LMFDB - Embedded newform 3042.2.b.l.1351.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3042,2,Mod(1351,3042)]

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, 1]))

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

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

chi := DirichletCharacter("3042.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3042 = 2 \cdot 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3042.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$24.2904922949$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 78)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.2
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	3042.1351
Dual form			3042.2.b.l.1351.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +1.73205i q^{5} +1.26795i q^{7} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} +1.73205i q^{5} +1.26795i q^{7} +1.00000i q^{8} +1.73205 q^{10} -1.26795i q^{11} +1.26795 q^{14} +1.00000 q^{16} +5.19615 q^{17} -4.73205i q^{19} -1.73205i q^{20} -1.26795 q^{22} -8.19615 q^{23} +2.00000 q^{25} -1.26795i q^{28} +3.00000 q^{29} -9.46410i q^{31} -1.00000i q^{32} -5.19615i q^{34} -2.19615 q^{35} +3.00000i q^{37} -4.73205 q^{38} -1.73205 q^{40} +6.46410i q^{41} +4.19615 q^{43} +1.26795i q^{44} +8.19615i q^{46} +4.73205i q^{47} +5.39230 q^{49} -2.00000i q^{50} -3.00000 q^{53} +2.19615 q^{55} -1.26795 q^{56} -3.00000i q^{58} +13.8564i q^{59} +15.1962 q^{61} -9.46410 q^{62} -1.00000 q^{64} -7.26795i q^{67} -5.19615 q^{68} +2.19615i q^{70} -2.19615i q^{71} +12.1244i q^{73} +3.00000 q^{74} +4.73205i q^{76} +1.60770 q^{77} +8.39230 q^{79} +1.73205i q^{80} +6.46410 q^{82} +5.66025i q^{83} +9.00000i q^{85} -4.19615i q^{86} +1.26795 q^{88} -9.46410i q^{89} +8.19615 q^{92} +4.73205 q^{94} +8.19615 q^{95} -6.00000i q^{97} -5.39230i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} + 12 q^{14} + 4 q^{16} - 12 q^{22} - 12 q^{23} + 8 q^{25} + 12 q^{29} + 12 q^{35} - 12 q^{38} - 4 q^{43} - 20 q^{49} - 12 q^{53} - 12 q^{55} - 12 q^{56} + 40 q^{61} - 24 q^{62} - 4 q^{64} + 12 q^{74} + 48 q^{77} - 8 q^{79} + 12 q^{82} + 12 q^{88} + 12 q^{92} + 12 q^{94} + 12 q^{95}+O(q^{100}) $ 4 * q - 4 * q^4 + 12 * q^14 + 4 * q^16 - 12 * q^22 - 12 * q^23 + 8 * q^25 + 12 * q^29 + 12 * q^35 - 12 * q^38 - 4 * q^43 - 20 * q^49 - 12 * q^53 - 12 * q^55 - 12 * q^56 + 40 * q^61 - 24 * q^62 - 4 * q^64 + 12 * q^74 + 48 * q^77 - 8 * q^79 + 12 * q^82 + 12 * q^88 + 12 * q^92 + 12 * q^94 + 12 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3042\mathbb{Z}\right)^\times$.

$n$	$677$	$847$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	1.73205i	0.774597i	0.921954	+	0.387298i	$0.126592\pi$
$5$	1.73205i	0.774597i	−0.921954	+	0.387298i	$0.873408\pi$
$6$	0	0
$7$	1.26795i	0.479240i	0.970867	+	0.239620i	$0.0770228\pi$
$7$	1.26795i	0.479240i	−0.970867	+	0.239620i	$0.922977\pi$
$8$	1.00000i	0.353553i
$9$	0	0
$10$	1.73205	0.547723
$11$	− 1.26795i	− 0.382301i	−0.981561	−	0.191151i	$-0.938778\pi$
$11$	− 1.26795i	− 0.382301i	0.981561	−	0.191151i	$-0.0612219\pi$
$12$	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	$-0.283003\pi$
$17$	5.19615	1.26025	0.630126	+	0.776493i	$0.283003\pi$
$18$	0	0
$19$	− 4.73205i	− 1.08561i	−0.839860	−	0.542803i	$-0.817363\pi$
$19$	− 4.73205i	− 1.08561i	0.839860	−	0.542803i	$-0.182637\pi$
$20$	− 1.73205i	− 0.387298i
$21$	0	0
$22$	−1.26795	−0.270328
$23$	−8.19615	−1.70902	−0.854508	−	0.519438i	$-0.826141\pi$
$23$	−8.19615	−1.70902	−0.854508	+	0.519438i	$0.826141\pi$
$24$	0	0
$25$	2.00000	0.400000
$26$	0	0
$27$	0	0
$28$	− 1.26795i	− 0.239620i
$29$	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
$31$	− 9.46410i	− 1.69980i	−0.526942	−	0.849901i	$-0.676661\pi$
$31$	− 9.46410i	− 1.69980i	0.526942	−	0.849901i	$-0.323339\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	− 5.19615i	− 0.891133i
$35$	−2.19615	−0.371218
$36$	0	0
$37$	3.00000i	0.493197i	0.969118	+	0.246598i	$0.0793129\pi$
$37$	3.00000i	0.493197i	−0.969118	+	0.246598i	$0.920687\pi$
$38$	−4.73205	−0.767640
$39$	0	0
$40$	−1.73205	−0.273861
$41$	6.46410i	1.00952i	0.863259	+	0.504762i	$0.168420\pi$
$41$	6.46410i	1.00952i	−0.863259	+	0.504762i	$0.831580\pi$
$42$	0	0
$43$	4.19615	0.639907	0.319954	−	0.947433i	$-0.396333\pi$
$43$	4.19615	0.639907	0.319954	+	0.947433i	$0.396333\pi$
$44$	1.26795i	0.191151i
$45$	0	0
$46$	8.19615i	1.20846i
$47$	4.73205i	0.690241i	0.938558	+	0.345120i	$0.112162\pi$
$47$	4.73205i	0.690241i	−0.938558	+	0.345120i	$0.887838\pi$
$48$	0	0
$49$	5.39230	0.770329
$50$	− 2.00000i	− 0.282843i
$51$	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
$58$	− 3.00000i	− 0.393919i
$59$	13.8564i	1.80395i	0.431788	+	0.901975i	$0.357883\pi$
$59$	13.8564i	1.80395i	−0.431788	+	0.901975i	$0.642117\pi$
$60$	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
$67$	− 7.26795i	− 0.887921i	−0.896046	−	0.443961i	$-0.853573\pi$
$67$	− 7.26795i	− 0.887921i	0.896046	−	0.443961i	$-0.146427\pi$
$68$	−5.19615	−0.630126
$69$	0	0
$70$	2.19615i	0.262490i
$71$	− 2.19615i	− 0.260635i	−0.991472	−	0.130318i	$-0.958400\pi$
$71$	− 2.19615i	− 0.260635i	0.991472	−	0.130318i	$-0.0415997\pi$
$72$	0	0
$73$	12.1244i	1.41905i	0.704681	+	0.709524i	$0.251090\pi$
$73$	12.1244i	1.41905i	−0.704681	+	0.709524i	$0.748910\pi$
$74$	3.00000	0.348743
$75$	0	0
$76$	4.73205i	0.542803i
$77$	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	$0.343495\pi$
$80$	1.73205i	0.193649i
$81$	0	0
$82$	6.46410	0.713841
$83$	5.66025i	0.621294i	0.950525	+	0.310647i	$0.100546\pi$
$83$	5.66025i	0.621294i	−0.950525	+	0.310647i	$0.899454\pi$
$84$	0	0
$85$	9.00000i	0.976187i
$86$	− 4.19615i	− 0.452483i
$87$	0	0
$88$	1.26795	0.135164
$89$	− 9.46410i	− 1.00319i	−0.865102	−	0.501596i	$-0.832746\pi$
$89$	− 9.46410i	− 1.00319i	0.865102	−	0.501596i	$-0.167254\pi$
$90$	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
$97$	− 6.00000i	− 0.609208i	−0.952479	−	0.304604i	$-0.901476\pi$
$97$	− 6.00000i	− 0.609208i	0.952479	−	0.304604i	$-0.0985241\pi$
$98$	− 5.39230i	− 0.544705i
$99$	0	0
$100$	−2.00000	−0.200000
$101$	19.3923	1.92961	0.964803	−	0.262973i	$-0.0847030\pi$
$101$	19.3923	1.92961	0.964803	+	0.262973i	$0.0847030\pi$
$102$	0	0
$103$	−6.19615	−0.610525	−0.305263	−	0.952268i	$-0.598744\pi$
$103$	−6.19615	−0.610525	−0.305263	+	0.952268i	$0.598744\pi$
$104$	0	0
$105$	0	0
$106$	3.00000i	0.291386i
$107$	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
$109$	4.39230i	0.420707i	0.977625	+	0.210353i	$0.0674614\pi$
$109$	4.39230i	0.420707i	−0.977625	+	0.210353i	$0.932539\pi$
$110$	− 2.19615i	− 0.209395i
$111$	0	0
$112$	1.26795i	0.119810i
$113$	−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
$115$	− 14.1962i	− 1.32380i
$116$	−3.00000	−0.278543
$117$	0	0
$118$	13.8564	1.27559
$119$	6.58846i	0.603963i
$120$	0	0
$121$	9.39230	0.853846
$122$	− 15.1962i	− 1.37579i
$123$	0	0
$124$	9.46410i	0.849901i
$125$	12.1244i	1.08444i
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	1.00000i	0.0883883i
$129$	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
$136$	5.19615i	0.445566i
$137$	9.00000i	0.768922i	0.923141	+	0.384461i	$0.125613\pi$
$137$	9.00000i	0.768922i	−0.923141	+	0.384461i	$0.874387\pi$
$138$	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
$145$	5.19615i	0.431517i
$146$	12.1244	1.00342
$147$	0	0
$148$	− 3.00000i	− 0.246598i
$149$	6.12436i	0.501727i	0.968023	+	0.250863i	$0.0807145\pi$
$149$	6.12436i	0.501727i	−0.968023	+	0.250863i	$0.919286\pi$
$150$	0	0
$151$	10.7321i	0.873362i	0.899616	+	0.436681i	$0.143846\pi$
$151$	10.7321i	0.873362i	−0.899616	+	0.436681i	$0.856154\pi$
$152$	4.73205	0.383820
$153$	0	0
$154$	− 1.60770i	− 0.129552i
$155$	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	$0.407290\pi$
$158$	− 8.39230i	− 0.667656i
$159$	0	0
$160$	1.73205	0.136931
$161$	− 10.3923i	− 0.819028i
$162$	0	0
$163$	2.53590i	0.198627i	0.995056	+	0.0993134i	$0.0316646\pi$
$163$	2.53590i	0.198627i	−0.995056	+	0.0993134i	$0.968335\pi$
$164$	− 6.46410i	− 0.504762i
$165$	0	0
$166$	5.66025	0.439321
$167$	9.46410i	0.732354i	0.930545	+	0.366177i	$0.119334\pi$
$167$	9.46410i	0.732354i	−0.930545	+	0.366177i	$0.880666\pi$
$168$	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	$-0.553398\pi$
$173$	−4.39230	−0.333941	−0.166970	+	0.985962i	$0.553398\pi$
$174$	0	0
$175$	2.53590i	0.191696i
$176$	− 1.26795i	− 0.0955753i
$177$	0	0
$178$	−9.46410	−0.709364
$179$	2.19615	0.164148	0.0820741	−	0.996626i	$-0.473846\pi$
$179$	2.19615	0.164148	0.0820741	+	0.996626i	$0.473846\pi$
$180$	0	0
$181$	−19.5885	−1.45600	−0.727999	−	0.685578i	$-0.759550\pi$
$181$	−19.5885	−1.45600	−0.727999	+	0.685578i	$0.759550\pi$
$182$	0	0
$183$	0	0
$184$	− 8.19615i	− 0.604228i
$185$	−5.19615	−0.382029
$186$	0	0
$187$	− 6.58846i	− 0.481796i
$188$	− 4.73205i	− 0.345120i
$189$	0	0
$190$	− 8.19615i	− 0.594611i
$191$	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
$193$	− 23.1962i	− 1.66970i	−0.550481	−	0.834848i	$-0.685556\pi$
$193$	− 23.1962i	− 1.66970i	0.550481	−	0.834848i	$-0.314444\pi$
$194$	−6.00000	−0.430775
$195$	0	0
$196$	−5.39230	−0.385165
$197$	− 6.92820i	− 0.493614i	−0.969065	−	0.246807i	$-0.920619\pi$
$197$	− 6.92820i	− 0.493614i	0.969065	−	0.246807i	$-0.0793814\pi$
$198$	0	0
$199$	22.5885	1.60125	0.800627	−	0.599164i	$-0.204500\pi$
$199$	22.5885	1.60125	0.800627	+	0.599164i	$0.204500\pi$
$200$	2.00000i	0.141421i
$201$	0	0
$202$	− 19.3923i	− 1.36444i
$203$	3.80385i	0.266978i
$204$	0	0
$205$	−11.1962	−0.781973
$206$	6.19615i	0.431706i
$207$	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
$214$	− 2.19615i	− 0.150126i
$215$	7.26795i	0.495670i
$216$	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
$223$	5.07180i	0.339633i	0.985476	+	0.169816i	$0.0543174\pi$
$223$	5.07180i	0.339633i	−0.985476	+	0.169816i	$0.945683\pi$
$224$	1.26795	0.0847184
$225$	0	0
$226$	0.803848i	0.0534711i
$227$	− 20.1962i	− 1.34047i	−0.742151	−	0.670233i	$-0.766194\pi$
$227$	− 20.1962i	− 1.34047i	0.742151	−	0.670233i	$-0.233806\pi$
$228$	0	0
$229$	− 7.85641i	− 0.519166i	−0.965721	−	0.259583i	$-0.916415\pi$
$229$	− 7.85641i	− 0.519166i	0.965721	−	0.259583i	$-0.0835851\pi$
$230$	−14.1962	−0.936067
$231$	0	0
$232$	3.00000i	0.196960i
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	−8.19615	−0.534658
$236$	− 13.8564i	− 0.901975i
$237$	0	0
$238$	6.58846	0.427066
$239$	6.58846i	0.426172i	0.977033	+	0.213086i	$0.0683514\pi$
$239$	6.58846i	0.426172i	−0.977033	+	0.213086i	$0.931649\pi$
$240$	0	0
$241$	11.1962i	0.721208i	0.932719	+	0.360604i	$0.117429\pi$
$241$	11.1962i	0.721208i	−0.932719	+	0.360604i	$0.882571\pi$
$242$	− 9.39230i	− 0.603760i
$243$	0	0
$244$	−15.1962	−0.972834
$245$	9.33975i	0.596694i
$246$	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	$-0.326924\pi$
$251$	16.3923	1.03467	0.517337	+	0.855782i	$0.326924\pi$
$252$	0	0
$253$	10.3923i	0.653359i
$254$	− 4.00000i	− 0.250982i
$255$	0	0
$256$	1.00000	0.0625000
$257$	23.1962	1.44694	0.723468	−	0.690358i	$-0.242547\pi$
$257$	23.1962	1.44694	0.723468	+	0.690358i	$0.242547\pi$
$258$	0	0
$259$	−3.80385	−0.236360
$260$	0	0
$261$	0	0
$262$	− 4.39230i	− 0.271357i
$263$	−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
$265$	− 5.19615i	− 0.319197i
$266$	− 6.00000i	− 0.367884i
$267$	0	0
$268$	7.26795i	0.443961i
$269$	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	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	5.19615	0.315063
$273$	0	0
$274$	9.00000	0.543710
$275$	− 2.53590i	− 0.152920i
$276$	0	0
$277$	−4.80385	−0.288635	−0.144318	−	0.989531i	$-0.546099\pi$
$277$	−4.80385	−0.288635	−0.144318	+	0.989531i	$0.546099\pi$
$278$	4.00000i	0.239904i
$279$	0	0
$280$	− 2.19615i	− 0.131245i
$281$	− 17.5359i	− 1.04610i	−0.852301	−	0.523052i	$-0.824793\pi$
$281$	− 17.5359i	− 1.04610i	0.852301	−	0.523052i	$-0.175207\pi$
$282$	0	0
$283$	19.8038	1.17722	0.588608	−	0.808418i	$-0.299676\pi$
$283$	19.8038	1.17722	0.588608	+	0.808418i	$0.299676\pi$
$284$	2.19615i	0.130318i
$285$	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
$292$	− 12.1244i	− 0.709524i
$293$	− 2.66025i	− 0.155414i	−0.996976	−	0.0777069i	$-0.975240\pi$
$293$	− 2.66025i	− 0.155414i	0.996976	−	0.0777069i	$-0.0247598\pi$
$294$	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
$301$	5.32051i	0.306669i
$302$	10.7321	0.617560
$303$	0	0
$304$	− 4.73205i	− 0.271402i
$305$	26.3205i	1.50711i
$306$	0	0
$307$	7.26795i	0.414804i	0.978256	+	0.207402i	$0.0665008\pi$
$307$	7.26795i	0.414804i	−0.978256	+	0.207402i	$0.933499\pi$
$308$	−1.60770	−0.0916069
$309$	0	0
$310$	− 16.3923i	− 0.931020i
$311$	−8.19615	−0.464761	−0.232381	−	0.972625i	$-0.574651\pi$
$311$	−8.19615	−0.464761	−0.232381	+	0.972625i	$0.574651\pi$
$312$	0	0
$313$	−3.60770	−0.203919	−0.101959	−	0.994789i	$-0.532511\pi$
$313$	−3.60770	−0.203919	−0.101959	+	0.994789i	$0.532511\pi$
$314$	− 7.19615i	− 0.406102i
$315$	0	0
$316$	−8.39230	−0.472104
$317$	− 18.1244i	− 1.01797i	−0.860777	−	0.508983i	$-0.830022\pi$
$317$	− 18.1244i	− 1.01797i	0.860777	−	0.508983i	$-0.169978\pi$
$318$	0	0
$319$	− 3.80385i	− 0.212975i
$320$	− 1.73205i	− 0.0968246i
$321$	0	0
$322$	−10.3923	−0.579141
$323$	− 24.5885i	− 1.36814i
$324$	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
$331$	− 12.0000i	− 0.659580i	−0.944054	−	0.329790i	$-0.893022\pi$
$331$	− 12.0000i	− 0.659580i	0.944054	−	0.329790i	$-0.106978\pi$
$332$	− 5.66025i	− 0.310647i
$333$	0	0
$334$	9.46410	0.517853
$335$	12.5885	0.687781
$336$	0	0
$337$	31.0000	1.68868	0.844339	−	0.535810i	$-0.179994\pi$
$337$	31.0000	1.68868	0.844339	+	0.535810i	$0.179994\pi$
$338$	0	0
$339$	0	0
$340$	− 9.00000i	− 0.488094i
$341$	−12.0000	−0.649836
$342$	0	0
$343$	15.7128i	0.848412i
$344$	4.19615i	0.226241i
$345$	0	0
$346$	4.39230i	0.236132i
$347$	−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
$349$	9.46410i	0.506602i	0.967388	+	0.253301i	$0.0815163\pi$
$349$	9.46410i	0.506602i	−0.967388	+	0.253301i	$0.918484\pi$
$350$	2.53590	0.135549
$351$	0	0
$352$	−1.26795	−0.0675819
$353$	− 35.7846i	− 1.90462i	−0.305128	−	0.952311i	$-0.598699\pi$
$353$	− 35.7846i	− 1.90462i	0.305128	−	0.952311i	$-0.401301\pi$
$354$	0	0
$355$	3.80385	0.201887
$356$	9.46410i	0.501596i
$357$	0	0
$358$	− 2.19615i	− 0.116070i
$359$	16.0526i	0.847222i	0.905844	+	0.423611i	$0.139238\pi$
$359$	16.0526i	0.847222i	−0.905844	+	0.423611i	$0.860762\pi$
$360$	0	0
$361$	−3.39230	−0.178542
$362$	19.5885i	1.02955i
$363$	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
$370$	5.19615i	0.270135i
$371$	− 3.80385i	− 0.197486i
$372$	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
$379$	− 30.2487i	− 1.55377i	−0.629641	−	0.776886i	$-0.716798\pi$
$379$	− 30.2487i	− 1.55377i	0.629641	−	0.776886i	$-0.283202\pi$
$380$	−8.19615	−0.420454
$381$	0	0
$382$	− 20.7846i	− 1.06343i
$383$	− 23.3205i	− 1.19162i	−0.803125	−	0.595811i	$-0.796831\pi$
$383$	− 23.3205i	− 1.19162i	0.803125	−	0.595811i	$-0.203169\pi$
$384$	0	0
$385$	2.78461i	0.141917i
$386$	−23.1962	−1.18065
$387$	0	0
$388$	6.00000i	0.304604i
$389$	7.39230	0.374805	0.187402	−	0.982283i	$-0.439993\pi$
$389$	7.39230	0.374805	0.187402	+	0.982283i	$0.439993\pi$
$390$	0	0
$391$	−42.5885	−2.15379
$392$	5.39230i	0.272353i
$393$	0	0
$394$	−6.92820	−0.349038
$395$	14.5359i	0.731380i
$396$	0	0
$397$	4.39230i	0.220443i	0.993907	+	0.110222i	$0.0351561\pi$
$397$	4.39230i	0.220443i	−0.993907	+	0.110222i	$0.964844\pi$
$398$	− 22.5885i	− 1.13226i
$399$	0	0
$400$	2.00000	0.100000
$401$	− 21.0000i	− 1.04869i	−0.851506	−	0.524345i	$-0.824310\pi$
$401$	− 21.0000i	− 1.04869i	0.851506	−	0.524345i	$-0.175690\pi$
$402$	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
$409$	20.6603i	1.02158i	0.859704	+	0.510792i	$0.170648\pi$
$409$	20.6603i	1.02158i	−0.859704	+	0.510792i	$0.829352\pi$
$410$	11.1962i	0.552939i
$411$	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
$418$	6.00000i	0.293470i
$419$	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
$421$	6.46410i	0.315041i	0.987516	+	0.157521i	$0.0503500\pi$
$421$	6.46410i	0.315041i	−0.987516	+	0.157521i	$0.949650\pi$
$422$	24.3923i	1.18740i
$423$	0	0
$424$	− 3.00000i	− 0.145693i
$425$	10.3923	0.504101
$426$	0	0
$427$	19.2679i	0.932441i
$428$	−2.19615	−0.106155
$429$	0	0
$430$	7.26795	0.350492
$431$	38.1962i	1.83984i	0.392101	+	0.919922i	$0.371748\pi$
$431$	38.1962i	1.83984i	−0.392101	+	0.919922i	$0.628252\pi$
$432$	0	0
$433$	−7.78461	−0.374104	−0.187052	−	0.982350i	$-0.559893\pi$
$433$	−7.78461	−0.374104	−0.187052	+	0.982350i	$0.559893\pi$
$434$	− 12.0000i	− 0.576018i
$435$	0	0
$436$	− 4.39230i	− 0.210353i
$437$	38.7846i	1.85532i
$438$	0	0
$439$	14.5885	0.696269	0.348135	−	0.937445i	$-0.386815\pi$
$439$	14.5885	0.696269	0.348135	+	0.937445i	$0.386815\pi$
$440$	2.19615i	0.104697i
$441$	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
$448$	− 1.26795i	− 0.0599050i
$449$	− 26.5359i	− 1.25231i	−0.779700	−	0.626153i	$-0.784628\pi$
$449$	− 26.5359i	− 1.25231i	0.779700	−	0.626153i	$-0.215372\pi$
$450$	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
$457$	31.9808i	1.49600i	0.663700	+	0.747998i	$0.268985\pi$
$457$	31.9808i	1.49600i	−0.663700	+	0.747998i	$0.731015\pi$
$458$	−7.85641	−0.367106
$459$	0	0
$460$	14.1962i	0.661899i
$461$	31.9808i	1.48949i	0.667348	+	0.744746i	$0.267430\pi$
$461$	31.9808i	1.48949i	−0.667348	+	0.744746i	$0.732570\pi$
$462$	0	0
$463$	− 15.8038i	− 0.734467i	−0.930129	−	0.367234i	$-0.880305\pi$
$463$	− 15.8038i	− 0.734467i	0.930129	−	0.367234i	$-0.119695\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	18.0000i	0.833834i
$467$	5.41154	0.250416	0.125208	−	0.992130i	$-0.460040\pi$
$467$	5.41154	0.250416	0.125208	+	0.992130i	$0.460040\pi$
$468$	0	0
$469$	9.21539	0.425527
$470$	8.19615i	0.378060i
$471$	0	0
$472$	−13.8564	−0.637793
$473$	− 5.32051i	− 0.244637i
$474$	0	0
$475$	− 9.46410i	− 0.434243i
$476$	− 6.58846i	− 0.301981i
$477$	0	0
$478$	6.58846	0.301349
$479$	0.679492i	0.0310468i	0.999880	+	0.0155234i	$0.00494145\pi$
$479$	0.679492i	0.0310468i	−0.999880	+	0.0155234i	$0.995059\pi$
$480$	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
$487$	15.1244i	0.685350i	0.939454	+	0.342675i	$0.111333\pi$
$487$	15.1244i	0.685350i	−0.939454	+	0.342675i	$0.888667\pi$
$488$	15.1962i	0.687897i
$489$	0	0
$490$	9.33975	0.421927
$491$	−30.5885	−1.38044	−0.690219	−	0.723601i	$-0.742486\pi$
$491$	−30.5885	−1.38044	−0.690219	+	0.723601i	$0.742486\pi$
$492$	0	0
$493$	15.5885	0.702069
$494$	0	0
$495$	0	0
$496$	− 9.46410i	− 0.424951i
$497$	2.78461	0.124907
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	− 12.1244i	− 0.542218i
$501$	0	0
$502$	− 16.3923i	− 0.731624i
$503$	−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
$505$	33.5885i	1.49467i
$506$	10.3923	0.461994
$507$	0	0
$508$	−4.00000	−0.177471
$509$	26.6603i	1.18169i	0.806783	+	0.590847i	$0.201207\pi$
$509$	26.6603i	1.18169i	−0.806783	+	0.590847i	$0.798793\pi$
$510$	0	0
$511$	−15.3731	−0.680064
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	− 23.1962i	− 1.02314i
$515$	− 10.7321i	− 0.472911i
$516$	0	0
$517$	6.00000	0.263880
$518$	3.80385i	0.167131i
$519$	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
$526$	8.19615i	0.357369i
$527$	− 49.1769i	− 2.14218i
$528$	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
$535$	3.80385i	0.164455i
$536$	7.26795	0.313928
$537$	0	0
$538$	− 7.60770i	− 0.327991i
$539$	− 6.83717i	− 0.294498i
$540$	0	0
$541$	− 10.8564i	− 0.466753i	−0.972386	−	0.233377i	$-0.925022\pi$
$541$	− 10.8564i	− 0.466753i	0.972386	−	0.233377i	$-0.0749775\pi$
$542$	0	0
$543$	0	0
$544$	− 5.19615i	− 0.222783i
$545$	−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	$0.471407\pi$
$548$	− 9.00000i	− 0.384461i
$549$	0	0
$550$	−2.53590	−0.108131
$551$	− 14.1962i	− 0.604776i
$552$	0	0
$553$	10.6410i	0.452502i
$554$	4.80385i	0.204096i
$555$	0	0
$556$	4.00000	0.169638
$557$	− 25.7321i	− 1.09030i	−0.838338	−	0.545151i	$-0.816472\pi$
$557$	− 25.7321i	− 1.09030i	0.838338	−	0.545151i	$-0.183528\pi$
$558$	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	$-0.742765\pi$
$563$	−32.7846	−1.38171	−0.690853	+	0.722995i	$0.742765\pi$
$564$	0	0
$565$	− 1.39230i	− 0.0585747i
$566$	− 19.8038i	− 0.832418i
$567$	0	0
$568$	2.19615	0.0921485
$569$	8.78461	0.368270	0.184135	−	0.982901i	$-0.441052\pi$
$569$	8.78461	0.368270	0.184135	+	0.982901i	$0.441052\pi$
$570$	0	0
$571$	24.1962	1.01258	0.506289	−	0.862364i	$-0.331017\pi$
$571$	24.1962	1.01258	0.506289	+	0.862364i	$0.331017\pi$
$572$	0	0
$573$	0	0
$574$	8.19615i	0.342101i
$575$	−16.3923	−0.683606
$576$	0	0
$577$	19.7321i	0.821456i	0.911758	+	0.410728i	$0.134725\pi$
$577$	19.7321i	0.821456i	−0.911758	+	0.410728i	$0.865275\pi$
$578$	− 10.0000i	− 0.415945i
$579$	0	0
$580$	− 5.19615i	− 0.215758i
$581$	−7.17691	−0.297749
$582$	0	0
$583$	3.80385i	0.157539i
$584$	−12.1244	−0.501709
$585$	0	0
$586$	−2.66025	−0.109894
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	−44.7846	−1.84532
$590$	24.0000i	0.988064i
$591$	0	0
$592$	3.00000i	0.123299i
$593$	19.1436i	0.786133i	0.919510	+	0.393067i	$0.128586\pi$
$593$	19.1436i	0.786133i	−0.919510	+	0.393067i	$0.871414\pi$
$594$	0	0
$595$	−11.4115	−0.467828
$596$	− 6.12436i	− 0.250863i
$597$	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
$604$	− 10.7321i	− 0.436681i
$605$	16.2679i	0.661386i
$606$	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
$613$	13.1436i	0.530865i	0.964129	+	0.265432i	$0.0855147\pi$
$613$	13.1436i	0.530865i	−0.964129	+	0.265432i	$0.914485\pi$
$614$	7.26795	0.293311
$615$	0	0
$616$	1.60770i	0.0647759i
$617$	31.3923i	1.26381i	0.775047	+	0.631903i	$0.217726\pi$
$617$	31.3923i	1.26381i	−0.775047	+	0.631903i	$0.782274\pi$
$618$	0	0
$619$	− 28.3923i	− 1.14118i	−0.821234	−	0.570592i	$-0.806714\pi$
$619$	− 28.3923i	− 1.14118i	0.821234	−	0.570592i	$-0.193286\pi$
$620$	−16.3923	−0.658331
$621$	0	0
$622$	8.19615i	0.328636i
$623$	12.0000	0.480770
$624$	0	0
$625$	−11.0000	−0.440000
$626$	3.60770i	0.144192i
$627$	0	0
$628$	−7.19615	−0.287158
$629$	15.5885i	0.621552i
$630$	0	0
$631$	− 1.85641i	− 0.0739024i	−0.999317	−	0.0369512i	$-0.988235\pi$
$631$	− 1.85641i	− 0.0739024i	0.999317	−	0.0369512i	$-0.0117646\pi$
$632$	8.39230i	0.333828i
$633$	0	0
$634$	−18.1244	−0.719810
$635$	6.92820i	0.274937i
$636$	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	$-0.802482\pi$
$641$	−41.1962	−1.62715	−0.813575	+	0.581460i	$0.802482\pi$
$642$	0	0
$643$	− 27.7128i	− 1.09289i	−0.837496	−	0.546443i	$-0.815981\pi$
$643$	− 27.7128i	− 1.09289i	0.837496	−	0.546443i	$-0.184019\pi$
$644$	10.3923i	0.409514i
$645$	0	0
$646$	−24.5885	−0.967420
$647$	−49.1769	−1.93334	−0.966672	−	0.256018i	$-0.917589\pi$
$647$	−49.1769	−1.93334	−0.966672	+	0.256018i	$0.917589\pi$
$648$	0	0
$649$	17.5692	0.689652
$650$	0	0
$651$	0	0
$652$	− 2.53590i	− 0.0993134i
$653$	−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
$655$	7.60770i	0.297257i
$656$	6.46410i	0.252381i
$657$	0	0
$658$	6.00000i	0.233904i
$659$	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
$661$	9.00000i	0.350059i	0.984563	+	0.175030i	$0.0560022\pi$
$661$	9.00000i	0.350059i	−0.984563	+	0.175030i	$0.943998\pi$
$662$	−12.0000	−0.466393
$663$	0	0
$664$	−5.66025	−0.219660
$665$	10.3923i	0.402996i
$666$	0	0
$667$	−24.5885	−0.952069
$668$	− 9.46410i	− 0.366177i
$669$	0	0
$670$	− 12.5885i	− 0.486335i
$671$	− 19.2679i	− 0.743831i
$672$	0	0
$673$	−1.00000	−0.0385472	−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000	−0.0385472	−0.0192736	+	0.999814i	$0.506135\pi$
$674$	− 31.0000i	− 1.19408i
$675$	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
$682$	12.0000i	0.459504i
$683$	27.7128i	1.06040i	0.847872	+	0.530201i	$0.177883\pi$
$683$	27.7128i	1.06040i	−0.847872	+	0.530201i	$0.822117\pi$
$684$	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
$691$	− 25.5167i	− 0.970700i	−0.874320	−	0.485350i	$-0.838692\pi$
$691$	− 25.5167i	− 0.970700i	0.874320	−	0.485350i	$-0.161308\pi$
$692$	4.39230	0.166970
$693$	0	0
$694$	18.5885i	0.705608i
$695$	− 6.92820i	− 0.262802i
$696$	0	0
$697$	33.5885i	1.27225i
$698$	9.46410	0.358222
$699$	0	0
$700$	− 2.53590i	− 0.0958479i
$701$	−16.3923	−0.619129	−0.309564	−	0.950878i	$-0.600183\pi$
$701$	−16.3923	−0.619129	−0.309564	+	0.950878i	$0.600183\pi$
$702$	0	0
$703$	14.1962	0.535418
$704$	1.26795i	0.0477876i
$705$	0	0
$706$	−35.7846	−1.34677
$707$	24.5885i	0.924744i
$708$	0	0
$709$	− 45.2487i	− 1.69935i	−0.527306	−	0.849676i	$-0.676798\pi$
$709$	− 45.2487i	− 1.69935i	0.527306	−	0.849676i	$-0.323202\pi$
$710$	− 3.80385i	− 0.142756i
$711$	0	0
$712$	9.46410	0.354682
$713$	77.5692i	2.90499i
$714$	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	$-0.299370\pi$
$719$	31.6077	1.17877	0.589384	+	0.807853i	$0.299370\pi$
$720$	0	0
$721$	− 7.85641i	− 0.292588i
$722$	3.39230i	0.126249i
$723$	0	0
$724$	19.5885	0.727999
$725$	6.00000	0.222834
$726$	0	0
$727$	13.8038	0.511956	0.255978	−	0.966683i	$-0.417602\pi$
$727$	13.8038	0.511956	0.255978	+	0.966683i	$0.417602\pi$
$728$	0	0
$729$	0	0
$730$	21.0000i	0.777245i
$731$	21.8038	0.806444
$732$	0	0
$733$	− 20.3205i	− 0.750555i	−0.926912	−	0.375278i	$-0.877547\pi$
$733$	− 20.3205i	− 0.750555i	0.926912	−	0.375278i	$-0.122453\pi$
$734$	13.8038i	0.509509i
$735$	0	0
$736$	8.19615i	0.302114i
$737$	−9.21539	−0.339453
$738$	0	0
$739$	− 5.07180i	− 0.186569i	−0.995639	−	0.0932845i	$-0.970263\pi$
$739$	− 5.07180i	− 0.186569i	0.995639	−	0.0932845i	$-0.0297366\pi$
$740$	5.19615	0.191014
$741$	0	0
$742$	−3.80385	−0.139644
$743$	− 16.3923i	− 0.601375i	−0.953723	−	0.300688i	$-0.902784\pi$
$743$	− 16.3923i	− 0.601375i	0.953723	−	0.300688i	$-0.0972162\pi$
$744$	0	0
$745$	−10.6077	−0.388636
$746$	27.9808i	1.02445i
$747$	0	0
$748$	6.58846i	0.240898i
$749$	2.78461i	0.101747i
$750$	0	0
$751$	−26.9808	−0.984542	−0.492271	−	0.870442i	$-0.663833\pi$
$751$	−26.9808	−0.984542	−0.492271	+	0.870442i	$0.663833\pi$
$752$	4.73205i	0.172560i
$753$	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
$760$	8.19615i	0.297306i
$761$	− 16.3923i	− 0.594221i	−0.954843	−	0.297110i	$-0.903977\pi$
$761$	− 16.3923i	− 0.594221i	0.954843	−	0.297110i	$-0.0960229\pi$
$762$	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
$769$	21.7128i	0.782984i	0.920181	+	0.391492i	$0.128041\pi$
$769$	21.7128i	0.782984i	−0.920181	+	0.391492i	$0.871959\pi$
$770$	2.78461	0.100350
$771$	0	0
$772$	23.1962i	0.834848i
$773$	− 9.21539i	− 0.331455i	−0.986172	−	0.165727i	$-0.947003\pi$
$773$	− 9.21539i	− 0.331455i	0.986172	−	0.165727i	$-0.0529971\pi$
$774$	0	0
$775$	− 18.9282i	− 0.679921i
$776$	6.00000	0.215387
$777$	0	0
$778$	− 7.39230i	− 0.265027i
$779$	30.5885	1.09595
$780$	0	0
$781$	−2.78461	−0.0996412
$782$	42.5885i	1.52296i
$783$	0	0
$784$	5.39230	0.192582
$785$	12.4641i	0.444863i
$786$	0	0
$787$	− 21.4641i	− 0.765113i	−0.923932	−	0.382556i	$-0.875044\pi$
$787$	− 21.4641i	− 0.765113i	0.923932	−	0.382556i	$-0.124956\pi$
$788$	6.92820i	0.246807i
$789$	0	0
$790$	14.5359	0.517164
$791$	− 1.01924i	− 0.0362399i
$792$	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	$-0.466111\pi$
$797$	6.00000	0.212531	0.106265	+	0.994338i	$0.466111\pi$
$798$	0	0
$799$	24.5885i	0.869877i
$800$	− 2.00000i	− 0.0707107i
$801$	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
$808$	19.3923i	0.682219i
$809$	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
$811$	16.3923i	0.575612i	0.957689	+	0.287806i	$0.0929258\pi$
$811$	16.3923i	0.575612i	−0.957689	+	0.287806i	$0.907074\pi$
$812$	− 3.80385i	− 0.133489i
$813$	0	0
$814$	− 3.80385i	− 0.133325i
$815$	−4.39230	−0.153856
$816$	0	0
$817$	− 19.8564i	− 0.694688i
$818$	20.6603	0.722369
$819$	0	0
$820$	11.1962	0.390987
$821$	28.6410i	0.999578i	0.866147	+	0.499789i	$0.166589\pi$
$821$	28.6410i	0.999578i	−0.866147	+	0.499789i	$0.833411\pi$
$822$	0	0
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	− 6.19615i	− 0.215853i
$825$	0	0
$826$	17.5692i	0.611311i
$827$	− 44.1051i	− 1.53369i	−0.641835	−	0.766843i	$-0.721827\pi$
$827$	− 44.1051i	− 1.53369i	0.641835	−	0.766843i	$-0.278173\pi$
$828$	0	0
$829$	−39.9808	−1.38859	−0.694295	−	0.719691i	$-0.744284\pi$
$829$	−39.9808	−1.38859	−0.694295	+	0.719691i	$0.744284\pi$
$830$	9.80385i	0.340297i
$831$	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
$838$	− 4.39230i	− 0.151730i
$839$	12.0000i	0.414286i	0.978311	+	0.207143i	$0.0664165\pi$
$839$	12.0000i	0.414286i	−0.978311	+	0.207143i	$0.933583\pi$
$840$	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
$847$	11.9090i	0.409197i
$848$	−3.00000	−0.103020
$849$	0	0
$850$	− 10.3923i	− 0.356453i
$851$	− 24.5885i	− 0.842881i
$852$	0	0
$853$	9.00000i	0.308154i	0.988059	+	0.154077i	$0.0492404\pi$
$853$	9.00000i	0.308154i	−0.988059	+	0.154077i	$0.950760\pi$
$854$	19.2679	0.659336
$855$	0	0
$856$	2.19615i	0.0750629i
$857$	18.3731	0.627612	0.313806	−	0.949487i	$-0.398396\pi$
$857$	18.3731	0.627612	0.313806	+	0.949487i	$0.398396\pi$
$858$	0	0
$859$	−20.5885	−0.702469	−0.351235	−	0.936288i	$-0.614238\pi$
$859$	−20.5885	−0.702469	−0.351235	+	0.936288i	$0.614238\pi$
$860$	− 7.26795i	− 0.247835i
$861$	0	0
$862$	38.1962	1.30097
$863$	49.5167i	1.68557i	0.538253	+	0.842783i	$0.319085\pi$
$863$	49.5167i	1.68557i	−0.538253	+	0.842783i	$0.680915\pi$
$864$	0	0
$865$	− 7.60770i	− 0.258669i
$866$	7.78461i	0.264532i
$867$	0	0
$868$	−12.0000	−0.407307
$869$	− 10.6410i	− 0.360972i
$870$	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
$877$	− 22.6077i	− 0.763408i	−0.924285	−	0.381704i	$-0.875337\pi$
$877$	− 22.6077i	− 0.763408i	0.924285	−	0.381704i	$-0.124663\pi$
$878$	− 14.5885i	− 0.492337i
$879$	0	0
$880$	2.19615	0.0740323
$881$	−13.9808	−0.471024	−0.235512	−	0.971871i	$-0.575677\pi$
$881$	−13.9808	−0.471024	−0.235512	+	0.971871i	$0.575677\pi$
$882$	0	0
$883$	−16.7846	−0.564847	−0.282424	−	0.959290i	$-0.591138\pi$
$883$	−16.7846	−0.564847	−0.282424	+	0.959290i	$0.591138\pi$
$884$	0	0
$885$	0	0
$886$	16.3923i	0.550710i
$887$	−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
$889$	5.07180i	0.170103i
$890$	− 16.3923i	− 0.549471i
$891$	0	0
$892$	− 5.07180i	− 0.169816i
$893$	22.3923	0.749330
$894$	0	0
$895$	3.80385i	0.127149i
$896$	−1.26795	−0.0423592
$897$	0	0
$898$	−26.5359	−0.885514
$899$	− 28.3923i	− 0.946936i
$900$	0	0
$901$	−15.5885	−0.519327
$902$	− 8.19615i	− 0.272902i
$903$	0	0
$904$	− 0.803848i	− 0.0267356i
$905$	− 33.9282i	− 1.12781i
$906$	0	0
$907$	−21.1769	−0.703168	−0.351584	−	0.936156i	$-0.614357\pi$
$907$	−21.1769	−0.703168	−0.351584	+	0.936156i	$0.614357\pi$
$908$	20.1962i	0.670233i
$909$	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
$916$	7.85641i	0.259583i
$917$	5.56922i	0.183912i
$918$	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
$925$	6.00000i	0.197279i
$926$	−15.8038	−0.519347
$927$	0	0
$928$	− 3.00000i	− 0.0984798i
$929$	− 55.3923i	− 1.81736i	−0.417491	−	0.908681i	$-0.637090\pi$
$929$	− 55.3923i	− 1.81736i	0.417491	−	0.908681i	$-0.362910\pi$
$930$	0	0
$931$	− 25.5167i	− 0.836275i
$932$	18.0000	0.589610
$933$	0	0
$934$	− 5.41154i	− 0.177071i
$935$	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	$0.580898\pi$
$938$	− 9.21539i	− 0.300893i
$939$	0	0
$940$	8.19615	0.267329
$941$	− 38.7846i	− 1.26434i	−0.774829	−	0.632171i	$-0.782164\pi$
$941$	− 38.7846i	− 1.26434i	0.774829	−	0.632171i	$-0.217836\pi$
$942$	0	0
$943$	− 52.9808i	− 1.72529i
$944$	13.8564i	0.450988i
$945$	0	0
$946$	−5.32051	−0.172985
$947$	29.0718i	0.944706i	0.881409	+	0.472353i	$0.156595\pi$
$947$	29.0718i	0.944706i	−0.881409	+	0.472353i	$0.843405\pi$
$948$	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	$-0.627082\pi$
$953$	−24.0000	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$954$	0	0
$955$	36.0000i	1.16493i
$956$	− 6.58846i	− 0.213086i
$957$	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
$964$	− 11.1962i	− 0.360604i
$965$	40.1769	1.29334
$966$	0	0
$967$	39.1244i	1.25815i	0.777343	+	0.629077i	$0.216567\pi$
$967$	39.1244i	1.25815i	−0.777343	+	0.629077i	$0.783433\pi$
$968$	9.39230i	0.301880i
$969$	0	0
$970$	− 10.3923i	− 0.333677i
$971$	−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
$973$	− 5.07180i	− 0.162594i
$974$	15.1244	0.484616
$975$	0	0
$976$	15.1962	0.486417
$977$	10.8564i	0.347327i	0.984805	+	0.173664i	$0.0555605\pi$
$977$	10.8564i	0.347327i	−0.984805	+	0.173664i	$0.944439\pi$
$978$	0	0
$979$	−12.0000	−0.383522
$980$	− 9.33975i	− 0.298347i
$981$	0	0
$982$	30.5885i	0.976117i
$983$	− 20.7846i	− 0.662926i	−0.943468	−	0.331463i	$-0.892458\pi$
$983$	− 20.7846i	− 0.662926i	0.943468	−	0.331463i	$-0.107542\pi$
$984$	0	0
$985$	12.0000	0.382352
$986$	− 15.5885i	− 0.496438i
$987$	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
$994$	− 2.78461i	− 0.0883225i
$995$	39.1244i	1.24033i
$996$	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.b.l.1351.2		4
3.2	odd	2		1014.2.b.d.337.3		4
13.3	even	3		234.2.l.a.199.2		4
13.4	even	6		234.2.l.a.127.2		4
13.5	odd	4		3042.2.a.s.1.2		2
13.8	odd	4		3042.2.a.v.1.1		2
13.12	even	2	inner	3042.2.b.l.1351.3		4
39.2	even	12		1014.2.e.h.529.1		4
39.5	even	4		1014.2.a.j.1.1		2
39.8	even	4		1014.2.a.h.1.2		2
39.11	even	12		1014.2.e.j.529.2		4
39.17	odd	6		78.2.i.b.49.1	yes	4
39.20	even	12		1014.2.e.j.991.2		4
39.23	odd	6		1014.2.i.f.823.2		4
39.29	odd	6		78.2.i.b.43.1	✓	4
39.32	even	12		1014.2.e.h.991.1		4
39.35	odd	6		1014.2.i.f.361.2		4
39.38	odd	2		1014.2.b.d.337.2		4
52.3	odd	6		1872.2.by.k.433.1		4
52.43	odd	6		1872.2.by.k.1297.1		4
156.47	odd	4		8112.2.a.bq.1.2		2
156.83	odd	4		8112.2.a.bx.1.1		2
156.95	even	6		624.2.bv.d.49.1		4
156.107	even	6		624.2.bv.d.433.1		4
195.17	even	12		1950.2.y.a.49.2		4
195.29	odd	6		1950.2.bc.c.901.2		4
195.68	even	12		1950.2.y.a.199.2		4
195.107	even	12		1950.2.y.h.199.1		4
195.134	odd	6		1950.2.bc.c.751.2		4
195.173	even	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	39.29	odd	6
78.2.i.b.49.1	yes	4	39.17	odd	6
234.2.l.a.127.2		4	13.4	even	6
234.2.l.a.199.2		4	13.3	even	3
624.2.bv.d.49.1		4	156.95	even	6
624.2.bv.d.433.1		4	156.107	even	6
1014.2.a.h.1.2		2	39.8	even	4
1014.2.a.j.1.1		2	39.5	even	4
1014.2.b.d.337.2		4	39.38	odd	2
1014.2.b.d.337.3		4	3.2	odd	2
1014.2.e.h.529.1		4	39.2	even	12
1014.2.e.h.991.1		4	39.32	even	12
1014.2.e.j.529.2		4	39.11	even	12
1014.2.e.j.991.2		4	39.20	even	12
1014.2.i.f.361.2		4	39.35	odd	6
1014.2.i.f.823.2		4	39.23	odd	6
1872.2.by.k.433.1		4	52.3	odd	6
1872.2.by.k.1297.1		4	52.43	odd	6
1950.2.y.a.49.2		4	195.17	even	12
1950.2.y.a.199.2		4	195.68	even	12
1950.2.y.h.49.1		4	195.173	even	12
1950.2.y.h.199.1		4	195.107	even	12
1950.2.bc.c.751.2		4	195.134	odd	6
1950.2.bc.c.901.2		4	195.29	odd	6
3042.2.a.s.1.2		2	13.5	odd	4
3042.2.a.v.1.1		2	13.8	odd	4
3042.2.b.l.1351.2		4	1.1	even	1	trivial
3042.2.b.l.1351.3		4	13.12	even	2	inner
8112.2.a.bq.1.2		2	156.47	odd	4
8112.2.a.bx.1.1		2	156.83	odd	4

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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.73205i q^{5} +1.26795i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.73205i q^{5} +1.26795i q^{7} +1.00000i q^{8} +1.73205 q^{10} -1.26795i q^{11} +1.26795 q^{14} +1.00000 q^{16} +5.19615 q^{17} -4.73205i q^{19} -1.73205i q^{20} -1.26795 q^{22} -8.19615 q^{23} +2.00000 q^{25} -1.26795i q^{28} +3.00000 q^{29} -9.46410i q^{31} -1.00000i q^{32} -5.19615i q^{34} -2.19615 q^{35} +3.00000i q^{37} -4.73205 q^{38} -1.73205 q^{40} +6.46410i q^{41} +4.19615 q^{43} +1.26795i q^{44} +8.19615i q^{46} +4.73205i q^{47} +5.39230 q^{49} -2.00000i q^{50} -3.00000 q^{53} +2.19615 q^{55} -1.26795 q^{56} -3.00000i q^{58} +13.8564i q^{59} +15.1962 q^{61} -9.46410 q^{62} -1.00000 q^{64} -7.26795i q^{67} -5.19615 q^{68} +2.19615i q^{70} -2.19615i q^{71} +12.1244i q^{73} +3.00000 q^{74} +4.73205i q^{76} +1.60770 q^{77} +8.39230 q^{79} +1.73205i q^{80} +6.46410 q^{82} +5.66025i q^{83} +9.00000i q^{85} -4.19615i q^{86} +1.26795 q^{88} -9.46410i q^{89} +8.19615 q^{92} +4.73205 q^{94} +8.19615 q^{95} -6.00000i q^{97} -5.39230i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} + 12 q^{14} + 4 q^{16} - 12 q^{22} - 12 q^{23} + 8 q^{25} + 12 q^{29} + 12 q^{35} - 12 q^{38} - 4 q^{43} - 20 q^{49} - 12 q^{53} - 12 q^{55} - 12 q^{56} + 40 q^{61} - 24 q^{62} - 4 q^{64} + 12 q^{74} + 48 q^{77} - 8 q^{79} + 12 q^{82} + 12 q^{88} + 12 q^{92} + 12 q^{94} + 12 q^{95}+O(q^{100}) \) 4 * q - 4 * q^4 + 12 * q^14 + 4 * q^16 - 12 * q^22 - 12 * q^23 + 8 * q^25 + 12 * q^29 + 12 * q^35 - 12 * q^38 - 4 * q^43 - 20 * q^49 - 12 * q^53 - 12 * q^55 - 12 * q^56 + 40 * q^61 - 24 * q^62 - 4 * q^64 + 12 * q^74 + 48 * q^77 - 8 * q^79 + 12 * q^82 + 12 * q^88 + 12 * q^92 + 12 * q^94 + 12 * q^95

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