4.00000	0.135147
$877$	−38.7395	−1.30814	−0.654071	−	0.756433i	$-0.726940\pi$
$877$	−38.7395	−1.30814	−0.654071	+	0.756433i	$0.726940\pi$
$878$	−30.5131	−1.02977
$879$	10.2265	0.344930
$880$	0	0
$881$	−6.03399	−0.203290	−0.101645	−	0.994821i	$-0.532411\pi$
$881$	−6.03399	−0.203290	−0.101645	+	0.994821i	$0.532411\pi$
$882$	0	0
$883$	7.77828	0.261760	0.130880	−	0.991398i	$-0.458220\pi$
$883$	7.77828	0.261760	0.130880	+	0.991398i	$0.458220\pi$
$884$	−16.5733	−0.557419
$885$	0	0
$886$	−9.50835	−0.319439
$887$	−24.8939	−0.835855	−0.417927	−	0.908480i	$-0.637243\pi$
$887$	−24.8939	−0.835855	−0.417927	+	0.908480i	$0.637243\pi$
$888$	7.17103	0.240644
$889$	0	0
$890$	0	0
$891$	−4.78426	−0.160279
$892$	14.2973	0.478711
$893$	−36.5083	−1.22171
$894$	22.3421	0.747230
$895$	0	0
$896$	0	0
$897$	−22.7395	−0.759251
$898$	17.6239	0.588119
$899$	9.95529	0.332027
$900$	0	0
$901$	−42.4159	−1.41308
$902$	9.83371	0.327427
$903$	0	0
$904$	6.34206	0.210934
$905$	0	0
$906$	16.5131	0.548611
$907$	−46.7056	−1.55083	−0.775416	−	0.631450i	$-0.782460\pi$
$907$	−46.7056	−1.55083	−0.775416	+	0.631450i	$0.782460\pi$
$908$	11.0554	0.366887
$909$	−8.00000	−0.265343
$910$	0	0
$911$	−6.00000	−0.198789	−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000	−0.198789	−0.0993944	+	0.995048i	$0.531691\pi$
$912$	−3.17103	−0.105003
$913$	16.2545	0.537944
$914$	0.386770	0.0127932
$915$	0	0
$916$	−18.4529	−0.609702
$917$	0	0
$918$	5.22646	0.172499
$919$	−34.2312	−1.12918	−0.564592	−	0.825370i	$-0.690966\pi$
$919$	−34.2312	−1.12918	−0.564592	+	0.825370i	$0.690966\pi$
$920$	0	0
$921$	−23.4577	−0.772956
$922$	−37.3682	−1.23066
$923$	−19.0262	−0.626254
$924$	0	0
$925$	0	0
$926$	24.3081	0.798813
$927$	−11.2265	−0.368725
$928$	2.38677	0.0783496
$929$	−27.6549	−0.907327	−0.453663	−	0.891173i	$-0.649883\pi$
$929$	−27.6549	−0.907327	−0.453663	+	0.891173i	$0.649883\pi$
$930$	0	0
$931$	0	0
$932$	−9.11560	−0.298591
$933$	−5.65794	−0.185233
$934$	28.5733	0.934946
$935$	0	0
$936$	−3.17103	−0.103648
$937$	−38.8397	−1.26884	−0.634419	−	0.772990i	$-0.718760\pi$
$937$	−38.8397	−1.26884	−0.634419	+	0.772990i	$0.718760\pi$
$938$	0	0
$939$	−16.3868	−0.534762
$940$	0	0
$941$	−19.7550	−0.643995	−0.321997	−	0.946741i	$-0.604354\pi$
$941$	−19.7550	−0.643995	−0.321997	+	0.946741i	$0.604354\pi$
$942$	9.94457	0.324012
$943$	−14.7395	−0.479986
$944$	−10.7288	−0.349194
$945$	0	0
$946$	9.56852	0.311099
$947$	−38.0524	−1.23654	−0.618268	−	0.785968i	$-0.712165\pi$
$947$	−38.0524	−1.23654	−0.618268	+	0.785968i	$0.712165\pi$
$948$	12.1710	0.395297
$949$	−12.6841	−0.411744
$950$	0	0
$951$	4.51309	0.146347
$952$	0	0
$953$	52.5947	1.70371	0.851855	−	0.523778i	$-0.175478\pi$
$953$	52.5947	1.70371	0.851855	+	0.523778i	$0.175478\pi$
$954$	−8.11560	−0.262752
$955$	0	0
$956$	13.1156	0.424189
$957$	−11.4189	−0.369122
$958$	−8.56378	−0.276683
$959$	0	0
$960$	0	0
$961$	−13.6025	−0.438791
$962$	−22.7395	−0.733152
$963$	−1.39749	−0.0450335
$964$	−5.34206	−0.172056
$965$	0	0
$966$	0	0
$967$	−4.61797	−0.148504	−0.0742520	−	0.997240i	$-0.523657\pi$
$967$	−4.61797	−0.148504	−0.0742520	+	0.997240i	$0.523657\pi$
$968$	11.8891	0.382131
$969$	−16.5733	−0.532410
$970$	0	0
$971$	38.9046	1.24851	0.624254	−	0.781221i	$-0.285403\pi$
$971$	38.9046	1.24851	0.624254	+	0.781221i	$0.285403\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−31.8444	−1.02036
$975$	0	0
$976$	7.11560	0.227765
$977$	−16.0214	−0.512571	−0.256286	−	0.966601i	$-0.582499\pi$
$977$	−16.0214	−0.512571	−0.256286	+	0.966601i	$0.582499\pi$
$978$	2.88440	0.0922329
$979$	−45.7783	−1.46308
$980$	0	0
$981$	12.4529	0.397591
$982$	4.49763	0.143525
$983$	−2.05543	−0.0655580	−0.0327790	−	0.999463i	$-0.510436\pi$
$983$	−2.05543	−0.0655580	−0.0327790	+	0.999463i	$0.510436\pi$
$984$	−2.05543	−0.0655247
$985$	0	0
$986$	12.4744	0.397264
$987$	0	0
$988$	10.0554	0.319906
$989$	−14.3421	−0.456051
$990$	0	0
$991$	−27.1972	−0.863948	−0.431974	−	0.901886i	$-0.642183\pi$
$991$	−27.1972	−0.863948	−0.431974	+	0.901886i	$0.642183\pi$
$992$	4.17103	0.132430
$993$	−13.0816	−0.415132
$994$	0	0
$995$	0	0
$996$	−3.39749	−0.107654
$997$	−18.4744	−0.585089	−0.292544	−	0.956252i	$-0.594502\pi$
$997$	−18.4744	−0.585089	−0.292544	+	0.956252i	$0.594502\pi$
$998$	−0.342060	−0.0108277
$999$	7.17103	0.226881

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7350.2.a.dq.1.1		3
5.2	odd	4		1470.2.g.i.589.5		6
5.3	odd	4		1470.2.g.i.589.2		6
5.4	even	2		7350.2.a.dn.1.1		3
7.2	even	3		1050.2.i.u.151.2		6
7.4	even	3		1050.2.i.u.751.2		6
7.6	odd	2		7350.2.a.dp.1.1		3
35.2	odd	12		210.2.n.b.109.4	yes	12
35.3	even	12		1470.2.n.j.79.6		12
35.4	even	6		1050.2.i.v.751.2		6
35.9	even	6		1050.2.i.v.151.2		6
35.12	even	12		1470.2.n.j.949.6		12
35.13	even	4		1470.2.g.h.589.2		6
35.17	even	12		1470.2.n.j.79.1		12
35.18	odd	12		210.2.n.b.79.4	yes	12
35.23	odd	12		210.2.n.b.109.3	yes	12
35.27	even	4		1470.2.g.h.589.5		6
35.32	odd	12		210.2.n.b.79.3	✓	12
35.33	even	12		1470.2.n.j.949.1		12
35.34	odd	2		7350.2.a.do.1.1		3
105.2	even	12		630.2.u.f.109.3		12
105.23	even	12		630.2.u.f.109.4		12
105.32	even	12		630.2.u.f.289.4		12
105.53	even	12		630.2.u.f.289.3		12
140.23	even	12		1680.2.di.c.529.6		12
140.67	even	12		1680.2.di.c.289.6		12
140.107	even	12		1680.2.di.c.529.1		12
140.123	even	12		1680.2.di.c.289.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.n.b.79.3	✓	12	35.32	odd	12
210.2.n.b.79.4	yes	12	35.18	odd	12
210.2.n.b.109.3	yes	12	35.23	odd	12
210.2.n.b.109.4	yes	12	35.2	odd	12
630.2.u.f.109.3		12	105.2	even	12
630.2.u.f.109.4		12	105.23	even	12
630.2.u.f.289.3		12	105.53	even	12
630.2.u.f.289.4		12	105.32	even	12
1050.2.i.u.151.2		6	7.2	even	3
1050.2.i.u.751.2		6	7.4	even	3
1050.2.i.v.151.2		6	35.9	even	6
1050.2.i.v.751.2		6	35.4	even	6
1470.2.g.h.589.2		6	35.13	even	4
1470.2.g.h.589.5		6	35.27	even	4
1470.2.g.i.589.2		6	5.3	odd	4
1470.2.g.i.589.5		6	5.2	odd	4
1470.2.n.j.79.1		12	35.17	even	12
1470.2.n.j.79.6		12	35.3	even	12
1470.2.n.j.949.1		12	35.33	even	12
1470.2.n.j.949.6		12	35.12	even	12
1680.2.di.c.289.1		12	140.123	even	12
1680.2.di.c.289.6		12	140.67	even	12
1680.2.di.c.529.1		12	140.107	even	12
1680.2.di.c.529.6		12	140.23	even	12
7350.2.a.dn.1.1		3	5.4	even	2
7350.2.a.do.1.1		3	35.34	odd	2
7350.2.a.dp.1.1		3	7.6	odd	2
7350.2.a.dq.1.1		3	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -4.78426 q^{11} +1.00000 q^{12} -3.17103 q^{13} +1.00000 q^{16} +5.22646 q^{17} +1.00000 q^{18} -3.17103 q^{19} -4.78426 q^{22} +7.17103 q^{23} +1.00000 q^{24} -3.17103 q^{26} +1.00000 q^{27} +2.38677 q^{29} +4.17103 q^{31} +1.00000 q^{32} -4.78426 q^{33} +5.22646 q^{34} +1.00000 q^{36} +7.17103 q^{37} -3.17103 q^{38} -3.17103 q^{39} -2.05543 q^{41} -2.00000 q^{43} -4.78426 q^{44} +7.17103 q^{46} +11.5131 q^{47} +1.00000 q^{48} +5.22646 q^{51} -3.17103 q^{52} -8.11560 q^{53} +1.00000 q^{54} -3.17103 q^{57} +2.38677 q^{58} -10.7288 q^{59} +7.11560 q^{61} +4.17103 q^{62} +1.00000 q^{64} -4.78426 q^{66} +9.56852 q^{67} +5.22646 q^{68} +7.17103 q^{69} +6.00000 q^{71} +1.00000 q^{72} +4.00000 q^{73} +7.17103 q^{74} -3.17103 q^{76} -3.17103 q^{78} +12.1710 q^{79} +1.00000 q^{81} -2.05543 q^{82} -3.39749 q^{83} -2.00000 q^{86} +2.38677 q^{87} -4.78426 q^{88} +9.56852 q^{89} +7.17103 q^{92} +4.17103 q^{93} +11.5131 q^{94} +1.00000 q^{96} -1.27117 q^{97} -4.78426 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^6 + 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{6} + 3 q^{8} + 3 q^{9} + 3 q^{11} + 3 q^{12} + 3 q^{13} + 3 q^{16} + 6 q^{17} + 3 q^{18} + 3 q^{19} + 3 q^{22} + 9 q^{23} + 3 q^{24} + 3 q^{26} + 3 q^{27} + 12 q^{29} + 3 q^{32} + 3 q^{33} + 6 q^{34} + 3 q^{36} + 9 q^{37} + 3 q^{38} + 3 q^{39} - 9 q^{41} - 6 q^{43} + 3 q^{44} + 9 q^{46} - 3 q^{47} + 3 q^{48} + 6 q^{51} + 3 q^{52} - 9 q^{53} + 3 q^{54} + 3 q^{57} + 12 q^{58} - 12 q^{59} + 6 q^{61} + 3 q^{64} + 3 q^{66} - 6 q^{67} + 6 q^{68} + 9 q^{69} + 18 q^{71} + 3 q^{72} + 12 q^{73} + 9 q^{74} + 3 q^{76} + 3 q^{78} + 24 q^{79} + 3 q^{81} - 9 q^{82} + 12 q^{83} - 6 q^{86} + 12 q^{87} + 3 q^{88} - 6 q^{89} + 9 q^{92} - 3 q^{94} + 3 q^{96} - 24 q^{97} + 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^6 + 3 * q^8 + 3 * q^9 + 3 * q^11 + 3 * q^12 + 3 * q^13 + 3 * q^16 + 6 * q^17 + 3 * q^18 + 3 * q^19 + 3 * q^22 + 9 * q^23 + 3 * q^24 + 3 * q^26 + 3 * q^27 + 12 * q^29 + 3 * q^32 + 3 * q^33 + 6 * q^34 + 3 * q^36 + 9 * q^37 + 3 * q^38 + 3 * q^39 - 9 * q^41 - 6 * q^43 + 3 * q^44 + 9 * q^46 - 3 * q^47 + 3 * q^48 + 6 * q^51 + 3 * q^52 - 9 * q^53 + 3 * q^54 + 3 * q^57 + 12 * q^58 - 12 * q^59 + 6 * q^61 + 3 * q^64 + 3 * q^66 - 6 * q^67 + 6 * q^68 + 9 * q^69 + 18 * q^71 + 3 * q^72 + 12 * q^73 + 9 * q^74 + 3 * q^76 + 3 * q^78 + 24 * q^79 + 3 * q^81 - 9 * q^82 + 12 * q^83 - 6 * q^86 + 12 * q^87 + 3 * q^88 - 6 * q^89 + 9 * q^92 - 3 * q^94 + 3 * q^96 - 24 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more