LMFDB - Embedded newform 3344.2.a.bb.1.5

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:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 22x^{7} + 22x^{6} + 152x^{5} - 136x^{4} - 341x^{3} + 169x^{2} + 196x + 32 $ x^9 - x^8 - 22x^7 + 22x^6 + 152x^5 - 136x^4 - 341x^3 + 169x^2 + 196*x + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1672)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.223788$ 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)

$f(q)$	$=$	$q+0.223788 q^{3} -4.30360 q^{5} +0.878402 q^{7} -2.94992 q^{9} +O(q^{10})$	$q+0.223788 q^{3} -4.30360 q^{5} +0.878402 q^{7} -2.94992 q^{9} -1.00000 q^{11} -4.33548 q^{13} -0.963093 q^{15} -5.89078 q^{17} +1.00000 q^{19} +0.196576 q^{21} -4.92481 q^{23} +13.5210 q^{25} -1.33152 q^{27} +3.35873 q^{29} -2.82544 q^{31} -0.223788 q^{33} -3.78029 q^{35} -1.07152 q^{37} -0.970226 q^{39} -0.852140 q^{41} +7.50229 q^{43} +12.6953 q^{45} -12.6072 q^{47} -6.22841 q^{49} -1.31828 q^{51} +13.1910 q^{53} +4.30360 q^{55} +0.223788 q^{57} +9.06585 q^{59} -4.02701 q^{61} -2.59122 q^{63} +18.6582 q^{65} +11.7510 q^{67} -1.10211 q^{69} -3.39679 q^{71} -4.50733 q^{73} +3.02583 q^{75} -0.878402 q^{77} +1.31971 q^{79} +8.55178 q^{81} -9.71715 q^{83} +25.3516 q^{85} +0.751643 q^{87} +0.835408 q^{89} -3.80829 q^{91} -0.632299 q^{93} -4.30360 q^{95} +11.0034 q^{97} +2.94992 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{3} + 6 q^{5} - 3 q^{7} + 18 q^{9}+O(q^{10}) $ 9 * q - q^3 + 6 * q^5 - 3 * q^7 + 18 * q^9	$ 9 q - q^{3} + 6 q^{5} - 3 q^{7} + 18 q^{9} - 9 q^{11} + q^{13} - 6 q^{15} + 7 q^{17} + 9 q^{19} + 3 q^{21} - 13 q^{23} + 31 q^{25} + 5 q^{27} + 9 q^{29} + 4 q^{31} + q^{33} + 4 q^{35} + 24 q^{37} - 13 q^{39} - 6 q^{41} + 14 q^{43} + 26 q^{45} - 24 q^{47} + 20 q^{49} + 33 q^{51} + 19 q^{53} - 6 q^{55} - q^{57} + 19 q^{59} + 28 q^{61} - 16 q^{63} + 16 q^{65} - 5 q^{67} + 35 q^{69} - 16 q^{71} + 15 q^{73} - 3 q^{75} + 3 q^{77} - 2 q^{79} + 37 q^{81} - 8 q^{83} + 20 q^{85} - 23 q^{87} + 12 q^{89} + 29 q^{91} + 44 q^{93} + 6 q^{95} - 4 q^{97} - 18 q^{99}+O(q^{100}) $ 9 * q - q^3 + 6 * q^5 - 3 * q^7 + 18 * q^9 - 9 * q^11 + q^13 - 6 * q^15 + 7 * q^17 + 9 * q^19 + 3 * q^21 - 13 * q^23 + 31 * q^25 + 5 * q^27 + 9 * q^29 + 4 * q^31 + q^33 + 4 * q^35 + 24 * q^37 - 13 * q^39 - 6 * q^41 + 14 * q^43 + 26 * q^45 - 24 * q^47 + 20 * q^49 + 33 * q^51 + 19 * q^53 - 6 * q^55 - q^57 + 19 * q^59 + 28 * q^61 - 16 * q^63 + 16 * q^65 - 5 * q^67 + 35 * q^69 - 16 * q^71 + 15 * q^73 - 3 * q^75 + 3 * q^77 - 2 * q^79 + 37 * q^81 - 8 * q^83 + 20 * q^85 - 23 * q^87 + 12 * q^89 + 29 * q^91 + 44 * q^93 + 6 * q^95 - 4 * q^97 - 18 * 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
$3$	0.223788	0.129204	0.0646020	−	0.997911i	$-0.479422\pi$
$3$	0.223788	0.129204	0.0646020	+	0.997911i	$0.479422\pi$
$4$	0	0
$5$	−4.30360	−1.92463	−0.962315	−	0.271938i	$-0.912335\pi$
$5$	−4.30360	−1.92463	−0.962315	+	0.271938i	$0.912335\pi$
$6$	0	0
$7$	0.878402	0.332005	0.166002	−	0.986125i	$-0.446914\pi$
$7$	0.878402	0.332005	0.166002	+	0.986125i	$0.446914\pi$
$8$	0	0
$9$	−2.94992	−0.983306
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−4.33548	−1.20244	−0.601222	−	0.799082i	$-0.705319\pi$
$13$	−4.33548	−1.20244	−0.601222	+	0.799082i	$0.705319\pi$
$14$	0	0
$15$	−0.963093	−0.248670
$16$	0	0
$17$	−5.89078	−1.42872	−0.714362	−	0.699776i	$-0.753283\pi$
$17$	−5.89078	−1.42872	−0.714362	+	0.699776i	$0.753283\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0.196576	0.0428963
$22$	0	0
$23$	−4.92481	−1.02689	−0.513447	−	0.858122i	$-0.671632\pi$
$23$	−4.92481	−1.02689	−0.513447	+	0.858122i	$0.671632\pi$
$24$	0	0
$25$	13.5210	2.70420
$26$	0	0
$27$	−1.33152	−0.256251
$28$	0	0
$29$	3.35873	0.623701	0.311850	−	0.950131i	$-0.399051\pi$
$29$	3.35873	0.623701	0.311850	+	0.950131i	$0.399051\pi$
$30$	0	0
$31$	−2.82544	−0.507464	−0.253732	−	0.967275i	$-0.581658\pi$
$31$	−2.82544	−0.507464	−0.253732	+	0.967275i	$0.581658\pi$
$32$	0	0
$33$	−0.223788	−0.0389564
$34$	0	0
$35$	−3.78029	−0.638986
$36$	0	0
$37$	−1.07152	−0.176156	−0.0880781	−	0.996114i	$-0.528073\pi$
$37$	−1.07152	−0.176156	−0.0880781	+	0.996114i	$0.528073\pi$
$38$	0	0
$39$	−0.970226	−0.155361
$40$	0	0
$41$	−0.852140	−0.133082	−0.0665410	−	0.997784i	$-0.521196\pi$
$41$	−0.852140	−0.133082	−0.0665410	+	0.997784i	$0.521196\pi$
$42$	0	0
$43$	7.50229	1.14409	0.572044	−	0.820223i	$-0.306150\pi$
$43$	7.50229	1.14409	0.572044	+	0.820223i	$0.306150\pi$
$44$	0	0
$45$	12.6953	1.89250
$46$	0	0
$47$	−12.6072	−1.83895	−0.919475	−	0.393149i	$-0.871386\pi$
$47$	−12.6072	−1.83895	−0.919475	+	0.393149i	$0.871386\pi$
$48$	0	0
$49$	−6.22841	−0.889773
$50$	0	0
$51$	−1.31828	−0.184597
$52$	0	0
$53$	13.1910	1.81192	0.905962	−	0.423358i	$-0.139149\pi$
$53$	13.1910	1.81192	0.905962	+	0.423358i	$0.139149\pi$
$54$	0	0
$55$	4.30360	0.580298
$56$	0	0
$57$	0.223788	0.0296414
$58$	0	0
$59$	9.06585	1.18027	0.590137	−	0.807303i	$-0.299074\pi$
$59$	9.06585	1.18027	0.590137	+	0.807303i	$0.299074\pi$
$60$	0	0
$61$	−4.02701	−0.515606	−0.257803	−	0.966197i	$-0.582999\pi$
$61$	−4.02701	−0.515606	−0.257803	+	0.966197i	$0.582999\pi$
$62$	0	0
$63$	−2.59122	−0.326463
$64$	0	0
$65$	18.6582	2.31426
$66$	0	0
$67$	11.7510	1.43561	0.717807	−	0.696242i	$-0.245146\pi$
$67$	11.7510	1.43561	0.717807	+	0.696242i	$0.245146\pi$
$68$	0	0
$69$	−1.10211	−0.132679
$70$	0	0
$71$	−3.39679	−0.403125	−0.201562	−	0.979476i	$-0.564602\pi$
$71$	−3.39679	−0.403125	−0.201562	+	0.979476i	$0.564602\pi$
$72$	0	0
$73$	−4.50733	−0.527543	−0.263771	−	0.964585i	$-0.584966\pi$
$73$	−4.50733	−0.527543	−0.263771	+	0.964585i	$0.584966\pi$
$74$	0	0
$75$	3.02583	0.349393
$76$	0	0
$77$	−0.878402	−0.100103
$78$	0	0
$79$	1.31971	0.148479	0.0742395	−	0.997240i	$-0.476347\pi$
$79$	1.31971	0.148479	0.0742395	+	0.997240i	$0.476347\pi$
$80$	0	0
$81$	8.55178	0.950198
$82$	0	0
$83$	−9.71715	−1.06660	−0.533298	−	0.845927i	$-0.679048\pi$
$83$	−9.71715	−1.06660	−0.533298	+	0.845927i	$0.679048\pi$
$84$	0	0
$85$	25.3516	2.74977
$86$	0	0
$87$	0.751643	0.0805846
$88$	0	0
$89$	0.835408	0.0885531	0.0442766	−	0.999019i	$-0.485902\pi$
$89$	0.835408	0.0885531	0.0442766	+	0.999019i	$0.485902\pi$
$90$	0	0
$91$	−3.80829	−0.399218
$92$	0	0
$93$	−0.632299	−0.0655663
$94$	0	0
$95$	−4.30360	−0.441540
$96$	0	0
$97$	11.0034	1.11722	0.558612	−	0.829429i	$-0.311334\pi$
$97$	11.0034	1.11722	0.558612	+	0.829429i	$0.311334\pi$
$98$	0	0
$99$	2.94992	0.296478
$100$	0	0
$101$	3.48900	0.347168	0.173584	−	0.984819i	$-0.444465\pi$
$101$	3.48900	0.347168	0.173584	+	0.984819i	$0.444465\pi$
$102$	0	0
$103$	−6.87409	−0.677324	−0.338662	−	0.940908i	$-0.609974\pi$
$103$	−6.87409	−0.677324	−0.338662	+	0.940908i	$0.609974\pi$
$104$	0	0
$105$	−0.845984	−0.0825595
$106$	0	0
$107$	−15.1783	−1.46734	−0.733670	−	0.679506i	$-0.762194\pi$
$107$	−15.1783	−1.46734	−0.733670	+	0.679506i	$0.762194\pi$
$108$	0	0
$109$	9.85668	0.944099	0.472050	−	0.881572i	$-0.343514\pi$
$109$	9.85668	0.944099	0.472050	+	0.881572i	$0.343514\pi$
$110$	0	0
$111$	−0.239792	−0.0227601
$112$	0	0
$113$	−7.47188	−0.702895	−0.351448	−	0.936208i	$-0.614310\pi$
$113$	−7.47188	−0.702895	−0.351448	+	0.936208i	$0.614310\pi$
$114$	0	0
$115$	21.1944	1.97639
$116$	0	0
$117$	12.7893	1.18237
$118$	0	0
$119$	−5.17448	−0.474344
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−0.190699	−0.0171947
$124$	0	0
$125$	−36.6710	−3.27995
$126$	0	0
$127$	−4.03541	−0.358085	−0.179042	−	0.983841i	$-0.557300\pi$
$127$	−4.03541	−0.358085	−0.179042	+	0.983841i	$0.557300\pi$
$128$	0	0
$129$	1.67892	0.147821
$130$	0	0
$131$	5.13452	0.448605	0.224302	−	0.974520i	$-0.427990\pi$
$131$	5.13452	0.448605	0.224302	+	0.974520i	$0.427990\pi$
$132$	0	0
$133$	0.878402	0.0761671
$134$	0	0
$135$	5.73033	0.493188
$136$	0	0
$137$	1.59587	0.136345	0.0681724	−	0.997674i	$-0.478283\pi$
$137$	1.59587	0.136345	0.0681724	+	0.997674i	$0.478283\pi$
$138$	0	0
$139$	18.5788	1.57583	0.787917	−	0.615782i	$-0.211160\pi$
$139$	18.5788	1.57583	0.787917	+	0.615782i	$0.211160\pi$
$140$	0	0
$141$	−2.82134	−0.237600
$142$	0	0
$143$	4.33548	0.362551
$144$	0	0
$145$	−14.4546	−1.20039
$146$	0	0
$147$	−1.39384	−0.114962
$148$	0	0
$149$	3.12995	0.256415	0.128208	−	0.991747i	$-0.459078\pi$
$149$	3.12995	0.256415	0.128208	+	0.991747i	$0.459078\pi$
$150$	0	0
$151$	0.838035	0.0681983	0.0340992	−	0.999418i	$-0.489144\pi$
$151$	0.838035	0.0681983	0.0340992	+	0.999418i	$0.489144\pi$
$152$	0	0
$153$	17.3773	1.40487
$154$	0	0
$155$	12.1596	0.976680
$156$	0	0
$157$	−3.84940	−0.307215	−0.153608	−	0.988132i	$-0.549089\pi$
$157$	−3.84940	−0.307215	−0.153608	+	0.988132i	$0.549089\pi$
$158$	0	0
$159$	2.95199	0.234108
$160$	0	0
$161$	−4.32596	−0.340934
$162$	0	0
$163$	−2.07384	−0.162436	−0.0812180	−	0.996696i	$-0.525881\pi$
$163$	−2.07384	−0.162436	−0.0812180	+	0.996696i	$0.525881\pi$
$164$	0	0
$165$	0.963093	0.0749767
$166$	0	0
$167$	13.0042	1.00629	0.503146	−	0.864201i	$-0.332176\pi$
$167$	13.0042	1.00629	0.503146	+	0.864201i	$0.332176\pi$
$168$	0	0
$169$	5.79635	0.445873
$170$	0	0
$171$	−2.94992	−0.225586
$172$	0	0
$173$	3.44201	0.261691	0.130846	−	0.991403i	$-0.458231\pi$
$173$	3.44201	0.261691	0.130846	+	0.991403i	$0.458231\pi$
$174$	0	0
$175$	11.8769	0.897807
$176$	0	0
$177$	2.02883	0.152496
$178$	0	0
$179$	−2.14307	−0.160181	−0.0800903	−	0.996788i	$-0.525521\pi$
$179$	−2.14307	−0.160181	−0.0800903	+	0.996788i	$0.525521\pi$
$180$	0	0
$181$	5.44990	0.405088	0.202544	−	0.979273i	$-0.435079\pi$
$181$	5.44990	0.405088	0.202544	+	0.979273i	$0.435079\pi$
$182$	0	0
$183$	−0.901196	−0.0666183
$184$	0	0
$185$	4.61138	0.339036
$186$	0	0
$187$	5.89078	0.430777
$188$	0	0
$189$	−1.16961	−0.0850766
$190$	0	0
$191$	5.94921	0.430470	0.215235	−	0.976562i	$-0.430948\pi$
$191$	5.94921	0.430470	0.215235	+	0.976562i	$0.430948\pi$
$192$	0	0
$193$	16.6003	1.19492	0.597459	−	0.801899i	$-0.296177\pi$
$193$	16.6003	1.19492	0.597459	+	0.801899i	$0.296177\pi$
$194$	0	0
$195$	4.17547	0.299011
$196$	0	0
$197$	7.98528	0.568928	0.284464	−	0.958687i	$-0.408184\pi$
$197$	7.98528	0.568928	0.284464	+	0.958687i	$0.408184\pi$
$198$	0	0
$199$	−5.65230	−0.400681	−0.200341	−	0.979726i	$-0.564205\pi$
$199$	−5.65230	−0.400681	−0.200341	+	0.979726i	$0.564205\pi$
$200$	0	0
$201$	2.62973	0.185487
$202$	0	0
$203$	2.95032	0.207072
$204$	0	0
$205$	3.66727	0.256133
$206$	0	0
$207$	14.5278	1.00975
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	27.8389	1.91651	0.958253	−	0.285922i	$-0.0923000\pi$
$211$	27.8389	1.91651	0.958253	+	0.285922i	$0.0923000\pi$
$212$	0	0
$213$	−0.760159	−0.0520853
$214$	0	0
$215$	−32.2869	−2.20195
$216$	0	0
$217$	−2.48187	−0.168480
$218$	0	0
$219$	−1.00868	−0.0681606
$220$	0	0
$221$	25.5393	1.71796
$222$	0	0
$223$	−24.0498	−1.61049	−0.805245	−	0.592942i	$-0.797966\pi$
$223$	−24.0498	−1.61049	−0.805245	+	0.592942i	$0.797966\pi$
$224$	0	0
$225$	−39.8858	−2.65906
$226$	0	0
$227$	−9.57797	−0.635712	−0.317856	−	0.948139i	$-0.602963\pi$
$227$	−9.57797	−0.635712	−0.317856	+	0.948139i	$0.602963\pi$
$228$	0	0
$229$	−19.4495	−1.28526	−0.642631	−	0.766176i	$-0.722157\pi$
$229$	−19.4495	−1.28526	−0.642631	+	0.766176i	$0.722157\pi$
$230$	0	0
$231$	−0.196576	−0.0129337
$232$	0	0
$233$	1.60569	0.105192	0.0525962	−	0.998616i	$-0.483250\pi$
$233$	1.60569	0.105192	0.0525962	+	0.998616i	$0.483250\pi$
$234$	0	0
$235$	54.2564	3.53930
$236$	0	0
$237$	0.295335	0.0191841
$238$	0	0
$239$	21.0297	1.36030	0.680150	−	0.733073i	$-0.261915\pi$
$239$	21.0297	1.36030	0.680150	+	0.733073i	$0.261915\pi$
$240$	0	0
$241$	8.80995	0.567499	0.283749	−	0.958898i	$-0.408422\pi$
$241$	8.80995	0.567499	0.283749	+	0.958898i	$0.408422\pi$
$242$	0	0
$243$	5.90834	0.379020
$244$	0	0
$245$	26.8046	1.71248
$246$	0	0
$247$	−4.33548	−0.275860
$248$	0	0
$249$	−2.17458	−0.137808
$250$	0	0
$251$	−5.13494	−0.324115	−0.162057	−	0.986781i	$-0.551813\pi$
$251$	−5.13494	−0.324115	−0.162057	+	0.986781i	$0.551813\pi$
$252$	0	0
$253$	4.92481	0.309620
$254$	0	0
$255$	5.67337	0.355280
$256$	0	0
$257$	28.4999	1.77777	0.888886	−	0.458128i	$-0.151480\pi$
$257$	28.4999	1.77777	0.888886	+	0.458128i	$0.151480\pi$
$258$	0	0
$259$	−0.941223	−0.0584847
$260$	0	0
$261$	−9.90798	−0.613289
$262$	0	0
$263$	−24.0190	−1.48107	−0.740537	−	0.672015i	$-0.765429\pi$
$263$	−24.0190	−1.48107	−0.740537	+	0.672015i	$0.765429\pi$
$264$	0	0
$265$	−56.7689	−3.48728
$266$	0	0
$267$	0.186954	0.0114414
$268$	0	0
$269$	−11.5387	−0.703527	−0.351764	−	0.936089i	$-0.614418\pi$
$269$	−11.5387	−0.703527	−0.351764	+	0.936089i	$0.614418\pi$
$270$	0	0
$271$	−24.7411	−1.50292	−0.751459	−	0.659780i	$-0.770649\pi$
$271$	−24.7411	−1.50292	−0.751459	+	0.659780i	$0.770649\pi$
$272$	0	0
$273$	−0.852249	−0.0515805
$274$	0	0
$275$	−13.5210	−0.815346
$276$	0	0
$277$	−24.8046	−1.49036	−0.745181	−	0.666862i	$-0.767637\pi$
$277$	−24.8046	−1.49036	−0.745181	+	0.666862i	$0.767637\pi$
$278$	0	0
$279$	8.33482	0.498992
$280$	0	0
$281$	16.6278	0.991933	0.495967	−	0.868342i	$-0.334814\pi$
$281$	16.6278	0.991933	0.495967	+	0.868342i	$0.334814\pi$
$282$	0	0
$283$	20.3460	1.20944	0.604722	−	0.796437i	$-0.293284\pi$
$283$	20.3460	1.20944	0.604722	+	0.796437i	$0.293284\pi$
$284$	0	0
$285$	−0.963093	−0.0570487
$286$	0	0
$287$	−0.748522	−0.0441839
$288$	0	0
$289$	17.7013	1.04125
$290$	0	0
$291$	2.46242	0.144350
$292$	0	0
$293$	−25.1473	−1.46912	−0.734561	−	0.678543i	$-0.762612\pi$
$293$	−25.1473	−1.46912	−0.734561	+	0.678543i	$0.762612\pi$
$294$	0	0
$295$	−39.0158	−2.27159
$296$	0	0
$297$	1.33152	0.0772626
$298$	0	0
$299$	21.3514	1.23478
$300$	0	0
$301$	6.59003	0.379843
$302$	0	0
$303$	0.780795	0.0448555
$304$	0	0
$305$	17.3307	0.992351
$306$	0	0
$307$	23.0581	1.31599	0.657997	−	0.753021i	$-0.271404\pi$
$307$	23.0581	1.31599	0.657997	+	0.753021i	$0.271404\pi$
$308$	0	0
$309$	−1.53834	−0.0875129
$310$	0	0
$311$	18.1775	1.03075	0.515375	−	0.856964i	$-0.327652\pi$
$311$	18.1775	1.03075	0.515375	+	0.856964i	$0.327652\pi$
$312$	0	0
$313$	−14.7299	−0.832585	−0.416293	−	0.909231i	$-0.636671\pi$
$313$	−14.7299	−0.832585	−0.416293	+	0.909231i	$0.636671\pi$
$314$	0	0
$315$	11.1516	0.628319
$316$	0	0
$317$	−6.61819	−0.371714	−0.185857	−	0.982577i	$-0.559506\pi$
$317$	−6.61819	−0.371714	−0.185857	+	0.982577i	$0.559506\pi$
$318$	0	0
$319$	−3.35873	−0.188053
$320$	0	0
$321$	−3.39671	−0.189586
$322$	0	0
$323$	−5.89078	−0.327772
$324$	0	0
$325$	−58.6199	−3.25165
$326$	0	0
$327$	2.20580	0.121981
$328$	0	0
$329$	−11.0742	−0.610540
$330$	0	0
$331$	−35.6562	−1.95984	−0.979920	−	0.199393i	$-0.936103\pi$
$331$	−35.6562	−1.95984	−0.979920	+	0.199393i	$0.936103\pi$
$332$	0	0
$333$	3.16089	0.173216
$334$	0	0
$335$	−50.5717	−2.76303
$336$	0	0
$337$	23.0805	1.25727	0.628637	−	0.777699i	$-0.283613\pi$
$337$	23.0805	1.25727	0.628637	+	0.777699i	$0.283613\pi$
$338$	0	0
$339$	−1.67211	−0.0908168
$340$	0	0
$341$	2.82544	0.153006
$342$	0	0
$343$	−11.6199	−0.627414
$344$	0	0
$345$	4.74305	0.255357
$346$	0	0
$347$	−28.9328	−1.55319	−0.776597	−	0.629997i	$-0.783056\pi$
$347$	−28.9328	−1.55319	−0.776597	+	0.629997i	$0.783056\pi$
$348$	0	0
$349$	17.9882	0.962884	0.481442	−	0.876478i	$-0.340113\pi$
$349$	17.9882	0.962884	0.481442	+	0.876478i	$0.340113\pi$
$350$	0	0
$351$	5.77277	0.308128
$352$	0	0
$353$	−8.99583	−0.478800	−0.239400	−	0.970921i	$-0.576951\pi$
$353$	−8.99583	−0.478800	−0.239400	+	0.970921i	$0.576951\pi$
$354$	0	0
$355$	14.6184	0.775865
$356$	0	0
$357$	−1.15798	−0.0612870
$358$	0	0
$359$	−9.97703	−0.526568	−0.263284	−	0.964718i	$-0.584806\pi$
$359$	−9.97703	−0.526568	−0.263284	+	0.964718i	$0.584806\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0.223788	0.0117458
$364$	0	0
$365$	19.3977	1.01532
$366$	0	0
$367$	18.3841	0.959644	0.479822	−	0.877366i	$-0.340701\pi$
$367$	18.3841	0.959644	0.479822	+	0.877366i	$0.340701\pi$
$368$	0	0
$369$	2.51374	0.130860
$370$	0	0
$371$	11.5870	0.601568
$372$	0	0
$373$	27.2170	1.40924	0.704621	−	0.709584i	$-0.251117\pi$
$373$	27.2170	1.40924	0.704621	+	0.709584i	$0.251117\pi$
$374$	0	0
$375$	−8.20651	−0.423782
$376$	0	0
$377$	−14.5617	−0.749966
$378$	0	0
$379$	17.2427	0.885700	0.442850	−	0.896596i	$-0.353967\pi$
$379$	17.2427	0.885700	0.442850	+	0.896596i	$0.353967\pi$
$380$	0	0
$381$	−0.903076	−0.0462660
$382$	0	0
$383$	21.4520	1.09614	0.548072	−	0.836431i	$-0.315362\pi$
$383$	21.4520	1.09614	0.548072	+	0.836431i	$0.315362\pi$
$384$	0	0
$385$	3.78029	0.192662
$386$	0	0
$387$	−22.1312	−1.12499
$388$	0	0
$389$	−36.5106	−1.85116	−0.925581	−	0.378549i	$-0.876423\pi$
$389$	−36.5106	−1.85116	−0.925581	+	0.378549i	$0.876423\pi$
$390$	0	0
$391$	29.0110	1.46715
$392$	0	0
$393$	1.14904	0.0579615
$394$	0	0
$395$	−5.67950	−0.285767
$396$	0	0
$397$	−14.2852	−0.716951	−0.358476	−	0.933539i	$-0.616703\pi$
$397$	−14.2852	−0.716951	−0.358476	+	0.933539i	$0.616703\pi$
$398$	0	0
$399$	0.196576	0.00984109
$400$	0	0
$401$	23.1603	1.15657	0.578285	−	0.815835i	$-0.303722\pi$
$401$	23.1603	1.15657	0.578285	+	0.815835i	$0.303722\pi$
$402$	0	0
$403$	12.2496	0.610197
$404$	0	0
$405$	−36.8035	−1.82878
$406$	0	0
$407$	1.07152	0.0531131
$408$	0	0
$409$	−18.9816	−0.938582	−0.469291	−	0.883044i	$-0.655490\pi$
$409$	−18.9816	−0.938582	−0.469291	+	0.883044i	$0.655490\pi$
$410$	0	0
$411$	0.357137	0.0176163
$412$	0	0
$413$	7.96347	0.391857
$414$	0	0
$415$	41.8188	2.05280
$416$	0	0
$417$	4.15771	0.203604
$418$	0	0
$419$	−31.0185	−1.51535	−0.757677	−	0.652630i	$-0.773666\pi$
$419$	−31.0185	−1.51535	−0.757677	+	0.652630i	$0.773666\pi$
$420$	0	0
$421$	29.5806	1.44167	0.720836	−	0.693105i	$-0.243758\pi$
$421$	29.5806	1.44167	0.720836	+	0.693105i	$0.243758\pi$
$422$	0	0
$423$	37.1902	1.80825
$424$	0	0
$425$	−79.6492	−3.86355
$426$	0	0
$427$	−3.53734	−0.171184
$428$	0	0
$429$	0.970226	0.0468430
$430$	0	0
$431$	16.9779	0.817797	0.408899	−	0.912580i	$-0.365913\pi$
$431$	16.9779	0.817797	0.408899	+	0.912580i	$0.365913\pi$
$432$	0	0
$433$	38.0673	1.82940	0.914699	−	0.404136i	$-0.132428\pi$
$433$	38.0673	1.82940	0.914699	+	0.404136i	$0.132428\pi$
$434$	0	0
$435$	−3.23477	−0.155095
$436$	0	0
$437$	−4.92481	−0.235585
$438$	0	0
$439$	−2.74711	−0.131112	−0.0655562	−	0.997849i	$-0.520882\pi$
$439$	−2.74711	−0.131112	−0.0655562	+	0.997849i	$0.520882\pi$
$440$	0	0
$441$	18.3733	0.874919
$442$	0	0
$443$	−30.1145	−1.43078	−0.715391	−	0.698724i	$-0.753752\pi$
$443$	−30.1145	−1.43078	−0.715391	+	0.698724i	$0.753752\pi$
$444$	0	0
$445$	−3.59527	−0.170432
$446$	0	0
$447$	0.700443	0.0331298
$448$	0	0
$449$	−8.82123	−0.416300	−0.208150	−	0.978097i	$-0.566744\pi$
$449$	−8.82123	−0.416300	−0.208150	+	0.978097i	$0.566744\pi$
$450$	0	0
$451$	0.852140	0.0401257
$452$	0	0
$453$	0.187542	0.00881149
$454$	0	0
$455$	16.3894	0.768346
$456$	0	0
$457$	−31.1927	−1.45913	−0.729566	−	0.683911i	$-0.760278\pi$
$457$	−31.1927	−1.45913	−0.729566	+	0.683911i	$0.760278\pi$
$458$	0	0
$459$	7.84369	0.366112
$460$	0	0
$461$	−5.96080	−0.277622	−0.138811	−	0.990319i	$-0.544328\pi$
$461$	−5.96080	−0.277622	−0.138811	+	0.990319i	$0.544328\pi$
$462$	0	0
$463$	−37.0942	−1.72392	−0.861958	−	0.506980i	$-0.830762\pi$
$463$	−37.0942	−1.72392	−0.861958	+	0.506980i	$0.830762\pi$
$464$	0	0
$465$	2.72116	0.126191
$466$	0	0
$467$	−22.7204	−1.05137	−0.525687	−	0.850678i	$-0.676192\pi$
$467$	−22.7204	−1.05137	−0.525687	+	0.850678i	$0.676192\pi$
$468$	0	0
$469$	10.3221	0.476631
$470$	0	0
$471$	−0.861447	−0.0396934
$472$	0	0
$473$	−7.50229	−0.344956
$474$	0	0
$475$	13.5210	0.620386
$476$	0	0
$477$	−38.9124	−1.78168
$478$	0	0
$479$	−29.3594	−1.34146	−0.670732	−	0.741700i	$-0.734020\pi$
$479$	−29.3594	−1.34146	−0.670732	+	0.741700i	$0.734020\pi$
$480$	0	0
$481$	4.64553	0.211818
$482$	0	0
$483$	−0.968097	−0.0440500
$484$	0	0
$485$	−47.3541	−2.15024
$486$	0	0
$487$	20.1152	0.911508	0.455754	−	0.890106i	$-0.349370\pi$
$487$	20.1152	0.911508	0.455754	+	0.890106i	$0.349370\pi$
$488$	0	0
$489$	−0.464101	−0.0209874
$490$	0	0
$491$	20.8517	0.941026	0.470513	−	0.882393i	$-0.344069\pi$
$491$	20.8517	0.941026	0.470513	+	0.882393i	$0.344069\pi$
$492$	0	0
$493$	−19.7856	−0.891097
$494$	0	0
$495$	−12.6953	−0.570610
$496$	0	0
$497$	−2.98375	−0.133839
$498$	0	0
$499$	−33.8159	−1.51381	−0.756904	−	0.653526i	$-0.773289\pi$
$499$	−33.8159	−1.51381	−0.756904	+	0.653526i	$0.773289\pi$
$500$	0	0
$501$	2.91017	0.130017
$502$	0	0
$503$	−21.4246	−0.955277	−0.477638	−	0.878557i	$-0.658507\pi$
$503$	−21.4246	−0.955277	−0.477638	+	0.878557i	$0.658507\pi$
$504$	0	0
$505$	−15.0153	−0.668170
$506$	0	0
$507$	1.29715	0.0576086
$508$	0	0
$509$	18.4647	0.818431	0.409216	−	0.912438i	$-0.365802\pi$
$509$	18.4647	0.818431	0.409216	+	0.912438i	$0.365802\pi$
$510$	0	0
$511$	−3.95925	−0.175147
$512$	0	0
$513$	−1.33152	−0.0587880
$514$	0	0
$515$	29.5833	1.30360
$516$	0	0
$517$	12.6072	0.554464
$518$	0	0
$519$	0.770281	0.0338116
$520$	0	0
$521$	30.1181	1.31950	0.659750	−	0.751485i	$-0.270662\pi$
$521$	30.1181	1.31950	0.659750	+	0.751485i	$0.270662\pi$
$522$	0	0
$523$	−15.7486	−0.688637	−0.344319	−	0.938853i	$-0.611890\pi$
$523$	−15.7486	−0.688637	−0.344319	+	0.938853i	$0.611890\pi$
$524$	0	0
$525$	2.65790	0.116000
$526$	0	0
$527$	16.6440	0.725026
$528$	0	0
$529$	1.25372	0.0545097
$530$	0	0
$531$	−26.7435	−1.16057
$532$	0	0
$533$	3.69443	0.160024
$534$	0	0
$535$	65.3213	2.82408
$536$	0	0
$537$	−0.479593	−0.0206959
$538$	0	0
$539$	6.22841	0.268277
$540$	0	0
$541$	12.8465	0.552312	0.276156	−	0.961113i	$-0.410939\pi$
$541$	12.8465	0.552312	0.276156	+	0.961113i	$0.410939\pi$
$542$	0	0
$543$	1.21962	0.0523390
$544$	0	0
$545$	−42.4192	−1.81704
$546$	0	0
$547$	33.6567	1.43906	0.719528	−	0.694464i	$-0.244358\pi$
$547$	33.6567	1.43906	0.719528	+	0.694464i	$0.244358\pi$
$548$	0	0
$549$	11.8794	0.506999
$550$	0	0
$551$	3.35873	0.143087
$552$	0	0
$553$	1.15924	0.0492957
$554$	0	0
$555$	1.03197	0.0438047
$556$	0	0
$557$	5.34432	0.226446	0.113223	−	0.993570i	$-0.463883\pi$
$557$	5.34432	0.226446	0.113223	+	0.993570i	$0.463883\pi$
$558$	0	0
$559$	−32.5260	−1.37570
$560$	0	0
$561$	1.31828	0.0556580
$562$	0	0
$563$	0.420327	0.0177147	0.00885735	−	0.999961i	$-0.497181\pi$
$563$	0.420327	0.0177147	0.00885735	+	0.999961i	$0.497181\pi$
$564$	0	0
$565$	32.1560	1.35281
$566$	0	0
$567$	7.51190	0.315470
$568$	0	0
$569$	−12.7067	−0.532693	−0.266346	−	0.963877i	$-0.585817\pi$
$569$	−12.7067	−0.532693	−0.266346	+	0.963877i	$0.585817\pi$
$570$	0	0
$571$	3.30457	0.138292	0.0691461	−	0.997607i	$-0.477973\pi$
$571$	3.30457	0.138292	0.0691461	+	0.997607i	$0.477973\pi$
$572$	0	0
$573$	1.33136	0.0556184
$574$	0	0
$575$	−66.5883	−2.77692
$576$	0	0
$577$	26.5484	1.10523	0.552613	−	0.833438i	$-0.313631\pi$
$577$	26.5484	1.10523	0.552613	+	0.833438i	$0.313631\pi$
$578$	0	0
$579$	3.71495	0.154388
$580$	0	0
$581$	−8.53557	−0.354115
$582$	0	0
$583$	−13.1910	−0.546316
$584$	0	0
$585$	−55.0401	−2.27563
$586$	0	0
$587$	33.4284	1.37974	0.689868	−	0.723935i	$-0.257669\pi$
$587$	33.4284	1.37974	0.689868	+	0.723935i	$0.257669\pi$
$588$	0	0
$589$	−2.82544	−0.116420
$590$	0	0
$591$	1.78701	0.0735077
$592$	0	0
$593$	23.9136	0.982015	0.491007	−	0.871155i	$-0.336629\pi$
$593$	23.9136	0.982015	0.491007	+	0.871155i	$0.336629\pi$
$594$	0	0
$595$	22.2689	0.912936
$596$	0	0
$597$	−1.26492	−0.0517696
$598$	0	0
$599$	19.4047	0.792854	0.396427	−	0.918066i	$-0.370250\pi$
$599$	19.4047	0.792854	0.396427	+	0.918066i	$0.370250\pi$
$600$	0	0
$601$	−46.7688	−1.90774	−0.953870	−	0.300221i	$-0.902939\pi$
$601$	−46.7688	−1.90774	−0.953870	+	0.300221i	$0.902939\pi$
$602$	0	0
$603$	−34.6645	−1.41165
$604$	0	0
$605$	−4.30360	−0.174966
$606$	0	0
$607$	23.5697	0.956666	0.478333	−	0.878178i	$-0.341241\pi$
$607$	23.5697	0.956666	0.478333	+	0.878178i	$0.341241\pi$
$608$	0	0
$609$	0.660245	0.0267545
$610$	0	0
$611$	54.6582	2.21124
$612$	0	0
$613$	−43.2688	−1.74761	−0.873806	−	0.486275i	$-0.838355\pi$
$613$	−43.2688	−1.74761	−0.873806	+	0.486275i	$0.838355\pi$
$614$	0	0
$615$	0.820691	0.0330934
$616$	0	0
$617$	24.7733	0.997336	0.498668	−	0.866793i	$-0.333823\pi$
$617$	24.7733	0.997336	0.498668	+	0.866793i	$0.333823\pi$
$618$	0	0
$619$	−11.6011	−0.466288	−0.233144	−	0.972442i	$-0.574901\pi$
$619$	−11.6011	−0.466288	−0.233144	+	0.972442i	$0.574901\pi$
$620$	0	0
$621$	6.55747	0.263142
$622$	0	0
$623$	0.733825	0.0294001
$624$	0	0
$625$	90.2123	3.60849
$626$	0	0
$627$	−0.223788	−0.00893722
$628$	0	0
$629$	6.31207	0.251679
$630$	0	0
$631$	−6.44818	−0.256698	−0.128349	−	0.991729i	$-0.540968\pi$
$631$	−6.44818	−0.256698	−0.128349	+	0.991729i	$0.540968\pi$
$632$	0	0
$633$	6.22999	0.247620
$634$	0	0
$635$	17.3668	0.689181
$636$	0	0
$637$	27.0031	1.06990
$638$	0	0
$639$	10.0202	0.396395
$640$	0	0
$641$	−13.9412	−0.550646	−0.275323	−	0.961352i	$-0.588785\pi$
$641$	−13.9412	−0.550646	−0.275323	+	0.961352i	$0.588785\pi$
$642$	0	0
$643$	−9.53544	−0.376041	−0.188020	−	0.982165i	$-0.560207\pi$
$643$	−9.53544	−0.376041	−0.188020	+	0.982165i	$0.560207\pi$
$644$	0	0
$645$	−7.22541	−0.284500
$646$	0	0
$647$	−20.3498	−0.800032	−0.400016	−	0.916508i	$-0.630995\pi$
$647$	−20.3498	−0.800032	−0.400016	+	0.916508i	$0.630995\pi$
$648$	0	0
$649$	−9.06585	−0.355866
$650$	0	0
$651$	−0.555413	−0.0217683
$652$	0	0
$653$	12.3099	0.481723	0.240861	−	0.970560i	$-0.422570\pi$
$653$	12.3099	0.481723	0.240861	+	0.970560i	$0.422570\pi$
$654$	0	0
$655$	−22.0969	−0.863398
$656$	0	0
$657$	13.2962	0.518736
$658$	0	0
$659$	−34.6544	−1.34994	−0.674972	−	0.737844i	$-0.735844\pi$
$659$	−34.6544	−1.34994	−0.674972	+	0.737844i	$0.735844\pi$
$660$	0	0
$661$	2.63635	0.102542	0.0512711	−	0.998685i	$-0.483673\pi$
$661$	2.63635	0.102542	0.0512711	+	0.998685i	$0.483673\pi$
$662$	0	0
$663$	5.71539	0.221967
$664$	0	0
$665$	−3.78029	−0.146594
$666$	0	0
$667$	−16.5411	−0.640474
$668$	0	0
$669$	−5.38204	−0.208082
$670$	0	0
$671$	4.02701	0.155461
$672$	0	0
$673$	30.5579	1.17792	0.588960	−	0.808162i	$-0.299538\pi$
$673$	30.5579	1.17792	0.588960	+	0.808162i	$0.299538\pi$
$674$	0	0
$675$	−18.0035	−0.692953
$676$	0	0
$677$	49.5682	1.90506	0.952530	−	0.304444i	$-0.0984705\pi$
$677$	49.5682	1.90506	0.952530	+	0.304444i	$0.0984705\pi$
$678$	0	0
$679$	9.66539	0.370924
$680$	0	0
$681$	−2.14343	−0.0821365
$682$	0	0
$683$	46.5635	1.78170	0.890852	−	0.454293i	$-0.150108\pi$
$683$	46.5635	1.78170	0.890852	+	0.454293i	$0.150108\pi$
$684$	0	0
$685$	−6.86801	−0.262413
$686$	0	0
$687$	−4.35257	−0.166061
$688$	0	0
$689$	−57.1893	−2.17874
$690$	0	0
$691$	−13.7098	−0.521545	−0.260772	−	0.965400i	$-0.583977\pi$
$691$	−13.7098	−0.521545	−0.260772	+	0.965400i	$0.583977\pi$
$692$	0	0
$693$	2.59122	0.0984322
$694$	0	0
$695$	−79.9558	−3.03290
$696$	0	0
$697$	5.01977	0.190137
$698$	0	0
$699$	0.359334	0.0135913
$700$	0	0
$701$	−13.6175	−0.514324	−0.257162	−	0.966368i	$-0.582787\pi$
$701$	−13.6175	−0.514324	−0.257162	+	0.966368i	$0.582787\pi$
$702$	0	0
$703$	−1.07152	−0.0404130
$704$	0	0
$705$	12.1419	0.457291
$706$	0	0
$707$	3.06474	0.115262
$708$	0	0
$709$	45.8545	1.72210	0.861051	−	0.508519i	$-0.169807\pi$
$709$	45.8545	1.72210	0.861051	+	0.508519i	$0.169807\pi$
$710$	0	0
$711$	−3.89304	−0.146000
$712$	0	0
$713$	13.9147	0.521111
$714$	0	0
$715$	−18.6582	−0.697776
$716$	0	0
$717$	4.70620	0.175756
$718$	0	0
$719$	9.48982	0.353910	0.176955	−	0.984219i	$-0.443375\pi$
$719$	9.48982	0.353910	0.176955	+	0.984219i	$0.443375\pi$
$720$	0	0
$721$	−6.03821	−0.224875
$722$	0	0
$723$	1.97156	0.0733231
$724$	0	0
$725$	45.4134	1.68661
$726$	0	0
$727$	12.5652	0.466017	0.233008	−	0.972475i	$-0.425143\pi$
$727$	12.5652	0.466017	0.233008	+	0.972475i	$0.425143\pi$
$728$	0	0
$729$	−24.3331	−0.901227
$730$	0	0
$731$	−44.1944	−1.63459
$732$	0	0
$733$	26.2129	0.968194	0.484097	−	0.875014i	$-0.339148\pi$
$733$	26.2129	0.968194	0.484097	+	0.875014i	$0.339148\pi$
$734$	0	0
$735$	5.99854	0.221259
$736$	0	0
$737$	−11.7510	−0.432854
$738$	0	0
$739$	−18.3636	−0.675515	−0.337758	−	0.941233i	$-0.609668\pi$
$739$	−18.3636	−0.675515	−0.337758	+	0.941233i	$0.609668\pi$
$740$	0	0
$741$	−0.970226	−0.0356422
$742$	0	0
$743$	−28.3746	−1.04096	−0.520481	−	0.853873i	$-0.674247\pi$
$743$	−28.3746	−1.04096	−0.520481	+	0.853873i	$0.674247\pi$
$744$	0	0
$745$	−13.4700	−0.493504
$746$	0	0
$747$	28.6648	1.04879
$748$	0	0
$749$	−13.3326	−0.487164
$750$	0	0
$751$	−24.4085	−0.890680	−0.445340	−	0.895362i	$-0.646917\pi$
$751$	−24.4085	−0.890680	−0.445340	+	0.895362i	$0.646917\pi$
$752$	0	0
$753$	−1.14914	−0.0418769
$754$	0	0
$755$	−3.60657	−0.131257
$756$	0	0
$757$	22.9354	0.833602	0.416801	−	0.908998i	$-0.363151\pi$
$757$	22.9354	0.833602	0.416801	+	0.908998i	$0.363151\pi$
$758$	0	0
$759$	1.10211	0.0400041
$760$	0	0
$761$	9.68821	0.351197	0.175599	−	0.984462i	$-0.443814\pi$
$761$	9.68821	0.351197	0.175599	+	0.984462i	$0.443814\pi$
$762$	0	0
$763$	8.65813	0.313446
$764$	0	0
$765$	−74.7851	−2.70386
$766$	0	0
$767$	−39.3048	−1.41921
$768$	0	0
$769$	27.1159	0.977824	0.488912	−	0.872333i	$-0.337394\pi$
$769$	27.1159	0.977824	0.488912	+	0.872333i	$0.337394\pi$
$770$	0	0
$771$	6.37792	0.229695
$772$	0	0
$773$	−12.5136	−0.450083	−0.225041	−	0.974349i	$-0.572252\pi$
$773$	−12.5136	−0.450083	−0.225041	+	0.974349i	$0.572252\pi$
$774$	0	0
$775$	−38.2027	−1.37228
$776$	0	0
$777$	−0.210634	−0.00755646
$778$	0	0
$779$	−0.852140	−0.0305311
$780$	0	0
$781$	3.39679	0.121547
$782$	0	0
$783$	−4.47221	−0.159824
$784$	0	0
$785$	16.5663	0.591275
$786$	0	0
$787$	−47.2864	−1.68558	−0.842790	−	0.538243i	$-0.819088\pi$
$787$	−47.2864	−1.68558	−0.842790	+	0.538243i	$0.819088\pi$
$788$	0	0
$789$	−5.37516	−0.191361
$790$	0	0
$791$	−6.56331	−0.233365
$792$	0	0
$793$	17.4590	0.619988
$794$	0	0
$795$	−12.7042	−0.450571
$796$	0	0
$797$	38.7521	1.37267	0.686335	−	0.727285i	$-0.259218\pi$
$797$	38.7521	1.37267	0.686335	+	0.727285i	$0.259218\pi$
$798$	0	0
$799$	74.2663	2.62735
$800$	0	0
$801$	−2.46439	−0.0870748
$802$	0	0
$803$	4.50733	0.159060
$804$	0	0
$805$	18.6172	0.656171
$806$	0	0
$807$	−2.58222	−0.0908985
$808$	0	0
$809$	−10.3475	−0.363799	−0.181899	−	0.983317i	$-0.558224\pi$
$809$	−10.3475	−0.363799	−0.181899	+	0.983317i	$0.558224\pi$
$810$	0	0
$811$	0.0573511	0.00201387	0.00100694	−	0.999999i	$-0.499679\pi$
$811$	0.0573511	0.00201387	0.00100694	+	0.999999i	$0.499679\pi$
$812$	0	0
$813$	−5.53676	−0.194183
$814$	0	0
$815$	8.92499	0.312629
$816$	0	0
$817$	7.50229	0.262472
$818$	0	0
$819$	11.2342	0.392553
$820$	0	0
$821$	35.7245	1.24679	0.623397	−	0.781906i	$-0.285752\pi$
$821$	35.7245	1.24679	0.623397	+	0.781906i	$0.285752\pi$
$822$	0	0
$823$	6.98284	0.243407	0.121703	−	0.992567i	$-0.461164\pi$
$823$	6.98284	0.243407	0.121703	+	0.992567i	$0.461164\pi$
$824$	0	0
$825$	−3.02583	−0.105346
$826$	0	0
$827$	−44.2279	−1.53795	−0.768977	−	0.639276i	$-0.779234\pi$
$827$	−44.2279	−1.53795	−0.768977	+	0.639276i	$0.779234\pi$
$828$	0	0
$829$	−2.86145	−0.0993823	−0.0496911	−	0.998765i	$-0.515824\pi$
$829$	−2.86145	−0.0993823	−0.0496911	+	0.998765i	$0.515824\pi$
$830$	0	0
$831$	−5.55096	−0.192561
$832$	0	0
$833$	36.6902	1.27124
$834$	0	0
$835$	−55.9648	−1.93674
$836$	0	0
$837$	3.76213	0.130038
$838$	0	0
$839$	23.6931	0.817976	0.408988	−	0.912540i	$-0.365882\pi$
$839$	23.6931	0.817976	0.408988	+	0.912540i	$0.365882\pi$
$840$	0	0
$841$	−17.7189	−0.610997
$842$	0	0
$843$	3.72110	0.128162
$844$	0	0
$845$	−24.9452	−0.858141
$846$	0	0
$847$	0.878402	0.0301823
$848$	0	0
$849$	4.55319	0.156265
$850$	0	0
$851$	5.27701	0.180894
$852$	0	0
$853$	−9.62879	−0.329683	−0.164842	−	0.986320i	$-0.552711\pi$
$853$	−9.62879	−0.329683	−0.164842	+	0.986320i	$0.552711\pi$
$854$	0	0
$855$	12.6953	0.434169
$856$	0	0
$857$	8.24206	0.281543	0.140772	−	0.990042i	$-0.455042\pi$
$857$	8.24206	0.281543	0.140772	+	0.990042i	$0.455042\pi$
$858$	0	0
$859$	−16.0972	−0.549230	−0.274615	−	0.961554i	$-0.588550\pi$
$859$	−16.0972	−0.549230	−0.274615	+	0.961554i	$0.588550\pi$
$860$	0	0
$861$	−0.167510	−0.00570873
$862$	0	0
$863$	−23.5053	−0.800129	−0.400064	−	0.916487i	$-0.631012\pi$
$863$	−23.5053	−0.800129	−0.400064	+	0.916487i	$0.631012\pi$
$864$	0	0
$865$	−14.8131	−0.503659
$866$	0	0
$867$	3.96134	0.134534
$868$	0	0
$869$	−1.31971	−0.0447681
$870$	0	0
$871$	−50.9462	−1.72625
$872$	0	0
$873$	−32.4591	−1.09857
$874$	0	0
$875$	−32.2119	−1.08896
$876$	0	0
$877$	−0.887794	−0.0299787	−0.0149893	−	0.999888i	$-0.504771\pi$
$877$	−0.887794	−0.0299787	−0.0149893	+	0.999888i	$0.504771\pi$
$878$	0	0
$879$	−5.62766	−0.189816
$880$	0	0
$881$	25.0213	0.842991	0.421495	−	0.906831i	$-0.361505\pi$
$881$	25.0213	0.842991	0.421495	+	0.906831i	$0.361505\pi$
$882$	0	0
$883$	36.3481	1.22321	0.611605	−	0.791163i	$-0.290524\pi$
$883$	36.3481	1.22321	0.611605	+	0.791163i	$0.290524\pi$
$884$	0	0
$885$	−8.73126	−0.293498
$886$	0	0
$887$	21.2102	0.712170	0.356085	−	0.934454i	$-0.384111\pi$
$887$	21.2102	0.712170	0.356085	+	0.934454i	$0.384111\pi$
$888$	0	0
$889$	−3.54472	−0.118886
$890$	0	0
$891$	−8.55178	−0.286495
$892$	0	0
$893$	−12.6072	−0.421884
$894$	0	0
$895$	9.22292	0.308288
$896$	0	0
$897$	4.77818	0.159539
$898$	0	0
$899$	−9.48989	−0.316506
$900$	0	0
$901$	−77.7054	−2.58874
$902$	0	0
$903$	1.47477	0.0490772
$904$	0	0
$905$	−23.4542	−0.779644
$906$	0	0
$907$	54.4639	1.80844	0.904222	−	0.427064i	$-0.140452\pi$
$907$	54.4639	1.80844	0.904222	+	0.427064i	$0.140452\pi$
$908$	0	0
$909$	−10.2923	−0.341373
$910$	0	0
$911$	−22.0402	−0.730224	−0.365112	−	0.930964i	$-0.618969\pi$
$911$	−22.0402	−0.730224	−0.365112	+	0.930964i	$0.618969\pi$
$912$	0	0
$913$	9.71715	0.321591
$914$	0	0
$915$	3.87839	0.128216
$916$	0	0
$917$	4.51017	0.148939
$918$	0	0
$919$	30.3739	1.00194	0.500972	−	0.865463i	$-0.332976\pi$
$919$	30.3739	1.00194	0.500972	+	0.865463i	$0.332976\pi$
$920$	0	0
$921$	5.16012	0.170032
$922$	0	0
$923$	14.7267	0.484735
$924$	0	0
$925$	−14.4880	−0.476361
$926$	0	0
$927$	20.2780	0.666017
$928$	0	0
$929$	48.1541	1.57989	0.789943	−	0.613181i	$-0.210110\pi$
$929$	48.1541	1.57989	0.789943	+	0.613181i	$0.210110\pi$
$930$	0	0
$931$	−6.22841	−0.204128
$932$	0	0
$933$	4.06790	0.133177
$934$	0	0
$935$	−25.3516	−0.829085
$936$	0	0
$937$	−4.55338	−0.148752	−0.0743761	−	0.997230i	$-0.523697\pi$
$937$	−4.55338	−0.148752	−0.0743761	+	0.997230i	$0.523697\pi$
$938$	0	0
$939$	−3.29638	−0.107573
$940$	0	0
$941$	16.6963	0.544285	0.272143	−	0.962257i	$-0.412268\pi$
$941$	16.6963	0.544285	0.272143	+	0.962257i	$0.412268\pi$
$942$	0	0
$943$	4.19663	0.136661
$944$	0	0
$945$	5.03353	0.163741
$946$	0	0
$947$	−7.22872	−0.234902	−0.117451	−	0.993079i	$-0.537472\pi$
$947$	−7.22872	−0.234902	−0.117451	+	0.993079i	$0.537472\pi$
$948$	0	0
$949$	19.5414	0.634341
$950$	0	0
$951$	−1.48107	−0.0480269
$952$	0	0
$953$	−57.9719	−1.87789	−0.938947	−	0.344063i	$-0.888197\pi$
$953$	−57.9719	−1.87789	−0.938947	+	0.344063i	$0.888197\pi$
$954$	0	0
$955$	−25.6030	−0.828495
$956$	0	0
$957$	−0.751643	−0.0242972
$958$	0	0
$959$	1.40182	0.0452671
$960$	0	0
$961$	−23.0169	−0.742480
$962$	0	0
$963$	44.7747	1.44284
$964$	0	0
$965$	−71.4413	−2.29978
$966$	0	0
$967$	−29.6896	−0.954754	−0.477377	−	0.878699i	$-0.658412\pi$
$967$	−29.6896	−0.954754	−0.477377	+	0.878699i	$0.658412\pi$
$968$	0	0
$969$	−1.31828	−0.0423494
$970$	0	0
$971$	2.71173	0.0870234	0.0435117	−	0.999053i	$-0.486145\pi$
$971$	2.71173	0.0870234	0.0435117	+	0.999053i	$0.486145\pi$
$972$	0	0
$973$	16.3197	0.523185
$974$	0	0
$975$	−13.1184	−0.420126
$976$	0	0
$977$	17.0111	0.544235	0.272117	−	0.962264i	$-0.412276\pi$
$977$	17.0111	0.544235	0.272117	+	0.962264i	$0.412276\pi$
$978$	0	0
$979$	−0.835408	−0.0266998
$980$	0	0
$981$	−29.0764	−0.928339
$982$	0	0
$983$	−31.6598	−1.00979	−0.504896	−	0.863180i	$-0.668469\pi$
$983$	−31.6598	−1.00979	−0.504896	+	0.863180i	$0.668469\pi$
$984$	0	0
$985$	−34.3655	−1.09498
$986$	0	0
$987$	−2.47827	−0.0788842
$988$	0	0
$989$	−36.9473	−1.17486
$990$	0	0
$991$	−38.7900	−1.23220	−0.616102	−	0.787666i	$-0.711289\pi$
$991$	−38.7900	−1.23220	−0.616102	+	0.787666i	$0.711289\pi$
$992$	0	0
$993$	−7.97941	−0.253219
$994$	0	0
$995$	24.3253	0.771163
$996$	0	0
$997$	52.8650	1.67425	0.837126	−	0.547010i	$-0.184234\pi$
$997$	52.8650	1.67425	0.837126	+	0.547010i	$0.184234\pi$
$998$	0	0
$999$	1.42674	0.0451402

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.bb.1.5		9
4.3	odd	2		1672.2.a.k.1.5	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.k.1.5	✓	9	4.3	odd	2
3344.2.a.bb.1.5		9	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q+0.223788 q^{3} -4.30360 q^{5} +0.878402 q^{7} -2.94992 q^{9} +O(q^{10})\)	\(q+0.223788 q^{3} -4.30360 q^{5} +0.878402 q^{7} -2.94992 q^{9} -1.00000 q^{11} -4.33548 q^{13} -0.963093 q^{15} -5.89078 q^{17} +1.00000 q^{19} +0.196576 q^{21} -4.92481 q^{23} +13.5210 q^{25} -1.33152 q^{27} +3.35873 q^{29} -2.82544 q^{31} -0.223788 q^{33} -3.78029 q^{35} -1.07152 q^{37} -0.970226 q^{39} -0.852140 q^{41} +7.50229 q^{43} +12.6953 q^{45} -12.6072 q^{47} -6.22841 q^{49} -1.31828 q^{51} +13.1910 q^{53} +4.30360 q^{55} +0.223788 q^{57} +9.06585 q^{59} -4.02701 q^{61} -2.59122 q^{63} +18.6582 q^{65} +11.7510 q^{67} -1.10211 q^{69} -3.39679 q^{71} -4.50733 q^{73} +3.02583 q^{75} -0.878402 q^{77} +1.31971 q^{79} +8.55178 q^{81} -9.71715 q^{83} +25.3516 q^{85} +0.751643 q^{87} +0.835408 q^{89} -3.80829 q^{91} -0.632299 q^{93} -4.30360 q^{95} +11.0034 q^{97} +2.94992 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{3} + 6 q^{5} - 3 q^{7} + 18 q^{9}+O(q^{10}) \) 9 * q - q^3 + 6 * q^5 - 3 * q^7 + 18 * q^9	\( 9 q - q^{3} + 6 q^{5} - 3 q^{7} + 18 q^{9} - 9 q^{11} + q^{13} - 6 q^{15} + 7 q^{17} + 9 q^{19} + 3 q^{21} - 13 q^{23} + 31 q^{25} + 5 q^{27} + 9 q^{29} + 4 q^{31} + q^{33} + 4 q^{35} + 24 q^{37} - 13 q^{39} - 6 q^{41} + 14 q^{43} + 26 q^{45} - 24 q^{47} + 20 q^{49} + 33 q^{51} + 19 q^{53} - 6 q^{55} - q^{57} + 19 q^{59} + 28 q^{61} - 16 q^{63} + 16 q^{65} - 5 q^{67} + 35 q^{69} - 16 q^{71} + 15 q^{73} - 3 q^{75} + 3 q^{77} - 2 q^{79} + 37 q^{81} - 8 q^{83} + 20 q^{85} - 23 q^{87} + 12 q^{89} + 29 q^{91} + 44 q^{93} + 6 q^{95} - 4 q^{97} - 18 q^{99}+O(q^{100}) \) 9 * q - q^3 + 6 * q^5 - 3 * q^7 + 18 * q^9 - 9 * q^11 + q^13 - 6 * q^15 + 7 * q^17 + 9 * q^19 + 3 * q^21 - 13 * q^23 + 31 * q^25 + 5 * q^27 + 9 * q^29 + 4 * q^31 + q^33 + 4 * q^35 + 24 * q^37 - 13 * q^39 - 6 * q^41 + 14 * q^43 + 26 * q^45 - 24 * q^47 + 20 * q^49 + 33 * q^51 + 19 * q^53 - 6 * q^55 - q^57 + 19 * q^59 + 28 * q^61 - 16 * q^63 + 16 * q^65 - 5 * q^67 + 35 * q^69 - 16 * q^71 + 15 * q^73 - 3 * q^75 + 3 * q^77 - 2 * q^79 + 37 * q^81 - 8 * q^83 + 20 * q^85 - 23 * q^87 + 12 * q^89 + 29 * q^91 + 44 * q^93 + 6 * q^95 - 4 * q^97 - 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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