LMFDB - Embedded newform 7350.2.a.dl.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7350, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7350.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$58.6900454856$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1470)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +2.24264 q^{11} +1.00000 q^{12} -5.65685 q^{13} +1.00000 q^{16} -2.58579 q^{17} +1.00000 q^{18} -6.82843 q^{19} +2.24264 q^{22} -3.17157 q^{23} +1.00000 q^{24} -5.65685 q^{26} +1.00000 q^{27} +2.58579 q^{29} -10.2426 q^{31} +1.00000 q^{32} +2.24264 q^{33} -2.58579 q^{34} +1.00000 q^{36} +0.242641 q^{37} -6.82843 q^{38} -5.65685 q^{39} +10.4853 q^{41} -9.07107 q^{43} +2.24264 q^{44} -3.17157 q^{46} +2.24264 q^{47} +1.00000 q^{48} -2.58579 q^{51} -5.65685 q^{52} -0.343146 q^{53} +1.00000 q^{54} -6.82843 q^{57} +2.58579 q^{58} -3.17157 q^{59} -11.6569 q^{61} -10.2426 q^{62} +1.00000 q^{64} +2.24264 q^{66} -9.07107 q^{67} -2.58579 q^{68} -3.17157 q^{69} -4.48528 q^{71} +1.00000 q^{72} +2.00000 q^{73} +0.242641 q^{74} -6.82843 q^{76} -5.65685 q^{78} +8.00000 q^{79} +1.00000 q^{81} +10.4853 q^{82} -5.17157 q^{83} -9.07107 q^{86} +2.58579 q^{87} +2.24264 q^{88} -0.828427 q^{89} -3.17157 q^{92} -10.2426 q^{93} +2.24264 q^{94} +1.00000 q^{96} +6.48528 q^{97} +2.24264 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{8} + 2 q^{9} - 4 q^{11} + 2 q^{12} + 2 q^{16} - 8 q^{17} + 2 q^{18} - 8 q^{19} - 4 q^{22} - 12 q^{23} + 2 q^{24} + 2 q^{27} + 8 q^{29} - 12 q^{31} + 2 q^{32} - 4 q^{33} - 8 q^{34} + 2 q^{36} - 8 q^{37} - 8 q^{38} + 4 q^{41} - 4 q^{43} - 4 q^{44} - 12 q^{46} - 4 q^{47} + 2 q^{48} - 8 q^{51} - 12 q^{53} + 2 q^{54} - 8 q^{57} + 8 q^{58} - 12 q^{59} - 12 q^{61} - 12 q^{62} + 2 q^{64} - 4 q^{66} - 4 q^{67} - 8 q^{68} - 12 q^{69} + 8 q^{71} + 2 q^{72} + 4 q^{73} - 8 q^{74} - 8 q^{76} + 16 q^{79} + 2 q^{81} + 4 q^{82} - 16 q^{83} - 4 q^{86} + 8 q^{87} - 4 q^{88} + 4 q^{89} - 12 q^{92} - 12 q^{93} - 4 q^{94} + 2 q^{96} - 4 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^8 + 2 * q^9 - 4 * q^11 + 2 * q^12 + 2 * q^16 - 8 * q^17 + 2 * q^18 - 8 * q^19 - 4 * q^22 - 12 * q^23 + 2 * q^24 + 2 * q^27 + 8 * q^29 - 12 * q^31 + 2 * q^32 - 4 * q^33 - 8 * q^34 + 2 * q^36 - 8 * q^37 - 8 * q^38 + 4 * q^41 - 4 * q^43 - 4 * q^44 - 12 * q^46 - 4 * q^47 + 2 * q^48 - 8 * q^51 - 12 * q^53 + 2 * q^54 - 8 * q^57 + 8 * q^58 - 12 * q^59 - 12 * q^61 - 12 * q^62 + 2 * q^64 - 4 * q^66 - 4 * q^67 - 8 * q^68 - 12 * q^69 + 8 * q^71 + 2 * q^72 + 4 * q^73 - 8 * q^74 - 8 * q^76 + 16 * q^79 + 2 * q^81 + 4 * q^82 - 16 * q^83 - 4 * q^86 + 8 * q^87 - 4 * q^88 + 4 * q^89 - 12 * q^92 - 12 * q^93 - 4 * q^94 + 2 * q^96 - 4 * q^97 - 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	2.24264	0.676182	0.338091	−	0.941113i	$-0.390219\pi$
$11$	2.24264	0.676182	0.338091	+	0.941113i	$0.390219\pi$
$12$	1.00000	0.288675
$13$	−5.65685	−1.56893	−0.784465	−	0.620174i	$-0.787062\pi$
$13$	−5.65685	−1.56893	−0.784465	+	0.620174i	$0.787062\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.58579	−0.627145	−0.313573	−	0.949564i	$-0.601526\pi$
$17$	−2.58579	−0.627145	−0.313573	+	0.949564i	$0.601526\pi$
$18$	1.00000	0.235702
$19$	−6.82843	−1.56655	−0.783274	−	0.621676i	$-0.786452\pi$
$19$	−6.82843	−1.56655	−0.783274	+	0.621676i	$0.786452\pi$
$20$	0	0
$21$	0	0
$22$	2.24264	0.478133
$23$	−3.17157	−0.661319	−0.330659	−	0.943750i	$-0.607271\pi$
$23$	−3.17157	−0.661319	−0.330659	+	0.943750i	$0.607271\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−5.65685	−1.10940
$27$	1.00000	0.192450
$28$	0	0
$29$	2.58579	0.480168	0.240084	−	0.970752i	$-0.422825\pi$
$29$	2.58579	0.480168	0.240084	+	0.970752i	$0.422825\pi$
$30$	0	0
$31$	−10.2426	−1.83963	−0.919816	−	0.392349i	$-0.871662\pi$
$31$	−10.2426	−1.83963	−0.919816	+	0.392349i	$0.871662\pi$
$32$	1.00000	0.176777
$33$	2.24264	0.390394
$34$	−2.58579	−0.443459
$35$	0	0
$36$	1.00000	0.166667
$37$	0.242641	0.0398899	0.0199449	−	0.999801i	$-0.493651\pi$
$37$	0.242641	0.0398899	0.0199449	+	0.999801i	$0.493651\pi$
$38$	−6.82843	−1.10772
$39$	−5.65685	−0.905822
$40$	0	0
$41$	10.4853	1.63753	0.818763	−	0.574132i	$-0.194660\pi$
$41$	10.4853	1.63753	0.818763	+	0.574132i	$0.194660\pi$
$42$	0	0
$43$	−9.07107	−1.38332	−0.691662	−	0.722221i	$-0.743121\pi$
$43$	−9.07107	−1.38332	−0.691662	+	0.722221i	$0.743121\pi$
$44$	2.24264	0.338091
$45$	0	0
$46$	−3.17157	−0.467623
$47$	2.24264	0.327123	0.163561	−	0.986533i	$-0.447702\pi$
$47$	2.24264	0.327123	0.163561	+	0.986533i	$0.447702\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	0	0
$51$	−2.58579	−0.362083
$52$	−5.65685	−0.784465
$53$	−0.343146	−0.0471347	−0.0235673	−	0.999722i	$-0.507502\pi$
$53$	−0.343146	−0.0471347	−0.0235673	+	0.999722i	$0.507502\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	0	0
$57$	−6.82843	−0.904447
$58$	2.58579	0.339530
$59$	−3.17157	−0.412904	−0.206452	−	0.978457i	$-0.566192\pi$
$59$	−3.17157	−0.412904	−0.206452	+	0.978457i	$0.566192\pi$
$60$	0	0
$61$	−11.6569	−1.49251	−0.746254	−	0.665662i	$-0.768149\pi$
$61$	−11.6569	−1.49251	−0.746254	+	0.665662i	$0.768149\pi$
$62$	−10.2426	−1.30082
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	2.24264	0.276050
$67$	−9.07107	−1.10821	−0.554104	−	0.832448i	$-0.686939\pi$
$67$	−9.07107	−1.10821	−0.554104	+	0.832448i	$0.686939\pi$
$68$	−2.58579	−0.313573
$69$	−3.17157	−0.381813
$70$	0	0
$71$	−4.48528	−0.532305	−0.266152	−	0.963931i	$-0.585752\pi$
$71$	−4.48528	−0.532305	−0.266152	+	0.963931i	$0.585752\pi$
$72$	1.00000	0.117851
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0.242641	0.0282064
$75$	0	0
$76$	−6.82843	−0.783274
$77$	0	0
$78$	−5.65685	−0.640513
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	10.4853	1.15791
$83$	−5.17157	−0.567654	−0.283827	−	0.958876i	$-0.591604\pi$
$83$	−5.17157	−0.567654	−0.283827	+	0.958876i	$0.591604\pi$
$84$	0	0
$85$	0	0
$86$	−9.07107	−0.978158
$87$	2.58579	0.277225
$88$	2.24264	0.239066
$89$	−0.828427	−0.0878131	−0.0439065	−	0.999036i	$-0.513980\pi$
$89$	−0.828427	−0.0878131	−0.0439065	+	0.999036i	$0.513980\pi$
$90$	0	0
$91$	0	0
$92$	−3.17157	−0.330659
$93$	−10.2426	−1.06211
$94$	2.24264	0.231311
$95$	0	0
$96$	1.00000	0.102062
$97$	6.48528	0.658481	0.329240	−	0.944246i	$-0.393207\pi$
$97$	6.48528	0.658481	0.329240	+	0.944246i	$0.393207\pi$
$98$	0	0
$99$	2.24264	0.225394
$100$	0	0
$101$	17.6569	1.75692	0.878461	−	0.477813i	$-0.158571\pi$
$101$	17.6569	1.75692	0.878461	+	0.477813i	$0.158571\pi$
$102$	−2.58579	−0.256031
$103$	−14.8284	−1.46109	−0.730544	−	0.682865i	$-0.760734\pi$
$103$	−14.8284	−1.46109	−0.730544	+	0.682865i	$0.760734\pi$
$104$	−5.65685	−0.554700
$105$	0	0
$106$	−0.343146	−0.0333293
$107$	−9.65685	−0.933563	−0.466782	−	0.884373i	$-0.654587\pi$
$107$	−9.65685	−0.933563	−0.466782	+	0.884373i	$0.654587\pi$
$108$	1.00000	0.0962250
$109$	−3.65685	−0.350263	−0.175132	−	0.984545i	$-0.556035\pi$
$109$	−3.65685	−0.350263	−0.175132	+	0.984545i	$0.556035\pi$
$110$	0	0
$111$	0.242641	0.0230304
$112$	0	0
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	−6.82843	−0.639541
$115$	0	0
$116$	2.58579	0.240084
$117$	−5.65685	−0.522976
$118$	−3.17157	−0.291967
$119$	0	0
$120$	0	0
$121$	−5.97056	−0.542778
$122$	−11.6569	−1.05536
$123$	10.4853	0.945426
$124$	−10.2426	−0.919816
$125$	0	0
$126$	0	0
$127$	−2.82843	−0.250982	−0.125491	−	0.992095i	$-0.540051\pi$
$127$	−2.82843	−0.250982	−0.125491	+	0.992095i	$0.540051\pi$
$128$	1.00000	0.0883883
$129$	−9.07107	−0.798663
$130$	0	0
$131$	−3.17157	−0.277102	−0.138551	−	0.990355i	$-0.544244\pi$
$131$	−3.17157	−0.277102	−0.138551	+	0.990355i	$0.544244\pi$
$132$	2.24264	0.195197
$133$	0	0
$134$	−9.07107	−0.783621
$135$	0	0
$136$	−2.58579	−0.221729
$137$	−13.6569	−1.16678	−0.583392	−	0.812191i	$-0.698275\pi$
$137$	−13.6569	−1.16678	−0.583392	+	0.812191i	$0.698275\pi$
$138$	−3.17157	−0.269982
$139$	−2.34315	−0.198743	−0.0993715	−	0.995050i	$-0.531683\pi$
$139$	−2.34315	−0.198743	−0.0993715	+	0.995050i	$0.531683\pi$
$140$	0	0
$141$	2.24264	0.188864
$142$	−4.48528	−0.376396
$143$	−12.6863	−1.06088
$144$	1.00000	0.0833333
$145$	0	0
$146$	2.00000	0.165521
$147$	0	0
$148$	0.242641	0.0199449
$149$	22.3848	1.83383	0.916916	−	0.399080i	$-0.130670\pi$
$149$	22.3848	1.83383	0.916916	+	0.399080i	$0.130670\pi$
$150$	0	0
$151$	20.1421	1.63914	0.819572	−	0.572976i	$-0.194211\pi$
$151$	20.1421	1.63914	0.819572	+	0.572976i	$0.194211\pi$
$152$	−6.82843	−0.553859
$153$	−2.58579	−0.209048
$154$	0	0
$155$	0	0
$156$	−5.65685	−0.452911
$157$	8.34315	0.665856	0.332928	−	0.942952i	$-0.391963\pi$
$157$	8.34315	0.665856	0.332928	+	0.942952i	$0.391963\pi$
$158$	8.00000	0.636446
$159$	−0.343146	−0.0272132
$160$	0	0
$161$	0	0
$162$	1.00000	0.0785674
$163$	17.0711	1.33711	0.668555	−	0.743663i	$-0.266913\pi$
$163$	17.0711	1.33711	0.668555	+	0.743663i	$0.266913\pi$
$164$	10.4853	0.818763
$165$	0	0
$166$	−5.17157	−0.401392
$167$	−2.24264	−0.173541	−0.0867704	−	0.996228i	$-0.527655\pi$
$167$	−2.24264	−0.173541	−0.0867704	+	0.996228i	$0.527655\pi$
$168$	0	0
$169$	19.0000	1.46154
$170$	0	0
$171$	−6.82843	−0.522183
$172$	−9.07107	−0.691662
$173$	−20.8284	−1.58356	−0.791778	−	0.610809i	$-0.790844\pi$
$173$	−20.8284	−1.58356	−0.791778	+	0.610809i	$0.790844\pi$
$174$	2.58579	0.196028
$175$	0	0
$176$	2.24264	0.169045
$177$	−3.17157	−0.238390
$178$	−0.828427	−0.0620932
$179$	26.2426	1.96147	0.980734	−	0.195350i	$-0.0625844\pi$
$179$	26.2426	1.96147	0.980734	+	0.195350i	$0.0625844\pi$
$180$	0	0
$181$	18.4853	1.37400	0.687000	−	0.726657i	$-0.258927\pi$
$181$	18.4853	1.37400	0.687000	+	0.726657i	$0.258927\pi$
$182$	0	0
$183$	−11.6569	−0.861699
$184$	−3.17157	−0.233811
$185$	0	0
$186$	−10.2426	−0.751027
$187$	−5.79899	−0.424064
$188$	2.24264	0.163561
$189$	0	0
$190$	0	0
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	1.00000	0.0721688
$193$	4.82843	0.347558	0.173779	−	0.984785i	$-0.444402\pi$
$193$	4.82843	0.347558	0.173779	+	0.984785i	$0.444402\pi$
$194$	6.48528	0.465616
$195$	0	0
$196$	0	0
$197$	25.7990	1.83810	0.919051	−	0.394139i	$-0.128957\pi$
$197$	25.7990	1.83810	0.919051	+	0.394139i	$0.128957\pi$
$198$	2.24264	0.159378
$199$	−5.75736	−0.408128	−0.204064	−	0.978958i	$-0.565415\pi$
$199$	−5.75736	−0.408128	−0.204064	+	0.978958i	$0.565415\pi$
$200$	0	0
$201$	−9.07107	−0.639824
$202$	17.6569	1.24233
$203$	0	0
$204$	−2.58579	−0.181041
$205$	0	0
$206$	−14.8284	−1.03315
$207$	−3.17157	−0.220440
$208$	−5.65685	−0.392232
$209$	−15.3137	−1.05927
$210$	0	0
$211$	−23.3137	−1.60498	−0.802491	−	0.596664i	$-0.796492\pi$
$211$	−23.3137	−1.60498	−0.802491	+	0.596664i	$0.796492\pi$
$212$	−0.343146	−0.0235673
$213$	−4.48528	−0.307326
$214$	−9.65685	−0.660129
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	−3.65685	−0.247673
$219$	2.00000	0.135147
$220$	0	0
$221$	14.6274	0.983947
$222$	0.242641	0.0162850
$223$	7.31371	0.489762	0.244881	−	0.969553i	$-0.421251\pi$
$223$	7.31371	0.489762	0.244881	+	0.969553i	$0.421251\pi$
$224$	0	0
$225$	0	0
$226$	−10.0000	−0.665190
$227$	24.9706	1.65735	0.828677	−	0.559727i	$-0.189094\pi$
$227$	24.9706	1.65735	0.828677	+	0.559727i	$0.189094\pi$
$228$	−6.82843	−0.452224
$229$	−16.1421	−1.06670	−0.533351	−	0.845894i	$-0.679068\pi$
$229$	−16.1421	−1.06670	−0.533351	+	0.845894i	$0.679068\pi$
$230$	0	0
$231$	0	0
$232$	2.58579	0.169765
$233$	−18.9706	−1.24280	−0.621401	−	0.783492i	$-0.713436\pi$
$233$	−18.9706	−1.24280	−0.621401	+	0.783492i	$0.713436\pi$
$234$	−5.65685	−0.369800
$235$	0	0
$236$	−3.17157	−0.206452
$237$	8.00000	0.519656
$238$	0	0
$239$	−19.3137	−1.24930	−0.624650	−	0.780905i	$-0.714758\pi$
$239$	−19.3137	−1.24930	−0.624650	+	0.780905i	$0.714758\pi$
$240$	0	0
$241$	−19.5563	−1.25974	−0.629868	−	0.776703i	$-0.716891\pi$
$241$	−19.5563	−1.25974	−0.629868	+	0.776703i	$0.716891\pi$
$242$	−5.97056	−0.383802
$243$	1.00000	0.0641500
$244$	−11.6569	−0.746254
$245$	0	0
$246$	10.4853	0.668517
$247$	38.6274	2.45780
$248$	−10.2426	−0.650408
$249$	−5.17157	−0.327735
$250$	0	0
$251$	−12.9706	−0.818695	−0.409347	−	0.912379i	$-0.634244\pi$
$251$	−12.9706	−0.818695	−0.409347	+	0.912379i	$0.634244\pi$
$252$	0	0
$253$	−7.11270	−0.447172
$254$	−2.82843	−0.177471
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.5858	1.15935	0.579675	−	0.814848i	$-0.303180\pi$
$257$	18.5858	1.15935	0.579675	+	0.814848i	$0.303180\pi$
$258$	−9.07107	−0.564740
$259$	0	0
$260$	0	0
$261$	2.58579	0.160056
$262$	−3.17157	−0.195940
$263$	12.1421	0.748716	0.374358	−	0.927284i	$-0.377863\pi$
$263$	12.1421	0.748716	0.374358	+	0.927284i	$0.377863\pi$
$264$	2.24264	0.138025
$265$	0	0
$266$	0	0
$267$	−0.828427	−0.0506989
$268$	−9.07107	−0.554104
$269$	12.3431	0.752575	0.376287	−	0.926503i	$-0.377201\pi$
$269$	12.3431	0.752575	0.376287	+	0.926503i	$0.377201\pi$
$270$	0	0
$271$	31.6985	1.92555	0.962773	−	0.270312i	$-0.0871267\pi$
$271$	31.6985	1.92555	0.962773	+	0.270312i	$0.0871267\pi$
$272$	−2.58579	−0.156786
$273$	0	0
$274$	−13.6569	−0.825041
$275$	0	0
$276$	−3.17157	−0.190906
$277$	15.5563	0.934690	0.467345	−	0.884075i	$-0.345211\pi$
$277$	15.5563	0.934690	0.467345	+	0.884075i	$0.345211\pi$
$278$	−2.34315	−0.140533
$279$	−10.2426	−0.613211
$280$	0	0
$281$	−23.4558	−1.39926	−0.699629	−	0.714506i	$-0.746651\pi$
$281$	−23.4558	−1.39926	−0.699629	+	0.714506i	$0.746651\pi$
$282$	2.24264	0.133547
$283$	−9.51472	−0.565591	−0.282796	−	0.959180i	$-0.591262\pi$
$283$	−9.51472	−0.565591	−0.282796	+	0.959180i	$0.591262\pi$
$284$	−4.48528	−0.266152
$285$	0	0
$286$	−12.6863	−0.750156
$287$	0	0
$288$	1.00000	0.0589256
$289$	−10.3137	−0.606689
$290$	0	0
$291$	6.48528	0.380174
$292$	2.00000	0.117041
$293$	−21.3137	−1.24516	−0.622580	−	0.782556i	$-0.713916\pi$
$293$	−21.3137	−1.24516	−0.622580	+	0.782556i	$0.713916\pi$
$294$	0	0
$295$	0	0
$296$	0.242641	0.0141032
$297$	2.24264	0.130131
$298$	22.3848	1.29672
$299$	17.9411	1.03756
$300$	0	0
$301$	0	0
$302$	20.1421	1.15905
$303$	17.6569	1.01436
$304$	−6.82843	−0.391637
$305$	0	0
$306$	−2.58579	−0.147820
$307$	7.31371	0.417415	0.208708	−	0.977978i	$-0.433074\pi$
$307$	7.31371	0.417415	0.208708	+	0.977978i	$0.433074\pi$
$308$	0	0
$309$	−14.8284	−0.843560
$310$	0	0
$311$	−10.8284	−0.614024	−0.307012	−	0.951706i	$-0.599329\pi$
$311$	−10.8284	−0.614024	−0.307012	+	0.951706i	$0.599329\pi$
$312$	−5.65685	−0.320256
$313$	−30.4853	−1.72313	−0.861565	−	0.507647i	$-0.830515\pi$
$313$	−30.4853	−1.72313	−0.861565	+	0.507647i	$0.830515\pi$
$314$	8.34315	0.470831
$315$	0	0
$316$	8.00000	0.450035
$317$	25.3137	1.42176	0.710880	−	0.703314i	$-0.248297\pi$
$317$	25.3137	1.42176	0.710880	+	0.703314i	$0.248297\pi$
$318$	−0.343146	−0.0192427
$319$	5.79899	0.324681
$320$	0	0
$321$	−9.65685	−0.538993
$322$	0	0
$323$	17.6569	0.982454
$324$	1.00000	0.0555556
$325$	0	0
$326$	17.0711	0.945479
$327$	−3.65685	−0.202225
$328$	10.4853	0.578953
$329$	0	0
$330$	0	0
$331$	−6.34315	−0.348651	−0.174325	−	0.984688i	$-0.555774\pi$
$331$	−6.34315	−0.348651	−0.174325	+	0.984688i	$0.555774\pi$
$332$	−5.17157	−0.283827
$333$	0.242641	0.0132966
$334$	−2.24264	−0.122712
$335$	0	0
$336$	0	0
$337$	−33.1127	−1.80376	−0.901882	−	0.431983i	$-0.857814\pi$
$337$	−33.1127	−1.80376	−0.901882	+	0.431983i	$0.857814\pi$
$338$	19.0000	1.03346
$339$	−10.0000	−0.543125
$340$	0	0
$341$	−22.9706	−1.24393
$342$	−6.82843	−0.369239
$343$	0	0
$344$	−9.07107	−0.489079
$345$	0	0
$346$	−20.8284	−1.11974
$347$	22.3431	1.19944	0.599721	−	0.800209i	$-0.295278\pi$
$347$	22.3431	1.19944	0.599721	+	0.800209i	$0.295278\pi$
$348$	2.58579	0.138613
$349$	23.4558	1.25556	0.627781	−	0.778390i	$-0.283963\pi$
$349$	23.4558	1.25556	0.627781	+	0.778390i	$0.283963\pi$
$350$	0	0
$351$	−5.65685	−0.301941
$352$	2.24264	0.119533
$353$	−16.0416	−0.853810	−0.426905	−	0.904297i	$-0.640396\pi$
$353$	−16.0416	−0.853810	−0.426905	+	0.904297i	$0.640396\pi$
$354$	−3.17157	−0.168567
$355$	0	0
$356$	−0.828427	−0.0439065
$357$	0	0
$358$	26.2426	1.38697
$359$	−28.4853	−1.50340	−0.751698	−	0.659508i	$-0.770765\pi$
$359$	−28.4853	−1.50340	−0.751698	+	0.659508i	$0.770765\pi$
$360$	0	0
$361$	27.6274	1.45407
$362$	18.4853	0.971565
$363$	−5.97056	−0.313373
$364$	0	0
$365$	0	0
$366$	−11.6569	−0.609314
$367$	10.8284	0.565239	0.282620	−	0.959232i	$-0.408797\pi$
$367$	10.8284	0.565239	0.282620	+	0.959232i	$0.408797\pi$
$368$	−3.17157	−0.165330
$369$	10.4853	0.545842
$370$	0	0
$371$	0	0
$372$	−10.2426	−0.531056
$373$	−6.58579	−0.340999	−0.170500	−	0.985358i	$-0.554538\pi$
$373$	−6.58579	−0.340999	−0.170500	+	0.985358i	$0.554538\pi$
$374$	−5.79899	−0.299859
$375$	0	0
$376$	2.24264	0.115655
$377$	−14.6274	−0.753350
$378$	0	0
$379$	0.485281	0.0249272	0.0124636	−	0.999922i	$-0.496033\pi$
$379$	0.485281	0.0249272	0.0124636	+	0.999922i	$0.496033\pi$
$380$	0	0
$381$	−2.82843	−0.144905
$382$	4.00000	0.204658
$383$	−19.4142	−0.992020	−0.496010	−	0.868317i	$-0.665202\pi$
$383$	−19.4142	−0.992020	−0.496010	+	0.868317i	$0.665202\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	4.82843	0.245760
$387$	−9.07107	−0.461108
$388$	6.48528	0.329240
$389$	−29.8995	−1.51596	−0.757982	−	0.652275i	$-0.773815\pi$
$389$	−29.8995	−1.51596	−0.757982	+	0.652275i	$0.773815\pi$
$390$	0	0
$391$	8.20101	0.414743
$392$	0	0
$393$	−3.17157	−0.159985
$394$	25.7990	1.29973
$395$	0	0
$396$	2.24264	0.112697
$397$	−6.68629	−0.335575	−0.167788	−	0.985823i	$-0.553662\pi$
$397$	−6.68629	−0.335575	−0.167788	+	0.985823i	$0.553662\pi$
$398$	−5.75736	−0.288590
$399$	0	0
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	−9.07107	−0.452424
$403$	57.9411	2.88625
$404$	17.6569	0.878461
$405$	0	0
$406$	0	0
$407$	0.544156	0.0269728
$408$	−2.58579	−0.128016
$409$	4.24264	0.209785	0.104893	−	0.994484i	$-0.466550\pi$
$409$	4.24264	0.209785	0.104893	+	0.994484i	$0.466550\pi$
$410$	0	0
$411$	−13.6569	−0.673643
$412$	−14.8284	−0.730544
$413$	0	0
$414$	−3.17157	−0.155874
$415$	0	0
$416$	−5.65685	−0.277350
$417$	−2.34315	−0.114744
$418$	−15.3137	−0.749018
$419$	−36.9706	−1.80613	−0.903065	−	0.429504i	$-0.858689\pi$
$419$	−36.9706	−1.80613	−0.903065	+	0.429504i	$0.858689\pi$
$420$	0	0
$421$	−3.65685	−0.178224	−0.0891121	−	0.996022i	$-0.528403\pi$
$421$	−3.65685	−0.178224	−0.0891121	+	0.996022i	$0.528403\pi$
$422$	−23.3137	−1.13489
$423$	2.24264	0.109041
$424$	−0.343146	−0.0166646
$425$	0	0
$426$	−4.48528	−0.217313
$427$	0	0
$428$	−9.65685	−0.466782
$429$	−12.6863	−0.612500
$430$	0	0
$431$	13.4558	0.648145	0.324073	−	0.946032i	$-0.394948\pi$
$431$	13.4558	0.648145	0.324073	+	0.946032i	$0.394948\pi$
$432$	1.00000	0.0481125
$433$	17.7990	0.855365	0.427682	−	0.903929i	$-0.359330\pi$
$433$	17.7990	0.855365	0.427682	+	0.903929i	$0.359330\pi$
$434$	0	0
$435$	0	0
$436$	−3.65685	−0.175132
$437$	21.6569	1.03599
$438$	2.00000	0.0955637
$439$	−23.6985	−1.13107	−0.565533	−	0.824725i	$-0.691330\pi$
$439$	−23.6985	−1.13107	−0.565533	+	0.824725i	$0.691330\pi$
$440$	0	0
$441$	0	0
$442$	14.6274	0.695755
$443$	−36.2843	−1.72392	−0.861959	−	0.506978i	$-0.830762\pi$
$443$	−36.2843	−1.72392	−0.861959	+	0.506978i	$0.830762\pi$
$444$	0.242641	0.0115152
$445$	0	0
$446$	7.31371	0.346314
$447$	22.3848	1.05876
$448$	0	0
$449$	−10.4853	−0.494831	−0.247416	−	0.968909i	$-0.579581\pi$
$449$	−10.4853	−0.494831	−0.247416	+	0.968909i	$0.579581\pi$
$450$	0	0
$451$	23.5147	1.10726
$452$	−10.0000	−0.470360
$453$	20.1421	0.946360
$454$	24.9706	1.17193
$455$	0	0
$456$	−6.82843	−0.319770
$457$	−2.48528	−0.116257	−0.0581283	−	0.998309i	$-0.518513\pi$
$457$	−2.48528	−0.116257	−0.0581283	+	0.998309i	$0.518513\pi$
$458$	−16.1421	−0.754272
$459$	−2.58579	−0.120694
$460$	0	0
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	0	0
$463$	25.6569	1.19238	0.596188	−	0.802845i	$-0.296681\pi$
$463$	25.6569	1.19238	0.596188	+	0.802845i	$0.296681\pi$
$464$	2.58579	0.120042
$465$	0	0
$466$	−18.9706	−0.878794
$467$	−12.4853	−0.577750	−0.288875	−	0.957367i	$-0.593281\pi$
$467$	−12.4853	−0.577750	−0.288875	+	0.957367i	$0.593281\pi$
$468$	−5.65685	−0.261488
$469$	0	0
$470$	0	0
$471$	8.34315	0.384432
$472$	−3.17157	−0.145983
$473$	−20.3431	−0.935379
$474$	8.00000	0.367452
$475$	0	0
$476$	0	0
$477$	−0.343146	−0.0157116
$478$	−19.3137	−0.883388
$479$	−8.48528	−0.387702	−0.193851	−	0.981031i	$-0.562098\pi$
$479$	−8.48528	−0.387702	−0.193851	+	0.981031i	$0.562098\pi$
$480$	0	0
$481$	−1.37258	−0.0625844
$482$	−19.5563	−0.890767
$483$	0	0
$484$	−5.97056	−0.271389
$485$	0	0
$486$	1.00000	0.0453609
$487$	−10.8284	−0.490683	−0.245341	−	0.969437i	$-0.578900\pi$
$487$	−10.8284	−0.490683	−0.245341	+	0.969437i	$0.578900\pi$
$488$	−11.6569	−0.527681
$489$	17.0711	0.771980
$490$	0	0
$491$	−17.0711	−0.770407	−0.385203	−	0.922832i	$-0.625869\pi$
$491$	−17.0711	−0.770407	−0.385203	+	0.922832i	$0.625869\pi$
$492$	10.4853	0.472713
$493$	−6.68629	−0.301135
$494$	38.6274	1.73793
$495$	0	0
$496$	−10.2426	−0.459908
$497$	0	0
$498$	−5.17157	−0.231744
$499$	6.82843	0.305682	0.152841	−	0.988251i	$-0.451158\pi$
$499$	6.82843	0.305682	0.152841	+	0.988251i	$0.451158\pi$
$500$	0	0
$501$	−2.24264	−0.100194
$502$	−12.9706	−0.578905
$503$	4.58579	0.204470	0.102235	−	0.994760i	$-0.467401\pi$
$503$	4.58579	0.204470	0.102235	+	0.994760i	$0.467401\pi$
$504$	0	0
$505$	0	0
$506$	−7.11270	−0.316198
$507$	19.0000	0.843820
$508$	−2.82843	−0.125491
$509$	−16.9706	−0.752207	−0.376103	−	0.926578i	$-0.622736\pi$
$509$	−16.9706	−0.752207	−0.376103	+	0.926578i	$0.622736\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	−6.82843	−0.301482
$514$	18.5858	0.819784
$515$	0	0
$516$	−9.07107	−0.399331
$517$	5.02944	0.221194
$518$	0	0
$519$	−20.8284	−0.914266
$520$	0	0
$521$	38.7696	1.69852	0.849262	−	0.527971i	$-0.177047\pi$
$521$	38.7696	1.69852	0.849262	+	0.527971i	$0.177047\pi$
$522$	2.58579	0.113177
$523$	−41.7990	−1.82774	−0.913871	−	0.406004i	$-0.866922\pi$
$523$	−41.7990	−1.82774	−0.913871	+	0.406004i	$0.866922\pi$
$524$	−3.17157	−0.138551
$525$	0	0
$526$	12.1421	0.529422
$527$	26.4853	1.15372
$528$	2.24264	0.0975984
$529$	−12.9411	−0.562658
$530$	0	0
$531$	−3.17157	−0.137635
$532$	0	0
$533$	−59.3137	−2.56916
$534$	−0.828427	−0.0358495
$535$	0	0
$536$	−9.07107	−0.391810
$537$	26.2426	1.13245
$538$	12.3431	0.532151
$539$	0	0
$540$	0	0
$541$	−29.7990	−1.28116	−0.640579	−	0.767892i	$-0.721306\pi$
$541$	−29.7990	−1.28116	−0.640579	+	0.767892i	$0.721306\pi$
$542$	31.6985	1.36157
$543$	18.4853	0.793279
$544$	−2.58579	−0.110865
$545$	0	0
$546$	0	0
$547$	28.3848	1.21365	0.606823	−	0.794837i	$-0.292444\pi$
$547$	28.3848	1.21365	0.606823	+	0.794837i	$0.292444\pi$
$548$	−13.6569	−0.583392
$549$	−11.6569	−0.497502
$550$	0	0
$551$	−17.6569	−0.752207
$552$	−3.17157	−0.134991
$553$	0	0
$554$	15.5563	0.660926
$555$	0	0
$556$	−2.34315	−0.0993715
$557$	38.0000	1.61011	0.805056	−	0.593199i	$-0.202135\pi$
$557$	38.0000	1.61011	0.805056	+	0.593199i	$0.202135\pi$
$558$	−10.2426	−0.433606
$559$	51.3137	2.17034
$560$	0	0
$561$	−5.79899	−0.244834
$562$	−23.4558	−0.989425
$563$	20.4853	0.863352	0.431676	−	0.902029i	$-0.357922\pi$
$563$	20.4853	0.863352	0.431676	+	0.902029i	$0.357922\pi$
$564$	2.24264	0.0944322
$565$	0	0
$566$	−9.51472	−0.399933
$567$	0	0
$568$	−4.48528	−0.188198
$569$	9.02944	0.378534	0.189267	−	0.981926i	$-0.439389\pi$
$569$	9.02944	0.378534	0.189267	+	0.981926i	$0.439389\pi$
$570$	0	0
$571$	4.68629	0.196115	0.0980576	−	0.995181i	$-0.468737\pi$
$571$	4.68629	0.196115	0.0980576	+	0.995181i	$0.468737\pi$
$572$	−12.6863	−0.530440
$573$	4.00000	0.167102
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	3.85786	0.160605	0.0803025	−	0.996771i	$-0.474411\pi$
$577$	3.85786	0.160605	0.0803025	+	0.996771i	$0.474411\pi$
$578$	−10.3137	−0.428994
$579$	4.82843	0.200663
$580$	0	0
$581$	0	0
$582$	6.48528	0.268824
$583$	−0.769553	−0.0318716
$584$	2.00000	0.0827606
$585$	0	0
$586$	−21.3137	−0.880461
$587$	5.17157	0.213454	0.106727	−	0.994288i	$-0.465963\pi$
$587$	5.17157	0.213454	0.106727	+	0.994288i	$0.465963\pi$
$588$	0	0
$589$	69.9411	2.88187
$590$	0	0
$591$	25.7990	1.06123
$592$	0.242641	0.00997247
$593$	41.2132	1.69242	0.846212	−	0.532847i	$-0.178878\pi$
$593$	41.2132	1.69242	0.846212	+	0.532847i	$0.178878\pi$
$594$	2.24264	0.0920167
$595$	0	0
$596$	22.3848	0.916916
$597$	−5.75736	−0.235633
$598$	17.9411	0.733667
$599$	4.00000	0.163436	0.0817178	−	0.996656i	$-0.473959\pi$
$599$	4.00000	0.163436	0.0817178	+	0.996656i	$0.473959\pi$
$600$	0	0
$601$	−31.7574	−1.29541	−0.647705	−	0.761891i	$-0.724271\pi$
$601$	−31.7574	−1.29541	−0.647705	+	0.761891i	$0.724271\pi$
$602$	0	0
$603$	−9.07107	−0.369402
$604$	20.1421	0.819572
$605$	0	0
$606$	17.6569	0.717261
$607$	−4.00000	−0.162355	−0.0811775	−	0.996700i	$-0.525868\pi$
$607$	−4.00000	−0.162355	−0.0811775	+	0.996700i	$0.525868\pi$
$608$	−6.82843	−0.276929
$609$	0	0
$610$	0	0
$611$	−12.6863	−0.513232
$612$	−2.58579	−0.104524
$613$	11.0711	0.447156	0.223578	−	0.974686i	$-0.428226\pi$
$613$	11.0711	0.447156	0.223578	+	0.974686i	$0.428226\pi$
$614$	7.31371	0.295157
$615$	0	0
$616$	0	0
$617$	−33.9411	−1.36642	−0.683209	−	0.730223i	$-0.739416\pi$
$617$	−33.9411	−1.36642	−0.683209	+	0.730223i	$0.739416\pi$
$618$	−14.8284	−0.596487
$619$	19.1127	0.768204	0.384102	−	0.923291i	$-0.374511\pi$
$619$	19.1127	0.768204	0.384102	+	0.923291i	$0.374511\pi$
$620$	0	0
$621$	−3.17157	−0.127271
$622$	−10.8284	−0.434180
$623$	0	0
$624$	−5.65685	−0.226455
$625$	0	0
$626$	−30.4853	−1.21844
$627$	−15.3137	−0.611571
$628$	8.34315	0.332928
$629$	−0.627417	−0.0250168
$630$	0	0
$631$	−8.14214	−0.324133	−0.162067	−	0.986780i	$-0.551816\pi$
$631$	−8.14214	−0.324133	−0.162067	+	0.986780i	$0.551816\pi$
$632$	8.00000	0.318223
$633$	−23.3137	−0.926637
$634$	25.3137	1.00534
$635$	0	0
$636$	−0.343146	−0.0136066
$637$	0	0
$638$	5.79899	0.229584
$639$	−4.48528	−0.177435
$640$	0	0
$641$	26.0000	1.02694	0.513469	−	0.858108i	$-0.328360\pi$
$641$	26.0000	1.02694	0.513469	+	0.858108i	$0.328360\pi$
$642$	−9.65685	−0.381126
$643$	−28.8284	−1.13688	−0.568441	−	0.822724i	$-0.692453\pi$
$643$	−28.8284	−1.13688	−0.568441	+	0.822724i	$0.692453\pi$
$644$	0	0
$645$	0	0
$646$	17.6569	0.694700
$647$	−8.87006	−0.348718	−0.174359	−	0.984682i	$-0.555785\pi$
$647$	−8.87006	−0.348718	−0.174359	+	0.984682i	$0.555785\pi$
$648$	1.00000	0.0392837
$649$	−7.11270	−0.279198
$650$	0	0
$651$	0	0
$652$	17.0711	0.668555
$653$	1.79899	0.0703999	0.0352000	−	0.999380i	$-0.488793\pi$
$653$	1.79899	0.0703999	0.0352000	+	0.999380i	$0.488793\pi$
$654$	−3.65685	−0.142994
$655$	0	0
$656$	10.4853	0.409381
$657$	2.00000	0.0780274
$658$	0	0
$659$	20.3848	0.794078	0.397039	−	0.917802i	$-0.370038\pi$
$659$	20.3848	0.794078	0.397039	+	0.917802i	$0.370038\pi$
$660$	0	0
$661$	−5.02944	−0.195622	−0.0978112	−	0.995205i	$-0.531184\pi$
$661$	−5.02944	−0.195622	−0.0978112	+	0.995205i	$0.531184\pi$
$662$	−6.34315	−0.246533
$663$	14.6274	0.568082
$664$	−5.17157	−0.200696
$665$	0	0
$666$	0.242641	0.00940214
$667$	−8.20101	−0.317544
$668$	−2.24264	−0.0867704
$669$	7.31371	0.282764
$670$	0	0
$671$	−26.1421	−1.00921
$672$	0	0
$673$	26.2843	1.01318	0.506592	−	0.862186i	$-0.330905\pi$
$673$	26.2843	1.01318	0.506592	+	0.862186i	$0.330905\pi$
$674$	−33.1127	−1.27545
$675$	0	0
$676$	19.0000	0.730769
$677$	44.4264	1.70745	0.853723	−	0.520728i	$-0.174339\pi$
$677$	44.4264	1.70745	0.853723	+	0.520728i	$0.174339\pi$
$678$	−10.0000	−0.384048
$679$	0	0
$680$	0	0
$681$	24.9706	0.956874
$682$	−22.9706	−0.879588
$683$	31.7990	1.21675	0.608377	−	0.793648i	$-0.291821\pi$
$683$	31.7990	1.21675	0.608377	+	0.793648i	$0.291821\pi$
$684$	−6.82843	−0.261091
$685$	0	0
$686$	0	0
$687$	−16.1421	−0.615861
$688$	−9.07107	−0.345831
$689$	1.94113	0.0739510
$690$	0	0
$691$	3.02944	0.115245	0.0576226	−	0.998338i	$-0.481648\pi$
$691$	3.02944	0.115245	0.0576226	+	0.998338i	$0.481648\pi$
$692$	−20.8284	−0.791778
$693$	0	0
$694$	22.3431	0.848134
$695$	0	0
$696$	2.58579	0.0980140
$697$	−27.1127	−1.02697
$698$	23.4558	0.887817
$699$	−18.9706	−0.717533
$700$	0	0
$701$	−17.2132	−0.650134	−0.325067	−	0.945691i	$-0.605387\pi$
$701$	−17.2132	−0.650134	−0.325067	+	0.945691i	$0.605387\pi$
$702$	−5.65685	−0.213504
$703$	−1.65685	−0.0624894
$704$	2.24264	0.0845227
$705$	0	0
$706$	−16.0416	−0.603735
$707$	0	0
$708$	−3.17157	−0.119195
$709$	11.8579	0.445331	0.222666	−	0.974895i	$-0.428524\pi$
$709$	11.8579	0.445331	0.222666	+	0.974895i	$0.428524\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	−0.828427	−0.0310466
$713$	32.4853	1.21658
$714$	0	0
$715$	0	0
$716$	26.2426	0.980734
$717$	−19.3137	−0.721284
$718$	−28.4853	−1.06306
$719$	17.9411	0.669091	0.334546	−	0.942380i	$-0.391417\pi$
$719$	17.9411	0.669091	0.334546	+	0.942380i	$0.391417\pi$
$720$	0	0
$721$	0	0
$722$	27.6274	1.02819
$723$	−19.5563	−0.727308
$724$	18.4853	0.687000
$725$	0	0
$726$	−5.97056	−0.221588
$727$	30.6274	1.13591	0.567954	−	0.823060i	$-0.307735\pi$
$727$	30.6274	1.13591	0.567954	+	0.823060i	$0.307735\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	23.4558	0.867546
$732$	−11.6569	−0.430850
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	10.8284	0.399685
$735$	0	0
$736$	−3.17157	−0.116906
$737$	−20.3431	−0.749349
$738$	10.4853	0.385969
$739$	11.1127	0.408787	0.204394	−	0.978889i	$-0.434478\pi$
$739$	11.1127	0.408787	0.204394	+	0.978889i	$0.434478\pi$
$740$	0	0
$741$	38.6274	1.41901
$742$	0	0
$743$	−28.2843	−1.03765	−0.518825	−	0.854881i	$-0.673630\pi$
$743$	−28.2843	−1.03765	−0.518825	+	0.854881i	$0.673630\pi$
$744$	−10.2426	−0.375513
$745$	0	0
$746$	−6.58579	−0.241123
$747$	−5.17157	−0.189218
$748$	−5.79899	−0.212032
$749$	0	0
$750$	0	0
$751$	21.7990	0.795456	0.397728	−	0.917503i	$-0.369799\pi$
$751$	21.7990	0.795456	0.397728	+	0.917503i	$0.369799\pi$
$752$	2.24264	0.0817807
$753$	−12.9706	−0.472674
$754$	−14.6274	−0.532699
$755$	0	0
$756$	0	0
$757$	−25.8995	−0.941333	−0.470667	−	0.882311i	$-0.655987\pi$
$757$	−25.8995	−0.941333	−0.470667	+	0.882311i	$0.655987\pi$
$758$	0.485281	0.0176262
$759$	−7.11270	−0.258175
$760$	0	0
$761$	−26.9706	−0.977682	−0.488841	−	0.872373i	$-0.662580\pi$
$761$	−26.9706	−0.977682	−0.488841	+	0.872373i	$0.662580\pi$
$762$	−2.82843	−0.102463
$763$	0	0
$764$	4.00000	0.144715
$765$	0	0
$766$	−19.4142	−0.701464
$767$	17.9411	0.647816
$768$	1.00000	0.0360844
$769$	4.72792	0.170493	0.0852466	−	0.996360i	$-0.472832\pi$
$769$	4.72792	0.170493	0.0852466	+	0.996360i	$0.472832\pi$
$770$	0	0
$771$	18.5858	0.669351
$772$	4.82843	0.173779
$773$	22.9706	0.826194	0.413097	−	0.910687i	$-0.364447\pi$
$773$	22.9706	0.826194	0.413097	+	0.910687i	$0.364447\pi$
$774$	−9.07107	−0.326053
$775$	0	0
$776$	6.48528	0.232808
$777$	0	0
$778$	−29.8995	−1.07195
$779$	−71.5980	−2.56526
$780$	0	0
$781$	−10.0589	−0.359935
$782$	8.20101	0.293268
$783$	2.58579	0.0924085
$784$	0	0
$785$	0	0
$786$	−3.17157	−0.113126
$787$	−38.3431	−1.36679	−0.683393	−	0.730051i	$-0.739496\pi$
$787$	−38.3431	−1.36679	−0.683393	+	0.730051i	$0.739496\pi$
$788$	25.7990	0.919051
$789$	12.1421	0.432271
$790$	0	0
$791$	0	0
$792$	2.24264	0.0796888
$793$	65.9411	2.34164
$794$	−6.68629	−0.237288
$795$	0	0
$796$	−5.75736	−0.204064
$797$	−0.142136	−0.00503470	−0.00251735	−	0.999997i	$-0.500801\pi$
$797$	−0.142136	−0.00503470	−0.00251735	+	0.999997i	$0.500801\pi$
$798$	0	0
$799$	−5.79899	−0.205154
$800$	0	0
$801$	−0.828427	−0.0292710
$802$	−6.00000	−0.211867
$803$	4.48528	0.158282
$804$	−9.07107	−0.319912
$805$	0	0
$806$	57.9411	2.04089
$807$	12.3431	0.434499
$808$	17.6569	0.621166
$809$	34.4853	1.21244	0.606219	−	0.795298i	$-0.292686\pi$
$809$	34.4853	1.21244	0.606219	+	0.795298i	$0.292686\pi$
$810$	0	0
$811$	10.3431	0.363197	0.181598	−	0.983373i	$-0.441873\pi$
$811$	10.3431	0.363197	0.181598	+	0.983373i	$0.441873\pi$
$812$	0	0
$813$	31.6985	1.11171
$814$	0.544156	0.0190727
$815$	0	0
$816$	−2.58579	−0.0905206
$817$	61.9411	2.16705
$818$	4.24264	0.148340
$819$	0	0
$820$	0	0
$821$	1.21320	0.0423411	0.0211705	−	0.999776i	$-0.493261\pi$
$821$	1.21320	0.0423411	0.0211705	+	0.999776i	$0.493261\pi$
$822$	−13.6569	−0.476337
$823$	−28.9706	−1.00985	−0.504925	−	0.863163i	$-0.668480\pi$
$823$	−28.9706	−1.00985	−0.504925	+	0.863163i	$0.668480\pi$
$824$	−14.8284	−0.516573
$825$	0	0
$826$	0	0
$827$	−35.5147	−1.23497	−0.617484	−	0.786584i	$-0.711848\pi$
$827$	−35.5147	−1.23497	−0.617484	+	0.786584i	$0.711848\pi$
$828$	−3.17157	−0.110220
$829$	33.3137	1.15703	0.578516	−	0.815671i	$-0.303632\pi$
$829$	33.3137	1.15703	0.578516	+	0.815671i	$0.303632\pi$
$830$	0	0
$831$	15.5563	0.539644
$832$	−5.65685	−0.196116
$833$	0	0
$834$	−2.34315	−0.0811365
$835$	0	0
$836$	−15.3137	−0.529636
$837$	−10.2426	−0.354037
$838$	−36.9706	−1.27713
$839$	8.48528	0.292944	0.146472	−	0.989215i	$-0.453208\pi$
$839$	8.48528	0.292944	0.146472	+	0.989215i	$0.453208\pi$
$840$	0	0
$841$	−22.3137	−0.769438
$842$	−3.65685	−0.126024
$843$	−23.4558	−0.807862
$844$	−23.3137	−0.802491
$845$	0	0
$846$	2.24264	0.0771036
$847$	0	0
$848$	−0.343146	−0.0117837
$849$	−9.51472	−0.326544
$850$	0	0
$851$	−0.769553	−0.0263799
$852$	−4.48528	−0.153663
$853$	40.6274	1.39106	0.695528	−	0.718499i	$-0.255170\pi$
$853$	40.6274	1.39106	0.695528	+	0.718499i	$0.255170\pi$
$854$	0	0
$855$	0	0
$856$	−9.65685	−0.330064
$857$	−24.7279	−0.844690	−0.422345	−	0.906435i	$-0.638793\pi$
$857$	−24.7279	−0.844690	−0.422345	+	0.906435i	$0.638793\pi$
$858$	−12.6863	−0.433103
$859$	−35.5980	−1.21459	−0.607294	−	0.794477i	$-0.707745\pi$
$859$	−35.5980	−1.21459	−0.607294	+	0.794477i	$0.707745\pi$
$860$	0	0
$861$	0	0
$862$	13.4558	0.458308
$863$	−45.6569	−1.55418	−0.777089	−	0.629391i	$-0.783304\pi$
$863$	−45.6569	−1.55418	−0.777089	+	0.629391i	$0.783304\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	17.7990	0.604834
$867$	−10.3137	−0.350272
$868$	0	0
$869$	17.9411	0.608611
$870$	0	0
$871$	51.3137	1.73870
$872$	−3.65685	−0.123837
$873$	6.48528	0.219494
$874$	21.6569	0.732554
$875$	0	0
$876$	2.00000	0.0675737
$877$	−4.24264	−0.143264	−0.0716319	−	0.997431i	$-0.522821\pi$
$877$	−4.24264	−0.143264	−0.0716319	+	0.997431i	$0.522821\pi$
$878$	−23.6985	−0.799785
$879$	−21.3137	−0.718894
$880$	0	0
$881$	26.0000	0.875962	0.437981	−	0.898984i	$-0.355694\pi$
$881$	26.0000	0.875962	0.437981	+	0.898984i	$0.355694\pi$
$882$	0	0
$883$	34.2426	1.15236	0.576178	−	0.817324i	$-0.304543\pi$
$883$	34.2426	1.15236	0.576178	+	0.817324i	$0.304543\pi$
$884$	14.6274	0.491973
$885$	0	0
$886$	−36.2843	−1.21899
$887$	32.8701	1.10367	0.551834	−	0.833954i	$-0.313928\pi$
$887$	32.8701	1.10367	0.551834	+	0.833954i	$0.313928\pi$
$888$	0.242641	0.00814249
$889$	0	0
$890$	0	0
$891$	2.24264	0.0751313
$892$	7.31371	0.244881
$893$	−15.3137	−0.512454
$894$	22.3848	0.748659
$895$	0	0
$896$	0	0
$897$	17.9411	0.599037
$898$	−10.4853	−0.349898
$899$	−26.4853	−0.883334
$900$	0	0
$901$	0.887302	0.0295603
$902$	23.5147	0.782954
$903$	0	0
$904$	−10.0000	−0.332595
$905$	0	0
$906$	20.1421	0.669178
$907$	16.8701	0.560161	0.280081	−	0.959977i	$-0.409639\pi$
$907$	16.8701	0.560161	0.280081	+	0.959977i	$0.409639\pi$
$908$	24.9706	0.828677
$909$	17.6569	0.585641
$910$	0	0
$911$	44.9706	1.48994	0.744971	−	0.667097i	$-0.232463\pi$
$911$	44.9706	1.48994	0.744971	+	0.667097i	$0.232463\pi$
$912$	−6.82843	−0.226112
$913$	−11.5980	−0.383837
$914$	−2.48528	−0.0822058
$915$	0	0
$916$	−16.1421	−0.533351
$917$	0	0
$918$	−2.58579	−0.0853437
$919$	−30.7696	−1.01499	−0.507497	−	0.861654i	$-0.669429\pi$
$919$	−30.7696	−1.01499	−0.507497	+	0.861654i	$0.669429\pi$
$920$	0	0
$921$	7.31371	0.240995
$922$	2.00000	0.0658665
$923$	25.3726	0.835149
$924$	0	0
$925$	0	0
$926$	25.6569	0.843137
$927$	−14.8284	−0.487029
$928$	2.58579	0.0848826
$929$	55.6569	1.82604	0.913021	−	0.407912i	$-0.133743\pi$
$929$	55.6569	1.82604	0.913021	+	0.407912i	$0.133743\pi$
$930$	0	0
$931$	0	0
$932$	−18.9706	−0.621401
$933$	−10.8284	−0.354507
$934$	−12.4853	−0.408531
$935$	0	0
$936$	−5.65685	−0.184900
$937$	−36.3431	−1.18728	−0.593639	−	0.804731i	$-0.702309\pi$
$937$	−36.3431	−1.18728	−0.593639	+	0.804731i	$0.702309\pi$
$938$	0	0
$939$	−30.4853	−0.994850
$940$	0	0
$941$	−28.2843	−0.922041	−0.461020	−	0.887390i	$-0.652517\pi$
$941$	−28.2843	−0.922041	−0.461020	+	0.887390i	$0.652517\pi$
$942$	8.34315	0.271834
$943$	−33.2548	−1.08293
$944$	−3.17157	−0.103226
$945$	0	0
$946$	−20.3431	−0.661413
$947$	9.17157	0.298036	0.149018	−	0.988834i	$-0.452389\pi$
$947$	9.17157	0.298036	0.149018	+	0.988834i	$0.452389\pi$
$948$	8.00000	0.259828
$949$	−11.3137	−0.367259
$950$	0	0
$951$	25.3137	0.820853
$952$	0	0
$953$	−4.68629	−0.151804	−0.0759019	−	0.997115i	$-0.524184\pi$
$953$	−4.68629	−0.151804	−0.0759019	+	0.997115i	$0.524184\pi$
$954$	−0.343146	−0.0111098
$955$	0	0
$956$	−19.3137	−0.624650
$957$	5.79899	0.187455
$958$	−8.48528	−0.274147
$959$	0	0
$960$	0	0
$961$	73.9117	2.38425
$962$	−1.37258	−0.0442539
$963$	−9.65685	−0.311188
$964$	−19.5563	−0.629868
$965$	0	0
$966$	0	0
$967$	10.3431	0.332613	0.166307	−	0.986074i	$-0.446816\pi$
$967$	10.3431	0.332613	0.166307	+	0.986074i	$0.446816\pi$
$968$	−5.97056	−0.191901
$969$	17.6569	0.567220
$970$	0	0
$971$	−4.97056	−0.159513	−0.0797565	−	0.996814i	$-0.525414\pi$
$971$	−4.97056	−0.159513	−0.0797565	+	0.996814i	$0.525414\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−10.8284	−0.346965
$975$	0	0
$976$	−11.6569	−0.373127
$977$	29.3137	0.937829	0.468914	−	0.883244i	$-0.344645\pi$
$977$	29.3137	0.937829	0.468914	+	0.883244i	$0.344645\pi$
$978$	17.0711	0.545873
$979$	−1.85786	−0.0593776
$980$	0	0
$981$	−3.65685	−0.116754
$982$	−17.0711	−0.544760
$983$	17.0711	0.544483	0.272241	−	0.962229i	$-0.412235\pi$
$983$	17.0711	0.544483	0.272241	+	0.962229i	$0.412235\pi$
$984$	10.4853	0.334259
$985$	0	0
$986$	−6.68629	−0.212935
$987$	0	0
$988$	38.6274	1.22890
$989$	28.7696	0.914819
$990$	0	0
$991$	−10.2010	−0.324046	−0.162023	−	0.986787i	$-0.551802\pi$
$991$	−10.2010	−0.324046	−0.162023	+	0.986787i	$0.551802\pi$
$992$	−10.2426	−0.325204
$993$	−6.34315	−0.201294
$994$	0	0
$995$	0	0
$996$	−5.17157	−0.163868
$997$	56.6274	1.79341	0.896704	−	0.442630i	$-0.145955\pi$
$997$	56.6274	1.79341	0.896704	+	0.442630i	$0.145955\pi$
$998$	6.82843	0.216150
$999$	0.242641	0.00767681

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7350.2.a.dl.1.2		2
5.4	even	2		1470.2.a.s.1.2	✓	2
7.6	odd	2		7350.2.a.dh.1.2		2
15.14	odd	2		4410.2.a.bw.1.1		2
35.4	even	6		1470.2.i.x.961.1		4
35.9	even	6		1470.2.i.x.361.1		4
35.19	odd	6		1470.2.i.w.361.1		4
35.24	odd	6		1470.2.i.w.961.1		4
35.34	odd	2		1470.2.a.t.1.2	yes	2
105.104	even	2		4410.2.a.bz.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1470.2.a.s.1.2	✓	2	5.4	even	2
1470.2.a.t.1.2	yes	2	35.34	odd	2
1470.2.i.w.361.1		4	35.19	odd	6
1470.2.i.w.961.1		4	35.24	odd	6
1470.2.i.x.361.1		4	35.9	even	6
1470.2.i.x.961.1		4	35.4	even	6
4410.2.a.bw.1.1		2	15.14	odd	2
4410.2.a.bz.1.1		2	105.104	even	2
7350.2.a.dh.1.2		2	7.6	odd	2
7350.2.a.dl.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +2.24264 q^{11} +1.00000 q^{12} -5.65685 q^{13} +1.00000 q^{16} -2.58579 q^{17} +1.00000 q^{18} -6.82843 q^{19} +2.24264 q^{22} -3.17157 q^{23} +1.00000 q^{24} -5.65685 q^{26} +1.00000 q^{27} +2.58579 q^{29} -10.2426 q^{31} +1.00000 q^{32} +2.24264 q^{33} -2.58579 q^{34} +1.00000 q^{36} +0.242641 q^{37} -6.82843 q^{38} -5.65685 q^{39} +10.4853 q^{41} -9.07107 q^{43} +2.24264 q^{44} -3.17157 q^{46} +2.24264 q^{47} +1.00000 q^{48} -2.58579 q^{51} -5.65685 q^{52} -0.343146 q^{53} +1.00000 q^{54} -6.82843 q^{57} +2.58579 q^{58} -3.17157 q^{59} -11.6569 q^{61} -10.2426 q^{62} +1.00000 q^{64} +2.24264 q^{66} -9.07107 q^{67} -2.58579 q^{68} -3.17157 q^{69} -4.48528 q^{71} +1.00000 q^{72} +2.00000 q^{73} +0.242641 q^{74} -6.82843 q^{76} -5.65685 q^{78} +8.00000 q^{79} +1.00000 q^{81} +10.4853 q^{82} -5.17157 q^{83} -9.07107 q^{86} +2.58579 q^{87} +2.24264 q^{88} -0.828427 q^{89} -3.17157 q^{92} -10.2426 q^{93} +2.24264 q^{94} +1.00000 q^{96} +6.48528 q^{97} +2.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{8} + 2 q^{9} - 4 q^{11} + 2 q^{12} + 2 q^{16} - 8 q^{17} + 2 q^{18} - 8 q^{19} - 4 q^{22} - 12 q^{23} + 2 q^{24} + 2 q^{27} + 8 q^{29} - 12 q^{31} + 2 q^{32} - 4 q^{33} - 8 q^{34} + 2 q^{36} - 8 q^{37} - 8 q^{38} + 4 q^{41} - 4 q^{43} - 4 q^{44} - 12 q^{46} - 4 q^{47} + 2 q^{48} - 8 q^{51} - 12 q^{53} + 2 q^{54} - 8 q^{57} + 8 q^{58} - 12 q^{59} - 12 q^{61} - 12 q^{62} + 2 q^{64} - 4 q^{66} - 4 q^{67} - 8 q^{68} - 12 q^{69} + 8 q^{71} + 2 q^{72} + 4 q^{73} - 8 q^{74} - 8 q^{76} + 16 q^{79} + 2 q^{81} + 4 q^{82} - 16 q^{83} - 4 q^{86} + 8 q^{87} - 4 q^{88} + 4 q^{89} - 12 q^{92} - 12 q^{93} - 4 q^{94} + 2 q^{96} - 4 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^8 + 2 * q^9 - 4 * q^11 + 2 * q^12 + 2 * q^16 - 8 * q^17 + 2 * q^18 - 8 * q^19 - 4 * q^22 - 12 * q^23 + 2 * q^24 + 2 * q^27 + 8 * q^29 - 12 * q^31 + 2 * q^32 - 4 * q^33 - 8 * q^34 + 2 * q^36 - 8 * q^37 - 8 * q^38 + 4 * q^41 - 4 * q^43 - 4 * q^44 - 12 * q^46 - 4 * q^47 + 2 * q^48 - 8 * q^51 - 12 * q^53 + 2 * q^54 - 8 * q^57 + 8 * q^58 - 12 * q^59 - 12 * q^61 - 12 * q^62 + 2 * q^64 - 4 * q^66 - 4 * q^67 - 8 * q^68 - 12 * q^69 + 8 * q^71 + 2 * q^72 + 4 * q^73 - 8 * q^74 - 8 * q^76 + 16 * q^79 + 2 * q^81 + 4 * q^82 - 16 * q^83 - 4 * q^86 + 8 * q^87 - 4 * q^88 + 4 * q^89 - 12 * q^92 - 12 * q^93 - 4 * q^94 + 2 * q^96 - 4 * q^97 - 4 * q^99

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