LMFDB - Embedded newform 3344.2.a.ba.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1,3344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, 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("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.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:	$26.7019744359$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 14x^{5} + 10x^{4} + 59x^{3} - 27x^{2} - 66x + 30 $ x^7 - x^6 - 14x^5 + 10x^4 + 59x^3 - 27x^2 - 66*x + 30
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 209)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.19313$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.16232 q^{3} +1.08235 q^{5} +1.30958 q^{7} +7.00029 q^{9} +O(q^{10})$	$q-3.16232 q^{3} +1.08235 q^{5} +1.30958 q^{7} +7.00029 q^{9} +1.00000 q^{11} +2.53997 q^{13} -3.42273 q^{15} -5.43892 q^{17} -1.00000 q^{19} -4.14130 q^{21} -3.87095 q^{23} -3.82853 q^{25} -12.6502 q^{27} -2.41412 q^{29} -3.03647 q^{31} -3.16232 q^{33} +1.41741 q^{35} +6.85067 q^{37} -8.03222 q^{39} -6.11344 q^{41} +2.95329 q^{43} +7.57673 q^{45} +12.0923 q^{47} -5.28501 q^{49} +17.1996 q^{51} -0.992927 q^{53} +1.08235 q^{55} +3.16232 q^{57} +14.2251 q^{59} -5.82518 q^{61} +9.16741 q^{63} +2.74913 q^{65} -8.79718 q^{67} +12.2412 q^{69} +2.44975 q^{71} +5.84063 q^{73} +12.1070 q^{75} +1.30958 q^{77} -17.0397 q^{79} +19.0032 q^{81} +7.01303 q^{83} -5.88679 q^{85} +7.63422 q^{87} -9.13103 q^{89} +3.32629 q^{91} +9.60229 q^{93} -1.08235 q^{95} -14.7950 q^{97} +7.00029 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 2 q^{3} + 2 q^{5} - 10 q^{7} + 11 q^{9}+O(q^{10}) $ 7 * q - 2 * q^3 + 2 * q^5 - 10 * q^7 + 11 * q^9	$ 7 q - 2 q^{3} + 2 q^{5} - 10 q^{7} + 11 q^{9} + 7 q^{11} - 4 q^{13} - 12 q^{15} + 2 q^{17} - 7 q^{19} - 14 q^{21} - 10 q^{23} + 9 q^{25} + 4 q^{27} - 18 q^{29} - 24 q^{31} - 2 q^{33} - 8 q^{35} - 24 q^{39} - 12 q^{41} - 2 q^{43} - 4 q^{45} - 8 q^{47} + 17 q^{49} + 24 q^{51} + 2 q^{53} + 2 q^{55} + 2 q^{57} + 10 q^{59} + 14 q^{61} - 14 q^{65} - 8 q^{67} - 6 q^{69} - 10 q^{71} - 6 q^{73} - 26 q^{75} - 10 q^{77} - 52 q^{79} - q^{81} + 10 q^{83} - 12 q^{85} - 6 q^{87} - 12 q^{91} + 2 q^{93} - 2 q^{95} - 24 q^{97} + 11 q^{99}+O(q^{100}) $ 7 * q - 2 * q^3 + 2 * q^5 - 10 * q^7 + 11 * q^9 + 7 * q^11 - 4 * q^13 - 12 * q^15 + 2 * q^17 - 7 * q^19 - 14 * q^21 - 10 * q^23 + 9 * q^25 + 4 * q^27 - 18 * q^29 - 24 * q^31 - 2 * q^33 - 8 * q^35 - 24 * q^39 - 12 * q^41 - 2 * q^43 - 4 * q^45 - 8 * q^47 + 17 * q^49 + 24 * q^51 + 2 * q^53 + 2 * q^55 + 2 * q^57 + 10 * q^59 + 14 * q^61 - 14 * q^65 - 8 * q^67 - 6 * q^69 - 10 * q^71 - 6 * q^73 - 26 * q^75 - 10 * q^77 - 52 * q^79 - q^81 + 10 * q^83 - 12 * q^85 - 6 * q^87 - 12 * q^91 + 2 * q^93 - 2 * q^95 - 24 * q^97 + 11 * 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
$3$	−3.16232	−1.82577	−0.912884	−	0.408218i	$-0.866150\pi$
$3$	−3.16232	−1.82577	−0.912884	+	0.408218i	$0.866150\pi$
$4$	0	0
$5$	1.08235	0.484040	0.242020	−	0.970271i	$-0.422190\pi$
$5$	1.08235	0.484040	0.242020	+	0.970271i	$0.422190\pi$
$6$	0	0
$7$	1.30958	0.494973	0.247487	−	0.968891i	$-0.420395\pi$
$7$	1.30958	0.494973	0.247487	+	0.968891i	$0.420395\pi$
$8$	0	0
$9$	7.00029	2.33343
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	2.53997	0.704462	0.352231	−	0.935913i	$-0.385423\pi$
$13$	2.53997	0.704462	0.352231	+	0.935913i	$0.385423\pi$
$14$	0	0
$15$	−3.42273	−0.883744
$16$	0	0
$17$	−5.43892	−1.31913	−0.659566	−	0.751646i	$-0.729260\pi$
$17$	−5.43892	−1.31913	−0.659566	+	0.751646i	$0.729260\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−4.14130	−0.903706
$22$	0	0
$23$	−3.87095	−0.807148	−0.403574	−	0.914947i	$-0.632232\pi$
$23$	−3.87095	−0.807148	−0.403574	+	0.914947i	$0.632232\pi$
$24$	0	0
$25$	−3.82853	−0.765706
$26$	0	0
$27$	−12.6502	−2.43454
$28$	0	0
$29$	−2.41412	−0.448290	−0.224145	−	0.974556i	$-0.571959\pi$
$29$	−2.41412	−0.448290	−0.224145	+	0.974556i	$0.571959\pi$
$30$	0	0
$31$	−3.03647	−0.545366	−0.272683	−	0.962104i	$-0.587911\pi$
$31$	−3.03647	−0.545366	−0.272683	+	0.962104i	$0.587911\pi$
$32$	0	0
$33$	−3.16232	−0.550490
$34$	0	0
$35$	1.41741	0.239587
$36$	0	0
$37$	6.85067	1.12624	0.563122	−	0.826374i	$-0.309600\pi$
$37$	6.85067	1.12624	0.563122	+	0.826374i	$0.309600\pi$
$38$	0	0
$39$	−8.03222	−1.28618
$40$	0	0
$41$	−6.11344	−0.954759	−0.477380	−	0.878697i	$-0.658413\pi$
$41$	−6.11344	−0.954759	−0.477380	+	0.878697i	$0.658413\pi$
$42$	0	0
$43$	2.95329	0.450373	0.225186	−	0.974316i	$-0.427701\pi$
$43$	2.95329	0.450373	0.225186	+	0.974316i	$0.427701\pi$
$44$	0	0
$45$	7.57673	1.12947
$46$	0	0
$47$	12.0923	1.76385	0.881925	−	0.471391i	$-0.156248\pi$
$47$	12.0923	1.76385	0.881925	+	0.471391i	$0.156248\pi$
$48$	0	0
$49$	−5.28501	−0.755002
$50$	0	0
$51$	17.1996	2.40843
$52$	0	0
$53$	−0.992927	−0.136389	−0.0681945	−	0.997672i	$-0.521724\pi$
$53$	−0.992927	−0.136389	−0.0681945	+	0.997672i	$0.521724\pi$
$54$	0	0
$55$	1.08235	0.145943
$56$	0	0
$57$	3.16232	0.418860
$58$	0	0
$59$	14.2251	1.85195	0.925977	−	0.377580i	$-0.123244\pi$
$59$	14.2251	1.85195	0.925977	+	0.377580i	$0.123244\pi$
$60$	0	0
$61$	−5.82518	−0.745838	−0.372919	−	0.927864i	$-0.621643\pi$
$61$	−5.82518	−0.745838	−0.372919	+	0.927864i	$0.621643\pi$
$62$	0	0
$63$	9.16741	1.15499
$64$	0	0
$65$	2.74913	0.340988
$66$	0	0
$67$	−8.79718	−1.07475	−0.537373	−	0.843345i	$-0.680583\pi$
$67$	−8.79718	−1.07475	−0.537373	+	0.843345i	$0.680583\pi$
$68$	0	0
$69$	12.2412	1.47367
$70$	0	0
$71$	2.44975	0.290732	0.145366	−	0.989378i	$-0.453564\pi$
$71$	2.44975	0.290732	0.145366	+	0.989378i	$0.453564\pi$
$72$	0	0
$73$	5.84063	0.683594	0.341797	−	0.939774i	$-0.388965\pi$
$73$	5.84063	0.683594	0.341797	+	0.939774i	$0.388965\pi$
$74$	0	0
$75$	12.1070	1.39800
$76$	0	0
$77$	1.30958	0.149240
$78$	0	0
$79$	−17.0397	−1.91711	−0.958557	−	0.284902i	$-0.908039\pi$
$79$	−17.0397	−1.91711	−0.958557	+	0.284902i	$0.908039\pi$
$80$	0	0
$81$	19.0032	2.11147
$82$	0	0
$83$	7.01303	0.769780	0.384890	−	0.922963i	$-0.374239\pi$
$83$	7.01303	0.769780	0.384890	+	0.922963i	$0.374239\pi$
$84$	0	0
$85$	−5.88679	−0.638512
$86$	0	0
$87$	7.63422	0.818475
$88$	0	0
$89$	−9.13103	−0.967887	−0.483944	−	0.875099i	$-0.660796\pi$
$89$	−9.13103	−0.967887	−0.483944	+	0.875099i	$0.660796\pi$
$90$	0	0
$91$	3.32629	0.348690
$92$	0	0
$93$	9.60229	0.995712
$94$	0	0
$95$	−1.08235	−0.111046
$96$	0	0
$97$	−14.7950	−1.50220	−0.751100	−	0.660188i	$-0.770476\pi$
$97$	−14.7950	−1.50220	−0.751100	+	0.660188i	$0.770476\pi$
$98$	0	0
$99$	7.00029	0.703556
$100$	0	0
$101$	19.0516	1.89570	0.947852	−	0.318711i	$-0.103250\pi$
$101$	19.0516	1.89570	0.947852	+	0.318711i	$0.103250\pi$
$102$	0	0
$103$	−10.4722	−1.03186	−0.515928	−	0.856632i	$-0.672553\pi$
$103$	−10.4722	−1.03186	−0.515928	+	0.856632i	$0.672553\pi$
$104$	0	0
$105$	−4.48232	−0.437430
$106$	0	0
$107$	−14.4916	−1.40095	−0.700477	−	0.713675i	$-0.747029\pi$
$107$	−14.4916	−1.40095	−0.700477	+	0.713675i	$0.747029\pi$
$108$	0	0
$109$	−6.39919	−0.612931	−0.306465	−	0.951882i	$-0.599146\pi$
$109$	−6.39919	−0.612931	−0.306465	+	0.951882i	$0.599146\pi$
$110$	0	0
$111$	−21.6640	−2.05626
$112$	0	0
$113$	−2.91702	−0.274410	−0.137205	−	0.990543i	$-0.543812\pi$
$113$	−2.91702	−0.274410	−0.137205	+	0.990543i	$0.543812\pi$
$114$	0	0
$115$	−4.18970	−0.390692
$116$	0	0
$117$	17.7806	1.64381
$118$	0	0
$119$	−7.12268	−0.652935
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	19.3327	1.74317
$124$	0	0
$125$	−9.55552	−0.854671
$126$	0	0
$127$	12.1995	1.08253	0.541267	−	0.840851i	$-0.317945\pi$
$127$	12.1995	1.08253	0.541267	+	0.840851i	$0.317945\pi$
$128$	0	0
$129$	−9.33926	−0.822276
$130$	0	0
$131$	−18.7468	−1.63792	−0.818959	−	0.573852i	$-0.805449\pi$
$131$	−18.7468	−1.63792	−0.818959	+	0.573852i	$0.805449\pi$
$132$	0	0
$133$	−1.30958	−0.113555
$134$	0	0
$135$	−13.6919	−1.17841
$136$	0	0
$137$	16.8291	1.43781	0.718904	−	0.695109i	$-0.244644\pi$
$137$	16.8291	1.43781	0.718904	+	0.695109i	$0.244644\pi$
$138$	0	0
$139$	−12.5906	−1.06792	−0.533959	−	0.845511i	$-0.679296\pi$
$139$	−12.5906	−1.06792	−0.533959	+	0.845511i	$0.679296\pi$
$140$	0	0
$141$	−38.2399	−3.22038
$142$	0	0
$143$	2.53997	0.212403
$144$	0	0
$145$	−2.61291	−0.216990
$146$	0	0
$147$	16.7129	1.37846
$148$	0	0
$149$	6.22689	0.510126	0.255063	−	0.966924i	$-0.417904\pi$
$149$	6.22689	0.510126	0.255063	+	0.966924i	$0.417904\pi$
$150$	0	0
$151$	−6.14114	−0.499759	−0.249879	−	0.968277i	$-0.580391\pi$
$151$	−6.14114	−0.499759	−0.249879	+	0.968277i	$0.580391\pi$
$152$	0	0
$153$	−38.0740	−3.07810
$154$	0	0
$155$	−3.28651	−0.263979
$156$	0	0
$157$	19.7489	1.57613	0.788065	−	0.615592i	$-0.211083\pi$
$157$	19.7489	1.57613	0.788065	+	0.615592i	$0.211083\pi$
$158$	0	0
$159$	3.13996	0.249015
$160$	0	0
$161$	−5.06930	−0.399517
$162$	0	0
$163$	5.70383	0.446759	0.223379	−	0.974732i	$-0.428291\pi$
$163$	5.70383	0.446759	0.223379	+	0.974732i	$0.428291\pi$
$164$	0	0
$165$	−3.42273	−0.266459
$166$	0	0
$167$	10.1612	0.786294	0.393147	−	0.919476i	$-0.371386\pi$
$167$	10.1612	0.786294	0.393147	+	0.919476i	$0.371386\pi$
$168$	0	0
$169$	−6.54853	−0.503733
$170$	0	0
$171$	−7.00029	−0.535326
$172$	0	0
$173$	2.22385	0.169076	0.0845382	−	0.996420i	$-0.473058\pi$
$173$	2.22385	0.169076	0.0845382	+	0.996420i	$0.473058\pi$
$174$	0	0
$175$	−5.01375	−0.379004
$176$	0	0
$177$	−44.9845	−3.38124
$178$	0	0
$179$	−12.0997	−0.904376	−0.452188	−	0.891923i	$-0.649356\pi$
$179$	−12.0997	−0.904376	−0.452188	+	0.891923i	$0.649356\pi$
$180$	0	0
$181$	−0.359088	−0.0266908	−0.0133454	−	0.999911i	$-0.504248\pi$
$181$	−0.359088	−0.0266908	−0.0133454	+	0.999911i	$0.504248\pi$
$182$	0	0
$183$	18.4211	1.36173
$184$	0	0
$185$	7.41479	0.545147
$186$	0	0
$187$	−5.43892	−0.397733
$188$	0	0
$189$	−16.5664	−1.20503
$190$	0	0
$191$	−21.3246	−1.54299	−0.771496	−	0.636235i	$-0.780491\pi$
$191$	−21.3246	−1.54299	−0.771496	+	0.636235i	$0.780491\pi$
$192$	0	0
$193$	−14.5451	−1.04698	−0.523491	−	0.852031i	$-0.675370\pi$
$193$	−14.5451	−1.04698	−0.523491	+	0.852031i	$0.675370\pi$
$194$	0	0
$195$	−8.69364	−0.622564
$196$	0	0
$197$	−4.71318	−0.335800	−0.167900	−	0.985804i	$-0.553699\pi$
$197$	−4.71318	−0.335800	−0.167900	+	0.985804i	$0.553699\pi$
$198$	0	0
$199$	1.77204	0.125616	0.0628081	−	0.998026i	$-0.479994\pi$
$199$	1.77204	0.125616	0.0628081	+	0.998026i	$0.479994\pi$
$200$	0	0
$201$	27.8195	1.96224
$202$	0	0
$203$	−3.16147	−0.221892
$204$	0	0
$205$	−6.61686	−0.462141
$206$	0	0
$207$	−27.0977	−1.88342
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−17.8069	−1.22588	−0.612940	−	0.790129i	$-0.710013\pi$
$211$	−17.8069	−1.22588	−0.612940	+	0.790129i	$0.710013\pi$
$212$	0	0
$213$	−7.74692	−0.530810
$214$	0	0
$215$	3.19648	0.217998
$216$	0	0
$217$	−3.97648	−0.269941
$218$	0	0
$219$	−18.4700	−1.24808
$220$	0	0
$221$	−13.8147	−0.929279
$222$	0	0
$223$	9.69081	0.648945	0.324472	−	0.945895i	$-0.394813\pi$
$223$	9.69081	0.648945	0.324472	+	0.945895i	$0.394813\pi$
$224$	0	0
$225$	−26.8008	−1.78672
$226$	0	0
$227$	6.02204	0.399697	0.199848	−	0.979827i	$-0.435955\pi$
$227$	6.02204	0.399697	0.199848	+	0.979827i	$0.435955\pi$
$228$	0	0
$229$	21.7503	1.43730	0.718649	−	0.695373i	$-0.244761\pi$
$229$	21.7503	1.43730	0.718649	+	0.695373i	$0.244761\pi$
$230$	0	0
$231$	−4.14130	−0.272478
$232$	0	0
$233$	−15.6866	−1.02766	−0.513831	−	0.857891i	$-0.671774\pi$
$233$	−15.6866	−1.02766	−0.513831	+	0.857891i	$0.671774\pi$
$234$	0	0
$235$	13.0881	0.853773
$236$	0	0
$237$	53.8850	3.50021
$238$	0	0
$239$	−21.8841	−1.41556	−0.707782	−	0.706430i	$-0.750304\pi$
$239$	−21.8841	−1.41556	−0.707782	+	0.706430i	$0.750304\pi$
$240$	0	0
$241$	5.98510	0.385534	0.192767	−	0.981245i	$-0.438254\pi$
$241$	5.98510	0.385534	0.192767	+	0.981245i	$0.438254\pi$
$242$	0	0
$243$	−22.1437	−1.42052
$244$	0	0
$245$	−5.72021	−0.365451
$246$	0	0
$247$	−2.53997	−0.161615
$248$	0	0
$249$	−22.1775	−1.40544
$250$	0	0
$251$	6.63439	0.418759	0.209379	−	0.977834i	$-0.432856\pi$
$251$	6.63439	0.418759	0.209379	+	0.977834i	$0.432856\pi$
$252$	0	0
$253$	−3.87095	−0.243364
$254$	0	0
$255$	18.6159	1.16578
$256$	0	0
$257$	−9.31591	−0.581111	−0.290555	−	0.956858i	$-0.593840\pi$
$257$	−9.31591	−0.581111	−0.290555	+	0.956858i	$0.593840\pi$
$258$	0	0
$259$	8.97148	0.557460
$260$	0	0
$261$	−16.8995	−1.04605
$262$	0	0
$263$	9.25004	0.570382	0.285191	−	0.958471i	$-0.407943\pi$
$263$	9.25004	0.570382	0.285191	+	0.958471i	$0.407943\pi$
$264$	0	0
$265$	−1.07469	−0.0660177
$266$	0	0
$267$	28.8753	1.76714
$268$	0	0
$269$	−17.0076	−1.03697	−0.518486	−	0.855086i	$-0.673504\pi$
$269$	−17.0076	−1.03697	−0.518486	+	0.855086i	$0.673504\pi$
$270$	0	0
$271$	−7.43527	−0.451660	−0.225830	−	0.974167i	$-0.572509\pi$
$271$	−7.43527	−0.451660	−0.225830	+	0.974167i	$0.572509\pi$
$272$	0	0
$273$	−10.5188	−0.636627
$274$	0	0
$275$	−3.82853	−0.230869
$276$	0	0
$277$	2.31338	0.138998	0.0694989	−	0.997582i	$-0.477860\pi$
$277$	2.31338	0.138998	0.0694989	+	0.997582i	$0.477860\pi$
$278$	0	0
$279$	−21.2562	−1.27257
$280$	0	0
$281$	−5.20248	−0.310354	−0.155177	−	0.987887i	$-0.549595\pi$
$281$	−5.20248	−0.310354	−0.155177	+	0.987887i	$0.549595\pi$
$282$	0	0
$283$	−26.7125	−1.58789	−0.793947	−	0.607987i	$-0.791977\pi$
$283$	−26.7125	−1.58789	−0.793947	+	0.607987i	$0.791977\pi$
$284$	0	0
$285$	3.42273	0.202745
$286$	0	0
$287$	−8.00602	−0.472580
$288$	0	0
$289$	12.5819	0.740110
$290$	0	0
$291$	46.7864	2.74267
$292$	0	0
$293$	−0.937617	−0.0547762	−0.0273881	−	0.999625i	$-0.508719\pi$
$293$	−0.937617	−0.0547762	−0.0273881	+	0.999625i	$0.508719\pi$
$294$	0	0
$295$	15.3965	0.896419
$296$	0	0
$297$	−12.6502	−0.734040
$298$	0	0
$299$	−9.83210	−0.568605
$300$	0	0
$301$	3.86756	0.222922
$302$	0	0
$303$	−60.2473	−3.46112
$304$	0	0
$305$	−6.30486	−0.361015
$306$	0	0
$307$	4.83167	0.275758	0.137879	−	0.990449i	$-0.455971\pi$
$307$	4.83167	0.275758	0.137879	+	0.990449i	$0.455971\pi$
$308$	0	0
$309$	33.1165	1.88393
$310$	0	0
$311$	14.3916	0.816071	0.408036	−	0.912966i	$-0.366214\pi$
$311$	14.3916	0.816071	0.408036	+	0.912966i	$0.366214\pi$
$312$	0	0
$313$	−29.3478	−1.65884	−0.829418	−	0.558628i	$-0.811328\pi$
$313$	−29.3478	−1.65884	−0.829418	+	0.558628i	$0.811328\pi$
$314$	0	0
$315$	9.92231	0.559059
$316$	0	0
$317$	−16.4027	−0.921267	−0.460633	−	0.887590i	$-0.652378\pi$
$317$	−16.4027	−0.921267	−0.460633	+	0.887590i	$0.652378\pi$
$318$	0	0
$319$	−2.41412	−0.135165
$320$	0	0
$321$	45.8271	2.55782
$322$	0	0
$323$	5.43892	0.302630
$324$	0	0
$325$	−9.72436	−0.539411
$326$	0	0
$327$	20.2363	1.11907
$328$	0	0
$329$	15.8358	0.873058
$330$	0	0
$331$	−14.1995	−0.780477	−0.390239	−	0.920714i	$-0.627608\pi$
$331$	−14.1995	−0.780477	−0.390239	+	0.920714i	$0.627608\pi$
$332$	0	0
$333$	47.9567	2.62801
$334$	0	0
$335$	−9.52159	−0.520220
$336$	0	0
$337$	−22.1449	−1.20631	−0.603154	−	0.797624i	$-0.706090\pi$
$337$	−22.1449	−1.20631	−0.603154	+	0.797624i	$0.706090\pi$
$338$	0	0
$339$	9.22456	0.501009
$340$	0	0
$341$	−3.03647	−0.164434
$342$	0	0
$343$	−16.0882	−0.868679
$344$	0	0
$345$	13.2492	0.713312
$346$	0	0
$347$	19.1475	1.02789	0.513946	−	0.857823i	$-0.328183\pi$
$347$	19.1475	1.02789	0.513946	+	0.857823i	$0.328183\pi$
$348$	0	0
$349$	−16.0709	−0.860255	−0.430127	−	0.902768i	$-0.641531\pi$
$349$	−16.0709	−0.860255	−0.430127	+	0.902768i	$0.641531\pi$
$350$	0	0
$351$	−32.1312	−1.71504
$352$	0	0
$353$	10.9142	0.580902	0.290451	−	0.956890i	$-0.406195\pi$
$353$	10.9142	0.580902	0.290451	+	0.956890i	$0.406195\pi$
$354$	0	0
$355$	2.65148	0.140726
$356$	0	0
$357$	22.5242	1.19211
$358$	0	0
$359$	35.4105	1.86889	0.934446	−	0.356104i	$-0.115895\pi$
$359$	35.4105	1.86889	0.934446	+	0.356104i	$0.115895\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−3.16232	−0.165979
$364$	0	0
$365$	6.32158	0.330886
$366$	0	0
$367$	23.3384	1.21825	0.609127	−	0.793073i	$-0.291520\pi$
$367$	23.3384	1.21825	0.609127	+	0.793073i	$0.291520\pi$
$368$	0	0
$369$	−42.7959	−2.22786
$370$	0	0
$371$	−1.30031	−0.0675089
$372$	0	0
$373$	−17.1972	−0.890440	−0.445220	−	0.895421i	$-0.646874\pi$
$373$	−17.1972	−0.890440	−0.445220	+	0.895421i	$0.646874\pi$
$374$	0	0
$375$	30.2176	1.56043
$376$	0	0
$377$	−6.13180	−0.315804
$378$	0	0
$379$	−25.2353	−1.29625	−0.648125	−	0.761534i	$-0.724447\pi$
$379$	−25.2353	−1.29625	−0.648125	+	0.761534i	$0.724447\pi$
$380$	0	0
$381$	−38.5789	−1.97646
$382$	0	0
$383$	−29.6247	−1.51375	−0.756877	−	0.653558i	$-0.773276\pi$
$383$	−29.6247	−1.51375	−0.756877	+	0.653558i	$0.773276\pi$
$384$	0	0
$385$	1.41741	0.0722381
$386$	0	0
$387$	20.6739	1.05091
$388$	0	0
$389$	7.70975	0.390900	0.195450	−	0.980714i	$-0.437383\pi$
$389$	7.70975	0.390900	0.195450	+	0.980714i	$0.437383\pi$
$390$	0	0
$391$	21.0538	1.06474
$392$	0	0
$393$	59.2836	2.99046
$394$	0	0
$395$	−18.4428	−0.927959
$396$	0	0
$397$	8.26973	0.415046	0.207523	−	0.978230i	$-0.433460\pi$
$397$	8.26973	0.415046	0.207523	+	0.978230i	$0.433460\pi$
$398$	0	0
$399$	4.14130	0.207324
$400$	0	0
$401$	1.98297	0.0990247	0.0495123	−	0.998774i	$-0.484233\pi$
$401$	1.98297	0.0990247	0.0495123	+	0.998774i	$0.484233\pi$
$402$	0	0
$403$	−7.71255	−0.384189
$404$	0	0
$405$	20.5680	1.02203
$406$	0	0
$407$	6.85067	0.339575
$408$	0	0
$409$	25.8761	1.27949	0.639746	−	0.768586i	$-0.279040\pi$
$409$	25.8761	1.27949	0.639746	+	0.768586i	$0.279040\pi$
$410$	0	0
$411$	−53.2191	−2.62510
$412$	0	0
$413$	18.6289	0.916667
$414$	0	0
$415$	7.59052	0.372604
$416$	0	0
$417$	39.8154	1.94977
$418$	0	0
$419$	−36.2262	−1.76977	−0.884883	−	0.465813i	$-0.845762\pi$
$419$	−36.2262	−1.76977	−0.884883	+	0.465813i	$0.845762\pi$
$420$	0	0
$421$	−5.91248	−0.288156	−0.144078	−	0.989566i	$-0.546022\pi$
$421$	−5.91248	−0.288156	−0.144078	+	0.989566i	$0.546022\pi$
$422$	0	0
$423$	84.6499	4.11582
$424$	0	0
$425$	20.8231	1.01007
$426$	0	0
$427$	−7.62852	−0.369170
$428$	0	0
$429$	−8.03222	−0.387799
$430$	0	0
$431$	11.1886	0.538935	0.269468	−	0.963009i	$-0.413152\pi$
$431$	11.1886	0.538935	0.269468	+	0.963009i	$0.413152\pi$
$432$	0	0
$433$	13.3903	0.643497	0.321748	−	0.946825i	$-0.395729\pi$
$433$	13.3903	0.643497	0.321748	+	0.946825i	$0.395729\pi$
$434$	0	0
$435$	8.26287	0.396174
$436$	0	0
$437$	3.87095	0.185172
$438$	0	0
$439$	−23.6338	−1.12798	−0.563989	−	0.825782i	$-0.690734\pi$
$439$	−23.6338	−1.12798	−0.563989	+	0.825782i	$0.690734\pi$
$440$	0	0
$441$	−36.9966	−1.76174
$442$	0	0
$443$	25.6914	1.22063	0.610317	−	0.792158i	$-0.291042\pi$
$443$	25.6914	1.22063	0.610317	+	0.792158i	$0.291042\pi$
$444$	0	0
$445$	−9.88293	−0.468496
$446$	0	0
$447$	−19.6914	−0.931373
$448$	0	0
$449$	−9.72223	−0.458820	−0.229410	−	0.973330i	$-0.573680\pi$
$449$	−9.72223	−0.458820	−0.229410	+	0.973330i	$0.573680\pi$
$450$	0	0
$451$	−6.11344	−0.287871
$452$	0	0
$453$	19.4203	0.912444
$454$	0	0
$455$	3.60019	0.168780
$456$	0	0
$457$	−2.04841	−0.0958207	−0.0479103	−	0.998852i	$-0.515256\pi$
$457$	−2.04841	−0.0958207	−0.0479103	+	0.998852i	$0.515256\pi$
$458$	0	0
$459$	68.8036	3.21148
$460$	0	0
$461$	−22.8905	−1.06612	−0.533059	−	0.846078i	$-0.678958\pi$
$461$	−22.8905	−1.06612	−0.533059	+	0.846078i	$0.678958\pi$
$462$	0	0
$463$	−34.9486	−1.62420	−0.812098	−	0.583520i	$-0.801675\pi$
$463$	−34.9486	−1.62420	−0.812098	+	0.583520i	$0.801675\pi$
$464$	0	0
$465$	10.3930	0.481964
$466$	0	0
$467$	−13.7353	−0.635595	−0.317797	−	0.948159i	$-0.602943\pi$
$467$	−13.7353	−0.635595	−0.317797	+	0.948159i	$0.602943\pi$
$468$	0	0
$469$	−11.5206	−0.531971
$470$	0	0
$471$	−62.4523	−2.87765
$472$	0	0
$473$	2.95329	0.135792
$474$	0	0
$475$	3.82853	0.175665
$476$	0	0
$477$	−6.95078	−0.318254
$478$	0	0
$479$	−3.13630	−0.143301	−0.0716506	−	0.997430i	$-0.522827\pi$
$479$	−3.13630	−0.143301	−0.0716506	+	0.997430i	$0.522827\pi$
$480$	0	0
$481$	17.4005	0.793396
$482$	0	0
$483$	16.0308	0.729425
$484$	0	0
$485$	−16.0133	−0.727124
$486$	0	0
$487$	−21.2263	−0.961854	−0.480927	−	0.876761i	$-0.659700\pi$
$487$	−21.2263	−0.961854	−0.480927	+	0.876761i	$0.659700\pi$
$488$	0	0
$489$	−18.0374	−0.815678
$490$	0	0
$491$	−28.5840	−1.28998	−0.644988	−	0.764193i	$-0.723138\pi$
$491$	−28.5840	−1.28998	−0.644988	+	0.764193i	$0.723138\pi$
$492$	0	0
$493$	13.1302	0.591354
$494$	0	0
$495$	7.57673	0.340549
$496$	0	0
$497$	3.20814	0.143905
$498$	0	0
$499$	20.1316	0.901212	0.450606	−	0.892723i	$-0.351208\pi$
$499$	20.1316	0.901212	0.450606	+	0.892723i	$0.351208\pi$
$500$	0	0
$501$	−32.1329	−1.43559
$502$	0	0
$503$	6.37170	0.284100	0.142050	−	0.989859i	$-0.454631\pi$
$503$	6.37170	0.284100	0.142050	+	0.989859i	$0.454631\pi$
$504$	0	0
$505$	20.6204	0.917596
$506$	0	0
$507$	20.7086	0.919700
$508$	0	0
$509$	−9.83799	−0.436061	−0.218031	−	0.975942i	$-0.569963\pi$
$509$	−9.83799	−0.436061	−0.218031	+	0.975942i	$0.569963\pi$
$510$	0	0
$511$	7.64874	0.338360
$512$	0	0
$513$	12.6502	0.558521
$514$	0	0
$515$	−11.3345	−0.499459
$516$	0	0
$517$	12.0923	0.531820
$518$	0	0
$519$	−7.03254	−0.308694
$520$	0	0
$521$	21.1840	0.928088	0.464044	−	0.885812i	$-0.346398\pi$
$521$	21.1840	0.928088	0.464044	+	0.885812i	$0.346398\pi$
$522$	0	0
$523$	−22.1326	−0.967790	−0.483895	−	0.875126i	$-0.660778\pi$
$523$	−22.1326	−0.967790	−0.483895	+	0.875126i	$0.660778\pi$
$524$	0	0
$525$	15.8551	0.691973
$526$	0	0
$527$	16.5151	0.719410
$528$	0	0
$529$	−8.01578	−0.348512
$530$	0	0
$531$	99.5800	4.32141
$532$	0	0
$533$	−15.5280	−0.672592
$534$	0	0
$535$	−15.6849	−0.678117
$536$	0	0
$537$	38.2633	1.65118
$538$	0	0
$539$	−5.28501	−0.227642
$540$	0	0
$541$	−1.83412	−0.0788551	−0.0394275	−	0.999222i	$-0.512553\pi$
$541$	−1.83412	−0.0788551	−0.0394275	+	0.999222i	$0.512553\pi$
$542$	0	0
$543$	1.13555	0.0487312
$544$	0	0
$545$	−6.92613	−0.296683
$546$	0	0
$547$	20.2583	0.866181	0.433091	−	0.901350i	$-0.357423\pi$
$547$	20.2583	0.866181	0.433091	+	0.901350i	$0.357423\pi$
$548$	0	0
$549$	−40.7780	−1.74036
$550$	0	0
$551$	2.41412	0.102845
$552$	0	0
$553$	−22.3147	−0.948920
$554$	0	0
$555$	−23.4480	−0.995311
$556$	0	0
$557$	15.1944	0.643805	0.321903	−	0.946773i	$-0.395678\pi$
$557$	15.1944	0.643805	0.321903	+	0.946773i	$0.395678\pi$
$558$	0	0
$559$	7.50128	0.317270
$560$	0	0
$561$	17.1996	0.726169
$562$	0	0
$563$	22.4513	0.946210	0.473105	−	0.881006i	$-0.343133\pi$
$563$	22.4513	0.946210	0.473105	+	0.881006i	$0.343133\pi$
$564$	0	0
$565$	−3.15722	−0.132825
$566$	0	0
$567$	24.8861	1.04512
$568$	0	0
$569$	−7.54832	−0.316442	−0.158221	−	0.987404i	$-0.550576\pi$
$569$	−7.54832	−0.316442	−0.158221	+	0.987404i	$0.550576\pi$
$570$	0	0
$571$	0.101602	0.00425192	0.00212596	−	0.999998i	$-0.499323\pi$
$571$	0.101602	0.00425192	0.00212596	+	0.999998i	$0.499323\pi$
$572$	0	0
$573$	67.4352	2.81715
$574$	0	0
$575$	14.8200	0.618038
$576$	0	0
$577$	−27.8194	−1.15814	−0.579068	−	0.815279i	$-0.696584\pi$
$577$	−27.8194	−1.15814	−0.579068	+	0.815279i	$0.696584\pi$
$578$	0	0
$579$	45.9964	1.91155
$580$	0	0
$581$	9.18409	0.381020
$582$	0	0
$583$	−0.992927	−0.0411228
$584$	0	0
$585$	19.2447	0.795671
$586$	0	0
$587$	−15.7385	−0.649596	−0.324798	−	0.945783i	$-0.605296\pi$
$587$	−15.7385	−0.649596	−0.324798	+	0.945783i	$0.605296\pi$
$588$	0	0
$589$	3.03647	0.125115
$590$	0	0
$591$	14.9046	0.613094
$592$	0	0
$593$	33.1616	1.36178	0.680892	−	0.732384i	$-0.261592\pi$
$593$	33.1616	1.36178	0.680892	+	0.732384i	$0.261592\pi$
$594$	0	0
$595$	−7.70920	−0.316046
$596$	0	0
$597$	−5.60375	−0.229346
$598$	0	0
$599$	5.80115	0.237029	0.118514	−	0.992952i	$-0.462187\pi$
$599$	5.80115	0.237029	0.118514	+	0.992952i	$0.462187\pi$
$600$	0	0
$601$	−21.7148	−0.885766	−0.442883	−	0.896580i	$-0.646044\pi$
$601$	−21.7148	−0.885766	−0.442883	+	0.896580i	$0.646044\pi$
$602$	0	0
$603$	−61.5828	−2.50785
$604$	0	0
$605$	1.08235	0.0440036
$606$	0	0
$607$	30.3992	1.23386	0.616932	−	0.787016i	$-0.288375\pi$
$607$	30.3992	1.23386	0.616932	+	0.787016i	$0.288375\pi$
$608$	0	0
$609$	9.99759	0.405123
$610$	0	0
$611$	30.7142	1.24256
$612$	0	0
$613$	8.70645	0.351650	0.175825	−	0.984421i	$-0.443741\pi$
$613$	8.70645	0.351650	0.175825	+	0.984421i	$0.443741\pi$
$614$	0	0
$615$	20.9246	0.843763
$616$	0	0
$617$	−14.9292	−0.601026	−0.300513	−	0.953778i	$-0.597158\pi$
$617$	−14.9292	−0.601026	−0.300513	+	0.953778i	$0.597158\pi$
$618$	0	0
$619$	16.5120	0.663673	0.331836	−	0.943337i	$-0.392332\pi$
$619$	16.5120	0.663673	0.331836	+	0.943337i	$0.392332\pi$
$620$	0	0
$621$	48.9683	1.96503
$622$	0	0
$623$	−11.9578	−0.479078
$624$	0	0
$625$	8.80027	0.352011
$626$	0	0
$627$	3.16232	0.126291
$628$	0	0
$629$	−37.2603	−1.48566
$630$	0	0
$631$	−33.1047	−1.31788	−0.658938	−	0.752197i	$-0.728994\pi$
$631$	−33.1047	−1.31788	−0.658938	+	0.752197i	$0.728994\pi$
$632$	0	0
$633$	56.3113	2.23817
$634$	0	0
$635$	13.2041	0.523989
$636$	0	0
$637$	−13.4238	−0.531870
$638$	0	0
$639$	17.1490	0.678404
$640$	0	0
$641$	9.80138	0.387131	0.193566	−	0.981087i	$-0.437995\pi$
$641$	9.80138	0.387131	0.193566	+	0.981087i	$0.437995\pi$
$642$	0	0
$643$	14.1775	0.559106	0.279553	−	0.960130i	$-0.409814\pi$
$643$	14.1775	0.559106	0.279553	+	0.960130i	$0.409814\pi$
$644$	0	0
$645$	−10.1083	−0.398014
$646$	0	0
$647$	−2.89498	−0.113813	−0.0569067	−	0.998379i	$-0.518124\pi$
$647$	−2.89498	−0.113813	−0.0569067	+	0.998379i	$0.518124\pi$
$648$	0	0
$649$	14.2251	0.558385
$650$	0	0
$651$	12.5749	0.492850
$652$	0	0
$653$	14.5415	0.569054	0.284527	−	0.958668i	$-0.408164\pi$
$653$	14.5415	0.569054	0.284527	+	0.958668i	$0.408164\pi$
$654$	0	0
$655$	−20.2906	−0.792817
$656$	0	0
$657$	40.8861	1.59512
$658$	0	0
$659$	−23.8625	−0.929550	−0.464775	−	0.885429i	$-0.653865\pi$
$659$	−23.8625	−0.929550	−0.464775	+	0.885429i	$0.653865\pi$
$660$	0	0
$661$	−5.72037	−0.222497	−0.111248	−	0.993793i	$-0.535485\pi$
$661$	−5.72037	−0.222497	−0.111248	+	0.993793i	$0.535485\pi$
$662$	0	0
$663$	43.6866	1.69665
$664$	0	0
$665$	−1.41741	−0.0549649
$666$	0	0
$667$	9.34492	0.361837
$668$	0	0
$669$	−30.6455	−1.18482
$670$	0	0
$671$	−5.82518	−0.224879
$672$	0	0
$673$	17.0169	0.655954	0.327977	−	0.944686i	$-0.393633\pi$
$673$	17.0169	0.655954	0.327977	+	0.944686i	$0.393633\pi$
$674$	0	0
$675$	48.4317	1.86414
$676$	0	0
$677$	30.2237	1.16159	0.580796	−	0.814049i	$-0.302741\pi$
$677$	30.2237	1.16159	0.580796	+	0.814049i	$0.302741\pi$
$678$	0	0
$679$	−19.3751	−0.743549
$680$	0	0
$681$	−19.0436	−0.729753
$682$	0	0
$683$	30.7218	1.17554	0.587768	−	0.809029i	$-0.300007\pi$
$683$	30.7218	1.17554	0.587768	+	0.809029i	$0.300007\pi$
$684$	0	0
$685$	18.2149	0.695956
$686$	0	0
$687$	−68.7814	−2.62417
$688$	0	0
$689$	−2.52201	−0.0960809
$690$	0	0
$691$	29.7093	1.13020	0.565098	−	0.825024i	$-0.308838\pi$
$691$	29.7093	1.13020	0.565098	+	0.825024i	$0.308838\pi$
$692$	0	0
$693$	9.16741	0.348241
$694$	0	0
$695$	−13.6273	−0.516914
$696$	0	0
$697$	33.2505	1.25945
$698$	0	0
$699$	49.6061	1.87627
$700$	0	0
$701$	9.54914	0.360666	0.180333	−	0.983606i	$-0.442282\pi$
$701$	9.54914	0.360666	0.180333	+	0.983606i	$0.442282\pi$
$702$	0	0
$703$	−6.85067	−0.258378
$704$	0	0
$705$	−41.3888	−1.55879
$706$	0	0
$707$	24.9495	0.938322
$708$	0	0
$709$	42.6564	1.60200	0.800998	−	0.598667i	$-0.204303\pi$
$709$	42.6564	1.60200	0.800998	+	0.598667i	$0.204303\pi$
$710$	0	0
$711$	−119.283	−4.47345
$712$	0	0
$713$	11.7540	0.440191
$714$	0	0
$715$	2.74913	0.102812
$716$	0	0
$717$	69.2046	2.58449
$718$	0	0
$719$	16.7679	0.625335	0.312668	−	0.949863i	$-0.398777\pi$
$719$	16.7679	0.625335	0.312668	+	0.949863i	$0.398777\pi$
$720$	0	0
$721$	−13.7141	−0.510741
$722$	0	0
$723$	−18.9268	−0.703896
$724$	0	0
$725$	9.24252	0.343259
$726$	0	0
$727$	−21.4858	−0.796864	−0.398432	−	0.917198i	$-0.630446\pi$
$727$	−21.4858	−0.796864	−0.398432	+	0.917198i	$0.630446\pi$
$728$	0	0
$729$	13.0158	0.482066
$730$	0	0
$731$	−16.0627	−0.594101
$732$	0	0
$733$	−34.1734	−1.26222	−0.631111	−	0.775692i	$-0.717401\pi$
$733$	−34.1734	−1.26222	−0.631111	+	0.775692i	$0.717401\pi$
$734$	0	0
$735$	18.0892	0.667228
$736$	0	0
$737$	−8.79718	−0.324048
$738$	0	0
$739$	11.5412	0.424548	0.212274	−	0.977210i	$-0.431913\pi$
$739$	11.5412	0.424548	0.212274	+	0.977210i	$0.431913\pi$
$740$	0	0
$741$	8.03222	0.295071
$742$	0	0
$743$	−24.7316	−0.907313	−0.453657	−	0.891177i	$-0.649881\pi$
$743$	−24.7316	−0.907313	−0.453657	+	0.891177i	$0.649881\pi$
$744$	0	0
$745$	6.73964	0.246921
$746$	0	0
$747$	49.0932	1.79623
$748$	0	0
$749$	−18.9778	−0.693435
$750$	0	0
$751$	−46.2026	−1.68596	−0.842978	−	0.537949i	$-0.819199\pi$
$751$	−46.2026	−1.68596	−0.842978	+	0.537949i	$0.819199\pi$
$752$	0	0
$753$	−20.9801	−0.764557
$754$	0	0
$755$	−6.64683	−0.241903
$756$	0	0
$757$	−43.9301	−1.59667	−0.798333	−	0.602216i	$-0.794285\pi$
$757$	−43.9301	−1.59667	−0.798333	+	0.602216i	$0.794285\pi$
$758$	0	0
$759$	12.2412	0.444327
$760$	0	0
$761$	−32.1299	−1.16471	−0.582355	−	0.812935i	$-0.697869\pi$
$761$	−32.1299	−1.16471	−0.582355	+	0.812935i	$0.697869\pi$
$762$	0	0
$763$	−8.38022	−0.303384
$764$	0	0
$765$	−41.2093	−1.48992
$766$	0	0
$767$	36.1315	1.30463
$768$	0	0
$769$	27.7514	1.00074	0.500371	−	0.865811i	$-0.333197\pi$
$769$	27.7514	1.00074	0.500371	+	0.865811i	$0.333197\pi$
$770$	0	0
$771$	29.4599	1.06097
$772$	0	0
$773$	−39.5999	−1.42431	−0.712154	−	0.702023i	$-0.752280\pi$
$773$	−39.5999	−1.42431	−0.712154	+	0.702023i	$0.752280\pi$
$774$	0	0
$775$	11.6252	0.417590
$776$	0	0
$777$	−28.3707	−1.01779
$778$	0	0
$779$	6.11344	0.219037
$780$	0	0
$781$	2.44975	0.0876591
$782$	0	0
$783$	30.5391	1.09138
$784$	0	0
$785$	21.3751	0.762910
$786$	0	0
$787$	26.9286	0.959902	0.479951	−	0.877295i	$-0.340654\pi$
$787$	26.9286	0.959902	0.479951	+	0.877295i	$0.340654\pi$
$788$	0	0
$789$	−29.2516	−1.04139
$790$	0	0
$791$	−3.82006	−0.135826
$792$	0	0
$793$	−14.7958	−0.525415
$794$	0	0
$795$	3.39852	0.120533
$796$	0	0
$797$	18.3883	0.651348	0.325674	−	0.945482i	$-0.394409\pi$
$797$	18.3883	0.651348	0.325674	+	0.945482i	$0.394409\pi$
$798$	0	0
$799$	−65.7693	−2.32675
$800$	0	0
$801$	−63.9199	−2.25850
$802$	0	0
$803$	5.84063	0.206111
$804$	0	0
$805$	−5.48673	−0.193382
$806$	0	0
$807$	53.7835	1.89327
$808$	0	0
$809$	47.5038	1.67014	0.835072	−	0.550141i	$-0.185426\pi$
$809$	47.5038	1.67014	0.835072	+	0.550141i	$0.185426\pi$
$810$	0	0
$811$	39.8438	1.39910	0.699552	−	0.714582i	$-0.253383\pi$
$811$	39.8438	1.39910	0.699552	+	0.714582i	$0.253383\pi$
$812$	0	0
$813$	23.5127	0.824627
$814$	0	0
$815$	6.17352	0.216249
$816$	0	0
$817$	−2.95329	−0.103323
$818$	0	0
$819$	23.2850	0.813643
$820$	0	0
$821$	−39.8464	−1.39065	−0.695325	−	0.718695i	$-0.744740\pi$
$821$	−39.8464	−1.39065	−0.695325	+	0.718695i	$0.744740\pi$
$822$	0	0
$823$	24.5258	0.854917	0.427459	−	0.904035i	$-0.359409\pi$
$823$	24.5258	0.854917	0.427459	+	0.904035i	$0.359409\pi$
$824$	0	0
$825$	12.1070	0.421513
$826$	0	0
$827$	38.5836	1.34168	0.670841	−	0.741601i	$-0.265933\pi$
$827$	38.5836	1.34168	0.670841	+	0.741601i	$0.265933\pi$
$828$	0	0
$829$	−5.07772	−0.176357	−0.0881783	−	0.996105i	$-0.528105\pi$
$829$	−5.07772	−0.176357	−0.0881783	+	0.996105i	$0.528105\pi$
$830$	0	0
$831$	−7.31567	−0.253778
$832$	0	0
$833$	28.7448	0.995947
$834$	0	0
$835$	10.9979	0.380597
$836$	0	0
$837$	38.4120	1.32771
$838$	0	0
$839$	44.8328	1.54780	0.773900	−	0.633308i	$-0.218303\pi$
$839$	44.8328	1.54780	0.773900	+	0.633308i	$0.218303\pi$
$840$	0	0
$841$	−23.1720	−0.799036
$842$	0	0
$843$	16.4519	0.566635
$844$	0	0
$845$	−7.08777	−0.243827
$846$	0	0
$847$	1.30958	0.0449976
$848$	0	0
$849$	84.4736	2.89913
$850$	0	0
$851$	−26.5186	−0.909045
$852$	0	0
$853$	−23.4081	−0.801477	−0.400738	−	0.916193i	$-0.631246\pi$
$853$	−23.4081	−0.801477	−0.400738	+	0.916193i	$0.631246\pi$
$854$	0	0
$855$	−7.57673	−0.259119
$856$	0	0
$857$	−12.1302	−0.414359	−0.207180	−	0.978303i	$-0.566428\pi$
$857$	−12.1302	−0.414359	−0.207180	+	0.978303i	$0.566428\pi$
$858$	0	0
$859$	4.25984	0.145344	0.0726720	−	0.997356i	$-0.476847\pi$
$859$	4.25984	0.145344	0.0726720	+	0.997356i	$0.476847\pi$
$860$	0	0
$861$	25.3176	0.862822
$862$	0	0
$863$	−19.0125	−0.647194	−0.323597	−	0.946195i	$-0.604892\pi$
$863$	−19.0125	−0.647194	−0.323597	+	0.946195i	$0.604892\pi$
$864$	0	0
$865$	2.40698	0.0818397
$866$	0	0
$867$	−39.7880	−1.35127
$868$	0	0
$869$	−17.0397	−0.578031
$870$	0	0
$871$	−22.3446	−0.757118
$872$	0	0
$873$	−103.569	−3.50528
$874$	0	0
$875$	−12.5137	−0.423039
$876$	0	0
$877$	−2.17273	−0.0733678	−0.0366839	−	0.999327i	$-0.511679\pi$
$877$	−2.17273	−0.0733678	−0.0366839	+	0.999327i	$0.511679\pi$
$878$	0	0
$879$	2.96505	0.100009
$880$	0	0
$881$	21.5283	0.725305	0.362653	−	0.931924i	$-0.381871\pi$
$881$	21.5283	0.725305	0.362653	+	0.931924i	$0.381871\pi$
$882$	0	0
$883$	5.14254	0.173060	0.0865302	−	0.996249i	$-0.472422\pi$
$883$	5.14254	0.173060	0.0865302	+	0.996249i	$0.472422\pi$
$884$	0	0
$885$	−48.6887	−1.63665
$886$	0	0
$887$	42.8806	1.43979	0.719895	−	0.694083i	$-0.244190\pi$
$887$	42.8806	1.43979	0.719895	+	0.694083i	$0.244190\pi$
$888$	0	0
$889$	15.9762	0.535825
$890$	0	0
$891$	19.0032	0.636632
$892$	0	0
$893$	−12.0923	−0.404655
$894$	0	0
$895$	−13.0961	−0.437754
$896$	0	0
$897$	31.0923	1.03814
$898$	0	0
$899$	7.33039	0.244482
$900$	0	0
$901$	5.40045	0.179915
$902$	0	0
$903$	−12.2305	−0.407005
$904$	0	0
$905$	−0.388657	−0.0129194
$906$	0	0
$907$	28.3695	0.941992	0.470996	−	0.882135i	$-0.343895\pi$
$907$	28.3695	0.941992	0.470996	+	0.882135i	$0.343895\pi$
$908$	0	0
$909$	133.367	4.42349
$910$	0	0
$911$	−40.0237	−1.32604	−0.663022	−	0.748600i	$-0.730727\pi$
$911$	−40.0237	−1.32604	−0.663022	+	0.748600i	$0.730727\pi$
$912$	0	0
$913$	7.01303	0.232097
$914$	0	0
$915$	19.9380	0.659130
$916$	0	0
$917$	−24.5504	−0.810726
$918$	0	0
$919$	−51.0340	−1.68345	−0.841727	−	0.539903i	$-0.818461\pi$
$919$	−51.0340	−1.68345	−0.841727	+	0.539903i	$0.818461\pi$
$920$	0	0
$921$	−15.2793	−0.503470
$922$	0	0
$923$	6.22231	0.204810
$924$	0	0
$925$	−26.2280	−0.862371
$926$	0	0
$927$	−73.3084	−2.40776
$928$	0	0
$929$	30.7910	1.01022	0.505110	−	0.863055i	$-0.331452\pi$
$929$	30.7910	1.01022	0.505110	+	0.863055i	$0.331452\pi$
$930$	0	0
$931$	5.28501	0.173209
$932$	0	0
$933$	−45.5108	−1.48996
$934$	0	0
$935$	−5.88679	−0.192519
$936$	0	0
$937$	25.9052	0.846287	0.423143	−	0.906063i	$-0.360927\pi$
$937$	25.9052	0.846287	0.423143	+	0.906063i	$0.360927\pi$
$938$	0	0
$939$	92.8073	3.02865
$940$	0	0
$941$	−31.0254	−1.01140	−0.505699	−	0.862710i	$-0.668765\pi$
$941$	−31.0254	−1.01140	−0.505699	+	0.862710i	$0.668765\pi$
$942$	0	0
$943$	23.6648	0.770632
$944$	0	0
$945$	−17.9306	−0.583282
$946$	0	0
$947$	−32.3742	−1.05202	−0.526009	−	0.850479i	$-0.676312\pi$
$947$	−32.3742	−1.05202	−0.526009	+	0.850479i	$0.676312\pi$
$948$	0	0
$949$	14.8350	0.481566
$950$	0	0
$951$	51.8706	1.68202
$952$	0	0
$953$	3.88933	0.125988	0.0629938	−	0.998014i	$-0.479935\pi$
$953$	3.88933	0.125988	0.0629938	+	0.998014i	$0.479935\pi$
$954$	0	0
$955$	−23.0805	−0.746869
$956$	0	0
$957$	7.63422	0.246779
$958$	0	0
$959$	22.0390	0.711676
$960$	0	0
$961$	−21.7799	−0.702576
$962$	0	0
$963$	−101.445	−3.26903
$964$	0	0
$965$	−15.7429	−0.506780
$966$	0	0
$967$	−15.6679	−0.503846	−0.251923	−	0.967747i	$-0.581063\pi$
$967$	−15.6679	−0.503846	−0.251923	+	0.967747i	$0.581063\pi$
$968$	0	0
$969$	−17.1996	−0.552532
$970$	0	0
$971$	−25.6584	−0.823417	−0.411708	−	0.911316i	$-0.635068\pi$
$971$	−25.6584	−0.823417	−0.411708	+	0.911316i	$0.635068\pi$
$972$	0	0
$973$	−16.4883	−0.528590
$974$	0	0
$975$	30.7516	0.984839
$976$	0	0
$977$	44.6708	1.42914	0.714572	−	0.699562i	$-0.246621\pi$
$977$	44.6708	1.42914	0.714572	+	0.699562i	$0.246621\pi$
$978$	0	0
$979$	−9.13103	−0.291829
$980$	0	0
$981$	−44.7962	−1.43023
$982$	0	0
$983$	−48.0898	−1.53383	−0.766913	−	0.641751i	$-0.778208\pi$
$983$	−48.0898	−1.53383	−0.766913	+	0.641751i	$0.778208\pi$
$984$	0	0
$985$	−5.10129	−0.162541
$986$	0	0
$987$	−50.0780	−1.59400
$988$	0	0
$989$	−11.4320	−0.363517
$990$	0	0
$991$	57.3200	1.82083	0.910415	−	0.413697i	$-0.135763\pi$
$991$	57.3200	1.82083	0.910415	+	0.413697i	$0.135763\pi$
$992$	0	0
$993$	44.9035	1.42497
$994$	0	0
$995$	1.91795	0.0608032
$996$	0	0
$997$	14.1271	0.447411	0.223705	−	0.974657i	$-0.428185\pi$
$997$	14.1271	0.447411	0.223705	+	0.974657i	$0.428185\pi$
$998$	0	0
$999$	−86.6625	−2.74188

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.ba.1.1		7
4.3	odd	2		209.2.a.d.1.3	✓	7
12.11	even	2		1881.2.a.p.1.5		7
20.19	odd	2		5225.2.a.n.1.5		7
44.43	even	2		2299.2.a.q.1.5		7
76.75	even	2		3971.2.a.i.1.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.a.d.1.3	✓	7	4.3	odd	2
1881.2.a.p.1.5		7	12.11	even	2
2299.2.a.q.1.5		7	44.43	even	2
3344.2.a.ba.1.1		7	1.1	even	1	trivial
3971.2.a.i.1.5		7	76.75	even	2
5225.2.a.n.1.5		7	20.19	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:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q-3.16232 q^{3} +1.08235 q^{5} +1.30958 q^{7} +7.00029 q^{9} +O(q^{10})\)	\(q-3.16232 q^{3} +1.08235 q^{5} +1.30958 q^{7} +7.00029 q^{9} +1.00000 q^{11} +2.53997 q^{13} -3.42273 q^{15} -5.43892 q^{17} -1.00000 q^{19} -4.14130 q^{21} -3.87095 q^{23} -3.82853 q^{25} -12.6502 q^{27} -2.41412 q^{29} -3.03647 q^{31} -3.16232 q^{33} +1.41741 q^{35} +6.85067 q^{37} -8.03222 q^{39} -6.11344 q^{41} +2.95329 q^{43} +7.57673 q^{45} +12.0923 q^{47} -5.28501 q^{49} +17.1996 q^{51} -0.992927 q^{53} +1.08235 q^{55} +3.16232 q^{57} +14.2251 q^{59} -5.82518 q^{61} +9.16741 q^{63} +2.74913 q^{65} -8.79718 q^{67} +12.2412 q^{69} +2.44975 q^{71} +5.84063 q^{73} +12.1070 q^{75} +1.30958 q^{77} -17.0397 q^{79} +19.0032 q^{81} +7.01303 q^{83} -5.88679 q^{85} +7.63422 q^{87} -9.13103 q^{89} +3.32629 q^{91} +9.60229 q^{93} -1.08235 q^{95} -14.7950 q^{97} +7.00029 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 2 q^{3} + 2 q^{5} - 10 q^{7} + 11 q^{9}+O(q^{10}) \) 7 * q - 2 * q^3 + 2 * q^5 - 10 * q^7 + 11 * q^9	\( 7 q - 2 q^{3} + 2 q^{5} - 10 q^{7} + 11 q^{9} + 7 q^{11} - 4 q^{13} - 12 q^{15} + 2 q^{17} - 7 q^{19} - 14 q^{21} - 10 q^{23} + 9 q^{25} + 4 q^{27} - 18 q^{29} - 24 q^{31} - 2 q^{33} - 8 q^{35} - 24 q^{39} - 12 q^{41} - 2 q^{43} - 4 q^{45} - 8 q^{47} + 17 q^{49} + 24 q^{51} + 2 q^{53} + 2 q^{55} + 2 q^{57} + 10 q^{59} + 14 q^{61} - 14 q^{65} - 8 q^{67} - 6 q^{69} - 10 q^{71} - 6 q^{73} - 26 q^{75} - 10 q^{77} - 52 q^{79} - q^{81} + 10 q^{83} - 12 q^{85} - 6 q^{87} - 12 q^{91} + 2 q^{93} - 2 q^{95} - 24 q^{97} + 11 q^{99}+O(q^{100}) \) 7 * q - 2 * q^3 + 2 * q^5 - 10 * q^7 + 11 * q^9 + 7 * q^11 - 4 * q^13 - 12 * q^15 + 2 * q^17 - 7 * q^19 - 14 * q^21 - 10 * q^23 + 9 * q^25 + 4 * q^27 - 18 * q^29 - 24 * q^31 - 2 * q^33 - 8 * q^35 - 24 * q^39 - 12 * q^41 - 2 * q^43 - 4 * q^45 - 8 * q^47 + 17 * q^49 + 24 * q^51 + 2 * q^53 + 2 * q^55 + 2 * q^57 + 10 * q^59 + 14 * q^61 - 14 * q^65 - 8 * q^67 - 6 * q^69 - 10 * q^71 - 6 * q^73 - 26 * q^75 - 10 * q^77 - 52 * q^79 - q^81 + 10 * q^83 - 12 * q^85 - 6 * q^87 - 12 * q^91 + 2 * q^93 - 2 * q^95 - 24 * q^97 + 11 * q^99

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