LMFDB - Embedded newform 2475.2.a.ba.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.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:	$19.7629745003$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.48119 q^{2} +0.193937 q^{4} +1.19394 q^{7} -2.67513 q^{8} +O(q^{10})$	$q+1.48119 q^{2} +0.193937 q^{4} +1.19394 q^{7} -2.67513 q^{8} -1.00000 q^{11} +0.806063 q^{13} +1.76845 q^{14} -4.35026 q^{16} -3.76845 q^{17} -5.35026 q^{19} -1.48119 q^{22} -4.00000 q^{23} +1.19394 q^{26} +0.231548 q^{28} +4.31265 q^{29} +0.962389 q^{31} -1.09332 q^{32} -5.58181 q^{34} +1.61213 q^{37} -7.92478 q^{38} -9.08840 q^{41} +4.41819 q^{43} -0.193937 q^{44} -5.92478 q^{46} -12.3127 q^{47} -5.57452 q^{49} +0.156325 q^{52} -1.42548 q^{53} -3.19394 q^{56} +6.38787 q^{58} -13.2750 q^{59} -0.0752228 q^{61} +1.42548 q^{62} +7.08110 q^{64} -2.70052 q^{67} -0.730841 q^{68} +14.0508 q^{71} -10.7308 q^{73} +2.38787 q^{74} -1.03761 q^{76} -1.19394 q^{77} +13.9756 q^{79} -13.4617 q^{82} +9.89446 q^{83} +6.54420 q^{86} +2.67513 q^{88} -16.8872 q^{89} +0.962389 q^{91} -0.775746 q^{92} -18.2374 q^{94} +11.4763 q^{97} -8.25694 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + q^{4} + 4 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - q^2 + q^4 + 4 * q^7 - 3 * q^8	$ 3 q - q^{2} + q^{4} + 4 q^{7} - 3 q^{8} - 3 q^{11} + 2 q^{13} - 6 q^{14} - 3 q^{16} - 6 q^{19} + q^{22} - 12 q^{23} + 4 q^{26} + 12 q^{28} - 8 q^{29} - 8 q^{31} + 3 q^{32} - 18 q^{34} + 4 q^{37} - 2 q^{38} - 8 q^{41} + 12 q^{43} - q^{44} + 4 q^{46} - 16 q^{47} - 5 q^{49} - 10 q^{52} - 16 q^{53} - 10 q^{56} + 20 q^{58} - 8 q^{59} - 22 q^{61} + 16 q^{62} - 11 q^{64} + 12 q^{67} + 20 q^{68} + 12 q^{71} - 10 q^{73} + 8 q^{74} - 14 q^{76} - 4 q^{77} - 10 q^{79} + 4 q^{82} + 10 q^{83} + 10 q^{86} + 3 q^{88} - 18 q^{89} - 8 q^{91} - 4 q^{92} - 12 q^{94} + 16 q^{97} - 21 q^{98}+O(q^{100}) $ 3 * q - q^2 + q^4 + 4 * q^7 - 3 * q^8 - 3 * q^11 + 2 * q^13 - 6 * q^14 - 3 * q^16 - 6 * q^19 + q^22 - 12 * q^23 + 4 * q^26 + 12 * q^28 - 8 * q^29 - 8 * q^31 + 3 * q^32 - 18 * q^34 + 4 * q^37 - 2 * q^38 - 8 * q^41 + 12 * q^43 - q^44 + 4 * q^46 - 16 * q^47 - 5 * q^49 - 10 * q^52 - 16 * q^53 - 10 * q^56 + 20 * q^58 - 8 * q^59 - 22 * q^61 + 16 * q^62 - 11 * q^64 + 12 * q^67 + 20 * q^68 + 12 * q^71 - 10 * q^73 + 8 * q^74 - 14 * q^76 - 4 * q^77 - 10 * q^79 + 4 * q^82 + 10 * q^83 + 10 * q^86 + 3 * q^88 - 18 * q^89 - 8 * q^91 - 4 * q^92 - 12 * q^94 + 16 * q^97 - 21 * q^98

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 $

$2$	1.48119	1.04736	0.523681	−	0.851914i	$-0.324558\pi$
$2$	1.48119	1.04736	0.523681	+	0.851914i	$0.324558\pi$
$3$	0	0
$3$	0	0
$4$	0.193937	0.0969683
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.19394	0.451266	0.225633	−	0.974212i	$-0.427555\pi$
$7$	1.19394	0.451266	0.225633	+	0.974212i	$0.427555\pi$
$8$	−2.67513	−0.945802
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.806063	0.223562	0.111781	−	0.993733i	$-0.464345\pi$
$13$	0.806063	0.223562	0.111781	+	0.993733i	$0.464345\pi$
$14$	1.76845	0.472639
$15$	0	0
$16$	−4.35026	−1.08757
$17$	−3.76845	−0.913984	−0.456992	−	0.889471i	$-0.651073\pi$
$17$	−3.76845	−0.913984	−0.456992	+	0.889471i	$0.651073\pi$
$18$	0	0
$19$	−5.35026	−1.22743	−0.613717	−	0.789526i	$-0.710326\pi$
$19$	−5.35026	−1.22743	−0.613717	+	0.789526i	$0.710326\pi$
$20$	0	0
$21$	0	0
$22$	−1.48119	−0.315792
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	0	0
$26$	1.19394	0.234150
$27$	0	0
$28$	0.231548	0.0437585
$29$	4.31265	0.800839	0.400420	−	0.916332i	$-0.368864\pi$
$29$	4.31265	0.800839	0.400420	+	0.916332i	$0.368864\pi$
$30$	0	0
$31$	0.962389	0.172850	0.0864250	−	0.996258i	$-0.472456\pi$
$31$	0.962389	0.172850	0.0864250	+	0.996258i	$0.472456\pi$
$32$	−1.09332	−0.193274
$33$	0	0
$34$	−5.58181	−0.957272
$35$	0	0
$36$	0	0
$37$	1.61213	0.265032	0.132516	−	0.991181i	$-0.457694\pi$
$37$	1.61213	0.265032	0.132516	+	0.991181i	$0.457694\pi$
$38$	−7.92478	−1.28557
$39$	0	0
$40$	0	0
$41$	−9.08840	−1.41937	−0.709685	−	0.704520i	$-0.751163\pi$
$41$	−9.08840	−1.41937	−0.709685	+	0.704520i	$0.751163\pi$
$42$	0	0
$43$	4.41819	0.673768	0.336884	−	0.941546i	$-0.390627\pi$
$43$	4.41819	0.673768	0.336884	+	0.941546i	$0.390627\pi$
$44$	−0.193937	−0.0292370
$45$	0	0
$46$	−5.92478	−0.873561
$47$	−12.3127	−1.79598	−0.897992	−	0.440011i	$-0.854974\pi$
$47$	−12.3127	−1.79598	−0.897992	+	0.440011i	$0.854974\pi$
$48$	0	0
$49$	−5.57452	−0.796359
$50$	0	0
$51$	0	0
$52$	0.156325	0.0216784
$53$	−1.42548	−0.195805	−0.0979027	−	0.995196i	$-0.531213\pi$
$53$	−1.42548	−0.195805	−0.0979027	+	0.995196i	$0.531213\pi$
$54$	0	0
$55$	0	0
$56$	−3.19394	−0.426808
$57$	0	0
$58$	6.38787	0.838769
$59$	−13.2750	−1.72826	−0.864131	−	0.503266i	$-0.832132\pi$
$59$	−13.2750	−1.72826	−0.864131	+	0.503266i	$0.832132\pi$
$60$	0	0
$61$	−0.0752228	−0.00963129	−0.00481565	−	0.999988i	$-0.501533\pi$
$61$	−0.0752228	−0.00963129	−0.00481565	+	0.999988i	$0.501533\pi$
$62$	1.42548	0.181037
$63$	0	0
$64$	7.08110	0.885138
$65$	0	0
$66$	0	0
$67$	−2.70052	−0.329921	−0.164961	−	0.986300i	$-0.552750\pi$
$67$	−2.70052	−0.329921	−0.164961	+	0.986300i	$0.552750\pi$
$68$	−0.730841	−0.0886274
$69$	0	0
$70$	0	0
$71$	14.0508	1.66752	0.833761	−	0.552126i	$-0.186183\pi$
$71$	14.0508	1.66752	0.833761	+	0.552126i	$0.186183\pi$
$72$	0	0
$73$	−10.7308	−1.25595	−0.627975	−	0.778234i	$-0.716116\pi$
$73$	−10.7308	−1.25595	−0.627975	+	0.778234i	$0.716116\pi$
$74$	2.38787	0.277585
$75$	0	0
$76$	−1.03761	−0.119022
$77$	−1.19394	−0.136062
$78$	0	0
$79$	13.9756	1.57237	0.786187	−	0.617989i	$-0.212052\pi$
$79$	13.9756	1.57237	0.786187	+	0.617989i	$0.212052\pi$
$80$	0	0
$81$	0	0
$82$	−13.4617	−1.48659
$83$	9.89446	1.08606	0.543029	−	0.839714i	$-0.317277\pi$
$83$	9.89446	1.08606	0.543029	+	0.839714i	$0.317277\pi$
$84$	0	0
$85$	0	0
$86$	6.54420	0.705679
$87$	0	0
$88$	2.67513	0.285170
$89$	−16.8872	−1.79004	−0.895018	−	0.446030i	$-0.852837\pi$
$89$	−16.8872	−1.79004	−0.895018	+	0.446030i	$0.852837\pi$
$90$	0	0
$91$	0.962389	0.100886
$92$	−0.775746	−0.0808771
$93$	0	0
$94$	−18.2374	−1.88105
$95$	0	0
$96$	0	0
$97$	11.4763	1.16524	0.582619	−	0.812745i	$-0.302028\pi$
$97$	11.4763	1.16524	0.582619	+	0.812745i	$0.302028\pi$
$98$	−8.25694	−0.834077
$99$	0	0
$100$	0	0
$101$	−10.7612	−1.07078	−0.535388	−	0.844606i	$-0.679834\pi$
$101$	−10.7612	−1.07078	−0.535388	+	0.844606i	$0.679834\pi$
$102$	0	0
$103$	16.9380	1.66895	0.834473	−	0.551049i	$-0.185772\pi$
$103$	16.9380	1.66895	0.834473	+	0.551049i	$0.185772\pi$
$104$	−2.15633	−0.211445
$105$	0	0
$106$	−2.11142	−0.205079
$107$	−8.28233	−0.800683	−0.400342	−	0.916366i	$-0.631109\pi$
$107$	−8.28233	−0.800683	−0.400342	+	0.916366i	$0.631109\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	0	0
$112$	−5.19394	−0.490781
$113$	−2.26187	−0.212778	−0.106389	−	0.994325i	$-0.533929\pi$
$113$	−2.26187	−0.212778	−0.106389	+	0.994325i	$0.533929\pi$
$114$	0	0
$115$	0	0
$116$	0.836381	0.0776560
$117$	0	0
$118$	−19.6629	−1.81012
$119$	−4.49929	−0.412449
$120$	0	0
$121$	1.00000	0.0909091
$122$	−0.111420	−0.0100875
$123$	0	0
$124$	0.186642	0.0167610
$125$	0	0
$126$	0	0
$127$	−13.8192	−1.22626	−0.613130	−	0.789982i	$-0.710090\pi$
$127$	−13.8192	−1.22626	−0.613130	+	0.789982i	$0.710090\pi$
$128$	12.6751	1.12033
$129$	0	0
$130$	0	0
$131$	5.92478	0.517650	0.258825	−	0.965924i	$-0.416665\pi$
$131$	5.92478	0.517650	0.258825	+	0.965924i	$0.416665\pi$
$132$	0	0
$133$	−6.38787	−0.553899
$134$	−4.00000	−0.345547
$135$	0	0
$136$	10.0811	0.864447
$137$	−3.35026	−0.286232	−0.143116	−	0.989706i	$-0.545712\pi$
$137$	−3.35026	−0.286232	−0.143116	+	0.989706i	$0.545712\pi$
$138$	0	0
$139$	−21.1998	−1.79814	−0.899072	−	0.437800i	$-0.855758\pi$
$139$	−21.1998	−1.79814	−0.899072	+	0.437800i	$0.855758\pi$
$140$	0	0
$141$	0	0
$142$	20.8119	1.74650
$143$	−0.806063	−0.0674064
$144$	0	0
$145$	0	0
$146$	−15.8945	−1.31543
$147$	0	0
$148$	0.312650	0.0256997
$149$	−6.38787	−0.523315	−0.261657	−	0.965161i	$-0.584269\pi$
$149$	−6.38787	−0.523315	−0.261657	+	0.965161i	$0.584269\pi$
$150$	0	0
$151$	−2.64974	−0.215633	−0.107816	−	0.994171i	$-0.534386\pi$
$151$	−2.64974	−0.215633	−0.107816	+	0.994171i	$0.534386\pi$
$152$	14.3127	1.16091
$153$	0	0
$154$	−1.76845	−0.142506
$155$	0	0
$156$	0	0
$157$	1.61213	0.128662	0.0643309	−	0.997929i	$-0.479509\pi$
$157$	1.61213	0.128662	0.0643309	+	0.997929i	$0.479509\pi$
$158$	20.7005	1.64685
$159$	0	0
$160$	0	0
$161$	−4.77575	−0.376382
$162$	0	0
$163$	−0.312650	−0.0244887	−0.0122443	−	0.999925i	$-0.503898\pi$
$163$	−0.312650	−0.0244887	−0.0122443	+	0.999925i	$0.503898\pi$
$164$	−1.76257	−0.137634
$165$	0	0
$166$	14.6556	1.13750
$167$	−0.493413	−0.0381815	−0.0190907	−	0.999818i	$-0.506077\pi$
$167$	−0.493413	−0.0381815	−0.0190907	+	0.999818i	$0.506077\pi$
$168$	0	0
$169$	−12.3503	−0.950020
$170$	0	0
$171$	0	0
$172$	0.856849	0.0653341
$173$	23.3054	1.77187	0.885937	−	0.463806i	$-0.153517\pi$
$173$	23.3054	1.77187	0.885937	+	0.463806i	$0.153517\pi$
$174$	0	0
$175$	0	0
$176$	4.35026	0.327913
$177$	0	0
$178$	−25.0132	−1.87482
$179$	10.7005	0.799795	0.399897	−	0.916560i	$-0.369046\pi$
$179$	10.7005	0.799795	0.399897	+	0.916560i	$0.369046\pi$
$180$	0	0
$181$	−7.79877	−0.579678	−0.289839	−	0.957075i	$-0.593602\pi$
$181$	−7.79877	−0.579678	−0.289839	+	0.957075i	$0.593602\pi$
$182$	1.42548	0.105664
$183$	0	0
$184$	10.7005	0.788853
$185$	0	0
$186$	0	0
$187$	3.76845	0.275577
$188$	−2.38787	−0.174154
$189$	0	0
$190$	0	0
$191$	−1.29948	−0.0940268	−0.0470134	−	0.998894i	$-0.514970\pi$
$191$	−1.29948	−0.0940268	−0.0470134	+	0.998894i	$0.514970\pi$
$192$	0	0
$193$	8.59498	0.618680	0.309340	−	0.950951i	$-0.399892\pi$
$193$	8.59498	0.618680	0.309340	+	0.950951i	$0.399892\pi$
$194$	16.9986	1.22043
$195$	0	0
$196$	−1.08110	−0.0772216
$197$	20.7064	1.47527	0.737635	−	0.675200i	$-0.235943\pi$
$197$	20.7064	1.47527	0.737635	+	0.675200i	$0.235943\pi$
$198$	0	0
$199$	5.55149	0.393535	0.196767	−	0.980450i	$-0.436956\pi$
$199$	5.55149	0.393535	0.196767	+	0.980450i	$0.436956\pi$
$200$	0	0
$201$	0	0
$202$	−15.9394	−1.12149
$203$	5.14903	0.361391
$204$	0	0
$205$	0	0
$206$	25.0884	1.74799
$207$	0	0
$208$	−3.50659	−0.243138
$209$	5.35026	0.370085
$210$	0	0
$211$	−18.4993	−1.27354	−0.636772	−	0.771052i	$-0.719731\pi$
$211$	−18.4993	−1.27354	−0.636772	+	0.771052i	$0.719731\pi$
$212$	−0.276454	−0.0189869
$213$	0	0
$214$	−12.2677	−0.838606
$215$	0	0
$216$	0	0
$217$	1.14903	0.0780013
$218$	−14.8119	−1.00319
$219$	0	0
$220$	0	0
$221$	−3.03761	−0.204332
$222$	0	0
$223$	17.6121	1.17940	0.589698	−	0.807624i	$-0.299247\pi$
$223$	17.6121	1.17940	0.589698	+	0.807624i	$0.299247\pi$
$224$	−1.30536	−0.0872178
$225$	0	0
$226$	−3.35026	−0.222856
$227$	17.4314	1.15696	0.578480	−	0.815696i	$-0.303646\pi$
$227$	17.4314	1.15696	0.578480	+	0.815696i	$0.303646\pi$
$228$	0	0
$229$	13.0738	0.863942	0.431971	−	0.901888i	$-0.357818\pi$
$229$	13.0738	0.863942	0.431971	+	0.901888i	$0.357818\pi$
$230$	0	0
$231$	0	0
$232$	−11.5369	−0.757435
$233$	13.8437	0.906929	0.453465	−	0.891274i	$-0.350188\pi$
$233$	13.8437	0.906929	0.453465	+	0.891274i	$0.350188\pi$
$234$	0	0
$235$	0	0
$236$	−2.57452	−0.167587
$237$	0	0
$238$	−6.66433	−0.431984
$239$	12.3733	0.800361	0.400181	−	0.916436i	$-0.368947\pi$
$239$	12.3733	0.800361	0.400181	+	0.916436i	$0.368947\pi$
$240$	0	0
$241$	−24.5501	−1.58141	−0.790705	−	0.612198i	$-0.790286\pi$
$241$	−24.5501	−1.58141	−0.790705	+	0.612198i	$0.790286\pi$
$242$	1.48119	0.0952148
$243$	0	0
$244$	−0.0145884	−0.000933930 0
$245$	0	0
$246$	0	0
$247$	−4.31265	−0.274407
$248$	−2.57452	−0.163482
$249$	0	0
$250$	0	0
$251$	−13.9003	−0.877382	−0.438691	−	0.898638i	$-0.644558\pi$
$251$	−13.9003	−0.877382	−0.438691	+	0.898638i	$0.644558\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	−20.4690	−1.28434
$255$	0	0
$256$	4.61213	0.288258
$257$	18.8872	1.17815	0.589075	−	0.808079i	$-0.299492\pi$
$257$	18.8872	1.17815	0.589075	+	0.808079i	$0.299492\pi$
$258$	0	0
$259$	1.92478	0.119600
$260$	0	0
$261$	0	0
$262$	8.77575	0.542167
$263$	−20.8061	−1.28296	−0.641478	−	0.767141i	$-0.721679\pi$
$263$	−20.8061	−1.28296	−0.641478	+	0.767141i	$0.721679\pi$
$264$	0	0
$265$	0	0
$266$	−9.46168	−0.580133
$267$	0	0
$268$	−0.523730	−0.0319919
$269$	32.3996	1.97544	0.987720	−	0.156233i	$-0.0499351\pi$
$269$	32.3996	1.97544	0.987720	+	0.156233i	$0.0499351\pi$
$270$	0	0
$271$	−16.8265	−1.02214	−0.511069	−	0.859539i	$-0.670751\pi$
$271$	−16.8265	−1.02214	−0.511069	+	0.859539i	$0.670751\pi$
$272$	16.3938	0.994017
$273$	0	0
$274$	−4.96239	−0.299789
$275$	0	0
$276$	0	0
$277$	16.9076	1.01588	0.507941	−	0.861392i	$-0.330407\pi$
$277$	16.9076	1.01588	0.507941	+	0.861392i	$0.330407\pi$
$278$	−31.4010	−1.88331
$279$	0	0
$280$	0	0
$281$	−5.61213	−0.334791	−0.167396	−	0.985890i	$-0.553536\pi$
$281$	−5.61213	−0.334791	−0.167396	+	0.985890i	$0.553536\pi$
$282$	0	0
$283$	5.81924	0.345918	0.172959	−	0.984929i	$-0.444667\pi$
$283$	5.81924	0.345918	0.172959	+	0.984929i	$0.444667\pi$
$284$	2.72496	0.161697
$285$	0	0
$286$	−1.19394	−0.0705989
$287$	−10.8510	−0.640512
$288$	0	0
$289$	−2.79877	−0.164633
$290$	0	0
$291$	0	0
$292$	−2.08110	−0.121787
$293$	8.29218	0.484434	0.242217	−	0.970222i	$-0.422125\pi$
$293$	8.29218	0.484434	0.242217	+	0.970222i	$0.422125\pi$
$294$	0	0
$295$	0	0
$296$	−4.31265	−0.250668
$297$	0	0
$298$	−9.46168	−0.548100
$299$	−3.22425	−0.186463
$300$	0	0
$301$	5.27504	0.304048
$302$	−3.92478	−0.225846
$303$	0	0
$304$	23.2750	1.33492
$305$	0	0
$306$	0	0
$307$	25.6688	1.46500	0.732498	−	0.680770i	$-0.238354\pi$
$307$	25.6688	1.46500	0.732498	+	0.680770i	$0.238354\pi$
$308$	−0.231548	−0.0131937
$309$	0	0
$310$	0	0
$311$	15.7235	0.891601	0.445800	−	0.895132i	$-0.352919\pi$
$311$	15.7235	0.891601	0.445800	+	0.895132i	$0.352919\pi$
$312$	0	0
$313$	−26.8627	−1.51837	−0.759186	−	0.650874i	$-0.774403\pi$
$313$	−26.8627	−1.51837	−0.759186	+	0.650874i	$0.774403\pi$
$314$	2.38787	0.134755
$315$	0	0
$316$	2.71037	0.152470
$317$	−0.710373	−0.0398985	−0.0199492	−	0.999801i	$-0.506350\pi$
$317$	−0.710373	−0.0398985	−0.0199492	+	0.999801i	$0.506350\pi$
$318$	0	0
$319$	−4.31265	−0.241462
$320$	0	0
$321$	0	0
$322$	−7.07381	−0.394208
$323$	20.1622	1.12186
$324$	0	0
$325$	0	0
$326$	−0.463096	−0.0256485
$327$	0	0
$328$	24.3127	1.34244
$329$	−14.7005	−0.810466
$330$	0	0
$331$	0.962389	0.0528977	0.0264488	−	0.999650i	$-0.491580\pi$
$331$	0.962389	0.0528977	0.0264488	+	0.999650i	$0.491580\pi$
$332$	1.91890	0.105313
$333$	0	0
$334$	−0.730841	−0.0399898
$335$	0	0
$336$	0	0
$337$	19.8192	1.07962	0.539811	−	0.841786i	$-0.318496\pi$
$337$	19.8192	1.07962	0.539811	+	0.841786i	$0.318496\pi$
$338$	−18.2931	−0.995015
$339$	0	0
$340$	0	0
$341$	−0.962389	−0.0521163
$342$	0	0
$343$	−15.0132	−0.810635
$344$	−11.8192	−0.637251
$345$	0	0
$346$	34.5198	1.85579
$347$	−6.20711	−0.333215	−0.166608	−	0.986023i	$-0.553281\pi$
$347$	−6.20711	−0.333215	−0.166608	+	0.986023i	$0.553281\pi$
$348$	0	0
$349$	4.44851	0.238123	0.119062	−	0.992887i	$-0.462011\pi$
$349$	4.44851	0.238123	0.119062	+	0.992887i	$0.462011\pi$
$350$	0	0
$351$	0	0
$352$	1.09332	0.0582742
$353$	−6.57452	−0.349926	−0.174963	−	0.984575i	$-0.555981\pi$
$353$	−6.57452	−0.349926	−0.174963	+	0.984575i	$0.555981\pi$
$354$	0	0
$355$	0	0
$356$	−3.27504	−0.173577
$357$	0	0
$358$	15.8496	0.837675
$359$	−8.62530	−0.455226	−0.227613	−	0.973752i	$-0.573092\pi$
$359$	−8.62530	−0.455226	−0.227613	+	0.973752i	$0.573092\pi$
$360$	0	0
$361$	9.62530	0.506595
$362$	−11.5515	−0.607133
$363$	0	0
$364$	0.186642	0.00978272
$365$	0	0
$366$	0	0
$367$	23.0132	1.20128	0.600639	−	0.799520i	$-0.294913\pi$
$367$	23.0132	1.20128	0.600639	+	0.799520i	$0.294913\pi$
$368$	17.4010	0.907092
$369$	0	0
$370$	0	0
$371$	−1.70194	−0.0883602
$372$	0	0
$373$	28.1925	1.45975	0.729877	−	0.683579i	$-0.239577\pi$
$373$	28.1925	1.45975	0.729877	+	0.683579i	$0.239577\pi$
$374$	5.58181	0.288629
$375$	0	0
$376$	32.9380	1.69865
$377$	3.47627	0.179037
$378$	0	0
$379$	3.74798	0.192521	0.0962605	−	0.995356i	$-0.469312\pi$
$379$	3.74798	0.192521	0.0962605	+	0.995356i	$0.469312\pi$
$380$	0	0
$381$	0	0
$382$	−1.92478	−0.0984802
$383$	−1.76257	−0.0900632	−0.0450316	−	0.998986i	$-0.514339\pi$
$383$	−1.76257	−0.0900632	−0.0450316	+	0.998986i	$0.514339\pi$
$384$	0	0
$385$	0	0
$386$	12.7308	0.647983
$387$	0	0
$388$	2.22567	0.112991
$389$	−6.52373	−0.330766	−0.165383	−	0.986229i	$-0.552886\pi$
$389$	−6.52373	−0.330766	−0.165383	+	0.986229i	$0.552886\pi$
$390$	0	0
$391$	15.0738	0.762315
$392$	14.9126	0.753198
$393$	0	0
$394$	30.6702	1.54514
$395$	0	0
$396$	0	0
$397$	−23.6991	−1.18942	−0.594712	−	0.803939i	$-0.702734\pi$
$397$	−23.6991	−1.18942	−0.594712	+	0.803939i	$0.702734\pi$
$398$	8.22284	0.412174
$399$	0	0
$400$	0	0
$401$	−8.88717	−0.443804	−0.221902	−	0.975069i	$-0.571226\pi$
$401$	−8.88717	−0.443804	−0.221902	+	0.975069i	$0.571226\pi$
$402$	0	0
$403$	0.775746	0.0386427
$404$	−2.08698	−0.103831
$405$	0	0
$406$	7.62672	0.378508
$407$	−1.61213	−0.0799102
$408$	0	0
$409$	−4.85097	−0.239865	−0.119932	−	0.992782i	$-0.538268\pi$
$409$	−4.85097	−0.239865	−0.119932	+	0.992782i	$0.538268\pi$
$410$	0	0
$411$	0	0
$412$	3.28489	0.161835
$413$	−15.8496	−0.779906
$414$	0	0
$415$	0	0
$416$	−0.881286	−0.0432086
$417$	0	0
$418$	7.92478	0.387614
$419$	10.7005	0.522755	0.261377	−	0.965237i	$-0.415823\pi$
$419$	10.7005	0.522755	0.261377	+	0.965237i	$0.415823\pi$
$420$	0	0
$421$	−30.6009	−1.49139	−0.745697	−	0.666285i	$-0.767884\pi$
$421$	−30.6009	−1.49139	−0.745697	+	0.666285i	$0.767884\pi$
$422$	−27.4010	−1.33386
$423$	0	0
$424$	3.81336	0.185193
$425$	0	0
$426$	0	0
$427$	−0.0898112	−0.00434627
$428$	−1.60625	−0.0776409
$429$	0	0
$430$	0	0
$431$	5.92478	0.285386	0.142693	−	0.989767i	$-0.454424\pi$
$431$	5.92478	0.285386	0.142693	+	0.989767i	$0.454424\pi$
$432$	0	0
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	1.70194	0.0816956
$435$	0	0
$436$	−1.93937	−0.0928788
$437$	21.4010	1.02375
$438$	0	0
$439$	5.35026	0.255354	0.127677	−	0.991816i	$-0.459248\pi$
$439$	5.35026	0.255354	0.127677	+	0.991816i	$0.459248\pi$
$440$	0	0
$441$	0	0
$442$	−4.49929	−0.214010
$443$	−19.6873	−0.935374	−0.467687	−	0.883894i	$-0.654913\pi$
$443$	−19.6873	−0.935374	−0.467687	+	0.883894i	$0.654913\pi$
$444$	0	0
$445$	0	0
$446$	26.0870	1.23525
$447$	0	0
$448$	8.45439	0.399432
$449$	31.3357	1.47882	0.739411	−	0.673254i	$-0.235104\pi$
$449$	31.3357	1.47882	0.739411	+	0.673254i	$0.235104\pi$
$450$	0	0
$451$	9.08840	0.427956
$452$	−0.438658	−0.0206328
$453$	0	0
$454$	25.8192	1.21176
$455$	0	0
$456$	0	0
$457$	−37.5936	−1.75855	−0.879276	−	0.476312i	$-0.841973\pi$
$457$	−37.5936	−1.75855	−0.879276	+	0.476312i	$0.841973\pi$
$458$	19.3649	0.904860
$459$	0	0
$460$	0	0
$461$	−17.9854	−0.837664	−0.418832	−	0.908064i	$-0.637560\pi$
$461$	−17.9854	−0.837664	−0.418832	+	0.908064i	$0.637560\pi$
$462$	0	0
$463$	39.0132	1.81310	0.906548	−	0.422103i	$-0.138708\pi$
$463$	39.0132	1.81310	0.906548	+	0.422103i	$0.138708\pi$
$464$	−18.7612	−0.870965
$465$	0	0
$466$	20.5052	0.949884
$467$	−14.5501	−0.673297	−0.336649	−	0.941630i	$-0.609293\pi$
$467$	−14.5501	−0.673297	−0.336649	+	0.941630i	$0.609293\pi$
$468$	0	0
$469$	−3.22425	−0.148882
$470$	0	0
$471$	0	0
$472$	35.5125	1.63459
$473$	−4.41819	−0.203149
$474$	0	0
$475$	0	0
$476$	−0.872577	−0.0399945
$477$	0	0
$478$	18.3272	0.838268
$479$	−28.6253	−1.30792	−0.653962	−	0.756528i	$-0.726894\pi$
$479$	−28.6253	−1.30792	−0.653962	+	0.756528i	$0.726894\pi$
$480$	0	0
$481$	1.29948	0.0592510
$482$	−36.3634	−1.65631
$483$	0	0
$484$	0.193937	0.00881530
$485$	0	0
$486$	0	0
$487$	1.44992	0.0657022	0.0328511	−	0.999460i	$-0.489541\pi$
$487$	1.44992	0.0657022	0.0328511	+	0.999460i	$0.489541\pi$
$488$	0.201231	0.00910929
$489$	0	0
$490$	0	0
$491$	8.00000	0.361035	0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000	0.361035	0.180517	+	0.983572i	$0.442223\pi$
$492$	0	0
$493$	−16.2520	−0.731954
$494$	−6.38787	−0.287404
$495$	0	0
$496$	−4.18664	−0.187986
$497$	16.7757	0.752495
$498$	0	0
$499$	30.7005	1.37434	0.687172	−	0.726495i	$-0.258852\pi$
$499$	30.7005	1.37434	0.687172	+	0.726495i	$0.258852\pi$
$500$	0	0
$501$	0	0
$502$	−20.5891	−0.918937
$503$	19.7586	0.880993	0.440496	−	0.897754i	$-0.354802\pi$
$503$	19.7586	0.880993	0.440496	+	0.897754i	$0.354802\pi$
$504$	0	0
$505$	0	0
$506$	5.92478	0.263388
$507$	0	0
$508$	−2.68006	−0.118908
$509$	−22.1016	−0.979635	−0.489817	−	0.871825i	$-0.662937\pi$
$509$	−22.1016	−0.979635	−0.489817	+	0.871825i	$0.662937\pi$
$510$	0	0
$511$	−12.8119	−0.566767
$512$	−18.5188	−0.818423
$513$	0	0
$514$	27.9756	1.23395
$515$	0	0
$516$	0	0
$517$	12.3127	0.541510
$518$	2.85097	0.125264
$519$	0	0
$520$	0	0
$521$	22.8119	0.999409	0.499705	−	0.866196i	$-0.333442\pi$
$521$	22.8119	0.999409	0.499705	+	0.866196i	$0.333442\pi$
$522$	0	0
$523$	−12.2677	−0.536431	−0.268216	−	0.963359i	$-0.586434\pi$
$523$	−12.2677	−0.536431	−0.268216	+	0.963359i	$0.586434\pi$
$524$	1.14903	0.0501957
$525$	0	0
$526$	−30.8178	−1.34372
$527$	−3.62672	−0.157982
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	−1.23884	−0.0537106
$533$	−7.32582	−0.317317
$534$	0	0
$535$	0	0
$536$	7.22425	0.312040
$537$	0	0
$538$	47.9902	2.06900
$539$	5.57452	0.240111
$540$	0	0
$541$	−5.22425	−0.224608	−0.112304	−	0.993674i	$-0.535823\pi$
$541$	−5.22425	−0.224608	−0.112304	+	0.993674i	$0.535823\pi$
$542$	−24.9234	−1.07055
$543$	0	0
$544$	4.12013	0.176649
$545$	0	0
$546$	0	0
$547$	−17.9697	−0.768328	−0.384164	−	0.923265i	$-0.625510\pi$
$547$	−17.9697	−0.768328	−0.384164	+	0.923265i	$0.625510\pi$
$548$	−0.649738	−0.0277554
$549$	0	0
$550$	0	0
$551$	−23.0738	−0.982977
$552$	0	0
$553$	16.6859	0.709558
$554$	25.0435	1.06400
$555$	0	0
$556$	−4.11142	−0.174363
$557$	−15.8700	−0.672434	−0.336217	−	0.941784i	$-0.609148\pi$
$557$	−15.8700	−0.672434	−0.336217	+	0.941784i	$0.609148\pi$
$558$	0	0
$559$	3.56134	0.150629
$560$	0	0
$561$	0	0
$562$	−8.31265	−0.350648
$563$	−31.6688	−1.33468	−0.667340	−	0.744753i	$-0.732567\pi$
$563$	−31.6688	−1.33468	−0.667340	+	0.744753i	$0.732567\pi$
$564$	0	0
$565$	0	0
$566$	8.61942	0.362301
$567$	0	0
$568$	−37.5877	−1.57714
$569$	24.3127	1.01924	0.509620	−	0.860400i	$-0.329786\pi$
$569$	24.3127	1.01924	0.509620	+	0.860400i	$0.329786\pi$
$570$	0	0
$571$	8.05079	0.336915	0.168457	−	0.985709i	$-0.446121\pi$
$571$	8.05079	0.336915	0.168457	+	0.985709i	$0.446121\pi$
$572$	−0.156325	−0.00653628
$573$	0	0
$574$	−16.0724	−0.670849
$575$	0	0
$576$	0	0
$577$	−44.5355	−1.85404	−0.927018	−	0.375016i	$-0.877637\pi$
$577$	−44.5355	−1.85404	−0.927018	+	0.375016i	$0.877637\pi$
$578$	−4.14552	−0.172431
$579$	0	0
$580$	0	0
$581$	11.8134	0.490101
$582$	0	0
$583$	1.42548	0.0590375
$584$	28.7064	1.18788
$585$	0	0
$586$	12.2823	0.507379
$587$	15.4763	0.638774	0.319387	−	0.947624i	$-0.396523\pi$
$587$	15.4763	0.638774	0.319387	+	0.947624i	$0.396523\pi$
$588$	0	0
$589$	−5.14903	−0.212162
$590$	0	0
$591$	0	0
$592$	−7.01317	−0.288240
$593$	9.53102	0.391392	0.195696	−	0.980665i	$-0.437303\pi$
$593$	9.53102	0.391392	0.195696	+	0.980665i	$0.437303\pi$
$594$	0	0
$595$	0	0
$596$	−1.23884	−0.0507450
$597$	0	0
$598$	−4.77575	−0.195295
$599$	25.5515	1.04401	0.522003	−	0.852944i	$-0.325185\pi$
$599$	25.5515	1.04401	0.522003	+	0.852944i	$0.325185\pi$
$600$	0	0
$601$	12.0263	0.490565	0.245282	−	0.969452i	$-0.421119\pi$
$601$	12.0263	0.490565	0.245282	+	0.969452i	$0.421119\pi$
$602$	7.81336	0.318449
$603$	0	0
$604$	−0.513881	−0.0209095
$605$	0	0
$606$	0	0
$607$	−6.86670	−0.278711	−0.139355	−	0.990242i	$-0.544503\pi$
$607$	−6.86670	−0.278711	−0.139355	+	0.990242i	$0.544503\pi$
$608$	5.84955	0.237231
$609$	0	0
$610$	0	0
$611$	−9.92478	−0.401514
$612$	0	0
$613$	−7.25457	−0.293009	−0.146505	−	0.989210i	$-0.546802\pi$
$613$	−7.25457	−0.293009	−0.146505	+	0.989210i	$0.546802\pi$
$614$	38.0205	1.53438
$615$	0	0
$616$	3.19394	0.128687
$617$	−38.3634	−1.54445	−0.772227	−	0.635347i	$-0.780857\pi$
$617$	−38.3634	−1.54445	−0.772227	+	0.635347i	$0.780857\pi$
$618$	0	0
$619$	−29.6893	−1.19331	−0.596656	−	0.802497i	$-0.703504\pi$
$619$	−29.6893	−1.19331	−0.596656	+	0.802497i	$0.703504\pi$
$620$	0	0
$621$	0	0
$622$	23.2896	0.933829
$623$	−20.1622	−0.807782
$624$	0	0
$625$	0	0
$626$	−39.7889	−1.59029
$627$	0	0
$628$	0.312650	0.0124761
$629$	−6.07522	−0.242235
$630$	0	0
$631$	−19.6991	−0.784209	−0.392105	−	0.919921i	$-0.628253\pi$
$631$	−19.6991	−0.784209	−0.392105	+	0.919921i	$0.628253\pi$
$632$	−37.3865	−1.48715
$633$	0	0
$634$	−1.05220	−0.0417882
$635$	0	0
$636$	0	0
$637$	−4.49341	−0.178036
$638$	−6.38787	−0.252898
$639$	0	0
$640$	0	0
$641$	31.4372	1.24170	0.620848	−	0.783931i	$-0.286788\pi$
$641$	31.4372	1.24170	0.620848	+	0.783931i	$0.286788\pi$
$642$	0	0
$643$	−34.4894	−1.36013	−0.680065	−	0.733151i	$-0.738049\pi$
$643$	−34.4894	−1.36013	−0.680065	+	0.733151i	$0.738049\pi$
$644$	−0.926192	−0.0364971
$645$	0	0
$646$	29.8641	1.17499
$647$	5.61213	0.220635	0.110318	−	0.993896i	$-0.464813\pi$
$647$	5.61213	0.220635	0.110318	+	0.993896i	$0.464813\pi$
$648$	0	0
$649$	13.2750	0.521091
$650$	0	0
$651$	0	0
$652$	−0.0606343	−0.00237462
$653$	−4.06537	−0.159090	−0.0795452	−	0.996831i	$-0.525347\pi$
$653$	−4.06537	−0.159090	−0.0795452	+	0.996831i	$0.525347\pi$
$654$	0	0
$655$	0	0
$656$	39.5369	1.54366
$657$	0	0
$658$	−21.7743	−0.848852
$659$	−11.3747	−0.443095	−0.221548	−	0.975150i	$-0.571111\pi$
$659$	−11.3747	−0.443095	−0.221548	+	0.975150i	$0.571111\pi$
$660$	0	0
$661$	−42.4749	−1.65208	−0.826040	−	0.563611i	$-0.809412\pi$
$661$	−42.4749	−1.65208	−0.826040	+	0.563611i	$0.809412\pi$
$662$	1.42548	0.0554030
$663$	0	0
$664$	−26.4690	−1.02720
$665$	0	0
$666$	0	0
$667$	−17.2506	−0.667946
$668$	−0.0956908	−0.00370239
$669$	0	0
$670$	0	0
$671$	0.0752228	0.00290394
$672$	0	0
$673$	−14.8813	−0.573631	−0.286816	−	0.957986i	$-0.592597\pi$
$673$	−14.8813	−0.573631	−0.286816	+	0.957986i	$0.592597\pi$
$674$	29.3561	1.13076
$675$	0	0
$676$	−2.39517	−0.0921218
$677$	27.7685	1.06723	0.533614	−	0.845728i	$-0.320833\pi$
$677$	27.7685	1.06723	0.533614	+	0.845728i	$0.320833\pi$
$678$	0	0
$679$	13.7019	0.525832
$680$	0	0
$681$	0	0
$682$	−1.42548	−0.0545846
$683$	25.4617	0.974264	0.487132	−	0.873328i	$-0.338043\pi$
$683$	25.4617	0.974264	0.487132	+	0.873328i	$0.338043\pi$
$684$	0	0
$685$	0	0
$686$	−22.2374	−0.849029
$687$	0	0
$688$	−19.2203	−0.732766
$689$	−1.14903	−0.0437746
$690$	0	0
$691$	43.6991	1.66239	0.831196	−	0.555979i	$-0.187657\pi$
$691$	43.6991	1.66239	0.831196	+	0.555979i	$0.187657\pi$
$692$	4.51976	0.171816
$693$	0	0
$694$	−9.19394	−0.348997
$695$	0	0
$696$	0	0
$697$	34.2492	1.29728
$698$	6.58910	0.249401
$699$	0	0
$700$	0	0
$701$	−7.01317	−0.264884	−0.132442	−	0.991191i	$-0.542282\pi$
$701$	−7.01317	−0.264884	−0.132442	+	0.991191i	$0.542282\pi$
$702$	0	0
$703$	−8.62530	−0.325309
$704$	−7.08110	−0.266879
$705$	0	0
$706$	−9.73813	−0.366500
$707$	−12.8481	−0.483204
$708$	0	0
$709$	45.6747	1.71535	0.857674	−	0.514194i	$-0.171909\pi$
$709$	45.6747	1.71535	0.857674	+	0.514194i	$0.171909\pi$
$710$	0	0
$711$	0	0
$712$	45.1754	1.69302
$713$	−3.84955	−0.144167
$714$	0	0
$715$	0	0
$716$	2.07522	0.0775547
$717$	0	0
$718$	−12.7757	−0.476787
$719$	−16.2520	−0.606098	−0.303049	−	0.952975i	$-0.598005\pi$
$719$	−16.2520	−0.606098	−0.303049	+	0.952975i	$0.598005\pi$
$720$	0	0
$721$	20.2228	0.753138
$722$	14.2569	0.530588
$723$	0	0
$724$	−1.51247	−0.0562104
$725$	0	0
$726$	0	0
$727$	15.2243	0.564636	0.282318	−	0.959321i	$-0.408897\pi$
$727$	15.2243	0.564636	0.282318	+	0.959321i	$0.408897\pi$
$728$	−2.57452	−0.0954179
$729$	0	0
$730$	0	0
$731$	−16.6497	−0.615813
$732$	0	0
$733$	−43.5066	−1.60695	−0.803476	−	0.595337i	$-0.797019\pi$
$733$	−43.5066	−1.60695	−0.803476	+	0.595337i	$0.797019\pi$
$734$	34.0870	1.25817
$735$	0	0
$736$	4.37328	0.161201
$737$	2.70052	0.0994751
$738$	0	0
$739$	−7.02302	−0.258346	−0.129173	−	0.991622i	$-0.541232\pi$
$739$	−7.02302	−0.258346	−0.129173	+	0.991622i	$0.541232\pi$
$740$	0	0
$741$	0	0
$742$	−2.52090	−0.0925452
$743$	2.94192	0.107929	0.0539643	−	0.998543i	$-0.482814\pi$
$743$	2.94192	0.107929	0.0539643	+	0.998543i	$0.482814\pi$
$744$	0	0
$745$	0	0
$746$	41.7586	1.52889
$747$	0	0
$748$	0.730841	0.0267222
$749$	−9.88858	−0.361321
$750$	0	0
$751$	24.1016	0.879479	0.439739	−	0.898125i	$-0.355071\pi$
$751$	24.1016	0.879479	0.439739	+	0.898125i	$0.355071\pi$
$752$	53.5633	1.95325
$753$	0	0
$754$	5.14903	0.187517
$755$	0	0
$756$	0	0
$757$	−16.3127	−0.592893	−0.296447	−	0.955049i	$-0.595802\pi$
$757$	−16.3127	−0.592893	−0.296447	+	0.955049i	$0.595802\pi$
$758$	5.55149	0.201639
$759$	0	0
$760$	0	0
$761$	5.08840	0.184454	0.0922271	−	0.995738i	$-0.470601\pi$
$761$	5.08840	0.184454	0.0922271	+	0.995738i	$0.470601\pi$
$762$	0	0
$763$	−11.9394	−0.432234
$764$	−0.252016	−0.00911762
$765$	0	0
$766$	−2.61071	−0.0943289
$767$	−10.7005	−0.386374
$768$	0	0
$769$	2.10157	0.0757846	0.0378923	−	0.999282i	$-0.487936\pi$
$769$	2.10157	0.0757846	0.0378923	+	0.999282i	$0.487936\pi$
$770$	0	0
$771$	0	0
$772$	1.66688	0.0599924
$773$	−46.0625	−1.65675	−0.828377	−	0.560171i	$-0.810736\pi$
$773$	−46.0625	−1.65675	−0.828377	+	0.560171i	$0.810736\pi$
$774$	0	0
$775$	0	0
$776$	−30.7005	−1.10208
$777$	0	0
$778$	−9.66291	−0.346432
$779$	48.6253	1.74218
$780$	0	0
$781$	−14.0508	−0.502777
$782$	22.3272	0.798420
$783$	0	0
$784$	24.2506	0.866093
$785$	0	0
$786$	0	0
$787$	−26.0303	−0.927881	−0.463940	−	0.885866i	$-0.653565\pi$
$787$	−26.0303	−0.927881	−0.463940	+	0.885866i	$0.653565\pi$
$788$	4.01573	0.143054
$789$	0	0
$790$	0	0
$791$	−2.70052	−0.0960196
$792$	0	0
$793$	−0.0606343	−0.00215319
$794$	−35.1030	−1.24576
$795$	0	0
$796$	1.07664	0.0381604
$797$	29.9902	1.06231	0.531153	−	0.847276i	$-0.321759\pi$
$797$	29.9902	1.06231	0.531153	+	0.847276i	$0.321759\pi$
$798$	0	0
$799$	46.3996	1.64150
$800$	0	0
$801$	0	0
$802$	−13.1636	−0.464824
$803$	10.7308	0.378683
$804$	0	0
$805$	0	0
$806$	1.14903	0.0404729
$807$	0	0
$808$	28.7875	1.01274
$809$	−39.7597	−1.39788	−0.698939	−	0.715181i	$-0.746344\pi$
$809$	−39.7597	−1.39788	−0.698939	+	0.715181i	$0.746344\pi$
$810$	0	0
$811$	−10.2765	−0.360855	−0.180428	−	0.983588i	$-0.557748\pi$
$811$	−10.2765	−0.360855	−0.180428	+	0.983588i	$0.557748\pi$
$812$	0.998585	0.0350435
$813$	0	0
$814$	−2.38787	−0.0836949
$815$	0	0
$816$	0	0
$817$	−23.6385	−0.827006
$818$	−7.18523	−0.251226
$819$	0	0
$820$	0	0
$821$	−50.0870	−1.74805	−0.874024	−	0.485883i	$-0.838498\pi$
$821$	−50.0870	−1.74805	−0.874024	+	0.485883i	$0.838498\pi$
$822$	0	0
$823$	29.7137	1.03575	0.517877	−	0.855455i	$-0.326722\pi$
$823$	29.7137	1.03575	0.517877	+	0.855455i	$0.326722\pi$
$824$	−45.3112	−1.57849
$825$	0	0
$826$	−23.4763	−0.816844
$827$	2.41819	0.0840887	0.0420444	−	0.999116i	$-0.486613\pi$
$827$	2.41819	0.0840887	0.0420444	+	0.999116i	$0.486613\pi$
$828$	0	0
$829$	−23.2750	−0.808376	−0.404188	−	0.914676i	$-0.632446\pi$
$829$	−23.2750	−0.808376	−0.404188	+	0.914676i	$0.632446\pi$
$830$	0	0
$831$	0	0
$832$	5.70782	0.197883
$833$	21.0073	0.727860
$834$	0	0
$835$	0	0
$836$	1.03761	0.0358865
$837$	0	0
$838$	15.8496	0.547514
$839$	18.8265	0.649964	0.324982	−	0.945720i	$-0.394642\pi$
$839$	18.8265	0.649964	0.324982	+	0.945720i	$0.394642\pi$
$840$	0	0
$841$	−10.4010	−0.358657
$842$	−45.3258	−1.56203
$843$	0	0
$844$	−3.58769	−0.123493
$845$	0	0
$846$	0	0
$847$	1.19394	0.0410241
$848$	6.20123	0.212951
$849$	0	0
$850$	0	0
$851$	−6.44851	−0.221052
$852$	0	0
$853$	−24.9076	−0.852821	−0.426411	−	0.904530i	$-0.640222\pi$
$853$	−24.9076	−0.852821	−0.426411	+	0.904530i	$0.640222\pi$
$854$	−0.133028	−0.00455212
$855$	0	0
$856$	22.1563	0.757288
$857$	7.60625	0.259824	0.129912	−	0.991525i	$-0.458530\pi$
$857$	7.60625	0.259824	0.129912	+	0.991525i	$0.458530\pi$
$858$	0	0
$859$	−2.14060	−0.0730362	−0.0365181	−	0.999333i	$-0.511627\pi$
$859$	−2.14060	−0.0730362	−0.0365181	+	0.999333i	$0.511627\pi$
$860$	0	0
$861$	0	0
$862$	8.77575	0.298903
$863$	−29.3112	−0.997766	−0.498883	−	0.866669i	$-0.666256\pi$
$863$	−29.3112	−0.997766	−0.498883	+	0.866669i	$0.666256\pi$
$864$	0	0
$865$	0	0
$866$	−23.6991	−0.805329
$867$	0	0
$868$	0.222839	0.00756365
$869$	−13.9756	−0.474088
$870$	0	0
$871$	−2.17679	−0.0737578
$872$	26.7513	0.905914
$873$	0	0
$874$	31.6991	1.07224
$875$	0	0
$876$	0	0
$877$	31.5183	1.06430	0.532149	−	0.846650i	$-0.321384\pi$
$877$	31.5183	1.06430	0.532149	+	0.846650i	$0.321384\pi$
$878$	7.92478	0.267448
$879$	0	0
$880$	0	0
$881$	−10.6253	−0.357975	−0.178988	−	0.983851i	$-0.557282\pi$
$881$	−10.6253	−0.357975	−0.178988	+	0.983851i	$0.557282\pi$
$882$	0	0
$883$	−6.29806	−0.211947	−0.105973	−	0.994369i	$-0.533796\pi$
$883$	−6.29806	−0.211947	−0.105973	+	0.994369i	$0.533796\pi$
$884$	−0.589104	−0.0198137
$885$	0	0
$886$	−29.1608	−0.979676
$887$	−29.1187	−0.977711	−0.488855	−	0.872365i	$-0.662585\pi$
$887$	−29.1187	−0.977711	−0.488855	+	0.872365i	$0.662585\pi$
$888$	0	0
$889$	−16.4993	−0.553369
$890$	0	0
$891$	0	0
$892$	3.41564	0.114364
$893$	65.8759	2.20445
$894$	0	0
$895$	0	0
$896$	15.1333	0.505568
$897$	0	0
$898$	46.4142	1.54886
$899$	4.15045	0.138425
$900$	0	0
$901$	5.37187	0.178963
$902$	13.4617	0.448225
$903$	0	0
$904$	6.05079	0.201246
$905$	0	0
$906$	0	0
$907$	−57.3112	−1.90299	−0.951494	−	0.307667i	$-0.900452\pi$
$907$	−57.3112	−1.90299	−0.951494	+	0.307667i	$0.900452\pi$
$908$	3.38058	0.112188
$909$	0	0
$910$	0	0
$911$	5.52232	0.182962	0.0914812	−	0.995807i	$-0.470840\pi$
$911$	5.52232	0.182962	0.0914812	+	0.995807i	$0.470840\pi$
$912$	0	0
$913$	−9.89446	−0.327459
$914$	−55.6834	−1.84184
$915$	0	0
$916$	2.53549	0.0837749
$917$	7.07381	0.233598
$918$	0	0
$919$	−5.75272	−0.189765	−0.0948824	−	0.995488i	$-0.530248\pi$
$919$	−5.75272	−0.189765	−0.0948824	+	0.995488i	$0.530248\pi$
$920$	0	0
$921$	0	0
$922$	−26.6399	−0.877338
$923$	11.3258	0.372794
$924$	0	0
$925$	0	0
$926$	57.7861	1.89897
$927$	0	0
$928$	−4.71511	−0.154781
$929$	−41.4274	−1.35919	−0.679594	−	0.733588i	$-0.737844\pi$
$929$	−41.4274	−1.35919	−0.679594	+	0.733588i	$0.737844\pi$
$930$	0	0
$931$	29.8251	0.977479
$932$	2.68479	0.0879434
$933$	0	0
$934$	−21.5515	−0.705186
$935$	0	0
$936$	0	0
$937$	−14.9222	−0.487488	−0.243744	−	0.969840i	$-0.578376\pi$
$937$	−14.9222	−0.487488	−0.243744	+	0.969840i	$0.578376\pi$
$938$	−4.77575	−0.155934
$939$	0	0
$940$	0	0
$941$	−57.0395	−1.85944	−0.929718	−	0.368273i	$-0.879949\pi$
$941$	−57.0395	−1.85944	−0.929718	+	0.368273i	$0.879949\pi$
$942$	0	0
$943$	36.3536	1.18384
$944$	57.7499	1.87960
$945$	0	0
$946$	−6.54420	−0.212770
$947$	−30.2374	−0.982584	−0.491292	−	0.870995i	$-0.663475\pi$
$947$	−30.2374	−0.982584	−0.491292	+	0.870995i	$0.663475\pi$
$948$	0	0
$949$	−8.64974	−0.280782
$950$	0	0
$951$	0	0
$952$	12.0362	0.390095
$953$	46.2697	1.49882	0.749411	−	0.662106i	$-0.230337\pi$
$953$	46.2697	1.49882	0.749411	+	0.662106i	$0.230337\pi$
$954$	0	0
$955$	0	0
$956$	2.39963	0.0776097
$957$	0	0
$958$	−42.3996	−1.36987
$959$	−4.00000	−0.129167
$960$	0	0
$961$	−30.0738	−0.970123
$962$	1.92478	0.0620573
$963$	0	0
$964$	−4.76116	−0.153347
$965$	0	0
$966$	0	0
$967$	49.5790	1.59435	0.797176	−	0.603747i	$-0.206326\pi$
$967$	49.5790	1.59435	0.797176	+	0.603747i	$0.206326\pi$
$968$	−2.67513	−0.0859820
$969$	0	0
$970$	0	0
$971$	3.12268	0.100212	0.0501058	−	0.998744i	$-0.484044\pi$
$971$	3.12268	0.100212	0.0501058	+	0.998744i	$0.484044\pi$
$972$	0	0
$973$	−25.3112	−0.811441
$974$	2.14762	0.0688141
$975$	0	0
$976$	0.327239	0.0104747
$977$	−5.15377	−0.164884	−0.0824419	−	0.996596i	$-0.526272\pi$
$977$	−5.15377	−0.164884	−0.0824419	+	0.996596i	$0.526272\pi$
$978$	0	0
$979$	16.8872	0.539716
$980$	0	0
$981$	0	0
$982$	11.8496	0.378134
$983$	−14.8627	−0.474047	−0.237024	−	0.971504i	$-0.576172\pi$
$983$	−14.8627	−0.474047	−0.237024	+	0.971504i	$0.576172\pi$
$984$	0	0
$985$	0	0
$986$	−24.0724	−0.766621
$987$	0	0
$988$	−0.836381	−0.0266088
$989$	−17.6728	−0.561961
$990$	0	0
$991$	4.43866	0.140999	0.0704993	−	0.997512i	$-0.477541\pi$
$991$	4.43866	0.140999	0.0704993	+	0.997512i	$0.477541\pi$
$992$	−1.05220	−0.0334074
$993$	0	0
$994$	24.8481	0.788135
$995$	0	0
$996$	0	0
$997$	−38.8324	−1.22983	−0.614917	−	0.788592i	$-0.710811\pi$
$997$	−38.8324	−1.22983	−0.614917	+	0.788592i	$0.710811\pi$
$998$	45.4734	1.43944
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.a.ba.1.3		3
3.2	odd	2		825.2.a.l.1.1		3
5.2	odd	4		495.2.c.e.199.5		6
5.3	odd	4		495.2.c.e.199.2		6
5.4	even	2		2475.2.a.bc.1.1		3
15.2	even	4		165.2.c.b.34.2	✓	6
15.8	even	4		165.2.c.b.34.5	yes	6
15.14	odd	2		825.2.a.j.1.3		3
33.32	even	2		9075.2.a.cg.1.3		3
60.23	odd	4		2640.2.d.h.529.1		6
60.47	odd	4		2640.2.d.h.529.4		6
165.32	odd	4		1815.2.c.e.364.5		6
165.98	odd	4		1815.2.c.e.364.2		6
165.164	even	2		9075.2.a.ch.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.c.b.34.2	✓	6	15.2	even	4
165.2.c.b.34.5	yes	6	15.8	even	4
495.2.c.e.199.2		6	5.3	odd	4
495.2.c.e.199.5		6	5.2	odd	4
825.2.a.j.1.3		3	15.14	odd	2
825.2.a.l.1.1		3	3.2	odd	2
1815.2.c.e.364.2		6	165.98	odd	4
1815.2.c.e.364.5		6	165.32	odd	4
2475.2.a.ba.1.3		3	1.1	even	1	trivial
2475.2.a.bc.1.1		3	5.4	even	2
2640.2.d.h.529.1		6	60.23	odd	4
2640.2.d.h.529.4		6	60.47	odd	4
9075.2.a.cg.1.3		3	33.32	even	2
9075.2.a.ch.1.1		3	165.164	even	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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q+1.48119 q^{2} +0.193937 q^{4} +1.19394 q^{7} -2.67513 q^{8} +O(q^{10})\)	\(q+1.48119 q^{2} +0.193937 q^{4} +1.19394 q^{7} -2.67513 q^{8} -1.00000 q^{11} +0.806063 q^{13} +1.76845 q^{14} -4.35026 q^{16} -3.76845 q^{17} -5.35026 q^{19} -1.48119 q^{22} -4.00000 q^{23} +1.19394 q^{26} +0.231548 q^{28} +4.31265 q^{29} +0.962389 q^{31} -1.09332 q^{32} -5.58181 q^{34} +1.61213 q^{37} -7.92478 q^{38} -9.08840 q^{41} +4.41819 q^{43} -0.193937 q^{44} -5.92478 q^{46} -12.3127 q^{47} -5.57452 q^{49} +0.156325 q^{52} -1.42548 q^{53} -3.19394 q^{56} +6.38787 q^{58} -13.2750 q^{59} -0.0752228 q^{61} +1.42548 q^{62} +7.08110 q^{64} -2.70052 q^{67} -0.730841 q^{68} +14.0508 q^{71} -10.7308 q^{73} +2.38787 q^{74} -1.03761 q^{76} -1.19394 q^{77} +13.9756 q^{79} -13.4617 q^{82} +9.89446 q^{83} +6.54420 q^{86} +2.67513 q^{88} -16.8872 q^{89} +0.962389 q^{91} -0.775746 q^{92} -18.2374 q^{94} +11.4763 q^{97} -8.25694 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + q^{4} + 4 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - q^2 + q^4 + 4 * q^7 - 3 * q^8	\( 3 q - q^{2} + q^{4} + 4 q^{7} - 3 q^{8} - 3 q^{11} + 2 q^{13} - 6 q^{14} - 3 q^{16} - 6 q^{19} + q^{22} - 12 q^{23} + 4 q^{26} + 12 q^{28} - 8 q^{29} - 8 q^{31} + 3 q^{32} - 18 q^{34} + 4 q^{37} - 2 q^{38} - 8 q^{41} + 12 q^{43} - q^{44} + 4 q^{46} - 16 q^{47} - 5 q^{49} - 10 q^{52} - 16 q^{53} - 10 q^{56} + 20 q^{58} - 8 q^{59} - 22 q^{61} + 16 q^{62} - 11 q^{64} + 12 q^{67} + 20 q^{68} + 12 q^{71} - 10 q^{73} + 8 q^{74} - 14 q^{76} - 4 q^{77} - 10 q^{79} + 4 q^{82} + 10 q^{83} + 10 q^{86} + 3 q^{88} - 18 q^{89} - 8 q^{91} - 4 q^{92} - 12 q^{94} + 16 q^{97} - 21 q^{98}+O(q^{100}) \) 3 * q - q^2 + q^4 + 4 * q^7 - 3 * q^8 - 3 * q^11 + 2 * q^13 - 6 * q^14 - 3 * q^16 - 6 * q^19 + q^22 - 12 * q^23 + 4 * q^26 + 12 * q^28 - 8 * q^29 - 8 * q^31 + 3 * q^32 - 18 * q^34 + 4 * q^37 - 2 * q^38 - 8 * q^41 + 12 * q^43 - q^44 + 4 * q^46 - 16 * q^47 - 5 * q^49 - 10 * q^52 - 16 * q^53 - 10 * q^56 + 20 * q^58 - 8 * q^59 - 22 * q^61 + 16 * q^62 - 11 * q^64 + 12 * q^67 + 20 * q^68 + 12 * q^71 - 10 * q^73 + 8 * q^74 - 14 * q^76 - 4 * q^77 - 10 * q^79 + 4 * q^82 + 10 * q^83 + 10 * q^86 + 3 * q^88 - 18 * q^89 - 8 * q^91 - 4 * q^92 - 12 * q^94 + 16 * q^97 - 21 * q^98

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