LMFDB - Embedded newform 1932.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1932,2,Mod(1,1932)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1932, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1932.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1932.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:	$15.4270976705$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1101.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 12 $ x^3 - x^2 - 9*x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.43163$ of defining polynomial
Character	$\chi$	$=$	1932.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.43163 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.43163 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +3.95044 q^{13} +1.43163 q^{15} +4.51882 q^{17} +3.00000 q^{19} -1.00000 q^{21} +1.00000 q^{23} -2.95044 q^{25} +1.00000 q^{27} -10.4197 q^{29} +0.344438 q^{31} +1.00000 q^{33} -1.43163 q^{35} -7.55645 q^{37} +3.95044 q^{39} +5.17438 q^{41} +7.08719 q^{43} +1.43163 q^{45} +7.51882 q^{47} +1.00000 q^{49} +4.51882 q^{51} -4.95044 q^{53} +1.43163 q^{55} +3.00000 q^{57} +4.77606 q^{59} +4.60601 q^{61} -1.00000 q^{63} +5.65556 q^{65} +10.4693 q^{67} +1.00000 q^{69} +12.1248 q^{71} +0.518817 q^{73} -2.95044 q^{75} -1.00000 q^{77} +1.38207 q^{79} +1.00000 q^{81} -6.41970 q^{83} +6.46926 q^{85} -10.4197 q^{87} -2.81370 q^{89} -3.95044 q^{91} +0.344438 q^{93} +4.29488 q^{95} -6.51882 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + q^5 - 3 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9} + 3 q^{11} - q^{13} + q^{15} + 4 q^{17} + 9 q^{19} - 3 q^{21} + 3 q^{23} + 4 q^{25} + 3 q^{27} + 4 q^{29} + 4 q^{31} + 3 q^{33} - q^{35} + 6 q^{37} - q^{39} + 3 q^{41} + 15 q^{43} + q^{45} + 13 q^{47} + 3 q^{49} + 4 q^{51} - 2 q^{53} + q^{55} + 9 q^{57} + 14 q^{59} - 2 q^{61} - 3 q^{63} + 14 q^{65} + 9 q^{67} + 3 q^{69} + 11 q^{71} - 8 q^{73} + 4 q^{75} - 3 q^{77} - 12 q^{79} + 3 q^{81} + 16 q^{83} - 3 q^{85} + 4 q^{87} + 11 q^{89} + q^{91} + 4 q^{93} + 3 q^{95} - 10 q^{97} + 3 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + q^5 - 3 * q^7 + 3 * q^9 + 3 * q^11 - q^13 + q^15 + 4 * q^17 + 9 * q^19 - 3 * q^21 + 3 * q^23 + 4 * q^25 + 3 * q^27 + 4 * q^29 + 4 * q^31 + 3 * q^33 - q^35 + 6 * q^37 - q^39 + 3 * q^41 + 15 * q^43 + q^45 + 13 * q^47 + 3 * q^49 + 4 * q^51 - 2 * q^53 + q^55 + 9 * q^57 + 14 * q^59 - 2 * q^61 - 3 * q^63 + 14 * q^65 + 9 * q^67 + 3 * q^69 + 11 * q^71 - 8 * q^73 + 4 * q^75 - 3 * q^77 - 12 * q^79 + 3 * q^81 + 16 * q^83 - 3 * q^85 + 4 * q^87 + 11 * q^89 + q^91 + 4 * q^93 + 3 * q^95 - 10 * q^97 + 3 * 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	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.43163	0.640243	0.320122	−	0.947376i	$-0.396276\pi$
$5$	1.43163	0.640243	0.320122	+	0.947376i	$0.396276\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	3.95044	1.09566	0.547828	−	0.836591i	$-0.315455\pi$
$13$	3.95044	1.09566	0.547828	+	0.836591i	$0.315455\pi$
$14$	0	0
$15$	1.43163	0.369645
$16$	0	0
$17$	4.51882	1.09597	0.547987	−	0.836487i	$-0.315394\pi$
$17$	4.51882	1.09597	0.547987	+	0.836487i	$0.315394\pi$
$18$	0	0
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−2.95044	−0.590089
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−10.4197	−1.93489	−0.967445	−	0.253080i	$-0.918556\pi$
$29$	−10.4197	−1.93489	−0.967445	+	0.253080i	$0.918556\pi$
$30$	0	0
$31$	0.344438	0.0618628	0.0309314	−	0.999522i	$-0.490153\pi$
$31$	0.344438	0.0618628	0.0309314	+	0.999522i	$0.490153\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	−1.43163	−0.241989
$36$	0	0
$37$	−7.55645	−1.24227	−0.621136	−	0.783703i	$-0.713329\pi$
$37$	−7.55645	−1.24227	−0.621136	+	0.783703i	$0.713329\pi$
$38$	0	0
$39$	3.95044	0.632577
$40$	0	0
$41$	5.17438	0.808102	0.404051	−	0.914736i	$-0.367602\pi$
$41$	5.17438	0.808102	0.404051	+	0.914736i	$0.367602\pi$
$42$	0	0
$43$	7.08719	1.08079	0.540393	−	0.841413i	$-0.318276\pi$
$43$	7.08719	1.08079	0.540393	+	0.841413i	$0.318276\pi$
$44$	0	0
$45$	1.43163	0.213414
$46$	0	0
$47$	7.51882	1.09673	0.548366	−	0.836238i	$-0.315250\pi$
$47$	7.51882	1.09673	0.548366	+	0.836238i	$0.315250\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.51882	0.632761
$52$	0	0
$53$	−4.95044	−0.679996	−0.339998	−	0.940426i	$-0.610426\pi$
$53$	−4.95044	−0.679996	−0.339998	+	0.940426i	$0.610426\pi$
$54$	0	0
$55$	1.43163	0.193041
$56$	0	0
$57$	3.00000	0.397360
$58$	0	0
$59$	4.77606	0.621791	0.310895	−	0.950444i	$-0.399371\pi$
$59$	4.77606	0.621791	0.310895	+	0.950444i	$0.399371\pi$
$60$	0	0
$61$	4.60601	0.589739	0.294869	−	0.955538i	$-0.404724\pi$
$61$	4.60601	0.589739	0.294869	+	0.955538i	$0.404724\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	5.65556	0.701486
$66$	0	0
$67$	10.4693	1.27902	0.639512	−	0.768781i	$-0.279136\pi$
$67$	10.4693	1.27902	0.639512	+	0.768781i	$0.279136\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	12.1248	1.43895	0.719476	−	0.694517i	$-0.244382\pi$
$71$	12.1248	1.43895	0.719476	+	0.694517i	$0.244382\pi$
$72$	0	0
$73$	0.518817	0.0607229	0.0303615	−	0.999539i	$-0.490334\pi$
$73$	0.518817	0.0607229	0.0303615	+	0.999539i	$0.490334\pi$
$74$	0	0
$75$	−2.95044	−0.340688
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	1.38207	0.155495	0.0777476	−	0.996973i	$-0.475227\pi$
$79$	1.38207	0.155495	0.0777476	+	0.996973i	$0.475227\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.41970	−0.704654	−0.352327	−	0.935877i	$-0.614610\pi$
$83$	−6.41970	−0.704654	−0.352327	+	0.935877i	$0.614610\pi$
$84$	0	0
$85$	6.46926	0.701690
$86$	0	0
$87$	−10.4197	−1.11711
$88$	0	0
$89$	−2.81370	−0.298251	−0.149126	−	0.988818i	$-0.547646\pi$
$89$	−2.81370	−0.298251	−0.149126	+	0.988818i	$0.547646\pi$
$90$	0	0
$91$	−3.95044	−0.414119
$92$	0	0
$93$	0.344438	0.0357165
$94$	0	0
$95$	4.29488	0.440646
$96$	0	0
$97$	−6.51882	−0.661886	−0.330943	−	0.943651i	$-0.607367\pi$
$97$	−6.51882	−0.661886	−0.330943	+	0.943651i	$0.607367\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	12.8513	1.27876	0.639378	−	0.768893i	$-0.279192\pi$
$101$	12.8513	1.27876	0.639378	+	0.768893i	$0.279192\pi$
$102$	0	0
$103$	−11.4197	−1.12522	−0.562608	−	0.826723i	$-0.690202\pi$
$103$	−11.4197	−1.12522	−0.562608	+	0.826723i	$0.690202\pi$
$104$	0	0
$105$	−1.43163	−0.139713
$106$	0	0
$107$	−1.95044	−0.188557	−0.0942783	−	0.995546i	$-0.530054\pi$
$107$	−1.95044	−0.188557	−0.0942783	+	0.995546i	$0.530054\pi$
$108$	0	0
$109$	−7.15814	−0.685625	−0.342813	−	0.939404i	$-0.611380\pi$
$109$	−7.15814	−0.685625	−0.342813	+	0.939404i	$0.611380\pi$
$110$	0	0
$111$	−7.55645	−0.717227
$112$	0	0
$113$	−0.469261	−0.0441443	−0.0220722	−	0.999756i	$-0.507026\pi$
$113$	−0.469261	−0.0441443	−0.0220722	+	0.999756i	$0.507026\pi$
$114$	0	0
$115$	1.43163	0.133500
$116$	0	0
$117$	3.95044	0.365219
$118$	0	0
$119$	−4.51882	−0.414239
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	5.17438	0.466558
$124$	0	0
$125$	−11.3821	−1.01804
$126$	0	0
$127$	2.70512	0.240040	0.120020	−	0.992771i	$-0.461704\pi$
$127$	2.70512	0.240040	0.120020	+	0.992771i	$0.461704\pi$
$128$	0	0
$129$	7.08719	0.623992
$130$	0	0
$131$	10.9009	0.952415	0.476207	−	0.879333i	$-0.342011\pi$
$131$	10.9009	0.952415	0.476207	+	0.879333i	$0.342011\pi$
$132$	0	0
$133$	−3.00000	−0.260133
$134$	0	0
$135$	1.43163	0.123215
$136$	0	0
$137$	8.31112	0.710067	0.355034	−	0.934854i	$-0.384469\pi$
$137$	8.31112	0.710067	0.355034	+	0.934854i	$0.384469\pi$
$138$	0	0
$139$	16.5445	1.40329	0.701644	−	0.712527i	$-0.252450\pi$
$139$	16.5445	1.40329	0.701644	+	0.712527i	$0.252450\pi$
$140$	0	0
$141$	7.51882	0.633199
$142$	0	0
$143$	3.95044	0.330353
$144$	0	0
$145$	−14.9171	−1.23880
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	13.5941	1.11367	0.556835	−	0.830623i	$-0.312015\pi$
$149$	13.5941	1.11367	0.556835	+	0.830623i	$0.312015\pi$
$150$	0	0
$151$	−10.5565	−0.859072	−0.429536	−	0.903050i	$-0.641323\pi$
$151$	−10.5565	−0.859072	−0.429536	+	0.903050i	$0.641323\pi$
$152$	0	0
$153$	4.51882	0.365325
$154$	0	0
$155$	0.493106	0.0396072
$156$	0	0
$157$	3.24533	0.259005	0.129503	−	0.991579i	$-0.458662\pi$
$157$	3.24533	0.259005	0.129503	+	0.991579i	$0.458662\pi$
$158$	0	0
$159$	−4.95044	−0.392596
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−17.1248	−1.34132	−0.670660	−	0.741765i	$-0.733989\pi$
$163$	−17.1248	−1.34132	−0.670660	+	0.741765i	$0.733989\pi$
$164$	0	0
$165$	1.43163	0.111452
$166$	0	0
$167$	−7.86325	−0.608477	−0.304238	−	0.952596i	$-0.598402\pi$
$167$	−7.86325	−0.608477	−0.304238	+	0.952596i	$0.598402\pi$
$168$	0	0
$169$	2.60601	0.200462
$170$	0	0
$171$	3.00000	0.229416
$172$	0	0
$173$	3.20769	0.243876	0.121938	−	0.992538i	$-0.461089\pi$
$173$	3.20769	0.243876	0.121938	+	0.992538i	$0.461089\pi$
$174$	0	0
$175$	2.95044	0.223033
$176$	0	0
$177$	4.77606	0.358991
$178$	0	0
$179$	−16.6436	−1.24400	−0.622002	−	0.783016i	$-0.713680\pi$
$179$	−16.6436	−1.24400	−0.622002	+	0.783016i	$0.713680\pi$
$180$	0	0
$181$	−4.79231	−0.356209	−0.178105	−	0.984012i	$-0.556997\pi$
$181$	−4.79231	−0.356209	−0.178105	+	0.984012i	$0.556997\pi$
$182$	0	0
$183$	4.60601	0.340486
$184$	0	0
$185$	−10.8180	−0.795357
$186$	0	0
$187$	4.51882	0.330449
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	5.24533	0.379538	0.189769	−	0.981829i	$-0.439226\pi$
$191$	5.24533	0.379538	0.189769	+	0.981829i	$0.439226\pi$
$192$	0	0
$193$	−18.2830	−1.31604	−0.658018	−	0.753002i	$-0.728605\pi$
$193$	−18.2830	−1.31604	−0.658018	+	0.753002i	$0.728605\pi$
$194$	0	0
$195$	5.65556	0.405003
$196$	0	0
$197$	5.77606	0.411528	0.205764	−	0.978602i	$-0.434032\pi$
$197$	5.77606	0.411528	0.205764	+	0.978602i	$0.434032\pi$
$198$	0	0
$199$	9.29920	0.659203	0.329601	−	0.944120i	$-0.393086\pi$
$199$	9.29920	0.659203	0.329601	+	0.944120i	$0.393086\pi$
$200$	0	0
$201$	10.4693	0.738445
$202$	0	0
$203$	10.4197	0.731320
$204$	0	0
$205$	7.40778	0.517382
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	3.00000	0.207514
$210$	0	0
$211$	−4.21201	−0.289967	−0.144983	−	0.989434i	$-0.546313\pi$
$211$	−4.21201	−0.289967	−0.144983	+	0.989434i	$0.546313\pi$
$212$	0	0
$213$	12.1248	0.830779
$214$	0	0
$215$	10.1462	0.691966
$216$	0	0
$217$	−0.344438	−0.0233819
$218$	0	0
$219$	0.518817	0.0350584
$220$	0	0
$221$	17.8513	1.20081
$222$	0	0
$223$	−22.0967	−1.47970	−0.739851	−	0.672771i	$-0.765104\pi$
$223$	−22.0967	−1.47970	−0.739851	+	0.672771i	$0.765104\pi$
$224$	0	0
$225$	−2.95044	−0.196696
$226$	0	0
$227$	2.49311	0.165473	0.0827366	−	0.996571i	$-0.473634\pi$
$227$	2.49311	0.165473	0.0827366	+	0.996571i	$0.473634\pi$
$228$	0	0
$229$	−23.0257	−1.52158	−0.760791	−	0.648997i	$-0.775189\pi$
$229$	−23.0257	−1.52158	−0.760791	+	0.648997i	$0.775189\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	−20.5445	−1.34592	−0.672958	−	0.739680i	$-0.734977\pi$
$233$	−20.5445	−1.34592	−0.672958	+	0.739680i	$0.734977\pi$
$234$	0	0
$235$	10.7641	0.702175
$236$	0	0
$237$	1.38207	0.0897752
$238$	0	0
$239$	13.6813	0.884968	0.442484	−	0.896776i	$-0.354097\pi$
$239$	13.6813	0.884968	0.442484	+	0.896776i	$0.354097\pi$
$240$	0	0
$241$	−18.6274	−1.19990	−0.599948	−	0.800039i	$-0.704812\pi$
$241$	−18.6274	−1.19990	−0.599948	+	0.800039i	$0.704812\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.43163	0.0914633
$246$	0	0
$247$	11.8513	0.754082
$248$	0	0
$249$	−6.41970	−0.406832
$250$	0	0
$251$	11.6274	0.733915	0.366957	−	0.930238i	$-0.380399\pi$
$251$	11.6274	0.733915	0.366957	+	0.930238i	$0.380399\pi$
$252$	0	0
$253$	1.00000	0.0628695
$254$	0	0
$255$	6.46926	0.405121
$256$	0	0
$257$	9.24533	0.576708	0.288354	−	0.957524i	$-0.406892\pi$
$257$	9.24533	0.576708	0.288354	+	0.957524i	$0.406892\pi$
$258$	0	0
$259$	7.55645	0.469535
$260$	0	0
$261$	−10.4197	−0.644964
$262$	0	0
$263$	−0.825621	−0.0509100	−0.0254550	−	0.999676i	$-0.508103\pi$
$263$	−0.825621	−0.0509100	−0.0254550	+	0.999676i	$0.508103\pi$
$264$	0	0
$265$	−7.08719	−0.435363
$266$	0	0
$267$	−2.81370	−0.172196
$268$	0	0
$269$	−15.2616	−0.930514	−0.465257	−	0.885176i	$-0.654038\pi$
$269$	−15.2616	−0.930514	−0.465257	+	0.885176i	$0.654038\pi$
$270$	0	0
$271$	−23.9718	−1.45619	−0.728093	−	0.685479i	$-0.759593\pi$
$271$	−23.9718	−1.45619	−0.728093	+	0.685479i	$0.759593\pi$
$272$	0	0
$273$	−3.95044	−0.239092
$274$	0	0
$275$	−2.95044	−0.177918
$276$	0	0
$277$	−5.29488	−0.318139	−0.159069	−	0.987267i	$-0.550849\pi$
$277$	−5.29488	−0.318139	−0.159069	+	0.987267i	$0.550849\pi$
$278$	0	0
$279$	0.344438	0.0206209
$280$	0	0
$281$	16.1744	0.964883	0.482441	−	0.875928i	$-0.339750\pi$
$281$	16.1744	0.964883	0.482441	+	0.875928i	$0.339750\pi$
$282$	0	0
$283$	4.29488	0.255304	0.127652	−	0.991819i	$-0.459256\pi$
$283$	4.29488	0.255304	0.127652	+	0.991819i	$0.459256\pi$
$284$	0	0
$285$	4.29488	0.254407
$286$	0	0
$287$	−5.17438	−0.305434
$288$	0	0
$289$	3.41970	0.201159
$290$	0	0
$291$	−6.51882	−0.382140
$292$	0	0
$293$	−5.72651	−0.334546	−0.167273	−	0.985911i	$-0.553496\pi$
$293$	−5.72651	−0.334546	−0.167273	+	0.985911i	$0.553496\pi$
$294$	0	0
$295$	6.83754	0.398097
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	3.95044	0.228460
$300$	0	0
$301$	−7.08719	−0.408499
$302$	0	0
$303$	12.8513	0.738290
$304$	0	0
$305$	6.59408	0.377576
$306$	0	0
$307$	−15.1838	−0.866588	−0.433294	−	0.901253i	$-0.642649\pi$
$307$	−15.1838	−0.866588	−0.433294	+	0.901253i	$0.642649\pi$
$308$	0	0
$309$	−11.4197	−0.649644
$310$	0	0
$311$	7.66749	0.434783	0.217392	−	0.976084i	$-0.430245\pi$
$311$	7.66749	0.434783	0.217392	+	0.976084i	$0.430245\pi$
$312$	0	0
$313$	26.8061	1.51517	0.757585	−	0.652736i	$-0.226379\pi$
$313$	26.8061	1.51517	0.757585	+	0.652736i	$0.226379\pi$
$314$	0	0
$315$	−1.43163	−0.0806630
$316$	0	0
$317$	2.05388	0.115357	0.0576786	−	0.998335i	$-0.481630\pi$
$317$	2.05388	0.115357	0.0576786	+	0.998335i	$0.481630\pi$
$318$	0	0
$319$	−10.4197	−0.583391
$320$	0	0
$321$	−1.95044	−0.108863
$322$	0	0
$323$	13.5565	0.754301
$324$	0	0
$325$	−11.6556	−0.646534
$326$	0	0
$327$	−7.15814	−0.395846
$328$	0	0
$329$	−7.51882	−0.414526
$330$	0	0
$331$	−28.8727	−1.58699	−0.793494	−	0.608578i	$-0.791740\pi$
$331$	−28.8727	−1.58699	−0.793494	+	0.608578i	$0.791740\pi$
$332$	0	0
$333$	−7.55645	−0.414091
$334$	0	0
$335$	14.9881	0.818886
$336$	0	0
$337$	−13.3368	−0.726504	−0.363252	−	0.931691i	$-0.618334\pi$
$337$	−13.3368	−0.726504	−0.363252	+	0.931691i	$0.618334\pi$
$338$	0	0
$339$	−0.469261	−0.0254867
$340$	0	0
$341$	0.344438	0.0186523
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	1.43163	0.0770762
$346$	0	0
$347$	21.2830	1.14253	0.571265	−	0.820766i	$-0.306453\pi$
$347$	21.2830	1.14253	0.571265	+	0.820766i	$0.306453\pi$
$348$	0	0
$349$	−31.7804	−1.70117	−0.850583	−	0.525842i	$-0.823750\pi$
$349$	−31.7804	−1.70117	−0.850583	+	0.525842i	$0.823750\pi$
$350$	0	0
$351$	3.95044	0.210859
$352$	0	0
$353$	22.3206	1.18801	0.594003	−	0.804463i	$-0.297547\pi$
$353$	22.3206	1.18801	0.594003	+	0.804463i	$0.297547\pi$
$354$	0	0
$355$	17.3582	0.921279
$356$	0	0
$357$	−4.51882	−0.239161
$358$	0	0
$359$	−31.0590	−1.63923	−0.819616	−	0.572913i	$-0.805813\pi$
$359$	−31.0590	−1.63923	−0.819616	+	0.572913i	$0.805813\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	0	0
$363$	−10.0000	−0.524864
$364$	0	0
$365$	0.742752	0.0388774
$366$	0	0
$367$	7.74275	0.404168	0.202084	−	0.979368i	$-0.435229\pi$
$367$	7.74275	0.404168	0.202084	+	0.979368i	$0.435229\pi$
$368$	0	0
$369$	5.17438	0.269367
$370$	0	0
$371$	4.95044	0.257014
$372$	0	0
$373$	−15.5565	−0.805482	−0.402741	−	0.915314i	$-0.631943\pi$
$373$	−15.5565	−0.805482	−0.402741	+	0.915314i	$0.631943\pi$
$374$	0	0
$375$	−11.3821	−0.587768
$376$	0	0
$377$	−41.1625	−2.11997
$378$	0	0
$379$	29.1129	1.49543	0.747715	−	0.664020i	$-0.231151\pi$
$379$	29.1129	1.49543	0.747715	+	0.664020i	$0.231151\pi$
$380$	0	0
$381$	2.70512	0.138587
$382$	0	0
$383$	−31.2591	−1.59727	−0.798633	−	0.601818i	$-0.794443\pi$
$383$	−31.2591	−1.59727	−0.798633	+	0.601818i	$0.794443\pi$
$384$	0	0
$385$	−1.43163	−0.0729625
$386$	0	0
$387$	7.08719	0.360262
$388$	0	0
$389$	−8.10343	−0.410860	−0.205430	−	0.978672i	$-0.565859\pi$
$389$	−8.10343	−0.410860	−0.205430	+	0.978672i	$0.565859\pi$
$390$	0	0
$391$	4.51882	0.228526
$392$	0	0
$393$	10.9009	0.549877
$394$	0	0
$395$	1.97861	0.0995547
$396$	0	0
$397$	−1.06148	−0.0532741	−0.0266371	−	0.999645i	$-0.508480\pi$
$397$	−1.06148	−0.0532741	−0.0266371	+	0.999645i	$0.508480\pi$
$398$	0	0
$399$	−3.00000	−0.150188
$400$	0	0
$401$	−25.3915	−1.26799	−0.633996	−	0.773336i	$-0.718587\pi$
$401$	−25.3915	−1.26799	−0.633996	+	0.773336i	$0.718587\pi$
$402$	0	0
$403$	1.36068	0.0677804
$404$	0	0
$405$	1.43163	0.0711381
$406$	0	0
$407$	−7.55645	−0.374559
$408$	0	0
$409$	−31.6556	−1.56527	−0.782633	−	0.622483i	$-0.786124\pi$
$409$	−31.6556	−1.56527	−0.782633	+	0.622483i	$0.786124\pi$
$410$	0	0
$411$	8.31112	0.409958
$412$	0	0
$413$	−4.77606	−0.235015
$414$	0	0
$415$	−9.19062	−0.451150
$416$	0	0
$417$	16.5445	0.810189
$418$	0	0
$419$	−2.84186	−0.138834	−0.0694171	−	0.997588i	$-0.522114\pi$
$419$	−2.84186	−0.138834	−0.0694171	+	0.997588i	$0.522114\pi$
$420$	0	0
$421$	24.5164	1.19485	0.597427	−	0.801923i	$-0.296190\pi$
$421$	24.5164	1.19485	0.597427	+	0.801923i	$0.296190\pi$
$422$	0	0
$423$	7.51882	0.365577
$424$	0	0
$425$	−13.3325	−0.646722
$426$	0	0
$427$	−4.60601	−0.222900
$428$	0	0
$429$	3.95044	0.190729
$430$	0	0
$431$	−3.74275	−0.180282	−0.0901410	−	0.995929i	$-0.528732\pi$
$431$	−3.74275	−0.180282	−0.0901410	+	0.995929i	$0.528732\pi$
$432$	0	0
$433$	32.7070	1.57180	0.785899	−	0.618355i	$-0.212201\pi$
$433$	32.7070	1.57180	0.785899	+	0.618355i	$0.212201\pi$
$434$	0	0
$435$	−14.9171	−0.715222
$436$	0	0
$437$	3.00000	0.143509
$438$	0	0
$439$	−38.1462	−1.82062	−0.910310	−	0.413928i	$-0.864157\pi$
$439$	−38.1462	−1.82062	−0.910310	+	0.413928i	$0.864157\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−40.1505	−1.90761	−0.953805	−	0.300427i	$-0.902871\pi$
$443$	−40.1505	−1.90761	−0.953805	+	0.300427i	$0.902871\pi$
$444$	0	0
$445$	−4.02817	−0.190953
$446$	0	0
$447$	13.5941	0.642978
$448$	0	0
$449$	23.2334	1.09645	0.548226	−	0.836330i	$-0.315303\pi$
$449$	23.2334	1.09645	0.548226	+	0.836330i	$0.315303\pi$
$450$	0	0
$451$	5.17438	0.243652
$452$	0	0
$453$	−10.5565	−0.495985
$454$	0	0
$455$	−5.65556	−0.265137
$456$	0	0
$457$	21.9975	1.02900	0.514501	−	0.857490i	$-0.327977\pi$
$457$	21.9975	1.02900	0.514501	+	0.857490i	$0.327977\pi$
$458$	0	0
$459$	4.51882	0.210920
$460$	0	0
$461$	−25.0590	−1.16712	−0.583558	−	0.812072i	$-0.698340\pi$
$461$	−25.0590	−1.16712	−0.583558	+	0.812072i	$0.698340\pi$
$462$	0	0
$463$	29.7641	1.38326	0.691628	−	0.722253i	$-0.256894\pi$
$463$	29.7641	1.38326	0.691628	+	0.722253i	$0.256894\pi$
$464$	0	0
$465$	0.493106	0.0228672
$466$	0	0
$467$	−3.30680	−0.153021	−0.0765103	−	0.997069i	$-0.524378\pi$
$467$	−3.30680	−0.153021	−0.0765103	+	0.997069i	$0.524378\pi$
$468$	0	0
$469$	−10.4693	−0.483426
$470$	0	0
$471$	3.24533	0.149537
$472$	0	0
$473$	7.08719	0.325869
$474$	0	0
$475$	−8.85133	−0.406127
$476$	0	0
$477$	−4.95044	−0.226665
$478$	0	0
$479$	31.2830	1.42935	0.714677	−	0.699454i	$-0.246574\pi$
$479$	31.2830	1.42935	0.714677	+	0.699454i	$0.246574\pi$
$480$	0	0
$481$	−29.8513	−1.36110
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	−9.33251	−0.423768
$486$	0	0
$487$	21.1086	0.956521	0.478261	−	0.878218i	$-0.341267\pi$
$487$	21.1086	0.956521	0.478261	+	0.878218i	$0.341267\pi$
$488$	0	0
$489$	−17.1248	−0.774411
$490$	0	0
$491$	2.32305	0.104838	0.0524188	−	0.998625i	$-0.483307\pi$
$491$	2.32305	0.104838	0.0524188	+	0.998625i	$0.483307\pi$
$492$	0	0
$493$	−47.0847	−2.12059
$494$	0	0
$495$	1.43163	0.0643469
$496$	0	0
$497$	−12.1248	−0.543873
$498$	0	0
$499$	−9.43163	−0.422218	−0.211109	−	0.977463i	$-0.567707\pi$
$499$	−9.43163	−0.422218	−0.211109	+	0.977463i	$0.567707\pi$
$500$	0	0
$501$	−7.86325	−0.351304
$502$	0	0
$503$	7.95476	0.354685	0.177343	−	0.984149i	$-0.443250\pi$
$503$	7.95476	0.354685	0.177343	+	0.984149i	$0.443250\pi$
$504$	0	0
$505$	18.3983	0.818714
$506$	0	0
$507$	2.60601	0.115737
$508$	0	0
$509$	−4.48118	−0.198625	−0.0993125	−	0.995056i	$-0.531664\pi$
$509$	−4.48118	−0.198625	−0.0993125	+	0.995056i	$0.531664\pi$
$510$	0	0
$511$	−0.518817	−0.0229511
$512$	0	0
$513$	3.00000	0.132453
$514$	0	0
$515$	−16.3488	−0.720412
$516$	0	0
$517$	7.51882	0.330677
$518$	0	0
$519$	3.20769	0.140802
$520$	0	0
$521$	32.4240	1.42052	0.710261	−	0.703938i	$-0.248577\pi$
$521$	32.4240	1.42052	0.710261	+	0.703938i	$0.248577\pi$
$522$	0	0
$523$	−25.4197	−1.11153	−0.555763	−	0.831341i	$-0.687574\pi$
$523$	−25.4197	−1.11153	−0.555763	+	0.831341i	$0.687574\pi$
$524$	0	0
$525$	2.95044	0.128768
$526$	0	0
$527$	1.55645	0.0678000
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	4.77606	0.207264
$532$	0	0
$533$	20.4411	0.885402
$534$	0	0
$535$	−2.79231	−0.120722
$536$	0	0
$537$	−16.6436	−0.718226
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−14.0095	−0.602314	−0.301157	−	0.953575i	$-0.597373\pi$
$541$	−14.0095	−0.602314	−0.301157	+	0.953575i	$0.597373\pi$
$542$	0	0
$543$	−4.79231	−0.205658
$544$	0	0
$545$	−10.2478	−0.438967
$546$	0	0
$547$	−3.15814	−0.135032	−0.0675161	−	0.997718i	$-0.521507\pi$
$547$	−3.15814	−0.135032	−0.0675161	+	0.997718i	$0.521507\pi$
$548$	0	0
$549$	4.60601	0.196580
$550$	0	0
$551$	−31.2591	−1.33168
$552$	0	0
$553$	−1.38207	−0.0587716
$554$	0	0
$555$	−10.8180	−0.459199
$556$	0	0
$557$	−18.2496	−0.773262	−0.386631	−	0.922234i	$-0.626361\pi$
$557$	−18.2496	−0.773262	−0.386631	+	0.922234i	$0.626361\pi$
$558$	0	0
$559$	27.9975	1.18417
$560$	0	0
$561$	4.51882	0.190785
$562$	0	0
$563$	17.8513	0.752344	0.376172	−	0.926550i	$-0.377240\pi$
$563$	17.8513	0.752344	0.376172	+	0.926550i	$0.377240\pi$
$564$	0	0
$565$	−0.671806	−0.0282631
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	9.69320	0.406360	0.203180	−	0.979141i	$-0.434872\pi$
$569$	9.69320	0.406360	0.203180	+	0.979141i	$0.434872\pi$
$570$	0	0
$571$	39.3821	1.64809	0.824044	−	0.566526i	$-0.191713\pi$
$571$	39.3821	1.64809	0.824044	+	0.566526i	$0.191713\pi$
$572$	0	0
$573$	5.24533	0.219127
$574$	0	0
$575$	−2.95044	−0.123042
$576$	0	0
$577$	−17.4530	−0.726579	−0.363289	−	0.931676i	$-0.618346\pi$
$577$	−17.4530	−0.726579	−0.363289	+	0.931676i	$0.618346\pi$
$578$	0	0
$579$	−18.2830	−0.759814
$580$	0	0
$581$	6.41970	0.266334
$582$	0	0
$583$	−4.95044	−0.205026
$584$	0	0
$585$	5.65556	0.233829
$586$	0	0
$587$	33.4693	1.38142	0.690712	−	0.723130i	$-0.257297\pi$
$587$	33.4693	1.38142	0.690712	+	0.723130i	$0.257297\pi$
$588$	0	0
$589$	1.03331	0.0425769
$590$	0	0
$591$	5.77606	0.237596
$592$	0	0
$593$	−2.09911	−0.0862002	−0.0431001	−	0.999071i	$-0.513723\pi$
$593$	−2.09911	−0.0862002	−0.0431001	+	0.999071i	$0.513723\pi$
$594$	0	0
$595$	−6.46926	−0.265214
$596$	0	0
$597$	9.29920	0.380591
$598$	0	0
$599$	−45.2805	−1.85011	−0.925056	−	0.379832i	$-0.875982\pi$
$599$	−45.2805	−1.85011	−0.925056	+	0.379832i	$0.875982\pi$
$600$	0	0
$601$	−28.9924	−1.18262	−0.591312	−	0.806443i	$-0.701390\pi$
$601$	−28.9924	−1.18262	−0.591312	+	0.806443i	$0.701390\pi$
$602$	0	0
$603$	10.4693	0.426341
$604$	0	0
$605$	−14.3163	−0.582039
$606$	0	0
$607$	2.29920	0.0933217	0.0466609	−	0.998911i	$-0.485142\pi$
$607$	2.29920	0.0933217	0.0466609	+	0.998911i	$0.485142\pi$
$608$	0	0
$609$	10.4197	0.422228
$610$	0	0
$611$	29.7027	1.20164
$612$	0	0
$613$	19.4573	0.785874	0.392937	−	0.919565i	$-0.371459\pi$
$613$	19.4573	0.785874	0.392937	+	0.919565i	$0.371459\pi$
$614$	0	0
$615$	7.40778	0.298711
$616$	0	0
$617$	8.73843	0.351796	0.175898	−	0.984408i	$-0.443717\pi$
$617$	8.73843	0.351796	0.175898	+	0.984408i	$0.443717\pi$
$618$	0	0
$619$	−36.0257	−1.44800	−0.723998	−	0.689802i	$-0.757697\pi$
$619$	−36.0257	−1.44800	−0.723998	+	0.689802i	$0.757697\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	2.81370	0.112728
$624$	0	0
$625$	−1.54266	−0.0617065
$626$	0	0
$627$	3.00000	0.119808
$628$	0	0
$629$	−34.1462	−1.36150
$630$	0	0
$631$	20.4435	0.813845	0.406922	−	0.913463i	$-0.366602\pi$
$631$	20.4435	0.813845	0.406922	+	0.913463i	$0.366602\pi$
$632$	0	0
$633$	−4.21201	−0.167412
$634$	0	0
$635$	3.87272	0.153684
$636$	0	0
$637$	3.95044	0.156522
$638$	0	0
$639$	12.1248	0.479651
$640$	0	0
$641$	27.8846	1.10138	0.550689	−	0.834711i	$-0.314365\pi$
$641$	27.8846	1.10138	0.550689	+	0.834711i	$0.314365\pi$
$642$	0	0
$643$	41.6864	1.64395	0.821976	−	0.569522i	$-0.192872\pi$
$643$	41.6864	1.64395	0.821976	+	0.569522i	$0.192872\pi$
$644$	0	0
$645$	10.1462	0.399507
$646$	0	0
$647$	17.2334	0.677515	0.338757	−	0.940874i	$-0.389993\pi$
$647$	17.2334	0.677515	0.338757	+	0.940874i	$0.389993\pi$
$648$	0	0
$649$	4.77606	0.187477
$650$	0	0
$651$	−0.344438	−0.0134996
$652$	0	0
$653$	−42.2710	−1.65419	−0.827097	−	0.562060i	$-0.810009\pi$
$653$	−42.2710	−1.65419	−0.827097	+	0.562060i	$0.810009\pi$
$654$	0	0
$655$	15.6060	0.609777
$656$	0	0
$657$	0.518817	0.0202410
$658$	0	0
$659$	12.6179	0.491525	0.245762	−	0.969330i	$-0.420962\pi$
$659$	12.6179	0.491525	0.245762	+	0.969330i	$0.420962\pi$
$660$	0	0
$661$	−40.4292	−1.57251	−0.786256	−	0.617901i	$-0.787983\pi$
$661$	−40.4292	−1.57251	−0.786256	+	0.617901i	$0.787983\pi$
$662$	0	0
$663$	17.8513	0.693288
$664$	0	0
$665$	−4.29488	−0.166548
$666$	0	0
$667$	−10.4197	−0.403453
$668$	0	0
$669$	−22.0967	−0.854306
$670$	0	0
$671$	4.60601	0.177813
$672$	0	0
$673$	−29.2258	−1.12657	−0.563286	−	0.826262i	$-0.690463\pi$
$673$	−29.2258	−1.12657	−0.563286	+	0.826262i	$0.690463\pi$
$674$	0	0
$675$	−2.95044	−0.113563
$676$	0	0
$677$	−2.83754	−0.109056	−0.0545278	−	0.998512i	$-0.517365\pi$
$677$	−2.83754	−0.109056	−0.0545278	+	0.998512i	$0.517365\pi$
$678$	0	0
$679$	6.51882	0.250169
$680$	0	0
$681$	2.49311	0.0955360
$682$	0	0
$683$	7.63172	0.292020	0.146010	−	0.989283i	$-0.453357\pi$
$683$	7.63172	0.292020	0.146010	+	0.989283i	$0.453357\pi$
$684$	0	0
$685$	11.8984	0.454616
$686$	0	0
$687$	−23.0257	−0.878486
$688$	0	0
$689$	−19.5565	−0.745041
$690$	0	0
$691$	−2.14867	−0.0817392	−0.0408696	−	0.999164i	$-0.513013\pi$
$691$	−2.14867	−0.0817392	−0.0408696	+	0.999164i	$0.513013\pi$
$692$	0	0
$693$	−1.00000	−0.0379869
$694$	0	0
$695$	23.6856	0.898446
$696$	0	0
$697$	23.3821	0.885659
$698$	0	0
$699$	−20.5445	−0.777065
$700$	0	0
$701$	−24.5112	−0.925776	−0.462888	−	0.886417i	$-0.653187\pi$
$701$	−24.5112	−0.925776	−0.462888	+	0.886417i	$0.653187\pi$
$702$	0	0
$703$	−22.6694	−0.854991
$704$	0	0
$705$	10.7641	0.405401
$706$	0	0
$707$	−12.8513	−0.483324
$708$	0	0
$709$	−44.5916	−1.67467	−0.837337	−	0.546687i	$-0.815889\pi$
$709$	−44.5916	−1.67467	−0.837337	+	0.546687i	$0.815889\pi$
$710$	0	0
$711$	1.38207	0.0518317
$712$	0	0
$713$	0.344438	0.0128993
$714$	0	0
$715$	5.65556	0.211506
$716$	0	0
$717$	13.6813	0.510937
$718$	0	0
$719$	−33.1086	−1.23474	−0.617371	−	0.786672i	$-0.711802\pi$
$719$	−33.1086	−1.23474	−0.617371	+	0.786672i	$0.711802\pi$
$720$	0	0
$721$	11.4197	0.425292
$722$	0	0
$723$	−18.6274	−0.692760
$724$	0	0
$725$	30.7428	1.14176
$726$	0	0
$727$	−12.8770	−0.477583	−0.238792	−	0.971071i	$-0.576751\pi$
$727$	−12.8770	−0.477583	−0.238792	+	0.971071i	$0.576751\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	32.0257	1.18451
$732$	0	0
$733$	4.17438	0.154184	0.0770921	−	0.997024i	$-0.475436\pi$
$733$	4.17438	0.154184	0.0770921	+	0.997024i	$0.475436\pi$
$734$	0	0
$735$	1.43163	0.0528064
$736$	0	0
$737$	10.4693	0.385640
$738$	0	0
$739$	37.5001	1.37946	0.689732	−	0.724065i	$-0.257728\pi$
$739$	37.5001	1.37946	0.689732	+	0.724065i	$0.257728\pi$
$740$	0	0
$741$	11.8513	0.435370
$742$	0	0
$743$	−41.1010	−1.50785	−0.753924	−	0.656961i	$-0.771841\pi$
$743$	−41.1010	−1.50785	−0.753924	+	0.656961i	$0.771841\pi$
$744$	0	0
$745$	19.4617	0.713020
$746$	0	0
$747$	−6.41970	−0.234885
$748$	0	0
$749$	1.95044	0.0712677
$750$	0	0
$751$	46.3701	1.69207	0.846035	−	0.533127i	$-0.178983\pi$
$751$	46.3701	1.69207	0.846035	+	0.533127i	$0.178983\pi$
$752$	0	0
$753$	11.6274	0.423726
$754$	0	0
$755$	−15.1129	−0.550015
$756$	0	0
$757$	−18.3872	−0.668295	−0.334147	−	0.942521i	$-0.608448\pi$
$757$	−18.3872	−0.668295	−0.334147	+	0.942521i	$0.608448\pi$
$758$	0	0
$759$	1.00000	0.0362977
$760$	0	0
$761$	−5.49497	−0.199193	−0.0995963	−	0.995028i	$-0.531755\pi$
$761$	−5.49497	−0.199193	−0.0995963	+	0.995028i	$0.531755\pi$
$762$	0	0
$763$	7.15814	0.259142
$764$	0	0
$765$	6.46926	0.233897
$766$	0	0
$767$	18.8676	0.681269
$768$	0	0
$769$	−33.7027	−1.21535	−0.607675	−	0.794186i	$-0.707897\pi$
$769$	−33.7027	−1.21535	−0.607675	+	0.794186i	$0.707897\pi$
$770$	0	0
$771$	9.24533	0.332962
$772$	0	0
$773$	30.4950	1.09683	0.548414	−	0.836207i	$-0.315232\pi$
$773$	30.4950	1.09683	0.548414	+	0.836207i	$0.315232\pi$
$774$	0	0
$775$	−1.01624	−0.0365045
$776$	0	0
$777$	7.55645	0.271086
$778$	0	0
$779$	15.5231	0.556174
$780$	0	0
$781$	12.1248	0.433860
$782$	0	0
$783$	−10.4197	−0.372370
$784$	0	0
$785$	4.64610	0.165826
$786$	0	0
$787$	21.9128	0.781107	0.390554	−	0.920580i	$-0.372284\pi$
$787$	21.9128	0.781107	0.390554	+	0.920580i	$0.372284\pi$
$788$	0	0
$789$	−0.825621	−0.0293929
$790$	0	0
$791$	0.469261	0.0166850
$792$	0	0
$793$	18.1958	0.646151
$794$	0	0
$795$	−7.08719	−0.251357
$796$	0	0
$797$	−27.2873	−0.966565	−0.483283	−	0.875464i	$-0.660556\pi$
$797$	−27.2873	−0.966565	−0.483283	+	0.875464i	$0.660556\pi$
$798$	0	0
$799$	33.9762	1.20199
$800$	0	0
$801$	−2.81370	−0.0994171
$802$	0	0
$803$	0.518817	0.0183086
$804$	0	0
$805$	−1.43163	−0.0504582
$806$	0	0
$807$	−15.2616	−0.537233
$808$	0	0
$809$	−16.5402	−0.581523	−0.290761	−	0.956796i	$-0.593909\pi$
$809$	−16.5402	−0.581523	−0.290761	+	0.956796i	$0.593909\pi$
$810$	0	0
$811$	−3.89657	−0.136827	−0.0684135	−	0.997657i	$-0.521794\pi$
$811$	−3.89657	−0.136827	−0.0684135	+	0.997657i	$0.521794\pi$
$812$	0	0
$813$	−23.9718	−0.840729
$814$	0	0
$815$	−24.5164	−0.858771
$816$	0	0
$817$	21.2616	0.743848
$818$	0	0
$819$	−3.95044	−0.138040
$820$	0	0
$821$	19.3539	0.675456	0.337728	−	0.941244i	$-0.390342\pi$
$821$	19.3539	0.675456	0.337728	+	0.941244i	$0.390342\pi$
$822$	0	0
$823$	6.16246	0.214810	0.107405	−	0.994215i	$-0.465746\pi$
$823$	6.16246	0.214810	0.107405	+	0.994215i	$0.465746\pi$
$824$	0	0
$825$	−2.95044	−0.102721
$826$	0	0
$827$	24.2377	0.842828	0.421414	−	0.906868i	$-0.361534\pi$
$827$	24.2377	0.842828	0.421414	+	0.906868i	$0.361534\pi$
$828$	0	0
$829$	−5.72219	−0.198740	−0.0993699	−	0.995051i	$-0.531683\pi$
$829$	−5.72219	−0.198740	−0.0993699	+	0.995051i	$0.531683\pi$
$830$	0	0
$831$	−5.29488	−0.183677
$832$	0	0
$833$	4.51882	0.156568
$834$	0	0
$835$	−11.2572	−0.389573
$836$	0	0
$837$	0.344438	0.0119055
$838$	0	0
$839$	−31.2052	−1.07732	−0.538662	−	0.842522i	$-0.681070\pi$
$839$	−31.2052	−1.07732	−0.538662	+	0.842522i	$0.681070\pi$
$840$	0	0
$841$	79.5702	2.74380
$842$	0	0
$843$	16.1744	0.557075
$844$	0	0
$845$	3.73083	0.128344
$846$	0	0
$847$	10.0000	0.343604
$848$	0	0
$849$	4.29488	0.147400
$850$	0	0
$851$	−7.55645	−0.259032
$852$	0	0
$853$	11.7590	0.402620	0.201310	−	0.979528i	$-0.435480\pi$
$853$	11.7590	0.402620	0.201310	+	0.979528i	$0.435480\pi$
$854$	0	0
$855$	4.29488	0.146882
$856$	0	0
$857$	14.1324	0.482754	0.241377	−	0.970431i	$-0.422401\pi$
$857$	14.1324	0.482754	0.241377	+	0.970431i	$0.422401\pi$
$858$	0	0
$859$	−31.3111	−1.06832	−0.534161	−	0.845383i	$-0.679372\pi$
$859$	−31.3111	−1.06832	−0.534161	+	0.845383i	$0.679372\pi$
$860$	0	0
$861$	−5.17438	−0.176342
$862$	0	0
$863$	19.8256	0.674872	0.337436	−	0.941348i	$-0.390440\pi$
$863$	19.8256	0.674872	0.337436	+	0.941348i	$0.390440\pi$
$864$	0	0
$865$	4.59222	0.156140
$866$	0	0
$867$	3.41970	0.116139
$868$	0	0
$869$	1.38207	0.0468835
$870$	0	0
$871$	41.3582	1.40137
$872$	0	0
$873$	−6.51882	−0.220629
$874$	0	0
$875$	11.3821	0.384784
$876$	0	0
$877$	6.04195	0.204022	0.102011	−	0.994783i	$-0.467472\pi$
$877$	6.04195	0.204022	0.102011	+	0.994783i	$0.467472\pi$
$878$	0	0
$879$	−5.72651	−0.193150
$880$	0	0
$881$	−12.5231	−0.421915	−0.210958	−	0.977495i	$-0.567658\pi$
$881$	−12.5231	−0.421915	−0.210958	+	0.977495i	$0.567658\pi$
$882$	0	0
$883$	−48.0771	−1.61792	−0.808962	−	0.587861i	$-0.799970\pi$
$883$	−48.0771	−1.61792	−0.808962	+	0.587861i	$0.799970\pi$
$884$	0	0
$885$	6.83754	0.229842
$886$	0	0
$887$	2.69074	0.0903462	0.0451731	−	0.998979i	$-0.485616\pi$
$887$	2.69074	0.0903462	0.0451731	+	0.998979i	$0.485616\pi$
$888$	0	0
$889$	−2.70512	−0.0907268
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	22.5565	0.754823
$894$	0	0
$895$	−23.8275	−0.796465
$896$	0	0
$897$	3.95044	0.131901
$898$	0	0
$899$	−3.58894	−0.119698
$900$	0	0
$901$	−22.3701	−0.745258
$902$	0	0
$903$	−7.08719	−0.235847
$904$	0	0
$905$	−6.86080	−0.228061
$906$	0	0
$907$	28.4931	0.946098	0.473049	−	0.881036i	$-0.343153\pi$
$907$	28.4931	0.946098	0.473049	+	0.881036i	$0.343153\pi$
$908$	0	0
$909$	12.8513	0.426252
$910$	0	0
$911$	−17.6274	−0.584022	−0.292011	−	0.956415i	$-0.594324\pi$
$911$	−17.6274	−0.584022	−0.292011	+	0.956415i	$0.594324\pi$
$912$	0	0
$913$	−6.41970	−0.212461
$914$	0	0
$915$	6.59408	0.217994
$916$	0	0
$917$	−10.9009	−0.359979
$918$	0	0
$919$	−8.10343	−0.267308	−0.133654	−	0.991028i	$-0.542671\pi$
$919$	−8.10343	−0.267308	−0.133654	+	0.991028i	$0.542671\pi$
$920$	0	0
$921$	−15.1838	−0.500325
$922$	0	0
$923$	47.8984	1.57660
$924$	0	0
$925$	22.2949	0.733051
$926$	0	0
$927$	−11.4197	−0.375072
$928$	0	0
$929$	14.0967	0.462496	0.231248	−	0.972895i	$-0.425719\pi$
$929$	14.0967	0.462496	0.231248	+	0.972895i	$0.425719\pi$
$930$	0	0
$931$	3.00000	0.0983210
$932$	0	0
$933$	7.66749	0.251022
$934$	0	0
$935$	6.46926	0.211567
$936$	0	0
$937$	−36.6036	−1.19579	−0.597893	−	0.801576i	$-0.703995\pi$
$937$	−36.6036	−1.19579	−0.597893	+	0.801576i	$0.703995\pi$
$938$	0	0
$939$	26.8061	0.874784
$940$	0	0
$941$	−14.4240	−0.470210	−0.235105	−	0.971970i	$-0.575543\pi$
$941$	−14.4240	−0.470210	−0.235105	+	0.971970i	$0.575543\pi$
$942$	0	0
$943$	5.17438	0.168501
$944$	0	0
$945$	−1.43163	−0.0465708
$946$	0	0
$947$	−17.7352	−0.576315	−0.288157	−	0.957583i	$-0.593043\pi$
$947$	−17.7352	−0.576315	−0.288157	+	0.957583i	$0.593043\pi$
$948$	0	0
$949$	2.04956	0.0665314
$950$	0	0
$951$	2.05388	0.0666015
$952$	0	0
$953$	44.1104	1.42888	0.714439	−	0.699698i	$-0.246682\pi$
$953$	44.1104	1.42888	0.714439	+	0.699698i	$0.246682\pi$
$954$	0	0
$955$	7.50935	0.242997
$956$	0	0
$957$	−10.4197	−0.336821
$958$	0	0
$959$	−8.31112	−0.268380
$960$	0	0
$961$	−30.8814	−0.996173
$962$	0	0
$963$	−1.95044	−0.0628522
$964$	0	0
$965$	−26.1744	−0.842583
$966$	0	0
$967$	16.6889	0.536678	0.268339	−	0.963325i	$-0.413525\pi$
$967$	16.6889	0.536678	0.268339	+	0.963325i	$0.413525\pi$
$968$	0	0
$969$	13.5565	0.435496
$970$	0	0
$971$	−44.1438	−1.41664	−0.708320	−	0.705891i	$-0.750547\pi$
$971$	−44.1438	−1.41664	−0.708320	+	0.705891i	$0.750547\pi$
$972$	0	0
$973$	−16.5445	−0.530393
$974$	0	0
$975$	−11.6556	−0.373277
$976$	0	0
$977$	25.7428	0.823584	0.411792	−	0.911278i	$-0.364903\pi$
$977$	25.7428	0.823584	0.411792	+	0.911278i	$0.364903\pi$
$978$	0	0
$979$	−2.81370	−0.0899262
$980$	0	0
$981$	−7.15814	−0.228542
$982$	0	0
$983$	48.8437	1.55787	0.778937	−	0.627103i	$-0.215759\pi$
$983$	48.8437	1.55787	0.778937	+	0.627103i	$0.215759\pi$
$984$	0	0
$985$	8.26917	0.263478
$986$	0	0
$987$	−7.51882	−0.239327
$988$	0	0
$989$	7.08719	0.225360
$990$	0	0
$991$	40.1343	1.27491	0.637454	−	0.770489i	$-0.279988\pi$
$991$	40.1343	1.27491	0.637454	+	0.770489i	$0.279988\pi$
$992$	0	0
$993$	−28.8727	−0.916248
$994$	0	0
$995$	13.3130	0.422050
$996$	0	0
$997$	11.8727	0.376013	0.188006	−	0.982168i	$-0.439797\pi$
$997$	11.8727	0.376013	0.188006	+	0.982168i	$0.439797\pi$
$998$	0	0
$999$	−7.55645	−0.239076

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1932.2.a.j.1.2	✓	3
3.2	odd	2		5796.2.a.o.1.2		3
4.3	odd	2		7728.2.a.br.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1932.2.a.j.1.2	✓	3	1.1	even	1	trivial
5796.2.a.o.1.2		3	3.2	odd	2
7728.2.a.br.1.2		3	4.3	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:	\( N \)	\(=\)	\( 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1932.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.43163 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.43163 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +3.95044 q^{13} +1.43163 q^{15} +4.51882 q^{17} +3.00000 q^{19} -1.00000 q^{21} +1.00000 q^{23} -2.95044 q^{25} +1.00000 q^{27} -10.4197 q^{29} +0.344438 q^{31} +1.00000 q^{33} -1.43163 q^{35} -7.55645 q^{37} +3.95044 q^{39} +5.17438 q^{41} +7.08719 q^{43} +1.43163 q^{45} +7.51882 q^{47} +1.00000 q^{49} +4.51882 q^{51} -4.95044 q^{53} +1.43163 q^{55} +3.00000 q^{57} +4.77606 q^{59} +4.60601 q^{61} -1.00000 q^{63} +5.65556 q^{65} +10.4693 q^{67} +1.00000 q^{69} +12.1248 q^{71} +0.518817 q^{73} -2.95044 q^{75} -1.00000 q^{77} +1.38207 q^{79} +1.00000 q^{81} -6.41970 q^{83} +6.46926 q^{85} -10.4197 q^{87} -2.81370 q^{89} -3.95044 q^{91} +0.344438 q^{93} +4.29488 q^{95} -6.51882 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + q^5 - 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9} + 3 q^{11} - q^{13} + q^{15} + 4 q^{17} + 9 q^{19} - 3 q^{21} + 3 q^{23} + 4 q^{25} + 3 q^{27} + 4 q^{29} + 4 q^{31} + 3 q^{33} - q^{35} + 6 q^{37} - q^{39} + 3 q^{41} + 15 q^{43} + q^{45} + 13 q^{47} + 3 q^{49} + 4 q^{51} - 2 q^{53} + q^{55} + 9 q^{57} + 14 q^{59} - 2 q^{61} - 3 q^{63} + 14 q^{65} + 9 q^{67} + 3 q^{69} + 11 q^{71} - 8 q^{73} + 4 q^{75} - 3 q^{77} - 12 q^{79} + 3 q^{81} + 16 q^{83} - 3 q^{85} + 4 q^{87} + 11 q^{89} + q^{91} + 4 q^{93} + 3 q^{95} - 10 q^{97} + 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + q^5 - 3 * q^7 + 3 * q^9 + 3 * q^11 - q^13 + q^15 + 4 * q^17 + 9 * q^19 - 3 * q^21 + 3 * q^23 + 4 * q^25 + 3 * q^27 + 4 * q^29 + 4 * q^31 + 3 * q^33 - q^35 + 6 * q^37 - q^39 + 3 * q^41 + 15 * q^43 + q^45 + 13 * q^47 + 3 * q^49 + 4 * q^51 - 2 * q^53 + q^55 + 9 * q^57 + 14 * q^59 - 2 * q^61 - 3 * q^63 + 14 * q^65 + 9 * q^67 + 3 * q^69 + 11 * q^71 - 8 * q^73 + 4 * q^75 - 3 * q^77 - 12 * q^79 + 3 * q^81 + 16 * q^83 - 3 * q^85 + 4 * q^87 + 11 * q^89 + q^91 + 4 * q^93 + 3 * q^95 - 10 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

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