LMFDB - Embedded newform 3525.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3525,2,Mod(1,3525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3525, 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("3525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3525 = 3 \cdot 5^{2} \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3525.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:	$28.1472667125$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.4352.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - 4x + 2 $ x^4 - 6x^2 - 4x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 705)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.68554$ of defining polynomial
Character	$\chi$	$=$	3525.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.27133 q^{2} +1.00000 q^{3} +3.15894 q^{4} -2.27133 q^{6} +0.797933 q^{7} -2.63234 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.27133 q^{2} +1.00000 q^{3} +3.15894 q^{4} -2.27133 q^{6} +0.797933 q^{7} -2.63234 q^{8} +1.00000 q^{9} +2.66949 q^{11} +3.15894 q^{12} -3.32453 q^{13} -1.81237 q^{14} -0.338971 q^{16} -2.55710 q^{17} -2.27133 q^{18} +4.89769 q^{19} +0.797933 q^{21} -6.06328 q^{22} -3.96951 q^{23} -2.63234 q^{24} +7.55112 q^{26} +1.00000 q^{27} +2.52062 q^{28} -7.42429 q^{29} -4.01786 q^{31} +6.03459 q^{32} +2.66949 q^{33} +5.80801 q^{34} +3.15894 q^{36} +10.0727 q^{37} -11.1243 q^{38} -3.32453 q^{39} -8.81580 q^{41} -1.81237 q^{42} -6.51054 q^{43} +8.43275 q^{44} +9.01606 q^{46} +1.00000 q^{47} -0.338971 q^{48} -6.36330 q^{49} -2.55710 q^{51} -10.5020 q^{52} -10.8319 q^{53} -2.27133 q^{54} -2.10043 q^{56} +4.89769 q^{57} +16.8630 q^{58} -7.40823 q^{59} +0.913749 q^{61} +9.12589 q^{62} +0.797933 q^{63} -13.0286 q^{64} -6.06328 q^{66} -5.01008 q^{67} -8.07772 q^{68} -3.96951 q^{69} +11.1835 q^{71} -2.63234 q^{72} +7.63899 q^{73} -22.8784 q^{74} +15.4715 q^{76} +2.13007 q^{77} +7.55112 q^{78} -2.38372 q^{79} +1.00000 q^{81} +20.0236 q^{82} +2.19951 q^{83} +2.52062 q^{84} +14.7876 q^{86} -7.42429 q^{87} -7.02699 q^{88} -10.8784 q^{89} -2.65275 q^{91} -12.5394 q^{92} -4.01786 q^{93} -2.27133 q^{94} +6.03459 q^{96} +11.5385 q^{97} +14.4532 q^{98} +2.66949 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{3} + 8 q^{4} - 4 q^{6} - 8 q^{7} - 12 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^3 + 8 * q^4 - 4 * q^6 - 8 * q^7 - 12 * q^8 + 4 * q^9	$ 4 q - 4 q^{2} + 4 q^{3} + 8 q^{4} - 4 q^{6} - 8 q^{7} - 12 q^{8} + 4 q^{9} + 4 q^{11} + 8 q^{12} + 4 q^{13} + 12 q^{16} - 4 q^{17} - 4 q^{18} - 8 q^{19} - 8 q^{21} + 16 q^{22} - 16 q^{23} - 12 q^{24} + 4 q^{27} - 20 q^{28} + 4 q^{29} - 28 q^{32} + 4 q^{33} - 16 q^{34} + 8 q^{36} + 4 q^{37} - 4 q^{38} + 4 q^{39} - 8 q^{41} - 24 q^{43} - 20 q^{44} + 32 q^{46} + 4 q^{47} + 12 q^{48} + 8 q^{49} - 4 q^{51} + 28 q^{52} - 12 q^{53} - 4 q^{54} + 40 q^{56} - 8 q^{57} + 4 q^{58} - 28 q^{61} - 12 q^{62} - 8 q^{63} + 24 q^{64} + 16 q^{66} + 8 q^{67} + 4 q^{68} - 16 q^{69} + 16 q^{71} - 12 q^{72} + 24 q^{73} - 56 q^{74} - 8 q^{76} - 4 q^{77} - 4 q^{79} + 4 q^{81} + 4 q^{82} - 24 q^{83} - 20 q^{84} - 8 q^{86} + 4 q^{87} + 40 q^{88} - 8 q^{89} - 40 q^{91} - 28 q^{92} - 4 q^{94} - 28 q^{96} + 32 q^{98} + 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^3 + 8 * q^4 - 4 * q^6 - 8 * q^7 - 12 * q^8 + 4 * q^9 + 4 * q^11 + 8 * q^12 + 4 * q^13 + 12 * q^16 - 4 * q^17 - 4 * q^18 - 8 * q^19 - 8 * q^21 + 16 * q^22 - 16 * q^23 - 12 * q^24 + 4 * q^27 - 20 * q^28 + 4 * q^29 - 28 * q^32 + 4 * q^33 - 16 * q^34 + 8 * q^36 + 4 * q^37 - 4 * q^38 + 4 * q^39 - 8 * q^41 - 24 * q^43 - 20 * q^44 + 32 * q^46 + 4 * q^47 + 12 * q^48 + 8 * q^49 - 4 * q^51 + 28 * q^52 - 12 * q^53 - 4 * q^54 + 40 * q^56 - 8 * q^57 + 4 * q^58 - 28 * q^61 - 12 * q^62 - 8 * q^63 + 24 * q^64 + 16 * q^66 + 8 * q^67 + 4 * q^68 - 16 * q^69 + 16 * q^71 - 12 * q^72 + 24 * q^73 - 56 * q^74 - 8 * q^76 - 4 * q^77 - 4 * q^79 + 4 * q^81 + 4 * q^82 - 24 * q^83 - 20 * q^84 - 8 * q^86 + 4 * q^87 + 40 * q^88 - 8 * q^89 - 40 * q^91 - 28 * q^92 - 4 * q^94 - 28 * q^96 + 32 * q^98 + 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.27133	−1.60607	−0.803037	−	0.595930i	$-0.796784\pi$
$2$	−2.27133	−1.60607	−0.803037	+	0.595930i	$0.796784\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	3.15894	1.57947
$5$	0	0
$5$	0	0
$6$	−2.27133	−0.927267
$7$	0.797933	0.301590	0.150795	−	0.988565i	$-0.451817\pi$
$7$	0.797933	0.301590	0.150795	+	0.988565i	$0.451817\pi$
$8$	−2.63234	−0.930673
$9$	1.00000	0.333333
$10$	0	0
$11$	2.66949	0.804880	0.402440	−	0.915446i	$-0.368162\pi$
$11$	2.66949	0.804880	0.402440	+	0.915446i	$0.368162\pi$
$12$	3.15894	0.911908
$13$	−3.32453	−0.922060	−0.461030	−	0.887385i	$-0.652520\pi$
$13$	−3.32453	−0.922060	−0.461030	+	0.887385i	$0.652520\pi$
$14$	−1.81237	−0.484376
$15$	0	0
$16$	−0.338971	−0.0847427
$17$	−2.55710	−0.620187	−0.310094	−	0.950706i	$-0.600360\pi$
$17$	−2.55710	−0.620187	−0.310094	+	0.950706i	$0.600360\pi$
$18$	−2.27133	−0.535358
$19$	4.89769	1.12361	0.561804	−	0.827271i	$-0.310108\pi$
$19$	4.89769	1.12361	0.561804	+	0.827271i	$0.310108\pi$
$20$	0	0
$21$	0.797933	0.174123
$22$	−6.06328	−1.29270
$23$	−3.96951	−0.827699	−0.413850	−	0.910345i	$-0.635816\pi$
$23$	−3.96951	−0.827699	−0.413850	+	0.910345i	$0.635816\pi$
$24$	−2.63234	−0.537324
$25$	0	0
$26$	7.55112	1.48090
$27$	1.00000	0.192450
$28$	2.52062	0.476353
$29$	−7.42429	−1.37866	−0.689328	−	0.724449i	$-0.742094\pi$
$29$	−7.42429	−1.37866	−0.689328	+	0.724449i	$0.742094\pi$
$30$	0	0
$31$	−4.01786	−0.721630	−0.360815	−	0.932637i	$-0.617501\pi$
$31$	−4.01786	−0.721630	−0.360815	+	0.932637i	$0.617501\pi$
$32$	6.03459	1.06678
$33$	2.66949	0.464698
$34$	5.80801	0.996066
$35$	0	0
$36$	3.15894	0.526490
$37$	10.0727	1.65594	0.827970	−	0.560772i	$-0.189496\pi$
$37$	10.0727	1.65594	0.827970	+	0.560772i	$0.189496\pi$
$38$	−11.1243	−1.80460
$39$	−3.32453	−0.532352
$40$	0	0
$41$	−8.81580	−1.37680	−0.688398	−	0.725333i	$-0.741686\pi$
$41$	−8.81580	−1.37680	−0.688398	+	0.725333i	$0.741686\pi$
$42$	−1.81237	−0.279655
$43$	−6.51054	−0.992849	−0.496424	−	0.868080i	$-0.665354\pi$
$43$	−6.51054	−0.992849	−0.496424	+	0.868080i	$0.665354\pi$
$44$	8.43275	1.27128
$45$	0	0
$46$	9.01606	1.32935
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	−0.338971	−0.0489262
$49$	−6.36330	−0.909043
$50$	0	0
$51$	−2.55710	−0.358065
$52$	−10.5020	−1.45637
$53$	−10.8319	−1.48787	−0.743935	−	0.668252i	$-0.767043\pi$
$53$	−10.8319	−1.48787	−0.743935	+	0.668252i	$0.767043\pi$
$54$	−2.27133	−0.309089
$55$	0	0
$56$	−2.10043	−0.280682
$57$	4.89769	0.648715
$58$	16.8630	2.21422
$59$	−7.40823	−0.964470	−0.482235	−	0.876042i	$-0.660175\pi$
$59$	−7.40823	−0.964470	−0.482235	+	0.876042i	$0.660175\pi$
$60$	0	0
$61$	0.913749	0.116994	0.0584968	−	0.998288i	$-0.481369\pi$
$61$	0.913749	0.116994	0.0584968	+	0.998288i	$0.481369\pi$
$62$	9.12589	1.15899
$63$	0.797933	0.100530
$64$	−13.0286	−1.62858
$65$	0	0
$66$	−6.06328	−0.746339
$67$	−5.01008	−0.612079	−0.306039	−	0.952019i	$-0.599004\pi$
$67$	−5.01008	−0.612079	−0.306039	+	0.952019i	$0.599004\pi$
$68$	−8.07772	−0.979567
$69$	−3.96951	−0.477872
$70$	0	0
$71$	11.1835	1.32723	0.663616	−	0.748073i	$-0.269021\pi$
$71$	11.1835	1.32723	0.663616	+	0.748073i	$0.269021\pi$
$72$	−2.63234	−0.310224
$73$	7.63899	0.894076	0.447038	−	0.894515i	$-0.352479\pi$
$73$	7.63899	0.894076	0.447038	+	0.894515i	$0.352479\pi$
$74$	−22.8784	−2.65956
$75$	0	0
$76$	15.4715	1.77470
$77$	2.13007	0.242744
$78$	7.55112	0.854996
$79$	−2.38372	−0.268189	−0.134095	−	0.990969i	$-0.542813\pi$
$79$	−2.38372	−0.268189	−0.134095	+	0.990969i	$0.542813\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	20.0236	2.21124
$83$	2.19951	0.241428	0.120714	−	0.992687i	$-0.461482\pi$
$83$	2.19951	0.241428	0.120714	+	0.992687i	$0.461482\pi$
$84$	2.52062	0.275022
$85$	0	0
$86$	14.7876	1.59459
$87$	−7.42429	−0.795968
$88$	−7.02699	−0.749080
$89$	−10.8784	−1.15311	−0.576554	−	0.817059i	$-0.695603\pi$
$89$	−10.8784	−1.15311	−0.576554	+	0.817059i	$0.695603\pi$
$90$	0	0
$91$	−2.65275	−0.278084
$92$	−12.5394	−1.30733
$93$	−4.01786	−0.416633
$94$	−2.27133	−0.234270
$95$	0	0
$96$	6.03459	0.615903
$97$	11.5385	1.17156	0.585778	−	0.810472i	$-0.300789\pi$
$97$	11.5385	1.17156	0.585778	+	0.810472i	$0.300789\pi$
$98$	14.4532	1.45999
$99$	2.66949	0.268293
$100$	0	0
$101$	−10.7243	−1.06711	−0.533554	−	0.845766i	$-0.679144\pi$
$101$	−10.7243	−1.06711	−0.533554	+	0.845766i	$0.679144\pi$
$102$	5.80801	0.575079
$103$	−7.62151	−0.750970	−0.375485	−	0.926828i	$-0.622524\pi$
$103$	−7.62151	−0.750970	−0.375485	+	0.926828i	$0.622524\pi$
$104$	8.75130	0.858136
$105$	0	0
$106$	24.6027	2.38963
$107$	−5.17157	−0.499955	−0.249977	−	0.968252i	$-0.580423\pi$
$107$	−5.17157	−0.499955	−0.249977	+	0.968252i	$0.580423\pi$
$108$	3.15894	0.303969
$109$	−0.928932	−0.0889756	−0.0444878	−	0.999010i	$-0.514166\pi$
$109$	−0.928932	−0.0889756	−0.0444878	+	0.999010i	$0.514166\pi$
$110$	0	0
$111$	10.0727	0.956057
$112$	−0.270476	−0.0255576
$113$	−8.49013	−0.798684	−0.399342	−	0.916802i	$-0.630761\pi$
$113$	−8.49013	−0.798684	−0.399342	+	0.916802i	$0.630761\pi$
$114$	−11.1243	−1.04188
$115$	0	0
$116$	−23.4529	−2.17755
$117$	−3.32453	−0.307353
$118$	16.8265	1.54901
$119$	−2.04039	−0.187042
$120$	0	0
$121$	−3.87385	−0.352168
$122$	−2.07542	−0.187900
$123$	−8.81580	−0.794894
$124$	−12.6922	−1.13979
$125$	0	0
$126$	−1.81237	−0.161459
$127$	8.02016	0.711674	0.355837	−	0.934548i	$-0.384196\pi$
$127$	8.02016	0.711674	0.355837	+	0.934548i	$0.384196\pi$
$128$	17.5231	1.54884
$129$	−6.51054	−0.573221
$130$	0	0
$131$	−9.95687	−0.869936	−0.434968	−	0.900446i	$-0.643240\pi$
$131$	−9.95687	−0.869936	−0.434968	+	0.900446i	$0.643240\pi$
$132$	8.43275	0.733977
$133$	3.90803	0.338869
$134$	11.3795	0.983043
$135$	0	0
$136$	6.73115	0.577191
$137$	−11.3711	−0.971498	−0.485749	−	0.874098i	$-0.661453\pi$
$137$	−11.3711	−0.971498	−0.485749	+	0.874098i	$0.661453\pi$
$138$	9.01606	0.767498
$139$	4.56880	0.387520	0.193760	−	0.981049i	$-0.437932\pi$
$139$	4.56880	0.387520	0.193760	+	0.981049i	$0.437932\pi$
$140$	0	0
$141$	1.00000	0.0842152
$142$	−25.4013	−2.13163
$143$	−8.87480	−0.742148
$144$	−0.338971	−0.0282476
$145$	0	0
$146$	−17.3507	−1.43595
$147$	−6.36330	−0.524836
$148$	31.8190	2.61551
$149$	6.42429	0.526299	0.263149	−	0.964755i	$-0.415239\pi$
$149$	6.42429	0.526299	0.263149	+	0.964755i	$0.415239\pi$
$150$	0	0
$151$	10.0693	0.819425	0.409713	−	0.912215i	$-0.365629\pi$
$151$	10.0693	0.819425	0.409713	+	0.912215i	$0.365629\pi$
$152$	−12.8924	−1.04571
$153$	−2.55710	−0.206729
$154$	−4.83809	−0.389865
$155$	0	0
$156$	−10.5020	−0.840834
$157$	1.87802	0.149882	0.0749412	−	0.997188i	$-0.476123\pi$
$157$	1.87802	0.149882	0.0749412	+	0.997188i	$0.476123\pi$
$158$	5.41421	0.430732
$159$	−10.8319	−0.859022
$160$	0	0
$161$	−3.16740	−0.249626
$162$	−2.27133	−0.178453
$163$	21.2884	1.66744	0.833720	−	0.552188i	$-0.186207\pi$
$163$	21.2884	1.66744	0.833720	+	0.552188i	$0.186207\pi$
$164$	−27.8486	−2.17461
$165$	0	0
$166$	−4.99583	−0.387751
$167$	16.9197	1.30928	0.654641	−	0.755940i	$-0.272820\pi$
$167$	16.9197	1.30928	0.654641	+	0.755940i	$0.272820\pi$
$168$	−2.10043	−0.162052
$169$	−1.94747	−0.149805
$170$	0	0
$171$	4.89769	0.374536
$172$	−20.5664	−1.56818
$173$	−20.7732	−1.57935	−0.789677	−	0.613523i	$-0.789752\pi$
$173$	−20.7732	−1.57935	−0.789677	+	0.613523i	$0.789752\pi$
$174$	16.8630	1.27838
$175$	0	0
$176$	−0.904878	−0.0682077
$177$	−7.40823	−0.556837
$178$	24.7085	1.85198
$179$	−1.27798	−0.0955209	−0.0477604	−	0.998859i	$-0.515208\pi$
$179$	−1.27798	−0.0955209	−0.0477604	+	0.998859i	$0.515208\pi$
$180$	0	0
$181$	0.181652	0.0135021	0.00675103	−	0.999977i	$-0.497851\pi$
$181$	0.181652	0.0135021	0.00675103	+	0.999977i	$0.497851\pi$
$182$	6.02528	0.446624
$183$	0.913749	0.0675462
$184$	10.4491	0.770317
$185$	0	0
$186$	9.12589	0.669143
$187$	−6.82613	−0.499176
$188$	3.15894	0.230389
$189$	0.797933	0.0580411
$190$	0	0
$191$	26.4779	1.91587	0.957936	−	0.286981i	$-0.0926518\pi$
$191$	26.4779	1.91587	0.957936	+	0.286981i	$0.0926518\pi$
$192$	−13.0286	−0.940259
$193$	8.00229	0.576018	0.288009	−	0.957628i	$-0.407007\pi$
$193$	8.00229	0.576018	0.288009	+	0.957628i	$0.407007\pi$
$194$	−26.2077	−1.88160
$195$	0	0
$196$	−20.1013	−1.43581
$197$	−19.2837	−1.37391	−0.686954	−	0.726701i	$-0.741052\pi$
$197$	−19.2837	−1.37391	−0.686954	+	0.726701i	$0.741052\pi$
$198$	−6.06328	−0.430899
$199$	4.26192	0.302120	0.151060	−	0.988525i	$-0.451731\pi$
$199$	4.26192	0.302120	0.151060	+	0.988525i	$0.451731\pi$
$200$	0	0
$201$	−5.01008	−0.353384
$202$	24.3585	1.71386
$203$	−5.92409	−0.415789
$204$	−8.07772	−0.565553
$205$	0	0
$206$	17.3110	1.20611
$207$	−3.96951	−0.275900
$208$	1.12692	0.0781379
$209$	13.0743	0.904369
$210$	0	0
$211$	0.785301	0.0540624	0.0270312	−	0.999635i	$-0.491395\pi$
$211$	0.785301	0.0540624	0.0270312	+	0.999635i	$0.491395\pi$
$212$	−34.2172	−2.35005
$213$	11.1835	0.766278
$214$	11.7464	0.802964
$215$	0	0
$216$	−2.63234	−0.179108
$217$	−3.20598	−0.217636
$218$	2.10991	0.142901
$219$	7.63899	0.516195
$220$	0	0
$221$	8.50116	0.571850
$222$	−22.8784	−1.53550
$223$	4.87155	0.326223	0.163112	−	0.986608i	$-0.447847\pi$
$223$	4.87155	0.326223	0.163112	+	0.986608i	$0.447847\pi$
$224$	4.81520	0.321729
$225$	0	0
$226$	19.2839	1.28275
$227$	3.75158	0.249001	0.124501	−	0.992220i	$-0.460267\pi$
$227$	3.75158	0.249001	0.124501	+	0.992220i	$0.460267\pi$
$228$	15.4715	1.02463
$229$	−13.6137	−0.899621	−0.449810	−	0.893124i	$-0.648508\pi$
$229$	−13.6137	−0.899621	−0.449810	+	0.893124i	$0.648508\pi$
$230$	0	0
$231$	2.13007	0.140148
$232$	19.5433	1.28308
$233$	−15.0243	−0.984277	−0.492138	−	0.870517i	$-0.663785\pi$
$233$	−15.0243	−0.984277	−0.492138	+	0.870517i	$0.663785\pi$
$234$	7.55112	0.493632
$235$	0	0
$236$	−23.4022	−1.52335
$237$	−2.38372	−0.154839
$238$	4.63440	0.300404
$239$	27.5275	1.78060	0.890302	−	0.455370i	$-0.150493\pi$
$239$	27.5275	1.78060	0.890302	+	0.455370i	$0.150493\pi$
$240$	0	0
$241$	1.19266	0.0768260	0.0384130	−	0.999262i	$-0.487770\pi$
$241$	1.19266	0.0768260	0.0384130	+	0.999262i	$0.487770\pi$
$242$	8.79879	0.565607
$243$	1.00000	0.0641500
$244$	2.88648	0.184788
$245$	0	0
$246$	20.0236	1.27666
$247$	−16.2825	−1.03603
$248$	10.5764	0.671601
$249$	2.19951	0.139389
$250$	0	0
$251$	9.38392	0.592308	0.296154	−	0.955140i	$-0.404296\pi$
$251$	9.38392	0.592308	0.296154	+	0.955140i	$0.404296\pi$
$252$	2.52062	0.158784
$253$	−10.5965	−0.666199
$254$	−18.2164	−1.14300
$255$	0	0
$256$	−13.7435	−0.858970
$257$	−3.59102	−0.224002	−0.112001	−	0.993708i	$-0.535726\pi$
$257$	−3.59102	−0.224002	−0.112001	+	0.993708i	$0.535726\pi$
$258$	14.7876	0.920636
$259$	8.03733	0.499415
$260$	0	0
$261$	−7.42429	−0.459552
$262$	22.6154	1.39718
$263$	−25.6458	−1.58139	−0.790695	−	0.612210i	$-0.790281\pi$
$263$	−25.6458	−1.58139	−0.790695	+	0.612210i	$0.790281\pi$
$264$	−7.02699	−0.432481
$265$	0	0
$266$	−8.87642	−0.544248
$267$	−10.8784	−0.665748
$268$	−15.8265	−0.966760
$269$	23.8311	1.45301	0.726504	−	0.687162i	$-0.241144\pi$
$269$	23.8311	1.45301	0.726504	+	0.687162i	$0.241144\pi$
$270$	0	0
$271$	−28.6018	−1.73744	−0.868718	−	0.495308i	$-0.835055\pi$
$271$	−28.6018	−1.73744	−0.868718	+	0.495308i	$0.835055\pi$
$272$	0.866782	0.0525563
$273$	−2.65275	−0.160552
$274$	25.8275	1.56030
$275$	0	0
$276$	−12.5394	−0.754785
$277$	7.39150	0.444112	0.222056	−	0.975034i	$-0.428723\pi$
$277$	7.39150	0.444112	0.222056	+	0.975034i	$0.428723\pi$
$278$	−10.3772	−0.622386
$279$	−4.01786	−0.240543
$280$	0	0
$281$	−2.17157	−0.129545	−0.0647726	−	0.997900i	$-0.520632\pi$
$281$	−2.17157	−0.129545	−0.0647726	+	0.997900i	$0.520632\pi$
$282$	−2.27133	−0.135256
$283$	−5.29680	−0.314862	−0.157431	−	0.987530i	$-0.550321\pi$
$283$	−5.29680	−0.314862	−0.157431	+	0.987530i	$0.550321\pi$
$284$	35.3279	2.09632
$285$	0	0
$286$	20.1576	1.19194
$287$	−7.03441	−0.415228
$288$	6.03459	0.355592
$289$	−10.4613	−0.615368
$290$	0	0
$291$	11.5385	0.676398
$292$	24.1311	1.41217
$293$	−3.97113	−0.231996	−0.115998	−	0.993249i	$-0.537007\pi$
$293$	−3.97113	−0.231996	−0.115998	+	0.993249i	$0.537007\pi$
$294$	14.4532	0.842926
$295$	0	0
$296$	−26.5147	−1.54114
$297$	2.66949	0.154899
$298$	−14.5917	−0.845274
$299$	13.1968	0.763188
$300$	0	0
$301$	−5.19498	−0.299433
$302$	−22.8706	−1.31606
$303$	−10.7243	−0.616096
$304$	−1.66017	−0.0952175
$305$	0	0
$306$	5.80801	0.332022
$307$	−6.07107	−0.346494	−0.173247	−	0.984878i	$-0.555426\pi$
$307$	−6.07107	−0.346494	−0.173247	+	0.984878i	$0.555426\pi$
$308$	6.72877	0.383407
$309$	−7.62151	−0.433573
$310$	0	0
$311$	−7.38279	−0.418640	−0.209320	−	0.977847i	$-0.567125\pi$
$311$	−7.38279	−0.418640	−0.209320	+	0.977847i	$0.567125\pi$
$312$	8.75130	0.495445
$313$	5.66028	0.319938	0.159969	−	0.987122i	$-0.448861\pi$
$313$	5.66028	0.319938	0.159969	+	0.987122i	$0.448861\pi$
$314$	−4.26561	−0.240722
$315$	0	0
$316$	−7.53003	−0.423597
$317$	32.1743	1.80709	0.903543	−	0.428497i	$-0.140957\pi$
$317$	32.1743	1.80709	0.903543	+	0.428497i	$0.140957\pi$
$318$	24.6027	1.37965
$319$	−19.8190	−1.10965
$320$	0	0
$321$	−5.17157	−0.288649
$322$	7.19421	0.400918
$323$	−12.5239	−0.696847
$324$	3.15894	0.175497
$325$	0	0
$326$	−48.3531	−2.67803
$327$	−0.928932	−0.0513701
$328$	23.2062	1.28135
$329$	0.797933	0.0439915
$330$	0	0
$331$	−11.2527	−0.618505	−0.309253	−	0.950980i	$-0.600079\pi$
$331$	−11.2527	−0.618505	−0.309253	+	0.950980i	$0.600079\pi$
$332$	6.94814	0.381329
$333$	10.0727	0.551980
$334$	−38.4301	−2.10280
$335$	0	0
$336$	−0.270476	−0.0147557
$337$	−28.6488	−1.56060	−0.780300	−	0.625405i	$-0.784934\pi$
$337$	−28.6488	−1.56060	−0.780300	+	0.625405i	$0.784934\pi$
$338$	4.42334	0.240598
$339$	−8.49013	−0.461120
$340$	0	0
$341$	−10.7256	−0.580825
$342$	−11.1243	−0.601532
$343$	−10.6630	−0.575749
$344$	17.1380	0.924017
$345$	0	0
$346$	47.1827	2.53656
$347$	−10.6289	−0.570590	−0.285295	−	0.958440i	$-0.592092\pi$
$347$	−10.6289	−0.570590	−0.285295	+	0.958440i	$0.592092\pi$
$348$	−23.4529	−1.25721
$349$	−24.1986	−1.29532	−0.647660	−	0.761930i	$-0.724252\pi$
$349$	−24.1986	−1.29532	−0.647660	+	0.761930i	$0.724252\pi$
$350$	0	0
$351$	−3.32453	−0.177451
$352$	16.1093	0.858626
$353$	−28.4779	−1.51572	−0.757862	−	0.652414i	$-0.773756\pi$
$353$	−28.4779	−1.51572	−0.757862	+	0.652414i	$0.773756\pi$
$354$	16.8265	0.894321
$355$	0	0
$356$	−34.3643	−1.82130
$357$	−2.04039	−0.107989
$358$	2.90272	0.153413
$359$	−8.34165	−0.440255	−0.220128	−	0.975471i	$-0.570647\pi$
$359$	−8.34165	−0.440255	−0.220128	+	0.975471i	$0.570647\pi$
$360$	0	0
$361$	4.98737	0.262493
$362$	−0.412591	−0.0216853
$363$	−3.87385	−0.203324
$364$	−8.37990	−0.439226
$365$	0	0
$366$	−2.07542	−0.108484
$367$	14.4110	0.752248	0.376124	−	0.926569i	$-0.377257\pi$
$367$	14.4110	0.752248	0.376124	+	0.926569i	$0.377257\pi$
$368$	1.34555	0.0701415
$369$	−8.81580	−0.458932
$370$	0	0
$371$	−8.64309	−0.448727
$372$	−12.6922	−0.658060
$373$	17.8674	0.925138	0.462569	−	0.886583i	$-0.346928\pi$
$373$	17.8674	0.925138	0.462569	+	0.886583i	$0.346928\pi$
$374$	15.5044	0.801714
$375$	0	0
$376$	−2.63234	−0.135753
$377$	24.6823	1.27120
$378$	−1.81237	−0.0932182
$379$	1.45574	0.0747762	0.0373881	−	0.999301i	$-0.488096\pi$
$379$	1.45574	0.0747762	0.0373881	+	0.999301i	$0.488096\pi$
$380$	0	0
$381$	8.02016	0.410885
$382$	−60.1400	−3.07703
$383$	17.6711	0.902951	0.451476	−	0.892283i	$-0.350898\pi$
$383$	17.6711	0.902951	0.451476	+	0.892283i	$0.350898\pi$
$384$	17.5231	0.894222
$385$	0	0
$386$	−18.1759	−0.925127
$387$	−6.51054	−0.330950
$388$	36.4494	1.85044
$389$	−3.63577	−0.184341	−0.0921703	−	0.995743i	$-0.529380\pi$
$389$	−3.63577	−0.184341	−0.0921703	+	0.995743i	$0.529380\pi$
$390$	0	0
$391$	10.1504	0.513328
$392$	16.7504	0.846022
$393$	−9.95687	−0.502258
$394$	43.7997	2.20660
$395$	0	0
$396$	8.43275	0.423762
$397$	−19.3206	−0.969671	−0.484836	−	0.874605i	$-0.661121\pi$
$397$	−19.3206	−0.969671	−0.484836	+	0.874605i	$0.661121\pi$
$398$	−9.68024	−0.485226
$399$	3.90803	0.195646
$400$	0	0
$401$	−16.2890	−0.813434	−0.406717	−	0.913554i	$-0.633327\pi$
$401$	−16.2890	−0.813434	−0.406717	+	0.913554i	$0.633327\pi$
$402$	11.3795	0.567560
$403$	13.3575	0.665386
$404$	−33.8775	−1.68547
$405$	0	0
$406$	13.4556	0.667788
$407$	26.8889	1.33283
$408$	6.73115	0.333241
$409$	−33.0205	−1.63276	−0.816380	−	0.577515i	$-0.804023\pi$
$409$	−33.0205	−1.63276	−0.816380	+	0.577515i	$0.804023\pi$
$410$	0	0
$411$	−11.3711	−0.560894
$412$	−24.0759	−1.18614
$413$	−5.91127	−0.290875
$414$	9.01606	0.443115
$415$	0	0
$416$	−20.0622	−0.983631
$417$	4.56880	0.223735
$418$	−29.6961	−1.45248
$419$	15.8661	0.775108	0.387554	−	0.921847i	$-0.373320\pi$
$419$	15.8661	0.775108	0.387554	+	0.921847i	$0.373320\pi$
$420$	0	0
$421$	33.1091	1.61364	0.806819	−	0.590799i	$-0.201187\pi$
$421$	33.1091	1.61364	0.806819	+	0.590799i	$0.201187\pi$
$422$	−1.78368	−0.0868281
$423$	1.00000	0.0486217
$424$	28.5131	1.38472
$425$	0	0
$426$	−25.4013	−1.23070
$427$	0.729110	0.0352841
$428$	−16.3367	−0.789664
$429$	−8.87480	−0.428479
$430$	0	0
$431$	11.3674	0.547549	0.273774	−	0.961794i	$-0.411728\pi$
$431$	11.3674	0.547549	0.273774	+	0.961794i	$0.411728\pi$
$432$	−0.338971	−0.0163087
$433$	−39.2974	−1.88851	−0.944257	−	0.329210i	$-0.893217\pi$
$433$	−39.2974	−1.88851	−0.944257	+	0.329210i	$0.893217\pi$
$434$	7.28185	0.349540
$435$	0	0
$436$	−2.93444	−0.140534
$437$	−19.4414	−0.930009
$438$	−17.3507	−0.829047
$439$	22.9754	1.09656	0.548278	−	0.836296i	$-0.315283\pi$
$439$	22.9754	1.09656	0.548278	+	0.836296i	$0.315283\pi$
$440$	0	0
$441$	−6.36330	−0.303014
$442$	−19.3089	−0.918432
$443$	15.3069	0.727252	0.363626	−	0.931545i	$-0.381539\pi$
$443$	15.3069	0.727252	0.363626	+	0.931545i	$0.381539\pi$
$444$	31.8190	1.51006
$445$	0	0
$446$	−11.0649	−0.523938
$447$	6.42429	0.303859
$448$	−10.3960	−0.491163
$449$	−31.8233	−1.50184	−0.750918	−	0.660396i	$-0.770389\pi$
$449$	−31.8233	−1.50184	−0.750918	+	0.660396i	$0.770389\pi$
$450$	0	0
$451$	−23.5336	−1.10816
$452$	−26.8198	−1.26150
$453$	10.0693	0.473095
$454$	−8.52108	−0.399914
$455$	0	0
$456$	−12.8924	−0.603741
$457$	−6.29262	−0.294356	−0.147178	−	0.989110i	$-0.547019\pi$
$457$	−6.29262	−0.294356	−0.147178	+	0.989110i	$0.547019\pi$
$458$	30.9213	1.44486
$459$	−2.55710	−0.119355
$460$	0	0
$461$	−12.8940	−0.600534	−0.300267	−	0.953855i	$-0.597076\pi$
$461$	−12.8940	−0.600534	−0.300267	+	0.953855i	$0.597076\pi$
$462$	−4.83809	−0.225088
$463$	−29.2774	−1.36064	−0.680319	−	0.732916i	$-0.738159\pi$
$463$	−29.2774	−1.36064	−0.680319	+	0.732916i	$0.738159\pi$
$464$	2.51662	0.116831
$465$	0	0
$466$	34.1252	1.58082
$467$	−38.4698	−1.78017	−0.890086	−	0.455793i	$-0.849356\pi$
$467$	−38.4698	−1.78017	−0.890086	+	0.455793i	$0.849356\pi$
$468$	−10.5020	−0.485456
$469$	−3.99771	−0.184597
$470$	0	0
$471$	1.87802	0.0865347
$472$	19.5010	0.897606
$473$	−17.3798	−0.799124
$474$	5.41421	0.248683
$475$	0	0
$476$	−6.44548	−0.295428
$477$	−10.8319	−0.495957
$478$	−62.5240	−2.85978
$479$	−27.6440	−1.26309	−0.631544	−	0.775340i	$-0.717578\pi$
$479$	−27.6440	−1.26309	−0.631544	+	0.775340i	$0.717578\pi$
$480$	0	0
$481$	−33.4870	−1.52688
$482$	−2.70893	−0.123388
$483$	−3.16740	−0.144122
$484$	−12.2373	−0.556239
$485$	0	0
$486$	−2.27133	−0.103030
$487$	21.2375	0.962365	0.481182	−	0.876621i	$-0.340208\pi$
$487$	21.2375	0.962365	0.481182	+	0.876621i	$0.340208\pi$
$488$	−2.40530	−0.108883
$489$	21.2884	0.962697
$490$	0	0
$491$	−27.0398	−1.22029	−0.610145	−	0.792290i	$-0.708889\pi$
$491$	−27.0398	−1.22029	−0.610145	+	0.792290i	$0.708889\pi$
$492$	−27.8486	−1.25551
$493$	18.9846	0.855025
$494$	36.9830	1.66395
$495$	0	0
$496$	1.36194	0.0611529
$497$	8.92365	0.400280
$498$	−4.99583	−0.223868
$499$	10.0297	0.448993	0.224497	−	0.974475i	$-0.427926\pi$
$499$	10.0297	0.448993	0.224497	+	0.974475i	$0.427926\pi$
$500$	0	0
$501$	16.9197	0.755914
$502$	−21.3140	−0.951290
$503$	−28.6507	−1.27747	−0.638736	−	0.769426i	$-0.720542\pi$
$503$	−28.6507	−1.27747	−0.638736	+	0.769426i	$0.720542\pi$
$504$	−2.10043	−0.0935606
$505$	0	0
$506$	24.0682	1.06996
$507$	−1.94747	−0.0864901
$508$	25.3352	1.12407
$509$	30.6362	1.35792	0.678962	−	0.734173i	$-0.262430\pi$
$509$	30.6362	1.35792	0.678962	+	0.734173i	$0.262430\pi$
$510$	0	0
$511$	6.09540	0.269645
$512$	−3.83013	−0.169269
$513$	4.89769	0.216238
$514$	8.15639	0.359763
$515$	0	0
$516$	−20.5664	−0.905387
$517$	2.66949	0.117404
$518$	−18.2554	−0.802097
$519$	−20.7732	−0.911841
$520$	0	0
$521$	18.8421	0.825489	0.412744	−	0.910847i	$-0.364570\pi$
$521$	18.8421	0.825489	0.412744	+	0.910847i	$0.364570\pi$
$522$	16.8630	0.738074
$523$	−31.8833	−1.39416	−0.697079	−	0.716995i	$-0.745517\pi$
$523$	−31.8833	−1.39416	−0.697079	+	0.716995i	$0.745517\pi$
$524$	−31.4532	−1.37404
$525$	0	0
$526$	58.2502	2.53983
$527$	10.2741	0.447545
$528$	−0.904878	−0.0393798
$529$	−7.24303	−0.314914
$530$	0	0
$531$	−7.40823	−0.321490
$532$	12.3452	0.535234
$533$	29.3084	1.26949
$534$	24.7085	1.06924
$535$	0	0
$536$	13.1882	0.569645
$537$	−1.27798	−0.0551490
$538$	−54.1283	−2.33364
$539$	−16.9867	−0.731671
$540$	0	0
$541$	−6.72496	−0.289128	−0.144564	−	0.989495i	$-0.546178\pi$
$541$	−6.72496	−0.289128	−0.144564	+	0.989495i	$0.546178\pi$
$542$	64.9641	2.79045
$543$	0.181652	0.00779542
$544$	−15.4310	−0.661600
$545$	0	0
$546$	6.02528	0.257858
$547$	−20.5495	−0.878634	−0.439317	−	0.898332i	$-0.644779\pi$
$547$	−20.5495	−0.878634	−0.439317	+	0.898332i	$0.644779\pi$
$548$	−35.9206	−1.53445
$549$	0.913749	0.0389978
$550$	0	0
$551$	−36.3619	−1.54907
$552$	10.4491	0.444743
$553$	−1.90205	−0.0808833
$554$	−16.7885	−0.713277
$555$	0	0
$556$	14.4326	0.612077
$557$	31.9660	1.35444	0.677222	−	0.735779i	$-0.263184\pi$
$557$	31.9660	1.35444	0.677222	+	0.735779i	$0.263184\pi$
$558$	9.12589	0.386330
$559$	21.6445	0.915466
$560$	0	0
$561$	−6.82613	−0.288200
$562$	4.93236	0.208059
$563$	27.0257	1.13900	0.569498	−	0.821993i	$-0.307138\pi$
$563$	27.0257	1.13900	0.569498	+	0.821993i	$0.307138\pi$
$564$	3.15894	0.133015
$565$	0	0
$566$	12.0308	0.505691
$567$	0.797933	0.0335100
$568$	−29.4387	−1.23522
$569$	4.29262	0.179956	0.0899780	−	0.995944i	$-0.471320\pi$
$569$	4.29262	0.179956	0.0899780	+	0.995944i	$0.471320\pi$
$570$	0	0
$571$	−45.7598	−1.91499	−0.957493	−	0.288456i	$-0.906858\pi$
$571$	−45.7598	−1.91499	−0.957493	+	0.288456i	$0.906858\pi$
$572$	−28.0350	−1.17220
$573$	26.4779	1.10613
$574$	15.9775	0.666887
$575$	0	0
$576$	−13.0286	−0.542859
$577$	−0.816286	−0.0339824	−0.0169912	−	0.999856i	$-0.505409\pi$
$577$	−0.816286	−0.0339824	−0.0169912	+	0.999856i	$0.505409\pi$
$578$	23.7610	0.988326
$579$	8.00229	0.332564
$580$	0	0
$581$	1.75506	0.0728124
$582$	−26.2077	−1.08634
$583$	−28.9155	−1.19756
$584$	−20.1084	−0.832092
$585$	0	0
$586$	9.01974	0.372602
$587$	27.8697	1.15030	0.575152	−	0.818046i	$-0.304943\pi$
$587$	27.8697	1.15030	0.575152	+	0.818046i	$0.304943\pi$
$588$	−20.1013	−0.828964
$589$	−19.6782	−0.810828
$590$	0	0
$591$	−19.2837	−0.793226
$592$	−3.41435	−0.140329
$593$	0.236479	0.00971102	0.00485551	−	0.999988i	$-0.498454\pi$
$593$	0.236479	0.00971102	0.00485551	+	0.999988i	$0.498454\pi$
$594$	−6.06328	−0.248780
$595$	0	0
$596$	20.2940	0.831273
$597$	4.26192	0.174429
$598$	−29.9742	−1.22574
$599$	−14.1353	−0.577552	−0.288776	−	0.957397i	$-0.593248\pi$
$599$	−14.1353	−0.577552	−0.288776	+	0.957397i	$0.593248\pi$
$600$	0	0
$601$	−27.9654	−1.14073	−0.570365	−	0.821391i	$-0.693198\pi$
$601$	−27.9654	−1.14073	−0.570365	+	0.821391i	$0.693198\pi$
$602$	11.7995	0.480912
$603$	−5.01008	−0.204026
$604$	31.8082	1.29426
$605$	0	0
$606$	24.3585	0.989495
$607$	−27.7821	−1.12764	−0.563820	−	0.825898i	$-0.690669\pi$
$607$	−27.7821	−1.12764	−0.563820	+	0.825898i	$0.690669\pi$
$608$	29.5556	1.19864
$609$	−5.92409	−0.240056
$610$	0	0
$611$	−3.32453	−0.134496
$612$	−8.07772	−0.326522
$613$	−39.6433	−1.60118	−0.800589	−	0.599214i	$-0.795480\pi$
$613$	−39.6433	−1.60118	−0.800589	+	0.599214i	$0.795480\pi$
$614$	13.7894	0.556495
$615$	0	0
$616$	−5.60707	−0.225915
$617$	−14.2206	−0.572501	−0.286250	−	0.958155i	$-0.592409\pi$
$617$	−14.2206	−0.572501	−0.286250	+	0.958155i	$0.592409\pi$
$618$	17.3110	0.696349
$619$	−5.29355	−0.212766	−0.106383	−	0.994325i	$-0.533927\pi$
$619$	−5.29355	−0.212766	−0.106383	+	0.994325i	$0.533927\pi$
$620$	0	0
$621$	−3.96951	−0.159291
$622$	16.7688	0.672366
$623$	−8.68024	−0.347766
$624$	1.12692	0.0451129
$625$	0	0
$626$	−12.8564	−0.513844
$627$	13.0743	0.522138
$628$	5.93256	0.236735
$629$	−25.7568	−1.02699
$630$	0	0
$631$	46.5330	1.85245	0.926225	−	0.376972i	$-0.123035\pi$
$631$	46.5330	1.85245	0.926225	+	0.376972i	$0.123035\pi$
$632$	6.27476	0.249596
$633$	0.785301	0.0312129
$634$	−73.0784	−2.90231
$635$	0	0
$636$	−34.2172	−1.35680
$637$	21.1550	0.838193
$638$	45.0156	1.78218
$639$	11.1835	0.442411
$640$	0	0
$641$	0.0999603	0.00394820	0.00197410	−	0.999998i	$-0.499372\pi$
$641$	0.0999603	0.00394820	0.00197410	+	0.999998i	$0.499372\pi$
$642$	11.7464	0.463591
$643$	19.3644	0.763656	0.381828	−	0.924233i	$-0.375295\pi$
$643$	19.3644	0.763656	0.381828	+	0.924233i	$0.375295\pi$
$644$	−10.0056	−0.394277
$645$	0	0
$646$	28.4458	1.11919
$647$	−10.4399	−0.410433	−0.205217	−	0.978717i	$-0.565790\pi$
$647$	−10.4399	−0.410433	−0.205217	+	0.978717i	$0.565790\pi$
$648$	−2.63234	−0.103408
$649$	−19.7762	−0.776283
$650$	0	0
$651$	−3.20598	−0.125652
$652$	67.2490	2.63367
$653$	−36.8066	−1.44035	−0.720177	−	0.693790i	$-0.755940\pi$
$653$	−36.8066	−1.44035	−0.720177	+	0.693790i	$0.755940\pi$
$654$	2.10991	0.0825041
$655$	0	0
$656$	2.98830	0.116673
$657$	7.63899	0.298025
$658$	−1.81237	−0.0706535
$659$	−4.16149	−0.162109	−0.0810544	−	0.996710i	$-0.525829\pi$
$659$	−4.16149	−0.162109	−0.0810544	+	0.996710i	$0.525829\pi$
$660$	0	0
$661$	6.17034	0.239998	0.119999	−	0.992774i	$-0.461711\pi$
$661$	6.17034	0.239998	0.119999	+	0.992774i	$0.461711\pi$
$662$	25.5586	0.993365
$663$	8.50116	0.330158
$664$	−5.78987	−0.224691
$665$	0	0
$666$	−22.8784	−0.886520
$667$	29.4708	1.14111
$668$	53.4482	2.06797
$669$	4.87155	0.188345
$670$	0	0
$671$	2.43924	0.0941658
$672$	4.81520	0.185750
$673$	30.2410	1.16570	0.582852	−	0.812579i	$-0.301937\pi$
$673$	30.2410	1.16570	0.582852	+	0.812579i	$0.301937\pi$
$674$	65.0709	2.50644
$675$	0	0
$676$	−6.15194	−0.236613
$677$	46.1536	1.77383	0.886913	−	0.461936i	$-0.152845\pi$
$677$	46.1536	1.77383	0.886913	+	0.461936i	$0.152845\pi$
$678$	19.2839	0.740593
$679$	9.20693	0.353330
$680$	0	0
$681$	3.75158	0.143761
$682$	24.3614	0.932848
$683$	12.4684	0.477089	0.238544	−	0.971132i	$-0.423330\pi$
$683$	12.4684	0.477089	0.238544	+	0.971132i	$0.423330\pi$
$684$	15.4715	0.591568
$685$	0	0
$686$	24.2192	0.924695
$687$	−13.6137	−0.519396
$688$	2.20689	0.0841367
$689$	36.0109	1.37191
$690$	0	0
$691$	44.4260	1.69004	0.845022	−	0.534732i	$-0.179587\pi$
$691$	44.4260	1.69004	0.845022	+	0.534732i	$0.179587\pi$
$692$	−65.6212	−2.49454
$693$	2.13007	0.0809147
$694$	24.1418	0.916409
$695$	0	0
$696$	19.5433	0.740785
$697$	22.5428	0.853871
$698$	54.9629	2.08038
$699$	−15.0243	−0.568272
$700$	0	0
$701$	11.5929	0.437859	0.218929	−	0.975741i	$-0.429744\pi$
$701$	11.5929	0.437859	0.218929	+	0.975741i	$0.429744\pi$
$702$	7.55112	0.284999
$703$	49.3329	1.86063
$704$	−34.7797	−1.31081
$705$	0	0
$706$	64.6827	2.43436
$707$	−8.55728	−0.321830
$708$	−23.4022	−0.879508
$709$	9.17292	0.344496	0.172248	−	0.985054i	$-0.444897\pi$
$709$	9.17292	0.344496	0.172248	+	0.985054i	$0.444897\pi$
$710$	0	0
$711$	−2.38372	−0.0893965
$712$	28.6357	1.07317
$713$	15.9489	0.597292
$714$	4.63440	0.173438
$715$	0	0
$716$	−4.03707	−0.150872
$717$	27.5275	1.02803
$718$	18.9466	0.707082
$719$	40.4142	1.50719	0.753597	−	0.657337i	$-0.228317\pi$
$719$	40.4142	1.50719	0.753597	+	0.657337i	$0.228317\pi$
$720$	0	0
$721$	−6.08145	−0.226485
$722$	−11.3280	−0.421583
$723$	1.19266	0.0443555
$724$	0.573827	0.0213261
$725$	0	0
$726$	8.79879	0.326554
$727$	−8.06460	−0.299099	−0.149550	−	0.988754i	$-0.547782\pi$
$727$	−8.06460	−0.299099	−0.149550	+	0.988754i	$0.547782\pi$
$728$	6.98295	0.258805
$729$	1.00000	0.0370370
$730$	0	0
$731$	16.6481	0.615752
$732$	2.88648	0.106687
$733$	15.3153	0.565684	0.282842	−	0.959167i	$-0.408723\pi$
$733$	15.3153	0.565684	0.282842	+	0.959167i	$0.408723\pi$
$734$	−32.7321	−1.20816
$735$	0	0
$736$	−23.9544	−0.882969
$737$	−13.3743	−0.492650
$738$	20.0236	0.737078
$739$	−44.0829	−1.62162	−0.810808	−	0.585313i	$-0.800972\pi$
$739$	−44.0829	−1.62162	−0.810808	+	0.585313i	$0.800972\pi$
$740$	0	0
$741$	−16.2825	−0.598154
$742$	19.6313	0.720688
$743$	9.37297	0.343861	0.171930	−	0.985109i	$-0.445000\pi$
$743$	9.37297	0.343861	0.171930	+	0.985109i	$0.445000\pi$
$744$	10.5764	0.387749
$745$	0	0
$746$	−40.5827	−1.48584
$747$	2.19951	0.0804760
$748$	−21.5634	−0.788434
$749$	−4.12657	−0.150781
$750$	0	0
$751$	−35.3123	−1.28856	−0.644282	−	0.764788i	$-0.722844\pi$
$751$	−35.3123	−1.28856	−0.644282	+	0.764788i	$0.722844\pi$
$752$	−0.338971	−0.0123610
$753$	9.38392	0.341969
$754$	−56.0617	−2.04165
$755$	0	0
$756$	2.52062	0.0916742
$757$	35.3211	1.28377	0.641884	−	0.766802i	$-0.278153\pi$
$757$	35.3211	1.28377	0.641884	+	0.766802i	$0.278153\pi$
$758$	−3.30646	−0.120096
$759$	−10.5965	−0.384630
$760$	0	0
$761$	28.3348	1.02714	0.513568	−	0.858049i	$-0.328323\pi$
$761$	28.3348	1.02714	0.513568	+	0.858049i	$0.328323\pi$
$762$	−18.2164	−0.659912
$763$	−0.741225	−0.0268342
$764$	83.6421	3.02606
$765$	0	0
$766$	−40.1369	−1.45021
$767$	24.6289	0.889299
$768$	−13.7435	−0.495927
$769$	50.4802	1.82036	0.910181	−	0.414211i	$-0.135942\pi$
$769$	50.4802	1.82036	0.910181	+	0.414211i	$0.135942\pi$
$770$	0	0
$771$	−3.59102	−0.129327
$772$	25.2788	0.909803
$773$	7.72983	0.278023	0.139011	−	0.990291i	$-0.455608\pi$
$773$	7.72983	0.278023	0.139011	+	0.990291i	$0.455608\pi$
$774$	14.7876	0.531529
$775$	0	0
$776$	−30.3732	−1.09033
$777$	8.03733	0.288338
$778$	8.25803	0.296065
$779$	−43.1770	−1.54698
$780$	0	0
$781$	29.8541	1.06826
$782$	−23.0549	−0.824443
$783$	−7.42429	−0.265323
$784$	2.15698	0.0770348
$785$	0	0
$786$	22.6154	0.806663
$787$	20.1923	0.719778	0.359889	−	0.932995i	$-0.382815\pi$
$787$	20.1923	0.719778	0.359889	+	0.932995i	$0.382815\pi$
$788$	−60.9161	−2.17005
$789$	−25.6458	−0.913016
$790$	0	0
$791$	−6.77455	−0.240875
$792$	−7.02699	−0.249693
$793$	−3.03779	−0.107875
$794$	43.8834	1.55736
$795$	0	0
$796$	13.4632	0.477189
$797$	21.8576	0.774236	0.387118	−	0.922030i	$-0.373471\pi$
$797$	21.8576	0.774236	0.387118	+	0.922030i	$0.373471\pi$
$798$	−8.87642	−0.314222
$799$	−2.55710	−0.0904636
$800$	0	0
$801$	−10.8784	−0.384370
$802$	36.9977	1.30644
$803$	20.3922	0.719624
$804$	−15.8265	−0.558159
$805$	0	0
$806$	−30.3394	−1.06866
$807$	23.8311	0.838895
$808$	28.2300	0.993129
$809$	18.2212	0.640622	0.320311	−	0.947312i	$-0.396213\pi$
$809$	18.2212	0.640622	0.320311	+	0.947312i	$0.396213\pi$
$810$	0	0
$811$	29.3086	1.02916	0.514582	−	0.857441i	$-0.327947\pi$
$811$	29.3086	1.02916	0.514582	+	0.857441i	$0.327947\pi$
$812$	−18.7138	−0.656727
$813$	−28.6018	−1.00311
$814$	−61.0736	−2.14063
$815$	0	0
$816$	0.866782	0.0303434
$817$	−31.8866	−1.11557
$818$	75.0006	2.62233
$819$	−2.65275	−0.0926948
$820$	0	0
$821$	44.9813	1.56986	0.784930	−	0.619585i	$-0.212699\pi$
$821$	44.9813	1.56986	0.784930	+	0.619585i	$0.212699\pi$
$822$	25.8275	0.900837
$823$	1.93256	0.0673649	0.0336824	−	0.999433i	$-0.489277\pi$
$823$	1.93256	0.0673649	0.0336824	+	0.999433i	$0.489277\pi$
$824$	20.0624	0.698907
$825$	0	0
$826$	13.4265	0.467166
$827$	40.2995	1.40135	0.700676	−	0.713480i	$-0.252882\pi$
$827$	40.2995	1.40135	0.700676	+	0.713480i	$0.252882\pi$
$828$	−12.5394	−0.435776
$829$	17.1568	0.595879	0.297940	−	0.954585i	$-0.403701\pi$
$829$	17.1568	0.595879	0.297940	+	0.954585i	$0.403701\pi$
$830$	0	0
$831$	7.39150	0.256408
$832$	43.3141	1.50165
$833$	16.2716	0.563777
$834$	−10.3772	−0.359335
$835$	0	0
$836$	41.3010	1.42842
$837$	−4.01786	−0.138878
$838$	−36.0371	−1.24488
$839$	−36.4600	−1.25874	−0.629370	−	0.777106i	$-0.716687\pi$
$839$	−36.4600	−1.25874	−0.629370	+	0.777106i	$0.716687\pi$
$840$	0	0
$841$	26.1201	0.900694
$842$	−75.2017	−2.59162
$843$	−2.17157	−0.0747929
$844$	2.48072	0.0853900
$845$	0	0
$846$	−2.27133	−0.0780899
$847$	−3.09107	−0.106210
$848$	3.67168	0.126086
$849$	−5.29680	−0.181786
$850$	0	0
$851$	−39.9836	−1.37062
$852$	35.3279	1.21031
$853$	−0.235241	−0.00805452	−0.00402726	−	0.999992i	$-0.501282\pi$
$853$	−0.235241	−0.00805452	−0.00402726	+	0.999992i	$0.501282\pi$
$854$	−1.65605	−0.0566688
$855$	0	0
$856$	13.6133	0.465294
$857$	35.5570	1.21461	0.607303	−	0.794471i	$-0.292252\pi$
$857$	35.5570	1.21461	0.607303	+	0.794471i	$0.292252\pi$
$858$	20.1576	0.688169
$859$	0.331627	0.0113150	0.00565748	−	0.999984i	$-0.498199\pi$
$859$	0.331627	0.0113150	0.00565748	+	0.999984i	$0.498199\pi$
$860$	0	0
$861$	−7.03441	−0.239732
$862$	−25.8191	−0.879403
$863$	−48.0449	−1.63547	−0.817734	−	0.575597i	$-0.804770\pi$
$863$	−48.0449	−1.63547	−0.817734	+	0.575597i	$0.804770\pi$
$864$	6.03459	0.205301
$865$	0	0
$866$	89.2574	3.03309
$867$	−10.4613	−0.355283
$868$	−10.1275	−0.343750
$869$	−6.36330	−0.215860
$870$	0	0
$871$	16.6562	0.564373
$872$	2.44526	0.0828071
$873$	11.5385	0.390519
$874$	44.1579	1.49366
$875$	0	0
$876$	24.1311	0.815315
$877$	34.1258	1.15235	0.576174	−	0.817327i	$-0.304545\pi$
$877$	34.1258	1.15235	0.576174	+	0.817327i	$0.304545\pi$
$878$	−52.1847	−1.76115
$879$	−3.97113	−0.133943
$880$	0	0
$881$	14.5381	0.489802	0.244901	−	0.969548i	$-0.421245\pi$
$881$	14.5381	0.489802	0.244901	+	0.969548i	$0.421245\pi$
$882$	14.4532	0.486663
$883$	20.8074	0.700224	0.350112	−	0.936708i	$-0.386144\pi$
$883$	20.8074	0.700224	0.350112	+	0.936708i	$0.386144\pi$
$884$	26.8547	0.903220
$885$	0	0
$886$	−34.7670	−1.16802
$887$	−45.5980	−1.53103	−0.765515	−	0.643418i	$-0.777516\pi$
$887$	−45.5980	−1.53103	−0.765515	+	0.643418i	$0.777516\pi$
$888$	−26.5147	−0.889776
$889$	6.39955	0.214634
$890$	0	0
$891$	2.66949	0.0894311
$892$	15.3890	0.515260
$893$	4.89769	0.163895
$894$	−14.5917	−0.488019
$895$	0	0
$896$	13.9823	0.467114
$897$	13.1968	0.440627
$898$	72.2813	2.41206
$899$	29.8298	0.994879
$900$	0	0
$901$	27.6981	0.922757
$902$	53.4527	1.77978
$903$	−5.19498	−0.172878
$904$	22.3489	0.743313
$905$	0	0
$906$	−22.8706	−0.759826
$907$	2.44670	0.0812412	0.0406206	−	0.999175i	$-0.487067\pi$
$907$	2.44670	0.0812412	0.0406206	+	0.999175i	$0.487067\pi$
$908$	11.8510	0.393290
$909$	−10.7243	−0.355703
$910$	0	0
$911$	56.0628	1.85744	0.928721	−	0.370779i	$-0.120909\pi$
$911$	56.0628	1.85744	0.928721	+	0.370779i	$0.120909\pi$
$912$	−1.66017	−0.0549739
$913$	5.87157	0.194321
$914$	14.2926	0.472758
$915$	0	0
$916$	−43.0050	−1.42092
$917$	−7.94492	−0.262364
$918$	5.80801	0.191693
$919$	51.1138	1.68609	0.843044	−	0.537845i	$-0.180762\pi$
$919$	51.1138	1.68609	0.843044	+	0.537845i	$0.180762\pi$
$920$	0	0
$921$	−6.07107	−0.200048
$922$	29.2865	0.964501
$923$	−37.1798	−1.22379
$924$	6.72877	0.221360
$925$	0	0
$926$	66.4987	2.18528
$927$	−7.62151	−0.250323
$928$	−44.8026	−1.47072
$929$	6.86663	0.225287	0.112643	−	0.993635i	$-0.464068\pi$
$929$	6.86663	0.225287	0.112643	+	0.993635i	$0.464068\pi$
$930$	0	0
$931$	−31.1655	−1.02141
$932$	−47.4610	−1.55464
$933$	−7.38279	−0.241702
$934$	87.3777	2.85909
$935$	0	0
$936$	8.75130	0.286045
$937$	−20.1146	−0.657115	−0.328557	−	0.944484i	$-0.606562\pi$
$937$	−20.1146	−0.657115	−0.328557	+	0.944484i	$0.606562\pi$
$938$	9.08011	0.296476
$939$	5.66028	0.184716
$940$	0	0
$941$	−42.2041	−1.37581	−0.687907	−	0.725799i	$-0.741470\pi$
$941$	−42.2041	−1.37581	−0.687907	+	0.725799i	$0.741470\pi$
$942$	−4.26561	−0.138981
$943$	34.9944	1.13957
$944$	2.51118	0.0817318
$945$	0	0
$946$	39.4753	1.28345
$947$	−40.1072	−1.30331	−0.651655	−	0.758516i	$-0.725925\pi$
$947$	−40.1072	−1.30331	−0.651655	+	0.758516i	$0.725925\pi$
$948$	−7.53003	−0.244564
$949$	−25.3961	−0.824392
$950$	0	0
$951$	32.1743	1.04332
$952$	5.37100	0.174075
$953$	−50.2995	−1.62936	−0.814680	−	0.579911i	$-0.803087\pi$
$953$	−50.2995	−1.62936	−0.814680	+	0.579911i	$0.803087\pi$
$954$	24.6027	0.796542
$955$	0	0
$956$	86.9577	2.81241
$957$	−19.8190	−0.640659
$958$	62.7887	2.02861
$959$	−9.07336	−0.292994
$960$	0	0
$961$	−14.8568	−0.479251
$962$	76.0601	2.45227
$963$	−5.17157	−0.166652
$964$	3.76754	0.121344
$965$	0	0
$966$	7.19421	0.231470
$967$	−39.5486	−1.27180	−0.635898	−	0.771773i	$-0.719370\pi$
$967$	−39.5486	−1.27180	−0.635898	+	0.771773i	$0.719370\pi$
$968$	10.1973	0.327753
$969$	−12.5239	−0.402325
$970$	0	0
$971$	−15.0922	−0.484331	−0.242165	−	0.970235i	$-0.577858\pi$
$971$	−15.0922	−0.484331	−0.242165	+	0.970235i	$0.577858\pi$
$972$	3.15894	0.101323
$973$	3.64559	0.116872
$974$	−48.2375	−1.54563
$975$	0	0
$976$	−0.309734	−0.00991435
$977$	−6.35893	−0.203440	−0.101720	−	0.994813i	$-0.532435\pi$
$977$	−6.35893	−0.203440	−0.101720	+	0.994813i	$0.532435\pi$
$978$	−48.3531	−1.54616
$979$	−29.0398	−0.928114
$980$	0	0
$981$	−0.928932	−0.0296585
$982$	61.4164	1.95988
$983$	13.3458	0.425666	0.212833	−	0.977089i	$-0.431731\pi$
$983$	13.3458	0.425666	0.212833	+	0.977089i	$0.431731\pi$
$984$	23.2062	0.739786
$985$	0	0
$986$	−43.1204	−1.37323
$987$	0.797933	0.0253985
$988$	−51.4356	−1.63638
$989$	25.8436	0.821780
$990$	0	0
$991$	43.8036	1.39147	0.695733	−	0.718300i	$-0.255080\pi$
$991$	43.8036	1.39147	0.695733	+	0.718300i	$0.255080\pi$
$992$	−24.2462	−0.769817
$993$	−11.2527	−0.357094
$994$	−20.2685	−0.642879
$995$	0	0
$996$	6.94814	0.220160
$997$	55.6878	1.76365	0.881826	−	0.471575i	$-0.156314\pi$
$997$	55.6878	1.76365	0.881826	+	0.471575i	$0.156314\pi$
$998$	−22.7809	−0.721116
$999$	10.0727	0.318686

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3525.2.a.t.1.2		4
5.4	even	2		705.2.a.k.1.3	✓	4
15.14	odd	2		2115.2.a.o.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
705.2.a.k.1.3	✓	4	5.4	even	2
2115.2.a.o.1.2		4	15.14	odd	2
3525.2.a.t.1.2		4	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:	\( N \)	\(=\)	\( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3525.a (trivial)

\(f(q)\)	\(=\)	\(q-2.27133 q^{2} +1.00000 q^{3} +3.15894 q^{4} -2.27133 q^{6} +0.797933 q^{7} -2.63234 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.27133 q^{2} +1.00000 q^{3} +3.15894 q^{4} -2.27133 q^{6} +0.797933 q^{7} -2.63234 q^{8} +1.00000 q^{9} +2.66949 q^{11} +3.15894 q^{12} -3.32453 q^{13} -1.81237 q^{14} -0.338971 q^{16} -2.55710 q^{17} -2.27133 q^{18} +4.89769 q^{19} +0.797933 q^{21} -6.06328 q^{22} -3.96951 q^{23} -2.63234 q^{24} +7.55112 q^{26} +1.00000 q^{27} +2.52062 q^{28} -7.42429 q^{29} -4.01786 q^{31} +6.03459 q^{32} +2.66949 q^{33} +5.80801 q^{34} +3.15894 q^{36} +10.0727 q^{37} -11.1243 q^{38} -3.32453 q^{39} -8.81580 q^{41} -1.81237 q^{42} -6.51054 q^{43} +8.43275 q^{44} +9.01606 q^{46} +1.00000 q^{47} -0.338971 q^{48} -6.36330 q^{49} -2.55710 q^{51} -10.5020 q^{52} -10.8319 q^{53} -2.27133 q^{54} -2.10043 q^{56} +4.89769 q^{57} +16.8630 q^{58} -7.40823 q^{59} +0.913749 q^{61} +9.12589 q^{62} +0.797933 q^{63} -13.0286 q^{64} -6.06328 q^{66} -5.01008 q^{67} -8.07772 q^{68} -3.96951 q^{69} +11.1835 q^{71} -2.63234 q^{72} +7.63899 q^{73} -22.8784 q^{74} +15.4715 q^{76} +2.13007 q^{77} +7.55112 q^{78} -2.38372 q^{79} +1.00000 q^{81} +20.0236 q^{82} +2.19951 q^{83} +2.52062 q^{84} +14.7876 q^{86} -7.42429 q^{87} -7.02699 q^{88} -10.8784 q^{89} -2.65275 q^{91} -12.5394 q^{92} -4.01786 q^{93} -2.27133 q^{94} +6.03459 q^{96} +11.5385 q^{97} +14.4532 q^{98} +2.66949 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{3} + 8 q^{4} - 4 q^{6} - 8 q^{7} - 12 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^3 + 8 * q^4 - 4 * q^6 - 8 * q^7 - 12 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} + 4 q^{3} + 8 q^{4} - 4 q^{6} - 8 q^{7} - 12 q^{8} + 4 q^{9} + 4 q^{11} + 8 q^{12} + 4 q^{13} + 12 q^{16} - 4 q^{17} - 4 q^{18} - 8 q^{19} - 8 q^{21} + 16 q^{22} - 16 q^{23} - 12 q^{24} + 4 q^{27} - 20 q^{28} + 4 q^{29} - 28 q^{32} + 4 q^{33} - 16 q^{34} + 8 q^{36} + 4 q^{37} - 4 q^{38} + 4 q^{39} - 8 q^{41} - 24 q^{43} - 20 q^{44} + 32 q^{46} + 4 q^{47} + 12 q^{48} + 8 q^{49} - 4 q^{51} + 28 q^{52} - 12 q^{53} - 4 q^{54} + 40 q^{56} - 8 q^{57} + 4 q^{58} - 28 q^{61} - 12 q^{62} - 8 q^{63} + 24 q^{64} + 16 q^{66} + 8 q^{67} + 4 q^{68} - 16 q^{69} + 16 q^{71} - 12 q^{72} + 24 q^{73} - 56 q^{74} - 8 q^{76} - 4 q^{77} - 4 q^{79} + 4 q^{81} + 4 q^{82} - 24 q^{83} - 20 q^{84} - 8 q^{86} + 4 q^{87} + 40 q^{88} - 8 q^{89} - 40 q^{91} - 28 q^{92} - 4 q^{94} - 28 q^{96} + 32 q^{98} + 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^3 + 8 * q^4 - 4 * q^6 - 8 * q^7 - 12 * q^8 + 4 * q^9 + 4 * q^11 + 8 * q^12 + 4 * q^13 + 12 * q^16 - 4 * q^17 - 4 * q^18 - 8 * q^19 - 8 * q^21 + 16 * q^22 - 16 * q^23 - 12 * q^24 + 4 * q^27 - 20 * q^28 + 4 * q^29 - 28 * q^32 + 4 * q^33 - 16 * q^34 + 8 * q^36 + 4 * q^37 - 4 * q^38 + 4 * q^39 - 8 * q^41 - 24 * q^43 - 20 * q^44 + 32 * q^46 + 4 * q^47 + 12 * q^48 + 8 * q^49 - 4 * q^51 + 28 * q^52 - 12 * q^53 - 4 * q^54 + 40 * q^56 - 8 * q^57 + 4 * q^58 - 28 * q^61 - 12 * q^62 - 8 * q^63 + 24 * q^64 + 16 * q^66 + 8 * q^67 + 4 * q^68 - 16 * q^69 + 16 * q^71 - 12 * q^72 + 24 * q^73 - 56 * q^74 - 8 * q^76 - 4 * q^77 - 4 * q^79 + 4 * q^81 + 4 * q^82 - 24 * q^83 - 20 * q^84 - 8 * q^86 + 4 * q^87 + 40 * q^88 - 8 * q^89 - 40 * q^91 - 28 * q^92 - 4 * q^94 - 28 * q^96 + 32 * q^98 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more