LMFDB - Embedded newform 3344.2.a.ba.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3344,2,Mod(1,3344)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [7,0,-2,0,2,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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			1.4
Root			$2.61330$ 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)

$f(q)$	$=$	$q-1.19599 q^{3} +4.07680 q^{5} -3.61829 q^{7} -1.56960 q^{9} +1.00000 q^{11} -1.47857 q^{13} -4.87582 q^{15} -3.27003 q^{17} -1.00000 q^{19} +4.32745 q^{21} +7.45793 q^{23} +11.6203 q^{25} +5.46521 q^{27} +1.02535 q^{29} -1.64921 q^{31} -1.19599 q^{33} -14.7511 q^{35} -6.71293 q^{37} +1.76836 q^{39} -3.92451 q^{41} -5.38113 q^{43} -6.39896 q^{45} +3.71597 q^{47} +6.09205 q^{49} +3.91093 q^{51} -0.102902 q^{53} +4.07680 q^{55} +1.19599 q^{57} -13.2986 q^{59} -6.49664 q^{61} +5.67929 q^{63} -6.02783 q^{65} +3.70989 q^{67} -8.91962 q^{69} -6.32968 q^{71} -1.37759 q^{73} -13.8978 q^{75} -3.61829 q^{77} -13.6725 q^{79} -1.82753 q^{81} -5.44061 q^{83} -13.3313 q^{85} -1.22631 q^{87} +12.1357 q^{89} +5.34990 q^{91} +1.97244 q^{93} -4.07680 q^{95} -13.7910 q^{97} -1.56960 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots + 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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.19599	−0.690506	−0.345253	−	0.938510i	$-0.612207\pi$
$3$	−1.19599	−0.690506	−0.345253	+	0.938510i	$0.612207\pi$
$4$	0	0
$5$	4.07680	1.82320	0.911600	−	0.411078i	$-0.134847\pi$
$5$	4.07680	1.82320	0.911600	+	0.411078i	$0.134847\pi$
$6$	0	0
$7$	−3.61829	−1.36759	−0.683793	−	0.729676i	$-0.739671\pi$
$7$	−3.61829	−1.36759	−0.683793	+	0.729676i	$0.739671\pi$
$8$	0	0
$9$	−1.56960	−0.523201
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.47857	−0.410081	−0.205041	−	0.978753i	$-0.565733\pi$
$13$	−1.47857	−0.410081	−0.205041	+	0.978753i	$0.565733\pi$
$14$	0	0
$15$	−4.87582	−1.25893
$16$	0	0
$17$	−3.27003	−0.793099	−0.396549	−	0.918013i	$-0.629792\pi$
$17$	−3.27003	−0.793099	−0.396549	+	0.918013i	$0.629792\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	4.32745	0.944327
$22$	0	0
$23$	7.45793	1.55509	0.777543	−	0.628830i	$-0.216466\pi$
$23$	7.45793	1.55509	0.777543	+	0.628830i	$0.216466\pi$
$24$	0	0
$25$	11.6203	2.32406
$26$	0	0
$27$	5.46521	1.05178
$28$	0	0
$29$	1.02535	0.190403	0.0952013	−	0.995458i	$-0.469651\pi$
$29$	1.02535	0.190403	0.0952013	+	0.995458i	$0.469651\pi$
$30$	0	0
$31$	−1.64921	−0.296207	−0.148104	−	0.988972i	$-0.547317\pi$
$31$	−1.64921	−0.296207	−0.148104	+	0.988972i	$0.547317\pi$
$32$	0	0
$33$	−1.19599	−0.208195
$34$	0	0
$35$	−14.7511	−2.49338
$36$	0	0
$37$	−6.71293	−1.10360	−0.551799	−	0.833977i	$-0.686059\pi$
$37$	−6.71293	−1.10360	−0.551799	+	0.833977i	$0.686059\pi$
$38$	0	0
$39$	1.76836	0.283164
$40$	0	0
$41$	−3.92451	−0.612905	−0.306453	−	0.951886i	$-0.599142\pi$
$41$	−3.92451	−0.612905	−0.306453	+	0.951886i	$0.599142\pi$
$42$	0	0
$43$	−5.38113	−0.820614	−0.410307	−	0.911947i	$-0.634578\pi$
$43$	−5.38113	−0.820614	−0.410307	+	0.911947i	$0.634578\pi$
$44$	0	0
$45$	−6.39896	−0.953901
$46$	0	0
$47$	3.71597	0.542030	0.271015	−	0.962575i	$-0.412641\pi$
$47$	3.71597	0.542030	0.271015	+	0.962575i	$0.412641\pi$
$48$	0	0
$49$	6.09205	0.870292
$50$	0	0
$51$	3.91093	0.547640
$52$	0	0
$53$	−0.102902	−0.0141347	−0.00706733	−	0.999975i	$-0.502250\pi$
$53$	−0.102902	−0.0141347	−0.00706733	+	0.999975i	$0.502250\pi$
$54$	0	0
$55$	4.07680	0.549716
$56$	0	0
$57$	1.19599	0.158413
$58$	0	0
$59$	−13.2986	−1.73134	−0.865668	−	0.500619i	$-0.833106\pi$
$59$	−13.2986	−1.73134	−0.865668	+	0.500619i	$0.833106\pi$
$60$	0	0
$61$	−6.49664	−0.831809	−0.415905	−	0.909408i	$-0.636535\pi$
$61$	−6.49664	−0.831809	−0.415905	+	0.909408i	$0.636535\pi$
$62$	0	0
$63$	5.67929	0.715523
$64$	0	0
$65$	−6.02783	−0.747661
$66$	0	0
$67$	3.70989	0.453235	0.226618	−	0.973984i	$-0.427233\pi$
$67$	3.70989	0.453235	0.226618	+	0.973984i	$0.427233\pi$
$68$	0	0
$69$	−8.91962	−1.07380
$70$	0	0
$71$	−6.32968	−0.751194	−0.375597	−	0.926783i	$-0.622562\pi$
$71$	−6.32968	−0.751194	−0.375597	+	0.926783i	$0.622562\pi$
$72$	0	0
$73$	−1.37759	−0.161235	−0.0806173	−	0.996745i	$-0.525689\pi$
$73$	−1.37759	−0.161235	−0.0806173	+	0.996745i	$0.525689\pi$
$74$	0	0
$75$	−13.8978	−1.60478
$76$	0	0
$77$	−3.61829	−0.412343
$78$	0	0
$79$	−13.6725	−1.53828	−0.769141	−	0.639079i	$-0.779316\pi$
$79$	−13.6725	−1.53828	−0.769141	+	0.639079i	$0.779316\pi$
$80$	0	0
$81$	−1.82753	−0.203059
$82$	0	0
$83$	−5.44061	−0.597184	−0.298592	−	0.954381i	$-0.596517\pi$
$83$	−5.44061	−0.597184	−0.298592	+	0.954381i	$0.596517\pi$
$84$	0	0
$85$	−13.3313	−1.44598
$86$	0	0
$87$	−1.22631	−0.131474
$88$	0	0
$89$	12.1357	1.28638	0.643191	−	0.765706i	$-0.277610\pi$
$89$	12.1357	1.28638	0.643191	+	0.765706i	$0.277610\pi$
$90$	0	0
$91$	5.34990	0.560822
$92$	0	0
$93$	1.97244	0.204533
$94$	0	0
$95$	−4.07680	−0.418271
$96$	0	0
$97$	−13.7910	−1.40026	−0.700131	−	0.714014i	$-0.746875\pi$
$97$	−13.7910	−1.40026	−0.700131	+	0.714014i	$0.746875\pi$
$98$	0	0
$99$	−1.56960	−0.157751
$100$	0	0
$101$	−11.0029	−1.09483	−0.547413	−	0.836863i	$-0.684387\pi$
$101$	−11.0029	−1.09483	−0.547413	+	0.836863i	$0.684387\pi$
$102$	0	0
$103$	4.99191	0.491867	0.245934	−	0.969287i	$-0.420906\pi$
$103$	4.99191	0.491867	0.245934	+	0.969287i	$0.420906\pi$
$104$	0	0
$105$	17.6421	1.72170
$106$	0	0
$107$	−7.31345	−0.707018	−0.353509	−	0.935431i	$-0.615012\pi$
$107$	−7.31345	−0.707018	−0.353509	+	0.935431i	$0.615012\pi$
$108$	0	0
$109$	−1.44482	−0.138389	−0.0691944	−	0.997603i	$-0.522043\pi$
$109$	−1.44482	−0.138389	−0.0691944	+	0.997603i	$0.522043\pi$
$110$	0	0
$111$	8.02861	0.762042
$112$	0	0
$113$	−12.0369	−1.13234	−0.566169	−	0.824289i	$-0.691575\pi$
$113$	−12.0369	−1.13234	−0.566169	+	0.824289i	$0.691575\pi$
$114$	0	0
$115$	30.4045	2.83523
$116$	0	0
$117$	2.32077	0.214555
$118$	0	0
$119$	11.8319	1.08463
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	4.69368	0.423215
$124$	0	0
$125$	26.9897	2.41403
$126$	0	0
$127$	4.69692	0.416784	0.208392	−	0.978045i	$-0.433177\pi$
$127$	4.69692	0.416784	0.208392	+	0.978045i	$0.433177\pi$
$128$	0	0
$129$	6.43578	0.566639
$130$	0	0
$131$	−3.74466	−0.327173	−0.163586	−	0.986529i	$-0.552306\pi$
$131$	−3.74466	−0.327173	−0.163586	+	0.986529i	$0.552306\pi$
$132$	0	0
$133$	3.61829	0.313746
$134$	0	0
$135$	22.2806	1.91761
$136$	0	0
$137$	−15.7595	−1.34643	−0.673213	−	0.739449i	$-0.735086\pi$
$137$	−15.7595	−1.34643	−0.673213	+	0.739449i	$0.735086\pi$
$138$	0	0
$139$	2.52822	0.214440	0.107220	−	0.994235i	$-0.465805\pi$
$139$	2.52822	0.214440	0.107220	+	0.994235i	$0.465805\pi$
$140$	0	0
$141$	−4.44427	−0.374275
$142$	0	0
$143$	−1.47857	−0.123644
$144$	0	0
$145$	4.18015	0.347142
$146$	0	0
$147$	−7.28604	−0.600942
$148$	0	0
$149$	1.84902	0.151477	0.0757387	−	0.997128i	$-0.475869\pi$
$149$	1.84902	0.151477	0.0757387	+	0.997128i	$0.475869\pi$
$150$	0	0
$151$	15.3184	1.24659	0.623296	−	0.781986i	$-0.285793\pi$
$151$	15.3184	1.24659	0.623296	+	0.781986i	$0.285793\pi$
$152$	0	0
$153$	5.13265	0.414950
$154$	0	0
$155$	−6.72351	−0.540045
$156$	0	0
$157$	24.6631	1.96833	0.984165	−	0.177254i	$-0.0567215\pi$
$157$	24.6631	1.96833	0.984165	+	0.177254i	$0.0567215\pi$
$158$	0	0
$159$	0.123070	0.00976006
$160$	0	0
$161$	−26.9850	−2.12671
$162$	0	0
$163$	3.72149	0.291490	0.145745	−	0.989322i	$-0.453442\pi$
$163$	3.72149	0.291490	0.145745	+	0.989322i	$0.453442\pi$
$164$	0	0
$165$	−4.87582	−0.379582
$166$	0	0
$167$	−2.64758	−0.204876	−0.102438	−	0.994739i	$-0.532664\pi$
$167$	−2.64758	−0.204876	−0.102438	+	0.994739i	$0.532664\pi$
$168$	0	0
$169$	−10.8138	−0.831833
$170$	0	0
$171$	1.56960	0.120031
$172$	0	0
$173$	7.34552	0.558469	0.279235	−	0.960223i	$-0.409919\pi$
$173$	7.34552	0.558469	0.279235	+	0.960223i	$0.409919\pi$
$174$	0	0
$175$	−42.0457	−3.17835
$176$	0	0
$177$	15.9051	1.19550
$178$	0	0
$179$	−9.55394	−0.714095	−0.357047	−	0.934086i	$-0.616217\pi$
$179$	−9.55394	−0.714095	−0.357047	+	0.934086i	$0.616217\pi$
$180$	0	0
$181$	6.02638	0.447937	0.223969	−	0.974596i	$-0.428099\pi$
$181$	6.02638	0.447937	0.223969	+	0.974596i	$0.428099\pi$
$182$	0	0
$183$	7.76993	0.574369
$184$	0	0
$185$	−27.3673	−2.01208
$186$	0	0
$187$	−3.27003	−0.239128
$188$	0	0
$189$	−19.7747	−1.43840
$190$	0	0
$191$	−17.7069	−1.28123	−0.640613	−	0.767864i	$-0.721320\pi$
$191$	−17.7069	−1.28123	−0.640613	+	0.767864i	$0.721320\pi$
$192$	0	0
$193$	3.69348	0.265863	0.132931	−	0.991125i	$-0.457561\pi$
$193$	3.69348	0.265863	0.132931	+	0.991125i	$0.457561\pi$
$194$	0	0
$195$	7.20924	0.516264
$196$	0	0
$197$	−25.7789	−1.83667	−0.918336	−	0.395802i	$-0.870467\pi$
$197$	−25.7789	−1.83667	−0.918336	+	0.395802i	$0.870467\pi$
$198$	0	0
$199$	−18.8953	−1.33945	−0.669726	−	0.742608i	$-0.733589\pi$
$199$	−18.8953	−1.33945	−0.669726	+	0.742608i	$0.733589\pi$
$200$	0	0
$201$	−4.43700	−0.312962
$202$	0	0
$203$	−3.71002	−0.260392
$204$	0	0
$205$	−15.9994	−1.11745
$206$	0	0
$207$	−11.7060	−0.813623
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	10.1993	0.702150	0.351075	−	0.936347i	$-0.385816\pi$
$211$	10.1993	0.702150	0.351075	+	0.936347i	$0.385816\pi$
$212$	0	0
$213$	7.57024	0.518704
$214$	0	0
$215$	−21.9378	−1.49614
$216$	0	0
$217$	5.96733	0.405089
$218$	0	0
$219$	1.64759	0.111334
$220$	0	0
$221$	4.83497	0.325235
$222$	0	0
$223$	0.262700	0.0175917	0.00879584	−	0.999961i	$-0.497200\pi$
$223$	0.262700	0.0175917	0.00879584	+	0.999961i	$0.497200\pi$
$224$	0	0
$225$	−18.2393	−1.21595
$226$	0	0
$227$	−27.3256	−1.81367	−0.906834	−	0.421489i	$-0.861508\pi$
$227$	−27.3256	−1.81367	−0.906834	+	0.421489i	$0.861508\pi$
$228$	0	0
$229$	−5.53171	−0.365546	−0.182773	−	0.983155i	$-0.558507\pi$
$229$	−5.53171	−0.365546	−0.182773	+	0.983155i	$0.558507\pi$
$230$	0	0
$231$	4.32745	0.284725
$232$	0	0
$233$	27.4733	1.79984	0.899918	−	0.436059i	$-0.143626\pi$
$233$	27.4733	1.79984	0.899918	+	0.436059i	$0.143626\pi$
$234$	0	0
$235$	15.1493	0.988229
$236$	0	0
$237$	16.3523	1.06219
$238$	0	0
$239$	−1.40339	−0.0907781	−0.0453890	−	0.998969i	$-0.514453\pi$
$239$	−1.40339	−0.0907781	−0.0453890	+	0.998969i	$0.514453\pi$
$240$	0	0
$241$	−20.7696	−1.33789	−0.668944	−	0.743312i	$-0.733254\pi$
$241$	−20.7696	−1.33789	−0.668944	+	0.743312i	$0.733254\pi$
$242$	0	0
$243$	−14.2099	−0.911566
$244$	0	0
$245$	24.8361	1.58672
$246$	0	0
$247$	1.47857	0.0940791
$248$	0	0
$249$	6.50692	0.412359
$250$	0	0
$251$	1.29936	0.0820148	0.0410074	−	0.999159i	$-0.486943\pi$
$251$	1.29936	0.0820148	0.0410074	+	0.999159i	$0.486943\pi$
$252$	0	0
$253$	7.45793	0.468876
$254$	0	0
$255$	15.9441	0.998457
$256$	0	0
$257$	3.41219	0.212847	0.106423	−	0.994321i	$-0.466060\pi$
$257$	3.41219	0.212847	0.106423	+	0.994321i	$0.466060\pi$
$258$	0	0
$259$	24.2893	1.50927
$260$	0	0
$261$	−1.60939	−0.0996189
$262$	0	0
$263$	−14.2418	−0.878189	−0.439094	−	0.898441i	$-0.644701\pi$
$263$	−14.2418	−0.878189	−0.439094	+	0.898441i	$0.644701\pi$
$264$	0	0
$265$	−0.419510	−0.0257703
$266$	0	0
$267$	−14.5142	−0.888254
$268$	0	0
$269$	5.39477	0.328925	0.164462	−	0.986383i	$-0.447411\pi$
$269$	5.39477	0.328925	0.164462	+	0.986383i	$0.447411\pi$
$270$	0	0
$271$	11.0624	0.671995	0.335998	−	0.941863i	$-0.390927\pi$
$271$	11.0624	0.671995	0.335998	+	0.941863i	$0.390927\pi$
$272$	0	0
$273$	−6.39843	−0.387251
$274$	0	0
$275$	11.6203	0.700731
$276$	0	0
$277$	14.0808	0.846036	0.423018	−	0.906121i	$-0.360971\pi$
$277$	14.0808	0.846036	0.423018	+	0.906121i	$0.360971\pi$
$278$	0	0
$279$	2.58861	0.154976
$280$	0	0
$281$	−10.8199	−0.645459	−0.322729	−	0.946491i	$-0.604600\pi$
$281$	−10.8199	−0.645459	−0.322729	+	0.946491i	$0.604600\pi$
$282$	0	0
$283$	−1.90947	−0.113506	−0.0567532	−	0.998388i	$-0.518075\pi$
$283$	−1.90947	−0.113506	−0.0567532	+	0.998388i	$0.518075\pi$
$284$	0	0
$285$	4.87582	0.288819
$286$	0	0
$287$	14.2000	0.838201
$288$	0	0
$289$	−6.30690	−0.370994
$290$	0	0
$291$	16.4939	0.966890
$292$	0	0
$293$	3.63550	0.212388	0.106194	−	0.994345i	$-0.466133\pi$
$293$	3.63550	0.212388	0.106194	+	0.994345i	$0.466133\pi$
$294$	0	0
$295$	−54.2159	−3.15657
$296$	0	0
$297$	5.46521	0.317124
$298$	0	0
$299$	−11.0271	−0.637712
$300$	0	0
$301$	19.4705	1.12226
$302$	0	0
$303$	13.1593	0.755984
$304$	0	0
$305$	−26.4855	−1.51656
$306$	0	0
$307$	23.5329	1.34310	0.671548	−	0.740961i	$-0.265630\pi$
$307$	23.5329	1.34310	0.671548	+	0.740961i	$0.265630\pi$
$308$	0	0
$309$	−5.97028	−0.339637
$310$	0	0
$311$	16.0026	0.907425	0.453713	−	0.891148i	$-0.350099\pi$
$311$	16.0026	0.907425	0.453713	+	0.891148i	$0.350099\pi$
$312$	0	0
$313$	16.0034	0.904566	0.452283	−	0.891875i	$-0.350610\pi$
$313$	16.0034	0.904566	0.452283	+	0.891875i	$0.350610\pi$
$314$	0	0
$315$	23.1533	1.30454
$316$	0	0
$317$	−20.8766	−1.17255	−0.586273	−	0.810113i	$-0.699405\pi$
$317$	−20.8766	−1.17255	−0.586273	+	0.810113i	$0.699405\pi$
$318$	0	0
$319$	1.02535	0.0574086
$320$	0	0
$321$	8.74683	0.488200
$322$	0	0
$323$	3.27003	0.181949
$324$	0	0
$325$	−17.1814	−0.953054
$326$	0	0
$327$	1.72800	0.0955584
$328$	0	0
$329$	−13.4455	−0.741273
$330$	0	0
$331$	−15.1136	−0.830721	−0.415360	−	0.909657i	$-0.636345\pi$
$331$	−15.1136	−0.830721	−0.415360	+	0.909657i	$0.636345\pi$
$332$	0	0
$333$	10.5366	0.577404
$334$	0	0
$335$	15.1245	0.826339
$336$	0	0
$337$	−12.2766	−0.668751	−0.334376	−	0.942440i	$-0.608525\pi$
$337$	−12.2766	−0.668751	−0.334376	+	0.942440i	$0.608525\pi$
$338$	0	0
$339$	14.3961	0.781886
$340$	0	0
$341$	−1.64921	−0.0893098
$342$	0	0
$343$	3.28525	0.177387
$344$	0	0
$345$	−36.3635	−1.95775
$346$	0	0
$347$	30.9067	1.65916	0.829580	−	0.558387i	$-0.188580\pi$
$347$	30.9067	1.65916	0.829580	+	0.558387i	$0.188580\pi$
$348$	0	0
$349$	−23.9024	−1.27947	−0.639733	−	0.768597i	$-0.720955\pi$
$349$	−23.9024	−1.27947	−0.639733	+	0.768597i	$0.720955\pi$
$350$	0	0
$351$	−8.08069	−0.431315
$352$	0	0
$353$	15.0158	0.799210	0.399605	−	0.916687i	$-0.369147\pi$
$353$	15.0158	0.799210	0.399605	+	0.916687i	$0.369147\pi$
$354$	0	0
$355$	−25.8048	−1.36958
$356$	0	0
$357$	−14.1509	−0.748945
$358$	0	0
$359$	−14.0826	−0.743251	−0.371626	−	0.928383i	$-0.621200\pi$
$359$	−14.0826	−0.743251	−0.371626	+	0.928383i	$0.621200\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−1.19599	−0.0627733
$364$	0	0
$365$	−5.61616	−0.293963
$366$	0	0
$367$	−21.3142	−1.11259	−0.556296	−	0.830984i	$-0.687778\pi$
$367$	−21.3142	−1.11259	−0.556296	+	0.830984i	$0.687778\pi$
$368$	0	0
$369$	6.15992	0.320673
$370$	0	0
$371$	0.372329	0.0193304
$372$	0	0
$373$	−2.42088	−0.125349	−0.0626743	−	0.998034i	$-0.519963\pi$
$373$	−2.42088	−0.125349	−0.0626743	+	0.998034i	$0.519963\pi$
$374$	0	0
$375$	−32.2794	−1.66690
$376$	0	0
$377$	−1.51605	−0.0780806
$378$	0	0
$379$	−8.87535	−0.455896	−0.227948	−	0.973673i	$-0.573202\pi$
$379$	−8.87535	−0.455896	−0.227948	+	0.973673i	$0.573202\pi$
$380$	0	0
$381$	−5.61748	−0.287792
$382$	0	0
$383$	3.54065	0.180919	0.0904595	−	0.995900i	$-0.471166\pi$
$383$	3.54065	0.180919	0.0904595	+	0.995900i	$0.471166\pi$
$384$	0	0
$385$	−14.7511	−0.751784
$386$	0	0
$387$	8.44624	0.429346
$388$	0	0
$389$	−16.7041	−0.846933	−0.423466	−	0.905912i	$-0.639187\pi$
$389$	−16.7041	−0.846933	−0.423466	+	0.905912i	$0.639187\pi$
$390$	0	0
$391$	−24.3877	−1.23334
$392$	0	0
$393$	4.47858	0.225915
$394$	0	0
$395$	−55.7403	−2.80460
$396$	0	0
$397$	28.2073	1.41568	0.707841	−	0.706372i	$-0.249669\pi$
$397$	28.2073	1.41568	0.707841	+	0.706372i	$0.249669\pi$
$398$	0	0
$399$	−4.32745	−0.216643
$400$	0	0
$401$	−36.2078	−1.80813	−0.904065	−	0.427395i	$-0.859431\pi$
$401$	−36.2078	−1.80813	−0.904065	+	0.427395i	$0.859431\pi$
$402$	0	0
$403$	2.43847	0.121469
$404$	0	0
$405$	−7.45049	−0.370218
$406$	0	0
$407$	−6.71293	−0.332748
$408$	0	0
$409$	36.9236	1.82576	0.912878	−	0.408233i	$-0.133855\pi$
$409$	36.9236	1.82576	0.912878	+	0.408233i	$0.133855\pi$
$410$	0	0
$411$	18.8482	0.929715
$412$	0	0
$413$	48.1184	2.36775
$414$	0	0
$415$	−22.1803	−1.08879
$416$	0	0
$417$	−3.02373	−0.148072
$418$	0	0
$419$	−18.0690	−0.882726	−0.441363	−	0.897329i	$-0.645505\pi$
$419$	−18.0690	−0.882726	−0.441363	+	0.897329i	$0.645505\pi$
$420$	0	0
$421$	−8.57629	−0.417983	−0.208991	−	0.977918i	$-0.567018\pi$
$421$	−8.57629	−0.417983	−0.208991	+	0.977918i	$0.567018\pi$
$422$	0	0
$423$	−5.83260	−0.283591
$424$	0	0
$425$	−37.9987	−1.84321
$426$	0	0
$427$	23.5067	1.13757
$428$	0	0
$429$	1.76836	0.0853771
$430$	0	0
$431$	−4.28147	−0.206231	−0.103116	−	0.994669i	$-0.532881\pi$
$431$	−4.28147	−0.206231	−0.103116	+	0.994669i	$0.532881\pi$
$432$	0	0
$433$	18.2035	0.874804	0.437402	−	0.899266i	$-0.355899\pi$
$433$	18.2035	0.874804	0.437402	+	0.899266i	$0.355899\pi$
$434$	0	0
$435$	−4.99942	−0.239704
$436$	0	0
$437$	−7.45793	−0.356761
$438$	0	0
$439$	−29.4442	−1.40529	−0.702647	−	0.711538i	$-0.747999\pi$
$439$	−29.4442	−1.40529	−0.702647	+	0.711538i	$0.747999\pi$
$440$	0	0
$441$	−9.56210	−0.455338
$442$	0	0
$443$	24.3633	1.15753	0.578767	−	0.815493i	$-0.303534\pi$
$443$	24.3633	1.15753	0.578767	+	0.815493i	$0.303534\pi$
$444$	0	0
$445$	49.4748	2.34533
$446$	0	0
$447$	−2.21141	−0.104596
$448$	0	0
$449$	38.7776	1.83003	0.915015	−	0.403421i	$-0.132179\pi$
$449$	38.7776	1.83003	0.915015	+	0.403421i	$0.132179\pi$
$450$	0	0
$451$	−3.92451	−0.184798
$452$	0	0
$453$	−18.3206	−0.860779
$454$	0	0
$455$	21.8105	1.02249
$456$	0	0
$457$	−7.47672	−0.349746	−0.174873	−	0.984591i	$-0.555952\pi$
$457$	−7.47672	−0.349746	−0.174873	+	0.984591i	$0.555952\pi$
$458$	0	0
$459$	−17.8714	−0.834165
$460$	0	0
$461$	15.9782	0.744181	0.372091	−	0.928196i	$-0.378641\pi$
$461$	15.9782	0.744181	0.372091	+	0.928196i	$0.378641\pi$
$462$	0	0
$463$	6.10221	0.283594	0.141797	−	0.989896i	$-0.454712\pi$
$463$	6.10221	0.283594	0.141797	+	0.989896i	$0.454712\pi$
$464$	0	0
$465$	8.04126	0.372904
$466$	0	0
$467$	26.6373	1.23263	0.616313	−	0.787502i	$-0.288626\pi$
$467$	26.6373	1.23263	0.616313	+	0.787502i	$0.288626\pi$
$468$	0	0
$469$	−13.4235	−0.619838
$470$	0	0
$471$	−29.4969	−1.35914
$472$	0	0
$473$	−5.38113	−0.247424
$474$	0	0
$475$	−11.6203	−0.533176
$476$	0	0
$477$	0.161515	0.00739527
$478$	0	0
$479$	16.2094	0.740626	0.370313	−	0.928907i	$-0.379250\pi$
$479$	16.2094	0.740626	0.370313	+	0.928907i	$0.379250\pi$
$480$	0	0
$481$	9.92553	0.452565
$482$	0	0
$483$	32.2738	1.46851
$484$	0	0
$485$	−56.2231	−2.55296
$486$	0	0
$487$	4.82448	0.218618	0.109309	−	0.994008i	$-0.465136\pi$
$487$	4.82448	0.218618	0.109309	+	0.994008i	$0.465136\pi$
$488$	0	0
$489$	−4.45088	−0.201276
$490$	0	0
$491$	8.53579	0.385215	0.192607	−	0.981276i	$-0.438306\pi$
$491$	8.53579	0.385215	0.192607	+	0.981276i	$0.438306\pi$
$492$	0	0
$493$	−3.35293	−0.151008
$494$	0	0
$495$	−6.39896	−0.287612
$496$	0	0
$497$	22.9026	1.02732
$498$	0	0
$499$	13.4325	0.601319	0.300660	−	0.953732i	$-0.402793\pi$
$499$	13.4325	0.601319	0.300660	+	0.953732i	$0.402793\pi$
$500$	0	0
$501$	3.16648	0.141468
$502$	0	0
$503$	1.66487	0.0742327	0.0371163	−	0.999311i	$-0.488183\pi$
$503$	1.66487	0.0742327	0.0371163	+	0.999311i	$0.488183\pi$
$504$	0	0
$505$	−44.8565	−1.99609
$506$	0	0
$507$	12.9333	0.574386
$508$	0	0
$509$	−14.9684	−0.663463	−0.331731	−	0.943374i	$-0.607633\pi$
$509$	−14.9684	−0.663463	−0.331731	+	0.943374i	$0.607633\pi$
$510$	0	0
$511$	4.98452	0.220502
$512$	0	0
$513$	−5.46521	−0.241295
$514$	0	0
$515$	20.3510	0.896772
$516$	0	0
$517$	3.71597	0.163428
$518$	0	0
$519$	−8.78518	−0.385627
$520$	0	0
$521$	−7.89123	−0.345721	−0.172861	−	0.984946i	$-0.555301\pi$
$521$	−7.89123	−0.345721	−0.172861	+	0.984946i	$0.555301\pi$
$522$	0	0
$523$	22.3062	0.975381	0.487690	−	0.873017i	$-0.337840\pi$
$523$	22.3062	0.975381	0.487690	+	0.873017i	$0.337840\pi$
$524$	0	0
$525$	50.2863	2.19467
$526$	0	0
$527$	5.39297	0.234922
$528$	0	0
$529$	32.6207	1.41829
$530$	0	0
$531$	20.8736	0.905837
$532$	0	0
$533$	5.80266	0.251341
$534$	0	0
$535$	−29.8155	−1.28904
$536$	0	0
$537$	11.4264	0.493087
$538$	0	0
$539$	6.09205	0.262403
$540$	0	0
$541$	38.9694	1.67542	0.837712	−	0.546112i	$-0.183893\pi$
$541$	38.9694	1.67542	0.837712	+	0.546112i	$0.183893\pi$
$542$	0	0
$543$	−7.20750	−0.309304
$544$	0	0
$545$	−5.89025	−0.252311
$546$	0	0
$547$	−37.7503	−1.61409	−0.807043	−	0.590492i	$-0.798934\pi$
$547$	−37.7503	−1.61409	−0.807043	+	0.590492i	$0.798934\pi$
$548$	0	0
$549$	10.1971	0.435204
$550$	0	0
$551$	−1.02535	−0.0436814
$552$	0	0
$553$	49.4713	2.10373
$554$	0	0
$555$	32.7310	1.38935
$556$	0	0
$557$	−3.17436	−0.134502	−0.0672511	−	0.997736i	$-0.521423\pi$
$557$	−3.17436	−0.134502	−0.0672511	+	0.997736i	$0.521423\pi$
$558$	0	0
$559$	7.95637	0.336519
$560$	0	0
$561$	3.91093	0.165120
$562$	0	0
$563$	19.9431	0.840503	0.420252	−	0.907408i	$-0.361942\pi$
$563$	19.9431	0.840503	0.420252	+	0.907408i	$0.361942\pi$
$564$	0	0
$565$	−49.0721	−2.06448
$566$	0	0
$567$	6.61255	0.277701
$568$	0	0
$569$	36.6424	1.53613	0.768064	−	0.640374i	$-0.221220\pi$
$569$	36.6424	1.53613	0.768064	+	0.640374i	$0.221220\pi$
$570$	0	0
$571$	−11.5300	−0.482515	−0.241258	−	0.970461i	$-0.577560\pi$
$571$	−11.5300	−0.482515	−0.241258	+	0.970461i	$0.577560\pi$
$572$	0	0
$573$	21.1773	0.884694
$574$	0	0
$575$	86.6634	3.61411
$576$	0	0
$577$	28.5590	1.18893	0.594463	−	0.804123i	$-0.297365\pi$
$577$	28.5590	1.18893	0.594463	+	0.804123i	$0.297365\pi$
$578$	0	0
$579$	−4.41737	−0.183580
$580$	0	0
$581$	19.6857	0.816701
$582$	0	0
$583$	−0.102902	−0.00426176
$584$	0	0
$585$	9.46131	0.391177
$586$	0	0
$587$	−18.1461	−0.748969	−0.374484	−	0.927233i	$-0.622180\pi$
$587$	−18.1461	−0.748969	−0.374484	+	0.927233i	$0.622180\pi$
$588$	0	0
$589$	1.64921	0.0679546
$590$	0	0
$591$	30.8314	1.26823
$592$	0	0
$593$	15.3085	0.628644	0.314322	−	0.949316i	$-0.398223\pi$
$593$	15.3085	0.628644	0.314322	+	0.949316i	$0.398223\pi$
$594$	0	0
$595$	48.2364	1.97750
$596$	0	0
$597$	22.5986	0.924900
$598$	0	0
$599$	−17.8806	−0.730580	−0.365290	−	0.930894i	$-0.619030\pi$
$599$	−17.8806	−0.730580	−0.365290	+	0.930894i	$0.619030\pi$
$600$	0	0
$601$	−32.5803	−1.32898	−0.664489	−	0.747298i	$-0.731351\pi$
$601$	−32.5803	−1.32898	−0.664489	+	0.747298i	$0.731351\pi$
$602$	0	0
$603$	−5.82306	−0.237133
$604$	0	0
$605$	4.07680	0.165746
$606$	0	0
$607$	−43.6494	−1.77167	−0.885837	−	0.463997i	$-0.846415\pi$
$607$	−43.6494	−1.77167	−0.885837	+	0.463997i	$0.846415\pi$
$608$	0	0
$609$	4.43715	0.179802
$610$	0	0
$611$	−5.49432	−0.222276
$612$	0	0
$613$	0.843061	0.0340509	0.0170254	−	0.999855i	$-0.494580\pi$
$613$	0.843061	0.0340509	0.0170254	+	0.999855i	$0.494580\pi$
$614$	0	0
$615$	19.1352	0.771606
$616$	0	0
$617$	4.89882	0.197219	0.0986094	−	0.995126i	$-0.468561\pi$
$617$	4.89882	0.197219	0.0986094	+	0.995126i	$0.468561\pi$
$618$	0	0
$619$	14.8704	0.597691	0.298846	−	0.954301i	$-0.403398\pi$
$619$	14.8704	0.597691	0.298846	+	0.954301i	$0.403398\pi$
$620$	0	0
$621$	40.7591	1.63561
$622$	0	0
$623$	−43.9105	−1.75924
$624$	0	0
$625$	51.9299	2.07720
$626$	0	0
$627$	1.19599	0.0477633
$628$	0	0
$629$	21.9515	0.875263
$630$	0	0
$631$	−2.83922	−0.113027	−0.0565137	−	0.998402i	$-0.517998\pi$
$631$	−2.83922	−0.113027	−0.0565137	+	0.998402i	$0.517998\pi$
$632$	0	0
$633$	−12.1983	−0.484839
$634$	0	0
$635$	19.1484	0.759881
$636$	0	0
$637$	−9.00751	−0.356891
$638$	0	0
$639$	9.93508	0.393026
$640$	0	0
$641$	20.6746	0.816599	0.408299	−	0.912848i	$-0.366122\pi$
$641$	20.6746	0.816599	0.408299	+	0.912848i	$0.366122\pi$
$642$	0	0
$643$	24.6254	0.971130	0.485565	−	0.874201i	$-0.338614\pi$
$643$	24.6254	0.971130	0.485565	+	0.874201i	$0.338614\pi$
$644$	0	0
$645$	26.2374	1.03310
$646$	0	0
$647$	−45.3626	−1.78339	−0.891693	−	0.452640i	$-0.850482\pi$
$647$	−45.3626	−1.78339	−0.891693	+	0.452640i	$0.850482\pi$
$648$	0	0
$649$	−13.2986	−0.522017
$650$	0	0
$651$	−7.13688	−0.279716
$652$	0	0
$653$	26.1012	1.02142	0.510710	−	0.859753i	$-0.329383\pi$
$653$	26.1012	1.02142	0.510710	+	0.859753i	$0.329383\pi$
$654$	0	0
$655$	−15.2662	−0.596501
$656$	0	0
$657$	2.16227	0.0843582
$658$	0	0
$659$	45.8507	1.78609	0.893045	−	0.449967i	$-0.148564\pi$
$659$	45.8507	1.78609	0.893045	+	0.449967i	$0.148564\pi$
$660$	0	0
$661$	−15.4225	−0.599864	−0.299932	−	0.953961i	$-0.596964\pi$
$661$	−15.4225	−0.599864	−0.299932	+	0.953961i	$0.596964\pi$
$662$	0	0
$663$	−5.78258	−0.224577
$664$	0	0
$665$	14.7511	0.572022
$666$	0	0
$667$	7.64698	0.296092
$668$	0	0
$669$	−0.314187	−0.0121472
$670$	0	0
$671$	−6.49664	−0.250800
$672$	0	0
$673$	−37.8633	−1.45952	−0.729762	−	0.683702i	$-0.760369\pi$
$673$	−37.8633	−1.45952	−0.729762	+	0.683702i	$0.760369\pi$
$674$	0	0
$675$	63.5074	2.44440
$676$	0	0
$677$	−25.3510	−0.974320	−0.487160	−	0.873313i	$-0.661967\pi$
$677$	−25.3510	−0.974320	−0.487160	+	0.873313i	$0.661967\pi$
$678$	0	0
$679$	49.8998	1.91498
$680$	0	0
$681$	32.6812	1.25235
$682$	0	0
$683$	21.7513	0.832290	0.416145	−	0.909298i	$-0.363381\pi$
$683$	21.7513	0.832290	0.416145	+	0.909298i	$0.363381\pi$
$684$	0	0
$685$	−64.2484	−2.45480
$686$	0	0
$687$	6.61588	0.252412
$688$	0	0
$689$	0.152147	0.00579636
$690$	0	0
$691$	−17.3051	−0.658318	−0.329159	−	0.944275i	$-0.606765\pi$
$691$	−17.3051	−0.658318	−0.329159	+	0.944275i	$0.606765\pi$
$692$	0	0
$693$	5.67929	0.215738
$694$	0	0
$695$	10.3070	0.390968
$696$	0	0
$697$	12.8333	0.486095
$698$	0	0
$699$	−32.8578	−1.24280
$700$	0	0
$701$	−29.6923	−1.12146	−0.560732	−	0.827997i	$-0.689480\pi$
$701$	−29.6923	−1.12146	−0.560732	+	0.827997i	$0.689480\pi$
$702$	0	0
$703$	6.71293	0.253183
$704$	0	0
$705$	−18.1184	−0.682378
$706$	0	0
$707$	39.8116	1.49727
$708$	0	0
$709$	−21.4898	−0.807067	−0.403534	−	0.914965i	$-0.632218\pi$
$709$	−21.4898	−0.807067	−0.403534	+	0.914965i	$0.632218\pi$
$710$	0	0
$711$	21.4605	0.804831
$712$	0	0
$713$	−12.2997	−0.460627
$714$	0	0
$715$	−6.02783	−0.225428
$716$	0	0
$717$	1.67845	0.0626828
$718$	0	0
$719$	−9.61388	−0.358537	−0.179269	−	0.983800i	$-0.557373\pi$
$719$	−9.61388	−0.358537	−0.179269	+	0.983800i	$0.557373\pi$
$720$	0	0
$721$	−18.0622	−0.672671
$722$	0	0
$723$	24.8403	0.923821
$724$	0	0
$725$	11.9149	0.442507
$726$	0	0
$727$	−6.84046	−0.253699	−0.126849	−	0.991922i	$-0.540486\pi$
$727$	−6.84046	−0.253699	−0.126849	+	0.991922i	$0.540486\pi$
$728$	0	0
$729$	22.4775	0.832501
$730$	0	0
$731$	17.5965	0.650828
$732$	0	0
$733$	38.2277	1.41197	0.705987	−	0.708225i	$-0.250504\pi$
$733$	38.2277	1.41197	0.705987	+	0.708225i	$0.250504\pi$
$734$	0	0
$735$	−29.7037	−1.09564
$736$	0	0
$737$	3.70989	0.136656
$738$	0	0
$739$	17.7157	0.651682	0.325841	−	0.945425i	$-0.394353\pi$
$739$	17.7157	0.651682	0.325841	+	0.945425i	$0.394353\pi$
$740$	0	0
$741$	−1.76836	−0.0649622
$742$	0	0
$743$	−21.3028	−0.781525	−0.390763	−	0.920491i	$-0.627789\pi$
$743$	−21.3028	−0.781525	−0.390763	+	0.920491i	$0.627789\pi$
$744$	0	0
$745$	7.53808	0.276174
$746$	0	0
$747$	8.53960	0.312447
$748$	0	0
$749$	26.4622	0.966908
$750$	0	0
$751$	−25.1552	−0.917926	−0.458963	−	0.888455i	$-0.651779\pi$
$751$	−25.1552	−0.917926	−0.458963	+	0.888455i	$0.651779\pi$
$752$	0	0
$753$	−1.55402	−0.0566317
$754$	0	0
$755$	62.4499	2.27279
$756$	0	0
$757$	15.5116	0.563780	0.281890	−	0.959447i	$-0.409039\pi$
$757$	15.5116	0.563780	0.281890	+	0.959447i	$0.409039\pi$
$758$	0	0
$759$	−8.91962	−0.323762
$760$	0	0
$761$	35.2085	1.27631	0.638154	−	0.769908i	$-0.279698\pi$
$761$	35.2085	1.27631	0.638154	+	0.769908i	$0.279698\pi$
$762$	0	0
$763$	5.22779	0.189259
$764$	0	0
$765$	20.9248	0.756538
$766$	0	0
$767$	19.6630	0.709988
$768$	0	0
$769$	−5.57667	−0.201100	−0.100550	−	0.994932i	$-0.532060\pi$
$769$	−5.57667	−0.201100	−0.100550	+	0.994932i	$0.532060\pi$
$770$	0	0
$771$	−4.08095	−0.146972
$772$	0	0
$773$	35.9175	1.29186	0.645931	−	0.763396i	$-0.276469\pi$
$773$	35.9175	1.29186	0.645931	+	0.763396i	$0.276469\pi$
$774$	0	0
$775$	−19.1643	−0.688403
$776$	0	0
$777$	−29.0499	−1.04216
$778$	0	0
$779$	3.92451	0.140610
$780$	0	0
$781$	−6.32968	−0.226494
$782$	0	0
$783$	5.60375	0.200262
$784$	0	0
$785$	100.547	3.58866
$786$	0	0
$787$	−7.53242	−0.268502	−0.134251	−	0.990947i	$-0.542863\pi$
$787$	−7.53242	−0.268502	−0.134251	+	0.990947i	$0.542863\pi$
$788$	0	0
$789$	17.0331	0.606395
$790$	0	0
$791$	43.5531	1.54857
$792$	0	0
$793$	9.60573	0.341110
$794$	0	0
$795$	0.501731	0.0177946
$796$	0	0
$797$	−49.3837	−1.74926	−0.874629	−	0.484792i	$-0.838895\pi$
$797$	−49.3837	−1.74926	−0.874629	+	0.484792i	$0.838895\pi$
$798$	0	0
$799$	−12.1513	−0.429883
$800$	0	0
$801$	−19.0482	−0.673036
$802$	0	0
$803$	−1.37759	−0.0486141
$804$	0	0
$805$	−110.012	−3.87743
$806$	0	0
$807$	−6.45210	−0.227125
$808$	0	0
$809$	34.4637	1.21168	0.605840	−	0.795587i	$-0.292837\pi$
$809$	34.4637	1.21168	0.605840	+	0.795587i	$0.292837\pi$
$810$	0	0
$811$	−20.0278	−0.703272	−0.351636	−	0.936137i	$-0.614375\pi$
$811$	−20.0278	−0.703272	−0.351636	+	0.936137i	$0.614375\pi$
$812$	0	0
$813$	−13.2306	−0.464017
$814$	0	0
$815$	15.1718	0.531444
$816$	0	0
$817$	5.38113	0.188262
$818$	0	0
$819$	−8.39722	−0.293423
$820$	0	0
$821$	6.45245	0.225192	0.112596	−	0.993641i	$-0.464083\pi$
$821$	6.45245	0.225192	0.112596	+	0.993641i	$0.464083\pi$
$822$	0	0
$823$	43.0190	1.49955	0.749774	−	0.661694i	$-0.230162\pi$
$823$	43.0190	1.49955	0.749774	+	0.661694i	$0.230162\pi$
$824$	0	0
$825$	−13.8978	−0.483859
$826$	0	0
$827$	43.1557	1.50067	0.750335	−	0.661058i	$-0.229892\pi$
$827$	43.1557	1.50067	0.750335	+	0.661058i	$0.229892\pi$
$828$	0	0
$829$	13.4937	0.468656	0.234328	−	0.972158i	$-0.424711\pi$
$829$	13.4937	0.468656	0.234328	+	0.972158i	$0.424711\pi$
$830$	0	0
$831$	−16.8406	−0.584193
$832$	0	0
$833$	−19.9212	−0.690228
$834$	0	0
$835$	−10.7937	−0.373530
$836$	0	0
$837$	−9.01329	−0.311545
$838$	0	0
$839$	14.6851	0.506988	0.253494	−	0.967337i	$-0.418420\pi$
$839$	14.6851	0.506988	0.253494	+	0.967337i	$0.418420\pi$
$840$	0	0
$841$	−27.9487	−0.963747
$842$	0	0
$843$	12.9405	0.445693
$844$	0	0
$845$	−44.0858	−1.51660
$846$	0	0
$847$	−3.61829	−0.124326
$848$	0	0
$849$	2.28372	0.0783769
$850$	0	0
$851$	−50.0645	−1.71619
$852$	0	0
$853$	−51.5775	−1.76598	−0.882990	−	0.469392i	$-0.844473\pi$
$853$	−51.5775	−1.76598	−0.882990	+	0.469392i	$0.844473\pi$
$854$	0	0
$855$	6.39896	0.218840
$856$	0	0
$857$	−46.6355	−1.59304	−0.796520	−	0.604612i	$-0.793328\pi$
$857$	−46.6355	−1.59304	−0.796520	+	0.604612i	$0.793328\pi$
$858$	0	0
$859$	18.6711	0.637048	0.318524	−	0.947915i	$-0.396813\pi$
$859$	18.6711	0.637048	0.318524	+	0.947915i	$0.396813\pi$
$860$	0	0
$861$	−16.9831	−0.578783
$862$	0	0
$863$	−43.0160	−1.46428	−0.732141	−	0.681153i	$-0.761479\pi$
$863$	−43.0160	−1.46428	−0.732141	+	0.681153i	$0.761479\pi$
$864$	0	0
$865$	29.9462	1.01820
$866$	0	0
$867$	7.54300	0.256174
$868$	0	0
$869$	−13.6725	−0.463809
$870$	0	0
$871$	−5.48533	−0.185863
$872$	0	0
$873$	21.6464	0.732619
$874$	0	0
$875$	−97.6565	−3.30139
$876$	0	0
$877$	56.0429	1.89243	0.946217	−	0.323534i	$-0.104871\pi$
$877$	56.0429	1.89243	0.946217	+	0.323534i	$0.104871\pi$
$878$	0	0
$879$	−4.34803	−0.146655
$880$	0	0
$881$	17.9947	0.606257	0.303128	−	0.952950i	$-0.401969\pi$
$881$	17.9947	0.606257	0.303128	+	0.952950i	$0.401969\pi$
$882$	0	0
$883$	−1.99550	−0.0671540	−0.0335770	−	0.999436i	$-0.510690\pi$
$883$	−1.99550	−0.0671540	−0.0335770	+	0.999436i	$0.510690\pi$
$884$	0	0
$885$	64.8418	2.17963
$886$	0	0
$887$	−7.20927	−0.242063	−0.121032	−	0.992649i	$-0.538620\pi$
$887$	−7.20927	−0.242063	−0.121032	+	0.992649i	$0.538620\pi$
$888$	0	0
$889$	−16.9948	−0.569988
$890$	0	0
$891$	−1.82753	−0.0612246
$892$	0	0
$893$	−3.71597	−0.124350
$894$	0	0
$895$	−38.9495	−1.30194
$896$	0	0
$897$	13.1883	0.440344
$898$	0	0
$899$	−1.69102	−0.0563986
$900$	0	0
$901$	0.336492	0.0112102
$902$	0	0
$903$	−23.2866	−0.774928
$904$	0	0
$905$	24.5684	0.816680
$906$	0	0
$907$	25.7832	0.856116	0.428058	−	0.903751i	$-0.359198\pi$
$907$	25.7832	0.856116	0.428058	+	0.903751i	$0.359198\pi$
$908$	0	0
$909$	17.2701	0.572814
$910$	0	0
$911$	−31.1334	−1.03149	−0.515747	−	0.856741i	$-0.672486\pi$
$911$	−31.1334	−1.03149	−0.515747	+	0.856741i	$0.672486\pi$
$912$	0	0
$913$	−5.44061	−0.180058
$914$	0	0
$915$	31.6764	1.04719
$916$	0	0
$917$	13.5493	0.447437
$918$	0	0
$919$	−5.42725	−0.179029	−0.0895143	−	0.995986i	$-0.528531\pi$
$919$	−5.42725	−0.179029	−0.0895143	+	0.995986i	$0.528531\pi$
$920$	0	0
$921$	−28.1452	−0.927416
$922$	0	0
$923$	9.35887	0.308051
$924$	0	0
$925$	−78.0063	−2.56483
$926$	0	0
$927$	−7.83531	−0.257345
$928$	0	0
$929$	−21.6025	−0.708756	−0.354378	−	0.935102i	$-0.615307\pi$
$929$	−21.6025	−0.708756	−0.354378	+	0.935102i	$0.615307\pi$
$930$	0	0
$931$	−6.09205	−0.199659
$932$	0	0
$933$	−19.1390	−0.626583
$934$	0	0
$935$	−13.3313	−0.435979
$936$	0	0
$937$	31.6840	1.03507	0.517536	−	0.855661i	$-0.326849\pi$
$937$	31.6840	1.03507	0.517536	+	0.855661i	$0.326849\pi$
$938$	0	0
$939$	−19.1399	−0.624608
$940$	0	0
$941$	−5.63987	−0.183854	−0.0919272	−	0.995766i	$-0.529303\pi$
$941$	−5.63987	−0.183854	−0.0919272	+	0.995766i	$0.529303\pi$
$942$	0	0
$943$	−29.2687	−0.953120
$944$	0	0
$945$	−80.6176	−2.62249
$946$	0	0
$947$	51.7369	1.68122	0.840611	−	0.541639i	$-0.182196\pi$
$947$	51.7369	1.68122	0.840611	+	0.541639i	$0.182196\pi$
$948$	0	0
$949$	2.03686	0.0661194
$950$	0	0
$951$	24.9682	0.809650
$952$	0	0
$953$	−16.2553	−0.526561	−0.263281	−	0.964719i	$-0.584804\pi$
$953$	−16.2553	−0.526561	−0.263281	+	0.964719i	$0.584804\pi$
$954$	0	0
$955$	−72.1874	−2.33593
$956$	0	0
$957$	−1.22631	−0.0396410
$958$	0	0
$959$	57.0225	1.84135
$960$	0	0
$961$	−28.2801	−0.912261
$962$	0	0
$963$	11.4792	0.369913
$964$	0	0
$965$	15.0576	0.484721
$966$	0	0
$967$	54.5004	1.75261	0.876307	−	0.481754i	$-0.160000\pi$
$967$	54.5004	1.75261	0.876307	+	0.481754i	$0.160000\pi$
$968$	0	0
$969$	−3.91093	−0.125637
$970$	0	0
$971$	34.2679	1.09971	0.549855	−	0.835260i	$-0.314683\pi$
$971$	34.2679	1.09971	0.549855	+	0.835260i	$0.314683\pi$
$972$	0	0
$973$	−9.14783	−0.293266
$974$	0	0
$975$	20.5488	0.658090
$976$	0	0
$977$	55.5644	1.77766	0.888831	−	0.458234i	$-0.151518\pi$
$977$	55.5644	1.77766	0.888831	+	0.458234i	$0.151518\pi$
$978$	0	0
$979$	12.1357	0.387858
$980$	0	0
$981$	2.26780	0.0724052
$982$	0	0
$983$	41.3971	1.32036	0.660181	−	0.751106i	$-0.270479\pi$
$983$	41.3971	1.32036	0.660181	+	0.751106i	$0.270479\pi$
$984$	0	0
$985$	−105.095	−3.34862
$986$	0	0
$987$	16.0807	0.511853
$988$	0	0
$989$	−40.1321	−1.27613
$990$	0	0
$991$	−60.8219	−1.93207	−0.966036	−	0.258406i	$-0.916803\pi$
$991$	−60.8219	−1.93207	−0.966036	+	0.258406i	$0.916803\pi$
$992$	0	0
$993$	18.0758	0.573618
$994$	0	0
$995$	−77.0324	−2.44209
$996$	0	0
$997$	26.0627	0.825414	0.412707	−	0.910864i	$-0.364583\pi$
$997$	26.0627	0.825414	0.412707	+	0.910864i	$0.364583\pi$
$998$	0	0
$999$	−36.6876	−1.16074

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.ba.1.4		7
4.3	odd	2		209.2.a.d.1.2	✓	7
12.11	even	2		1881.2.a.p.1.6		7
20.19	odd	2		5225.2.a.n.1.6		7
44.43	even	2		2299.2.a.q.1.6		7
76.75	even	2		3971.2.a.i.1.6		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.a.d.1.2	✓	7	4.3	odd	2
1881.2.a.p.1.6		7	12.11	even	2
2299.2.a.q.1.6		7	44.43	even	2
3344.2.a.ba.1.4		7	1.1	even	1	trivial
3971.2.a.i.1.6		7	76.75	even	2
5225.2.a.n.1.6		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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-1.19599 q^{3} +4.07680 q^{5} -3.61829 q^{7} -1.56960 q^{9} +1.00000 q^{11} -1.47857 q^{13} -4.87582 q^{15} -3.27003 q^{17} -1.00000 q^{19} +4.32745 q^{21} +7.45793 q^{23} +11.6203 q^{25} +5.46521 q^{27} +1.02535 q^{29} -1.64921 q^{31} -1.19599 q^{33} -14.7511 q^{35} -6.71293 q^{37} +1.76836 q^{39} -3.92451 q^{41} -5.38113 q^{43} -6.39896 q^{45} +3.71597 q^{47} +6.09205 q^{49} +3.91093 q^{51} -0.102902 q^{53} +4.07680 q^{55} +1.19599 q^{57} -13.2986 q^{59} -6.49664 q^{61} +5.67929 q^{63} -6.02783 q^{65} +3.70989 q^{67} -8.91962 q^{69} -6.32968 q^{71} -1.37759 q^{73} -13.8978 q^{75} -3.61829 q^{77} -13.6725 q^{79} -1.82753 q^{81} -5.44061 q^{83} -13.3313 q^{85} -1.22631 q^{87} +12.1357 q^{89} +5.34990 q^{91} +1.97244 q^{93} -4.07680 q^{95} -13.7910 q^{97} -1.56960 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 2 q^{3} + 2 q^{5} - 10 q^{7} + 11 q^{9} + 7 q^{11} - 4 q^{13} - 12 q^{15} + 2 q^{17} - 7 q^{19} - 14 q^{21} - 10 q^{23} + 9 q^{25} + 4 q^{27} - 18 q^{29} - 24 q^{31} - 2 q^{33} - 8 q^{35} - 24 q^{39}+ \cdots + 11 q^{99}+O(q^{100}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more