LMFDB - Embedded newform 7350.2.a.dt.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:	$1$
Dimension:	$4$
Coefficient field:	4.4.10304.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 8x + 8 $ x^4 - 2x^3 - 7x^2 + 8*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1470)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.16053$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -3.05545 q^{11} -1.00000 q^{12} -1.64124 q^{13} +1.00000 q^{16} +2.90685 q^{17} +1.00000 q^{18} -2.21016 q^{19} -3.05545 q^{22} +1.78984 q^{23} -1.00000 q^{24} -1.64124 q^{26} -1.00000 q^{27} +5.58667 q^{29} -0.944547 q^{31} +1.00000 q^{32} +3.05545 q^{33} +2.90685 q^{34} +1.00000 q^{36} -10.7123 q^{37} -2.21016 q^{38} +1.64124 q^{39} -6.67982 q^{41} +10.5021 q^{43} -3.05545 q^{44} +1.78984 q^{46} -11.3765 q^{47} -1.00000 q^{48} -2.90685 q^{51} -1.64124 q^{52} +5.78984 q^{53} -1.00000 q^{54} +2.21016 q^{57} +5.58667 q^{58} +5.97792 q^{59} -0.445811 q^{61} -0.944547 q^{62} +1.00000 q^{64} +3.05545 q^{66} +1.26561 q^{67} +2.90685 q^{68} -1.78984 q^{69} -14.8523 q^{71} +1.00000 q^{72} -7.91636 q^{73} -10.7123 q^{74} -2.21016 q^{76} +1.64124 q^{78} +14.2990 q^{79} +1.00000 q^{81} -6.67982 q^{82} -0.874366 q^{83} +10.5021 q^{86} -5.58667 q^{87} -3.05545 q^{88} -17.5083 q^{89} +1.78984 q^{92} +0.944547 q^{93} -11.3765 q^{94} -1.00000 q^{96} +10.4476 q^{97} -3.05545 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{6} + 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^6 + 4 * q^8 + 4 * q^9	$ 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{6} + 4 q^{8} + 4 q^{9} - 4 q^{12} + 4 q^{16} - 4 q^{17} + 4 q^{18} - 12 q^{19} + 4 q^{23} - 4 q^{24} - 4 q^{27} - 8 q^{29} - 16 q^{31} + 4 q^{32} - 4 q^{34} + 4 q^{36} - 8 q^{37} - 12 q^{38} - 12 q^{41} + 4 q^{43} + 4 q^{46} - 12 q^{47} - 4 q^{48} + 4 q^{51} + 20 q^{53} - 4 q^{54} + 12 q^{57} - 8 q^{58} - 20 q^{59} - 12 q^{61} - 16 q^{62} + 4 q^{64} - 4 q^{67} - 4 q^{68} - 4 q^{69} - 20 q^{71} + 4 q^{72} + 12 q^{73} - 8 q^{74} - 12 q^{76} - 8 q^{79} + 4 q^{81} - 12 q^{82} - 8 q^{83} + 4 q^{86} + 8 q^{87} - 44 q^{89} + 4 q^{92} + 16 q^{93} - 12 q^{94} - 4 q^{96} - 20 q^{97}+O(q^{100}) $ 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^6 + 4 * q^8 + 4 * q^9 - 4 * q^12 + 4 * q^16 - 4 * q^17 + 4 * q^18 - 12 * q^19 + 4 * q^23 - 4 * q^24 - 4 * q^27 - 8 * q^29 - 16 * q^31 + 4 * q^32 - 4 * q^34 + 4 * q^36 - 8 * q^37 - 12 * q^38 - 12 * q^41 + 4 * q^43 + 4 * q^46 - 12 * q^47 - 4 * q^48 + 4 * q^51 + 20 * q^53 - 4 * q^54 + 12 * q^57 - 8 * q^58 - 20 * q^59 - 12 * q^61 - 16 * q^62 + 4 * q^64 - 4 * q^67 - 4 * q^68 - 4 * q^69 - 20 * q^71 + 4 * q^72 + 12 * q^73 - 8 * q^74 - 12 * q^76 - 8 * q^79 + 4 * q^81 - 12 * q^82 - 8 * q^83 + 4 * q^86 + 8 * q^87 - 44 * q^89 + 4 * q^92 + 16 * q^93 - 12 * q^94 - 4 * q^96 - 20 * q^97

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
$11$	−3.05545	−0.921254	−0.460627	−	0.887594i	$-0.652375\pi$
$11$	−3.05545	−0.921254	−0.460627	+	0.887594i	$0.652375\pi$
$12$	−1.00000	−0.288675
$13$	−1.64124	−0.455198	−0.227599	−	0.973755i	$-0.573088\pi$
$13$	−1.64124	−0.455198	−0.227599	+	0.973755i	$0.573088\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	2.90685	0.705014	0.352507	−	0.935809i	$-0.385329\pi$
$17$	2.90685	0.705014	0.352507	+	0.935809i	$0.385329\pi$
$18$	1.00000	0.235702
$19$	−2.21016	−0.507045	−0.253522	−	0.967330i	$-0.581589\pi$
$19$	−2.21016	−0.507045	−0.253522	+	0.967330i	$0.581589\pi$
$20$	0	0
$21$	0	0
$22$	−3.05545	−0.651425
$23$	1.78984	0.373208	0.186604	−	0.982435i	$-0.440252\pi$
$23$	1.78984	0.373208	0.186604	+	0.982435i	$0.440252\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−1.64124	−0.321873
$27$	−1.00000	−0.192450
$28$	0	0
$29$	5.58667	1.03742	0.518710	−	0.854951i	$-0.326413\pi$
$29$	5.58667	1.03742	0.518710	+	0.854951i	$0.326413\pi$
$30$	0	0
$31$	−0.944547	−0.169646	−0.0848228	−	0.996396i	$-0.527032\pi$
$31$	−0.944547	−0.169646	−0.0848228	+	0.996396i	$0.527032\pi$
$32$	1.00000	0.176777
$33$	3.05545	0.531886
$34$	2.90685	0.498521
$35$	0	0
$36$	1.00000	0.166667
$37$	−10.7123	−1.76109	−0.880546	−	0.473960i	$-0.842824\pi$
$37$	−10.7123	−1.76109	−0.880546	+	0.473960i	$0.842824\pi$
$38$	−2.21016	−0.358535
$39$	1.64124	0.262809
$40$	0	0
$41$	−6.67982	−1.04321	−0.521607	−	0.853186i	$-0.674667\pi$
$41$	−6.67982	−1.04321	−0.521607	+	0.853186i	$0.674667\pi$
$42$	0	0
$43$	10.5021	1.60156	0.800781	−	0.598957i	$-0.204418\pi$
$43$	10.5021	1.60156	0.800781	+	0.598957i	$0.204418\pi$
$44$	−3.05545	−0.460627
$45$	0	0
$46$	1.78984	0.263898
$47$	−11.3765	−1.65944	−0.829718	−	0.558183i	$-0.811499\pi$
$47$	−11.3765	−1.65944	−0.829718	+	0.558183i	$0.811499\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	0	0
$51$	−2.90685	−0.407040
$52$	−1.64124	−0.227599
$53$	5.78984	0.795296	0.397648	−	0.917538i	$-0.369827\pi$
$53$	5.78984	0.795296	0.397648	+	0.917538i	$0.369827\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	2.21016	0.292742
$58$	5.58667	0.733566
$59$	5.97792	0.778259	0.389129	−	0.921183i	$-0.372776\pi$
$59$	5.97792	0.778259	0.389129	+	0.921183i	$0.372776\pi$
$60$	0	0
$61$	−0.445811	−0.0570802	−0.0285401	−	0.999593i	$-0.509086\pi$
$61$	−0.445811	−0.0570802	−0.0285401	+	0.999593i	$0.509086\pi$
$62$	−0.944547	−0.119958
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	3.05545	0.376100
$67$	1.26561	0.154619	0.0773094	−	0.997007i	$-0.475367\pi$
$67$	1.26561	0.154619	0.0773094	+	0.997007i	$0.475367\pi$
$68$	2.90685	0.352507
$69$	−1.78984	−0.215472
$70$	0	0
$71$	−14.8523	−1.76264	−0.881321	−	0.472518i	$-0.843345\pi$
$71$	−14.8523	−1.76264	−0.881321	+	0.472518i	$0.843345\pi$
$72$	1.00000	0.117851
$73$	−7.91636	−0.926540	−0.463270	−	0.886217i	$-0.653324\pi$
$73$	−7.91636	−0.926540	−0.463270	+	0.886217i	$0.653324\pi$
$74$	−10.7123	−1.24528
$75$	0	0
$76$	−2.21016	−0.253522
$77$	0	0
$78$	1.64124	0.185834
$79$	14.2990	1.60876	0.804380	−	0.594115i	$-0.202497\pi$
$79$	14.2990	1.60876	0.804380	+	0.594115i	$0.202497\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−6.67982	−0.737663
$83$	−0.874366	−0.0959741	−0.0479871	−	0.998848i	$-0.515281\pi$
$83$	−0.874366	−0.0959741	−0.0479871	+	0.998848i	$0.515281\pi$
$84$	0	0
$85$	0	0
$86$	10.5021	1.13248
$87$	−5.58667	−0.598954
$88$	−3.05545	−0.325712
$89$	−17.5083	−1.85587	−0.927935	−	0.372741i	$-0.878418\pi$
$89$	−17.5083	−1.85587	−0.927935	+	0.372741i	$0.878418\pi$
$90$	0	0
$91$	0	0
$92$	1.78984	0.186604
$93$	0.944547	0.0979450
$94$	−11.3765	−1.17340
$95$	0	0
$96$	−1.00000	−0.102062
$97$	10.4476	1.06079	0.530396	−	0.847750i	$-0.322043\pi$
$97$	10.4476	1.06079	0.530396	+	0.847750i	$0.322043\pi$
$98$	0	0
$99$	−3.05545	−0.307085
$100$	0	0
$101$	−15.4549	−1.53782	−0.768912	−	0.639355i	$-0.779202\pi$
$101$	−15.4549	−1.53782	−0.768912	+	0.639355i	$0.779202\pi$
$102$	−2.90685	−0.287821
$103$	−3.12563	−0.307978	−0.153989	−	0.988073i	$-0.549212\pi$
$103$	−3.12563	−0.307978	−0.153989	+	0.988073i	$0.549212\pi$
$104$	−1.64124	−0.160937
$105$	0	0
$106$	5.78984	0.562359
$107$	−4.78249	−0.462341	−0.231170	−	0.972913i	$-0.574255\pi$
$107$	−4.78249	−0.462341	−0.231170	+	0.972913i	$0.574255\pi$
$108$	−1.00000	−0.0962250
$109$	−9.10355	−0.871962	−0.435981	−	0.899956i	$-0.643599\pi$
$109$	−9.10355	−0.871962	−0.435981	+	0.899956i	$0.643599\pi$
$110$	0	0
$111$	10.7123	1.01677
$112$	0	0
$113$	14.3982	1.35447	0.677236	−	0.735766i	$-0.263178\pi$
$113$	14.3982	1.35447	0.677236	+	0.735766i	$0.263178\pi$
$114$	2.21016	0.207000
$115$	0	0
$116$	5.58667	0.518710
$117$	−1.64124	−0.151733
$118$	5.97792	0.550312
$119$	0	0
$120$	0	0
$121$	−1.66421	−0.151292
$122$	−0.445811	−0.0403618
$123$	6.67982	0.602299
$124$	−0.944547	−0.0848228
$125$	0	0
$126$	0	0
$127$	−8.45405	−0.750176	−0.375088	−	0.926989i	$-0.622387\pi$
$127$	−8.45405	−0.750176	−0.375088	+	0.926989i	$0.622387\pi$
$128$	1.00000	0.0883883
$129$	−10.5021	−0.924663
$130$	0	0
$131$	−8.32106	−0.727015	−0.363507	−	0.931591i	$-0.618421\pi$
$131$	−8.32106	−0.727015	−0.363507	+	0.931591i	$0.618421\pi$
$132$	3.05545	0.265943
$133$	0	0
$134$	1.26561	0.109332
$135$	0	0
$136$	2.90685	0.249260
$137$	9.86701	0.842996	0.421498	−	0.906829i	$-0.361505\pi$
$137$	9.86701	0.842996	0.421498	+	0.906829i	$0.361505\pi$
$138$	−1.78984	−0.152362
$139$	−16.9632	−1.43880	−0.719399	−	0.694597i	$-0.755583\pi$
$139$	−16.9632	−1.43880	−0.719399	+	0.694597i	$0.755583\pi$
$140$	0	0
$141$	11.3765	0.958075
$142$	−14.8523	−1.24638
$143$	5.01473	0.419353
$144$	1.00000	0.0833333
$145$	0	0
$146$	−7.91636	−0.655163
$147$	0	0
$148$	−10.7123	−0.880546
$149$	8.04810	0.659326	0.329663	−	0.944099i	$-0.393065\pi$
$149$	8.04810	0.659326	0.329663	+	0.944099i	$0.393065\pi$
$150$	0	0
$151$	−9.97792	−0.811991	−0.405996	−	0.913875i	$-0.633075\pi$
$151$	−9.97792	−0.811991	−0.405996	+	0.913875i	$0.633075\pi$
$152$	−2.21016	−0.179267
$153$	2.90685	0.235005
$154$	0	0
$155$	0	0
$156$	1.64124	0.131404
$157$	12.6676	1.01099	0.505493	−	0.862831i	$-0.331311\pi$
$157$	12.6676	1.01099	0.505493	+	0.862831i	$0.331311\pi$
$158$	14.2990	1.13757
$159$	−5.78984	−0.459164
$160$	0	0
$161$	0	0
$162$	1.00000	0.0785674
$163$	−23.5646	−1.84572	−0.922861	−	0.385134i	$-0.874155\pi$
$163$	−23.5646	−1.84572	−0.922861	+	0.385134i	$0.874155\pi$
$164$	−6.67982	−0.521607
$165$	0	0
$166$	−0.874366	−0.0678639
$167$	−18.0818	−1.39921	−0.699607	−	0.714528i	$-0.746642\pi$
$167$	−18.0818	−1.39921	−0.699607	+	0.714528i	$0.746642\pi$
$168$	0	0
$169$	−10.3063	−0.792795
$170$	0	0
$171$	−2.21016	−0.169015
$172$	10.5021	0.800781
$173$	1.93845	0.147377	0.0736887	−	0.997281i	$-0.476523\pi$
$173$	1.93845	0.147377	0.0736887	+	0.997281i	$0.476523\pi$
$174$	−5.58667	−0.423525
$175$	0	0
$176$	−3.05545	−0.230313
$177$	−5.97792	−0.449328
$178$	−17.5083	−1.31230
$179$	−24.4463	−1.82720	−0.913602	−	0.406609i	$-0.866711\pi$
$179$	−24.4463	−1.82720	−0.913602	+	0.406609i	$0.866711\pi$
$180$	0	0
$181$	16.9788	1.26202	0.631012	−	0.775773i	$-0.282640\pi$
$181$	16.9788	1.26202	0.631012	+	0.775773i	$0.282640\pi$
$182$	0	0
$183$	0.445811	0.0329553
$184$	1.78984	0.131949
$185$	0	0
$186$	0.944547	0.0692576
$187$	−8.88174	−0.649497
$188$	−11.3765	−0.829718
$189$	0	0
$190$	0	0
$191$	0.741377	0.0536442	0.0268221	−	0.999640i	$-0.491461\pi$
$191$	0.741377	0.0536442	0.0268221	+	0.999640i	$0.491461\pi$
$192$	−1.00000	−0.0721688
$193$	−6.11091	−0.439873	−0.219936	−	0.975514i	$-0.570585\pi$
$193$	−6.11091	−0.439873	−0.219936	+	0.975514i	$0.570585\pi$
$194$	10.4476	0.750093
$195$	0	0
$196$	0	0
$197$	20.6832	1.47362	0.736810	−	0.676100i	$-0.236331\pi$
$197$	20.6832	1.47362	0.736810	+	0.676100i	$0.236331\pi$
$198$	−3.05545	−0.217142
$199$	−12.0702	−0.855632	−0.427816	−	0.903866i	$-0.640717\pi$
$199$	−12.0702	−0.855632	−0.427816	+	0.903866i	$0.640717\pi$
$200$	0	0
$201$	−1.26561	−0.0892692
$202$	−15.4549	−1.08741
$203$	0	0
$204$	−2.90685	−0.203520
$205$	0	0
$206$	−3.12563	−0.217773
$207$	1.78984	0.124403
$208$	−1.64124	−0.113799
$209$	6.75303	0.467117
$210$	0	0
$211$	2.95153	0.203192	0.101596	−	0.994826i	$-0.467605\pi$
$211$	2.95153	0.203192	0.101596	+	0.994826i	$0.467605\pi$
$212$	5.78984	0.397648
$213$	14.8523	1.01766
$214$	−4.78249	−0.326924
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−9.10355	−0.616570
$219$	7.91636	0.534938
$220$	0	0
$221$	−4.77083	−0.320921
$222$	10.7123	0.718963
$223$	11.7678	0.788027	0.394014	−	0.919105i	$-0.371086\pi$
$223$	11.7678	0.788027	0.394014	+	0.919105i	$0.371086\pi$
$224$	0	0
$225$	0	0
$226$	14.3982	0.957756
$227$	−13.2365	−0.878540	−0.439270	−	0.898355i	$-0.644763\pi$
$227$	−13.2365	−0.878540	−0.439270	+	0.898355i	$0.644763\pi$
$228$	2.21016	0.146371
$229$	−25.2448	−1.66822	−0.834111	−	0.551597i	$-0.814019\pi$
$229$	−25.2448	−1.66822	−0.834111	+	0.551597i	$0.814019\pi$
$230$	0	0
$231$	0	0
$232$	5.58667	0.366783
$233$	3.32842	0.218052	0.109026	−	0.994039i	$-0.465227\pi$
$233$	3.32842	0.218052	0.109026	+	0.994039i	$0.465227\pi$
$234$	−1.64124	−0.107291
$235$	0	0
$236$	5.97792	0.389129
$237$	−14.2990	−0.928819
$238$	0	0
$239$	24.8713	1.60879	0.804396	−	0.594094i	$-0.202489\pi$
$239$	24.8713	1.60879	0.804396	+	0.594094i	$0.202489\pi$
$240$	0	0
$241$	−10.3535	−0.666931	−0.333465	−	0.942762i	$-0.608218\pi$
$241$	−10.3535	−0.666931	−0.333465	+	0.942762i	$0.608218\pi$
$242$	−1.66421	−0.106979
$243$	−1.00000	−0.0641500
$244$	−0.445811	−0.0285401
$245$	0	0
$246$	6.67982	0.425890
$247$	3.62740	0.230806
$248$	−0.944547	−0.0599788
$249$	0.874366	0.0554107
$250$	0	0
$251$	−16.5502	−1.04464	−0.522321	−	0.852749i	$-0.674934\pi$
$251$	−16.5502	−1.04464	−0.522321	+	0.852749i	$0.674934\pi$
$252$	0	0
$253$	−5.46878	−0.343819
$254$	−8.45405	−0.530454
$255$	0	0
$256$	1.00000	0.0625000
$257$	24.4546	1.52543	0.762717	−	0.646732i	$-0.223865\pi$
$257$	24.4546	1.52543	0.762717	+	0.646732i	$0.223865\pi$
$258$	−10.5021	−0.653835
$259$	0	0
$260$	0	0
$261$	5.58667	0.345806
$262$	−8.32106	−0.514077
$263$	−12.5091	−0.771346	−0.385673	−	0.922635i	$-0.626031\pi$
$263$	−12.5091	−0.771346	−0.385673	+	0.922635i	$0.626031\pi$
$264$	3.05545	0.188050
$265$	0	0
$266$	0	0
$267$	17.5083	1.07149
$268$	1.26561	0.0773094
$269$	18.9666	1.15641	0.578207	−	0.815890i	$-0.303753\pi$
$269$	18.9666	1.15641	0.578207	+	0.815890i	$0.303753\pi$
$270$	0	0
$271$	−19.6638	−1.19449	−0.597247	−	0.802058i	$-0.703739\pi$
$271$	−19.6638	−1.19449	−0.597247	+	0.802058i	$0.703739\pi$
$272$	2.90685	0.176254
$273$	0	0
$274$	9.86701	0.596088
$275$	0	0
$276$	−1.78984	−0.107736
$277$	−6.13998	−0.368915	−0.184458	−	0.982840i	$-0.559053\pi$
$277$	−6.13998	−0.368915	−0.184458	+	0.982840i	$0.559053\pi$
$278$	−16.9632	−1.01738
$279$	−0.944547	−0.0565486
$280$	0	0
$281$	−12.1330	−0.723793	−0.361897	−	0.932218i	$-0.617871\pi$
$281$	−12.1330	−0.723793	−0.361897	+	0.932218i	$0.617871\pi$
$282$	11.3765	0.677462
$283$	17.4172	1.03535	0.517674	−	0.855578i	$-0.326798\pi$
$283$	17.4172	1.03535	0.517674	+	0.855578i	$0.326798\pi$
$284$	−14.8523	−0.881321
$285$	0	0
$286$	5.01473	0.296527
$287$	0	0
$288$	1.00000	0.0589256
$289$	−8.55023	−0.502955
$290$	0	0
$291$	−10.4476	−0.612448
$292$	−7.91636	−0.463270
$293$	23.7521	1.38762	0.693808	−	0.720160i	$-0.255932\pi$
$293$	23.7521	1.38762	0.693808	+	0.720160i	$0.255932\pi$
$294$	0	0
$295$	0	0
$296$	−10.7123	−0.622640
$297$	3.05545	0.177295
$298$	8.04810	0.466214
$299$	−2.93756	−0.169883
$300$	0	0
$301$	0	0
$302$	−9.97792	−0.574165
$303$	15.4549	0.887863
$304$	−2.21016	−0.126761
$305$	0	0
$306$	2.90685	0.166174
$307$	7.90812	0.451340	0.225670	−	0.974204i	$-0.427543\pi$
$307$	7.90812	0.451340	0.225670	+	0.974204i	$0.427543\pi$
$308$	0	0
$309$	3.12563	0.177811
$310$	0	0
$311$	10.0772	0.571424	0.285712	−	0.958316i	$-0.407770\pi$
$311$	10.0772	0.571424	0.285712	+	0.958316i	$0.407770\pi$
$312$	1.64124	0.0929169
$313$	−20.3844	−1.15219	−0.576097	−	0.817381i	$-0.695425\pi$
$313$	−20.3844	−1.15219	−0.576097	+	0.817381i	$0.695425\pi$
$314$	12.6676	0.714875
$315$	0	0
$316$	14.2990	0.804380
$317$	−0.321063	−0.0180327	−0.00901634	−	0.999959i	$-0.502870\pi$
$317$	−0.321063	−0.0180327	−0.00901634	+	0.999959i	$0.502870\pi$
$318$	−5.78984	−0.324678
$319$	−17.0698	−0.955726
$320$	0	0
$321$	4.78249	0.266932
$322$	0	0
$323$	−6.42459	−0.357474
$324$	1.00000	0.0555556
$325$	0	0
$326$	−23.5646	−1.30512
$327$	9.10355	0.503428
$328$	−6.67982	−0.368832
$329$	0	0
$330$	0	0
$331$	−16.6084	−0.912880	−0.456440	−	0.889754i	$-0.650876\pi$
$331$	−16.6084	−0.912880	−0.456440	+	0.889754i	$0.650876\pi$
$332$	−0.874366	−0.0479871
$333$	−10.7123	−0.587031
$334$	−18.0818	−0.989394
$335$	0	0
$336$	0	0
$337$	−1.91118	−0.104108	−0.0520542	−	0.998644i	$-0.516577\pi$
$337$	−1.91118	−0.104108	−0.0520542	+	0.998644i	$0.516577\pi$
$338$	−10.3063	−0.560591
$339$	−14.3982	−0.782005
$340$	0	0
$341$	2.88602	0.156287
$342$	−2.21016	−0.119512
$343$	0	0
$344$	10.5021	0.566238
$345$	0	0
$346$	1.93845	0.104212
$347$	−7.90812	−0.424530	−0.212265	−	0.977212i	$-0.568084\pi$
$347$	−7.90812	−0.424530	−0.212265	+	0.977212i	$0.568084\pi$
$348$	−5.58667	−0.299477
$349$	23.5273	1.25939	0.629693	−	0.776844i	$-0.283181\pi$
$349$	23.5273	1.25939	0.629693	+	0.776844i	$0.283181\pi$
$350$	0	0
$351$	1.64124	0.0876029
$352$	−3.05545	−0.162856
$353$	22.9718	1.22267	0.611333	−	0.791373i	$-0.290634\pi$
$353$	22.9718	1.22267	0.611333	+	0.791373i	$0.290634\pi$
$354$	−5.97792	−0.317723
$355$	0	0
$356$	−17.5083	−0.927935
$357$	0	0
$358$	−24.4463	−1.29203
$359$	11.0264	0.581950	0.290975	−	0.956731i	$-0.406020\pi$
$359$	11.0264	0.581950	0.290975	+	0.956731i	$0.406020\pi$
$360$	0	0
$361$	−14.1152	−0.742906
$362$	16.9788	0.892386
$363$	1.66421	0.0873483
$364$	0	0
$365$	0	0
$366$	0.445811	0.0233029
$367$	−10.1446	−0.529546	−0.264773	−	0.964311i	$-0.585297\pi$
$367$	−10.1446	−0.529546	−0.264773	+	0.964311i	$0.585297\pi$
$368$	1.78984	0.0933020
$369$	−6.67982	−0.347738
$370$	0	0
$371$	0	0
$372$	0.944547	0.0489725
$373$	−26.5021	−1.37223	−0.686115	−	0.727493i	$-0.740685\pi$
$373$	−26.5021	−1.37223	−0.686115	+	0.727493i	$0.740685\pi$
$374$	−8.88174	−0.459264
$375$	0	0
$376$	−11.3765	−0.586699
$377$	−9.16907	−0.472231
$378$	0	0
$379$	29.4142	1.51091	0.755453	−	0.655203i	$-0.227417\pi$
$379$	29.4142	1.51091	0.755453	+	0.655203i	$0.227417\pi$
$380$	0	0
$381$	8.45405	0.433114
$382$	0.741377	0.0379322
$383$	−3.64249	−0.186123	−0.0930613	−	0.995660i	$-0.529665\pi$
$383$	−3.64249	−0.186123	−0.0930613	+	0.995660i	$0.529665\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−6.11091	−0.311037
$387$	10.5021	0.533854
$388$	10.4476	0.530396
$389$	−23.7938	−1.20639	−0.603196	−	0.797593i	$-0.706106\pi$
$389$	−23.7938	−1.20639	−0.603196	+	0.797593i	$0.706106\pi$
$390$	0	0
$391$	5.20280	0.263117
$392$	0	0
$393$	8.32106	0.419742
$394$	20.6832	1.04201
$395$	0	0
$396$	−3.05545	−0.153542
$397$	−9.76129	−0.489905	−0.244953	−	0.969535i	$-0.578772\pi$
$397$	−9.76129	−0.489905	−0.244953	+	0.969535i	$0.578772\pi$
$398$	−12.0702	−0.605024
$399$	0	0
$400$	0	0
$401$	9.97792	0.498273	0.249137	−	0.968468i	$-0.419853\pi$
$401$	9.97792	0.498273	0.249137	+	0.968468i	$0.419853\pi$
$402$	−1.26561	−0.0631229
$403$	1.55023	0.0772224
$404$	−15.4549	−0.768912
$405$	0	0
$406$	0	0
$407$	32.7309	1.62241
$408$	−2.90685	−0.143910
$409$	−35.1837	−1.73972	−0.869862	−	0.493295i	$-0.835792\pi$
$409$	−35.1837	−1.73972	−0.869862	+	0.493295i	$0.835792\pi$
$410$	0	0
$411$	−9.86701	−0.486704
$412$	−3.12563	−0.153989
$413$	0	0
$414$	1.78984	0.0879660
$415$	0	0
$416$	−1.64124	−0.0804684
$417$	16.9632	0.830691
$418$	6.75303	0.330302
$419$	−18.6274	−0.910009	−0.455004	−	0.890489i	$-0.650362\pi$
$419$	−18.6274	−0.910009	−0.455004	+	0.890489i	$0.650362\pi$
$420$	0	0
$421$	31.6464	1.54235	0.771176	−	0.636622i	$-0.219669\pi$
$421$	31.6464	1.54235	0.771176	+	0.636622i	$0.219669\pi$
$422$	2.95153	0.143678
$423$	−11.3765	−0.553145
$424$	5.78984	0.281180
$425$	0	0
$426$	14.8523	0.719595
$427$	0	0
$428$	−4.78249	−0.231170
$429$	−5.01473	−0.242113
$430$	0	0
$431$	−14.8046	−0.713111	−0.356556	−	0.934274i	$-0.616049\pi$
$431$	−14.8046	−0.713111	−0.356556	+	0.934274i	$0.616049\pi$
$432$	−1.00000	−0.0481125
$433$	−4.24731	−0.204113	−0.102056	−	0.994779i	$-0.532542\pi$
$433$	−4.24731	−0.204113	−0.102056	+	0.994779i	$0.532542\pi$
$434$	0	0
$435$	0	0
$436$	−9.10355	−0.435981
$437$	−3.95583	−0.189233
$438$	7.91636	0.378258
$439$	2.22880	0.106375	0.0531874	−	0.998585i	$-0.483062\pi$
$439$	2.22880	0.106375	0.0531874	+	0.998585i	$0.483062\pi$
$440$	0	0
$441$	0	0
$442$	−4.77083	−0.226925
$443$	−4.64213	−0.220554	−0.110277	−	0.993901i	$-0.535174\pi$
$443$	−4.64213	−0.220554	−0.110277	+	0.993901i	$0.535174\pi$
$444$	10.7123	0.508384
$445$	0	0
$446$	11.7678	0.557220
$447$	−8.04810	−0.380662
$448$	0	0
$449$	−3.69059	−0.174170	−0.0870849	−	0.996201i	$-0.527755\pi$
$449$	−3.69059	−0.174170	−0.0870849	+	0.996201i	$0.527755\pi$
$450$	0	0
$451$	20.4099	0.961064
$452$	14.3982	0.677236
$453$	9.97792	0.468803
$454$	−13.2365	−0.621222
$455$	0	0
$456$	2.21016	0.103500
$457$	−2.72972	−0.127691	−0.0638455	−	0.997960i	$-0.520336\pi$
$457$	−2.72972	−0.127691	−0.0638455	+	0.997960i	$0.520336\pi$
$458$	−25.2448	−1.17961
$459$	−2.90685	−0.135680
$460$	0	0
$461$	−11.8072	−0.549918	−0.274959	−	0.961456i	$-0.588664\pi$
$461$	−11.8072	−0.549918	−0.274959	+	0.961456i	$0.588664\pi$
$462$	0	0
$463$	2.40561	0.111798	0.0558990	−	0.998436i	$-0.482198\pi$
$463$	2.40561	0.111798	0.0558990	+	0.998436i	$0.482198\pi$
$464$	5.58667	0.259355
$465$	0	0
$466$	3.32842	0.154186
$467$	14.6612	0.678437	0.339219	−	0.940708i	$-0.389837\pi$
$467$	14.6612	0.678437	0.339219	+	0.940708i	$0.389837\pi$
$468$	−1.64124	−0.0758663
$469$	0	0
$470$	0	0
$471$	−12.6676	−0.583693
$472$	5.97792	0.275156
$473$	−32.0888	−1.47545
$474$	−14.2990	−0.656774
$475$	0	0
$476$	0	0
$477$	5.78984	0.265099
$478$	24.8713	1.13759
$479$	−11.8934	−0.543422	−0.271711	−	0.962379i	$-0.587590\pi$
$479$	−11.8934	−0.543422	−0.271711	+	0.962379i	$0.587590\pi$
$480$	0	0
$481$	17.5815	0.801645
$482$	−10.3535	−0.471591
$483$	0	0
$484$	−1.66421	−0.0756458
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−35.5018	−1.60874	−0.804370	−	0.594129i	$-0.797497\pi$
$487$	−35.5018	−1.60874	−0.804370	+	0.594129i	$0.797497\pi$
$488$	−0.445811	−0.0201809
$489$	23.5646	1.06563
$490$	0	0
$491$	−0.978284	−0.0441493	−0.0220747	−	0.999756i	$-0.507027\pi$
$491$	−0.978284	−0.0441493	−0.0220747	+	0.999756i	$0.507027\pi$
$492$	6.67982	0.301150
$493$	16.2396	0.731395
$494$	3.62740	0.163204
$495$	0	0
$496$	−0.944547	−0.0424114
$497$	0	0
$498$	0.874366	0.0391813
$499$	11.7340	0.525287	0.262644	−	0.964893i	$-0.415406\pi$
$499$	11.7340	0.525287	0.262644	+	0.964893i	$0.415406\pi$
$500$	0	0
$501$	18.0818	0.807837
$502$	−16.5502	−0.738674
$503$	−17.3618	−0.774125	−0.387062	−	0.922054i	$-0.626510\pi$
$503$	−17.3618	−0.774125	−0.387062	+	0.922054i	$0.626510\pi$
$504$	0	0
$505$	0	0
$506$	−5.46878	−0.243117
$507$	10.3063	0.457720
$508$	−8.45405	−0.375088
$509$	−25.8293	−1.14486	−0.572432	−	0.819952i	$-0.694000\pi$
$509$	−25.8293	−1.14486	−0.572432	+	0.819952i	$0.694000\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	2.21016	0.0975808
$514$	24.4546	1.07864
$515$	0	0
$516$	−10.5021	−0.462331
$517$	34.7604	1.52876
$518$	0	0
$519$	−1.93845	−0.0850884
$520$	0	0
$521$	24.7466	1.08417	0.542083	−	0.840325i	$-0.317636\pi$
$521$	24.7466	1.08417	0.542083	+	0.840325i	$0.317636\pi$
$522$	5.58667	0.244522
$523$	1.61574	0.0706515	0.0353258	−	0.999376i	$-0.488753\pi$
$523$	1.61574	0.0706515	0.0353258	+	0.999376i	$0.488753\pi$
$524$	−8.32106	−0.363507
$525$	0	0
$526$	−12.5091	−0.545424
$527$	−2.74566	−0.119603
$528$	3.05545	0.132972
$529$	−19.7965	−0.860716
$530$	0	0
$531$	5.97792	0.259420
$532$	0	0
$533$	10.9632	0.474868
$534$	17.5083	0.757656
$535$	0	0
$536$	1.26561	0.0546660
$537$	24.4463	1.05494
$538$	18.9666	0.817708
$539$	0	0
$540$	0	0
$541$	24.7530	1.06422	0.532108	−	0.846677i	$-0.321400\pi$
$541$	24.7530	1.06422	0.532108	+	0.846677i	$0.321400\pi$
$542$	−19.6638	−0.844634
$543$	−16.9788	−0.728630
$544$	2.90685	0.124630
$545$	0	0
$546$	0	0
$547$	−44.0524	−1.88354	−0.941772	−	0.336253i	$-0.890840\pi$
$547$	−44.0524	−1.88354	−0.941772	+	0.336253i	$0.890840\pi$
$548$	9.86701	0.421498
$549$	−0.445811	−0.0190267
$550$	0	0
$551$	−12.3474	−0.526018
$552$	−1.78984	−0.0761808
$553$	0	0
$554$	−6.13998	−0.260863
$555$	0	0
$556$	−16.9632	−0.719399
$557$	−2.06979	−0.0877000	−0.0438500	−	0.999038i	$-0.513962\pi$
$557$	−2.06979	−0.0877000	−0.0438500	+	0.999038i	$0.513962\pi$
$558$	−0.944547	−0.0399859
$559$	−17.2365	−0.729028
$560$	0	0
$561$	8.88174	0.374987
$562$	−12.1330	−0.511799
$563$	−9.84493	−0.414914	−0.207457	−	0.978244i	$-0.566519\pi$
$563$	−9.84493	−0.414914	−0.207457	+	0.978244i	$0.566519\pi$
$564$	11.3765	0.479038
$565$	0	0
$566$	17.4172	0.732101
$567$	0	0
$568$	−14.8523	−0.623188
$569$	13.7561	0.576686	0.288343	−	0.957527i	$-0.406896\pi$
$569$	13.7561	0.576686	0.288343	+	0.957527i	$0.406896\pi$
$570$	0	0
$571$	19.2071	0.803791	0.401896	−	0.915685i	$-0.368351\pi$
$571$	19.2071	0.803791	0.401896	+	0.915685i	$0.368351\pi$
$572$	5.01473	0.209676
$573$	−0.741377	−0.0309715
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	37.3072	1.55312	0.776560	−	0.630043i	$-0.216963\pi$
$577$	37.3072	1.55312	0.776560	+	0.630043i	$0.216963\pi$
$578$	−8.55023	−0.355643
$579$	6.11091	0.253961
$580$	0	0
$581$	0	0
$582$	−10.4476	−0.433066
$583$	−17.6906	−0.732669
$584$	−7.91636	−0.327581
$585$	0	0
$586$	23.7521	0.981192
$587$	19.5459	0.806748	0.403374	−	0.915035i	$-0.367837\pi$
$587$	19.5459	0.806748	0.403374	+	0.915035i	$0.367837\pi$
$588$	0	0
$589$	2.08760	0.0860180
$590$	0	0
$591$	−20.6832	−0.850795
$592$	−10.7123	−0.440273
$593$	28.9424	1.18852	0.594260	−	0.804273i	$-0.297445\pi$
$593$	28.9424	1.18852	0.594260	+	0.804273i	$0.297445\pi$
$594$	3.05545	0.125367
$595$	0	0
$596$	8.04810	0.329663
$597$	12.0702	0.494000
$598$	−2.93756	−0.120126
$599$	−9.73170	−0.397627	−0.198813	−	0.980037i	$-0.563709\pi$
$599$	−9.73170	−0.397627	−0.198813	+	0.980037i	$0.563709\pi$
$600$	0	0
$601$	−33.4038	−1.36257	−0.681284	−	0.732019i	$-0.738578\pi$
$601$	−33.4038	−1.36257	−0.681284	+	0.732019i	$0.738578\pi$
$602$	0	0
$603$	1.26561	0.0515396
$604$	−9.97792	−0.405996
$605$	0	0
$606$	15.4549	0.627814
$607$	−43.2358	−1.75489	−0.877444	−	0.479680i	$-0.840753\pi$
$607$	−43.2358	−1.75489	−0.877444	+	0.479680i	$0.840753\pi$
$608$	−2.21016	−0.0896337
$609$	0	0
$610$	0	0
$611$	18.6716	0.755371
$612$	2.90685	0.117502
$613$	−1.79023	−0.0723067	−0.0361534	−	0.999346i	$-0.511510\pi$
$613$	−1.79023	−0.0723067	−0.0361534	+	0.999346i	$0.511510\pi$
$614$	7.90812	0.319146
$615$	0	0
$616$	0	0
$617$	−38.2665	−1.54055	−0.770275	−	0.637712i	$-0.779881\pi$
$617$	−38.2665	−1.54055	−0.770275	+	0.637712i	$0.779881\pi$
$618$	3.12563	0.125731
$619$	−0.132989	−0.00534526	−0.00267263	−	0.999996i	$-0.500851\pi$
$619$	−0.132989	−0.00534526	−0.00267263	+	0.999996i	$0.500851\pi$
$620$	0	0
$621$	−1.78984	−0.0718239
$622$	10.0772	0.404058
$623$	0	0
$624$	1.64124	0.0657022
$625$	0	0
$626$	−20.3844	−0.814724
$627$	−6.75303	−0.269690
$628$	12.6676	0.505493
$629$	−31.1391	−1.24160
$630$	0	0
$631$	−25.1850	−1.00260	−0.501299	−	0.865274i	$-0.667145\pi$
$631$	−25.1850	−1.00260	−0.501299	+	0.865274i	$0.667145\pi$
$632$	14.2990	0.568783
$633$	−2.95153	−0.117313
$634$	−0.321063	−0.0127510
$635$	0	0
$636$	−5.78984	−0.229582
$637$	0	0
$638$	−17.0698	−0.675800
$639$	−14.8523	−0.587547
$640$	0	0
$641$	−24.5906	−0.971270	−0.485635	−	0.874162i	$-0.661412\pi$
$641$	−24.5906	−0.971270	−0.485635	+	0.874162i	$0.661412\pi$
$642$	4.78249	0.188750
$643$	−9.15126	−0.360891	−0.180445	−	0.983585i	$-0.557754\pi$
$643$	−9.15126	−0.360891	−0.180445	+	0.983585i	$0.557754\pi$
$644$	0	0
$645$	0	0
$646$	−6.42459	−0.252772
$647$	31.9268	1.25517	0.627585	−	0.778548i	$-0.284043\pi$
$647$	31.9268	1.25517	0.627585	+	0.778548i	$0.284043\pi$
$648$	1.00000	0.0392837
$649$	−18.2652	−0.716973
$650$	0	0
$651$	0	0
$652$	−23.5646	−0.922861
$653$	3.06244	0.119843	0.0599213	−	0.998203i	$-0.480915\pi$
$653$	3.06244	0.119843	0.0599213	+	0.998203i	$0.480915\pi$
$654$	9.10355	0.355977
$655$	0	0
$656$	−6.67982	−0.260803
$657$	−7.91636	−0.308847
$658$	0	0
$659$	−9.16636	−0.357071	−0.178535	−	0.983934i	$-0.557136\pi$
$659$	−9.16636	−0.357071	−0.178535	+	0.983934i	$0.557136\pi$
$660$	0	0
$661$	−31.4181	−1.22202	−0.611012	−	0.791621i	$-0.709237\pi$
$661$	−31.4181	−1.22202	−0.611012	+	0.791621i	$0.709237\pi$
$662$	−16.6084	−0.645503
$663$	4.77083	0.185284
$664$	−0.874366	−0.0339320
$665$	0	0
$666$	−10.7123	−0.415093
$667$	9.99927	0.387173
$668$	−18.0818	−0.699607
$669$	−11.7678	−0.454968
$670$	0	0
$671$	1.36215	0.0525854
$672$	0	0
$673$	6.45100	0.248668	0.124334	−	0.992240i	$-0.460321\pi$
$673$	6.45100	0.248668	0.124334	+	0.992240i	$0.460321\pi$
$674$	−1.91118	−0.0736158
$675$	0	0
$676$	−10.3063	−0.396397
$677$	6.01384	0.231131	0.115565	−	0.993300i	$-0.463132\pi$
$677$	6.01384	0.231131	0.115565	+	0.993300i	$0.463132\pi$
$678$	−14.3982	−0.552961
$679$	0	0
$680$	0	0
$681$	13.2365	0.507225
$682$	2.88602	0.110511
$683$	8.92208	0.341394	0.170697	−	0.985324i	$-0.445398\pi$
$683$	8.92208	0.341394	0.170697	+	0.985324i	$0.445398\pi$
$684$	−2.21016	−0.0845075
$685$	0	0
$686$	0	0
$687$	25.2448	0.963148
$688$	10.5021	0.400391
$689$	−9.50252	−0.362017
$690$	0	0
$691$	−47.6581	−1.81300	−0.906499	−	0.422207i	$-0.861255\pi$
$691$	−47.6581	−1.81300	−0.906499	+	0.422207i	$0.861255\pi$
$692$	1.93845	0.0736887
$693$	0	0
$694$	−7.90812	−0.300188
$695$	0	0
$696$	−5.58667	−0.211762
$697$	−19.4172	−0.735480
$698$	23.5273	0.890521
$699$	−3.32842	−0.125892
$700$	0	0
$701$	31.0788	1.17383	0.586914	−	0.809649i	$-0.300343\pi$
$701$	31.0788	1.17383	0.586914	+	0.809649i	$0.300343\pi$
$702$	1.64124	0.0619446
$703$	23.6759	0.892953
$704$	−3.05545	−0.115157
$705$	0	0
$706$	22.9718	0.864556
$707$	0	0
$708$	−5.97792	−0.224664
$709$	−12.1815	−0.457485	−0.228742	−	0.973487i	$-0.573461\pi$
$709$	−12.1815	−0.457485	−0.228742	+	0.973487i	$0.573461\pi$
$710$	0	0
$711$	14.2990	0.536254
$712$	−17.5083	−0.656149
$713$	−1.69059	−0.0633131
$714$	0	0
$715$	0	0
$716$	−24.4463	−0.913602
$717$	−24.8713	−0.928836
$718$	11.0264	0.411501
$719$	31.9896	1.19301	0.596505	−	0.802609i	$-0.296556\pi$
$719$	31.9896	1.19301	0.596505	+	0.802609i	$0.296556\pi$
$720$	0	0
$721$	0	0
$722$	−14.1152	−0.525314
$723$	10.3535	0.385053
$724$	16.9788	0.631012
$725$	0	0
$726$	1.66421	0.0617646
$727$	22.9992	0.852995	0.426497	−	0.904489i	$-0.359747\pi$
$727$	22.9992	0.852995	0.426497	+	0.904489i	$0.359747\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	30.5282	1.12912
$732$	0.445811	0.0164776
$733$	17.6001	0.650076	0.325038	−	0.945701i	$-0.394623\pi$
$733$	17.6001	0.650076	0.325038	+	0.945701i	$0.394623\pi$
$734$	−10.1446	−0.374446
$735$	0	0
$736$	1.78984	0.0659745
$737$	−3.86701	−0.142443
$738$	−6.67982	−0.245888
$739$	−0.174100	−0.00640437	−0.00320218	−	0.999995i	$-0.501019\pi$
$739$	−0.174100	−0.00640437	−0.00320218	+	0.999995i	$0.501019\pi$
$740$	0	0
$741$	−3.62740	−0.133256
$742$	0	0
$743$	53.6360	1.96771	0.983857	−	0.178957i	$-0.0572723\pi$
$743$	53.6360	1.96771	0.983857	+	0.178957i	$0.0572723\pi$
$744$	0.944547	0.0346288
$745$	0	0
$746$	−26.5021	−0.970313
$747$	−0.874366	−0.0319914
$748$	−8.88174	−0.324749
$749$	0	0
$750$	0	0
$751$	29.0931	1.06162	0.530812	−	0.847490i	$-0.321887\pi$
$751$	29.0931	1.06162	0.530812	+	0.847490i	$0.321887\pi$
$752$	−11.3765	−0.414859
$753$	16.5502	0.603125
$754$	−9.16907	−0.333918
$755$	0	0
$756$	0	0
$757$	−34.0842	−1.23881	−0.619405	−	0.785072i	$-0.712626\pi$
$757$	−34.0842	−1.23881	−0.619405	+	0.785072i	$0.712626\pi$
$758$	29.4142	1.06837
$759$	5.46878	0.198504
$760$	0	0
$761$	7.62959	0.276572	0.138286	−	0.990392i	$-0.455841\pi$
$761$	7.62959	0.276572	0.138286	+	0.990392i	$0.455841\pi$
$762$	8.45405	0.306258
$763$	0	0
$764$	0.741377	0.0268221
$765$	0	0
$766$	−3.64249	−0.131609
$767$	−9.81119	−0.354262
$768$	−1.00000	−0.0360844
$769$	37.8241	1.36397	0.681986	−	0.731365i	$-0.261117\pi$
$769$	37.8241	1.36397	0.681986	+	0.731365i	$0.261117\pi$
$770$	0	0
$771$	−24.4546	−0.880710
$772$	−6.11091	−0.219936
$773$	25.2349	0.907636	0.453818	−	0.891094i	$-0.350062\pi$
$773$	25.2349	0.907636	0.453818	+	0.891094i	$0.350062\pi$
$774$	10.5021	0.377492
$775$	0	0
$776$	10.4476	0.375046
$777$	0	0
$778$	−23.7938	−0.853048
$779$	14.7635	0.528956
$780$	0	0
$781$	45.3804	1.62384
$782$	5.20280	0.186052
$783$	−5.58667	−0.199651
$784$	0	0
$785$	0	0
$786$	8.32106	0.296802
$787$	22.7193	0.809855	0.404928	−	0.914349i	$-0.367297\pi$
$787$	22.7193	0.809855	0.404928	+	0.914349i	$0.367297\pi$
$788$	20.6832	0.736810
$789$	12.5091	0.445337
$790$	0	0
$791$	0	0
$792$	−3.05545	−0.108571
$793$	0.731682	0.0259828
$794$	−9.76129	−0.346415
$795$	0	0
$796$	−12.0702	−0.427816
$797$	10.1057	0.357963	0.178981	−	0.983852i	$-0.442720\pi$
$797$	10.1057	0.357963	0.178981	+	0.983852i	$0.442720\pi$
$798$	0	0
$799$	−33.0698	−1.16993
$800$	0	0
$801$	−17.5083	−0.618624
$802$	9.97792	0.352332
$803$	24.1881	0.853579
$804$	−1.26561	−0.0446346
$805$	0	0
$806$	1.55023	0.0546045
$807$	−18.9666	−0.667656
$808$	−15.4549	−0.543703
$809$	46.3180	1.62845	0.814227	−	0.580546i	$-0.197161\pi$
$809$	46.3180	1.62845	0.814227	+	0.580546i	$0.197161\pi$
$810$	0	0
$811$	42.0012	1.47486	0.737431	−	0.675422i	$-0.236039\pi$
$811$	42.0012	1.47486	0.737431	+	0.675422i	$0.236039\pi$
$812$	0	0
$813$	19.6638	0.689641
$814$	32.7309	1.14722
$815$	0	0
$816$	−2.90685	−0.101760
$817$	−23.2114	−0.812064
$818$	−35.1837	−1.23017
$819$	0	0
$820$	0	0
$821$	16.8981	0.589746	0.294873	−	0.955536i	$-0.404723\pi$
$821$	16.8981	0.589746	0.294873	+	0.955536i	$0.404723\pi$
$822$	−9.86701	−0.344152
$823$	−45.2401	−1.57697	−0.788485	−	0.615054i	$-0.789134\pi$
$823$	−45.2401	−1.57697	−0.788485	+	0.615054i	$0.789134\pi$
$824$	−3.12563	−0.108887
$825$	0	0
$826$	0	0
$827$	34.5452	1.20125	0.600627	−	0.799529i	$-0.294918\pi$
$827$	34.5452	1.20125	0.600627	+	0.799529i	$0.294918\pi$
$828$	1.78984	0.0622013
$829$	9.99604	0.347177	0.173588	−	0.984818i	$-0.444464\pi$
$829$	9.99604	0.347177	0.173588	+	0.984818i	$0.444464\pi$
$830$	0	0
$831$	6.13998	0.212993
$832$	−1.64124	−0.0568997
$833$	0	0
$834$	16.9632	0.587387
$835$	0	0
$836$	6.75303	0.233558
$837$	0.944547	0.0326483
$838$	−18.6274	−0.643473
$839$	−32.3180	−1.11574	−0.557871	−	0.829928i	$-0.688382\pi$
$839$	−32.3180	−1.11574	−0.557871	+	0.829928i	$0.688382\pi$
$840$	0	0
$841$	2.21091	0.0762383
$842$	31.6464	1.09061
$843$	12.1330	0.417882
$844$	2.95153	0.101596
$845$	0	0
$846$	−11.3765	−0.391133
$847$	0	0
$848$	5.78984	0.198824
$849$	−17.4172	−0.597758
$850$	0	0
$851$	−19.1733	−0.657254
$852$	14.8523	0.508831
$853$	13.7577	0.471056	0.235528	−	0.971868i	$-0.424318\pi$
$853$	13.7577	0.471056	0.235528	+	0.971868i	$0.424318\pi$
$854$	0	0
$855$	0	0
$856$	−4.78249	−0.163462
$857$	−7.45710	−0.254730	−0.127365	−	0.991856i	$-0.540652\pi$
$857$	−7.45710	−0.254730	−0.127365	+	0.991856i	$0.540652\pi$
$858$	−5.01473	−0.171200
$859$	33.8860	1.15618	0.578088	−	0.815974i	$-0.303799\pi$
$859$	33.8860	1.15618	0.578088	+	0.815974i	$0.303799\pi$
$860$	0	0
$861$	0	0
$862$	−14.8046	−0.504246
$863$	41.2843	1.40533	0.702666	−	0.711520i	$-0.251993\pi$
$863$	41.2843	1.40533	0.702666	+	0.711520i	$0.251993\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−4.24731	−0.144329
$867$	8.55023	0.290381
$868$	0	0
$869$	−43.6899	−1.48208
$870$	0	0
$871$	−2.07717	−0.0703822
$872$	−9.10355	−0.308285
$873$	10.4476	0.353597
$874$	−3.95583	−0.133808
$875$	0	0
$876$	7.91636	0.267469
$877$	−26.7938	−0.904761	−0.452380	−	0.891825i	$-0.649425\pi$
$877$	−26.7938	−0.904761	−0.452380	+	0.891825i	$0.649425\pi$
$878$	2.22880	0.0752183
$879$	−23.7521	−0.801140
$880$	0	0
$881$	−31.0247	−1.04525	−0.522625	−	0.852563i	$-0.675047\pi$
$881$	−31.0247	−1.04525	−0.522625	+	0.852563i	$0.675047\pi$
$882$	0	0
$883$	39.6123	1.33306	0.666530	−	0.745478i	$-0.267779\pi$
$883$	39.6123	1.33306	0.666530	+	0.745478i	$0.267779\pi$
$884$	−4.77083	−0.160461
$885$	0	0
$886$	−4.64213	−0.155955
$887$	11.8453	0.397727	0.198863	−	0.980027i	$-0.436275\pi$
$887$	11.8453	0.397727	0.198863	+	0.980027i	$0.436275\pi$
$888$	10.7123	0.359481
$889$	0	0
$890$	0	0
$891$	−3.05545	−0.102362
$892$	11.7678	0.394014
$893$	25.1439	0.841408
$894$	−8.04810	−0.269169
$895$	0	0
$896$	0	0
$897$	2.93756	0.0980823
$898$	−3.69059	−0.123157
$899$	−5.27688	−0.175994
$900$	0	0
$901$	16.8302	0.560695
$902$	20.4099	0.679575
$903$	0	0
$904$	14.3982	0.478878
$905$	0	0
$906$	9.97792	0.331494
$907$	16.9849	0.563974	0.281987	−	0.959418i	$-0.409007\pi$
$907$	16.9849	0.563974	0.281987	+	0.959418i	$0.409007\pi$
$908$	−13.2365	−0.439270
$909$	−15.4549	−0.512608
$910$	0	0
$911$	15.5723	0.515934	0.257967	−	0.966154i	$-0.416947\pi$
$911$	15.5723	0.515934	0.257967	+	0.966154i	$0.416947\pi$
$912$	2.21016	0.0731856
$913$	2.67158	0.0884165
$914$	−2.72972	−0.0902912
$915$	0	0
$916$	−25.2448	−0.834111
$917$	0	0
$918$	−2.90685	−0.0959403
$919$	−36.9632	−1.21930	−0.609652	−	0.792670i	$-0.708691\pi$
$919$	−36.9632	−1.21930	−0.609652	+	0.792670i	$0.708691\pi$
$920$	0	0
$921$	−7.90812	−0.260582
$922$	−11.8072	−0.388850
$923$	24.3761	0.802351
$924$	0	0
$925$	0	0
$926$	2.40561	0.0790531
$927$	−3.12563	−0.102659
$928$	5.58667	0.183392
$929$	−33.9831	−1.11495	−0.557475	−	0.830194i	$-0.688230\pi$
$929$	−33.9831	−1.11495	−0.557475	+	0.830194i	$0.688230\pi$
$930$	0	0
$931$	0	0
$932$	3.32842	0.109026
$933$	−10.0772	−0.329912
$934$	14.6612	0.479728
$935$	0	0
$936$	−1.64124	−0.0536456
$937$	43.1719	1.41037	0.705183	−	0.709026i	$-0.250865\pi$
$937$	43.1719	1.41037	0.705183	+	0.709026i	$0.250865\pi$
$938$	0	0
$939$	20.3844	0.665219
$940$	0	0
$941$	33.1186	1.07964	0.539818	−	0.841782i	$-0.318493\pi$
$941$	33.1186	1.07964	0.539818	+	0.841782i	$0.318493\pi$
$942$	−12.6676	−0.412733
$943$	−11.9558	−0.389336
$944$	5.97792	0.194565
$945$	0	0
$946$	−32.0888	−1.04330
$947$	42.2408	1.37264	0.686321	−	0.727298i	$-0.259224\pi$
$947$	42.2408	1.37264	0.686321	+	0.727298i	$0.259224\pi$
$948$	−14.2990	−0.464409
$949$	12.9926	0.421759
$950$	0	0
$951$	0.321063	0.0104112
$952$	0	0
$953$	−16.2358	−0.525929	−0.262964	−	0.964806i	$-0.584700\pi$
$953$	−16.2358	−0.525929	−0.262964	+	0.964806i	$0.584700\pi$
$954$	5.78984	0.187453
$955$	0	0
$956$	24.8713	0.804396
$957$	17.0698	0.551789
$958$	−11.8934	−0.384257
$959$	0	0
$960$	0	0
$961$	−30.1078	−0.971220
$962$	17.5815	0.566849
$963$	−4.78249	−0.154114
$964$	−10.3535	−0.333465
$965$	0	0
$966$	0	0
$967$	−56.9706	−1.83205	−0.916025	−	0.401121i	$-0.868621\pi$
$967$	−56.9706	−1.83205	−0.916025	+	0.401121i	$0.868621\pi$
$968$	−1.66421	−0.0534897
$969$	6.42459	0.206388
$970$	0	0
$971$	48.3995	1.55321	0.776606	−	0.629986i	$-0.216940\pi$
$971$	48.3995	1.55321	0.776606	+	0.629986i	$0.216940\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−35.5018	−1.13755
$975$	0	0
$976$	−0.445811	−0.0142701
$977$	39.2401	1.25540	0.627701	−	0.778455i	$-0.283996\pi$
$977$	39.2401	1.25540	0.627701	+	0.778455i	$0.283996\pi$
$978$	23.5646	0.753512
$979$	53.4956	1.70973
$980$	0	0
$981$	−9.10355	−0.290654
$982$	−0.978284	−0.0312183
$983$	−11.8453	−0.377807	−0.188903	−	0.981996i	$-0.560493\pi$
$983$	−11.8453	−0.377807	−0.188903	+	0.981996i	$0.560493\pi$
$984$	6.67982	0.212945
$985$	0	0
$986$	16.2396	0.517175
$987$	0	0
$988$	3.62740	0.115403
$989$	18.7972	0.597716
$990$	0	0
$991$	6.15202	0.195425	0.0977126	−	0.995215i	$-0.468847\pi$
$991$	6.15202	0.195425	0.0977126	+	0.995215i	$0.468847\pi$
$992$	−0.944547	−0.0299894
$993$	16.6084	0.527051
$994$	0	0
$995$	0	0
$996$	0.874366	0.0277053
$997$	49.7613	1.57596	0.787978	−	0.615703i	$-0.211128\pi$
$997$	49.7613	1.57596	0.787978	+	0.615703i	$0.211128\pi$
$998$	11.7340	0.371434
$999$	10.7123	0.338922

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7350.2.a.dt.1.1		4
5.2	odd	4		1470.2.g.j.589.7	yes	8
5.3	odd	4		1470.2.g.j.589.3	✓	8
5.4	even	2		7350.2.a.ds.1.1		4
7.6	odd	2		7350.2.a.du.1.1		4
35.2	odd	12		1470.2.n.l.949.5		16
35.3	even	12		1470.2.n.k.79.8		16
35.12	even	12		1470.2.n.k.949.8		16
35.13	even	4		1470.2.g.k.589.2	yes	8
35.17	even	12		1470.2.n.k.79.1		16
35.18	odd	12		1470.2.n.l.79.5		16
35.23	odd	12		1470.2.n.l.949.4		16
35.27	even	4		1470.2.g.k.589.6	yes	8
35.32	odd	12		1470.2.n.l.79.4		16
35.33	even	12		1470.2.n.k.949.1		16
35.34	odd	2		7350.2.a.dr.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1470.2.g.j.589.3	✓	8	5.3	odd	4
1470.2.g.j.589.7	yes	8	5.2	odd	4
1470.2.g.k.589.2	yes	8	35.13	even	4
1470.2.g.k.589.6	yes	8	35.27	even	4
1470.2.n.k.79.1		16	35.17	even	12
1470.2.n.k.79.8		16	35.3	even	12
1470.2.n.k.949.1		16	35.33	even	12
1470.2.n.k.949.8		16	35.12	even	12
1470.2.n.l.79.4		16	35.32	odd	12
1470.2.n.l.79.5		16	35.18	odd	12
1470.2.n.l.949.4		16	35.23	odd	12
1470.2.n.l.949.5		16	35.2	odd	12
7350.2.a.dr.1.1		4	35.34	odd	2
7350.2.a.ds.1.1		4	5.4	even	2
7350.2.a.dt.1.1		4	1.1	even	1	trivial
7350.2.a.du.1.1		4	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7350.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -3.05545 q^{11} -1.00000 q^{12} -1.64124 q^{13} +1.00000 q^{16} +2.90685 q^{17} +1.00000 q^{18} -2.21016 q^{19} -3.05545 q^{22} +1.78984 q^{23} -1.00000 q^{24} -1.64124 q^{26} -1.00000 q^{27} +5.58667 q^{29} -0.944547 q^{31} +1.00000 q^{32} +3.05545 q^{33} +2.90685 q^{34} +1.00000 q^{36} -10.7123 q^{37} -2.21016 q^{38} +1.64124 q^{39} -6.67982 q^{41} +10.5021 q^{43} -3.05545 q^{44} +1.78984 q^{46} -11.3765 q^{47} -1.00000 q^{48} -2.90685 q^{51} -1.64124 q^{52} +5.78984 q^{53} -1.00000 q^{54} +2.21016 q^{57} +5.58667 q^{58} +5.97792 q^{59} -0.445811 q^{61} -0.944547 q^{62} +1.00000 q^{64} +3.05545 q^{66} +1.26561 q^{67} +2.90685 q^{68} -1.78984 q^{69} -14.8523 q^{71} +1.00000 q^{72} -7.91636 q^{73} -10.7123 q^{74} -2.21016 q^{76} +1.64124 q^{78} +14.2990 q^{79} +1.00000 q^{81} -6.67982 q^{82} -0.874366 q^{83} +10.5021 q^{86} -5.58667 q^{87} -3.05545 q^{88} -17.5083 q^{89} +1.78984 q^{92} +0.944547 q^{93} -11.3765 q^{94} -1.00000 q^{96} +10.4476 q^{97} -3.05545 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{6} + 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^6 + 4 * q^8 + 4 * q^9	\( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{6} + 4 q^{8} + 4 q^{9} - 4 q^{12} + 4 q^{16} - 4 q^{17} + 4 q^{18} - 12 q^{19} + 4 q^{23} - 4 q^{24} - 4 q^{27} - 8 q^{29} - 16 q^{31} + 4 q^{32} - 4 q^{34} + 4 q^{36} - 8 q^{37} - 12 q^{38} - 12 q^{41} + 4 q^{43} + 4 q^{46} - 12 q^{47} - 4 q^{48} + 4 q^{51} + 20 q^{53} - 4 q^{54} + 12 q^{57} - 8 q^{58} - 20 q^{59} - 12 q^{61} - 16 q^{62} + 4 q^{64} - 4 q^{67} - 4 q^{68} - 4 q^{69} - 20 q^{71} + 4 q^{72} + 12 q^{73} - 8 q^{74} - 12 q^{76} - 8 q^{79} + 4 q^{81} - 12 q^{82} - 8 q^{83} + 4 q^{86} + 8 q^{87} - 44 q^{89} + 4 q^{92} + 16 q^{93} - 12 q^{94} - 4 q^{96} - 20 q^{97}+O(q^{100}) \) 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^6 + 4 * q^8 + 4 * q^9 - 4 * q^12 + 4 * q^16 - 4 * q^17 + 4 * q^18 - 12 * q^19 + 4 * q^23 - 4 * q^24 - 4 * q^27 - 8 * q^29 - 16 * q^31 + 4 * q^32 - 4 * q^34 + 4 * q^36 - 8 * q^37 - 12 * q^38 - 12 * q^41 + 4 * q^43 + 4 * q^46 - 12 * q^47 - 4 * q^48 + 4 * q^51 + 20 * q^53 - 4 * q^54 + 12 * q^57 - 8 * q^58 - 20 * q^59 - 12 * q^61 - 16 * q^62 + 4 * q^64 - 4 * q^67 - 4 * q^68 - 4 * q^69 - 20 * q^71 + 4 * q^72 + 12 * q^73 - 8 * q^74 - 12 * q^76 - 8 * q^79 + 4 * q^81 - 12 * q^82 - 8 * q^83 + 4 * q^86 + 8 * q^87 - 44 * q^89 + 4 * q^92 + 16 * q^93 - 12 * q^94 - 4 * q^96 - 20 * q^97

Introduction

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