LMFDB - Embedded newform 7350.2.a.dp.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:	$0$
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			1.2
Root			$-2.80560$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.67701 q^{11} -1.00000 q^{12} -4.48261 q^{13} +1.00000 q^{16} -7.61121 q^{17} +1.00000 q^{18} -4.48261 q^{19} +1.67701 q^{22} -0.482613 q^{23} -1.00000 q^{24} -4.48261 q^{26} -1.00000 q^{27} +1.19440 q^{29} +3.48261 q^{31} +1.00000 q^{32} -1.67701 q^{33} -7.61121 q^{34} +1.00000 q^{36} -0.482613 q^{37} -4.48261 q^{38} +4.48261 q^{39} +12.0938 q^{41} -2.00000 q^{43} +1.67701 q^{44} -0.482613 q^{46} +11.4478 q^{47} -1.00000 q^{48} +7.61121 q^{51} -4.48261 q^{52} +9.57643 q^{53} -1.00000 q^{54} +4.48261 q^{57} +1.19440 q^{58} -5.77083 q^{59} +10.5764 q^{61} +3.48261 q^{62} +1.00000 q^{64} -1.67701 q^{66} -3.35402 q^{67} -7.61121 q^{68} +0.482613 q^{69} +6.00000 q^{71} +1.00000 q^{72} -4.00000 q^{73} -0.482613 q^{74} -4.48261 q^{76} +4.48261 q^{78} +4.51739 q^{79} +1.00000 q^{81} +12.0938 q^{82} -1.87141 q^{83} -2.00000 q^{86} -1.19440 q^{87} +1.67701 q^{88} +3.35402 q^{89} -0.482613 q^{92} -3.48261 q^{93} +11.4478 q^{94} -1.00000 q^{96} +17.7708 q^{97} +1.67701 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$11$	1.67701	0.505638	0.252819	−	0.967514i	$-0.418642\pi$
$11$	1.67701	0.505638	0.252819	+	0.967514i	$0.418642\pi$
$12$	−1.00000	−0.288675
$13$	−4.48261	−1.24325	−0.621627	−	0.783314i	$-0.713528\pi$
$13$	−4.48261	−1.24325	−0.621627	+	0.783314i	$0.713528\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−7.61121	−1.84599	−0.922994	−	0.384814i	$-0.874266\pi$
$17$	−7.61121	−1.84599	−0.922994	+	0.384814i	$0.874266\pi$
$18$	1.00000	0.235702
$19$	−4.48261	−1.02838	−0.514191	−	0.857676i	$-0.671908\pi$
$19$	−4.48261	−1.02838	−0.514191	+	0.857676i	$0.671908\pi$
$20$	0	0
$21$	0	0
$22$	1.67701	0.357540
$23$	−0.482613	−0.100632	−0.0503159	−	0.998733i	$-0.516023\pi$
$23$	−0.482613	−0.100632	−0.0503159	+	0.998733i	$0.516023\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−4.48261	−0.879113
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.19440	0.221794	0.110897	−	0.993832i	$-0.464628\pi$
$29$	1.19440	0.221794	0.110897	+	0.993832i	$0.464628\pi$
$30$	0	0
$31$	3.48261	0.625496	0.312748	−	0.949836i	$-0.398751\pi$
$31$	3.48261	0.625496	0.312748	+	0.949836i	$0.398751\pi$
$32$	1.00000	0.176777
$33$	−1.67701	−0.291930
$34$	−7.61121	−1.30531
$35$	0	0
$36$	1.00000	0.166667
$37$	−0.482613	−0.0793411	−0.0396705	−	0.999213i	$-0.512631\pi$
$37$	−0.482613	−0.0793411	−0.0396705	+	0.999213i	$0.512631\pi$
$38$	−4.48261	−0.727176
$39$	4.48261	0.717793
$40$	0	0
$41$	12.0938	1.88874	0.944369	−	0.328889i	$-0.106674\pi$
$41$	12.0938	1.88874	0.944369	+	0.328889i	$0.106674\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	1.67701	0.252819
$45$	0	0
$46$	−0.482613	−0.0711574
$47$	11.4478	1.66984	0.834919	−	0.550372i	$-0.185514\pi$
$47$	11.4478	1.66984	0.834919	+	0.550372i	$0.185514\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	0	0
$51$	7.61121	1.06578
$52$	−4.48261	−0.621627
$53$	9.57643	1.31542	0.657712	−	0.753269i	$-0.271524\pi$
$53$	9.57643	1.31542	0.657712	+	0.753269i	$0.271524\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	4.48261	0.593737
$58$	1.19440	0.156832
$59$	−5.77083	−0.751298	−0.375649	−	0.926762i	$-0.622580\pi$
$59$	−5.77083	−0.751298	−0.375649	+	0.926762i	$0.622580\pi$
$60$	0	0
$61$	10.5764	1.35417	0.677087	−	0.735903i	$-0.263242\pi$
$61$	10.5764	1.35417	0.677087	+	0.735903i	$0.263242\pi$
$62$	3.48261	0.442292
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.67701	−0.206426
$67$	−3.35402	−0.409759	−0.204879	−	0.978787i	$-0.565680\pi$
$67$	−3.35402	−0.409759	−0.204879	+	0.978787i	$0.565680\pi$
$68$	−7.61121	−0.922994
$69$	0.482613	0.0580998
$70$	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	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	−0.482613	−0.0561026
$75$	0	0
$76$	−4.48261	−0.514191
$77$	0	0
$78$	4.48261	0.507556
$79$	4.51739	0.508246	0.254123	−	0.967172i	$-0.418213\pi$
$79$	4.51739	0.508246	0.254123	+	0.967172i	$0.418213\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	12.0938	1.33554
$83$	−1.87141	−0.205414	−0.102707	−	0.994712i	$-0.532750\pi$
$83$	−1.87141	−0.205414	−0.102707	+	0.994712i	$0.532750\pi$
$84$	0	0
$85$	0	0
$86$	−2.00000	−0.215666
$87$	−1.19440	−0.128053
$88$	1.67701	0.178770
$89$	3.35402	0.355525	0.177763	−	0.984073i	$-0.443114\pi$
$89$	3.35402	0.355525	0.177763	+	0.984073i	$0.443114\pi$
$90$	0	0
$91$	0	0
$92$	−0.482613	−0.0503159
$93$	−3.48261	−0.361130
$94$	11.4478	1.18075
$95$	0	0
$96$	−1.00000	−0.102062
$97$	17.7708	1.80435	0.902177	−	0.431366i	$-0.141968\pi$
$97$	17.7708	1.80435	0.902177	+	0.431366i	$0.141968\pi$
$98$	0	0
$99$	1.67701	0.168546
$100$	0	0
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	7.61121	0.753622
$103$	13.6112	1.34115	0.670576	−	0.741841i	$-0.266047\pi$
$103$	13.6112	1.34115	0.670576	+	0.741841i	$0.266047\pi$
$104$	−4.48261	−0.439556
$105$	0	0
$106$	9.57643	0.930145
$107$	3.87141	0.374263	0.187132	−	0.982335i	$-0.440081\pi$
$107$	3.87141	0.374263	0.187132	+	0.982335i	$0.440081\pi$
$108$	−1.00000	−0.0962250
$109$	17.2224	1.64961	0.824804	−	0.565419i	$-0.191285\pi$
$109$	17.2224	1.64961	0.824804	+	0.565419i	$0.191285\pi$
$110$	0	0
$111$	0.482613	0.0458076
$112$	0	0
$113$	−8.96523	−0.843378	−0.421689	−	0.906741i	$-0.638563\pi$
$113$	−8.96523	−0.843378	−0.421689	+	0.906741i	$0.638563\pi$
$114$	4.48261	0.419835
$115$	0	0
$116$	1.19440	0.110897
$117$	−4.48261	−0.414418
$118$	−5.77083	−0.531248
$119$	0	0
$120$	0	0
$121$	−8.18764	−0.744331
$122$	10.5764	0.957545
$123$	−12.0938	−1.09046
$124$	3.48261	0.312748
$125$	0	0
$126$	0	0
$127$	13.9342	1.23646	0.618230	−	0.785997i	$-0.287850\pi$
$127$	13.9342	1.23646	0.618230	+	0.785997i	$0.287850\pi$
$128$	1.00000	0.0883883
$129$	2.00000	0.176090
$130$	0	0
$131$	−3.28822	−0.287293	−0.143646	−	0.989629i	$-0.545883\pi$
$131$	−3.28822	−0.287293	−0.143646	+	0.989629i	$0.545883\pi$
$132$	−1.67701	−0.145965
$133$	0	0
$134$	−3.35402	−0.289743
$135$	0	0
$136$	−7.61121	−0.652656
$137$	−4.25719	−0.363716	−0.181858	−	0.983325i	$-0.558211\pi$
$137$	−4.25719	−0.363716	−0.181858	+	0.983325i	$0.558211\pi$
$138$	0.482613	0.0410827
$139$	−18.9652	−1.60861	−0.804305	−	0.594217i	$-0.797462\pi$
$139$	−18.9652	−1.60861	−0.804305	+	0.594217i	$0.797462\pi$
$140$	0	0
$141$	−11.4478	−0.964082
$142$	6.00000	0.503509
$143$	−7.51739	−0.628635
$144$	1.00000	0.0833333
$145$	0	0
$146$	−4.00000	−0.331042
$147$	0	0
$148$	−0.482613	−0.0396705
$149$	7.03477	0.576311	0.288156	−	0.957584i	$-0.406958\pi$
$149$	7.03477	0.576311	0.288156	+	0.957584i	$0.406958\pi$
$150$	0	0
$151$	−6.44784	−0.524718	−0.262359	−	0.964970i	$-0.584500\pi$
$151$	−6.44784	−0.524718	−0.262359	+	0.964970i	$0.584500\pi$
$152$	−4.48261	−0.363588
$153$	−7.61121	−0.615330
$154$	0	0
$155$	0	0
$156$	4.48261	0.358896
$157$	0.0938186	0.00748754	0.00374377	−	0.999993i	$-0.498808\pi$
$157$	0.0938186	0.00748754	0.00374377	+	0.999993i	$0.498808\pi$
$158$	4.51739	0.359384
$159$	−9.57643	−0.759460
$160$	0	0
$161$	0	0
$162$	1.00000	0.0785674
$163$	20.5764	1.61167	0.805835	−	0.592140i	$-0.201717\pi$
$163$	20.5764	1.61167	0.805835	+	0.592140i	$0.201717\pi$
$164$	12.0938	0.944369
$165$	0	0
$166$	−1.87141	−0.145249
$167$	−14.0938	−1.09061	−0.545306	−	0.838237i	$-0.683587\pi$
$167$	−14.0938	−1.09061	−0.545306	+	0.838237i	$0.683587\pi$
$168$	0	0
$169$	7.09382	0.545678
$170$	0	0
$171$	−4.48261	−0.342794
$172$	−2.00000	−0.152499
$173$	−1.12859	−0.0858053	−0.0429027	−	0.999079i	$-0.513661\pi$
$173$	−1.12859	−0.0858053	−0.0429027	+	0.999079i	$0.513661\pi$
$174$	−1.19440	−0.0905470
$175$	0	0
$176$	1.67701	0.126409
$177$	5.77083	0.433762
$178$	3.35402	0.251394
$179$	−3.44784	−0.257704	−0.128852	−	0.991664i	$-0.541129\pi$
$179$	−3.44784	−0.257704	−0.128852	+	0.991664i	$0.541129\pi$
$180$	0	0
$181$	−22.8957	−1.70182	−0.850911	−	0.525310i	$-0.823950\pi$
$181$	−22.8957	−1.70182	−0.850911	+	0.525310i	$0.823950\pi$
$182$	0	0
$183$	−10.5764	−0.781832
$184$	−0.482613	−0.0355787
$185$	0	0
$186$	−3.48261	−0.255358
$187$	−12.7641	−0.933401
$188$	11.4478	0.834919
$189$	0	0
$190$	0	0
$191$	25.1529	1.82000	0.909999	−	0.414611i	$-0.136082\pi$
$191$	25.1529	1.82000	0.909999	+	0.414611i	$0.136082\pi$
$192$	−1.00000	−0.0721688
$193$	−12.8056	−0.921767	−0.460884	−	0.887461i	$-0.652467\pi$
$193$	−12.8056	−0.921767	−0.460884	+	0.887461i	$0.652467\pi$
$194$	17.7708	1.27587
$195$	0	0
$196$	0	0
$197$	2.87141	0.204579	0.102290	−	0.994755i	$-0.467383\pi$
$197$	2.87141	0.204579	0.102290	+	0.994755i	$0.467383\pi$
$198$	1.67701	0.119180
$199$	3.03477	0.215129	0.107565	−	0.994198i	$-0.465695\pi$
$199$	3.03477	0.215129	0.107565	+	0.994198i	$0.465695\pi$
$200$	0	0
$201$	3.35402	0.236574
$202$	8.00000	0.562878
$203$	0	0
$204$	7.61121	0.532891
$205$	0	0
$206$	13.6112	0.948338
$207$	−0.482613	−0.0335439
$208$	−4.48261	−0.310813
$209$	−7.51739	−0.519989
$210$	0	0
$211$	−10.4826	−0.721653	−0.360826	−	0.932633i	$-0.617505\pi$
$211$	−10.4826	−0.721653	−0.360826	+	0.932633i	$0.617505\pi$
$212$	9.57643	0.657712
$213$	−6.00000	−0.411113
$214$	3.87141	0.264644
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	17.2224	1.16645
$219$	4.00000	0.270295
$220$	0	0
$221$	34.1181	2.29503
$222$	0.482613	0.0323909
$223$	15.1248	1.01283	0.506417	−	0.862288i	$-0.330970\pi$
$223$	15.1248	1.01283	0.506417	+	0.862288i	$0.330970\pi$
$224$	0	0
$225$	0	0
$226$	−8.96523	−0.596358
$227$	−21.0938	−1.40005	−0.700023	−	0.714120i	$-0.746827\pi$
$227$	−21.0938	−1.40005	−0.700023	+	0.714120i	$0.746827\pi$
$228$	4.48261	0.296868
$229$	23.2224	1.53458	0.767290	−	0.641300i	$-0.221605\pi$
$229$	23.2224	1.53458	0.767290	+	0.641300i	$0.221605\pi$
$230$	0	0
$231$	0	0
$232$	1.19440	0.0784160
$233$	8.57643	0.561861	0.280930	−	0.959728i	$-0.409357\pi$
$233$	8.57643	0.561861	0.280930	+	0.959728i	$0.409357\pi$
$234$	−4.48261	−0.293038
$235$	0	0
$236$	−5.77083	−0.375649
$237$	−4.51739	−0.293436
$238$	0	0
$239$	−4.57643	−0.296025	−0.148012	−	0.988986i	$-0.547288\pi$
$239$	−4.57643	−0.296025	−0.148012	+	0.988986i	$0.547288\pi$
$240$	0	0
$241$	−9.96523	−0.641917	−0.320958	−	0.947093i	$-0.604005\pi$
$241$	−9.96523	−0.641917	−0.320958	+	0.947093i	$0.604005\pi$
$242$	−8.18764	−0.526321
$243$	−1.00000	−0.0641500
$244$	10.5764	0.677087
$245$	0	0
$246$	−12.0938	−0.771074
$247$	20.0938	1.27854
$248$	3.48261	0.221146
$249$	1.87141	0.118596
$250$	0	0
$251$	4.38505	0.276782	0.138391	−	0.990378i	$-0.455807\pi$
$251$	4.38505	0.276782	0.138391	+	0.990378i	$0.455807\pi$
$252$	0	0
$253$	−0.809347	−0.0508832
$254$	13.9342	0.874309
$255$	0	0
$256$	1.00000	0.0625000
$257$	−4.70804	−0.293679	−0.146840	−	0.989160i	$-0.546910\pi$
$257$	−4.70804	−0.293679	−0.146840	+	0.989160i	$0.546910\pi$
$258$	2.00000	0.124515
$259$	0	0
$260$	0	0
$261$	1.19440	0.0739313
$262$	−3.28822	−0.203147
$263$	−16.3192	−1.00629	−0.503144	−	0.864203i	$-0.667823\pi$
$263$	−16.3192	−1.00629	−0.503144	+	0.864203i	$0.667823\pi$
$264$	−1.67701	−0.103213
$265$	0	0
$266$	0	0
$267$	−3.35402	−0.205263
$268$	−3.35402	−0.204879
$269$	19.7708	1.20545	0.602724	−	0.797949i	$-0.294082\pi$
$269$	19.7708	1.20545	0.602724	+	0.797949i	$0.294082\pi$
$270$	0	0
$271$	2.26020	0.137297	0.0686487	−	0.997641i	$-0.478131\pi$
$271$	2.26020	0.137297	0.0686487	+	0.997641i	$0.478131\pi$
$272$	−7.61121	−0.461497
$273$	0	0
$274$	−4.25719	−0.257186
$275$	0	0
$276$	0.482613	0.0290499
$277$	−18.2572	−1.09697	−0.548484	−	0.836161i	$-0.684795\pi$
$277$	−18.2572	−1.09697	−0.548484	+	0.836161i	$0.684795\pi$
$278$	−18.9652	−1.13746
$279$	3.48261	0.208499
$280$	0	0
$281$	10.4826	0.625340	0.312670	−	0.949862i	$-0.398777\pi$
$281$	10.4826	0.625340	0.312670	+	0.949862i	$0.398777\pi$
$282$	−11.4478	−0.681709
$283$	−0.708040	−0.0420886	−0.0210443	−	0.999779i	$-0.506699\pi$
$283$	−0.708040	−0.0420886	−0.0210443	+	0.999779i	$0.506699\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	−7.51739	−0.444512
$287$	0	0
$288$	1.00000	0.0589256
$289$	40.9305	2.40767
$290$	0	0
$291$	−17.7708	−1.04174
$292$	−4.00000	−0.234082
$293$	−12.6112	−0.736755	−0.368377	−	0.929676i	$-0.620087\pi$
$293$	−12.6112	−0.736755	−0.368377	+	0.929676i	$0.620087\pi$
$294$	0	0
$295$	0	0
$296$	−0.482613	−0.0280513
$297$	−1.67701	−0.0973100
$298$	7.03477	0.407514
$299$	2.16337	0.125111
$300$	0	0
$301$	0	0
$302$	−6.44784	−0.371031
$303$	−8.00000	−0.459588
$304$	−4.48261	−0.257095
$305$	0	0
$306$	−7.61121	−0.435104
$307$	−9.54166	−0.544571	−0.272286	−	0.962216i	$-0.587780\pi$
$307$	−9.54166	−0.544571	−0.272286	+	0.962216i	$0.587780\pi$
$308$	0	0
$309$	−13.6112	−0.774314
$310$	0	0
$311$	20.9652	1.18883	0.594414	−	0.804159i	$-0.297384\pi$
$311$	20.9652	1.18883	0.594414	+	0.804159i	$0.297384\pi$
$312$	4.48261	0.253778
$313$	15.1944	0.858838	0.429419	−	0.903105i	$-0.358718\pi$
$313$	15.1944	0.858838	0.429419	+	0.903105i	$0.358718\pi$
$314$	0.0938186	0.00529449
$315$	0	0
$316$	4.51739	0.254123
$317$	−18.4478	−1.03613	−0.518067	−	0.855340i	$-0.673348\pi$
$317$	−18.4478	−1.03613	−0.518067	+	0.855340i	$0.673348\pi$
$318$	−9.57643	−0.537020
$319$	2.00302	0.112147
$320$	0	0
$321$	−3.87141	−0.216081
$322$	0	0
$323$	34.1181	1.89838
$324$	1.00000	0.0555556
$325$	0	0
$326$	20.5764	1.13962
$327$	−17.2224	−0.952402
$328$	12.0938	0.667769
$329$	0	0
$330$	0	0
$331$	22.8019	1.25330	0.626652	−	0.779299i	$-0.284425\pi$
$331$	22.8019	1.25330	0.626652	+	0.779299i	$0.284425\pi$
$332$	−1.87141	−0.102707
$333$	−0.482613	−0.0264470
$334$	−14.0938	−0.771179
$335$	0	0
$336$	0	0
$337$	−9.12485	−0.497062	−0.248531	−	0.968624i	$-0.579948\pi$
$337$	−9.12485	−0.497062	−0.248531	+	0.968624i	$0.579948\pi$
$338$	7.09382	0.385853
$339$	8.96523	0.486924
$340$	0	0
$341$	5.84038	0.316274
$342$	−4.48261	−0.242392
$343$	0	0
$344$	−2.00000	−0.107833
$345$	0	0
$346$	−1.12859	−0.0606735
$347$	19.2224	1.03191	0.515957	−	0.856615i	$-0.327437\pi$
$347$	19.2224	1.03191	0.515957	+	0.856615i	$0.327437\pi$
$348$	−1.19440	−0.0640264
$349$	6.18764	0.331217	0.165608	−	0.986192i	$-0.447041\pi$
$349$	6.18764	0.331217	0.165608	+	0.986192i	$0.447041\pi$
$350$	0	0
$351$	4.48261	0.239264
$352$	1.67701	0.0893849
$353$	11.2224	0.597309	0.298654	−	0.954361i	$-0.403462\pi$
$353$	11.2224	0.597309	0.298654	+	0.954361i	$0.403462\pi$
$354$	5.77083	0.306716
$355$	0	0
$356$	3.35402	0.177763
$357$	0	0
$358$	−3.44784	−0.182224
$359$	9.61121	0.507260	0.253630	−	0.967301i	$-0.418375\pi$
$359$	9.61121	0.507260	0.253630	+	0.967301i	$0.418375\pi$
$360$	0	0
$361$	1.09382	0.0575694
$362$	−22.8957	−1.20337
$363$	8.18764	0.429740
$364$	0	0
$365$	0	0
$366$	−10.5764	−0.552839
$367$	26.2534	1.37042	0.685209	−	0.728346i	$-0.259711\pi$
$367$	26.2534	1.37042	0.685209	+	0.728346i	$0.259711\pi$
$368$	−0.482613	−0.0251579
$369$	12.0938	0.629579
$370$	0	0
$371$	0	0
$372$	−3.48261	−0.180565
$373$	−17.4796	−0.905059	−0.452530	−	0.891749i	$-0.649478\pi$
$373$	−17.4796	−0.905059	−0.452530	+	0.891749i	$0.649478\pi$
$374$	−12.7641	−0.660014
$375$	0	0
$376$	11.4478	0.590377
$377$	−5.35402	−0.275746
$378$	0	0
$379$	32.2815	1.65819	0.829094	−	0.559110i	$-0.188857\pi$
$379$	32.2815	1.65819	0.829094	+	0.559110i	$0.188857\pi$
$380$	0	0
$381$	−13.9342	−0.713870
$382$	25.1529	1.28693
$383$	18.0938	0.924551	0.462275	−	0.886736i	$-0.347033\pi$
$383$	18.0938	0.924551	0.462275	+	0.886736i	$0.347033\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−12.8056	−0.651788
$387$	−2.00000	−0.101666
$388$	17.7708	0.902177
$389$	−8.18764	−0.415130	−0.207565	−	0.978221i	$-0.566554\pi$
$389$	−8.18764	−0.415130	−0.207565	+	0.978221i	$0.566554\pi$
$390$	0	0
$391$	3.67327	0.185765
$392$	0	0
$393$	3.28822	0.165869
$394$	2.87141	0.144659
$395$	0	0
$396$	1.67701	0.0842729
$397$	36.1876	1.81621	0.908103	−	0.418747i	$-0.137531\pi$
$397$	36.1876	1.81621	0.908103	+	0.418747i	$0.137531\pi$
$398$	3.03477	0.152119
$399$	0	0
$400$	0	0
$401$	−30.9895	−1.54754	−0.773771	−	0.633466i	$-0.781632\pi$
$401$	−30.9895	−1.54754	−0.773771	+	0.633466i	$0.781632\pi$
$402$	3.35402	0.167283
$403$	−15.6112	−0.777650
$404$	8.00000	0.398015
$405$	0	0
$406$	0	0
$407$	−0.809347	−0.0401178
$408$	7.61121	0.376811
$409$	−13.0938	−0.647448	−0.323724	−	0.946152i	$-0.604935\pi$
$409$	−13.0938	−0.647448	−0.323724	+	0.946152i	$0.604935\pi$
$410$	0	0
$411$	4.25719	0.209991
$412$	13.6112	0.670576
$413$	0	0
$414$	−0.482613	−0.0237191
$415$	0	0
$416$	−4.48261	−0.219778
$417$	18.9652	0.928731
$418$	−7.51739	−0.367687
$419$	−33.3783	−1.63064	−0.815318	−	0.579013i	$-0.803438\pi$
$419$	−33.3783	−1.63064	−0.815318	+	0.579013i	$0.803438\pi$
$420$	0	0
$421$	−6.90317	−0.336440	−0.168220	−	0.985750i	$-0.553802\pi$
$421$	−6.90317	−0.336440	−0.168220	+	0.985750i	$0.553802\pi$
$422$	−10.4826	−0.510286
$423$	11.4478	0.556613
$424$	9.57643	0.465073
$425$	0	0
$426$	−6.00000	−0.290701
$427$	0	0
$428$	3.87141	0.187132
$429$	7.51739	0.362943
$430$	0	0
$431$	−26.1181	−1.25806	−0.629032	−	0.777379i	$-0.716549\pi$
$431$	−26.1181	−1.25806	−0.629032	+	0.777379i	$0.716549\pi$
$432$	−1.00000	−0.0481125
$433$	25.9305	1.24614	0.623069	−	0.782167i	$-0.285886\pi$
$433$	25.9305	1.24614	0.623069	+	0.782167i	$0.285886\pi$
$434$	0	0
$435$	0	0
$436$	17.2224	0.824804
$437$	2.16337	0.103488
$438$	4.00000	0.191127
$439$	7.55216	0.360445	0.180222	−	0.983626i	$-0.442318\pi$
$439$	7.55216	0.360445	0.180222	+	0.983626i	$0.442318\pi$
$440$	0	0
$441$	0	0
$442$	34.1181	1.62283
$443$	−24.3162	−1.15530	−0.577649	−	0.816285i	$-0.696030\pi$
$443$	−24.3162	−1.15530	−0.577649	+	0.816285i	$0.696030\pi$
$444$	0.482613	0.0229038
$445$	0	0
$446$	15.1248	0.716182
$447$	−7.03477	−0.332733
$448$	0	0
$449$	14.7398	0.695614	0.347807	−	0.937566i	$-0.386926\pi$
$449$	14.7398	0.695614	0.347807	+	0.937566i	$0.386926\pi$
$450$	0	0
$451$	20.2815	0.955016
$452$	−8.96523	−0.421689
$453$	6.44784	0.302946
$454$	−21.0938	−0.989982
$455$	0	0
$456$	4.48261	0.209918
$457$	−0.805603	−0.0376845	−0.0188423	−	0.999822i	$-0.505998\pi$
$457$	−0.805603	−0.0376845	−0.0188423	+	0.999822i	$0.505998\pi$
$458$	23.2224	1.08511
$459$	7.61121	0.355261
$460$	0	0
$461$	−23.8609	−1.11131	−0.555657	−	0.831412i	$-0.687533\pi$
$461$	−23.8609	−1.11131	−0.555657	+	0.831412i	$0.687533\pi$
$462$	0	0
$463$	−9.19065	−0.427126	−0.213563	−	0.976929i	$-0.568507\pi$
$463$	−9.19065	−0.427126	−0.213563	+	0.976929i	$0.568507\pi$
$464$	1.19440	0.0554485
$465$	0	0
$466$	8.57643	0.397296
$467$	22.1181	1.02350	0.511752	−	0.859133i	$-0.328997\pi$
$467$	22.1181	1.02350	0.511752	+	0.859133i	$0.328997\pi$
$468$	−4.48261	−0.207209
$469$	0	0
$470$	0	0
$471$	−0.0938186	−0.00432293
$472$	−5.77083	−0.265624
$473$	−3.35402	−0.154218
$474$	−4.51739	−0.207490
$475$	0	0
$476$	0	0
$477$	9.57643	0.438475
$478$	−4.57643	−0.209321
$479$	33.4100	1.52654	0.763272	−	0.646077i	$-0.223592\pi$
$479$	33.4100	1.52654	0.763272	+	0.646077i	$0.223592\pi$
$480$	0	0
$481$	2.16337	0.0986410
$482$	−9.96523	−0.453904
$483$	0	0
$484$	−8.18764	−0.372165
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	2.34726	0.106365	0.0531823	−	0.998585i	$-0.483064\pi$
$487$	2.34726	0.106365	0.0531823	+	0.998585i	$0.483064\pi$
$488$	10.5764	0.478773
$489$	−20.5764	−0.930498
$490$	0	0
$491$	23.3820	1.05522	0.527608	−	0.849488i	$-0.323089\pi$
$491$	23.3820	1.05522	0.527608	+	0.849488i	$0.323089\pi$
$492$	−12.0938	−0.545231
$493$	−9.09080	−0.409429
$494$	20.0938	0.904064
$495$	0	0
$496$	3.48261	0.156374
$497$	0	0
$498$	1.87141	0.0838598
$499$	14.9652	0.669936	0.334968	−	0.942230i	$-0.391275\pi$
$499$	14.9652	0.669936	0.334968	+	0.942230i	$0.391275\pi$
$500$	0	0
$501$	14.0938	0.629665
$502$	4.38505	0.195714
$503$	−5.35402	−0.238724	−0.119362	−	0.992851i	$-0.538085\pi$
$503$	−5.35402	−0.238724	−0.119362	+	0.992851i	$0.538085\pi$
$504$	0	0
$505$	0	0
$506$	−0.809347	−0.0359799
$507$	−7.09382	−0.315048
$508$	13.9342	0.618230
$509$	20.9237	0.927426	0.463713	−	0.885985i	$-0.346517\pi$
$509$	20.9237	0.927426	0.463713	+	0.885985i	$0.346517\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	4.48261	0.197912
$514$	−4.70804	−0.207663
$515$	0	0
$516$	2.00000	0.0880451
$517$	19.1981	0.844333
$518$	0	0
$519$	1.12859	0.0495397
$520$	0	0
$521$	−4.35100	−0.190621	−0.0953105	−	0.995448i	$-0.530384\pi$
$521$	−4.35100	−0.190621	−0.0953105	+	0.995448i	$0.530384\pi$
$522$	1.19440	0.0522773
$523$	39.2845	1.71779	0.858895	−	0.512152i	$-0.171151\pi$
$523$	39.2845	1.71779	0.858895	+	0.512152i	$0.171151\pi$
$524$	−3.28822	−0.143646
$525$	0	0
$526$	−16.3192	−0.711553
$527$	−26.5069	−1.15466
$528$	−1.67701	−0.0729825
$529$	−22.7671	−0.989873
$530$	0	0
$531$	−5.77083	−0.250433
$532$	0	0
$533$	−54.2119	−2.34818
$534$	−3.35402	−0.145143
$535$	0	0
$536$	−3.35402	−0.144872
$537$	3.44784	0.148785
$538$	19.7708	0.852381
$539$	0	0
$540$	0	0
$541$	14.9032	0.640737	0.320369	−	0.947293i	$-0.396193\pi$
$541$	14.9032	0.640737	0.320369	+	0.947293i	$0.396193\pi$
$542$	2.26020	0.0970840
$543$	22.8957	0.982548
$544$	−7.61121	−0.326328
$545$	0	0
$546$	0	0
$547$	−43.9865	−1.88073	−0.940363	−	0.340173i	$-0.889515\pi$
$547$	−43.9865	−1.88073	−0.940363	+	0.340173i	$0.889515\pi$
$548$	−4.25719	−0.181858
$549$	10.5764	0.451391
$550$	0	0
$551$	−5.35402	−0.228089
$552$	0.482613	0.0205414
$553$	0	0
$554$	−18.2572	−0.775673
$555$	0	0
$556$	−18.9652	−0.804305
$557$	4.86839	0.206280	0.103140	−	0.994667i	$-0.467111\pi$
$557$	4.86839	0.206280	0.103140	+	0.994667i	$0.467111\pi$
$558$	3.48261	0.147431
$559$	8.96523	0.379189
$560$	0	0
$561$	12.7641	0.538899
$562$	10.4826	0.442182
$563$	−5.35100	−0.225518	−0.112759	−	0.993622i	$-0.535969\pi$
$563$	−5.35100	−0.225518	−0.112759	+	0.993622i	$0.535969\pi$
$564$	−11.4478	−0.482041
$565$	0	0
$566$	−0.708040	−0.0297612
$567$	0	0
$568$	6.00000	0.251754
$569$	8.15588	0.341912	0.170956	−	0.985279i	$-0.445314\pi$
$569$	8.15588	0.341912	0.170956	+	0.985279i	$0.445314\pi$
$570$	0	0
$571$	−5.61121	−0.234822	−0.117411	−	0.993083i	$-0.537459\pi$
$571$	−5.61121	−0.234822	−0.117411	+	0.993083i	$0.537459\pi$
$572$	−7.51739	−0.314318
$573$	−25.1529	−1.05078
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	−4.67399	−0.194581	−0.0972905	−	0.995256i	$-0.531018\pi$
$577$	−4.67399	−0.194581	−0.0972905	+	0.995256i	$0.531018\pi$
$578$	40.9305	1.70248
$579$	12.8056	0.532183
$580$	0	0
$581$	0	0
$582$	−17.7708	−0.736625
$583$	16.0598	0.665128
$584$	−4.00000	−0.165521
$585$	0	0
$586$	−12.6112	−0.520964
$587$	−18.9062	−0.780342	−0.390171	−	0.920742i	$-0.627584\pi$
$587$	−18.9062	−0.780342	−0.390171	+	0.920742i	$0.627584\pi$
$588$	0	0
$589$	−15.6112	−0.643249
$590$	0	0
$591$	−2.87141	−0.118114
$592$	−0.482613	−0.0198353
$593$	−21.5417	−0.884610	−0.442305	−	0.896865i	$-0.645839\pi$
$593$	−21.5417	−0.884610	−0.442305	+	0.896865i	$0.645839\pi$
$594$	−1.67701	−0.0688086
$595$	0	0
$596$	7.03477	0.288156
$597$	−3.03477	−0.124205
$598$	2.16337	0.0884667
$599$	34.7641	1.42042	0.710211	−	0.703989i	$-0.248600\pi$
$599$	34.7641	1.42042	0.710211	+	0.703989i	$0.248600\pi$
$600$	0	0
$601$	−4.19814	−0.171246	−0.0856229	−	0.996328i	$-0.527288\pi$
$601$	−4.19814	−0.171246	−0.0856229	+	0.996328i	$0.527288\pi$
$602$	0	0
$603$	−3.35402	−0.136586
$604$	−6.44784	−0.262359
$605$	0	0
$606$	−8.00000	−0.324978
$607$	42.1218	1.70967	0.854836	−	0.518898i	$-0.173658\pi$
$607$	42.1218	1.70967	0.854836	+	0.518898i	$0.173658\pi$
$608$	−4.48261	−0.181794
$609$	0	0
$610$	0	0
$611$	−51.3162	−2.07603
$612$	−7.61121	−0.307665
$613$	−18.5522	−0.749315	−0.374657	−	0.927163i	$-0.622240\pi$
$613$	−18.5522	−0.749315	−0.374657	+	0.927163i	$0.622240\pi$
$614$	−9.54166	−0.385070
$615$	0	0
$616$	0	0
$617$	−13.4236	−0.540413	−0.270206	−	0.962802i	$-0.587092\pi$
$617$	−13.4236	−0.540413	−0.270206	+	0.962802i	$0.587092\pi$
$618$	−13.6112	−0.547523
$619$	31.1211	1.25086	0.625431	−	0.780279i	$-0.284923\pi$
$619$	31.1211	1.25086	0.625431	+	0.780279i	$0.284923\pi$
$620$	0	0
$621$	0.482613	0.0193666
$622$	20.9652	0.840629
$623$	0	0
$624$	4.48261	0.179448
$625$	0	0
$626$	15.1944	0.607290
$627$	7.51739	0.300216
$628$	0.0938186	0.00374377
$629$	3.67327	0.146463
$630$	0	0
$631$	−1.15588	−0.0460148	−0.0230074	−	0.999735i	$-0.507324\pi$
$631$	−1.15588	−0.0460148	−0.0230074	+	0.999735i	$0.507324\pi$
$632$	4.51739	0.179692
$633$	10.4826	0.416646
$634$	−18.4478	−0.732657
$635$	0	0
$636$	−9.57643	−0.379730
$637$	0	0
$638$	2.00302	0.0793002
$639$	6.00000	0.237356
$640$	0	0
$641$	−15.0590	−0.594796	−0.297398	−	0.954754i	$-0.596119\pi$
$641$	−15.0590	−0.594796	−0.297398	+	0.954754i	$0.596119\pi$
$642$	−3.87141	−0.152792
$643$	−35.2149	−1.38874	−0.694371	−	0.719618i	$-0.744317\pi$
$643$	−35.2149	−1.38874	−0.694371	+	0.719618i	$0.744317\pi$
$644$	0	0
$645$	0	0
$646$	34.1181	1.34236
$647$	−4.41306	−0.173495	−0.0867477	−	0.996230i	$-0.527647\pi$
$647$	−4.41306	−0.173495	−0.0867477	+	0.996230i	$0.527647\pi$
$648$	1.00000	0.0392837
$649$	−9.67774	−0.379884
$650$	0	0
$651$	0	0
$652$	20.5764	0.805835
$653$	−18.0968	−0.708184	−0.354092	−	0.935211i	$-0.615210\pi$
$653$	−18.0968	−0.708184	−0.354092	+	0.935211i	$0.615210\pi$
$654$	−17.2224	−0.673450
$655$	0	0
$656$	12.0938	0.472184
$657$	−4.00000	−0.156055
$658$	0	0
$659$	25.6112	0.997671	0.498835	−	0.866697i	$-0.333761\pi$
$659$	25.6112	0.997671	0.498835	+	0.866697i	$0.333761\pi$
$660$	0	0
$661$	−22.2497	−0.865413	−0.432706	−	0.901535i	$-0.642441\pi$
$661$	−22.2497	−0.865413	−0.432706	+	0.901535i	$0.642441\pi$
$662$	22.8019	0.886219
$663$	−34.1181	−1.32504
$664$	−1.87141	−0.0726247
$665$	0	0
$666$	−0.482613	−0.0187009
$667$	−0.576432	−0.0223195
$668$	−14.0938	−0.545306
$669$	−15.1248	−0.584760
$670$	0	0
$671$	17.7368	0.684721
$672$	0	0
$673$	36.5349	1.40832	0.704158	−	0.710043i	$-0.251324\pi$
$673$	36.5349	1.40832	0.704158	+	0.710043i	$0.251324\pi$
$674$	−9.12485	−0.351476
$675$	0	0
$676$	7.09382	0.272839
$677$	−28.6112	−1.09962	−0.549809	−	0.835290i	$-0.685299\pi$
$677$	−28.6112	−1.09962	−0.549809	+	0.835290i	$0.685299\pi$
$678$	8.96523	0.344307
$679$	0	0
$680$	0	0
$681$	21.0938	0.808317
$682$	5.84038	0.223640
$683$	12.5099	0.478678	0.239339	−	0.970936i	$-0.423069\pi$
$683$	12.5099	0.478678	0.239339	+	0.970936i	$0.423069\pi$
$684$	−4.48261	−0.171397
$685$	0	0
$686$	0	0
$687$	−23.2224	−0.885990
$688$	−2.00000	−0.0762493
$689$	−42.9274	−1.63541
$690$	0	0
$691$	−18.4448	−0.701674	−0.350837	−	0.936437i	$-0.614103\pi$
$691$	−18.4448	−0.701674	−0.350837	+	0.936437i	$0.614103\pi$
$692$	−1.12859	−0.0429027
$693$	0	0
$694$	19.2224	0.729673
$695$	0	0
$696$	−1.19440	−0.0452735
$697$	−92.0485	−3.48659
$698$	6.18764	0.234206
$699$	−8.57643	−0.324390
$700$	0	0
$701$	−2.15962	−0.0815678	−0.0407839	−	0.999168i	$-0.512986\pi$
$701$	−2.15962	−0.0815678	−0.0407839	+	0.999168i	$0.512986\pi$
$702$	4.48261	0.169185
$703$	2.16337	0.0815929
$704$	1.67701	0.0632047
$705$	0	0
$706$	11.2224	0.422361
$707$	0	0
$708$	5.77083	0.216881
$709$	−19.6037	−0.736233	−0.368117	−	0.929780i	$-0.619997\pi$
$709$	−19.6037	−0.736233	−0.368117	+	0.929780i	$0.619997\pi$
$710$	0	0
$711$	4.51739	0.169415
$712$	3.35402	0.125697
$713$	−1.68075	−0.0629447
$714$	0	0
$715$	0	0
$716$	−3.44784	−0.128852
$717$	4.57643	0.170910
$718$	9.61121	0.358687
$719$	−53.0213	−1.97736	−0.988680	−	0.150042i	$-0.952059\pi$
$719$	−53.0213	−1.97736	−0.988680	+	0.150042i	$0.952059\pi$
$720$	0	0
$721$	0	0
$722$	1.09382	0.0407077
$723$	9.96523	0.370611
$724$	−22.8957	−0.850911
$725$	0	0
$726$	8.18764	0.303872
$727$	28.3155	1.05016	0.525082	−	0.851052i	$-0.324035\pi$
$727$	28.3155	1.05016	0.525082	+	0.851052i	$0.324035\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	15.2224	0.563021
$732$	−10.5764	−0.390916
$733$	−24.9895	−0.923008	−0.461504	−	0.887138i	$-0.652690\pi$
$733$	−24.9895	−0.923008	−0.461504	+	0.887138i	$0.652690\pi$
$734$	26.2534	0.969032
$735$	0	0
$736$	−0.482613	−0.0177893
$737$	−5.62473	−0.207189
$738$	12.0938	0.445180
$739$	43.2467	1.59085	0.795427	−	0.606049i	$-0.207246\pi$
$739$	43.2467	1.59085	0.795427	+	0.606049i	$0.207246\pi$
$740$	0	0
$741$	−20.0938	−0.738165
$742$	0	0
$743$	−21.9062	−0.803660	−0.401830	−	0.915714i	$-0.631626\pi$
$743$	−21.9062	−0.803660	−0.401830	+	0.915714i	$0.631626\pi$
$744$	−3.48261	−0.127679
$745$	0	0
$746$	−17.4796	−0.639974
$747$	−1.87141	−0.0684712
$748$	−12.7641	−0.466701
$749$	0	0
$750$	0	0
$751$	26.8927	0.981327	0.490664	−	0.871349i	$-0.336754\pi$
$751$	26.8927	0.981327	0.490664	+	0.871349i	$0.336754\pi$
$752$	11.4478	0.417460
$753$	−4.38505	−0.159800
$754$	−5.35402	−0.194982
$755$	0	0
$756$	0	0
$757$	11.1529	0.405358	0.202679	−	0.979245i	$-0.435035\pi$
$757$	11.1529	0.405358	0.202679	+	0.979245i	$0.435035\pi$
$758$	32.2815	1.17252
$759$	0.809347	0.0293774
$760$	0	0
$761$	−1.58694	−0.0575264	−0.0287632	−	0.999586i	$-0.509157\pi$
$761$	−1.58694	−0.0575264	−0.0287632	+	0.999586i	$0.509157\pi$
$762$	−13.9342	−0.504783
$763$	0	0
$764$	25.1529	0.909999
$765$	0	0
$766$	18.0938	0.653756
$767$	25.8684	0.934053
$768$	−1.00000	−0.0360844
$769$	−19.2572	−0.694432	−0.347216	−	0.937785i	$-0.612873\pi$
$769$	−19.2572	−0.694432	−0.347216	+	0.937785i	$0.612873\pi$
$770$	0	0
$771$	4.70804	0.169556
$772$	−12.8056	−0.460884
$773$	−30.2815	−1.08915	−0.544574	−	0.838713i	$-0.683309\pi$
$773$	−30.2815	−1.08915	−0.544574	+	0.838713i	$0.683309\pi$
$774$	−2.00000	−0.0718885
$775$	0	0
$776$	17.7708	0.637936
$777$	0	0
$778$	−8.18764	−0.293541
$779$	−54.2119	−1.94234
$780$	0	0
$781$	10.0621	0.360049
$782$	3.67327	0.131356
$783$	−1.19440	−0.0426843
$784$	0	0
$785$	0	0
$786$	3.28822	0.117287
$787$	33.2224	1.18425	0.592126	−	0.805846i	$-0.298289\pi$
$787$	33.2224	1.18425	0.592126	+	0.805846i	$0.298289\pi$
$788$	2.87141	0.102290
$789$	16.3192	0.580981
$790$	0	0
$791$	0	0
$792$	1.67701	0.0595900
$793$	−47.4100	−1.68358
$794$	36.1876	1.28425
$795$	0	0
$796$	3.03477	0.107565
$797$	41.7883	1.48022	0.740109	−	0.672486i	$-0.234774\pi$
$797$	41.7883	1.48022	0.740109	+	0.672486i	$0.234774\pi$
$798$	0	0
$799$	−87.1319	−3.08250
$800$	0	0
$801$	3.35402	0.118508
$802$	−30.9895	−1.09428
$803$	−6.70804	−0.236722
$804$	3.35402	0.118287
$805$	0	0
$806$	−15.6112	−0.549881
$807$	−19.7708	−0.695966
$808$	8.00000	0.281439
$809$	5.05904	0.177867	0.0889333	−	0.996038i	$-0.471654\pi$
$809$	5.05904	0.177867	0.0889333	+	0.996038i	$0.471654\pi$
$810$	0	0
$811$	33.5039	1.17648	0.588240	−	0.808686i	$-0.299821\pi$
$811$	33.5039	1.17648	0.588240	+	0.808686i	$0.299821\pi$
$812$	0	0
$813$	−2.26020	−0.0792687
$814$	−0.809347	−0.0283676
$815$	0	0
$816$	7.61121	0.266445
$817$	8.96523	0.313654
$818$	−13.0938	−0.457815
$819$	0	0
$820$	0	0
$821$	29.7013	1.03658	0.518291	−	0.855204i	$-0.326568\pi$
$821$	29.7013	1.03658	0.518291	+	0.855204i	$0.326568\pi$
$822$	4.25719	0.148486
$823$	−38.2497	−1.33330	−0.666650	−	0.745371i	$-0.732273\pi$
$823$	−38.2497	−1.33330	−0.666650	+	0.745371i	$0.732273\pi$
$824$	13.6112	0.474169
$825$	0	0
$826$	0	0
$827$	13.3510	0.464260	0.232130	−	0.972685i	$-0.425431\pi$
$827$	13.3510	0.464260	0.232130	+	0.972685i	$0.425431\pi$
$828$	−0.482613	−0.0167720
$829$	3.61121	0.125422	0.0627112	−	0.998032i	$-0.480025\pi$
$829$	3.61121	0.125422	0.0627112	+	0.998032i	$0.480025\pi$
$830$	0	0
$831$	18.2572	0.633335
$832$	−4.48261	−0.155407
$833$	0	0
$834$	18.9652	0.656712
$835$	0	0
$836$	−7.51739	−0.259994
$837$	−3.48261	−0.120377
$838$	−33.3783	−1.15303
$839$	−1.68075	−0.0580261	−0.0290130	−	0.999579i	$-0.509236\pi$
$839$	−1.68075	−0.0580261	−0.0290130	+	0.999579i	$0.509236\pi$
$840$	0	0
$841$	−27.5734	−0.950807
$842$	−6.90317	−0.237899
$843$	−10.4826	−0.361040
$844$	−10.4826	−0.360826
$845$	0	0
$846$	11.4478	0.393585
$847$	0	0
$848$	9.57643	0.328856
$849$	0.708040	0.0242999
$850$	0	0
$851$	0.232915	0.00798423
$852$	−6.00000	−0.205557
$853$	−1.70502	−0.0583789	−0.0291895	−	0.999574i	$-0.509293\pi$
$853$	−1.70502	−0.0583789	−0.0291895	+	0.999574i	$0.509293\pi$
$854$	0	0
$855$	0	0
$856$	3.87141	0.132322
$857$	28.6945	0.980186	0.490093	−	0.871670i	$-0.336963\pi$
$857$	28.6945	0.980186	0.490093	+	0.871670i	$0.336963\pi$
$858$	7.51739	0.256639
$859$	26.7641	0.913178	0.456589	−	0.889678i	$-0.349071\pi$
$859$	26.7641	0.913178	0.456589	+	0.889678i	$0.349071\pi$
$860$	0	0
$861$	0	0
$862$	−26.1181	−0.889586
$863$	−32.0243	−1.09012	−0.545059	−	0.838397i	$-0.683493\pi$
$863$	−32.0243	−1.09012	−0.545059	+	0.838397i	$0.683493\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	25.9305	0.881153
$867$	−40.9305	−1.39007
$868$	0	0
$869$	7.57570	0.256988
$870$	0	0
$871$	15.0348	0.509434
$872$	17.2224	0.583224
$873$	17.7708	0.601451
$874$	2.16337	0.0731770
$875$	0	0
$876$	4.00000	0.135147
$877$	−18.1634	−0.613333	−0.306667	−	0.951817i	$-0.599214\pi$
$877$	−18.1634	−0.613333	−0.306667	+	0.951817i	$0.599214\pi$
$878$	7.55216	0.254873
$879$	12.6112	0.425365
$880$	0	0
$881$	24.2254	0.816175	0.408088	−	0.912943i	$-0.366196\pi$
$881$	24.2254	0.816175	0.408088	+	0.912943i	$0.366196\pi$
$882$	0	0
$883$	−32.3753	−1.08951	−0.544757	−	0.838594i	$-0.683378\pi$
$883$	−32.3753	−1.08951	−0.544757	+	0.838594i	$0.683378\pi$
$884$	34.1181	1.14752
$885$	0	0
$886$	−24.3162	−0.816920
$887$	−32.9517	−1.10641	−0.553205	−	0.833045i	$-0.686595\pi$
$887$	−32.9517	−1.10641	−0.553205	+	0.833045i	$0.686595\pi$
$888$	0.482613	0.0161954
$889$	0	0
$890$	0	0
$891$	1.67701	0.0561820
$892$	15.1248	0.506417
$893$	−51.3162	−1.71723
$894$	−7.03477	−0.235278
$895$	0	0
$896$	0	0
$897$	−2.16337	−0.0722327
$898$	14.7398	0.491873
$899$	4.15962	0.138731
$900$	0	0
$901$	−72.8882	−2.42826
$902$	20.2815	0.675299
$903$	0	0
$904$	−8.96523	−0.298179
$905$	0	0
$906$	6.44784	0.214215
$907$	−7.93794	−0.263575	−0.131787	−	0.991278i	$-0.542072\pi$
$907$	−7.93794	−0.263575	−0.131787	+	0.991278i	$0.542072\pi$
$908$	−21.0938	−0.700023
$909$	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	$0.531691\pi$
$912$	4.48261	0.148434
$913$	−3.13837	−0.103865
$914$	−0.805603	−0.0266470
$915$	0	0
$916$	23.2224	0.767290
$917$	0	0
$918$	7.61121	0.251207
$919$	1.15286	0.0380294	0.0190147	−	0.999819i	$-0.493947\pi$
$919$	1.15286	0.0380294	0.0190147	+	0.999819i	$0.493947\pi$
$920$	0	0
$921$	9.54166	0.314408
$922$	−23.8609	−0.785817
$923$	−26.8957	−0.885282
$924$	0	0
$925$	0	0
$926$	−9.19065	−0.302024
$927$	13.6112	0.447051
$928$	1.19440	0.0392080
$929$	−58.9199	−1.93310	−0.966550	−	0.256477i	$-0.917438\pi$
$929$	−58.9199	−1.93310	−0.966550	+	0.256477i	$0.917438\pi$
$930$	0	0
$931$	0	0
$932$	8.57643	0.280930
$933$	−20.9652	−0.686371
$934$	22.1181	0.723726
$935$	0	0
$936$	−4.48261	−0.146519
$937$	42.4168	1.38570	0.692848	−	0.721083i	$-0.256356\pi$
$937$	42.4168	1.38570	0.692848	+	0.721083i	$0.256356\pi$
$938$	0	0
$939$	−15.1944	−0.495850
$940$	0	0
$941$	−42.6665	−1.39089	−0.695444	−	0.718580i	$-0.744792\pi$
$941$	−42.6665	−1.39089	−0.695444	+	0.718580i	$0.744792\pi$
$942$	−0.0938186	−0.00305677
$943$	−5.83663	−0.190067
$944$	−5.77083	−0.187824
$945$	0	0
$946$	−3.35402	−0.109049
$947$	53.7914	1.74798	0.873992	−	0.485940i	$-0.161523\pi$
$947$	53.7914	1.74798	0.873992	+	0.485940i	$0.161523\pi$
$948$	−4.51739	−0.146718
$949$	17.9305	0.582047
$950$	0	0
$951$	18.4478	0.598212
$952$	0	0
$953$	−6.24970	−0.202448	−0.101224	−	0.994864i	$-0.532276\pi$
$953$	−6.24970	−0.202448	−0.101224	+	0.994864i	$0.532276\pi$
$954$	9.57643	0.310048
$955$	0	0
$956$	−4.57643	−0.148012
$957$	−2.00302	−0.0647483
$958$	33.4100	1.07943
$959$	0	0
$960$	0	0
$961$	−18.8714	−0.608755
$962$	2.16337	0.0697497
$963$	3.87141	0.124754
$964$	−9.96523	−0.320958
$965$	0	0
$966$	0	0
$967$	31.9585	1.02771	0.513857	−	0.857876i	$-0.328216\pi$
$967$	31.9585	1.02771	0.513857	+	0.857876i	$0.328216\pi$
$968$	−8.18764	−0.263161
$969$	−34.1181	−1.09603
$970$	0	0
$971$	23.0175	0.738667	0.369334	−	0.929297i	$-0.379586\pi$
$971$	23.0175	0.738667	0.369334	+	0.929297i	$0.379586\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	2.34726	0.0752111
$975$	0	0
$976$	10.5764	0.338543
$977$	−7.86839	−0.251732	−0.125866	−	0.992047i	$-0.540171\pi$
$977$	−7.86839	−0.251732	−0.125866	+	0.992047i	$0.540171\pi$
$978$	−20.5764	−0.657962
$979$	5.62473	0.179767
$980$	0	0
$981$	17.2224	0.549869
$982$	23.3820	0.746151
$983$	12.0938	0.385733	0.192866	−	0.981225i	$-0.438222\pi$
$983$	12.0938	0.385733	0.192866	+	0.981225i	$0.438222\pi$
$984$	−12.0938	−0.385537
$985$	0	0
$986$	−9.09080	−0.289510
$987$	0	0
$988$	20.0938	0.639270
$989$	0.965226	0.0306924
$990$	0	0
$991$	26.3783	0.837934	0.418967	−	0.908001i	$-0.362392\pi$
$991$	26.3783	0.837934	0.418967	+	0.908001i	$0.362392\pi$
$992$	3.48261	0.110573
$993$	−22.8019	−0.723595
$994$	0	0
$995$	0	0
$996$	1.87141	0.0592978
$997$	15.0908	0.477931	0.238965	−	0.971028i	$-0.423192\pi$
$997$	15.0908	0.477931	0.238965	+	0.971028i	$0.423192\pi$
$998$	14.9652	0.473716
$999$	0.482613	0.0152692

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7350.2.a.dp.1.2		3
5.2	odd	4		1470.2.g.h.589.6		6
5.3	odd	4		1470.2.g.h.589.3		6
5.4	even	2		7350.2.a.do.1.2		3
7.3	odd	6		1050.2.i.u.751.3		6
7.5	odd	6		1050.2.i.u.151.3		6
7.6	odd	2		7350.2.a.dq.1.2		3
35.2	odd	12		1470.2.n.j.949.4		12
35.3	even	12		210.2.n.b.79.6	yes	12
35.12	even	12		210.2.n.b.109.6	yes	12
35.13	even	4		1470.2.g.i.589.1		6
35.17	even	12		210.2.n.b.79.2	✓	12
35.18	odd	12		1470.2.n.j.79.4		12
35.19	odd	6		1050.2.i.v.151.1		6
35.23	odd	12		1470.2.n.j.949.2		12
35.24	odd	6		1050.2.i.v.751.1		6
35.27	even	4		1470.2.g.i.589.4		6
35.32	odd	12		1470.2.n.j.79.2		12
35.33	even	12		210.2.n.b.109.2	yes	12
35.34	odd	2		7350.2.a.dn.1.2		3
105.17	odd	12		630.2.u.f.289.5		12
105.38	odd	12		630.2.u.f.289.1		12
105.47	odd	12		630.2.u.f.109.1		12
105.68	odd	12		630.2.u.f.109.5		12
140.3	odd	12		1680.2.di.c.289.3		12
140.47	odd	12		1680.2.di.c.529.3		12
140.87	odd	12		1680.2.di.c.289.5		12
140.103	odd	12		1680.2.di.c.529.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.n.b.79.2	✓	12	35.17	even	12
210.2.n.b.79.6	yes	12	35.3	even	12
210.2.n.b.109.2	yes	12	35.33	even	12
210.2.n.b.109.6	yes	12	35.12	even	12
630.2.u.f.109.1		12	105.47	odd	12
630.2.u.f.109.5		12	105.68	odd	12
630.2.u.f.289.1		12	105.38	odd	12
630.2.u.f.289.5		12	105.17	odd	12
1050.2.i.u.151.3		6	7.5	odd	6
1050.2.i.u.751.3		6	7.3	odd	6
1050.2.i.v.151.1		6	35.19	odd	6
1050.2.i.v.751.1		6	35.24	odd	6
1470.2.g.h.589.3		6	5.3	odd	4
1470.2.g.h.589.6		6	5.2	odd	4
1470.2.g.i.589.1		6	35.13	even	4
1470.2.g.i.589.4		6	35.27	even	4
1470.2.n.j.79.2		12	35.32	odd	12
1470.2.n.j.79.4		12	35.18	odd	12
1470.2.n.j.949.2		12	35.23	odd	12
1470.2.n.j.949.4		12	35.2	odd	12
1680.2.di.c.289.3		12	140.3	odd	12
1680.2.di.c.289.5		12	140.87	odd	12
1680.2.di.c.529.3		12	140.47	odd	12
1680.2.di.c.529.5		12	140.103	odd	12
7350.2.a.dn.1.2		3	35.34	odd	2
7350.2.a.do.1.2		3	5.4	even	2
7350.2.a.dp.1.2		3	1.1	even	1	trivial
7350.2.a.dq.1.2		3	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.67701 q^{11} -1.00000 q^{12} -4.48261 q^{13} +1.00000 q^{16} -7.61121 q^{17} +1.00000 q^{18} -4.48261 q^{19} +1.67701 q^{22} -0.482613 q^{23} -1.00000 q^{24} -4.48261 q^{26} -1.00000 q^{27} +1.19440 q^{29} +3.48261 q^{31} +1.00000 q^{32} -1.67701 q^{33} -7.61121 q^{34} +1.00000 q^{36} -0.482613 q^{37} -4.48261 q^{38} +4.48261 q^{39} +12.0938 q^{41} -2.00000 q^{43} +1.67701 q^{44} -0.482613 q^{46} +11.4478 q^{47} -1.00000 q^{48} +7.61121 q^{51} -4.48261 q^{52} +9.57643 q^{53} -1.00000 q^{54} +4.48261 q^{57} +1.19440 q^{58} -5.77083 q^{59} +10.5764 q^{61} +3.48261 q^{62} +1.00000 q^{64} -1.67701 q^{66} -3.35402 q^{67} -7.61121 q^{68} +0.482613 q^{69} +6.00000 q^{71} +1.00000 q^{72} -4.00000 q^{73} -0.482613 q^{74} -4.48261 q^{76} +4.48261 q^{78} +4.51739 q^{79} +1.00000 q^{81} +12.0938 q^{82} -1.87141 q^{83} -2.00000 q^{86} -1.19440 q^{87} +1.67701 q^{88} +3.35402 q^{89} -0.482613 q^{92} -3.48261 q^{93} +11.4478 q^{94} -1.00000 q^{96} +17.7708 q^{97} +1.67701 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 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

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