LMFDB - Embedded newform 3042.2.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3042,2,Mod(1,3042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3042, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3042 = 2 \cdot 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3042.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:	$24.2904922949$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 78)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	3042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.73205 q^{5} -1.26795 q^{7} -1.00000 q^{8} -1.73205 q^{10} +1.26795 q^{11} +1.26795 q^{14} +1.00000 q^{16} -5.19615 q^{17} -4.73205 q^{19} +1.73205 q^{20} -1.26795 q^{22} +8.19615 q^{23} -2.00000 q^{25} -1.26795 q^{28} +3.00000 q^{29} -9.46410 q^{31} -1.00000 q^{32} +5.19615 q^{34} -2.19615 q^{35} -3.00000 q^{37} +4.73205 q^{38} -1.73205 q^{40} +6.46410 q^{41} -4.19615 q^{43} +1.26795 q^{44} -8.19615 q^{46} -4.73205 q^{47} -5.39230 q^{49} +2.00000 q^{50} -3.00000 q^{53} +2.19615 q^{55} +1.26795 q^{56} -3.00000 q^{58} -13.8564 q^{59} +15.1962 q^{61} +9.46410 q^{62} +1.00000 q^{64} -7.26795 q^{67} -5.19615 q^{68} +2.19615 q^{70} -2.19615 q^{71} -12.1244 q^{73} +3.00000 q^{74} -4.73205 q^{76} -1.60770 q^{77} +8.39230 q^{79} +1.73205 q^{80} -6.46410 q^{82} +5.66025 q^{83} -9.00000 q^{85} +4.19615 q^{86} -1.26795 q^{88} +9.46410 q^{89} +8.19615 q^{92} +4.73205 q^{94} -8.19615 q^{95} -6.00000 q^{97} +5.39230 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 6 q^{7} - 2 q^{8} + 6 q^{11} + 6 q^{14} + 2 q^{16} - 6 q^{19} - 6 q^{22} + 6 q^{23} - 4 q^{25} - 6 q^{28} + 6 q^{29} - 12 q^{31} - 2 q^{32} + 6 q^{35} - 6 q^{37} + 6 q^{38} + 6 q^{41}+ \cdots - 10 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 6 * q^7 - 2 * q^8 + 6 * q^11 + 6 * q^14 + 2 * q^16 - 6 * q^19 - 6 * q^22 + 6 * q^23 - 4 * q^25 - 6 * q^28 + 6 * q^29 - 12 * q^31 - 2 * q^32 + 6 * q^35 - 6 * q^37 + 6 * q^38 + 6 * q^41 + 2 * q^43 + 6 * q^44 - 6 * q^46 - 6 * q^47 + 10 * q^49 + 4 * q^50 - 6 * q^53 - 6 * q^55 + 6 * q^56 - 6 * q^58 + 20 * q^61 + 12 * q^62 + 2 * q^64 - 18 * q^67 - 6 * q^70 + 6 * q^71 + 6 * q^74 - 6 * q^76 - 24 * q^77 - 4 * q^79 - 6 * q^82 - 6 * q^83 - 18 * q^85 - 2 * q^86 - 6 * q^88 + 12 * q^89 + 6 * q^92 + 6 * q^94 - 6 * q^95 - 12 * q^97 - 10 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.73205	0.774597	0.387298	−	0.921954i	$-0.373408\pi$
$5$	1.73205	0.774597	0.387298	+	0.921954i	$0.373408\pi$
$6$	0	0
$7$	−1.26795	−0.479240	−0.239620	−	0.970867i	$-0.577023\pi$
$7$	−1.26795	−0.479240	−0.239620	+	0.970867i	$0.577023\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.73205	−0.547723
$11$	1.26795	0.382301	0.191151	−	0.981561i	$-0.438778\pi$
$11$	1.26795	0.382301	0.191151	+	0.981561i	$0.438778\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	1.26795	0.338874
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.19615	−1.26025	−0.630126	−	0.776493i	$-0.716997\pi$
$17$	−5.19615	−1.26025	−0.630126	+	0.776493i	$0.716997\pi$
$18$	0	0
$19$	−4.73205	−1.08561	−0.542803	−	0.839860i	$-0.682637\pi$
$19$	−4.73205	−1.08561	−0.542803	+	0.839860i	$0.682637\pi$
$20$	1.73205	0.387298
$21$	0	0
$22$	−1.26795	−0.270328
$23$	8.19615	1.70902	0.854508	−	0.519438i	$-0.173859\pi$
$23$	8.19615	1.70902	0.854508	+	0.519438i	$0.173859\pi$
$24$	0	0
$25$	−2.00000	−0.400000
$26$	0	0
$27$	0	0
$28$	−1.26795	−0.239620
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	−9.46410	−1.69980	−0.849901	−	0.526942i	$-0.823339\pi$
$31$	−9.46410	−1.69980	−0.849901	+	0.526942i	$0.823339\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	5.19615	0.891133
$35$	−2.19615	−0.371218
$36$	0	0
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	4.73205	0.767640
$39$	0	0
$40$	−1.73205	−0.273861
$41$	6.46410	1.00952	0.504762	−	0.863259i	$-0.331580\pi$
$41$	6.46410	1.00952	0.504762	+	0.863259i	$0.331580\pi$
$42$	0	0
$43$	−4.19615	−0.639907	−0.319954	−	0.947433i	$-0.603667\pi$
$43$	−4.19615	−0.639907	−0.319954	+	0.947433i	$0.603667\pi$
$44$	1.26795	0.191151
$45$	0	0
$46$	−8.19615	−1.20846
$47$	−4.73205	−0.690241	−0.345120	−	0.938558i	$-0.612162\pi$
$47$	−4.73205	−0.690241	−0.345120	+	0.938558i	$0.612162\pi$
$48$	0	0
$49$	−5.39230	−0.770329
$50$	2.00000	0.282843
$51$	0	0
$52$	0	0
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	0	0
$55$	2.19615	0.296129
$56$	1.26795	0.169437
$57$	0	0
$58$	−3.00000	−0.393919
$59$	−13.8564	−1.80395	−0.901975	−	0.431788i	$-0.857883\pi$
$59$	−13.8564	−1.80395	−0.901975	+	0.431788i	$0.857883\pi$
$60$	0	0
$61$	15.1962	1.94567	0.972834	−	0.231504i	$-0.0743646\pi$
$61$	15.1962	1.94567	0.972834	+	0.231504i	$0.0743646\pi$
$62$	9.46410	1.20194
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−7.26795	−0.887921	−0.443961	−	0.896046i	$-0.646427\pi$
$67$	−7.26795	−0.887921	−0.443961	+	0.896046i	$0.646427\pi$
$68$	−5.19615	−0.630126
$69$	0	0
$70$	2.19615	0.262490
$71$	−2.19615	−0.260635	−0.130318	−	0.991472i	$-0.541600\pi$
$71$	−2.19615	−0.260635	−0.130318	+	0.991472i	$0.541600\pi$
$72$	0	0
$73$	−12.1244	−1.41905	−0.709524	−	0.704681i	$-0.751090\pi$
$73$	−12.1244	−1.41905	−0.709524	+	0.704681i	$0.751090\pi$
$74$	3.00000	0.348743
$75$	0	0
$76$	−4.73205	−0.542803
$77$	−1.60770	−0.183214
$78$	0	0
$79$	8.39230	0.944208	0.472104	−	0.881543i	$-0.343495\pi$
$79$	8.39230	0.944208	0.472104	+	0.881543i	$0.343495\pi$
$80$	1.73205	0.193649
$81$	0	0
$82$	−6.46410	−0.713841
$83$	5.66025	0.621294	0.310647	−	0.950525i	$-0.399454\pi$
$83$	5.66025	0.621294	0.310647	+	0.950525i	$0.399454\pi$
$84$	0	0
$85$	−9.00000	−0.976187
$86$	4.19615	0.452483
$87$	0	0
$88$	−1.26795	−0.135164
$89$	9.46410	1.00319	0.501596	−	0.865102i	$-0.332746\pi$
$89$	9.46410	1.00319	0.501596	+	0.865102i	$0.332746\pi$
$90$	0	0
$91$	0	0
$92$	8.19615	0.854508
$93$	0	0
$94$	4.73205	0.488074
$95$	−8.19615	−0.840907
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	5.39230	0.544705
$99$	0	0
$100$	−2.00000	−0.200000
$101$	−19.3923	−1.92961	−0.964803	−	0.262973i	$-0.915297\pi$
$101$	−19.3923	−1.92961	−0.964803	+	0.262973i	$0.915297\pi$
$102$	0	0
$103$	6.19615	0.610525	0.305263	−	0.952268i	$-0.401256\pi$
$103$	6.19615	0.610525	0.305263	+	0.952268i	$0.401256\pi$
$104$	0	0
$105$	0	0
$106$	3.00000	0.291386
$107$	2.19615	0.212310	0.106155	−	0.994350i	$-0.466146\pi$
$107$	2.19615	0.212310	0.106155	+	0.994350i	$0.466146\pi$
$108$	0	0
$109$	4.39230	0.420707	0.210353	−	0.977625i	$-0.432539\pi$
$109$	4.39230	0.420707	0.210353	+	0.977625i	$0.432539\pi$
$110$	−2.19615	−0.209395
$111$	0	0
$112$	−1.26795	−0.119810
$113$	−0.803848	−0.0756196	−0.0378098	−	0.999285i	$-0.512038\pi$
$113$	−0.803848	−0.0756196	−0.0378098	+	0.999285i	$0.512038\pi$
$114$	0	0
$115$	14.1962	1.32380
$116$	3.00000	0.278543
$117$	0	0
$118$	13.8564	1.27559
$119$	6.58846	0.603963
$120$	0	0
$121$	−9.39230	−0.853846
$122$	−15.1962	−1.37579
$123$	0	0
$124$	−9.46410	−0.849901
$125$	−12.1244	−1.08444
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	4.39230	0.383757	0.191879	−	0.981419i	$-0.438542\pi$
$131$	4.39230	0.383757	0.191879	+	0.981419i	$0.438542\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	7.26795	0.627855
$135$	0	0
$136$	5.19615	0.445566
$137$	−9.00000	−0.768922	−0.384461	−	0.923141i	$-0.625613\pi$
$137$	−9.00000	−0.768922	−0.384461	+	0.923141i	$0.625613\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	−2.19615	−0.185609
$141$	0	0
$142$	2.19615	0.184297
$143$	0	0
$144$	0	0
$145$	5.19615	0.431517
$146$	12.1244	1.00342
$147$	0	0
$148$	−3.00000	−0.246598
$149$	6.12436	0.501727	0.250863	−	0.968023i	$-0.419286\pi$
$149$	6.12436	0.501727	0.250863	+	0.968023i	$0.419286\pi$
$150$	0	0
$151$	−10.7321	−0.873362	−0.436681	−	0.899616i	$-0.643846\pi$
$151$	−10.7321	−0.873362	−0.436681	+	0.899616i	$0.643846\pi$
$152$	4.73205	0.383820
$153$	0	0
$154$	1.60770	0.129552
$155$	−16.3923	−1.31666
$156$	0	0
$157$	7.19615	0.574315	0.287158	−	0.957883i	$-0.407290\pi$
$157$	7.19615	0.574315	0.287158	+	0.957883i	$0.407290\pi$
$158$	−8.39230	−0.667656
$159$	0	0
$160$	−1.73205	−0.136931
$161$	−10.3923	−0.819028
$162$	0	0
$163$	−2.53590	−0.198627	−0.0993134	−	0.995056i	$-0.531665\pi$
$163$	−2.53590	−0.198627	−0.0993134	+	0.995056i	$0.531665\pi$
$164$	6.46410	0.504762
$165$	0	0
$166$	−5.66025	−0.439321
$167$	−9.46410	−0.732354	−0.366177	−	0.930545i	$-0.619334\pi$
$167$	−9.46410	−0.732354	−0.366177	+	0.930545i	$0.619334\pi$
$168$	0	0
$169$	0	0
$170$	9.00000	0.690268
$171$	0	0
$172$	−4.19615	−0.319954
$173$	4.39230	0.333941	0.166970	−	0.985962i	$-0.446602\pi$
$173$	4.39230	0.333941	0.166970	+	0.985962i	$0.446602\pi$
$174$	0	0
$175$	2.53590	0.191696
$176$	1.26795	0.0955753
$177$	0	0
$178$	−9.46410	−0.709364
$179$	−2.19615	−0.164148	−0.0820741	−	0.996626i	$-0.526154\pi$
$179$	−2.19615	−0.164148	−0.0820741	+	0.996626i	$0.526154\pi$
$180$	0	0
$181$	19.5885	1.45600	0.727999	−	0.685578i	$-0.240450\pi$
$181$	19.5885	1.45600	0.727999	+	0.685578i	$0.240450\pi$
$182$	0	0
$183$	0	0
$184$	−8.19615	−0.604228
$185$	−5.19615	−0.382029
$186$	0	0
$187$	−6.58846	−0.481796
$188$	−4.73205	−0.345120
$189$	0	0
$190$	8.19615	0.594611
$191$	20.7846	1.50392	0.751961	−	0.659208i	$-0.229108\pi$
$191$	20.7846	1.50392	0.751961	+	0.659208i	$0.229108\pi$
$192$	0	0
$193$	23.1962	1.66970	0.834848	−	0.550481i	$-0.185556\pi$
$193$	23.1962	1.66970	0.834848	+	0.550481i	$0.185556\pi$
$194$	6.00000	0.430775
$195$	0	0
$196$	−5.39230	−0.385165
$197$	−6.92820	−0.493614	−0.246807	−	0.969065i	$-0.579381\pi$
$197$	−6.92820	−0.493614	−0.246807	+	0.969065i	$0.579381\pi$
$198$	0	0
$199$	−22.5885	−1.60125	−0.800627	−	0.599164i	$-0.795500\pi$
$199$	−22.5885	−1.60125	−0.800627	+	0.599164i	$0.795500\pi$
$200$	2.00000	0.141421
$201$	0	0
$202$	19.3923	1.36444
$203$	−3.80385	−0.266978
$204$	0	0
$205$	11.1962	0.781973
$206$	−6.19615	−0.431706
$207$	0	0
$208$	0	0
$209$	−6.00000	−0.415029
$210$	0	0
$211$	−24.3923	−1.67924	−0.839618	−	0.543178i	$-0.817221\pi$
$211$	−24.3923	−1.67924	−0.839618	+	0.543178i	$0.817221\pi$
$212$	−3.00000	−0.206041
$213$	0	0
$214$	−2.19615	−0.150126
$215$	−7.26795	−0.495670
$216$	0	0
$217$	12.0000	0.814613
$218$	−4.39230	−0.297484
$219$	0	0
$220$	2.19615	0.148065
$221$	0	0
$222$	0	0
$223$	5.07180	0.339633	0.169816	−	0.985476i	$-0.445683\pi$
$223$	5.07180	0.339633	0.169816	+	0.985476i	$0.445683\pi$
$224$	1.26795	0.0847184
$225$	0	0
$226$	0.803848	0.0534711
$227$	−20.1962	−1.34047	−0.670233	−	0.742151i	$-0.733806\pi$
$227$	−20.1962	−1.34047	−0.670233	+	0.742151i	$0.733806\pi$
$228$	0	0
$229$	7.85641	0.519166	0.259583	−	0.965721i	$-0.416415\pi$
$229$	7.85641	0.519166	0.259583	+	0.965721i	$0.416415\pi$
$230$	−14.1962	−0.936067
$231$	0	0
$232$	−3.00000	−0.196960
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	−8.19615	−0.534658
$236$	−13.8564	−0.901975
$237$	0	0
$238$	−6.58846	−0.427066
$239$	6.58846	0.426172	0.213086	−	0.977033i	$-0.431649\pi$
$239$	6.58846	0.426172	0.213086	+	0.977033i	$0.431649\pi$
$240$	0	0
$241$	−11.1962	−0.721208	−0.360604	−	0.932719i	$-0.617429\pi$
$241$	−11.1962	−0.721208	−0.360604	+	0.932719i	$0.617429\pi$
$242$	9.39230	0.603760
$243$	0	0
$244$	15.1962	0.972834
$245$	−9.33975	−0.596694
$246$	0	0
$247$	0	0
$248$	9.46410	0.600971
$249$	0	0
$250$	12.1244	0.766812
$251$	−16.3923	−1.03467	−0.517337	−	0.855782i	$-0.673076\pi$
$251$	−16.3923	−1.03467	−0.517337	+	0.855782i	$0.673076\pi$
$252$	0	0
$253$	10.3923	0.653359
$254$	4.00000	0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	−23.1962	−1.44694	−0.723468	−	0.690358i	$-0.757453\pi$
$257$	−23.1962	−1.44694	−0.723468	+	0.690358i	$0.757453\pi$
$258$	0	0
$259$	3.80385	0.236360
$260$	0	0
$261$	0	0
$262$	−4.39230	−0.271357
$263$	−8.19615	−0.505396	−0.252698	−	0.967545i	$-0.581318\pi$
$263$	−8.19615	−0.505396	−0.252698	+	0.967545i	$0.581318\pi$
$264$	0	0
$265$	−5.19615	−0.319197
$266$	−6.00000	−0.367884
$267$	0	0
$268$	−7.26795	−0.443961
$269$	7.60770	0.463849	0.231925	−	0.972734i	$-0.425498\pi$
$269$	7.60770	0.463849	0.231925	+	0.972734i	$0.425498\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	−5.19615	−0.315063
$273$	0	0
$274$	9.00000	0.543710
$275$	−2.53590	−0.152920
$276$	0	0
$277$	4.80385	0.288635	0.144318	−	0.989531i	$-0.453901\pi$
$277$	4.80385	0.288635	0.144318	+	0.989531i	$0.453901\pi$
$278$	4.00000	0.239904
$279$	0	0
$280$	2.19615	0.131245
$281$	17.5359	1.04610	0.523052	−	0.852301i	$-0.324793\pi$
$281$	17.5359	1.04610	0.523052	+	0.852301i	$0.324793\pi$
$282$	0	0
$283$	−19.8038	−1.17722	−0.588608	−	0.808418i	$-0.700324\pi$
$283$	−19.8038	−1.17722	−0.588608	+	0.808418i	$0.700324\pi$
$284$	−2.19615	−0.130318
$285$	0	0
$286$	0	0
$287$	−8.19615	−0.483804
$288$	0	0
$289$	10.0000	0.588235
$290$	−5.19615	−0.305129
$291$	0	0
$292$	−12.1244	−0.709524
$293$	2.66025	0.155414	0.0777069	−	0.996976i	$-0.475240\pi$
$293$	2.66025	0.155414	0.0777069	+	0.996976i	$0.475240\pi$
$294$	0	0
$295$	−24.0000	−1.39733
$296$	3.00000	0.174371
$297$	0	0
$298$	−6.12436	−0.354774
$299$	0	0
$300$	0	0
$301$	5.32051	0.306669
$302$	10.7321	0.617560
$303$	0	0
$304$	−4.73205	−0.271402
$305$	26.3205	1.50711
$306$	0	0
$307$	−7.26795	−0.414804	−0.207402	−	0.978256i	$-0.566501\pi$
$307$	−7.26795	−0.414804	−0.207402	+	0.978256i	$0.566501\pi$
$308$	−1.60770	−0.0916069
$309$	0	0
$310$	16.3923	0.931020
$311$	8.19615	0.464761	0.232381	−	0.972625i	$-0.425349\pi$
$311$	8.19615	0.464761	0.232381	+	0.972625i	$0.425349\pi$
$312$	0	0
$313$	−3.60770	−0.203919	−0.101959	−	0.994789i	$-0.532511\pi$
$313$	−3.60770	−0.203919	−0.101959	+	0.994789i	$0.532511\pi$
$314$	−7.19615	−0.406102
$315$	0	0
$316$	8.39230	0.472104
$317$	−18.1244	−1.01797	−0.508983	−	0.860777i	$-0.669978\pi$
$317$	−18.1244	−1.01797	−0.508983	+	0.860777i	$0.669978\pi$
$318$	0	0
$319$	3.80385	0.212975
$320$	1.73205	0.0968246
$321$	0	0
$322$	10.3923	0.579141
$323$	24.5885	1.36814
$324$	0	0
$325$	0	0
$326$	2.53590	0.140450
$327$	0	0
$328$	−6.46410	−0.356920
$329$	6.00000	0.330791
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	5.66025	0.310647
$333$	0	0
$334$	9.46410	0.517853
$335$	−12.5885	−0.687781
$336$	0	0
$337$	−31.0000	−1.68868	−0.844339	−	0.535810i	$-0.820006\pi$
$337$	−31.0000	−1.68868	−0.844339	+	0.535810i	$0.820006\pi$
$338$	0	0
$339$	0	0
$340$	−9.00000	−0.488094
$341$	−12.0000	−0.649836
$342$	0	0
$343$	15.7128	0.848412
$344$	4.19615	0.226241
$345$	0	0
$346$	−4.39230	−0.236132
$347$	−18.5885	−0.997881	−0.498940	−	0.866636i	$-0.666277\pi$
$347$	−18.5885	−0.997881	−0.498940	+	0.866636i	$0.666277\pi$
$348$	0	0
$349$	−9.46410	−0.506602	−0.253301	−	0.967388i	$-0.581516\pi$
$349$	−9.46410	−0.506602	−0.253301	+	0.967388i	$0.581516\pi$
$350$	−2.53590	−0.135549
$351$	0	0
$352$	−1.26795	−0.0675819
$353$	−35.7846	−1.90462	−0.952311	−	0.305128i	$-0.901301\pi$
$353$	−35.7846	−1.90462	−0.952311	+	0.305128i	$0.901301\pi$
$354$	0	0
$355$	−3.80385	−0.201887
$356$	9.46410	0.501596
$357$	0	0
$358$	2.19615	0.116070
$359$	−16.0526	−0.847222	−0.423611	−	0.905844i	$-0.639238\pi$
$359$	−16.0526	−0.847222	−0.423611	+	0.905844i	$0.639238\pi$
$360$	0	0
$361$	3.39230	0.178542
$362$	−19.5885	−1.02955
$363$	0	0
$364$	0	0
$365$	−21.0000	−1.09919
$366$	0	0
$367$	−13.8038	−0.720555	−0.360277	−	0.932845i	$-0.617318\pi$
$367$	−13.8038	−0.720555	−0.360277	+	0.932845i	$0.617318\pi$
$368$	8.19615	0.427254
$369$	0	0
$370$	5.19615	0.270135
$371$	3.80385	0.197486
$372$	0	0
$373$	−27.9808	−1.44879	−0.724394	−	0.689386i	$-0.757881\pi$
$373$	−27.9808	−1.44879	−0.724394	+	0.689386i	$0.757881\pi$
$374$	6.58846	0.340681
$375$	0	0
$376$	4.73205	0.244037
$377$	0	0
$378$	0	0
$379$	−30.2487	−1.55377	−0.776886	−	0.629641i	$-0.783202\pi$
$379$	−30.2487	−1.55377	−0.776886	+	0.629641i	$0.783202\pi$
$380$	−8.19615	−0.420454
$381$	0	0
$382$	−20.7846	−1.06343
$383$	−23.3205	−1.19162	−0.595811	−	0.803125i	$-0.703169\pi$
$383$	−23.3205	−1.19162	−0.595811	+	0.803125i	$0.703169\pi$
$384$	0	0
$385$	−2.78461	−0.141917
$386$	−23.1962	−1.18065
$387$	0	0
$388$	−6.00000	−0.304604
$389$	−7.39230	−0.374805	−0.187402	−	0.982283i	$-0.560007\pi$
$389$	−7.39230	−0.374805	−0.187402	+	0.982283i	$0.560007\pi$
$390$	0	0
$391$	−42.5885	−2.15379
$392$	5.39230	0.272353
$393$	0	0
$394$	6.92820	0.349038
$395$	14.5359	0.731380
$396$	0	0
$397$	−4.39230	−0.220443	−0.110222	−	0.993907i	$-0.535156\pi$
$397$	−4.39230	−0.220443	−0.110222	+	0.993907i	$0.535156\pi$
$398$	22.5885	1.13226
$399$	0	0
$400$	−2.00000	−0.100000
$401$	21.0000	1.04869	0.524345	−	0.851506i	$-0.324310\pi$
$401$	21.0000	1.04869	0.524345	+	0.851506i	$0.324310\pi$
$402$	0	0
$403$	0	0
$404$	−19.3923	−0.964803
$405$	0	0
$406$	3.80385	0.188782
$407$	−3.80385	−0.188550
$408$	0	0
$409$	20.6603	1.02158	0.510792	−	0.859704i	$-0.329352\pi$
$409$	20.6603	1.02158	0.510792	+	0.859704i	$0.329352\pi$
$410$	−11.1962	−0.552939
$411$	0	0
$412$	6.19615	0.305263
$413$	17.5692	0.864525
$414$	0	0
$415$	9.80385	0.481252
$416$	0	0
$417$	0	0
$418$	6.00000	0.293470
$419$	4.39230	0.214578	0.107289	−	0.994228i	$-0.465783\pi$
$419$	4.39230	0.214578	0.107289	+	0.994228i	$0.465783\pi$
$420$	0	0
$421$	6.46410	0.315041	0.157521	−	0.987516i	$-0.449650\pi$
$421$	6.46410	0.315041	0.157521	+	0.987516i	$0.449650\pi$
$422$	24.3923	1.18740
$423$	0	0
$424$	3.00000	0.145693
$425$	10.3923	0.504101
$426$	0	0
$427$	−19.2679	−0.932441
$428$	2.19615	0.106155
$429$	0	0
$430$	7.26795	0.350492
$431$	38.1962	1.83984	0.919922	−	0.392101i	$-0.128252\pi$
$431$	38.1962	1.83984	0.919922	+	0.392101i	$0.128252\pi$
$432$	0	0
$433$	7.78461	0.374104	0.187052	−	0.982350i	$-0.440107\pi$
$433$	7.78461	0.374104	0.187052	+	0.982350i	$0.440107\pi$
$434$	−12.0000	−0.576018
$435$	0	0
$436$	4.39230	0.210353
$437$	−38.7846	−1.85532
$438$	0	0
$439$	−14.5885	−0.696269	−0.348135	−	0.937445i	$-0.613185\pi$
$439$	−14.5885	−0.696269	−0.348135	+	0.937445i	$0.613185\pi$
$440$	−2.19615	−0.104697
$441$	0	0
$442$	0	0
$443$	−16.3923	−0.778822	−0.389411	−	0.921064i	$-0.627321\pi$
$443$	−16.3923	−0.778822	−0.389411	+	0.921064i	$0.627321\pi$
$444$	0	0
$445$	16.3923	0.777070
$446$	−5.07180	−0.240157
$447$	0	0
$448$	−1.26795	−0.0599050
$449$	26.5359	1.25231	0.626153	−	0.779700i	$-0.284628\pi$
$449$	26.5359	1.25231	0.626153	+	0.779700i	$0.284628\pi$
$450$	0	0
$451$	8.19615	0.385942
$452$	−0.803848	−0.0378098
$453$	0	0
$454$	20.1962	0.947852
$455$	0	0
$456$	0	0
$457$	31.9808	1.49600	0.747998	−	0.663700i	$-0.231015\pi$
$457$	31.9808	1.49600	0.747998	+	0.663700i	$0.231015\pi$
$458$	−7.85641	−0.367106
$459$	0	0
$460$	14.1962	0.661899
$461$	31.9808	1.48949	0.744746	−	0.667348i	$-0.232570\pi$
$461$	31.9808	1.48949	0.744746	+	0.667348i	$0.232570\pi$
$462$	0	0
$463$	15.8038	0.734467	0.367234	−	0.930129i	$-0.380305\pi$
$463$	15.8038	0.734467	0.367234	+	0.930129i	$0.380305\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	−18.0000	−0.833834
$467$	−5.41154	−0.250416	−0.125208	−	0.992130i	$-0.539960\pi$
$467$	−5.41154	−0.250416	−0.125208	+	0.992130i	$0.539960\pi$
$468$	0	0
$469$	9.21539	0.425527
$470$	8.19615	0.378060
$471$	0	0
$472$	13.8564	0.637793
$473$	−5.32051	−0.244637
$474$	0	0
$475$	9.46410	0.434243
$476$	6.58846	0.301981
$477$	0	0
$478$	−6.58846	−0.301349
$479$	−0.679492	−0.0310468	−0.0155234	−	0.999880i	$-0.504941\pi$
$479$	−0.679492	−0.0310468	−0.0155234	+	0.999880i	$0.504941\pi$
$480$	0	0
$481$	0	0
$482$	11.1962	0.509971
$483$	0	0
$484$	−9.39230	−0.426923
$485$	−10.3923	−0.471890
$486$	0	0
$487$	15.1244	0.685350	0.342675	−	0.939454i	$-0.388667\pi$
$487$	15.1244	0.685350	0.342675	+	0.939454i	$0.388667\pi$
$488$	−15.1962	−0.687897
$489$	0	0
$490$	9.33975	0.421927
$491$	30.5885	1.38044	0.690219	−	0.723601i	$-0.257514\pi$
$491$	30.5885	1.38044	0.690219	+	0.723601i	$0.257514\pi$
$492$	0	0
$493$	−15.5885	−0.702069
$494$	0	0
$495$	0	0
$496$	−9.46410	−0.424951
$497$	2.78461	0.124907
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	−12.1244	−0.542218
$501$	0	0
$502$	16.3923	0.731624
$503$	−12.5885	−0.561292	−0.280646	−	0.959811i	$-0.590549\pi$
$503$	−12.5885	−0.561292	−0.280646	+	0.959811i	$0.590549\pi$
$504$	0	0
$505$	−33.5885	−1.49467
$506$	−10.3923	−0.461994
$507$	0	0
$508$	−4.00000	−0.177471
$509$	26.6603	1.18169	0.590847	−	0.806783i	$-0.298793\pi$
$509$	26.6603	1.18169	0.590847	+	0.806783i	$0.298793\pi$
$510$	0	0
$511$	15.3731	0.680064
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	23.1962	1.02314
$515$	10.7321	0.472911
$516$	0	0
$517$	−6.00000	−0.263880
$518$	−3.80385	−0.167131
$519$	0	0
$520$	0	0
$521$	29.1962	1.27911	0.639553	−	0.768747i	$-0.279119\pi$
$521$	29.1962	1.27911	0.639553	+	0.768747i	$0.279119\pi$
$522$	0	0
$523$	32.5885	1.42499	0.712497	−	0.701675i	$-0.247564\pi$
$523$	32.5885	1.42499	0.712497	+	0.701675i	$0.247564\pi$
$524$	4.39230	0.191879
$525$	0	0
$526$	8.19615	0.357369
$527$	49.1769	2.14218
$528$	0	0
$529$	44.1769	1.92074
$530$	5.19615	0.225706
$531$	0	0
$532$	6.00000	0.260133
$533$	0	0
$534$	0	0
$535$	3.80385	0.164455
$536$	7.26795	0.313928
$537$	0	0
$538$	−7.60770	−0.327991
$539$	−6.83717	−0.294498
$540$	0	0
$541$	10.8564	0.466753	0.233377	−	0.972386i	$-0.425022\pi$
$541$	10.8564	0.466753	0.233377	+	0.972386i	$0.425022\pi$
$542$	0	0
$543$	0	0
$544$	5.19615	0.222783
$545$	7.60770	0.325878
$546$	0	0
$547$	4.19615	0.179415	0.0897073	−	0.995968i	$-0.471407\pi$
$547$	4.19615	0.179415	0.0897073	+	0.995968i	$0.471407\pi$
$548$	−9.00000	−0.384461
$549$	0	0
$550$	2.53590	0.108131
$551$	−14.1962	−0.604776
$552$	0	0
$553$	−10.6410	−0.452502
$554$	−4.80385	−0.204096
$555$	0	0
$556$	−4.00000	−0.169638
$557$	25.7321	1.09030	0.545151	−	0.838338i	$-0.316472\pi$
$557$	25.7321	1.09030	0.545151	+	0.838338i	$0.316472\pi$
$558$	0	0
$559$	0	0
$560$	−2.19615	−0.0928044
$561$	0	0
$562$	−17.5359	−0.739707
$563$	32.7846	1.38171	0.690853	−	0.722995i	$-0.257235\pi$
$563$	32.7846	1.38171	0.690853	+	0.722995i	$0.257235\pi$
$564$	0	0
$565$	−1.39230	−0.0585747
$566$	19.8038	0.832418
$567$	0	0
$568$	2.19615	0.0921485
$569$	−8.78461	−0.368270	−0.184135	−	0.982901i	$-0.558948\pi$
$569$	−8.78461	−0.368270	−0.184135	+	0.982901i	$0.558948\pi$
$570$	0	0
$571$	−24.1962	−1.01258	−0.506289	−	0.862364i	$-0.668983\pi$
$571$	−24.1962	−1.01258	−0.506289	+	0.862364i	$0.668983\pi$
$572$	0	0
$573$	0	0
$574$	8.19615	0.342101
$575$	−16.3923	−0.683606
$576$	0	0
$577$	19.7321	0.821456	0.410728	−	0.911758i	$-0.365275\pi$
$577$	19.7321	0.821456	0.410728	+	0.911758i	$0.365275\pi$
$578$	−10.0000	−0.415945
$579$	0	0
$580$	5.19615	0.215758
$581$	−7.17691	−0.297749
$582$	0	0
$583$	−3.80385	−0.157539
$584$	12.1244	0.501709
$585$	0	0
$586$	−2.66025	−0.109894
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	44.7846	1.84532
$590$	24.0000	0.988064
$591$	0	0
$592$	−3.00000	−0.123299
$593$	−19.1436	−0.786133	−0.393067	−	0.919510i	$-0.628586\pi$
$593$	−19.1436	−0.786133	−0.393067	+	0.919510i	$0.628586\pi$
$594$	0	0
$595$	11.4115	0.467828
$596$	6.12436	0.250863
$597$	0	0
$598$	0	0
$599$	−16.3923	−0.669771	−0.334886	−	0.942259i	$-0.608698\pi$
$599$	−16.3923	−0.669771	−0.334886	+	0.942259i	$0.608698\pi$
$600$	0	0
$601$	−19.7846	−0.807031	−0.403516	−	0.914973i	$-0.632212\pi$
$601$	−19.7846	−0.807031	−0.403516	+	0.914973i	$0.632212\pi$
$602$	−5.32051	−0.216848
$603$	0	0
$604$	−10.7321	−0.436681
$605$	−16.2679	−0.661386
$606$	0	0
$607$	−7.21539	−0.292864	−0.146432	−	0.989221i	$-0.546779\pi$
$607$	−7.21539	−0.292864	−0.146432	+	0.989221i	$0.546779\pi$
$608$	4.73205	0.191910
$609$	0	0
$610$	−26.3205	−1.06569
$611$	0	0
$612$	0	0
$613$	13.1436	0.530865	0.265432	−	0.964129i	$-0.414485\pi$
$613$	13.1436	0.530865	0.265432	+	0.964129i	$0.414485\pi$
$614$	7.26795	0.293311
$615$	0	0
$616$	1.60770	0.0647759
$617$	31.3923	1.26381	0.631903	−	0.775047i	$-0.282274\pi$
$617$	31.3923	1.26381	0.631903	+	0.775047i	$0.282274\pi$
$618$	0	0
$619$	28.3923	1.14118	0.570592	−	0.821234i	$-0.306714\pi$
$619$	28.3923	1.14118	0.570592	+	0.821234i	$0.306714\pi$
$620$	−16.3923	−0.658331
$621$	0	0
$622$	−8.19615	−0.328636
$623$	−12.0000	−0.480770
$624$	0	0
$625$	−11.0000	−0.440000
$626$	3.60770	0.144192
$627$	0	0
$628$	7.19615	0.287158
$629$	15.5885	0.621552
$630$	0	0
$631$	1.85641	0.0739024	0.0369512	−	0.999317i	$-0.488235\pi$
$631$	1.85641	0.0739024	0.0369512	+	0.999317i	$0.488235\pi$
$632$	−8.39230	−0.333828
$633$	0	0
$634$	18.1244	0.719810
$635$	−6.92820	−0.274937
$636$	0	0
$637$	0	0
$638$	−3.80385	−0.150596
$639$	0	0
$640$	−1.73205	−0.0684653
$641$	41.1962	1.62715	0.813575	−	0.581460i	$-0.197518\pi$
$641$	41.1962	1.62715	0.813575	+	0.581460i	$0.197518\pi$
$642$	0	0
$643$	−27.7128	−1.09289	−0.546443	−	0.837496i	$-0.684019\pi$
$643$	−27.7128	−1.09289	−0.546443	+	0.837496i	$0.684019\pi$
$644$	−10.3923	−0.409514
$645$	0	0
$646$	−24.5885	−0.967420
$647$	49.1769	1.93334	0.966672	−	0.256018i	$-0.0824107\pi$
$647$	49.1769	1.93334	0.966672	+	0.256018i	$0.0824107\pi$
$648$	0	0
$649$	−17.5692	−0.689652
$650$	0	0
$651$	0	0
$652$	−2.53590	−0.0993134
$653$	−13.1769	−0.515653	−0.257826	−	0.966191i	$-0.583006\pi$
$653$	−13.1769	−0.515653	−0.257826	+	0.966191i	$0.583006\pi$
$654$	0	0
$655$	7.60770	0.297257
$656$	6.46410	0.252381
$657$	0	0
$658$	−6.00000	−0.233904
$659$	37.1769	1.44821	0.724103	−	0.689691i	$-0.242254\pi$
$659$	37.1769	1.44821	0.724103	+	0.689691i	$0.242254\pi$
$660$	0	0
$661$	−9.00000	−0.350059	−0.175030	−	0.984563i	$-0.556002\pi$
$661$	−9.00000	−0.350059	−0.175030	+	0.984563i	$0.556002\pi$
$662$	12.0000	0.466393
$663$	0	0
$664$	−5.66025	−0.219660
$665$	10.3923	0.402996
$666$	0	0
$667$	24.5885	0.952069
$668$	−9.46410	−0.366177
$669$	0	0
$670$	12.5885	0.486335
$671$	19.2679	0.743831
$672$	0	0
$673$	1.00000	0.0385472	0.0192736	−	0.999814i	$-0.493865\pi$
$673$	1.00000	0.0385472	0.0192736	+	0.999814i	$0.493865\pi$
$674$	31.0000	1.19408
$675$	0	0
$676$	0	0
$677$	16.3923	0.630007	0.315004	−	0.949090i	$-0.397994\pi$
$677$	16.3923	0.630007	0.315004	+	0.949090i	$0.397994\pi$
$678$	0	0
$679$	7.60770	0.291957
$680$	9.00000	0.345134
$681$	0	0
$682$	12.0000	0.459504
$683$	−27.7128	−1.06040	−0.530201	−	0.847872i	$-0.677883\pi$
$683$	−27.7128	−1.06040	−0.530201	+	0.847872i	$0.677883\pi$
$684$	0	0
$685$	−15.5885	−0.595604
$686$	−15.7128	−0.599918
$687$	0	0
$688$	−4.19615	−0.159977
$689$	0	0
$690$	0	0
$691$	−25.5167	−0.970700	−0.485350	−	0.874320i	$-0.661308\pi$
$691$	−25.5167	−0.970700	−0.485350	+	0.874320i	$0.661308\pi$
$692$	4.39230	0.166970
$693$	0	0
$694$	18.5885	0.705608
$695$	−6.92820	−0.262802
$696$	0	0
$697$	−33.5885	−1.27225
$698$	9.46410	0.358222
$699$	0	0
$700$	2.53590	0.0958479
$701$	16.3923	0.619129	0.309564	−	0.950878i	$-0.399817\pi$
$701$	16.3923	0.619129	0.309564	+	0.950878i	$0.399817\pi$
$702$	0	0
$703$	14.1962	0.535418
$704$	1.26795	0.0477876
$705$	0	0
$706$	35.7846	1.34677
$707$	24.5885	0.924744
$708$	0	0
$709$	45.2487	1.69935	0.849676	−	0.527306i	$-0.176798\pi$
$709$	45.2487	1.69935	0.849676	+	0.527306i	$0.176798\pi$
$710$	3.80385	0.142756
$711$	0	0
$712$	−9.46410	−0.354682
$713$	−77.5692	−2.90499
$714$	0	0
$715$	0	0
$716$	−2.19615	−0.0820741
$717$	0	0
$718$	16.0526	0.599076
$719$	−31.6077	−1.17877	−0.589384	−	0.807853i	$-0.700630\pi$
$719$	−31.6077	−1.17877	−0.589384	+	0.807853i	$0.700630\pi$
$720$	0	0
$721$	−7.85641	−0.292588
$722$	−3.39230	−0.126249
$723$	0	0
$724$	19.5885	0.727999
$725$	−6.00000	−0.222834
$726$	0	0
$727$	−13.8038	−0.511956	−0.255978	−	0.966683i	$-0.582398\pi$
$727$	−13.8038	−0.511956	−0.255978	+	0.966683i	$0.582398\pi$
$728$	0	0
$729$	0	0
$730$	21.0000	0.777245
$731$	21.8038	0.806444
$732$	0	0
$733$	−20.3205	−0.750555	−0.375278	−	0.926912i	$-0.622453\pi$
$733$	−20.3205	−0.750555	−0.375278	+	0.926912i	$0.622453\pi$
$734$	13.8038	0.509509
$735$	0	0
$736$	−8.19615	−0.302114
$737$	−9.21539	−0.339453
$738$	0	0
$739$	5.07180	0.186569	0.0932845	−	0.995639i	$-0.470263\pi$
$739$	5.07180	0.186569	0.0932845	+	0.995639i	$0.470263\pi$
$740$	−5.19615	−0.191014
$741$	0	0
$742$	−3.80385	−0.139644
$743$	−16.3923	−0.601375	−0.300688	−	0.953723i	$-0.597216\pi$
$743$	−16.3923	−0.601375	−0.300688	+	0.953723i	$0.597216\pi$
$744$	0	0
$745$	10.6077	0.388636
$746$	27.9808	1.02445
$747$	0	0
$748$	−6.58846	−0.240898
$749$	−2.78461	−0.101747
$750$	0	0
$751$	26.9808	0.984542	0.492271	−	0.870442i	$-0.336167\pi$
$751$	26.9808	0.984542	0.492271	+	0.870442i	$0.336167\pi$
$752$	−4.73205	−0.172560
$753$	0	0
$754$	0	0
$755$	−18.5885	−0.676503
$756$	0	0
$757$	−22.7846	−0.828121	−0.414060	−	0.910249i	$-0.635890\pi$
$757$	−22.7846	−0.828121	−0.414060	+	0.910249i	$0.635890\pi$
$758$	30.2487	1.09868
$759$	0	0
$760$	8.19615	0.297306
$761$	16.3923	0.594221	0.297110	−	0.954843i	$-0.403977\pi$
$761$	16.3923	0.594221	0.297110	+	0.954843i	$0.403977\pi$
$762$	0	0
$763$	−5.56922	−0.201619
$764$	20.7846	0.751961
$765$	0	0
$766$	23.3205	0.842604
$767$	0	0
$768$	0	0
$769$	21.7128	0.782984	0.391492	−	0.920181i	$-0.371959\pi$
$769$	21.7128	0.782984	0.391492	+	0.920181i	$0.371959\pi$
$770$	2.78461	0.100350
$771$	0	0
$772$	23.1962	0.834848
$773$	−9.21539	−0.331455	−0.165727	−	0.986172i	$-0.552997\pi$
$773$	−9.21539	−0.331455	−0.165727	+	0.986172i	$0.552997\pi$
$774$	0	0
$775$	18.9282	0.679921
$776$	6.00000	0.215387
$777$	0	0
$778$	7.39230	0.265027
$779$	−30.5885	−1.09595
$780$	0	0
$781$	−2.78461	−0.0996412
$782$	42.5885	1.52296
$783$	0	0
$784$	−5.39230	−0.192582
$785$	12.4641	0.444863
$786$	0	0
$787$	21.4641	0.765113	0.382556	−	0.923932i	$-0.375044\pi$
$787$	21.4641	0.765113	0.382556	+	0.923932i	$0.375044\pi$
$788$	−6.92820	−0.246807
$789$	0	0
$790$	−14.5359	−0.517164
$791$	1.01924	0.0362399
$792$	0	0
$793$	0	0
$794$	4.39230	0.155877
$795$	0	0
$796$	−22.5885	−0.800627
$797$	−6.00000	−0.212531	−0.106265	−	0.994338i	$-0.533889\pi$
$797$	−6.00000	−0.212531	−0.106265	+	0.994338i	$0.533889\pi$
$798$	0	0
$799$	24.5885	0.869877
$800$	2.00000	0.0707107
$801$	0	0
$802$	−21.0000	−0.741536
$803$	−15.3731	−0.542504
$804$	0	0
$805$	−18.0000	−0.634417
$806$	0	0
$807$	0	0
$808$	19.3923	0.682219
$809$	36.8038	1.29395	0.646977	−	0.762509i	$-0.276033\pi$
$809$	36.8038	1.29395	0.646977	+	0.762509i	$0.276033\pi$
$810$	0	0
$811$	16.3923	0.575612	0.287806	−	0.957689i	$-0.407074\pi$
$811$	16.3923	0.575612	0.287806	+	0.957689i	$0.407074\pi$
$812$	−3.80385	−0.133489
$813$	0	0
$814$	3.80385	0.133325
$815$	−4.39230	−0.153856
$816$	0	0
$817$	19.8564	0.694688
$818$	−20.6603	−0.722369
$819$	0	0
$820$	11.1962	0.390987
$821$	28.6410	0.999578	0.499789	−	0.866147i	$-0.333411\pi$
$821$	28.6410	0.999578	0.499789	+	0.866147i	$0.333411\pi$
$822$	0	0
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	−6.19615	−0.215853
$825$	0	0
$826$	−17.5692	−0.611311
$827$	44.1051	1.53369	0.766843	−	0.641835i	$-0.221827\pi$
$827$	44.1051	1.53369	0.766843	+	0.641835i	$0.221827\pi$
$828$	0	0
$829$	39.9808	1.38859	0.694295	−	0.719691i	$-0.255716\pi$
$829$	39.9808	1.38859	0.694295	+	0.719691i	$0.255716\pi$
$830$	−9.80385	−0.340297
$831$	0	0
$832$	0	0
$833$	28.0192	0.970809
$834$	0	0
$835$	−16.3923	−0.567279
$836$	−6.00000	−0.207514
$837$	0	0
$838$	−4.39230	−0.151730
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	−6.46410	−0.222768
$843$	0	0
$844$	−24.3923	−0.839618
$845$	0	0
$846$	0	0
$847$	11.9090	0.409197
$848$	−3.00000	−0.103020
$849$	0	0
$850$	−10.3923	−0.356453
$851$	−24.5885	−0.842881
$852$	0	0
$853$	−9.00000	−0.308154	−0.154077	−	0.988059i	$-0.549240\pi$
$853$	−9.00000	−0.308154	−0.154077	+	0.988059i	$0.549240\pi$
$854$	19.2679	0.659336
$855$	0	0
$856$	−2.19615	−0.0750629
$857$	−18.3731	−0.627612	−0.313806	−	0.949487i	$-0.601604\pi$
$857$	−18.3731	−0.627612	−0.313806	+	0.949487i	$0.601604\pi$
$858$	0	0
$859$	−20.5885	−0.702469	−0.351235	−	0.936288i	$-0.614238\pi$
$859$	−20.5885	−0.702469	−0.351235	+	0.936288i	$0.614238\pi$
$860$	−7.26795	−0.247835
$861$	0	0
$862$	−38.1962	−1.30097
$863$	49.5167	1.68557	0.842783	−	0.538253i	$-0.180915\pi$
$863$	49.5167	1.68557	0.842783	+	0.538253i	$0.180915\pi$
$864$	0	0
$865$	7.60770	0.258669
$866$	−7.78461	−0.264532
$867$	0	0
$868$	12.0000	0.407307
$869$	10.6410	0.360972
$870$	0	0
$871$	0	0
$872$	−4.39230	−0.148742
$873$	0	0
$874$	38.7846	1.31191
$875$	15.3731	0.519705
$876$	0	0
$877$	−22.6077	−0.763408	−0.381704	−	0.924285i	$-0.624663\pi$
$877$	−22.6077	−0.763408	−0.381704	+	0.924285i	$0.624663\pi$
$878$	14.5885	0.492337
$879$	0	0
$880$	2.19615	0.0740323
$881$	13.9808	0.471024	0.235512	−	0.971871i	$-0.424323\pi$
$881$	13.9808	0.471024	0.235512	+	0.971871i	$0.424323\pi$
$882$	0	0
$883$	16.7846	0.564847	0.282424	−	0.959290i	$-0.408862\pi$
$883$	16.7846	0.564847	0.282424	+	0.959290i	$0.408862\pi$
$884$	0	0
$885$	0	0
$886$	16.3923	0.550710
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	0	0
$889$	5.07180	0.170103
$890$	−16.3923	−0.549471
$891$	0	0
$892$	5.07180	0.169816
$893$	22.3923	0.749330
$894$	0	0
$895$	−3.80385	−0.127149
$896$	1.26795	0.0423592
$897$	0	0
$898$	−26.5359	−0.885514
$899$	−28.3923	−0.946936
$900$	0	0
$901$	15.5885	0.519327
$902$	−8.19615	−0.272902
$903$	0	0
$904$	0.803848	0.0267356
$905$	33.9282	1.12781
$906$	0	0
$907$	21.1769	0.703168	0.351584	−	0.936156i	$-0.385643\pi$
$907$	21.1769	0.703168	0.351584	+	0.936156i	$0.385643\pi$
$908$	−20.1962	−0.670233
$909$	0	0
$910$	0	0
$911$	25.1769	0.834148	0.417074	−	0.908872i	$-0.363056\pi$
$911$	25.1769	0.834148	0.417074	+	0.908872i	$0.363056\pi$
$912$	0	0
$913$	7.17691	0.237521
$914$	−31.9808	−1.05783
$915$	0	0
$916$	7.85641	0.259583
$917$	−5.56922	−0.183912
$918$	0	0
$919$	11.6077	0.382903	0.191451	−	0.981502i	$-0.438681\pi$
$919$	11.6077	0.382903	0.191451	+	0.981502i	$0.438681\pi$
$920$	−14.1962	−0.468033
$921$	0	0
$922$	−31.9808	−1.05323
$923$	0	0
$924$	0	0
$925$	6.00000	0.197279
$926$	−15.8038	−0.519347
$927$	0	0
$928$	−3.00000	−0.0984798
$929$	−55.3923	−1.81736	−0.908681	−	0.417491i	$-0.862910\pi$
$929$	−55.3923	−1.81736	−0.908681	+	0.417491i	$0.862910\pi$
$930$	0	0
$931$	25.5167	0.836275
$932$	18.0000	0.589610
$933$	0	0
$934$	5.41154	0.177071
$935$	−11.4115	−0.373197
$936$	0	0
$937$	−15.3923	−0.502845	−0.251422	−	0.967877i	$-0.580898\pi$
$937$	−15.3923	−0.502845	−0.251422	+	0.967877i	$0.580898\pi$
$938$	−9.21539	−0.300893
$939$	0	0
$940$	−8.19615	−0.267329
$941$	−38.7846	−1.26434	−0.632171	−	0.774829i	$-0.717836\pi$
$941$	−38.7846	−1.26434	−0.632171	+	0.774829i	$0.717836\pi$
$942$	0	0
$943$	52.9808	1.72529
$944$	−13.8564	−0.450988
$945$	0	0
$946$	5.32051	0.172985
$947$	−29.0718	−0.944706	−0.472353	−	0.881409i	$-0.656595\pi$
$947$	−29.0718	−0.944706	−0.472353	+	0.881409i	$0.656595\pi$
$948$	0	0
$949$	0	0
$950$	−9.46410	−0.307056
$951$	0	0
$952$	−6.58846	−0.213533
$953$	24.0000	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$954$	0	0
$955$	36.0000	1.16493
$956$	6.58846	0.213086
$957$	0	0
$958$	0.679492	0.0219534
$959$	11.4115	0.368498
$960$	0	0
$961$	58.5692	1.88933
$962$	0	0
$963$	0	0
$964$	−11.1962	−0.360604
$965$	40.1769	1.29334
$966$	0	0
$967$	39.1244	1.25815	0.629077	−	0.777343i	$-0.283433\pi$
$967$	39.1244	1.25815	0.629077	+	0.777343i	$0.283433\pi$
$968$	9.39230	0.301880
$969$	0	0
$970$	10.3923	0.333677
$971$	−49.1769	−1.57816	−0.789081	−	0.614289i	$-0.789443\pi$
$971$	−49.1769	−1.57816	−0.789081	+	0.614289i	$0.789443\pi$
$972$	0	0
$973$	5.07180	0.162594
$974$	−15.1244	−0.484616
$975$	0	0
$976$	15.1962	0.486417
$977$	10.8564	0.347327	0.173664	−	0.984805i	$-0.444439\pi$
$977$	10.8564	0.347327	0.173664	+	0.984805i	$0.444439\pi$
$978$	0	0
$979$	12.0000	0.383522
$980$	−9.33975	−0.298347
$981$	0	0
$982$	−30.5885	−0.976117
$983$	20.7846	0.662926	0.331463	−	0.943468i	$-0.392458\pi$
$983$	20.7846	0.662926	0.331463	+	0.943468i	$0.392458\pi$
$984$	0	0
$985$	−12.0000	−0.382352
$986$	15.5885	0.496438
$987$	0	0
$988$	0	0
$989$	−34.3923	−1.09361
$990$	0	0
$991$	−43.3731	−1.37779	−0.688895	−	0.724861i	$-0.741904\pi$
$991$	−43.3731	−1.37779	−0.688895	+	0.724861i	$0.741904\pi$
$992$	9.46410	0.300486
$993$	0	0
$994$	−2.78461	−0.0883225
$995$	−39.1244	−1.24033
$996$	0	0
$997$	2.80385	0.0887987	0.0443994	−	0.999014i	$-0.485863\pi$
$997$	2.80385	0.0887987	0.0443994	+	0.999014i	$0.485863\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.2.a.s.1.2		2
3.2	odd	2		1014.2.a.j.1.1		2
12.11	even	2		8112.2.a.bx.1.1		2
13.5	odd	4		3042.2.b.l.1351.3		4
13.6	odd	12		234.2.l.a.127.2		4
13.8	odd	4		3042.2.b.l.1351.2		4
13.11	odd	12		234.2.l.a.199.2		4
13.12	even	2		3042.2.a.v.1.1		2
39.2	even	12		1014.2.i.f.823.2		4
39.5	even	4		1014.2.b.d.337.2		4
39.8	even	4		1014.2.b.d.337.3		4
39.11	even	12		78.2.i.b.43.1	✓	4
39.17	odd	6		1014.2.e.j.991.2		4
39.20	even	12		1014.2.i.f.361.2		4
39.23	odd	6		1014.2.e.j.529.2		4
39.29	odd	6		1014.2.e.h.529.1		4
39.32	even	12		78.2.i.b.49.1	yes	4
39.35	odd	6		1014.2.e.h.991.1		4
39.38	odd	2		1014.2.a.h.1.2		2
52.11	even	12		1872.2.by.k.433.1		4
52.19	even	12		1872.2.by.k.1297.1		4
156.11	odd	12		624.2.bv.d.433.1		4
156.71	odd	12		624.2.bv.d.49.1		4
156.155	even	2		8112.2.a.bq.1.2		2
195.32	odd	12		1950.2.y.a.49.2		4
195.89	even	12		1950.2.bc.c.901.2		4
195.128	odd	12		1950.2.y.a.199.2		4
195.149	even	12		1950.2.bc.c.751.2		4
195.167	odd	12		1950.2.y.h.199.1		4
195.188	odd	12		1950.2.y.h.49.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.2.i.b.43.1	✓	4	39.11	even	12
78.2.i.b.49.1	yes	4	39.32	even	12
234.2.l.a.127.2		4	13.6	odd	12
234.2.l.a.199.2		4	13.11	odd	12
624.2.bv.d.49.1		4	156.71	odd	12
624.2.bv.d.433.1		4	156.11	odd	12
1014.2.a.h.1.2		2	39.38	odd	2
1014.2.a.j.1.1		2	3.2	odd	2
1014.2.b.d.337.2		4	39.5	even	4
1014.2.b.d.337.3		4	39.8	even	4
1014.2.e.h.529.1		4	39.29	odd	6
1014.2.e.h.991.1		4	39.35	odd	6
1014.2.e.j.529.2		4	39.23	odd	6
1014.2.e.j.991.2		4	39.17	odd	6
1014.2.i.f.361.2		4	39.20	even	12
1014.2.i.f.823.2		4	39.2	even	12
1872.2.by.k.433.1		4	52.11	even	12
1872.2.by.k.1297.1		4	52.19	even	12
1950.2.y.a.49.2		4	195.32	odd	12
1950.2.y.a.199.2		4	195.128	odd	12
1950.2.y.h.49.1		4	195.188	odd	12
1950.2.y.h.199.1		4	195.167	odd	12
1950.2.bc.c.751.2		4	195.149	even	12
1950.2.bc.c.901.2		4	195.89	even	12
3042.2.a.s.1.2		2	1.1	even	1	trivial
3042.2.a.v.1.1		2	13.12	even	2
3042.2.b.l.1351.2		4	13.8	odd	4
3042.2.b.l.1351.3		4	13.5	odd	4
8112.2.a.bq.1.2		2	156.155	even	2
8112.2.a.bx.1.1		2	12.11	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:	\( N \)	\(=\)	\( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3042.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.73205 q^{5} -1.26795 q^{7} -1.00000 q^{8} -1.73205 q^{10} +1.26795 q^{11} +1.26795 q^{14} +1.00000 q^{16} -5.19615 q^{17} -4.73205 q^{19} +1.73205 q^{20} -1.26795 q^{22} +8.19615 q^{23} -2.00000 q^{25} -1.26795 q^{28} +3.00000 q^{29} -9.46410 q^{31} -1.00000 q^{32} +5.19615 q^{34} -2.19615 q^{35} -3.00000 q^{37} +4.73205 q^{38} -1.73205 q^{40} +6.46410 q^{41} -4.19615 q^{43} +1.26795 q^{44} -8.19615 q^{46} -4.73205 q^{47} -5.39230 q^{49} +2.00000 q^{50} -3.00000 q^{53} +2.19615 q^{55} +1.26795 q^{56} -3.00000 q^{58} -13.8564 q^{59} +15.1962 q^{61} +9.46410 q^{62} +1.00000 q^{64} -7.26795 q^{67} -5.19615 q^{68} +2.19615 q^{70} -2.19615 q^{71} -12.1244 q^{73} +3.00000 q^{74} -4.73205 q^{76} -1.60770 q^{77} +8.39230 q^{79} +1.73205 q^{80} -6.46410 q^{82} +5.66025 q^{83} -9.00000 q^{85} +4.19615 q^{86} -1.26795 q^{88} +9.46410 q^{89} +8.19615 q^{92} +4.73205 q^{94} -8.19615 q^{95} -6.00000 q^{97} +5.39230 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 6 q^{7} - 2 q^{8} + 6 q^{11} + 6 q^{14} + 2 q^{16} - 6 q^{19} - 6 q^{22} + 6 q^{23} - 4 q^{25} - 6 q^{28} + 6 q^{29} - 12 q^{31} - 2 q^{32} + 6 q^{35} - 6 q^{37} + 6 q^{38} + 6 q^{41}+ \cdots - 10 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 6 * q^7 - 2 * q^8 + 6 * q^11 + 6 * q^14 + 2 * q^16 - 6 * q^19 - 6 * q^22 + 6 * q^23 - 4 * q^25 - 6 * q^28 + 6 * q^29 - 12 * q^31 - 2 * q^32 + 6 * q^35 - 6 * q^37 + 6 * q^38 + 6 * q^41 + 2 * q^43 + 6 * q^44 - 6 * q^46 - 6 * q^47 + 10 * q^49 + 4 * q^50 - 6 * q^53 - 6 * q^55 + 6 * q^56 - 6 * q^58 + 20 * q^61 + 12 * q^62 + 2 * q^64 - 18 * q^67 - 6 * q^70 + 6 * q^71 + 6 * q^74 - 6 * q^76 - 24 * q^77 - 4 * q^79 - 6 * q^82 - 6 * q^83 - 18 * q^85 - 2 * q^86 - 6 * q^88 + 12 * q^89 + 6 * q^92 + 6 * q^94 - 6 * q^95 - 12 * q^97 - 10 * q^98

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