LMFDB - Embedded newform 7350.2.a.dg.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("7350.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$58.6900454856$
Analytic rank:	$0$
Dimension:	$2$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +0.585786 q^{11} +1.00000 q^{12} +3.00000 q^{13} +1.00000 q^{16} +0.171573 q^{17} -1.00000 q^{18} -1.41421 q^{19} -0.585786 q^{22} +2.41421 q^{23} -1.00000 q^{24} -3.00000 q^{26} +1.00000 q^{27} +1.00000 q^{29} +0.414214 q^{31} -1.00000 q^{32} +0.585786 q^{33} -0.171573 q^{34} +1.00000 q^{36} -7.07107 q^{37} +1.41421 q^{38} +3.00000 q^{39} -0.171573 q^{41} +4.41421 q^{43} +0.585786 q^{44} -2.41421 q^{46} +9.65685 q^{47} +1.00000 q^{48} +0.171573 q^{51} +3.00000 q^{52} +1.34315 q^{53} -1.00000 q^{54} -1.41421 q^{57} -1.00000 q^{58} -1.24264 q^{59} -5.82843 q^{61} -0.414214 q^{62} +1.00000 q^{64} -0.585786 q^{66} +7.31371 q^{67} +0.171573 q^{68} +2.41421 q^{69} +13.3137 q^{71} -1.00000 q^{72} +0.928932 q^{73} +7.07107 q^{74} -1.41421 q^{76} -3.00000 q^{78} -13.0711 q^{79} +1.00000 q^{81} +0.171573 q^{82} +14.0711 q^{83} -4.41421 q^{86} +1.00000 q^{87} -0.585786 q^{88} +4.00000 q^{89} +2.41421 q^{92} +0.414214 q^{93} -9.65685 q^{94} -1.00000 q^{96} -11.6569 q^{97} +0.585786 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9} + 4 q^{11} + 2 q^{12} + 6 q^{13} + 2 q^{16} + 6 q^{17} - 2 q^{18} - 4 q^{22} + 2 q^{23} - 2 q^{24} - 6 q^{26} + 2 q^{27} + 2 q^{29} - 2 q^{31} - 2 q^{32} + 4 q^{33} - 6 q^{34} + 2 q^{36} + 6 q^{39} - 6 q^{41} + 6 q^{43} + 4 q^{44} - 2 q^{46} + 8 q^{47} + 2 q^{48} + 6 q^{51} + 6 q^{52} + 14 q^{53} - 2 q^{54} - 2 q^{58} + 6 q^{59} - 6 q^{61} + 2 q^{62} + 2 q^{64} - 4 q^{66} - 8 q^{67} + 6 q^{68} + 2 q^{69} + 4 q^{71} - 2 q^{72} + 16 q^{73} - 6 q^{78} - 12 q^{79} + 2 q^{81} + 6 q^{82} + 14 q^{83} - 6 q^{86} + 2 q^{87} - 4 q^{88} + 8 q^{89} + 2 q^{92} - 2 q^{93} - 8 q^{94} - 2 q^{96} - 12 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^8 + 2 * q^9 + 4 * q^11 + 2 * q^12 + 6 * q^13 + 2 * q^16 + 6 * q^17 - 2 * q^18 - 4 * q^22 + 2 * q^23 - 2 * q^24 - 6 * q^26 + 2 * q^27 + 2 * q^29 - 2 * q^31 - 2 * q^32 + 4 * q^33 - 6 * q^34 + 2 * q^36 + 6 * q^39 - 6 * q^41 + 6 * q^43 + 4 * q^44 - 2 * q^46 + 8 * q^47 + 2 * q^48 + 6 * q^51 + 6 * q^52 + 14 * q^53 - 2 * q^54 - 2 * q^58 + 6 * q^59 - 6 * q^61 + 2 * q^62 + 2 * q^64 - 4 * q^66 - 8 * q^67 + 6 * q^68 + 2 * q^69 + 4 * q^71 - 2 * q^72 + 16 * q^73 - 6 * q^78 - 12 * q^79 + 2 * q^81 + 6 * q^82 + 14 * q^83 - 6 * q^86 + 2 * q^87 - 4 * q^88 + 8 * q^89 + 2 * q^92 - 2 * q^93 - 8 * q^94 - 2 * q^96 - 12 * 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
$11$	0.585786	0.176621	0.0883106	−	0.996093i	$-0.471853\pi$
$11$	0.585786	0.176621	0.0883106	+	0.996093i	$0.471853\pi$
$12$	1.00000	0.288675
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	0.171573	0.0416125	0.0208063	−	0.999784i	$-0.493377\pi$
$17$	0.171573	0.0416125	0.0208063	+	0.999784i	$0.493377\pi$
$18$	−1.00000	−0.235702
$19$	−1.41421	−0.324443	−0.162221	−	0.986754i	$-0.551866\pi$
$19$	−1.41421	−0.324443	−0.162221	+	0.986754i	$0.551866\pi$
$20$	0	0
$21$	0	0
$22$	−0.585786	−0.124890
$23$	2.41421	0.503398	0.251699	−	0.967806i	$-0.419011\pi$
$23$	2.41421	0.503398	0.251699	+	0.967806i	$0.419011\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−3.00000	−0.588348
$27$	1.00000	0.192450
$28$	0	0
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	0.414214	0.0743950	0.0371975	−	0.999308i	$-0.488157\pi$
$31$	0.414214	0.0743950	0.0371975	+	0.999308i	$0.488157\pi$
$32$	−1.00000	−0.176777
$33$	0.585786	0.101972
$34$	−0.171573	−0.0294245
$35$	0	0
$36$	1.00000	0.166667
$37$	−7.07107	−1.16248	−0.581238	−	0.813733i	$-0.697432\pi$
$37$	−7.07107	−1.16248	−0.581238	+	0.813733i	$0.697432\pi$
$38$	1.41421	0.229416
$39$	3.00000	0.480384
$40$	0	0
$41$	−0.171573	−0.0267952	−0.0133976	−	0.999910i	$-0.504265\pi$
$41$	−0.171573	−0.0267952	−0.0133976	+	0.999910i	$0.504265\pi$
$42$	0	0
$43$	4.41421	0.673161	0.336581	−	0.941655i	$-0.390730\pi$
$43$	4.41421	0.673161	0.336581	+	0.941655i	$0.390730\pi$
$44$	0.585786	0.0883106
$45$	0	0
$46$	−2.41421	−0.355956
$47$	9.65685	1.40860	0.704298	−	0.709904i	$-0.251262\pi$
$47$	9.65685	1.40860	0.704298	+	0.709904i	$0.251262\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	0	0
$51$	0.171573	0.0240250
$52$	3.00000	0.416025
$53$	1.34315	0.184495	0.0922476	−	0.995736i	$-0.470595\pi$
$53$	1.34315	0.184495	0.0922476	+	0.995736i	$0.470595\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	−1.41421	−0.187317
$58$	−1.00000	−0.131306
$59$	−1.24264	−0.161778	−0.0808890	−	0.996723i	$-0.525776\pi$
$59$	−1.24264	−0.161778	−0.0808890	+	0.996723i	$0.525776\pi$
$60$	0	0
$61$	−5.82843	−0.746254	−0.373127	−	0.927780i	$-0.621714\pi$
$61$	−5.82843	−0.746254	−0.373127	+	0.927780i	$0.621714\pi$
$62$	−0.414214	−0.0526052
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	−0.585786	−0.0721053
$67$	7.31371	0.893512	0.446756	−	0.894656i	$-0.352579\pi$
$67$	7.31371	0.893512	0.446756	+	0.894656i	$0.352579\pi$
$68$	0.171573	0.0208063
$69$	2.41421	0.290637
$70$	0	0
$71$	13.3137	1.58005	0.790023	−	0.613077i	$-0.210068\pi$
$71$	13.3137	1.58005	0.790023	+	0.613077i	$0.210068\pi$
$72$	−1.00000	−0.117851
$73$	0.928932	0.108723	0.0543616	−	0.998521i	$-0.482688\pi$
$73$	0.928932	0.108723	0.0543616	+	0.998521i	$0.482688\pi$
$74$	7.07107	0.821995
$75$	0	0
$76$	−1.41421	−0.162221
$77$	0	0
$78$	−3.00000	−0.339683
$79$	−13.0711	−1.47061	−0.735305	−	0.677736i	$-0.762961\pi$
$79$	−13.0711	−1.47061	−0.735305	+	0.677736i	$0.762961\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0.171573	0.0189471
$83$	14.0711	1.54450	0.772250	−	0.635319i	$-0.219131\pi$
$83$	14.0711	1.54450	0.772250	+	0.635319i	$0.219131\pi$
$84$	0	0
$85$	0	0
$86$	−4.41421	−0.475997
$87$	1.00000	0.107211
$88$	−0.585786	−0.0624450
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	0	0
$91$	0	0
$92$	2.41421	0.251699
$93$	0.414214	0.0429519
$94$	−9.65685	−0.996028
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−11.6569	−1.18357	−0.591787	−	0.806094i	$-0.701577\pi$
$97$	−11.6569	−1.18357	−0.591787	+	0.806094i	$0.701577\pi$
$98$	0	0
$99$	0.585786	0.0588738
$100$	0	0
$101$	−5.65685	−0.562878	−0.281439	−	0.959579i	$-0.590812\pi$
$101$	−5.65685	−0.562878	−0.281439	+	0.959579i	$0.590812\pi$
$102$	−0.171573	−0.0169882
$103$	14.0711	1.38646	0.693232	−	0.720715i	$-0.256186\pi$
$103$	14.0711	1.38646	0.693232	+	0.720715i	$0.256186\pi$
$104$	−3.00000	−0.294174
$105$	0	0
$106$	−1.34315	−0.130458
$107$	−10.4853	−1.01365	−0.506825	−	0.862049i	$-0.669181\pi$
$107$	−10.4853	−1.01365	−0.506825	+	0.862049i	$0.669181\pi$
$108$	1.00000	0.0962250
$109$	11.3137	1.08366	0.541828	−	0.840489i	$-0.317732\pi$
$109$	11.3137	1.08366	0.541828	+	0.840489i	$0.317732\pi$
$110$	0	0
$111$	−7.07107	−0.671156
$112$	0	0
$113$	9.41421	0.885615	0.442807	−	0.896617i	$-0.353983\pi$
$113$	9.41421	0.885615	0.442807	+	0.896617i	$0.353983\pi$
$114$	1.41421	0.132453
$115$	0	0
$116$	1.00000	0.0928477
$117$	3.00000	0.277350
$118$	1.24264	0.114394
$119$	0	0
$120$	0	0
$121$	−10.6569	−0.968805
$122$	5.82843	0.527681
$123$	−0.171573	−0.0154702
$124$	0.414214	0.0371975
$125$	0	0
$126$	0	0
$127$	−8.82843	−0.783396	−0.391698	−	0.920094i	$-0.628112\pi$
$127$	−8.82843	−0.783396	−0.391698	+	0.920094i	$0.628112\pi$
$128$	−1.00000	−0.0883883
$129$	4.41421	0.388650
$130$	0	0
$131$	−13.1716	−1.15081	−0.575403	−	0.817870i	$-0.695155\pi$
$131$	−13.1716	−1.15081	−0.575403	+	0.817870i	$0.695155\pi$
$132$	0.585786	0.0509862
$133$	0	0
$134$	−7.31371	−0.631808
$135$	0	0
$136$	−0.171573	−0.0147123
$137$	19.2132	1.64149	0.820747	−	0.571291i	$-0.193557\pi$
$137$	19.2132	1.64149	0.820747	+	0.571291i	$0.193557\pi$
$138$	−2.41421	−0.205512
$139$	15.5563	1.31947	0.659736	−	0.751497i	$-0.270668\pi$
$139$	15.5563	1.31947	0.659736	+	0.751497i	$0.270668\pi$
$140$	0	0
$141$	9.65685	0.813254
$142$	−13.3137	−1.11726
$143$	1.75736	0.146958
$144$	1.00000	0.0833333
$145$	0	0
$146$	−0.928932	−0.0768790
$147$	0	0
$148$	−7.07107	−0.581238
$149$	−6.65685	−0.545351	−0.272675	−	0.962106i	$-0.587908\pi$
$149$	−6.65685	−0.545351	−0.272675	+	0.962106i	$0.587908\pi$
$150$	0	0
$151$	−11.8995	−0.968367	−0.484184	−	0.874966i	$-0.660883\pi$
$151$	−11.8995	−0.968367	−0.484184	+	0.874966i	$0.660883\pi$
$152$	1.41421	0.114708
$153$	0.171573	0.0138708
$154$	0	0
$155$	0	0
$156$	3.00000	0.240192
$157$	19.7990	1.58013	0.790066	−	0.613022i	$-0.210046\pi$
$157$	19.7990	1.58013	0.790066	+	0.613022i	$0.210046\pi$
$158$	13.0711	1.03988
$159$	1.34315	0.106518
$160$	0	0
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−3.24264	−0.253983	−0.126992	−	0.991904i	$-0.540532\pi$
$163$	−3.24264	−0.253983	−0.126992	+	0.991904i	$0.540532\pi$
$164$	−0.171573	−0.0133976
$165$	0	0
$166$	−14.0711	−1.09213
$167$	−3.17157	−0.245424	−0.122712	−	0.992442i	$-0.539159\pi$
$167$	−3.17157	−0.245424	−0.122712	+	0.992442i	$0.539159\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	−1.41421	−0.108148
$172$	4.41421	0.336581
$173$	5.75736	0.437724	0.218862	−	0.975756i	$-0.429766\pi$
$173$	5.75736	0.437724	0.218862	+	0.975756i	$0.429766\pi$
$174$	−1.00000	−0.0758098
$175$	0	0
$176$	0.585786	0.0441553
$177$	−1.24264	−0.0934026
$178$	−4.00000	−0.299813
$179$	9.07107	0.678003	0.339002	−	0.940786i	$-0.389911\pi$
$179$	9.07107	0.678003	0.339002	+	0.940786i	$0.389911\pi$
$180$	0	0
$181$	−0.686292	−0.0510116	−0.0255058	−	0.999675i	$-0.508120\pi$
$181$	−0.686292	−0.0510116	−0.0255058	+	0.999675i	$0.508120\pi$
$182$	0	0
$183$	−5.82843	−0.430850
$184$	−2.41421	−0.177978
$185$	0	0
$186$	−0.414214	−0.0303716
$187$	0.100505	0.00734966
$188$	9.65685	0.704298
$189$	0	0
$190$	0	0
$191$	1.92893	0.139573	0.0697863	−	0.997562i	$-0.477768\pi$
$191$	1.92893	0.139573	0.0697863	+	0.997562i	$0.477768\pi$
$192$	1.00000	0.0721688
$193$	−1.17157	−0.0843317	−0.0421658	−	0.999111i	$-0.513426\pi$
$193$	−1.17157	−0.0843317	−0.0421658	+	0.999111i	$0.513426\pi$
$194$	11.6569	0.836913
$195$	0	0
$196$	0	0
$197$	−5.14214	−0.366362	−0.183181	−	0.983079i	$-0.558639\pi$
$197$	−5.14214	−0.366362	−0.183181	+	0.983079i	$0.558639\pi$
$198$	−0.585786	−0.0416300
$199$	8.34315	0.591430	0.295715	−	0.955276i	$-0.404442\pi$
$199$	8.34315	0.591430	0.295715	+	0.955276i	$0.404442\pi$
$200$	0	0
$201$	7.31371	0.515869
$202$	5.65685	0.398015
$203$	0	0
$204$	0.171573	0.0120125
$205$	0	0
$206$	−14.0711	−0.980378
$207$	2.41421	0.167799
$208$	3.00000	0.208013
$209$	−0.828427	−0.0573035
$210$	0	0
$211$	4.55635	0.313672	0.156836	−	0.987625i	$-0.449871\pi$
$211$	4.55635	0.313672	0.156836	+	0.987625i	$0.449871\pi$
$212$	1.34315	0.0922476
$213$	13.3137	0.912240
$214$	10.4853	0.716759
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−11.3137	−0.766261
$219$	0.928932	0.0627714
$220$	0	0
$221$	0.514719	0.0346237
$222$	7.07107	0.474579
$223$	−6.55635	−0.439046	−0.219523	−	0.975607i	$-0.570450\pi$
$223$	−6.55635	−0.439046	−0.219523	+	0.975607i	$0.570450\pi$
$224$	0	0
$225$	0	0
$226$	−9.41421	−0.626224
$227$	−11.3848	−0.755634	−0.377817	−	0.925880i	$-0.623325\pi$
$227$	−11.3848	−0.755634	−0.377817	+	0.925880i	$0.623325\pi$
$228$	−1.41421	−0.0936586
$229$	6.68629	0.441843	0.220921	−	0.975292i	$-0.429094\pi$
$229$	6.68629	0.441843	0.220921	+	0.975292i	$0.429094\pi$
$230$	0	0
$231$	0	0
$232$	−1.00000	−0.0656532
$233$	5.75736	0.377177	0.188589	−	0.982056i	$-0.439609\pi$
$233$	5.75736	0.377177	0.188589	+	0.982056i	$0.439609\pi$
$234$	−3.00000	−0.196116
$235$	0	0
$236$	−1.24264	−0.0808890
$237$	−13.0711	−0.849057
$238$	0	0
$239$	11.3137	0.731823	0.365911	−	0.930650i	$-0.380757\pi$
$239$	11.3137	0.731823	0.365911	+	0.930650i	$0.380757\pi$
$240$	0	0
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	10.6569	0.685049
$243$	1.00000	0.0641500
$244$	−5.82843	−0.373127
$245$	0	0
$246$	0.171573	0.0109391
$247$	−4.24264	−0.269953
$248$	−0.414214	−0.0263026
$249$	14.0711	0.891718
$250$	0	0
$251$	−16.0711	−1.01440	−0.507198	−	0.861829i	$-0.669319\pi$
$251$	−16.0711	−1.01440	−0.507198	+	0.861829i	$0.669319\pi$
$252$	0	0
$253$	1.41421	0.0889108
$254$	8.82843	0.553945
$255$	0	0
$256$	1.00000	0.0625000
$257$	24.6569	1.53805	0.769026	−	0.639217i	$-0.220742\pi$
$257$	24.6569	1.53805	0.769026	+	0.639217i	$0.220742\pi$
$258$	−4.41421	−0.274817
$259$	0	0
$260$	0	0
$261$	1.00000	0.0618984
$262$	13.1716	0.813742
$263$	20.5563	1.26756	0.633779	−	0.773514i	$-0.281503\pi$
$263$	20.5563	1.26756	0.633779	+	0.773514i	$0.281503\pi$
$264$	−0.585786	−0.0360527
$265$	0	0
$266$	0	0
$267$	4.00000	0.244796
$268$	7.31371	0.446756
$269$	31.5563	1.92402	0.962012	−	0.273006i	$-0.0880179\pi$
$269$	31.5563	1.92402	0.962012	+	0.273006i	$0.0880179\pi$
$270$	0	0
$271$	−28.4853	−1.73036	−0.865179	−	0.501463i	$-0.832795\pi$
$271$	−28.4853	−1.73036	−0.865179	+	0.501463i	$0.832795\pi$
$272$	0.171573	0.0104031
$273$	0	0
$274$	−19.2132	−1.16071
$275$	0	0
$276$	2.41421	0.145319
$277$	24.0416	1.44452	0.722261	−	0.691621i	$-0.243103\pi$
$277$	24.0416	1.44452	0.722261	+	0.691621i	$0.243103\pi$
$278$	−15.5563	−0.933008
$279$	0.414214	0.0247983
$280$	0	0
$281$	20.0000	1.19310	0.596550	−	0.802576i	$-0.296538\pi$
$281$	20.0000	1.19310	0.596550	+	0.802576i	$0.296538\pi$
$282$	−9.65685	−0.575057
$283$	8.24264	0.489974	0.244987	−	0.969526i	$-0.421216\pi$
$283$	8.24264	0.489974	0.244987	+	0.969526i	$0.421216\pi$
$284$	13.3137	0.790023
$285$	0	0
$286$	−1.75736	−0.103915
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−16.9706	−0.998268
$290$	0	0
$291$	−11.6569	−0.683337
$292$	0.928932	0.0543616
$293$	11.3137	0.660954	0.330477	−	0.943814i	$-0.392790\pi$
$293$	11.3137	0.660954	0.330477	+	0.943814i	$0.392790\pi$
$294$	0	0
$295$	0	0
$296$	7.07107	0.410997
$297$	0.585786	0.0339908
$298$	6.65685	0.385621
$299$	7.24264	0.418853
$300$	0	0
$301$	0	0
$302$	11.8995	0.684739
$303$	−5.65685	−0.324978
$304$	−1.41421	−0.0811107
$305$	0	0
$306$	−0.171573	−0.00980817
$307$	11.4142	0.651444	0.325722	−	0.945466i	$-0.394393\pi$
$307$	11.4142	0.651444	0.325722	+	0.945466i	$0.394393\pi$
$308$	0	0
$309$	14.0711	0.800475
$310$	0	0
$311$	−8.58579	−0.486855	−0.243428	−	0.969919i	$-0.578272\pi$
$311$	−8.58579	−0.486855	−0.243428	+	0.969919i	$0.578272\pi$
$312$	−3.00000	−0.169842
$313$	32.0416	1.81110	0.905550	−	0.424240i	$-0.139459\pi$
$313$	32.0416	1.81110	0.905550	+	0.424240i	$0.139459\pi$
$314$	−19.7990	−1.11732
$315$	0	0
$316$	−13.0711	−0.735305
$317$	−13.9706	−0.784665	−0.392332	−	0.919823i	$-0.628332\pi$
$317$	−13.9706	−0.784665	−0.392332	+	0.919823i	$0.628332\pi$
$318$	−1.34315	−0.0753199
$319$	0.585786	0.0327977
$320$	0	0
$321$	−10.4853	−0.585231
$322$	0	0
$323$	−0.242641	−0.0135009
$324$	1.00000	0.0555556
$325$	0	0
$326$	3.24264	0.179593
$327$	11.3137	0.625650
$328$	0.171573	0.00947353
$329$	0	0
$330$	0	0
$331$	−9.92893	−0.545743	−0.272872	−	0.962050i	$-0.587973\pi$
$331$	−9.92893	−0.545743	−0.272872	+	0.962050i	$0.587973\pi$
$332$	14.0711	0.772250
$333$	−7.07107	−0.387492
$334$	3.17157	0.173541
$335$	0	0
$336$	0	0
$337$	−34.7990	−1.89562	−0.947811	−	0.318833i	$-0.896709\pi$
$337$	−34.7990	−1.89562	−0.947811	+	0.318833i	$0.896709\pi$
$338$	4.00000	0.217571
$339$	9.41421	0.511310
$340$	0	0
$341$	0.242641	0.0131397
$342$	1.41421	0.0764719
$343$	0	0
$344$	−4.41421	−0.237998
$345$	0	0
$346$	−5.75736	−0.309518
$347$	29.7990	1.59969	0.799847	−	0.600204i	$-0.204914\pi$
$347$	29.7990	1.59969	0.799847	+	0.600204i	$0.204914\pi$
$348$	1.00000	0.0536056
$349$	13.3431	0.714242	0.357121	−	0.934058i	$-0.383758\pi$
$349$	13.3431	0.714242	0.357121	+	0.934058i	$0.383758\pi$
$350$	0	0
$351$	3.00000	0.160128
$352$	−0.585786	−0.0312225
$353$	3.51472	0.187070	0.0935348	−	0.995616i	$-0.470183\pi$
$353$	3.51472	0.187070	0.0935348	+	0.995616i	$0.470183\pi$
$354$	1.24264	0.0660456
$355$	0	0
$356$	4.00000	0.212000
$357$	0	0
$358$	−9.07107	−0.479421
$359$	−19.8701	−1.04870	−0.524351	−	0.851502i	$-0.675692\pi$
$359$	−19.8701	−1.04870	−0.524351	+	0.851502i	$0.675692\pi$
$360$	0	0
$361$	−17.0000	−0.894737
$362$	0.686292	0.0360707
$363$	−10.6569	−0.559340
$364$	0	0
$365$	0	0
$366$	5.82843	0.304657
$367$	9.24264	0.482462	0.241231	−	0.970468i	$-0.422449\pi$
$367$	9.24264	0.482462	0.241231	+	0.970468i	$0.422449\pi$
$368$	2.41421	0.125850
$369$	−0.171573	−0.00893173
$370$	0	0
$371$	0	0
$372$	0.414214	0.0214760
$373$	13.6569	0.707125	0.353563	−	0.935411i	$-0.384970\pi$
$373$	13.6569	0.707125	0.353563	+	0.935411i	$0.384970\pi$
$374$	−0.100505	−0.00519699
$375$	0	0
$376$	−9.65685	−0.498014
$377$	3.00000	0.154508
$378$	0	0
$379$	8.21320	0.421884	0.210942	−	0.977499i	$-0.432347\pi$
$379$	8.21320	0.421884	0.210942	+	0.977499i	$0.432347\pi$
$380$	0	0
$381$	−8.82843	−0.452294
$382$	−1.92893	−0.0986928
$383$	−3.55635	−0.181721	−0.0908605	−	0.995864i	$-0.528962\pi$
$383$	−3.55635	−0.181721	−0.0908605	+	0.995864i	$0.528962\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	1.17157	0.0596315
$387$	4.41421	0.224387
$388$	−11.6569	−0.591787
$389$	−21.6569	−1.09805	−0.549023	−	0.835807i	$-0.685000\pi$
$389$	−21.6569	−1.09805	−0.549023	+	0.835807i	$0.685000\pi$
$390$	0	0
$391$	0.414214	0.0209477
$392$	0	0
$393$	−13.1716	−0.664418
$394$	5.14214	0.259057
$395$	0	0
$396$	0.585786	0.0294369
$397$	7.68629	0.385764	0.192882	−	0.981222i	$-0.438217\pi$
$397$	7.68629	0.385764	0.192882	+	0.981222i	$0.438217\pi$
$398$	−8.34315	−0.418204
$399$	0	0
$400$	0	0
$401$	31.8995	1.59298	0.796492	−	0.604649i	$-0.206686\pi$
$401$	31.8995	1.59298	0.796492	+	0.604649i	$0.206686\pi$
$402$	−7.31371	−0.364775
$403$	1.24264	0.0619003
$404$	−5.65685	−0.281439
$405$	0	0
$406$	0	0
$407$	−4.14214	−0.205318
$408$	−0.171573	−0.00849412
$409$	25.4142	1.25665	0.628326	−	0.777950i	$-0.283740\pi$
$409$	25.4142	1.25665	0.628326	+	0.777950i	$0.283740\pi$
$410$	0	0
$411$	19.2132	0.947717
$412$	14.0711	0.693232
$413$	0	0
$414$	−2.41421	−0.118652
$415$	0	0
$416$	−3.00000	−0.147087
$417$	15.5563	0.761798
$418$	0.828427	0.0405197
$419$	−6.55635	−0.320299	−0.160149	−	0.987093i	$-0.551198\pi$
$419$	−6.55635	−0.320299	−0.160149	+	0.987093i	$0.551198\pi$
$420$	0	0
$421$	−24.3431	−1.18641	−0.593206	−	0.805051i	$-0.702138\pi$
$421$	−24.3431	−1.18641	−0.593206	+	0.805051i	$0.702138\pi$
$422$	−4.55635	−0.221800
$423$	9.65685	0.469532
$424$	−1.34315	−0.0652289
$425$	0	0
$426$	−13.3137	−0.645051
$427$	0	0
$428$	−10.4853	−0.506825
$429$	1.75736	0.0848461
$430$	0	0
$431$	−15.0416	−0.724530	−0.362265	−	0.932075i	$-0.617996\pi$
$431$	−15.0416	−0.724530	−0.362265	+	0.932075i	$0.617996\pi$
$432$	1.00000	0.0481125
$433$	−16.6274	−0.799063	−0.399531	−	0.916720i	$-0.630827\pi$
$433$	−16.6274	−0.799063	−0.399531	+	0.916720i	$0.630827\pi$
$434$	0	0
$435$	0	0
$436$	11.3137	0.541828
$437$	−3.41421	−0.163324
$438$	−0.928932	−0.0443861
$439$	4.41421	0.210679	0.105339	−	0.994436i	$-0.466407\pi$
$439$	4.41421	0.210679	0.105339	+	0.994436i	$0.466407\pi$
$440$	0	0
$441$	0	0
$442$	−0.514719	−0.0244827
$443$	−25.6569	−1.21899	−0.609497	−	0.792788i	$-0.708629\pi$
$443$	−25.6569	−1.21899	−0.609497	+	0.792788i	$0.708629\pi$
$444$	−7.07107	−0.335578
$445$	0	0
$446$	6.55635	0.310452
$447$	−6.65685	−0.314858
$448$	0	0
$449$	−10.9289	−0.515768	−0.257884	−	0.966176i	$-0.583025\pi$
$449$	−10.9289	−0.515768	−0.257884	+	0.966176i	$0.583025\pi$
$450$	0	0
$451$	−0.100505	−0.00473260
$452$	9.41421	0.442807
$453$	−11.8995	−0.559087
$454$	11.3848	0.534314
$455$	0	0
$456$	1.41421	0.0662266
$457$	1.82843	0.0855302	0.0427651	−	0.999085i	$-0.486383\pi$
$457$	1.82843	0.0855302	0.0427651	+	0.999085i	$0.486383\pi$
$458$	−6.68629	−0.312430
$459$	0.171573	0.00800834
$460$	0	0
$461$	20.0000	0.931493	0.465746	−	0.884918i	$-0.345786\pi$
$461$	20.0000	0.931493	0.465746	+	0.884918i	$0.345786\pi$
$462$	0	0
$463$	10.1005	0.469410	0.234705	−	0.972067i	$-0.424588\pi$
$463$	10.1005	0.469410	0.234705	+	0.972067i	$0.424588\pi$
$464$	1.00000	0.0464238
$465$	0	0
$466$	−5.75736	−0.266705
$467$	7.24264	0.335149	0.167575	−	0.985859i	$-0.446406\pi$
$467$	7.24264	0.335149	0.167575	+	0.985859i	$0.446406\pi$
$468$	3.00000	0.138675
$469$	0	0
$470$	0	0
$471$	19.7990	0.912289
$472$	1.24264	0.0571972
$473$	2.58579	0.118895
$474$	13.0711	0.600374
$475$	0	0
$476$	0	0
$477$	1.34315	0.0614984
$478$	−11.3137	−0.517477
$479$	11.7574	0.537207	0.268604	−	0.963251i	$-0.413438\pi$
$479$	11.7574	0.537207	0.268604	+	0.963251i	$0.413438\pi$
$480$	0	0
$481$	−21.2132	−0.967239
$482$	6.00000	0.273293
$483$	0	0
$484$	−10.6569	−0.484402
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	22.4853	1.01891	0.509453	−	0.860499i	$-0.329848\pi$
$487$	22.4853	1.01891	0.509453	+	0.860499i	$0.329848\pi$
$488$	5.82843	0.263840
$489$	−3.24264	−0.146637
$490$	0	0
$491$	28.1421	1.27004	0.635018	−	0.772497i	$-0.280993\pi$
$491$	28.1421	1.27004	0.635018	+	0.772497i	$0.280993\pi$
$492$	−0.171573	−0.00773510
$493$	0.171573	0.00772725
$494$	4.24264	0.190885
$495$	0	0
$496$	0.414214	0.0185987
$497$	0	0
$498$	−14.0711	−0.630540
$499$	−11.9289	−0.534012	−0.267006	−	0.963695i	$-0.586034\pi$
$499$	−11.9289	−0.534012	−0.267006	+	0.963695i	$0.586034\pi$
$500$	0	0
$501$	−3.17157	−0.141695
$502$	16.0711	0.717287
$503$	28.8701	1.28725	0.643626	−	0.765340i	$-0.277429\pi$
$503$	28.8701	1.28725	0.643626	+	0.765340i	$0.277429\pi$
$504$	0	0
$505$	0	0
$506$	−1.41421	−0.0628695
$507$	−4.00000	−0.177646
$508$	−8.82843	−0.391698
$509$	42.0416	1.86346	0.931731	−	0.363149i	$-0.118298\pi$
$509$	42.0416	1.86346	0.931731	+	0.363149i	$0.118298\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−1.41421	−0.0624391
$514$	−24.6569	−1.08757
$515$	0	0
$516$	4.41421	0.194325
$517$	5.65685	0.248788
$518$	0	0
$519$	5.75736	0.252720
$520$	0	0
$521$	0.656854	0.0287773	0.0143887	−	0.999896i	$-0.495420\pi$
$521$	0.656854	0.0287773	0.0143887	+	0.999896i	$0.495420\pi$
$522$	−1.00000	−0.0437688
$523$	32.2843	1.41169	0.705846	−	0.708365i	$-0.250567\pi$
$523$	32.2843	1.41169	0.705846	+	0.708365i	$0.250567\pi$
$524$	−13.1716	−0.575403
$525$	0	0
$526$	−20.5563	−0.896299
$527$	0.0710678	0.00309576
$528$	0.585786	0.0254931
$529$	−17.1716	−0.746590
$530$	0	0
$531$	−1.24264	−0.0539260
$532$	0	0
$533$	−0.514719	−0.0222949
$534$	−4.00000	−0.173097
$535$	0	0
$536$	−7.31371	−0.315904
$537$	9.07107	0.391445
$538$	−31.5563	−1.36049
$539$	0	0
$540$	0	0
$541$	−25.7574	−1.10740	−0.553698	−	0.832718i	$-0.686784\pi$
$541$	−25.7574	−1.10740	−0.553698	+	0.832718i	$0.686784\pi$
$542$	28.4853	1.22355
$543$	−0.686292	−0.0294516
$544$	−0.171573	−0.00735613
$545$	0	0
$546$	0	0
$547$	−19.8701	−0.849582	−0.424791	−	0.905291i	$-0.639652\pi$
$547$	−19.8701	−0.849582	−0.424791	+	0.905291i	$0.639652\pi$
$548$	19.2132	0.820747
$549$	−5.82843	−0.248751
$550$	0	0
$551$	−1.41421	−0.0602475
$552$	−2.41421	−0.102756
$553$	0	0
$554$	−24.0416	−1.02143
$555$	0	0
$556$	15.5563	0.659736
$557$	3.31371	0.140406	0.0702032	−	0.997533i	$-0.477635\pi$
$557$	3.31371	0.140406	0.0702032	+	0.997533i	$0.477635\pi$
$558$	−0.414214	−0.0175351
$559$	13.2426	0.560104
$560$	0	0
$561$	0.100505	0.00424333
$562$	−20.0000	−0.843649
$563$	−9.38478	−0.395521	−0.197761	−	0.980250i	$-0.563367\pi$
$563$	−9.38478	−0.395521	−0.197761	+	0.980250i	$0.563367\pi$
$564$	9.65685	0.406627
$565$	0	0
$566$	−8.24264	−0.346464
$567$	0	0
$568$	−13.3137	−0.558631
$569$	−19.7574	−0.828272	−0.414136	−	0.910215i	$-0.635916\pi$
$569$	−19.7574	−0.828272	−0.414136	+	0.910215i	$0.635916\pi$
$570$	0	0
$571$	−46.0122	−1.92555	−0.962775	−	0.270303i	$-0.912876\pi$
$571$	−46.0122	−1.92555	−0.962775	+	0.270303i	$0.912876\pi$
$572$	1.75736	0.0734789
$573$	1.92893	0.0805823
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	8.34315	0.347330	0.173665	−	0.984805i	$-0.444439\pi$
$577$	8.34315	0.347330	0.173665	+	0.984805i	$0.444439\pi$
$578$	16.9706	0.705882
$579$	−1.17157	−0.0486889
$580$	0	0
$581$	0	0
$582$	11.6569	0.483192
$583$	0.786797	0.0325858
$584$	−0.928932	−0.0384395
$585$	0	0
$586$	−11.3137	−0.467365
$587$	23.5269	0.971060	0.485530	−	0.874220i	$-0.338627\pi$
$587$	23.5269	0.971060	0.485530	+	0.874220i	$0.338627\pi$
$588$	0	0
$589$	−0.585786	−0.0241369
$590$	0	0
$591$	−5.14214	−0.211519
$592$	−7.07107	−0.290619
$593$	−26.0000	−1.06769	−0.533846	−	0.845582i	$-0.679254\pi$
$593$	−26.0000	−1.06769	−0.533846	+	0.845582i	$0.679254\pi$
$594$	−0.585786	−0.0240351
$595$	0	0
$596$	−6.65685	−0.272675
$597$	8.34315	0.341462
$598$	−7.24264	−0.296174
$599$	−27.5858	−1.12712	−0.563562	−	0.826074i	$-0.690570\pi$
$599$	−27.5858	−1.12712	−0.563562	+	0.826074i	$0.690570\pi$
$600$	0	0
$601$	26.7279	1.09025	0.545127	−	0.838353i	$-0.316481\pi$
$601$	26.7279	1.09025	0.545127	+	0.838353i	$0.316481\pi$
$602$	0	0
$603$	7.31371	0.297837
$604$	−11.8995	−0.484184
$605$	0	0
$606$	5.65685	0.229794
$607$	−27.9411	−1.13410	−0.567048	−	0.823685i	$-0.691914\pi$
$607$	−27.9411	−1.13410	−0.567048	+	0.823685i	$0.691914\pi$
$608$	1.41421	0.0573539
$609$	0	0
$610$	0	0
$611$	28.9706	1.17202
$612$	0.171573	0.00693542
$613$	−22.0000	−0.888572	−0.444286	−	0.895885i	$-0.646543\pi$
$613$	−22.0000	−0.888572	−0.444286	+	0.895885i	$0.646543\pi$
$614$	−11.4142	−0.460640
$615$	0	0
$616$	0	0
$617$	−14.3848	−0.579109	−0.289555	−	0.957161i	$-0.593507\pi$
$617$	−14.3848	−0.579109	−0.289555	+	0.957161i	$0.593507\pi$
$618$	−14.0711	−0.566021
$619$	16.1421	0.648807	0.324404	−	0.945919i	$-0.394836\pi$
$619$	16.1421	0.648807	0.324404	+	0.945919i	$0.394836\pi$
$620$	0	0
$621$	2.41421	0.0968791
$622$	8.58579	0.344259
$623$	0	0
$624$	3.00000	0.120096
$625$	0	0
$626$	−32.0416	−1.28064
$627$	−0.828427	−0.0330842
$628$	19.7990	0.790066
$629$	−1.21320	−0.0483736
$630$	0	0
$631$	−35.1716	−1.40016	−0.700079	−	0.714065i	$-0.746852\pi$
$631$	−35.1716	−1.40016	−0.700079	+	0.714065i	$0.746852\pi$
$632$	13.0711	0.519939
$633$	4.55635	0.181099
$634$	13.9706	0.554842
$635$	0	0
$636$	1.34315	0.0532592
$637$	0	0
$638$	−0.585786	−0.0231915
$639$	13.3137	0.526682
$640$	0	0
$641$	40.2426	1.58949	0.794744	−	0.606944i	$-0.207605\pi$
$641$	40.2426	1.58949	0.794744	+	0.606944i	$0.207605\pi$
$642$	10.4853	0.413821
$643$	−22.2426	−0.877164	−0.438582	−	0.898691i	$-0.644519\pi$
$643$	−22.2426	−0.877164	−0.438582	+	0.898691i	$0.644519\pi$
$644$	0	0
$645$	0	0
$646$	0.242641	0.00954657
$647$	−48.8701	−1.92128	−0.960640	−	0.277796i	$-0.910396\pi$
$647$	−48.8701	−1.92128	−0.960640	+	0.277796i	$0.910396\pi$
$648$	−1.00000	−0.0392837
$649$	−0.727922	−0.0285734
$650$	0	0
$651$	0	0
$652$	−3.24264	−0.126992
$653$	−35.3137	−1.38193	−0.690966	−	0.722887i	$-0.742815\pi$
$653$	−35.3137	−1.38193	−0.690966	+	0.722887i	$0.742815\pi$
$654$	−11.3137	−0.442401
$655$	0	0
$656$	−0.171573	−0.00669880
$657$	0.928932	0.0362411
$658$	0	0
$659$	8.82843	0.343907	0.171953	−	0.985105i	$-0.444992\pi$
$659$	8.82843	0.343907	0.171953	+	0.985105i	$0.444992\pi$
$660$	0	0
$661$	45.6569	1.77585	0.887923	−	0.459992i	$-0.152148\pi$
$661$	45.6569	1.77585	0.887923	+	0.459992i	$0.152148\pi$
$662$	9.92893	0.385899
$663$	0.514719	0.0199900
$664$	−14.0711	−0.546063
$665$	0	0
$666$	7.07107	0.273998
$667$	2.41421	0.0934787
$668$	−3.17157	−0.122712
$669$	−6.55635	−0.253483
$670$	0	0
$671$	−3.41421	−0.131804
$672$	0	0
$673$	38.7990	1.49559	0.747796	−	0.663929i	$-0.231112\pi$
$673$	38.7990	1.49559	0.747796	+	0.663929i	$0.231112\pi$
$674$	34.7990	1.34041
$675$	0	0
$676$	−4.00000	−0.153846
$677$	32.3848	1.24465	0.622324	−	0.782760i	$-0.286189\pi$
$677$	32.3848	1.24465	0.622324	+	0.782760i	$0.286189\pi$
$678$	−9.41421	−0.361551
$679$	0	0
$680$	0	0
$681$	−11.3848	−0.436266
$682$	−0.242641	−0.00929119
$683$	10.9289	0.418184	0.209092	−	0.977896i	$-0.432949\pi$
$683$	10.9289	0.418184	0.209092	+	0.977896i	$0.432949\pi$
$684$	−1.41421	−0.0540738
$685$	0	0
$686$	0	0
$687$	6.68629	0.255098
$688$	4.41421	0.168290
$689$	4.02944	0.153509
$690$	0	0
$691$	−29.1127	−1.10750	−0.553750	−	0.832683i	$-0.686804\pi$
$691$	−29.1127	−1.10750	−0.553750	+	0.832683i	$0.686804\pi$
$692$	5.75736	0.218862
$693$	0	0
$694$	−29.7990	−1.13115
$695$	0	0
$696$	−1.00000	−0.0379049
$697$	−0.0294373	−0.00111502
$698$	−13.3431	−0.505046
$699$	5.75736	0.217763
$700$	0	0
$701$	13.6863	0.516924	0.258462	−	0.966021i	$-0.416784\pi$
$701$	13.6863	0.516924	0.258462	+	0.966021i	$0.416784\pi$
$702$	−3.00000	−0.113228
$703$	10.0000	0.377157
$704$	0.585786	0.0220777
$705$	0	0
$706$	−3.51472	−0.132278
$707$	0	0
$708$	−1.24264	−0.0467013
$709$	0.100505	0.00377455	0.00188727	−	0.999998i	$-0.499399\pi$
$709$	0.100505	0.00377455	0.00188727	+	0.999998i	$0.499399\pi$
$710$	0	0
$711$	−13.0711	−0.490203
$712$	−4.00000	−0.149906
$713$	1.00000	0.0374503
$714$	0	0
$715$	0	0
$716$	9.07107	0.339002
$717$	11.3137	0.422518
$718$	19.8701	0.741544
$719$	33.0711	1.23334	0.616671	−	0.787221i	$-0.288481\pi$
$719$	33.0711	1.23334	0.616671	+	0.787221i	$0.288481\pi$
$720$	0	0
$721$	0	0
$722$	17.0000	0.632674
$723$	−6.00000	−0.223142
$724$	−0.686292	−0.0255058
$725$	0	0
$726$	10.6569	0.395513
$727$	15.8701	0.588588	0.294294	−	0.955715i	$-0.404916\pi$
$727$	15.8701	0.588588	0.294294	+	0.955715i	$0.404916\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0.757359	0.0280119
$732$	−5.82843	−0.215425
$733$	−17.4853	−0.645834	−0.322917	−	0.946427i	$-0.604663\pi$
$733$	−17.4853	−0.645834	−0.322917	+	0.946427i	$0.604663\pi$
$734$	−9.24264	−0.341152
$735$	0	0
$736$	−2.41421	−0.0889891
$737$	4.28427	0.157813
$738$	0.171573	0.00631568
$739$	0.129942	0.00478001	0.00239000	−	0.999997i	$-0.499239\pi$
$739$	0.129942	0.00478001	0.00239000	+	0.999997i	$0.499239\pi$
$740$	0	0
$741$	−4.24264	−0.155857
$742$	0	0
$743$	−5.44365	−0.199708	−0.0998541	−	0.995002i	$-0.531838\pi$
$743$	−5.44365	−0.199708	−0.0998541	+	0.995002i	$0.531838\pi$
$744$	−0.414214	−0.0151858
$745$	0	0
$746$	−13.6569	−0.500013
$747$	14.0711	0.514833
$748$	0.100505	0.00367483
$749$	0	0
$750$	0	0
$751$	20.9706	0.765227	0.382613	−	0.923909i	$-0.375024\pi$
$751$	20.9706	0.765227	0.382613	+	0.923909i	$0.375024\pi$
$752$	9.65685	0.352149
$753$	−16.0711	−0.585662
$754$	−3.00000	−0.109254
$755$	0	0
$756$	0	0
$757$	−35.7574	−1.29962	−0.649812	−	0.760095i	$-0.725152\pi$
$757$	−35.7574	−1.29962	−0.649812	+	0.760095i	$0.725152\pi$
$758$	−8.21320	−0.298317
$759$	1.41421	0.0513327
$760$	0	0
$761$	48.9706	1.77518	0.887591	−	0.460633i	$-0.152378\pi$
$761$	48.9706	1.77518	0.887591	+	0.460633i	$0.152378\pi$
$762$	8.82843	0.319820
$763$	0	0
$764$	1.92893	0.0697863
$765$	0	0
$766$	3.55635	0.128496
$767$	−3.72792	−0.134607
$768$	1.00000	0.0360844
$769$	7.31371	0.263739	0.131870	−	0.991267i	$-0.457902\pi$
$769$	7.31371	0.263739	0.131870	+	0.991267i	$0.457902\pi$
$770$	0	0
$771$	24.6569	0.887995
$772$	−1.17157	−0.0421658
$773$	3.31371	0.119186	0.0595929	−	0.998223i	$-0.481020\pi$
$773$	3.31371	0.119186	0.0595929	+	0.998223i	$0.481020\pi$
$774$	−4.41421	−0.158666
$775$	0	0
$776$	11.6569	0.418457
$777$	0	0
$778$	21.6569	0.776436
$779$	0.242641	0.00869350
$780$	0	0
$781$	7.79899	0.279070
$782$	−0.414214	−0.0148122
$783$	1.00000	0.0357371
$784$	0	0
$785$	0	0
$786$	13.1716	0.469814
$787$	7.45584	0.265772	0.132886	−	0.991131i	$-0.457576\pi$
$787$	7.45584	0.265772	0.132886	+	0.991131i	$0.457576\pi$
$788$	−5.14214	−0.183181
$789$	20.5563	0.731825
$790$	0	0
$791$	0	0
$792$	−0.585786	−0.0208150
$793$	−17.4853	−0.620921
$794$	−7.68629	−0.272776
$795$	0	0
$796$	8.34315	0.295715
$797$	7.75736	0.274780	0.137390	−	0.990517i	$-0.456129\pi$
$797$	7.75736	0.274780	0.137390	+	0.990517i	$0.456129\pi$
$798$	0	0
$799$	1.65685	0.0586153
$800$	0	0
$801$	4.00000	0.141333
$802$	−31.8995	−1.12641
$803$	0.544156	0.0192028
$804$	7.31371	0.257935
$805$	0	0
$806$	−1.24264	−0.0437702
$807$	31.5563	1.11084
$808$	5.65685	0.199007
$809$	10.6274	0.373640	0.186820	−	0.982394i	$-0.440182\pi$
$809$	10.6274	0.373640	0.186820	+	0.982394i	$0.440182\pi$
$810$	0	0
$811$	42.2426	1.48334	0.741670	−	0.670765i	$-0.234034\pi$
$811$	42.2426	1.48334	0.741670	+	0.670765i	$0.234034\pi$
$812$	0	0
$813$	−28.4853	−0.999022
$814$	4.14214	0.145182
$815$	0	0
$816$	0.171573	0.00600625
$817$	−6.24264	−0.218402
$818$	−25.4142	−0.888587
$819$	0	0
$820$	0	0
$821$	18.8284	0.657117	0.328558	−	0.944484i	$-0.393437\pi$
$821$	18.8284	0.657117	0.328558	+	0.944484i	$0.393437\pi$
$822$	−19.2132	−0.670137
$823$	31.4142	1.09503	0.547515	−	0.836796i	$-0.315574\pi$
$823$	31.4142	1.09503	0.547515	+	0.836796i	$0.315574\pi$
$824$	−14.0711	−0.490189
$825$	0	0
$826$	0	0
$827$	7.89949	0.274692	0.137346	−	0.990523i	$-0.456143\pi$
$827$	7.89949	0.274692	0.137346	+	0.990523i	$0.456143\pi$
$828$	2.41421	0.0838997
$829$	38.7990	1.34754	0.673772	−	0.738939i	$-0.264673\pi$
$829$	38.7990	1.34754	0.673772	+	0.738939i	$0.264673\pi$
$830$	0	0
$831$	24.0416	0.833995
$832$	3.00000	0.104006
$833$	0	0
$834$	−15.5563	−0.538672
$835$	0	0
$836$	−0.828427	−0.0286518
$837$	0.414214	0.0143173
$838$	6.55635	0.226485
$839$	14.5858	0.503557	0.251779	−	0.967785i	$-0.418985\pi$
$839$	14.5858	0.503557	0.251779	+	0.967785i	$0.418985\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	24.3431	0.838920
$843$	20.0000	0.688837
$844$	4.55635	0.156836
$845$	0	0
$846$	−9.65685	−0.332009
$847$	0	0
$848$	1.34315	0.0461238
$849$	8.24264	0.282887
$850$	0	0
$851$	−17.0711	−0.585189
$852$	13.3137	0.456120
$853$	−8.31371	−0.284656	−0.142328	−	0.989820i	$-0.545459\pi$
$853$	−8.31371	−0.284656	−0.142328	+	0.989820i	$0.545459\pi$
$854$	0	0
$855$	0	0
$856$	10.4853	0.358380
$857$	−21.4558	−0.732918	−0.366459	−	0.930434i	$-0.619430\pi$
$857$	−21.4558	−0.732918	−0.366459	+	0.930434i	$0.619430\pi$
$858$	−1.75736	−0.0599953
$859$	−45.9411	−1.56749	−0.783745	−	0.621082i	$-0.786693\pi$
$859$	−45.9411	−1.56749	−0.783745	+	0.621082i	$0.786693\pi$
$860$	0	0
$861$	0	0
$862$	15.0416	0.512320
$863$	−8.97056	−0.305362	−0.152681	−	0.988276i	$-0.548791\pi$
$863$	−8.97056	−0.305362	−0.152681	+	0.988276i	$0.548791\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	16.6274	0.565023
$867$	−16.9706	−0.576351
$868$	0	0
$869$	−7.65685	−0.259741
$870$	0	0
$871$	21.9411	0.743447
$872$	−11.3137	−0.383131
$873$	−11.6569	−0.394525
$874$	3.41421	0.115487
$875$	0	0
$876$	0.928932	0.0313857
$877$	−37.6569	−1.27158	−0.635791	−	0.771861i	$-0.719326\pi$
$877$	−37.6569	−1.27158	−0.635791	+	0.771861i	$0.719326\pi$
$878$	−4.41421	−0.148972
$879$	11.3137	0.381602
$880$	0	0
$881$	−3.62742	−0.122211	−0.0611054	−	0.998131i	$-0.519463\pi$
$881$	−3.62742	−0.122211	−0.0611054	+	0.998131i	$0.519463\pi$
$882$	0	0
$883$	33.5269	1.12827	0.564135	−	0.825682i	$-0.309210\pi$
$883$	33.5269	1.12827	0.564135	+	0.825682i	$0.309210\pi$
$884$	0.514719	0.0173119
$885$	0	0
$886$	25.6569	0.861959
$887$	−34.2426	−1.14976	−0.574878	−	0.818239i	$-0.694950\pi$
$887$	−34.2426	−1.14976	−0.574878	+	0.818239i	$0.694950\pi$
$888$	7.07107	0.237289
$889$	0	0
$890$	0	0
$891$	0.585786	0.0196246
$892$	−6.55635	−0.219523
$893$	−13.6569	−0.457009
$894$	6.65685	0.222639
$895$	0	0
$896$	0	0
$897$	7.24264	0.241825
$898$	10.9289	0.364703
$899$	0.414214	0.0138148
$900$	0	0
$901$	0.230447	0.00767732
$902$	0.100505	0.00334645
$903$	0	0
$904$	−9.41421	−0.313112
$905$	0	0
$906$	11.8995	0.395334
$907$	−18.6985	−0.620873	−0.310436	−	0.950594i	$-0.600475\pi$
$907$	−18.6985	−0.620873	−0.310436	+	0.950594i	$0.600475\pi$
$908$	−11.3848	−0.377817
$909$	−5.65685	−0.187626
$910$	0	0
$911$	7.92893	0.262697	0.131349	−	0.991336i	$-0.458069\pi$
$911$	7.92893	0.262697	0.131349	+	0.991336i	$0.458069\pi$
$912$	−1.41421	−0.0468293
$913$	8.24264	0.272792
$914$	−1.82843	−0.0604790
$915$	0	0
$916$	6.68629	0.220921
$917$	0	0
$918$	−0.171573	−0.00566275
$919$	35.5563	1.17290	0.586448	−	0.809987i	$-0.300526\pi$
$919$	35.5563	1.17290	0.586448	+	0.809987i	$0.300526\pi$
$920$	0	0
$921$	11.4142	0.376111
$922$	−20.0000	−0.658665
$923$	39.9411	1.31468
$924$	0	0
$925$	0	0
$926$	−10.1005	−0.331923
$927$	14.0711	0.462155
$928$	−1.00000	−0.0328266
$929$	2.65685	0.0871686	0.0435843	−	0.999050i	$-0.486122\pi$
$929$	2.65685	0.0871686	0.0435843	+	0.999050i	$0.486122\pi$
$930$	0	0
$931$	0	0
$932$	5.75736	0.188589
$933$	−8.58579	−0.281086
$934$	−7.24264	−0.236986
$935$	0	0
$936$	−3.00000	−0.0980581
$937$	−8.24264	−0.269275	−0.134638	−	0.990895i	$-0.542987\pi$
$937$	−8.24264	−0.269275	−0.134638	+	0.990895i	$0.542987\pi$
$938$	0	0
$939$	32.0416	1.04564
$940$	0	0
$941$	28.6863	0.935146	0.467573	−	0.883954i	$-0.345128\pi$
$941$	28.6863	0.935146	0.467573	+	0.883954i	$0.345128\pi$
$942$	−19.7990	−0.645086
$943$	−0.414214	−0.0134886
$944$	−1.24264	−0.0404445
$945$	0	0
$946$	−2.58579	−0.0840712
$947$	17.2132	0.559354	0.279677	−	0.960094i	$-0.409773\pi$
$947$	17.2132	0.559354	0.279677	+	0.960094i	$0.409773\pi$
$948$	−13.0711	−0.424529
$949$	2.78680	0.0904632
$950$	0	0
$951$	−13.9706	−0.453027
$952$	0	0
$953$	47.5980	1.54185	0.770925	−	0.636926i	$-0.219794\pi$
$953$	47.5980	1.54185	0.770925	+	0.636926i	$0.219794\pi$
$954$	−1.34315	−0.0434859
$955$	0	0
$956$	11.3137	0.365911
$957$	0.585786	0.0189358
$958$	−11.7574	−0.379863
$959$	0	0
$960$	0	0
$961$	−30.8284	−0.994465
$962$	21.2132	0.683941
$963$	−10.4853	−0.337883
$964$	−6.00000	−0.193247
$965$	0	0
$966$	0	0
$967$	4.04163	0.129970	0.0649850	−	0.997886i	$-0.479300\pi$
$967$	4.04163	0.129970	0.0649850	+	0.997886i	$0.479300\pi$
$968$	10.6569	0.342524
$969$	−0.242641	−0.00779474
$970$	0	0
$971$	−11.0294	−0.353951	−0.176976	−	0.984215i	$-0.556631\pi$
$971$	−11.0294	−0.353951	−0.176976	+	0.984215i	$0.556631\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−22.4853	−0.720475
$975$	0	0
$976$	−5.82843	−0.186563
$977$	−39.5563	−1.26552	−0.632760	−	0.774348i	$-0.718078\pi$
$977$	−39.5563	−1.26552	−0.632760	+	0.774348i	$0.718078\pi$
$978$	3.24264	0.103688
$979$	2.34315	0.0748873
$980$	0	0
$981$	11.3137	0.361219
$982$	−28.1421	−0.898052
$983$	51.3137	1.63665	0.818327	−	0.574754i	$-0.194902\pi$
$983$	51.3137	1.63665	0.818327	+	0.574754i	$0.194902\pi$
$984$	0.171573	0.00546954
$985$	0	0
$986$	−0.171573	−0.00546399
$987$	0	0
$988$	−4.24264	−0.134976
$989$	10.6569	0.338868
$990$	0	0
$991$	−29.6569	−0.942081	−0.471041	−	0.882112i	$-0.656121\pi$
$991$	−29.6569	−0.942081	−0.471041	+	0.882112i	$0.656121\pi$
$992$	−0.414214	−0.0131513
$993$	−9.92893	−0.315085
$994$	0	0
$995$	0	0
$996$	14.0711	0.445859
$997$	−2.68629	−0.0850757	−0.0425379	−	0.999095i	$-0.513544\pi$
$997$	−2.68629	−0.0850757	−0.0425379	+	0.999095i	$0.513544\pi$
$998$	11.9289	0.377604
$999$	−7.07107	−0.223719

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7350.2.a.dg.1.1	yes	2
5.4	even	2		7350.2.a.dj.1.1	yes	2
7.6	odd	2		7350.2.a.dc.1.1	✓	2
35.34	odd	2		7350.2.a.dm.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7350.2.a.dc.1.1	✓	2	7.6	odd	2
7350.2.a.dg.1.1	yes	2	1.1	even	1	trivial
7350.2.a.dj.1.1	yes	2	5.4	even	2
7350.2.a.dm.1.1	yes	2	35.34	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +0.585786 q^{11} +1.00000 q^{12} +3.00000 q^{13} +1.00000 q^{16} +0.171573 q^{17} -1.00000 q^{18} -1.41421 q^{19} -0.585786 q^{22} +2.41421 q^{23} -1.00000 q^{24} -3.00000 q^{26} +1.00000 q^{27} +1.00000 q^{29} +0.414214 q^{31} -1.00000 q^{32} +0.585786 q^{33} -0.171573 q^{34} +1.00000 q^{36} -7.07107 q^{37} +1.41421 q^{38} +3.00000 q^{39} -0.171573 q^{41} +4.41421 q^{43} +0.585786 q^{44} -2.41421 q^{46} +9.65685 q^{47} +1.00000 q^{48} +0.171573 q^{51} +3.00000 q^{52} +1.34315 q^{53} -1.00000 q^{54} -1.41421 q^{57} -1.00000 q^{58} -1.24264 q^{59} -5.82843 q^{61} -0.414214 q^{62} +1.00000 q^{64} -0.585786 q^{66} +7.31371 q^{67} +0.171573 q^{68} +2.41421 q^{69} +13.3137 q^{71} -1.00000 q^{72} +0.928932 q^{73} +7.07107 q^{74} -1.41421 q^{76} -3.00000 q^{78} -13.0711 q^{79} +1.00000 q^{81} +0.171573 q^{82} +14.0711 q^{83} -4.41421 q^{86} +1.00000 q^{87} -0.585786 q^{88} +4.00000 q^{89} +2.41421 q^{92} +0.414214 q^{93} -9.65685 q^{94} -1.00000 q^{96} -11.6569 q^{97} +0.585786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9} + 4 q^{11} + 2 q^{12} + 6 q^{13} + 2 q^{16} + 6 q^{17} - 2 q^{18} - 4 q^{22} + 2 q^{23} - 2 q^{24} - 6 q^{26} + 2 q^{27} + 2 q^{29} - 2 q^{31} - 2 q^{32} + 4 q^{33} - 6 q^{34} + 2 q^{36} + 6 q^{39} - 6 q^{41} + 6 q^{43} + 4 q^{44} - 2 q^{46} + 8 q^{47} + 2 q^{48} + 6 q^{51} + 6 q^{52} + 14 q^{53} - 2 q^{54} - 2 q^{58} + 6 q^{59} - 6 q^{61} + 2 q^{62} + 2 q^{64} - 4 q^{66} - 8 q^{67} + 6 q^{68} + 2 q^{69} + 4 q^{71} - 2 q^{72} + 16 q^{73} - 6 q^{78} - 12 q^{79} + 2 q^{81} + 6 q^{82} + 14 q^{83} - 6 q^{86} + 2 q^{87} - 4 q^{88} + 8 q^{89} + 2 q^{92} - 2 q^{93} - 8 q^{94} - 2 q^{96} - 12 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^8 + 2 * q^9 + 4 * q^11 + 2 * q^12 + 6 * q^13 + 2 * q^16 + 6 * q^17 - 2 * q^18 - 4 * q^22 + 2 * q^23 - 2 * q^24 - 6 * q^26 + 2 * q^27 + 2 * q^29 - 2 * q^31 - 2 * q^32 + 4 * q^33 - 6 * q^34 + 2 * q^36 + 6 * q^39 - 6 * q^41 + 6 * q^43 + 4 * q^44 - 2 * q^46 + 8 * q^47 + 2 * q^48 + 6 * q^51 + 6 * q^52 + 14 * q^53 - 2 * q^54 - 2 * q^58 + 6 * q^59 - 6 * q^61 + 2 * q^62 + 2 * q^64 - 4 * q^66 - 8 * q^67 + 6 * q^68 + 2 * q^69 + 4 * q^71 - 2 * q^72 + 16 * q^73 - 6 * q^78 - 12 * q^79 + 2 * q^81 + 6 * q^82 + 14 * q^83 - 6 * q^86 + 2 * q^87 - 4 * q^88 + 8 * q^89 + 2 * q^92 - 2 * q^93 - 8 * q^94 - 2 * q^96 - 12 * q^97 + 4 * q^99

Introduction

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