LMFDB - Embedded newform 9025.2.a.bt.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9025,2,Mod(1,9025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9025 = 5^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9025.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:	$72.0649878242$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.4227136.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 6x^{4} + 7x^{2} - 1 $ x^6 - 6x^4 + 7x^2 - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.11917$ of defining polynomial
Character	$\chi$	$=$	9025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.471884 q^{2} +1.04022 q^{3} -1.77733 q^{4} -0.490864 q^{6} -1.17540 q^{7} +1.78246 q^{8} -1.91794 q^{9} +O(q^{10})$	\(q-0.471884 q^{2} +1.04022 q^{3} -1.77733 q^{4} -0.490864 q^{6} -1.17540 q^{7} +1.78246 q^{8} -1.91794 q^{9} +0.713538 q^{11} -1.84881 q^{12} +4.10315 q^{13} +0.554651 q^{14} +2.71354 q^{16} -2.55233 q^{17} +0.905045 q^{18} -1.22267 q^{21} -0.336707 q^{22} +0.607061 q^{23} +1.85415 q^{24} -1.93621 q^{26} -5.11575 q^{27} +2.08907 q^{28} +0.859386 q^{29} -2.50914 q^{31} -4.84539 q^{32} +0.742237 q^{33} +1.20440 q^{34} +3.40880 q^{36} +9.38171 q^{37} +4.26819 q^{39} +4.12234 q^{41} +0.576960 q^{42} -10.1239 q^{43} -1.26819 q^{44} -0.286462 q^{46} -10.5184 q^{47} +2.82268 q^{48} -5.61844 q^{49} -2.65498 q^{51} -7.29264 q^{52} +4.98057 q^{53} +2.41404 q^{54} -2.09510 q^{56} -0.405530 q^{58} +6.24992 q^{59} -4.55465 q^{61} +1.18402 q^{62} +2.25434 q^{63} -3.14061 q^{64} -0.350250 q^{66} -5.18210 q^{67} +4.53632 q^{68} +0.631477 q^{69} +13.1679 q^{71} -3.41865 q^{72} +12.3783 q^{73} -4.42708 q^{74} -0.838691 q^{77} -2.01409 q^{78} +5.96345 q^{79} +0.432310 q^{81} -1.94527 q^{82} -13.8603 q^{83} +2.17309 q^{84} +4.77733 q^{86} +0.893952 q^{87} +1.27185 q^{88} -15.9635 q^{89} -4.82284 q^{91} -1.07894 q^{92} -2.61006 q^{93} +4.96345 q^{94} -5.04028 q^{96} +16.7020 q^{97} +2.65125 q^{98} -1.36852 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{4} + 12 q^{6} + 8 q^{9}+O(q^{10}) $ 6 * q + 2 * q^4 + 12 * q^6 + 8 * q^9	$ 6 q + 2 q^{4} + 12 q^{6} + 8 q^{9} + 2 q^{11} - 22 q^{14} + 14 q^{16} - 20 q^{21} + 2 q^{24} - 22 q^{26} + 12 q^{29} - 30 q^{31} - 10 q^{34} - 14 q^{36} - 2 q^{39} - 12 q^{41} + 20 q^{44} - 4 q^{46} + 2 q^{49} - 40 q^{51} - 4 q^{54} - 46 q^{56} - 20 q^{59} - 2 q^{61} - 12 q^{64} + 6 q^{66} - 18 q^{69} + 2 q^{71} - 22 q^{74} - 24 q^{79} + 14 q^{81} - 48 q^{84} + 16 q^{86} - 36 q^{89} + 24 q^{91} - 30 q^{94} + 26 q^{96} - 30 q^{99}+O(q^{100}) $ 6 * q + 2 * q^4 + 12 * q^6 + 8 * q^9 + 2 * q^11 - 22 * q^14 + 14 * q^16 - 20 * q^21 + 2 * q^24 - 22 * q^26 + 12 * q^29 - 30 * q^31 - 10 * q^34 - 14 * q^36 - 2 * q^39 - 12 * q^41 + 20 * q^44 - 4 * q^46 + 2 * q^49 - 40 * q^51 - 4 * q^54 - 46 * q^56 - 20 * q^59 - 2 * q^61 - 12 * q^64 + 6 * q^66 - 18 * q^69 + 2 * q^71 - 22 * q^74 - 24 * q^79 + 14 * q^81 - 48 * q^84 + 16 * q^86 - 36 * q^89 + 24 * q^91 - 30 * q^94 + 26 * q^96 - 30 * 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 $

$2$	−0.471884	−0.333672	−0.166836	−	0.985985i	$-0.553355\pi$
$2$	−0.471884	−0.333672	−0.166836	+	0.985985i	$0.553355\pi$
$3$	1.04022	0.600572	0.300286	−	0.953849i	$-0.402918\pi$
$3$	1.04022	0.600572	0.300286	+	0.953849i	$0.402918\pi$
$4$	−1.77733	−0.888663
$5$	0	0
$5$	0	0
$6$	−0.490864	−0.200394
$7$	−1.17540	−0.444259	−0.222129	−	0.975017i	$-0.571301\pi$
$7$	−1.17540	−0.444259	−0.222129	+	0.975017i	$0.571301\pi$
$8$	1.78246	0.630194
$9$	−1.91794	−0.639313
$10$	0	0
$11$	0.713538	0.215140	0.107570	−	0.994198i	$-0.465693\pi$
$11$	0.713538	0.215140	0.107570	+	0.994198i	$0.465693\pi$
$12$	−1.84881	−0.533706
$13$	4.10315	1.13801	0.569005	−	0.822334i	$-0.307328\pi$
$13$	4.10315	1.13801	0.569005	+	0.822334i	$0.307328\pi$
$14$	0.554651	0.148237
$15$	0	0
$16$	2.71354	0.678384
$17$	−2.55233	−0.619030	−0.309515	−	0.950895i	$-0.600167\pi$
$17$	−2.55233	−0.619030	−0.309515	+	0.950895i	$0.600167\pi$
$18$	0.905045	0.213321
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−1.22267	−0.266809
$22$	−0.336707	−0.0717862
$23$	0.607061	0.126581	0.0632904	−	0.997995i	$-0.479841\pi$
$23$	0.607061	0.126581	0.0632904	+	0.997995i	$0.479841\pi$
$24$	1.85415	0.378477
$25$	0	0
$26$	−1.93621	−0.379722
$27$	−5.11575	−0.984526
$28$	2.08907	0.394796
$29$	0.859386	0.159584	0.0797920	−	0.996812i	$-0.474574\pi$
$29$	0.859386	0.159584	0.0797920	+	0.996812i	$0.474574\pi$
$30$	0	0
$31$	−2.50914	−0.450654	−0.225327	−	0.974283i	$-0.572345\pi$
$31$	−2.50914	−0.450654	−0.225327	+	0.974283i	$0.572345\pi$
$32$	−4.84539	−0.856552
$33$	0.742237	0.129207
$34$	1.20440	0.206553
$35$	0	0
$36$	3.40880	0.568134
$37$	9.38171	1.54234	0.771172	−	0.636627i	$-0.219671\pi$
$37$	9.38171	1.54234	0.771172	+	0.636627i	$0.219671\pi$
$38$	0	0
$39$	4.26819	0.683457
$40$	0	0
$41$	4.12234	0.643802	0.321901	−	0.946773i	$-0.395678\pi$
$41$	4.12234	0.643802	0.321901	+	0.946773i	$0.395678\pi$
$42$	0.576960	0.0890269
$43$	−10.1239	−1.54389	−0.771944	−	0.635691i	$-0.780715\pi$
$43$	−10.1239	−1.54389	−0.771944	+	0.635691i	$0.780715\pi$
$44$	−1.26819	−0.191187
$45$	0	0
$46$	−0.286462	−0.0422365
$47$	−10.5184	−1.53426	−0.767132	−	0.641489i	$-0.778317\pi$
$47$	−10.5184	−1.53426	−0.767132	+	0.641489i	$0.778317\pi$
$48$	2.82268	0.407419
$49$	−5.61844	−0.802634
$50$	0	0
$51$	−2.65498	−0.371772
$52$	−7.29264	−1.01131
$53$	4.98057	0.684134	0.342067	−	0.939676i	$-0.388873\pi$
$53$	4.98057	0.684134	0.342067	+	0.939676i	$0.388873\pi$
$54$	2.41404	0.328509
$55$	0	0
$56$	−2.09510	−0.279969
$57$	0	0
$58$	−0.405530	−0.0532488
$59$	6.24992	0.813670	0.406835	−	0.913502i	$-0.366632\pi$
$59$	6.24992	0.813670	0.406835	+	0.913502i	$0.366632\pi$
$60$	0	0
$61$	−4.55465	−0.583163	−0.291582	−	0.956546i	$-0.594182\pi$
$61$	−4.55465	−0.583163	−0.291582	+	0.956546i	$0.594182\pi$
$62$	1.18402	0.150371
$63$	2.25434	0.284020
$64$	−3.14061	−0.392577
$65$	0	0
$66$	−0.350250	−0.0431128
$67$	−5.18210	−0.633094	−0.316547	−	0.948577i	$-0.602524\pi$
$67$	−5.18210	−0.633094	−0.316547	+	0.948577i	$0.602524\pi$
$68$	4.53632	0.550109
$69$	0.631477	0.0760209
$70$	0	0
$71$	13.1679	1.56274	0.781368	−	0.624070i	$-0.214522\pi$
$71$	13.1679	1.56274	0.781368	+	0.624070i	$0.214522\pi$
$72$	−3.41865	−0.402892
$73$	12.3783	1.44877	0.724384	−	0.689396i	$-0.242124\pi$
$73$	12.3783	1.44877	0.724384	+	0.689396i	$0.242124\pi$
$74$	−4.42708	−0.514637
$75$	0	0
$76$	0	0
$77$	−0.838691	−0.0955777
$78$	−2.01409	−0.228051
$79$	5.96345	0.670941	0.335471	−	0.942051i	$-0.391105\pi$
$79$	5.96345	0.670941	0.335471	+	0.942051i	$0.391105\pi$
$80$	0	0
$81$	0.432310	0.0480345
$82$	−1.94527	−0.214819
$83$	−13.8603	−1.52136	−0.760682	−	0.649124i	$-0.775136\pi$
$83$	−13.8603	−1.52136	−0.760682	+	0.649124i	$0.775136\pi$
$84$	2.17309	0.237104
$85$	0	0
$86$	4.77733	0.515152
$87$	0.893952	0.0958417
$88$	1.27185	0.135580
$89$	−15.9635	−1.69212	−0.846061	−	0.533085i	$-0.821032\pi$
$89$	−15.9635	−1.69212	−0.846061	+	0.533085i	$0.821032\pi$
$90$	0	0
$91$	−4.82284	−0.505571
$92$	−1.07894	−0.112488
$93$	−2.61006	−0.270650
$94$	4.96345	0.511941
$95$	0	0
$96$	−5.04028	−0.514421
$97$	16.7020	1.69583	0.847915	−	0.530133i	$-0.177858\pi$
$97$	16.7020	1.69583	0.847915	+	0.530133i	$0.177858\pi$
$98$	2.65125	0.267817
$99$	−1.36852	−0.137542
$100$	0	0
$101$	14.0272	1.39576	0.697881	−	0.716213i	$-0.254126\pi$
$101$	14.0272	1.39576	0.697881	+	0.716213i	$0.254126\pi$
$102$	1.25284	0.124050
$103$	3.55382	0.350169	0.175084	−	0.984553i	$-0.443980\pi$
$103$	3.55382	0.350169	0.175084	+	0.984553i	$0.443980\pi$
$104$	7.31370	0.717168
$105$	0	0
$106$	−2.35025	−0.228276
$107$	3.63127	0.351048	0.175524	−	0.984475i	$-0.443838\pi$
$107$	3.63127	0.351048	0.175524	+	0.984475i	$0.443838\pi$
$108$	9.09235	0.874911
$109$	6.22267	0.596024	0.298012	−	0.954562i	$-0.403676\pi$
$109$	6.22267	0.596024	0.298012	+	0.954562i	$0.403676\pi$
$110$	0	0
$111$	9.75905	0.926288
$112$	−3.18949	−0.301378
$113$	−12.2707	−1.15433	−0.577167	−	0.816626i	$-0.695842\pi$
$113$	−12.2707	−1.15433	−0.577167	+	0.816626i	$0.695842\pi$
$114$	0	0
$115$	0	0
$116$	−1.52741	−0.141816
$117$	−7.86960	−0.727545
$118$	−2.94923	−0.271499
$119$	3.00000	0.275010
$120$	0	0
$121$	−10.4909	−0.953715
$122$	2.14927	0.194585
$123$	4.28815	0.386649
$124$	4.45955	0.400480
$125$	0	0
$126$	−1.06379	−0.0947697
$127$	2.15789	0.191482	0.0957408	−	0.995406i	$-0.469478\pi$
$127$	2.15789	0.191482	0.0957408	+	0.995406i	$0.469478\pi$
$128$	11.1728	0.987544
$129$	−10.5311	−0.927216
$130$	0	0
$131$	−17.5494	−1.53330	−0.766650	−	0.642065i	$-0.778078\pi$
$131$	−17.5494	−1.53330	−0.766650	+	0.642065i	$0.778078\pi$
$132$	−1.31920	−0.114821
$133$	0	0
$134$	2.44535	0.211246
$135$	0	0
$136$	−4.54942	−0.390109
$137$	−5.07455	−0.433548	−0.216774	−	0.976222i	$-0.569554\pi$
$137$	−5.07455	−0.433548	−0.216774	+	0.976222i	$0.569554\pi$
$138$	−0.297984	−0.0253661
$139$	5.98173	0.507363	0.253682	−	0.967288i	$-0.418358\pi$
$139$	5.98173	0.507363	0.253682	+	0.967288i	$0.418358\pi$
$140$	0	0
$141$	−10.9414	−0.921436
$142$	−6.21370	−0.521442
$143$	2.92776	0.244831
$144$	−5.20440	−0.433700
$145$	0	0
$146$	−5.84111	−0.483414
$147$	−5.84442	−0.482040
$148$	−16.6744	−1.37062
$149$	9.60017	0.786476	0.393238	−	0.919437i	$-0.371355\pi$
$149$	9.60017	0.786476	0.393238	+	0.919437i	$0.371355\pi$
$150$	0	0
$151$	−7.53638	−0.613302	−0.306651	−	0.951822i	$-0.599208\pi$
$151$	−7.53638	−0.613302	−0.306651	+	0.951822i	$0.599208\pi$
$152$	0	0
$153$	4.89521	0.395754
$154$	0.395765	0.0318916
$155$	0	0
$156$	−7.58596	−0.607363
$157$	9.85606	0.786599	0.393300	−	0.919410i	$-0.371333\pi$
$157$	9.85606	0.786599	0.393300	+	0.919410i	$0.371333\pi$
$158$	−2.81406	−0.223874
$159$	5.18089	0.410872
$160$	0	0
$161$	−0.713538	−0.0562347
$162$	−0.204000	−0.0160278
$163$	0.0688234	0.00539067	0.00269533	−	0.999996i	$-0.499142\pi$
$163$	0.0688234	0.00539067	0.00269533	+	0.999996i	$0.499142\pi$
$164$	−7.32674	−0.572122
$165$	0	0
$166$	6.54045	0.507637
$167$	−16.6442	−1.28797	−0.643985	−	0.765038i	$-0.722720\pi$
$167$	−16.6442	−1.28797	−0.643985	+	0.765038i	$0.722720\pi$
$168$	−2.17937	−0.168142
$169$	3.83588	0.295068
$170$	0	0
$171$	0	0
$172$	17.9935	1.37200
$173$	12.1356	0.922650	0.461325	−	0.887231i	$-0.347374\pi$
$173$	12.1356	0.922650	0.461325	+	0.887231i	$0.347374\pi$
$174$	−0.421841	−0.0319797
$175$	0	0
$176$	1.93621	0.145947
$177$	6.50130	0.488667
$178$	7.53290	0.564614
$179$	−11.7538	−0.878522	−0.439261	−	0.898360i	$-0.644760\pi$
$179$	−11.7538	−0.878522	−0.439261	+	0.898360i	$0.644760\pi$
$180$	0	0
$181$	8.83588	0.656766	0.328383	−	0.944545i	$-0.393496\pi$
$181$	8.83588	0.656766	0.328383	+	0.944545i	$0.393496\pi$
$182$	2.27582	0.168695
$183$	−4.73785	−0.350232
$184$	1.08206	0.0797706
$185$	0	0
$186$	1.23164	0.0903085
$187$	−1.82118	−0.133178
$188$	18.6946	1.36344
$189$	6.01304	0.437384
$190$	0	0
$191$	14.2447	1.03071	0.515355	−	0.856977i	$-0.327660\pi$
$191$	14.2447	1.03071	0.515355	+	0.856977i	$0.327660\pi$
$192$	−3.26693	−0.235771
$193$	−19.5057	−1.40405	−0.702024	−	0.712153i	$-0.747720\pi$
$193$	−19.5057	−1.40405	−0.702024	+	0.712153i	$0.747720\pi$
$194$	−7.88139	−0.565851
$195$	0	0
$196$	9.98580	0.713271
$197$	−4.33232	−0.308665	−0.154332	−	0.988019i	$-0.549323\pi$
$197$	−4.33232	−0.308665	−0.154332	+	0.988019i	$0.549323\pi$
$198$	0.645784	0.0458938
$199$	−12.6315	−0.895422	−0.447711	−	0.894178i	$-0.647761\pi$
$199$	−12.6315	−0.895422	−0.447711	+	0.894178i	$0.647761\pi$
$200$	0	0
$201$	−5.39053	−0.380219
$202$	−6.61923	−0.465727
$203$	−1.01012	−0.0708966
$204$	4.71877	0.330380
$205$	0	0
$206$	−1.67699	−0.116842
$207$	−1.16431	−0.0809248
$208$	11.1341	0.772009
$209$	0	0
$210$	0	0
$211$	−22.2447	−1.53139	−0.765694	−	0.643206i	$-0.777604\pi$
$211$	−22.2447	−1.53139	−0.765694	+	0.643206i	$0.777604\pi$
$212$	−8.85209	−0.607964
$213$	13.6975	0.938536
$214$	−1.71354	−0.117135
$215$	0	0
$216$	−9.11861	−0.620443
$217$	2.94923	0.200207
$218$	−2.93638	−0.198877
$219$	12.8762	0.870090
$220$	0	0
$221$	−10.4726	−0.704463
$222$	−4.60514	−0.309077
$223$	−13.6225	−0.912231	−0.456115	−	0.889921i	$-0.650760\pi$
$223$	−13.6225	−0.912231	−0.456115	+	0.889921i	$0.650760\pi$
$224$	5.69527	0.380531
$225$	0	0
$226$	5.79036	0.385169
$227$	−6.58913	−0.437336	−0.218668	−	0.975799i	$-0.570171\pi$
$227$	−6.58913	−0.437336	−0.218668	+	0.975799i	$0.570171\pi$
$228$	0	0
$229$	−0.585962	−0.0387215	−0.0193607	−	0.999813i	$-0.506163\pi$
$229$	−0.585962	−0.0387215	−0.0193607	+	0.999813i	$0.506163\pi$
$230$	0	0
$231$	−0.872425	−0.0574013
$232$	1.53182	0.100569
$233$	7.15747	0.468901	0.234451	−	0.972128i	$-0.424671\pi$
$233$	7.15747	0.468901	0.234451	+	0.972128i	$0.424671\pi$
$234$	3.71354	0.242762
$235$	0	0
$236$	−11.1081	−0.723078
$237$	6.20331	0.402948
$238$	−1.41565	−0.0917631
$239$	3.65498	0.236421	0.118211	−	0.992989i	$-0.462284\pi$
$239$	3.65498	0.236421	0.118211	+	0.992989i	$0.462284\pi$
$240$	0	0
$241$	−17.2630	−1.11200	−0.556002	−	0.831181i	$-0.687665\pi$
$241$	−17.2630	−1.11200	−0.556002	+	0.831181i	$0.687665\pi$
$242$	4.95047	0.318228
$243$	15.7969	1.01337
$244$	8.09510	0.518236
$245$	0	0
$246$	−2.02351	−0.129014
$247$	0	0
$248$	−4.47243	−0.284000
$249$	−14.4178	−0.913689
$250$	0	0
$251$	−19.4140	−1.22540	−0.612702	−	0.790314i	$-0.709917\pi$
$251$	−19.4140	−1.22540	−0.612702	+	0.790314i	$0.709917\pi$
$252$	−4.00670	−0.252398
$253$	0.433161	0.0272326
$254$	−1.01827	−0.0638921
$255$	0	0
$256$	1.00897	0.0630606
$257$	13.8929	0.866613	0.433306	−	0.901247i	$-0.357347\pi$
$257$	13.8929	0.866613	0.433306	+	0.901247i	$0.357347\pi$
$258$	4.96948	0.309386
$259$	−11.0272	−0.685199
$260$	0	0
$261$	−1.64825	−0.102024
$262$	8.28129	0.511620
$263$	−14.9883	−0.924221	−0.462110	−	0.886822i	$-0.652908\pi$
$263$	−14.9883	−0.924221	−0.462110	+	0.886822i	$0.652908\pi$
$264$	1.32301	0.0814255
$265$	0	0
$266$	0	0
$267$	−16.6055	−1.01624
$268$	9.21028	0.562607
$269$	−19.3723	−1.18115	−0.590574	−	0.806984i	$-0.701098\pi$
$269$	−19.3723	−1.18115	−0.590574	+	0.806984i	$0.701098\pi$
$270$	0	0
$271$	11.9179	0.723963	0.361982	−	0.932185i	$-0.382100\pi$
$271$	11.9179	0.723963	0.361982	+	0.932185i	$0.382100\pi$
$272$	−6.92583	−0.419940
$273$	−5.01682	−0.303632
$274$	2.39460	0.144663
$275$	0	0
$276$	−1.12234	−0.0675570
$277$	10.9824	0.659866	0.329933	−	0.944004i	$-0.392974\pi$
$277$	10.9824	0.659866	0.329933	+	0.944004i	$0.392974\pi$
$278$	−2.82268	−0.169293
$279$	4.81237	0.288109
$280$	0	0
$281$	−14.6498	−0.873931	−0.436965	−	0.899478i	$-0.643947\pi$
$281$	−14.6498	−0.873931	−0.436965	+	0.899478i	$0.643947\pi$
$282$	5.16309	0.307458
$283$	−18.9287	−1.12519	−0.562597	−	0.826731i	$-0.690198\pi$
$283$	−18.9287	−1.12519	−0.562597	+	0.826731i	$0.690198\pi$
$284$	−23.4036	−1.38875
$285$	0	0
$286$	−1.38156	−0.0816934
$287$	−4.84539	−0.286014
$288$	9.29317	0.547605
$289$	−10.4856	−0.616802
$290$	0	0
$291$	17.3738	1.01847
$292$	−22.0002	−1.28747
$293$	−5.59625	−0.326937	−0.163468	−	0.986549i	$-0.552268\pi$
$293$	−5.59625	−0.326937	−0.163468	+	0.986549i	$0.552268\pi$
$294$	2.75789	0.160843
$295$	0	0
$296$	16.7225	0.971976
$297$	−3.65028	−0.211811
$298$	−4.53016	−0.262425
$299$	2.49086	0.144050
$300$	0	0
$301$	11.8997	0.685885
$302$	3.55629	0.204642
$303$	14.5914	0.838256
$304$	0	0
$305$	0	0
$306$	−2.30997	−0.132052
$307$	20.6319	1.17753	0.588764	−	0.808305i	$-0.299615\pi$
$307$	20.6319	1.17753	0.588764	+	0.808305i	$0.299615\pi$
$308$	1.49063	0.0849364
$309$	3.69676	0.210302
$310$	0	0
$311$	−24.6587	−1.39827	−0.699134	−	0.714991i	$-0.746431\pi$
$311$	−24.6587	−1.39827	−0.699134	+	0.714991i	$0.746431\pi$
$312$	7.60787	0.430711
$313$	−3.49856	−0.197751	−0.0988753	−	0.995100i	$-0.531524\pi$
$313$	−3.49856	−0.197751	−0.0988753	+	0.995100i	$0.531524\pi$
$314$	−4.65092	−0.262466
$315$	0	0
$316$	−10.5990	−0.596240
$317$	31.3709	1.76196	0.880982	−	0.473150i	$-0.156883\pi$
$317$	31.3709	1.76196	0.880982	+	0.473150i	$0.156883\pi$
$318$	−2.44478	−0.137096
$319$	0.613205	0.0343329
$320$	0	0
$321$	3.77733	0.210830
$322$	0.336707	0.0187639
$323$	0	0
$324$	−0.768356	−0.0426865
$325$	0	0
$326$	−0.0324767	−0.00179872
$327$	6.47296	0.357955
$328$	7.34790	0.405720
$329$	12.3633	0.681610
$330$	0	0
$331$	−35.7497	−1.96498	−0.982492	−	0.186305i	$-0.940349\pi$
$331$	−35.7497	−1.96498	−0.982492	+	0.186305i	$0.940349\pi$
$332$	24.6343	1.35198
$333$	−17.9935	−0.986040
$334$	7.85415	0.429760
$335$	0	0
$336$	−3.31777	−0.180999
$337$	26.4892	1.44296	0.721480	−	0.692435i	$-0.243462\pi$
$337$	26.4892	1.44296	0.721480	+	0.692435i	$0.243462\pi$
$338$	−1.81009	−0.0984559
$339$	−12.7643	−0.693261
$340$	0	0
$341$	−1.79036	−0.0969536
$342$	0	0
$343$	14.8317	0.800836
$344$	−18.0455	−0.972949
$345$	0	0
$346$	−5.72658	−0.307863
$347$	−34.3398	−1.84346	−0.921729	−	0.387835i	$-0.873223\pi$
$347$	−34.3398	−1.84346	−0.921729	+	0.387835i	$0.873223\pi$
$348$	−1.58884	−0.0851710
$349$	−21.4308	−1.14717	−0.573583	−	0.819148i	$-0.694447\pi$
$349$	−21.4308	−1.14717	−0.573583	+	0.819148i	$0.694447\pi$
$350$	0	0
$351$	−20.9907	−1.12040
$352$	−3.45737	−0.184279
$353$	−19.5659	−1.04139	−0.520693	−	0.853744i	$-0.674326\pi$
$353$	−19.5659	−1.04139	−0.520693	+	0.853744i	$0.674326\pi$
$354$	−3.06786	−0.163055
$355$	0	0
$356$	28.3723	1.50373
$357$	3.12066	0.165163
$358$	5.54644	0.293138
$359$	−16.4726	−0.869390	−0.434695	−	0.900578i	$-0.643144\pi$
$359$	−16.4726	−0.869390	−0.434695	+	0.900578i	$0.643144\pi$
$360$	0	0
$361$	0	0
$362$	−4.16951	−0.219144
$363$	−10.9128	−0.572775
$364$	8.57176	0.449282
$365$	0	0
$366$	2.23571	0.116863
$367$	−12.7347	−0.664746	−0.332373	−	0.943148i	$-0.607849\pi$
$367$	−12.7347	−0.664746	−0.332373	+	0.943148i	$0.607849\pi$
$368$	1.64728	0.0858705
$369$	−7.90640	−0.411591
$370$	0	0
$371$	−5.85415	−0.303932
$372$	4.63892	0.240517
$373$	−19.3342	−1.00109	−0.500544	−	0.865711i	$-0.666867\pi$
$373$	−19.3342	−1.00109	−0.500544	+	0.865711i	$0.666867\pi$
$374$	0.859386	0.0444378
$375$	0	0
$376$	−18.7486	−0.966884
$377$	3.52619	0.181608
$378$	−2.83746	−0.145943
$379$	−9.85789	−0.506366	−0.253183	−	0.967418i	$-0.581477\pi$
$379$	−9.85789	−0.506366	−0.253183	+	0.967418i	$0.581477\pi$
$380$	0	0
$381$	2.24468	0.114999
$382$	−6.72183	−0.343919
$383$	−20.3333	−1.03898	−0.519490	−	0.854476i	$-0.673878\pi$
$383$	−20.3333	−1.03898	−0.519490	+	0.854476i	$0.673878\pi$
$384$	11.6222	0.593092
$385$	0	0
$386$	9.20440	0.468492
$387$	19.4171	0.987027
$388$	−29.6849	−1.50702
$389$	−1.00523	−0.0509674	−0.0254837	−	0.999675i	$-0.508113\pi$
$389$	−1.00523	−0.0509674	−0.0254837	+	0.999675i	$0.508113\pi$
$390$	0	0
$391$	−1.54942	−0.0783574
$392$	−10.0146	−0.505816
$393$	−18.2553	−0.920857
$394$	2.04435	0.102993
$395$	0	0
$396$	2.43231	0.122228
$397$	9.86221	0.494970	0.247485	−	0.968892i	$-0.420396\pi$
$397$	9.86221	0.494970	0.247485	+	0.968892i	$0.420396\pi$
$398$	5.96059	0.298777
$399$	0	0
$400$	0	0
$401$	−29.8318	−1.48973	−0.744865	−	0.667215i	$-0.767486\pi$
$401$	−29.8318	−1.48973	−0.744865	+	0.667215i	$0.767486\pi$
$402$	2.54370	0.126868
$403$	−10.2954	−0.512849
$404$	−24.9310	−1.24036
$405$	0	0
$406$	0.476660	0.0236562
$407$	6.69420	0.331819
$408$	−4.73240	−0.234289
$409$	−27.6442	−1.36692	−0.683458	−	0.729989i	$-0.739525\pi$
$409$	−27.6442	−1.36692	−0.683458	+	0.729989i	$0.739525\pi$
$410$	0	0
$411$	−5.27866	−0.260377
$412$	−6.31630	−0.311182
$413$	−7.34614	−0.361480
$414$	0.549417	0.0270024
$415$	0	0
$416$	−19.8814	−0.974766
$417$	6.22232	0.304708
$418$	0	0
$419$	−11.2902	−0.551562	−0.275781	−	0.961220i	$-0.588936\pi$
$419$	−11.2902	−0.551562	−0.275781	+	0.961220i	$0.588936\pi$
$420$	0	0
$421$	−4.54942	−0.221725	−0.110863	−	0.993836i	$-0.535361\pi$
$421$	−4.54942	−0.221725	−0.110863	+	0.993836i	$0.535361\pi$
$422$	10.4969	0.510981
$423$	20.1736	0.980875
$424$	8.87766	0.431137
$425$	0	0
$426$	−6.46362	−0.313163
$427$	5.35353	0.259075
$428$	−6.45395	−0.311964
$429$	3.04551	0.147039
$430$	0	0
$431$	−32.9034	−1.58490	−0.792451	−	0.609936i	$-0.791195\pi$
$431$	−32.9034	−1.58490	−0.792451	+	0.609936i	$0.791195\pi$
$432$	−13.8818	−0.667887
$433$	−22.3093	−1.07212	−0.536059	−	0.844181i	$-0.680088\pi$
$433$	−22.3093	−1.07212	−0.536059	+	0.844181i	$0.680088\pi$
$434$	−1.39170	−0.0668035
$435$	0	0
$436$	−11.0597	−0.529664
$437$	0	0
$438$	−6.07605	−0.290325
$439$	−5.70573	−0.272320	−0.136160	−	0.990687i	$-0.543476\pi$
$439$	−5.70573	−0.272320	−0.136160	+	0.990687i	$0.543476\pi$
$440$	0	0
$441$	10.7758	0.513135
$442$	4.94185	0.235060
$443$	11.5776	0.550069	0.275034	−	0.961434i	$-0.411311\pi$
$443$	11.5776	0.550069	0.275034	+	0.961434i	$0.411311\pi$
$444$	−17.3450	−0.823158
$445$	0	0
$446$	6.42824	0.304386
$447$	9.98630	0.472336
$448$	3.69147	0.174406
$449$	25.4726	1.20213	0.601063	−	0.799202i	$-0.294744\pi$
$449$	25.4726	1.20213	0.601063	+	0.799202i	$0.294744\pi$
$450$	0	0
$451$	2.94145	0.138507
$452$	21.8091	1.02581
$453$	−7.83950	−0.368332
$454$	3.10930	0.145927
$455$	0	0
$456$	0	0
$457$	−6.92830	−0.324092	−0.162046	−	0.986783i	$-0.551809\pi$
$457$	−6.92830	−0.324092	−0.162046	+	0.986783i	$0.551809\pi$
$458$	0.276506	0.0129203
$459$	13.0571	0.609451
$460$	0	0
$461$	32.4946	1.51342	0.756712	−	0.653748i	$-0.226804\pi$
$461$	32.4946	1.51342	0.756712	+	0.653748i	$0.226804\pi$
$462$	0.411683	0.0191532
$463$	21.5062	0.999477	0.499738	−	0.866176i	$-0.333429\pi$
$463$	21.5062	0.999477	0.499738	+	0.866176i	$0.333429\pi$
$464$	2.33198	0.108259
$465$	0	0
$466$	−3.37749	−0.156459
$467$	−16.9509	−0.784392	−0.392196	−	0.919882i	$-0.628285\pi$
$467$	−16.9509	−0.784392	−0.392196	+	0.919882i	$0.628285\pi$
$468$	13.9868	0.646542
$469$	6.09103	0.281258
$470$	0	0
$471$	10.2525	0.472410
$472$	11.1402	0.512770
$473$	−7.22382	−0.332152
$474$	−2.92724	−0.134453
$475$	0	0
$476$	−5.33198	−0.244391
$477$	−9.55243	−0.437376
$478$	−1.72473	−0.0788872
$479$	8.85789	0.404727	0.202364	−	0.979310i	$-0.435138\pi$
$479$	8.85789	0.404727	0.202364	+	0.979310i	$0.435138\pi$
$480$	0	0
$481$	38.4946	1.75520
$482$	8.14611	0.371045
$483$	−0.742237	−0.0337730
$484$	18.6457	0.847531
$485$	0	0
$486$	−7.45432	−0.338135
$487$	30.0628	1.36227	0.681137	−	0.732156i	$-0.261486\pi$
$487$	30.0628	1.36227	0.681137	+	0.732156i	$0.261486\pi$
$488$	−8.11848	−0.367506
$489$	0.0715916	0.00323748
$490$	0	0
$491$	3.10033	0.139916	0.0699580	−	0.997550i	$-0.477713\pi$
$491$	3.10033	0.139916	0.0699580	+	0.997550i	$0.477713\pi$
$492$	−7.62143	−0.343601
$493$	−2.19343	−0.0987873
$494$	0	0
$495$	0	0
$496$	−6.80864	−0.305717
$497$	−15.4775	−0.694260
$498$	6.80351	0.304873
$499$	32.1391	1.43874	0.719372	−	0.694625i	$-0.244430\pi$
$499$	32.1391	1.43874	0.719372	+	0.694625i	$0.244430\pi$
$500$	0	0
$501$	−17.3137	−0.773519
$502$	9.16117	0.408883
$503$	−8.41822	−0.375350	−0.187675	−	0.982231i	$-0.560095\pi$
$503$	−8.41822	−0.375350	−0.187675	+	0.982231i	$0.560095\pi$
$504$	4.01827	0.178988
$505$	0	0
$506$	−0.204402	−0.00908676
$507$	3.99016	0.177209
$508$	−3.83527	−0.170163
$509$	−20.3450	−0.901777	−0.450888	−	0.892580i	$-0.648893\pi$
$509$	−20.3450	−0.901777	−0.450888	+	0.892580i	$0.648893\pi$
$510$	0	0
$511$	−14.5494	−0.643628
$512$	−22.8217	−1.00859
$513$	0	0
$514$	−6.55582	−0.289165
$515$	0	0
$516$	18.7173	0.823982
$517$	−7.50527	−0.330081
$518$	5.20358	0.228632
$519$	12.6237	0.554118
$520$	0	0
$521$	−15.3502	−0.672507	−0.336253	−	0.941772i	$-0.609160\pi$
$521$	−15.3502	−0.672507	−0.336253	+	0.941772i	$0.609160\pi$
$522$	0.777783	0.0340426
$523$	13.2367	0.578800	0.289400	−	0.957208i	$-0.406544\pi$
$523$	13.2367	0.578800	0.289400	+	0.957208i	$0.406544\pi$
$524$	31.1910	1.36259
$525$	0	0
$526$	7.07276	0.308387
$527$	6.40414	0.278969
$528$	2.01409	0.0876520
$529$	−22.6315	−0.983977
$530$	0	0
$531$	−11.9870	−0.520190
$532$	0	0
$533$	16.9146	0.732653
$534$	7.83588	0.339092
$535$	0	0
$536$	−9.23688	−0.398972
$537$	−12.2266	−0.527616
$538$	9.14145	0.394116
$539$	−4.00897	−0.172679
$540$	0	0
$541$	−27.7408	−1.19267	−0.596335	−	0.802736i	$-0.703377\pi$
$541$	−27.7408	−1.19267	−0.596335	+	0.802736i	$0.703377\pi$
$542$	−5.62388	−0.241566
$543$	9.19127	0.394435
$544$	12.3670	0.530232
$545$	0	0
$546$	2.36736	0.101314
$547$	17.0602	0.729440	0.364720	−	0.931117i	$-0.381165\pi$
$547$	17.0602	0.729440	0.364720	+	0.931117i	$0.381165\pi$
$548$	9.01913	0.385278
$549$	8.73555	0.372824
$550$	0	0
$551$	0	0
$552$	1.12558	0.0479080
$553$	−7.00943	−0.298071
$554$	−5.18239	−0.220179
$555$	0	0
$556$	−10.6315	−0.450875
$557$	26.5415	1.12460	0.562300	−	0.826933i	$-0.309917\pi$
$557$	26.5415	1.12460	0.562300	+	0.826933i	$0.309917\pi$
$558$	−2.27088	−0.0961340
$559$	−41.5401	−1.75696
$560$	0	0
$561$	−1.89443	−0.0799830
$562$	6.91298	0.291606
$563$	35.5594	1.49865	0.749325	−	0.662203i	$-0.230378\pi$
$563$	35.5594	1.49865	0.749325	+	0.662203i	$0.230378\pi$
$564$	19.4465	0.818846
$565$	0	0
$566$	8.93214	0.375446
$567$	−0.508137	−0.0213397
$568$	23.4712	0.984828
$569$	14.9713	0.627628	0.313814	−	0.949485i	$-0.398393\pi$
$569$	14.9713	0.627628	0.313814	+	0.949485i	$0.398393\pi$
$570$	0	0
$571$	5.92724	0.248047	0.124024	−	0.992279i	$-0.460420\pi$
$571$	5.92724	0.248047	0.124024	+	0.992279i	$0.460420\pi$
$572$	−5.20358	−0.217572
$573$	14.8176	0.619015
$574$	2.28646	0.0954351
$575$	0	0
$576$	6.02351	0.250979
$577$	−14.5559	−0.605970	−0.302985	−	0.952995i	$-0.597983\pi$
$577$	−14.5559	−0.605970	−0.302985	+	0.952995i	$0.597983\pi$
$578$	4.94800	0.205810
$579$	−20.2902	−0.843232
$580$	0	0
$581$	16.2914	0.675880
$582$	−8.19839	−0.339834
$583$	3.55382	0.147184
$584$	22.0638	0.913006
$585$	0	0
$586$	2.64078	0.109090
$587$	20.5839	0.849588	0.424794	−	0.905290i	$-0.360347\pi$
$587$	20.5839	0.849588	0.424794	+	0.905290i	$0.360347\pi$
$588$	10.3874	0.428371
$589$	0	0
$590$	0	0
$591$	−4.50657	−0.185375
$592$	25.4576	1.04630
$593$	14.3322	0.588552	0.294276	−	0.955720i	$-0.404922\pi$
$593$	14.3322	0.588552	0.294276	+	0.955720i	$0.404922\pi$
$594$	1.72251	0.0706753
$595$	0	0
$596$	−17.0626	−0.698912
$597$	−13.1395	−0.537765
$598$	−1.17540	−0.0480656
$599$	−2.50017	−0.102154	−0.0510770	−	0.998695i	$-0.516265\pi$
$599$	−2.50017	−0.102154	−0.0510770	+	0.998695i	$0.516265\pi$
$600$	0	0
$601$	−14.5259	−0.592524	−0.296262	−	0.955107i	$-0.595740\pi$
$601$	−14.5259	−0.592524	−0.296262	+	0.955107i	$0.595740\pi$
$602$	−5.61526	−0.228861
$603$	9.93895	0.404745
$604$	13.3946	0.545019
$605$	0	0
$606$	−6.88546	−0.279703
$607$	−31.8704	−1.29358	−0.646789	−	0.762669i	$-0.723889\pi$
$607$	−31.8704	−1.29358	−0.646789	+	0.762669i	$0.723889\pi$
$608$	0	0
$609$	−1.05075	−0.0425785
$610$	0	0
$611$	−43.1586	−1.74601
$612$	−8.70038	−0.351692
$613$	−20.0488	−0.809765	−0.404882	−	0.914369i	$-0.632688\pi$
$613$	−20.0488	−0.809765	−0.404882	+	0.914369i	$0.632688\pi$
$614$	−9.73588	−0.392908
$615$	0	0
$616$	−1.49493	−0.0602325
$617$	20.0819	0.808467	0.404234	−	0.914656i	$-0.367538\pi$
$617$	20.0819	0.808467	0.404234	+	0.914656i	$0.367538\pi$
$618$	−1.74444	−0.0701718
$619$	5.62217	0.225974	0.112987	−	0.993596i	$-0.463958\pi$
$619$	5.62217	0.225974	0.112987	+	0.993596i	$0.463958\pi$
$620$	0	0
$621$	−3.10557	−0.124622
$622$	11.6361	0.466563
$623$	18.7634	0.751740
$624$	11.5819	0.463647
$625$	0	0
$626$	1.65092	0.0659839
$627$	0	0
$628$	−17.5174	−0.699022
$629$	−23.9452	−0.954757
$630$	0	0
$631$	−4.84368	−0.192824	−0.0964120	−	0.995342i	$-0.530737\pi$
$631$	−4.84368	−0.192824	−0.0964120	+	0.995342i	$0.530737\pi$
$632$	10.6296	0.422823
$633$	−23.1394	−0.919708
$634$	−14.8034	−0.587918
$635$	0	0
$636$	−9.20814	−0.365126
$637$	−23.0533	−0.913406
$638$	−0.289361	−0.0114559
$639$	−25.2552	−0.999078
$640$	0	0
$641$	25.3007	0.999316	0.499658	−	0.866223i	$-0.333459\pi$
$641$	25.3007	0.999316	0.499658	+	0.866223i	$0.333459\pi$
$642$	−1.78246	−0.0703480
$643$	−13.9457	−0.549963	−0.274981	−	0.961450i	$-0.588672\pi$
$643$	−13.9457	−0.549963	−0.274981	+	0.961450i	$0.588672\pi$
$644$	1.26819	0.0499737
$645$	0	0
$646$	0	0
$647$	2.51360	0.0988200	0.0494100	−	0.998779i	$-0.484266\pi$
$647$	2.51360	0.0988200	0.0494100	+	0.998779i	$0.484266\pi$
$648$	0.770575	0.0302711
$649$	4.45955	0.175053
$650$	0	0
$651$	3.06786	0.120239
$652$	−0.122322	−0.00479049
$653$	−1.92873	−0.0754770	−0.0377385	−	0.999288i	$-0.512015\pi$
$653$	−1.92873	−0.0754770	−0.0377385	+	0.999288i	$0.512015\pi$
$654$	−3.05448	−0.119440
$655$	0	0
$656$	11.1861	0.436745
$657$	−23.7408	−0.926217
$658$	−5.83404	−0.227434
$659$	−15.8008	−0.615513	−0.307757	−	0.951465i	$-0.599578\pi$
$659$	−15.8008	−0.615513	−0.307757	+	0.951465i	$0.599578\pi$
$660$	0	0
$661$	−16.4036	−0.638025	−0.319012	−	0.947751i	$-0.603351\pi$
$661$	−16.4036	−0.638025	−0.319012	+	0.947751i	$0.603351\pi$
$662$	16.8697	0.655661
$663$	−10.8938	−0.423081
$664$	−24.7054	−0.958755
$665$	0	0
$666$	8.49086	0.329014
$667$	0.521700	0.0202003
$668$	29.5823	1.14457
$669$	−14.1704	−0.547860
$670$	0	0
$671$	−3.24992	−0.125462
$672$	5.92434	0.228536
$673$	17.0878	0.658686	0.329343	−	0.944210i	$-0.393173\pi$
$673$	17.0878	0.658686	0.329343	+	0.944210i	$0.393173\pi$
$674$	−12.4998	−0.481476
$675$	0	0
$676$	−6.81761	−0.262216
$677$	−4.57680	−0.175901	−0.0879504	−	0.996125i	$-0.528032\pi$
$677$	−4.57680	−0.175901	−0.0879504	+	0.996125i	$0.528032\pi$
$678$	6.02326	0.231322
$679$	−19.6315	−0.753387
$680$	0	0
$681$	−6.85415	−0.262652
$682$	0.844844	0.0323507
$683$	−29.0692	−1.11230	−0.556151	−	0.831082i	$-0.687722\pi$
$683$	−29.0692	−1.11230	−0.556151	+	0.831082i	$0.687722\pi$
$684$	0	0
$685$	0	0
$686$	−6.99883	−0.267217
$687$	−0.609531	−0.0232550
$688$	−27.4717	−1.04735
$689$	20.4360	0.778551
$690$	0	0
$691$	−6.07833	−0.231230	−0.115615	−	0.993294i	$-0.536884\pi$
$691$	−6.07833	−0.231230	−0.115615	+	0.993294i	$0.536884\pi$
$692$	−21.5689	−0.819925
$693$	1.60856	0.0611041
$694$	16.2044	0.615111
$695$	0	0
$696$	1.59343	0.0603989
$697$	−10.5216	−0.398533
$698$	10.1129	0.382777
$699$	7.44535	0.281609
$700$	0	0
$701$	10.4114	0.393232	0.196616	−	0.980481i	$-0.437005\pi$
$701$	10.4114	0.393232	0.196616	+	0.980481i	$0.437005\pi$
$702$	9.90517	0.373847
$703$	0	0
$704$	−2.24095	−0.0844589
$705$	0	0
$706$	9.23281	0.347481
$707$	−16.4876	−0.620080
$708$	−11.5549	−0.434261
$709$	−0.531144	−0.0199475	−0.00997377	−	0.999950i	$-0.503175\pi$
$709$	−0.531144	−0.0199475	−0.00997377	+	0.999950i	$0.503175\pi$
$710$	0	0
$711$	−11.4375	−0.428941
$712$	−28.4542	−1.06637
$713$	−1.52320	−0.0570442
$714$	−1.47259	−0.0551103
$715$	0	0
$716$	20.8904	0.780710
$717$	3.80199	0.141988
$718$	7.77315	0.290091
$719$	22.4271	0.836389	0.418194	−	0.908358i	$-0.362663\pi$
$719$	22.4271	0.836389	0.418194	+	0.908358i	$0.362663\pi$
$720$	0	0
$721$	−4.17716	−0.155566
$722$	0	0
$723$	−17.9573	−0.667839
$724$	−15.7042	−0.583643
$725$	0	0
$726$	5.14958	0.191119
$727$	45.9734	1.70506	0.852529	−	0.522679i	$-0.175067\pi$
$727$	45.9734	1.70506	0.852529	+	0.522679i	$0.175067\pi$
$728$	−8.59651	−0.318608
$729$	15.1354	0.560570
$730$	0	0
$731$	25.8396	0.955713
$732$	8.42069	0.311238
$733$	−42.5178	−1.57043	−0.785215	−	0.619223i	$-0.787448\pi$
$733$	−42.5178	−1.57043	−0.785215	+	0.619223i	$0.787448\pi$
$734$	6.00930	0.221807
$735$	0	0
$736$	−2.94145	−0.108423
$737$	−3.69762	−0.136204
$738$	3.73090	0.137336
$739$	−3.41777	−0.125725	−0.0628624	−	0.998022i	$-0.520023\pi$
$739$	−3.41777	−0.125725	−0.0628624	+	0.998022i	$0.520023\pi$
$740$	0	0
$741$	0	0
$742$	2.76248	0.101414
$743$	−9.72088	−0.356625	−0.178312	−	0.983974i	$-0.557064\pi$
$743$	−9.72088	−0.356625	−0.178312	+	0.983974i	$0.557064\pi$
$744$	−4.65232	−0.170562
$745$	0	0
$746$	9.12351	0.334035
$747$	26.5832	0.972629
$748$	3.23683	0.118350
$749$	−4.26819	−0.155956
$750$	0	0
$751$	11.3853	0.415455	0.207728	−	0.978187i	$-0.433393\pi$
$751$	11.3853	0.415455	0.207728	+	0.978187i	$0.433393\pi$
$752$	−28.5420	−1.04082
$753$	−20.1949	−0.735943
$754$	−1.66395	−0.0605976
$755$	0	0
$756$	−10.6871	−0.388687
$757$	−22.7842	−0.828105	−0.414053	−	0.910253i	$-0.635887\pi$
$757$	−22.7842	−0.828105	−0.414053	+	0.910253i	$0.635887\pi$
$758$	4.65178	0.168960
$759$	0.450583	0.0163551
$760$	0	0
$761$	−36.7665	−1.33279	−0.666393	−	0.745601i	$-0.732163\pi$
$761$	−36.7665	−1.33279	−0.666393	+	0.745601i	$0.732163\pi$
$762$	−1.05923	−0.0383718
$763$	−7.31412	−0.264789
$764$	−25.3174	−0.915953
$765$	0	0
$766$	9.59493	0.346679
$767$	25.6444	0.925965
$768$	1.04955	0.0378724
$769$	−20.8008	−0.750097	−0.375049	−	0.927005i	$-0.622374\pi$
$769$	−20.8008	−0.750097	−0.375049	+	0.927005i	$0.622374\pi$
$770$	0	0
$771$	14.4517	0.520464
$772$	34.6679	1.24772
$773$	21.0091	0.755646	0.377823	−	0.925878i	$-0.376673\pi$
$773$	21.0091	0.755646	0.377823	+	0.925878i	$0.376673\pi$
$774$	−9.16262	−0.329344
$775$	0	0
$776$	29.7706	1.06870
$777$	−11.4708	−0.411512
$778$	0.474354	0.0170064
$779$	0	0
$780$	0	0
$781$	9.39576	0.336207
$782$	0.731145	0.0261457
$783$	−4.39640	−0.157115
$784$	−15.2458	−0.544495
$785$	0	0
$786$	8.61437	0.307264
$787$	−18.0606	−0.643791	−0.321896	−	0.946775i	$-0.604320\pi$
$787$	−18.0606	−0.643791	−0.321896	+	0.946775i	$0.604320\pi$
$788$	7.69994	0.274299
$789$	−15.5912	−0.555061
$790$	0	0
$791$	14.4230	0.512823
$792$	−2.43934	−0.0866780
$793$	−18.6884	−0.663646
$794$	−4.65382	−0.165158
$795$	0	0
$796$	22.4502	0.795728
$797$	24.2571	0.859229	0.429615	−	0.903012i	$-0.358649\pi$
$797$	24.2571	0.859229	0.429615	+	0.903012i	$0.358649\pi$
$798$	0	0
$799$	26.8463	0.949756
$800$	0	0
$801$	30.6169	1.08180
$802$	14.0771	0.497081
$803$	8.83238	0.311688
$804$	9.58073	0.337886
$805$	0	0
$806$	4.85822	0.171124
$807$	−20.1514	−0.709364
$808$	25.0030	0.879602
$809$	−4.27192	−0.150193	−0.0750964	−	0.997176i	$-0.523926\pi$
$809$	−4.27192	−0.150193	−0.0750964	+	0.997176i	$0.523926\pi$
$810$	0	0
$811$	−10.1824	−0.357552	−0.178776	−	0.983890i	$-0.557214\pi$
$811$	−10.1824	−0.357552	−0.178776	+	0.983890i	$0.557214\pi$
$812$	1.79531	0.0630032
$813$	12.3973	0.434792
$814$	−3.15889	−0.110719
$815$	0	0
$816$	−7.20440	−0.252205
$817$	0	0
$818$	13.0448	0.456102
$819$	9.24992	0.323218
$820$	0	0
$821$	−56.0217	−1.95517	−0.977585	−	0.210541i	$-0.932477\pi$
$821$	−56.0217	−1.95517	−0.977585	+	0.210541i	$0.932477\pi$
$822$	2.49091	0.0868806
$823$	33.5416	1.16919	0.584593	−	0.811326i	$-0.301254\pi$
$823$	33.5416	1.16919	0.584593	+	0.811326i	$0.301254\pi$
$824$	6.33455	0.220674
$825$	0	0
$826$	3.46652	0.120616
$827$	0.0362530	0.00126064	0.000630320	−	1.00000i	$-0.499799\pi$
$827$	0.0362530	0.00126064	0.000630320		1.00000i	$0.499799\pi$
$828$	2.06935	0.0719149
$829$	−39.5662	−1.37419	−0.687095	−	0.726567i	$-0.741115\pi$
$829$	−39.5662	−1.37419	−0.687095	+	0.726567i	$0.741115\pi$
$830$	0	0
$831$	11.4241	0.396297
$832$	−12.8864	−0.446756
$833$	14.3401	0.496855
$834$	−2.93621	−0.101673
$835$	0	0
$836$	0	0
$837$	12.8361	0.443681
$838$	5.32766	0.184041
$839$	−40.1623	−1.38656	−0.693278	−	0.720670i	$-0.743834\pi$
$839$	−40.1623	−1.38656	−0.693278	+	0.720670i	$0.743834\pi$
$840$	0	0
$841$	−28.2615	−0.974533
$842$	2.14680	0.0739835
$843$	−15.2390	−0.524858
$844$	39.5360	1.36089
$845$	0	0
$846$	−9.51961	−0.327291
$847$	12.3309	0.423696
$848$	13.5150	0.464106
$849$	−19.6900	−0.675760
$850$	0	0
$851$	5.69527	0.195231
$852$	−24.3449	−0.834042
$853$	16.4617	0.563639	0.281819	−	0.959467i	$-0.409062\pi$
$853$	16.4617	0.563639	0.281819	+	0.959467i	$0.409062\pi$
$854$	−2.52624	−0.0864463
$855$	0	0
$856$	6.47259	0.221229
$857$	−5.60664	−0.191519	−0.0957595	−	0.995404i	$-0.530528\pi$
$857$	−5.60664	−0.191519	−0.0957595	+	0.995404i	$0.530528\pi$
$858$	−1.43713	−0.0490628
$859$	25.0298	0.854006	0.427003	−	0.904250i	$-0.359569\pi$
$859$	25.0298	0.854006	0.427003	+	0.904250i	$0.359569\pi$
$860$	0	0
$861$	−5.04028	−0.171772
$862$	15.5266	0.528837
$863$	42.4307	1.44436	0.722178	−	0.691707i	$-0.243141\pi$
$863$	42.4307	1.44436	0.722178	+	0.691707i	$0.243141\pi$
$864$	24.7878	0.843298
$865$	0	0
$866$	10.5274	0.357736
$867$	−10.9074	−0.370434
$868$	−5.24175	−0.177917
$869$	4.25515	0.144346
$870$	0	0
$871$	−21.2630	−0.720468
$872$	11.0917	0.375611
$873$	−32.0334	−1.08417
$874$	0	0
$875$	0	0
$876$	−22.8851	−0.773217
$877$	−0.557245	−0.0188168	−0.00940841	−	0.999956i	$-0.502995\pi$
$877$	−0.557245	−0.0188168	−0.00940841	+	0.999956i	$0.502995\pi$
$878$	2.69244	0.0908656
$879$	−5.82134	−0.196349
$880$	0	0
$881$	8.35432	0.281464	0.140732	−	0.990048i	$-0.455054\pi$
$881$	8.35432	0.281464	0.140732	+	0.990048i	$0.455054\pi$
$882$	−5.08494	−0.171219
$883$	19.4842	0.655695	0.327847	−	0.944731i	$-0.393677\pi$
$883$	19.4842	0.655695	0.327847	+	0.944731i	$0.393677\pi$
$884$	18.6132	0.626030
$885$	0	0
$886$	−5.46329	−0.183543
$887$	−35.9373	−1.20666	−0.603328	−	0.797493i	$-0.706159\pi$
$887$	−35.9373	−1.20666	−0.603328	+	0.797493i	$0.706159\pi$
$888$	17.3951	0.583742
$889$	−2.53638	−0.0850674
$890$	0	0
$891$	0.308470	0.0103341
$892$	24.2116	0.810666
$893$	0	0
$894$	−4.71237	−0.157605
$895$	0	0
$896$	−13.1325	−0.438725
$897$	2.59105	0.0865126
$898$	−12.0201	−0.401116
$899$	−2.15632	−0.0719172
$900$	0	0
$901$	−12.7120	−0.423499
$902$	−1.38802	−0.0462160
$903$	12.3783	0.411924
$904$	−21.8721	−0.727455
$905$	0	0
$906$	3.69933	0.122902
$907$	−48.9503	−1.62537	−0.812683	−	0.582706i	$-0.801994\pi$
$907$	−48.9503	−1.62537	−0.812683	+	0.582706i	$0.801994\pi$
$908$	11.7110	0.388644
$909$	−26.9034	−0.892330
$910$	0	0
$911$	19.7811	0.655376	0.327688	−	0.944786i	$-0.393731\pi$
$911$	19.7811	0.655376	0.327688	+	0.944786i	$0.393731\pi$
$912$	0	0
$913$	−9.88984	−0.327306
$914$	3.26935	0.108141
$915$	0	0
$916$	1.04145	0.0344103
$917$	20.6276	0.681182
$918$	−6.16141	−0.203357
$919$	26.0582	0.859581	0.429791	−	0.902929i	$-0.358587\pi$
$919$	26.0582	0.859581	0.429791	+	0.902929i	$0.358587\pi$
$920$	0	0
$921$	21.4618	0.707190
$922$	−15.3337	−0.504988
$923$	54.0298	1.77841
$924$	1.55058	0.0510104
$925$	0	0
$926$	−10.1484	−0.333498
$927$	−6.81602	−0.223868
$928$	−4.16406	−0.136692
$929$	−7.97799	−0.261749	−0.130875	−	0.991399i	$-0.541779\pi$
$929$	−7.97799	−0.261749	−0.130875	+	0.991399i	$0.541779\pi$
$930$	0	0
$931$	0	0
$932$	−12.7211	−0.416695
$933$	−25.6505	−0.839761
$934$	7.99883	0.261730
$935$	0	0
$936$	−14.0272	−0.458495
$937$	−46.6542	−1.52413	−0.762063	−	0.647502i	$-0.775814\pi$
$937$	−46.6542	−1.52413	−0.762063	+	0.647502i	$0.775814\pi$
$938$	−2.87426	−0.0938479
$939$	−3.63928	−0.118763
$940$	0	0
$941$	1.40100	0.0456713	0.0228356	−	0.999739i	$-0.492731\pi$
$941$	1.40100	0.0456713	0.0228356	+	0.999739i	$0.492731\pi$
$942$	−4.83798	−0.157630
$943$	2.50251	0.0814930
$944$	16.9594	0.551981
$945$	0	0
$946$	3.40880	0.110830
$947$	−38.2501	−1.24296	−0.621480	−	0.783430i	$-0.713468\pi$
$947$	−38.2501	−1.24296	−0.621480	+	0.783430i	$0.713468\pi$
$948$	−11.0253	−0.358085
$949$	50.7900	1.64871
$950$	0	0
$951$	32.6326	1.05819
$952$	5.34738	0.173309
$953$	54.4390	1.76345	0.881726	−	0.471763i	$-0.156382\pi$
$953$	54.4390	1.76345	0.881726	+	0.471763i	$0.156382\pi$
$954$	4.50764	0.145940
$955$	0	0
$956$	−6.49610	−0.210099
$957$	0.637869	0.0206194
$958$	−4.17989	−0.135046
$959$	5.96462	0.192608
$960$	0	0
$961$	−24.7042	−0.796911
$962$	−18.1650	−0.585662
$963$	−6.96456	−0.224430
$964$	30.6819	0.988197
$965$	0	0
$966$	0.350250	0.0112691
$967$	−4.98919	−0.160442	−0.0802208	−	0.996777i	$-0.525563\pi$
$967$	−4.98919	−0.160442	−0.0802208	+	0.996777i	$0.525563\pi$
$968$	−18.6995	−0.601026
$969$	0	0
$970$	0	0
$971$	14.5401	0.466614	0.233307	−	0.972403i	$-0.425045\pi$
$971$	14.5401	0.466614	0.233307	+	0.972403i	$0.425045\pi$
$972$	−28.0763	−0.900548
$973$	−7.03091	−0.225401
$974$	−14.1861	−0.454553
$975$	0	0
$976$	−12.3592	−0.395609
$977$	26.1353	0.836141	0.418071	−	0.908415i	$-0.362706\pi$
$977$	26.1353	0.836141	0.418071	+	0.908415i	$0.362706\pi$
$978$	−0.0337829	−0.00108026
$979$	−11.3905	−0.364043
$980$	0	0
$981$	−11.9347	−0.381046
$982$	−1.46300	−0.0466861
$983$	46.1160	1.47087	0.735436	−	0.677595i	$-0.236978\pi$
$983$	46.1160	1.47087	0.735436	+	0.677595i	$0.236978\pi$
$984$	7.64345	0.243664
$985$	0	0
$986$	1.03505	0.0329626
$987$	12.8606	0.409356
$988$	0	0
$989$	−6.14585	−0.195427
$990$	0	0
$991$	31.3279	0.995164	0.497582	−	0.867417i	$-0.334221\pi$
$991$	31.3279	0.995164	0.497582	+	0.867417i	$0.334221\pi$
$992$	12.1577	0.386009
$993$	−37.1877	−1.18011
$994$	7.30357	0.231655
$995$	0	0
$996$	25.6251	0.811962
$997$	21.6144	0.684536	0.342268	−	0.939602i	$-0.388805\pi$
$997$	21.6144	0.684536	0.342268	+	0.939602i	$0.388805\pi$
$998$	−15.1659	−0.480069
$999$	−47.9944	−1.51848

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.bt.1.3		6
5.2	odd	4		1805.2.b.g.1084.3		6
5.3	odd	4		1805.2.b.g.1084.4		6
5.4	even	2	inner	9025.2.a.bt.1.4		6
19.8	odd	6		475.2.e.g.26.3		12
19.12	odd	6		475.2.e.g.201.3		12
19.18	odd	2		9025.2.a.bu.1.4		6
95.8	even	12		95.2.i.b.64.3	yes	12
95.12	even	12		95.2.i.b.49.3	✓	12
95.18	even	4		1805.2.b.f.1084.3		6
95.27	even	12		95.2.i.b.64.4	yes	12
95.37	even	4		1805.2.b.f.1084.4		6
95.69	odd	6		475.2.e.g.201.4		12
95.84	odd	6		475.2.e.g.26.4		12
95.88	even	12		95.2.i.b.49.4	yes	12
95.94	odd	2		9025.2.a.bu.1.3		6
285.8	odd	12		855.2.be.d.64.4		12
285.107	odd	12		855.2.be.d.334.4		12
285.122	odd	12		855.2.be.d.64.3		12
285.278	odd	12		855.2.be.d.334.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.i.b.49.3	✓	12	95.12	even	12
95.2.i.b.49.4	yes	12	95.88	even	12
95.2.i.b.64.3	yes	12	95.8	even	12
95.2.i.b.64.4	yes	12	95.27	even	12
475.2.e.g.26.3		12	19.8	odd	6
475.2.e.g.26.4		12	95.84	odd	6
475.2.e.g.201.3		12	19.12	odd	6
475.2.e.g.201.4		12	95.69	odd	6
855.2.be.d.64.3		12	285.122	odd	12
855.2.be.d.64.4		12	285.8	odd	12
855.2.be.d.334.3		12	285.278	odd	12
855.2.be.d.334.4		12	285.107	odd	12
1805.2.b.f.1084.3		6	95.18	even	4
1805.2.b.f.1084.4		6	95.37	even	4
1805.2.b.g.1084.3		6	5.2	odd	4
1805.2.b.g.1084.4		6	5.3	odd	4
9025.2.a.bt.1.3		6	1.1	even	1	trivial
9025.2.a.bt.1.4		6	5.4	even	2	inner
9025.2.a.bu.1.3		6	95.94	odd	2
9025.2.a.bu.1.4		6	19.18	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:	\( N \)	\(=\)	\( 9025 = 5^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9025.a (trivial)

\(f(q)\)	\(=\)	\(q-0.471884 q^{2} +1.04022 q^{3} -1.77733 q^{4} -0.490864 q^{6} -1.17540 q^{7} +1.78246 q^{8} -1.91794 q^{9} +O(q^{10})\)	\(q-0.471884 q^{2} +1.04022 q^{3} -1.77733 q^{4} -0.490864 q^{6} -1.17540 q^{7} +1.78246 q^{8} -1.91794 q^{9} +0.713538 q^{11} -1.84881 q^{12} +4.10315 q^{13} +0.554651 q^{14} +2.71354 q^{16} -2.55233 q^{17} +0.905045 q^{18} -1.22267 q^{21} -0.336707 q^{22} +0.607061 q^{23} +1.85415 q^{24} -1.93621 q^{26} -5.11575 q^{27} +2.08907 q^{28} +0.859386 q^{29} -2.50914 q^{31} -4.84539 q^{32} +0.742237 q^{33} +1.20440 q^{34} +3.40880 q^{36} +9.38171 q^{37} +4.26819 q^{39} +4.12234 q^{41} +0.576960 q^{42} -10.1239 q^{43} -1.26819 q^{44} -0.286462 q^{46} -10.5184 q^{47} +2.82268 q^{48} -5.61844 q^{49} -2.65498 q^{51} -7.29264 q^{52} +4.98057 q^{53} +2.41404 q^{54} -2.09510 q^{56} -0.405530 q^{58} +6.24992 q^{59} -4.55465 q^{61} +1.18402 q^{62} +2.25434 q^{63} -3.14061 q^{64} -0.350250 q^{66} -5.18210 q^{67} +4.53632 q^{68} +0.631477 q^{69} +13.1679 q^{71} -3.41865 q^{72} +12.3783 q^{73} -4.42708 q^{74} -0.838691 q^{77} -2.01409 q^{78} +5.96345 q^{79} +0.432310 q^{81} -1.94527 q^{82} -13.8603 q^{83} +2.17309 q^{84} +4.77733 q^{86} +0.893952 q^{87} +1.27185 q^{88} -15.9635 q^{89} -4.82284 q^{91} -1.07894 q^{92} -2.61006 q^{93} +4.96345 q^{94} -5.04028 q^{96} +16.7020 q^{97} +2.65125 q^{98} -1.36852 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{4} + 12 q^{6} + 8 q^{9}+O(q^{10}) \) 6 * q + 2 * q^4 + 12 * q^6 + 8 * q^9	\( 6 q + 2 q^{4} + 12 q^{6} + 8 q^{9} + 2 q^{11} - 22 q^{14} + 14 q^{16} - 20 q^{21} + 2 q^{24} - 22 q^{26} + 12 q^{29} - 30 q^{31} - 10 q^{34} - 14 q^{36} - 2 q^{39} - 12 q^{41} + 20 q^{44} - 4 q^{46} + 2 q^{49} - 40 q^{51} - 4 q^{54} - 46 q^{56} - 20 q^{59} - 2 q^{61} - 12 q^{64} + 6 q^{66} - 18 q^{69} + 2 q^{71} - 22 q^{74} - 24 q^{79} + 14 q^{81} - 48 q^{84} + 16 q^{86} - 36 q^{89} + 24 q^{91} - 30 q^{94} + 26 q^{96} - 30 q^{99}+O(q^{100}) \) 6 * q + 2 * q^4 + 12 * q^6 + 8 * q^9 + 2 * q^11 - 22 * q^14 + 14 * q^16 - 20 * q^21 + 2 * q^24 - 22 * q^26 + 12 * q^29 - 30 * q^31 - 10 * q^34 - 14 * q^36 - 2 * q^39 - 12 * q^41 + 20 * q^44 - 4 * q^46 + 2 * q^49 - 40 * q^51 - 4 * q^54 - 46 * q^56 - 20 * q^59 - 2 * q^61 - 12 * q^64 + 6 * q^66 - 18 * q^69 + 2 * q^71 - 22 * q^74 - 24 * q^79 + 14 * q^81 - 48 * q^84 + 16 * q^86 - 36 * q^89 + 24 * q^91 - 30 * q^94 + 26 * q^96 - 30 * q^99

Introduction

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