LMFDB - Embedded newform 7623.2.a.cy.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7623,2,Mod(1,7623)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7623, 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("7623.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7623 = 3^{2} \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7623.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:	$60.8699614608$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 19x^{8} - x^{7} + 124x^{6} + 6x^{5} - 316x^{4} + 17x^{3} + 253x^{2} - 70x - 11 $ x^10 - 19x^8 - x^7 + 124x^6 + 6x^5 - 316x^4 + 17x^3 + 253x^2 - 70*x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.09767$ of defining polynomial
Character	$\chi$	$=$	7623.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.09767 q^{2} +2.40021 q^{4} -3.15947 q^{5} +1.00000 q^{7} -0.839503 q^{8} +O(q^{10})$	$q-2.09767 q^{2} +2.40021 q^{4} -3.15947 q^{5} +1.00000 q^{7} -0.839503 q^{8} +6.62751 q^{10} -2.12862 q^{13} -2.09767 q^{14} -3.03942 q^{16} -4.79430 q^{17} -1.53847 q^{19} -7.58338 q^{20} -5.11509 q^{23} +4.98224 q^{25} +4.46514 q^{26} +2.40021 q^{28} -0.958246 q^{29} -7.22037 q^{31} +8.05469 q^{32} +10.0569 q^{34} -3.15947 q^{35} +2.39901 q^{37} +3.22720 q^{38} +2.65238 q^{40} +0.266914 q^{41} -9.28678 q^{43} +10.7298 q^{46} -10.3245 q^{47} +1.00000 q^{49} -10.4511 q^{50} -5.10914 q^{52} +0.945600 q^{53} -0.839503 q^{56} +2.01008 q^{58} +9.55077 q^{59} -8.55756 q^{61} +15.1459 q^{62} -10.8172 q^{64} +6.72532 q^{65} +13.2710 q^{67} -11.5073 q^{68} +6.62751 q^{70} -14.1564 q^{71} -7.72198 q^{73} -5.03232 q^{74} -3.69265 q^{76} -9.31226 q^{79} +9.60295 q^{80} -0.559897 q^{82} -16.8536 q^{83} +15.1475 q^{85} +19.4806 q^{86} +11.2670 q^{89} -2.12862 q^{91} -12.2773 q^{92} +21.6574 q^{94} +4.86075 q^{95} +7.08789 q^{97} -2.09767 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 18 q^{4} - 5 q^{5} + 10 q^{7} + 3 q^{8}+O(q^{10}) $ 10 * q + 18 * q^4 - 5 * q^5 + 10 * q^7 + 3 * q^8	$ 10 q + 18 q^{4} - 5 q^{5} + 10 q^{7} + 3 q^{8} - 6 q^{10} + 6 q^{13} + 38 q^{16} - 8 q^{17} - 7 q^{20} + 31 q^{25} - q^{26} + 18 q^{28} + 14 q^{29} + 26 q^{31} + 41 q^{32} + 21 q^{34} - 5 q^{35} + 24 q^{37} - 8 q^{38} - 5 q^{40} - 19 q^{41} - 6 q^{43} - q^{46} - 15 q^{47} + 10 q^{49} + q^{50} - 25 q^{52} + q^{53} + 3 q^{56} + 11 q^{58} - 23 q^{59} - 11 q^{62} + 53 q^{64} + 29 q^{65} + 38 q^{67} - 87 q^{68} - 6 q^{70} - 26 q^{71} - q^{73} + 39 q^{74} - 2 q^{76} + 5 q^{79} - 6 q^{80} + 5 q^{82} - 6 q^{83} - q^{85} + 41 q^{86} + 9 q^{89} + 6 q^{91} + 48 q^{92} + 42 q^{95} + 24 q^{97}+O(q^{100}) $ 10 * q + 18 * q^4 - 5 * q^5 + 10 * q^7 + 3 * q^8 - 6 * q^10 + 6 * q^13 + 38 * q^16 - 8 * q^17 - 7 * q^20 + 31 * q^25 - q^26 + 18 * q^28 + 14 * q^29 + 26 * q^31 + 41 * q^32 + 21 * q^34 - 5 * q^35 + 24 * q^37 - 8 * q^38 - 5 * q^40 - 19 * q^41 - 6 * q^43 - q^46 - 15 * q^47 + 10 * q^49 + q^50 - 25 * q^52 + q^53 + 3 * q^56 + 11 * q^58 - 23 * q^59 - 11 * q^62 + 53 * q^64 + 29 * q^65 + 38 * q^67 - 87 * q^68 - 6 * q^70 - 26 * q^71 - q^73 + 39 * q^74 - 2 * q^76 + 5 * q^79 - 6 * q^80 + 5 * q^82 - 6 * q^83 - q^85 + 41 * q^86 + 9 * q^89 + 6 * q^91 + 48 * q^92 + 42 * q^95 + 24 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.09767	−1.48327	−0.741637	−	0.670801i	$-0.765950\pi$
$2$	−2.09767	−1.48327	−0.741637	+	0.670801i	$0.765950\pi$
$3$	0	0
$3$	0	0
$4$	2.40021	1.20010
$5$	−3.15947	−1.41296	−0.706479	−	0.707734i	$-0.749717\pi$
$5$	−3.15947	−1.41296	−0.706479	+	0.707734i	$0.749717\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−0.839503	−0.296809
$9$	0	0
$10$	6.62751	2.09580
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−2.12862	−0.590374	−0.295187	−	0.955440i	$-0.595382\pi$
$13$	−2.12862	−0.590374	−0.295187	+	0.955440i	$0.595382\pi$
$14$	−2.09767	−0.560625
$15$	0	0
$16$	−3.03942	−0.759855
$17$	−4.79430	−1.16279	−0.581395	−	0.813622i	$-0.697493\pi$
$17$	−4.79430	−1.16279	−0.581395	+	0.813622i	$0.697493\pi$
$18$	0	0
$19$	−1.53847	−0.352949	−0.176475	−	0.984305i	$-0.556469\pi$
$19$	−1.53847	−0.352949	−0.176475	+	0.984305i	$0.556469\pi$
$20$	−7.58338	−1.69570
$21$	0	0
$22$	0	0
$23$	−5.11509	−1.06657	−0.533285	−	0.845935i	$-0.679043\pi$
$23$	−5.11509	−1.06657	−0.533285	+	0.845935i	$0.679043\pi$
$24$	0	0
$25$	4.98224	0.996449
$26$	4.46514	0.875687
$27$	0	0
$28$	2.40021	0.453597
$29$	−0.958246	−0.177942	−0.0889709	−	0.996034i	$-0.528358\pi$
$29$	−0.958246	−0.177942	−0.0889709	+	0.996034i	$0.528358\pi$
$30$	0	0
$31$	−7.22037	−1.29682	−0.648409	−	0.761292i	$-0.724565\pi$
$31$	−7.22037	−1.29682	−0.648409	+	0.761292i	$0.724565\pi$
$32$	8.05469	1.42388
$33$	0	0
$34$	10.0569	1.72474
$35$	−3.15947	−0.534048
$36$	0	0
$37$	2.39901	0.394394	0.197197	−	0.980364i	$-0.436816\pi$
$37$	2.39901	0.394394	0.197197	+	0.980364i	$0.436816\pi$
$38$	3.22720	0.523521
$39$	0	0
$40$	2.65238	0.419378
$41$	0.266914	0.0416850	0.0208425	−	0.999783i	$-0.493365\pi$
$41$	0.266914	0.0416850	0.0208425	+	0.999783i	$0.493365\pi$
$42$	0	0
$43$	−9.28678	−1.41622	−0.708110	−	0.706102i	$-0.750452\pi$
$43$	−9.28678	−1.41622	−0.708110	+	0.706102i	$0.750452\pi$
$44$	0	0
$45$	0	0
$46$	10.7298	1.58202
$47$	−10.3245	−1.50599	−0.752993	−	0.658029i	$-0.771391\pi$
$47$	−10.3245	−1.50599	−0.752993	+	0.658029i	$0.771391\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−10.4511	−1.47801
$51$	0	0
$52$	−5.10914	−0.708510
$53$	0.945600	0.129888	0.0649441	−	0.997889i	$-0.479313\pi$
$53$	0.945600	0.129888	0.0649441	+	0.997889i	$0.479313\pi$
$54$	0	0
$55$	0	0
$56$	−0.839503	−0.112183
$57$	0	0
$58$	2.01008	0.263937
$59$	9.55077	1.24340	0.621702	−	0.783254i	$-0.286442\pi$
$59$	9.55077	1.24340	0.621702	+	0.783254i	$0.286442\pi$
$60$	0	0
$61$	−8.55756	−1.09568	−0.547841	−	0.836582i	$-0.684550\pi$
$61$	−8.55756	−1.09568	−0.547841	+	0.836582i	$0.684550\pi$
$62$	15.1459	1.92354
$63$	0	0
$64$	−10.8172	−1.35215
$65$	6.72532	0.834173
$66$	0	0
$67$	13.2710	1.62131	0.810657	−	0.585521i	$-0.199110\pi$
$67$	13.2710	1.62131	0.810657	+	0.585521i	$0.199110\pi$
$68$	−11.5073	−1.39547
$69$	0	0
$70$	6.62751	0.792139
$71$	−14.1564	−1.68006	−0.840030	−	0.542540i	$-0.817463\pi$
$71$	−14.1564	−1.68006	−0.840030	+	0.542540i	$0.817463\pi$
$72$	0	0
$73$	−7.72198	−0.903789	−0.451895	−	0.892071i	$-0.649252\pi$
$73$	−7.72198	−0.903789	−0.451895	+	0.892071i	$0.649252\pi$
$74$	−5.03232	−0.584995
$75$	0	0
$76$	−3.69265	−0.423576
$77$	0	0
$78$	0	0
$79$	−9.31226	−1.04771	−0.523856	−	0.851807i	$-0.675507\pi$
$79$	−9.31226	−1.04771	−0.523856	+	0.851807i	$0.675507\pi$
$80$	9.60295	1.07364
$81$	0	0
$82$	−0.559897	−0.0618302
$83$	−16.8536	−1.84992	−0.924961	−	0.380061i	$-0.875903\pi$
$83$	−16.8536	−1.84992	−0.924961	+	0.380061i	$0.875903\pi$
$84$	0	0
$85$	15.1475	1.64297
$86$	19.4806	2.10064
$87$	0	0
$88$	0	0
$89$	11.2670	1.19430	0.597150	−	0.802130i	$-0.296300\pi$
$89$	11.2670	1.19430	0.597150	+	0.802130i	$0.296300\pi$
$90$	0	0
$91$	−2.12862	−0.223140
$92$	−12.2773	−1.28000
$93$	0	0
$94$	21.6574	2.23379
$95$	4.86075	0.498702
$96$	0	0
$97$	7.08789	0.719666	0.359833	−	0.933017i	$-0.382834\pi$
$97$	7.08789	0.719666	0.359833	+	0.933017i	$0.382834\pi$
$98$	−2.09767	−0.211896
$99$	0	0
$100$	11.9584	1.19584
$101$	8.34609	0.830467	0.415234	−	0.909715i	$-0.363700\pi$
$101$	8.34609	0.830467	0.415234	+	0.909715i	$0.363700\pi$
$102$	0	0
$103$	−7.53555	−0.742500	−0.371250	−	0.928533i	$-0.621071\pi$
$103$	−7.53555	−0.742500	−0.371250	+	0.928533i	$0.621071\pi$
$104$	1.78698	0.175228
$105$	0	0
$106$	−1.98355	−0.192660
$107$	15.3887	1.48768	0.743841	−	0.668357i	$-0.233002\pi$
$107$	15.3887	1.48768	0.743841	+	0.668357i	$0.233002\pi$
$108$	0	0
$109$	−20.2581	−1.94038	−0.970188	−	0.242353i	$-0.922081\pi$
$109$	−20.2581	−1.94038	−0.970188	+	0.242353i	$0.922081\pi$
$110$	0	0
$111$	0	0
$112$	−3.03942	−0.287198
$113$	18.1019	1.70288	0.851440	−	0.524452i	$-0.175730\pi$
$113$	18.1019	1.70288	0.851440	+	0.524452i	$0.175730\pi$
$114$	0	0
$115$	16.1610	1.50702
$116$	−2.29999	−0.213549
$117$	0	0
$118$	−20.0343	−1.84431
$119$	−4.79430	−0.439493
$120$	0	0
$121$	0	0
$122$	17.9509	1.62520
$123$	0	0
$124$	−17.3304	−1.55632
$125$	0.0560984	0.00501759
$126$	0	0
$127$	15.3711	1.36396	0.681981	−	0.731369i	$-0.261118\pi$
$127$	15.3711	1.36396	0.681981	+	0.731369i	$0.261118\pi$
$128$	6.58156	0.581733
$129$	0	0
$130$	−14.1075	−1.23731
$131$	−10.7418	−0.938516	−0.469258	−	0.883061i	$-0.655478\pi$
$131$	−10.7418	−0.938516	−0.469258	+	0.883061i	$0.655478\pi$
$132$	0	0
$133$	−1.53847	−0.133402
$134$	−27.8382	−2.40485
$135$	0	0
$136$	4.02483	0.345126
$137$	4.42612	0.378149	0.189075	−	0.981963i	$-0.439451\pi$
$137$	4.42612	0.378149	0.189075	+	0.981963i	$0.439451\pi$
$138$	0	0
$139$	−4.52505	−0.383810	−0.191905	−	0.981414i	$-0.561467\pi$
$139$	−4.52505	−0.383810	−0.191905	+	0.981414i	$0.561467\pi$
$140$	−7.58338	−0.640913
$141$	0	0
$142$	29.6955	2.49199
$143$	0	0
$144$	0	0
$145$	3.02755	0.251424
$146$	16.1981	1.34057
$147$	0	0
$148$	5.75812	0.473314
$149$	2.99471	0.245336	0.122668	−	0.992448i	$-0.460855\pi$
$149$	2.99471	0.245336	0.122668	+	0.992448i	$0.460855\pi$
$150$	0	0
$151$	−8.81260	−0.717159	−0.358579	−	0.933499i	$-0.616739\pi$
$151$	−8.81260	−0.717159	−0.358579	+	0.933499i	$0.616739\pi$
$152$	1.29155	0.104759
$153$	0	0
$154$	0	0
$155$	22.8125	1.83235
$156$	0	0
$157$	1.56655	0.125024	0.0625122	−	0.998044i	$-0.480089\pi$
$157$	1.56655	0.125024	0.0625122	+	0.998044i	$0.480089\pi$
$158$	19.5340	1.55404
$159$	0	0
$160$	−25.4486	−2.01188
$161$	−5.11509	−0.403126
$162$	0	0
$163$	−6.80568	−0.533062	−0.266531	−	0.963826i	$-0.585877\pi$
$163$	−6.80568	−0.533062	−0.266531	+	0.963826i	$0.585877\pi$
$164$	0.640649	0.0500263
$165$	0	0
$166$	35.3532	2.74394
$167$	−12.2497	−0.947913	−0.473957	−	0.880548i	$-0.657175\pi$
$167$	−12.2497	−0.947913	−0.473957	+	0.880548i	$0.657175\pi$
$168$	0	0
$169$	−8.46896	−0.651459
$170$	−31.7743	−2.43698
$171$	0	0
$172$	−22.2902	−1.69961
$173$	−19.8500	−1.50917	−0.754583	−	0.656204i	$-0.772161\pi$
$173$	−19.8500	−1.50917	−0.754583	+	0.656204i	$0.772161\pi$
$174$	0	0
$175$	4.98224	0.376622
$176$	0	0
$177$	0	0
$178$	−23.6344	−1.77147
$179$	−8.22231	−0.614564	−0.307282	−	0.951618i	$-0.599420\pi$
$179$	−8.22231	−0.614564	−0.307282	+	0.951618i	$0.599420\pi$
$180$	0	0
$181$	10.4317	0.775381	0.387691	−	0.921790i	$-0.373273\pi$
$181$	10.4317	0.775381	0.387691	+	0.921790i	$0.373273\pi$
$182$	4.46514	0.330978
$183$	0	0
$184$	4.29413	0.316568
$185$	−7.57959	−0.557263
$186$	0	0
$187$	0	0
$188$	−24.7810	−1.80734
$189$	0	0
$190$	−10.1962	−0.739713
$191$	−14.2162	−1.02865	−0.514325	−	0.857595i	$-0.671957\pi$
$191$	−14.2162	−1.02865	−0.514325	+	0.857595i	$0.671957\pi$
$192$	0	0
$193$	7.01633	0.505046	0.252523	−	0.967591i	$-0.418740\pi$
$193$	7.01633	0.505046	0.252523	+	0.967591i	$0.418740\pi$
$194$	−14.8680	−1.06746
$195$	0	0
$196$	2.40021	0.171443
$197$	−3.60302	−0.256705	−0.128352	−	0.991729i	$-0.540969\pi$
$197$	−3.60302	−0.256705	−0.128352	+	0.991729i	$0.540969\pi$
$198$	0	0
$199$	−3.85136	−0.273015	−0.136508	−	0.990639i	$-0.543588\pi$
$199$	−3.85136	−0.273015	−0.136508	+	0.990639i	$0.543588\pi$
$200$	−4.18261	−0.295755
$201$	0	0
$202$	−17.5073	−1.23181
$203$	−0.958246	−0.0672557
$204$	0	0
$205$	−0.843306	−0.0588991
$206$	15.8071	1.10133
$207$	0	0
$208$	6.46978	0.448598
$209$	0	0
$210$	0	0
$211$	−11.1325	−0.766391	−0.383196	−	0.923667i	$-0.625176\pi$
$211$	−11.1325	−0.766391	−0.383196	+	0.923667i	$0.625176\pi$
$212$	2.26964	0.155879
$213$	0	0
$214$	−32.2804	−2.20664
$215$	29.3413	2.00106
$216$	0	0
$217$	−7.22037	−0.490151
$218$	42.4948	2.87811
$219$	0	0
$220$	0	0
$221$	10.2053	0.686481
$222$	0	0
$223$	−5.27119	−0.352985	−0.176493	−	0.984302i	$-0.556475\pi$
$223$	−5.27119	−0.352985	−0.176493	+	0.984302i	$0.556475\pi$
$224$	8.05469	0.538177
$225$	0	0
$226$	−37.9717	−2.52584
$227$	−14.8204	−0.983661	−0.491831	−	0.870691i	$-0.663672\pi$
$227$	−14.8204	−0.983661	−0.491831	+	0.870691i	$0.663672\pi$
$228$	0	0
$229$	7.60478	0.502538	0.251269	−	0.967917i	$-0.419152\pi$
$229$	7.60478	0.502538	0.251269	+	0.967917i	$0.419152\pi$
$230$	−33.9004	−2.23532
$231$	0	0
$232$	0.804450	0.0528147
$233$	−1.50544	−0.0986248	−0.0493124	−	0.998783i	$-0.515703\pi$
$233$	−1.50544	−0.0986248	−0.0493124	+	0.998783i	$0.515703\pi$
$234$	0	0
$235$	32.6200	2.12789
$236$	22.9238	1.49221
$237$	0	0
$238$	10.0569	0.651889
$239$	−5.02336	−0.324934	−0.162467	−	0.986714i	$-0.551945\pi$
$239$	−5.02336	−0.324934	−0.162467	+	0.986714i	$0.551945\pi$
$240$	0	0
$241$	−16.1502	−1.04033	−0.520164	−	0.854066i	$-0.674129\pi$
$241$	−16.1502	−1.04033	−0.520164	+	0.854066i	$0.674129\pi$
$242$	0	0
$243$	0	0
$244$	−20.5399	−1.31493
$245$	−3.15947	−0.201851
$246$	0	0
$247$	3.27482	0.208372
$248$	6.06152	0.384907
$249$	0	0
$250$	−0.117676	−0.00744246
$251$	−25.7765	−1.62700	−0.813500	−	0.581565i	$-0.802440\pi$
$251$	−25.7765	−1.62700	−0.813500	+	0.581565i	$0.802440\pi$
$252$	0	0
$253$	0	0
$254$	−32.2434	−2.02313
$255$	0	0
$256$	7.82853	0.489283
$257$	−17.6678	−1.10208	−0.551042	−	0.834477i	$-0.685770\pi$
$257$	−17.6678	−1.10208	−0.551042	+	0.834477i	$0.685770\pi$
$258$	0	0
$259$	2.39901	0.149067
$260$	16.1422	1.00109
$261$	0	0
$262$	22.5327	1.39208
$263$	18.6325	1.14893	0.574465	−	0.818529i	$-0.305210\pi$
$263$	18.6325	1.14893	0.574465	+	0.818529i	$0.305210\pi$
$264$	0	0
$265$	−2.98759	−0.183526
$266$	3.22720	0.197872
$267$	0	0
$268$	31.8532	1.94575
$269$	20.7126	1.26287	0.631435	−	0.775429i	$-0.282466\pi$
$269$	20.7126	1.26287	0.631435	+	0.775429i	$0.282466\pi$
$270$	0	0
$271$	−14.8438	−0.901694	−0.450847	−	0.892601i	$-0.648878\pi$
$271$	−14.8438	−0.901694	−0.450847	+	0.892601i	$0.648878\pi$
$272$	14.5719	0.883551
$273$	0	0
$274$	−9.28453	−0.560899
$275$	0	0
$276$	0	0
$277$	3.07309	0.184644	0.0923221	−	0.995729i	$-0.470571\pi$
$277$	3.07309	0.184644	0.0923221	+	0.995729i	$0.470571\pi$
$278$	9.49206	0.569296
$279$	0	0
$280$	2.65238	0.158510
$281$	4.09698	0.244405	0.122203	−	0.992505i	$-0.461004\pi$
$281$	4.09698	0.244405	0.122203	+	0.992505i	$0.461004\pi$
$282$	0	0
$283$	7.39800	0.439766	0.219883	−	0.975526i	$-0.429432\pi$
$283$	7.39800	0.439766	0.219883	+	0.975526i	$0.429432\pi$
$284$	−33.9784	−2.01625
$285$	0	0
$286$	0	0
$287$	0.266914	0.0157554
$288$	0	0
$289$	5.98536	0.352080
$290$	−6.35079	−0.372931
$291$	0	0
$292$	−18.5344	−1.08464
$293$	−10.5489	−0.616273	−0.308137	−	0.951342i	$-0.599705\pi$
$293$	−10.5489	−0.616273	−0.308137	+	0.951342i	$0.599705\pi$
$294$	0	0
$295$	−30.1754	−1.75688
$296$	−2.01397	−0.117060
$297$	0	0
$298$	−6.28190	−0.363901
$299$	10.8881	0.629675
$300$	0	0
$301$	−9.28678	−0.535281
$302$	18.4859	1.06374
$303$	0	0
$304$	4.67605	0.268190
$305$	27.0373	1.54815
$306$	0	0
$307$	−1.74955	−0.0998522	−0.0499261	−	0.998753i	$-0.515899\pi$
$307$	−1.74955	−0.0998522	−0.0499261	+	0.998753i	$0.515899\pi$
$308$	0	0
$309$	0	0
$310$	−47.8531	−2.71788
$311$	23.0075	1.30463	0.652317	−	0.757946i	$-0.273797\pi$
$311$	23.0075	1.30463	0.652317	+	0.757946i	$0.273797\pi$
$312$	0	0
$313$	18.2568	1.03193	0.515967	−	0.856608i	$-0.327433\pi$
$313$	18.2568	1.03193	0.515967	+	0.856608i	$0.327433\pi$
$314$	−3.28610	−0.185446
$315$	0	0
$316$	−22.3514	−1.25736
$317$	−32.6206	−1.83216	−0.916079	−	0.400999i	$-0.868663\pi$
$317$	−32.6206	−1.83216	−0.916079	+	0.400999i	$0.868663\pi$
$318$	0	0
$319$	0	0
$320$	34.1767	1.91054
$321$	0	0
$322$	10.7298	0.597946
$323$	7.37589	0.410406
$324$	0	0
$325$	−10.6053	−0.588277
$326$	14.2760	0.790677
$327$	0	0
$328$	−0.224075	−0.0123725
$329$	−10.3245	−0.569209
$330$	0	0
$331$	6.80537	0.374057	0.187029	−	0.982354i	$-0.440114\pi$
$331$	6.80537	0.374057	0.187029	+	0.982354i	$0.440114\pi$
$332$	−40.4521	−2.22010
$333$	0	0
$334$	25.6959	1.40602
$335$	−41.9294	−2.29085
$336$	0	0
$337$	−0.372041	−0.0202664	−0.0101332	−	0.999949i	$-0.503226\pi$
$337$	−0.372041	−0.0202664	−0.0101332	+	0.999949i	$0.503226\pi$
$338$	17.7651	0.966292
$339$	0	0
$340$	36.3570	1.97174
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	7.79627	0.420347
$345$	0	0
$346$	41.6387	2.23851
$347$	5.75075	0.308716	0.154358	−	0.988015i	$-0.450669\pi$
$347$	5.75075	0.308716	0.154358	+	0.988015i	$0.450669\pi$
$348$	0	0
$349$	0.646523	0.0346076	0.0173038	−	0.999850i	$-0.494492\pi$
$349$	0.646523	0.0346076	0.0173038	+	0.999850i	$0.494492\pi$
$350$	−10.4511	−0.558634
$351$	0	0
$352$	0	0
$353$	18.3601	0.977209	0.488604	−	0.872505i	$-0.337506\pi$
$353$	18.3601	0.977209	0.488604	+	0.872505i	$0.337506\pi$
$354$	0	0
$355$	44.7268	2.37385
$356$	27.0431	1.43328
$357$	0	0
$358$	17.2477	0.911567
$359$	26.9753	1.42370	0.711852	−	0.702330i	$-0.247857\pi$
$359$	26.9753	1.42370	0.711852	+	0.702330i	$0.247857\pi$
$360$	0	0
$361$	−16.6331	−0.875427
$362$	−21.8822	−1.15010
$363$	0	0
$364$	−5.10914	−0.267792
$365$	24.3974	1.27702
$366$	0	0
$367$	28.3215	1.47837	0.739186	−	0.673501i	$-0.235210\pi$
$367$	28.3215	1.47837	0.739186	+	0.673501i	$0.235210\pi$
$368$	15.5469	0.810439
$369$	0	0
$370$	15.8995	0.826574
$371$	0.945600	0.0490931
$372$	0	0
$373$	12.9215	0.669050	0.334525	−	0.942387i	$-0.391424\pi$
$373$	12.9215	0.669050	0.334525	+	0.942387i	$0.391424\pi$
$374$	0	0
$375$	0	0
$376$	8.66746	0.446990
$377$	2.03974	0.105052
$378$	0	0
$379$	20.9480	1.07603	0.538014	−	0.842936i	$-0.319175\pi$
$379$	20.9480	1.07603	0.538014	+	0.842936i	$0.319175\pi$
$380$	11.6668	0.598495
$381$	0	0
$382$	29.8209	1.52577
$383$	−24.9764	−1.27623	−0.638117	−	0.769939i	$-0.720286\pi$
$383$	−24.9764	−1.27623	−0.638117	+	0.769939i	$0.720286\pi$
$384$	0	0
$385$	0	0
$386$	−14.7179	−0.749123
$387$	0	0
$388$	17.0124	0.863674
$389$	13.5780	0.688430	0.344215	−	0.938891i	$-0.388145\pi$
$389$	13.5780	0.688430	0.344215	+	0.938891i	$0.388145\pi$
$390$	0	0
$391$	24.5233	1.24020
$392$	−0.839503	−0.0424013
$393$	0	0
$394$	7.55794	0.380763
$395$	29.4218	1.48037
$396$	0	0
$397$	−4.26450	−0.214029	−0.107014	−	0.994257i	$-0.534129\pi$
$397$	−4.26450	−0.214029	−0.107014	+	0.994257i	$0.534129\pi$
$398$	8.07886	0.404957
$399$	0	0
$400$	−15.1431	−0.757156
$401$	32.4423	1.62009	0.810045	−	0.586367i	$-0.199442\pi$
$401$	32.4423	1.62009	0.810045	+	0.586367i	$0.199442\pi$
$402$	0	0
$403$	15.3695	0.765607
$404$	20.0324	0.996647
$405$	0	0
$406$	2.01008	0.0997587
$407$	0	0
$408$	0	0
$409$	24.8877	1.23062	0.615308	−	0.788287i	$-0.289032\pi$
$409$	24.8877	1.23062	0.615308	+	0.788287i	$0.289032\pi$
$410$	1.76898	0.0873635
$411$	0	0
$412$	−18.0869	−0.891077
$413$	9.55077	0.469963
$414$	0	0
$415$	53.2484	2.61386
$416$	−17.1454	−0.840623
$417$	0	0
$418$	0	0
$419$	−10.5948	−0.517588	−0.258794	−	0.965933i	$-0.583325\pi$
$419$	−10.5948	−0.517588	−0.258794	+	0.965933i	$0.583325\pi$
$420$	0	0
$421$	−37.8632	−1.84534	−0.922670	−	0.385591i	$-0.873998\pi$
$421$	−37.8632	−1.84534	−0.922670	+	0.385591i	$0.873998\pi$
$422$	23.3522	1.13677
$423$	0	0
$424$	−0.793833	−0.0385520
$425$	−23.8864	−1.15866
$426$	0	0
$427$	−8.55756	−0.414129
$428$	36.9361	1.78537
$429$	0	0
$430$	−61.5483	−2.96812
$431$	5.64489	0.271905	0.135952	−	0.990715i	$-0.456591\pi$
$431$	5.64489	0.271905	0.135952	+	0.990715i	$0.456591\pi$
$432$	0	0
$433$	−2.69528	−0.129527	−0.0647635	−	0.997901i	$-0.520629\pi$
$433$	−2.69528	−0.129527	−0.0647635	+	0.997901i	$0.520629\pi$
$434$	15.1459	0.727028
$435$	0	0
$436$	−48.6237	−2.32865
$437$	7.86942	0.376445
$438$	0	0
$439$	−30.5668	−1.45888	−0.729438	−	0.684047i	$-0.760218\pi$
$439$	−30.5668	−1.45888	−0.729438	+	0.684047i	$0.760218\pi$
$440$	0	0
$441$	0	0
$442$	−21.4073	−1.01824
$443$	24.6371	1.17054	0.585272	−	0.810837i	$-0.300988\pi$
$443$	24.6371	1.17054	0.585272	+	0.810837i	$0.300988\pi$
$444$	0	0
$445$	−35.5977	−1.68749
$446$	11.0572	0.523574
$447$	0	0
$448$	−10.8172	−0.511066
$449$	15.3755	0.725617	0.362808	−	0.931864i	$-0.381818\pi$
$449$	15.3755	0.725617	0.362808	+	0.931864i	$0.381818\pi$
$450$	0	0
$451$	0	0
$452$	43.4482	2.04363
$453$	0	0
$454$	31.0882	1.45904
$455$	6.72532	0.315288
$456$	0	0
$457$	−14.2362	−0.665943	−0.332972	−	0.942937i	$-0.608051\pi$
$457$	−14.2362	−0.665943	−0.332972	+	0.942937i	$0.608051\pi$
$458$	−15.9523	−0.745402
$459$	0	0
$460$	38.7897	1.80858
$461$	−36.7148	−1.70998	−0.854990	−	0.518645i	$-0.826437\pi$
$461$	−36.7148	−1.70998	−0.854990	+	0.518645i	$0.826437\pi$
$462$	0	0
$463$	−34.7682	−1.61582	−0.807908	−	0.589308i	$-0.799400\pi$
$463$	−34.7682	−1.61582	−0.807908	+	0.589308i	$0.799400\pi$
$464$	2.91251	0.135210
$465$	0	0
$466$	3.15792	0.146288
$467$	−13.5536	−0.627184	−0.313592	−	0.949558i	$-0.601532\pi$
$467$	−13.5536	−0.627184	−0.313592	+	0.949558i	$0.601532\pi$
$468$	0	0
$469$	13.2710	0.612799
$470$	−68.4259	−3.15625
$471$	0	0
$472$	−8.01789	−0.369053
$473$	0	0
$474$	0	0
$475$	−7.66503	−0.351696
$476$	−11.5073	−0.527437
$477$	0	0
$478$	10.5373	0.481967
$479$	27.5973	1.26095	0.630476	−	0.776208i	$-0.282860\pi$
$479$	27.5973	1.26095	0.630476	+	0.776208i	$0.282860\pi$
$480$	0	0
$481$	−5.10658	−0.232840
$482$	33.8778	1.54309
$483$	0	0
$484$	0	0
$485$	−22.3940	−1.01686
$486$	0	0
$487$	−20.4778	−0.927936	−0.463968	−	0.885852i	$-0.653575\pi$
$487$	−20.4778	−0.927936	−0.463968	+	0.885852i	$0.653575\pi$
$488$	7.18409	0.325208
$489$	0	0
$490$	6.62751	0.299401
$491$	28.8664	1.30272	0.651361	−	0.758768i	$-0.274198\pi$
$491$	28.8664	1.30272	0.651361	+	0.758768i	$0.274198\pi$
$492$	0	0
$493$	4.59412	0.206909
$494$	−6.86949	−0.309073
$495$	0	0
$496$	21.9457	0.985393
$497$	−14.1564	−0.635003
$498$	0	0
$499$	−21.4940	−0.962202	−0.481101	−	0.876665i	$-0.659763\pi$
$499$	−21.4940	−0.962202	−0.481101	+	0.876665i	$0.659763\pi$
$500$	0.134648	0.00602163
$501$	0	0
$502$	54.0706	2.41329
$503$	−8.60198	−0.383543	−0.191772	−	0.981440i	$-0.561423\pi$
$503$	−8.60198	−0.383543	−0.191772	+	0.981440i	$0.561423\pi$
$504$	0	0
$505$	−26.3692	−1.17342
$506$	0	0
$507$	0	0
$508$	36.8938	1.63690
$509$	−11.0437	−0.489505	−0.244752	−	0.969586i	$-0.578707\pi$
$509$	−11.0437	−0.489505	−0.244752	+	0.969586i	$0.578707\pi$
$510$	0	0
$511$	−7.72198	−0.341600
$512$	−29.5848	−1.30748
$513$	0	0
$514$	37.0611	1.63469
$515$	23.8083	1.04912
$516$	0	0
$517$	0	0
$518$	−5.03232	−0.221107
$519$	0	0
$520$	−5.64592	−0.247590
$521$	12.6215	0.552960	0.276480	−	0.961020i	$-0.410832\pi$
$521$	12.6215	0.552960	0.276480	+	0.961020i	$0.410832\pi$
$522$	0	0
$523$	43.9100	1.92005	0.960025	−	0.279913i	$-0.0903055\pi$
$523$	43.9100	1.92005	0.960025	+	0.279913i	$0.0903055\pi$
$524$	−25.7826	−1.12632
$525$	0	0
$526$	−39.0848	−1.70418
$527$	34.6167	1.50793
$528$	0	0
$529$	3.16418	0.137573
$530$	6.26698	0.272220
$531$	0	0
$532$	−3.69265	−0.160097
$533$	−0.568159	−0.0246097
$534$	0	0
$535$	−48.6201	−2.10203
$536$	−11.1411	−0.481221
$537$	0	0
$538$	−43.4482	−1.87318
$539$	0	0
$540$	0	0
$541$	27.3128	1.17427	0.587134	−	0.809490i	$-0.300256\pi$
$541$	27.3128	1.17427	0.587134	+	0.809490i	$0.300256\pi$
$542$	31.1373	1.33746
$543$	0	0
$544$	−38.6167	−1.65568
$545$	64.0049	2.74167
$546$	0	0
$547$	−18.3092	−0.782845	−0.391423	−	0.920211i	$-0.628017\pi$
$547$	−18.3092	−0.782845	−0.391423	+	0.920211i	$0.628017\pi$
$548$	10.6236	0.453818
$549$	0	0
$550$	0	0
$551$	1.47423	0.0628044
$552$	0	0
$553$	−9.31226	−0.395998
$554$	−6.44632	−0.273878
$555$	0	0
$556$	−10.8611	−0.460612
$557$	−3.82733	−0.162169	−0.0810845	−	0.996707i	$-0.525838\pi$
$557$	−3.82733	−0.162169	−0.0810845	+	0.996707i	$0.525838\pi$
$558$	0	0
$559$	19.7681	0.836099
$560$	9.60295	0.405799
$561$	0	0
$562$	−8.59411	−0.362520
$563$	−36.1468	−1.52340	−0.761702	−	0.647927i	$-0.775636\pi$
$563$	−36.1468	−1.52340	−0.761702	+	0.647927i	$0.775636\pi$
$564$	0	0
$565$	−57.1923	−2.40610
$566$	−15.5186	−0.652293
$567$	0	0
$568$	11.8844	0.498657
$569$	−9.55602	−0.400609	−0.200305	−	0.979734i	$-0.564193\pi$
$569$	−9.55602	−0.400609	−0.200305	+	0.979734i	$0.564193\pi$
$570$	0	0
$571$	31.8446	1.33265	0.666327	−	0.745659i	$-0.267865\pi$
$571$	31.8446	1.33265	0.666327	+	0.745659i	$0.267865\pi$
$572$	0	0
$573$	0	0
$574$	−0.559897	−0.0233696
$575$	−25.4846	−1.06278
$576$	0	0
$577$	8.65987	0.360515	0.180258	−	0.983619i	$-0.442307\pi$
$577$	8.65987	0.360515	0.180258	+	0.983619i	$0.442307\pi$
$578$	−12.5553	−0.522231
$579$	0	0
$580$	7.26674	0.301735
$581$	−16.8536	−0.699205
$582$	0	0
$583$	0	0
$584$	6.48262	0.268253
$585$	0	0
$586$	22.1281	0.914103
$587$	5.32917	0.219959	0.109979	−	0.993934i	$-0.464922\pi$
$587$	5.32917	0.219959	0.109979	+	0.993934i	$0.464922\pi$
$588$	0	0
$589$	11.1083	0.457711
$590$	63.2978	2.60593
$591$	0	0
$592$	−7.29159	−0.299682
$593$	11.0464	0.453622	0.226811	−	0.973939i	$-0.427170\pi$
$593$	11.0464	0.453622	0.226811	+	0.973939i	$0.427170\pi$
$594$	0	0
$595$	15.1475	0.620985
$596$	7.18792	0.294429
$597$	0	0
$598$	−22.8396	−0.933982
$599$	−26.2832	−1.07390	−0.536951	−	0.843614i	$-0.680424\pi$
$599$	−26.2832	−1.07390	−0.536951	+	0.843614i	$0.680424\pi$
$600$	0	0
$601$	−12.9669	−0.528929	−0.264465	−	0.964395i	$-0.585195\pi$
$601$	−12.9669	−0.528929	−0.264465	+	0.964395i	$0.585195\pi$
$602$	19.4806	0.793969
$603$	0	0
$604$	−21.1521	−0.860665
$605$	0	0
$606$	0	0
$607$	22.6090	0.917672	0.458836	−	0.888521i	$-0.348266\pi$
$607$	22.6090	0.917672	0.458836	+	0.888521i	$0.348266\pi$
$608$	−12.3919	−0.502558
$609$	0	0
$610$	−56.7153	−2.29634
$611$	21.9770	0.889095
$612$	0	0
$613$	22.1412	0.894276	0.447138	−	0.894465i	$-0.352443\pi$
$613$	22.1412	0.894276	0.447138	+	0.894465i	$0.352443\pi$
$614$	3.66998	0.148108
$615$	0	0
$616$	0	0
$617$	20.4747	0.824281	0.412140	−	0.911120i	$-0.364781\pi$
$617$	20.4747	0.824281	0.412140	+	0.911120i	$0.364781\pi$
$618$	0	0
$619$	40.1013	1.61181	0.805903	−	0.592047i	$-0.201680\pi$
$619$	40.1013	1.61181	0.805903	+	0.592047i	$0.201680\pi$
$620$	54.7549	2.19901
$621$	0	0
$622$	−48.2620	−1.93513
$623$	11.2670	0.451403
$624$	0	0
$625$	−25.0885	−1.00354
$626$	−38.2967	−1.53064
$627$	0	0
$628$	3.76005	0.150042
$629$	−11.5016	−0.458598
$630$	0	0
$631$	29.1072	1.15874	0.579370	−	0.815064i	$-0.303298\pi$
$631$	29.1072	1.15874	0.579370	+	0.815064i	$0.303298\pi$
$632$	7.81767	0.310970
$633$	0	0
$634$	68.4272	2.71759
$635$	−48.5645	−1.92722
$636$	0	0
$637$	−2.12862	−0.0843391
$638$	0	0
$639$	0	0
$640$	−20.7942	−0.821964
$641$	−3.75720	−0.148400	−0.0742001	−	0.997243i	$-0.523640\pi$
$641$	−3.75720	−0.148400	−0.0742001	+	0.997243i	$0.523640\pi$
$642$	0	0
$643$	−9.96068	−0.392811	−0.196405	−	0.980523i	$-0.562927\pi$
$643$	−9.96068	−0.392811	−0.196405	+	0.980523i	$0.562927\pi$
$644$	−12.2773	−0.483793
$645$	0	0
$646$	−15.4722	−0.608744
$647$	11.5017	0.452180	0.226090	−	0.974106i	$-0.427406\pi$
$647$	11.5017	0.452180	0.226090	+	0.974106i	$0.427406\pi$
$648$	0	0
$649$	0	0
$650$	22.2464	0.872577
$651$	0	0
$652$	−16.3350	−0.639729
$653$	5.08048	0.198815	0.0994073	−	0.995047i	$-0.468305\pi$
$653$	5.08048	0.198815	0.0994073	+	0.995047i	$0.468305\pi$
$654$	0	0
$655$	33.9384	1.32608
$656$	−0.811263	−0.0316745
$657$	0	0
$658$	21.6574	0.844293
$659$	−42.3777	−1.65080	−0.825401	−	0.564547i	$-0.809051\pi$
$659$	−42.3777	−1.65080	−0.825401	+	0.564547i	$0.809051\pi$
$660$	0	0
$661$	−12.7151	−0.494559	−0.247280	−	0.968944i	$-0.579537\pi$
$661$	−12.7151	−0.494559	−0.247280	+	0.968944i	$0.579537\pi$
$662$	−14.2754	−0.554829
$663$	0	0
$664$	14.1486	0.549074
$665$	4.86075	0.188492
$666$	0	0
$667$	4.90152	0.189788
$668$	−29.4019	−1.13759
$669$	0	0
$670$	87.9539	3.39796
$671$	0	0
$672$	0	0
$673$	28.0236	1.08023	0.540115	−	0.841591i	$-0.318381\pi$
$673$	28.0236	1.08023	0.540115	+	0.841591i	$0.318381\pi$
$674$	0.780419	0.0300606
$675$	0	0
$676$	−20.3273	−0.781818
$677$	12.3532	0.474773	0.237387	−	0.971415i	$-0.423709\pi$
$677$	12.3532	0.474773	0.237387	+	0.971415i	$0.423709\pi$
$678$	0	0
$679$	7.08789	0.272008
$680$	−12.7163	−0.487649
$681$	0	0
$682$	0	0
$683$	−4.43874	−0.169844	−0.0849219	−	0.996388i	$-0.527064\pi$
$683$	−4.43874	−0.169844	−0.0849219	+	0.996388i	$0.527064\pi$
$684$	0	0
$685$	−13.9842	−0.534309
$686$	−2.09767	−0.0800893
$687$	0	0
$688$	28.2264	1.07612
$689$	−2.01283	−0.0766825
$690$	0	0
$691$	33.1423	1.26079	0.630395	−	0.776274i	$-0.282893\pi$
$691$	33.1423	1.26079	0.630395	+	0.776274i	$0.282893\pi$
$692$	−47.6441	−1.81116
$693$	0	0
$694$	−12.0632	−0.457911
$695$	14.2968	0.542307
$696$	0	0
$697$	−1.27967	−0.0484708
$698$	−1.35619	−0.0513325
$699$	0	0
$700$	11.9584	0.451986
$701$	4.62594	0.174719	0.0873596	−	0.996177i	$-0.472157\pi$
$701$	4.62594	0.174719	0.0873596	+	0.996177i	$0.472157\pi$
$702$	0	0
$703$	−3.69080	−0.139201
$704$	0	0
$705$	0	0
$706$	−38.5133	−1.44947
$707$	8.34609	0.313887
$708$	0	0
$709$	14.9159	0.560178	0.280089	−	0.959974i	$-0.409636\pi$
$709$	14.9159	0.560178	0.280089	+	0.959974i	$0.409636\pi$
$710$	−93.8220	−3.52108
$711$	0	0
$712$	−9.45868	−0.354479
$713$	36.9329	1.38315
$714$	0	0
$715$	0	0
$716$	−19.7352	−0.737541
$717$	0	0
$718$	−56.5852	−2.11174
$719$	−33.0569	−1.23282	−0.616408	−	0.787427i	$-0.711413\pi$
$719$	−33.0569	−1.23282	−0.616408	+	0.787427i	$0.711413\pi$
$720$	0	0
$721$	−7.53555	−0.280639
$722$	34.8907	1.29850
$723$	0	0
$724$	25.0382	0.930538
$725$	−4.77422	−0.177310
$726$	0	0
$727$	6.06384	0.224895	0.112448	−	0.993658i	$-0.464131\pi$
$727$	6.06384	0.224895	0.112448	+	0.993658i	$0.464131\pi$
$728$	1.78698	0.0662301
$729$	0	0
$730$	−51.1775	−1.89417
$731$	44.5236	1.64677
$732$	0	0
$733$	0.926026	0.0342035	0.0171018	−	0.999854i	$-0.494556\pi$
$733$	0.926026	0.0342035	0.0171018	+	0.999854i	$0.494556\pi$
$734$	−59.4092	−2.19283
$735$	0	0
$736$	−41.2005	−1.51867
$737$	0	0
$738$	0	0
$739$	30.4962	1.12182	0.560910	−	0.827877i	$-0.310451\pi$
$739$	30.4962	1.12182	0.560910	+	0.827877i	$0.310451\pi$
$740$	−18.1926	−0.668773
$741$	0	0
$742$	−1.98355	−0.0728185
$743$	17.8520	0.654925	0.327463	−	0.944864i	$-0.393806\pi$
$743$	17.8520	0.654925	0.327463	+	0.944864i	$0.393806\pi$
$744$	0	0
$745$	−9.46169	−0.346649
$746$	−27.1050	−0.992386
$747$	0	0
$748$	0	0
$749$	15.3887	0.562291
$750$	0	0
$751$	35.7837	1.30577	0.652884	−	0.757458i	$-0.273559\pi$
$751$	35.7837	1.30577	0.652884	+	0.757458i	$0.273559\pi$
$752$	31.3805	1.14433
$753$	0	0
$754$	−4.27870	−0.155821
$755$	27.8431	1.01332
$756$	0	0
$757$	42.5125	1.54514	0.772571	−	0.634928i	$-0.218970\pi$
$757$	42.5125	1.54514	0.772571	+	0.634928i	$0.218970\pi$
$758$	−43.9420	−1.59605
$759$	0	0
$760$	−4.08061	−0.148019
$761$	9.10570	0.330081	0.165041	−	0.986287i	$-0.447224\pi$
$761$	9.10570	0.330081	0.165041	+	0.986287i	$0.447224\pi$
$762$	0	0
$763$	−20.2581	−0.733393
$764$	−34.1219	−1.23449
$765$	0	0
$766$	52.3922	1.89301
$767$	−20.3300	−0.734073
$768$	0	0
$769$	6.93312	0.250015	0.125007	−	0.992156i	$-0.460105\pi$
$769$	6.93312	0.250015	0.125007	+	0.992156i	$0.460105\pi$
$770$	0	0
$771$	0	0
$772$	16.8406	0.606108
$773$	18.2629	0.656872	0.328436	−	0.944526i	$-0.393478\pi$
$773$	18.2629	0.656872	0.328436	+	0.944526i	$0.393478\pi$
$774$	0	0
$775$	−35.9737	−1.29221
$776$	−5.95030	−0.213603
$777$	0	0
$778$	−28.4820	−1.02113
$779$	−0.410639	−0.0147127
$780$	0	0
$781$	0	0
$782$	−51.4418	−1.83955
$783$	0	0
$784$	−3.03942	−0.108551
$785$	−4.94947	−0.176654
$786$	0	0
$787$	12.2571	0.436918	0.218459	−	0.975846i	$-0.429897\pi$
$787$	12.2571	0.436918	0.218459	+	0.975846i	$0.429897\pi$
$788$	−8.64800	−0.308072
$789$	0	0
$790$	−61.7171	−2.19580
$791$	18.1019	0.643628
$792$	0	0
$793$	18.2158	0.646862
$794$	8.94549	0.317464
$795$	0	0
$796$	−9.24405	−0.327647
$797$	22.8561	0.809606	0.404803	−	0.914404i	$-0.367340\pi$
$797$	22.8561	0.809606	0.404803	+	0.914404i	$0.367340\pi$
$798$	0	0
$799$	49.4989	1.75114
$800$	40.1305	1.41883
$801$	0	0
$802$	−68.0531	−2.40304
$803$	0	0
$804$	0	0
$805$	16.1610	0.569600
$806$	−32.2400	−1.13561
$807$	0	0
$808$	−7.00657	−0.246490
$809$	−43.6342	−1.53410	−0.767048	−	0.641589i	$-0.778275\pi$
$809$	−43.6342	−1.53410	−0.767048	+	0.641589i	$0.778275\pi$
$810$	0	0
$811$	13.8319	0.485705	0.242853	−	0.970063i	$-0.421917\pi$
$811$	13.8319	0.485705	0.242853	+	0.970063i	$0.421917\pi$
$812$	−2.29999	−0.0807138
$813$	0	0
$814$	0	0
$815$	21.5023	0.753194
$816$	0	0
$817$	14.2874	0.499854
$818$	−52.2060	−1.82534
$819$	0	0
$820$	−2.02411	−0.0706850
$821$	−19.5987	−0.684000	−0.342000	−	0.939700i	$-0.611104\pi$
$821$	−19.5987	−0.684000	−0.342000	+	0.939700i	$0.611104\pi$
$822$	0	0
$823$	38.0461	1.32620	0.663101	−	0.748530i	$-0.269240\pi$
$823$	38.0461	1.32620	0.663101	+	0.748530i	$0.269240\pi$
$824$	6.32612	0.220381
$825$	0	0
$826$	−20.0343	−0.697084
$827$	−2.70341	−0.0940068	−0.0470034	−	0.998895i	$-0.514967\pi$
$827$	−2.70341	−0.0940068	−0.0470034	+	0.998895i	$0.514967\pi$
$828$	0	0
$829$	−17.6913	−0.614444	−0.307222	−	0.951638i	$-0.599399\pi$
$829$	−17.6913	−0.614444	−0.307222	+	0.951638i	$0.599399\pi$
$830$	−111.697	−3.87708
$831$	0	0
$832$	23.0258	0.798276
$833$	−4.79430	−0.166113
$834$	0	0
$835$	38.7027	1.33936
$836$	0	0
$837$	0	0
$838$	22.2243	0.767725
$839$	−0.168659	−0.00582276	−0.00291138	−	0.999996i	$-0.500927\pi$
$839$	−0.168659	−0.00582276	−0.00291138	+	0.999996i	$0.500927\pi$
$840$	0	0
$841$	−28.0818	−0.968337
$842$	79.4244	2.73715
$843$	0	0
$844$	−26.7203	−0.919749
$845$	26.7574	0.920484
$846$	0	0
$847$	0	0
$848$	−2.87407	−0.0986961
$849$	0	0
$850$	50.1057	1.71861
$851$	−12.2711	−0.420650
$852$	0	0
$853$	42.9770	1.47151	0.735753	−	0.677250i	$-0.236829\pi$
$853$	42.9770	1.47151	0.735753	+	0.677250i	$0.236829\pi$
$854$	17.9509	0.614267
$855$	0	0
$856$	−12.9189	−0.441557
$857$	−3.31550	−0.113255	−0.0566276	−	0.998395i	$-0.518035\pi$
$857$	−3.31550	−0.113255	−0.0566276	+	0.998395i	$0.518035\pi$
$858$	0	0
$859$	12.5924	0.429646	0.214823	−	0.976653i	$-0.431083\pi$
$859$	12.5924	0.429646	0.214823	+	0.976653i	$0.431083\pi$
$860$	70.4252	2.40148
$861$	0	0
$862$	−11.8411	−0.403309
$863$	55.8374	1.90073	0.950363	−	0.311143i	$-0.100712\pi$
$863$	55.8374	1.90073	0.950363	+	0.311143i	$0.100712\pi$
$864$	0	0
$865$	62.7154	2.13239
$866$	5.65381	0.192124
$867$	0	0
$868$	−17.3304	−0.588232
$869$	0	0
$870$	0	0
$871$	−28.2490	−0.957182
$872$	17.0067	0.575921
$873$	0	0
$874$	−16.5074	−0.558372
$875$	0.0560984	0.00189647
$876$	0	0
$877$	−8.26736	−0.279169	−0.139584	−	0.990210i	$-0.544577\pi$
$877$	−8.26736	−0.279169	−0.139584	+	0.990210i	$0.544577\pi$
$878$	64.1191	2.16391
$879$	0	0
$880$	0	0
$881$	−26.0843	−0.878803	−0.439401	−	0.898291i	$-0.644809\pi$
$881$	−26.0843	−0.878803	−0.439401	+	0.898291i	$0.644809\pi$
$882$	0	0
$883$	0.915087	0.0307951	0.0153976	−	0.999881i	$-0.495099\pi$
$883$	0.915087	0.0307951	0.0153976	+	0.999881i	$0.495099\pi$
$884$	24.4948	0.823848
$885$	0	0
$886$	−51.6804	−1.73624
$887$	−56.0572	−1.88222	−0.941108	−	0.338107i	$-0.890214\pi$
$887$	−56.0572	−1.88222	−0.941108	+	0.338107i	$0.890214\pi$
$888$	0	0
$889$	15.3711	0.515530
$890$	74.6722	2.50302
$891$	0	0
$892$	−12.6520	−0.423619
$893$	15.8840	0.531537
$894$	0	0
$895$	25.9781	0.868353
$896$	6.58156	0.219875
$897$	0	0
$898$	−32.2528	−1.07629
$899$	6.91889	0.230758
$900$	0	0
$901$	−4.53349	−0.151033
$902$	0	0
$903$	0	0
$904$	−15.1966	−0.505430
$905$	−32.9586	−1.09558
$906$	0	0
$907$	9.24637	0.307021	0.153510	−	0.988147i	$-0.450942\pi$
$907$	9.24637	0.307021	0.153510	+	0.988147i	$0.450942\pi$
$908$	−35.5719	−1.18050
$909$	0	0
$910$	−14.1075	−0.467658
$911$	37.2957	1.23566	0.617831	−	0.786311i	$-0.288011\pi$
$911$	37.2957	1.23566	0.617831	+	0.786311i	$0.288011\pi$
$912$	0	0
$913$	0	0
$914$	29.8629	0.987777
$915$	0	0
$916$	18.2531	0.603098
$917$	−10.7418	−0.354726
$918$	0	0
$919$	14.6385	0.482880	0.241440	−	0.970416i	$-0.422380\pi$
$919$	14.6385	0.482880	0.241440	+	0.970416i	$0.422380\pi$
$920$	−13.5672	−0.447297
$921$	0	0
$922$	77.0155	2.53637
$923$	30.1337	0.991864
$924$	0	0
$925$	11.9524	0.392994
$926$	72.9322	2.39670
$927$	0	0
$928$	−7.71838	−0.253368
$929$	7.80689	0.256136	0.128068	−	0.991765i	$-0.459122\pi$
$929$	7.80689	0.256136	0.128068	+	0.991765i	$0.459122\pi$
$930$	0	0
$931$	−1.53847	−0.0504213
$932$	−3.61337	−0.118360
$933$	0	0
$934$	28.4309	0.930287
$935$	0	0
$936$	0	0
$937$	53.1013	1.73474	0.867372	−	0.497661i	$-0.165808\pi$
$937$	53.1013	1.73474	0.867372	+	0.497661i	$0.165808\pi$
$938$	−27.8382	−0.908950
$939$	0	0
$940$	78.2948	2.55369
$941$	−13.0587	−0.425702	−0.212851	−	0.977085i	$-0.568275\pi$
$941$	−13.0587	−0.425702	−0.212851	+	0.977085i	$0.568275\pi$
$942$	0	0
$943$	−1.36529	−0.0444599
$944$	−29.0288	−0.944806
$945$	0	0
$946$	0	0
$947$	39.8291	1.29427	0.647136	−	0.762375i	$-0.275967\pi$
$947$	39.8291	1.29427	0.647136	+	0.762375i	$0.275967\pi$
$948$	0	0
$949$	16.4372	0.533574
$950$	16.0787	0.521662
$951$	0	0
$952$	4.02483	0.130446
$953$	3.79128	0.122812	0.0614058	−	0.998113i	$-0.480442\pi$
$953$	3.79128	0.122812	0.0614058	+	0.998113i	$0.480442\pi$
$954$	0	0
$955$	44.9157	1.45344
$956$	−12.0571	−0.389955
$957$	0	0
$958$	−57.8899	−1.87034
$959$	4.42612	0.142927
$960$	0	0
$961$	21.1338	0.681735
$962$	10.7119	0.345366
$963$	0	0
$964$	−38.7639	−1.24850
$965$	−22.1679	−0.713609
$966$	0	0
$967$	−5.63563	−0.181230	−0.0906149	−	0.995886i	$-0.528883\pi$
$967$	−5.63563	−0.181230	−0.0906149	+	0.995886i	$0.528883\pi$
$968$	0	0
$969$	0	0
$970$	46.9751	1.50828
$971$	2.41724	0.0775729	0.0387865	−	0.999248i	$-0.487651\pi$
$971$	2.41724	0.0775729	0.0387865	+	0.999248i	$0.487651\pi$
$972$	0	0
$973$	−4.52505	−0.145067
$974$	42.9555	1.37638
$975$	0	0
$976$	26.0100	0.832560
$977$	−46.6426	−1.49223	−0.746114	−	0.665818i	$-0.768083\pi$
$977$	−46.6426	−1.49223	−0.746114	+	0.665818i	$0.768083\pi$
$978$	0	0
$979$	0	0
$980$	−7.58338	−0.242242
$981$	0	0
$982$	−60.5521	−1.93230
$983$	−21.6905	−0.691821	−0.345910	−	0.938268i	$-0.612430\pi$
$983$	−21.6905	−0.691821	−0.345910	+	0.938268i	$0.612430\pi$
$984$	0	0
$985$	11.3836	0.362713
$986$	−9.63694	−0.306903
$987$	0	0
$988$	7.86026	0.250068
$989$	47.5027	1.51050
$990$	0	0
$991$	33.7715	1.07279	0.536394	−	0.843968i	$-0.319786\pi$
$991$	33.7715	1.07279	0.536394	+	0.843968i	$0.319786\pi$
$992$	−58.1579	−1.84652
$993$	0	0
$994$	29.6955	0.941884
$995$	12.1682	0.385759
$996$	0	0
$997$	−0.624922	−0.0197915	−0.00989575	−	0.999951i	$-0.503150\pi$
$997$	−0.624922	−0.0197915	−0.00989575	+	0.999951i	$0.503150\pi$
$998$	45.0872	1.42721
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cy.1.3		10
3.2	odd	2		2541.2.a.br.1.8		10
11.7	odd	10		693.2.m.j.379.4		20
11.8	odd	10		693.2.m.j.64.4		20
11.10	odd	2		7623.2.a.cx.1.8		10
33.8	even	10		231.2.j.g.64.2	✓	20
33.29	even	10		231.2.j.g.148.2	yes	20
33.32	even	2		2541.2.a.bq.1.3		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.j.g.64.2	✓	20	33.8	even	10
231.2.j.g.148.2	yes	20	33.29	even	10
693.2.m.j.64.4		20	11.8	odd	10
693.2.m.j.379.4		20	11.7	odd	10
2541.2.a.bq.1.3		10	33.32	even	2
2541.2.a.br.1.8		10	3.2	odd	2
7623.2.a.cx.1.8		10	11.10	odd	2
7623.2.a.cy.1.3		10	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:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

\(f(q)\)	\(=\)	\(q-2.09767 q^{2} +2.40021 q^{4} -3.15947 q^{5} +1.00000 q^{7} -0.839503 q^{8} +O(q^{10})\)	\(q-2.09767 q^{2} +2.40021 q^{4} -3.15947 q^{5} +1.00000 q^{7} -0.839503 q^{8} +6.62751 q^{10} -2.12862 q^{13} -2.09767 q^{14} -3.03942 q^{16} -4.79430 q^{17} -1.53847 q^{19} -7.58338 q^{20} -5.11509 q^{23} +4.98224 q^{25} +4.46514 q^{26} +2.40021 q^{28} -0.958246 q^{29} -7.22037 q^{31} +8.05469 q^{32} +10.0569 q^{34} -3.15947 q^{35} +2.39901 q^{37} +3.22720 q^{38} +2.65238 q^{40} +0.266914 q^{41} -9.28678 q^{43} +10.7298 q^{46} -10.3245 q^{47} +1.00000 q^{49} -10.4511 q^{50} -5.10914 q^{52} +0.945600 q^{53} -0.839503 q^{56} +2.01008 q^{58} +9.55077 q^{59} -8.55756 q^{61} +15.1459 q^{62} -10.8172 q^{64} +6.72532 q^{65} +13.2710 q^{67} -11.5073 q^{68} +6.62751 q^{70} -14.1564 q^{71} -7.72198 q^{73} -5.03232 q^{74} -3.69265 q^{76} -9.31226 q^{79} +9.60295 q^{80} -0.559897 q^{82} -16.8536 q^{83} +15.1475 q^{85} +19.4806 q^{86} +11.2670 q^{89} -2.12862 q^{91} -12.2773 q^{92} +21.6574 q^{94} +4.86075 q^{95} +7.08789 q^{97} -2.09767 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 18 q^{4} - 5 q^{5} + 10 q^{7} + 3 q^{8}+O(q^{10}) \) 10 * q + 18 * q^4 - 5 * q^5 + 10 * q^7 + 3 * q^8	\( 10 q + 18 q^{4} - 5 q^{5} + 10 q^{7} + 3 q^{8} - 6 q^{10} + 6 q^{13} + 38 q^{16} - 8 q^{17} - 7 q^{20} + 31 q^{25} - q^{26} + 18 q^{28} + 14 q^{29} + 26 q^{31} + 41 q^{32} + 21 q^{34} - 5 q^{35} + 24 q^{37} - 8 q^{38} - 5 q^{40} - 19 q^{41} - 6 q^{43} - q^{46} - 15 q^{47} + 10 q^{49} + q^{50} - 25 q^{52} + q^{53} + 3 q^{56} + 11 q^{58} - 23 q^{59} - 11 q^{62} + 53 q^{64} + 29 q^{65} + 38 q^{67} - 87 q^{68} - 6 q^{70} - 26 q^{71} - q^{73} + 39 q^{74} - 2 q^{76} + 5 q^{79} - 6 q^{80} + 5 q^{82} - 6 q^{83} - q^{85} + 41 q^{86} + 9 q^{89} + 6 q^{91} + 48 q^{92} + 42 q^{95} + 24 q^{97}+O(q^{100}) \) 10 * q + 18 * q^4 - 5 * q^5 + 10 * q^7 + 3 * q^8 - 6 * q^10 + 6 * q^13 + 38 * q^16 - 8 * q^17 - 7 * q^20 + 31 * q^25 - q^26 + 18 * q^28 + 14 * q^29 + 26 * q^31 + 41 * q^32 + 21 * q^34 - 5 * q^35 + 24 * q^37 - 8 * q^38 - 5 * q^40 - 19 * q^41 - 6 * q^43 - q^46 - 15 * q^47 + 10 * q^49 + q^50 - 25 * q^52 + q^53 + 3 * q^56 + 11 * q^58 - 23 * q^59 - 11 * q^62 + 53 * q^64 + 29 * q^65 + 38 * q^67 - 87 * q^68 - 6 * q^70 - 26 * q^71 - q^73 + 39 * q^74 - 2 * q^76 + 5 * q^79 - 6 * q^80 + 5 * q^82 - 6 * q^83 - q^85 + 41 * q^86 + 9 * q^89 + 6 * q^91 + 48 * q^92 + 42 * q^95 + 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more