LMFDB - Embedded newform 3344.2.a.w.1.6

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 = [6,0,-1,0,5,0,2] 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:	$0$
Dimension:	$6$
Coefficient field:	6.6.744786576.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 17x^{4} + 13x^{3} + 69x^{2} - 21x - 30 $ x^6 - x^5 - 17x^4 + 13x^3 + 69x^2 - 21x - 30
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 836)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-3.31717$ 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)

$f(q)$	$=$	$q+3.31717 q^{3} +1.60797 q^{5} +4.11176 q^{7} +8.00361 q^{9} +1.00000 q^{11} -0.192283 q^{13} +5.33391 q^{15} -7.62265 q^{17} +1.00000 q^{19} +13.6394 q^{21} -6.73080 q^{23} -2.41443 q^{25} +16.5978 q^{27} +6.80988 q^{29} -7.33391 q^{31} +3.31717 q^{33} +6.61158 q^{35} +0.734914 q^{37} -0.637837 q^{39} +0.976345 q^{41} +5.55470 q^{43} +12.8696 q^{45} -8.83859 q^{47} +9.90653 q^{49} -25.2856 q^{51} +11.8503 q^{53} +1.60797 q^{55} +3.31717 q^{57} -3.30268 q^{59} +9.41840 q^{61} +32.9089 q^{63} -0.309186 q^{65} -12.5025 q^{67} -22.3272 q^{69} -4.12228 q^{71} -6.15694 q^{73} -8.00909 q^{75} +4.11176 q^{77} +2.98831 q^{79} +31.0469 q^{81} -6.33876 q^{83} -12.2570 q^{85} +22.5895 q^{87} -13.5694 q^{89} -0.790623 q^{91} -24.3278 q^{93} +1.60797 q^{95} -4.33130 q^{97} +8.00361 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{3} + 5 q^{5} + 2 q^{7} + 17 q^{9} + 6 q^{11} + 8 q^{13} - 9 q^{15} - 2 q^{17} + 6 q^{19} + 18 q^{21} - 5 q^{23} + 13 q^{25} - 7 q^{27} + 10 q^{29} - 3 q^{31} - q^{33} + 4 q^{35} + 7 q^{37} + 10 q^{39}+ \cdots + 17 q^{99}+O(q^{100}) $ 6 * q - q^3 + 5 * q^5 + 2 * q^7 + 17 * q^9 + 6 * q^11 + 8 * q^13 - 9 * q^15 - 2 * q^17 + 6 * q^19 + 18 * q^21 - 5 * q^23 + 13 * q^25 - 7 * q^27 + 10 * q^29 - 3 * q^31 - q^33 + 4 * q^35 + 7 * q^37 + 10 * q^39 + 6 * q^41 - 16 * q^43 + 42 * q^45 + 12 * q^49 - 8 * q^51 + 20 * q^53 + 5 * q^55 - q^57 - 15 * q^59 + 24 * q^61 + 20 * q^63 - 28 * q^65 - 25 * q^67 - 33 * q^69 + 9 * q^71 - 26 * q^73 - 28 * q^75 + 2 * q^77 + 16 * q^79 + 58 * q^81 + 2 * q^83 + 12 * q^85 + 36 * q^87 + 7 * q^89 - 8 * q^91 - 55 * q^93 + 5 * q^95 + 37 * q^97 + 17 * 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
$3$	3.31717	1.91517	0.957584	−	0.288154i	$-0.0930415\pi$
$3$	3.31717	1.91517	0.957584	+	0.288154i	$0.0930415\pi$
$4$	0	0
$5$	1.60797	0.719106	0.359553	−	0.933125i	$-0.382929\pi$
$5$	1.60797	0.719106	0.359553	+	0.933125i	$0.382929\pi$
$6$	0	0
$7$	4.11176	1.55410	0.777049	−	0.629441i	$-0.216716\pi$
$7$	4.11176	1.55410	0.777049	+	0.629441i	$0.216716\pi$
$8$	0	0
$9$	8.00361	2.66787
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−0.192283	−0.0533298	−0.0266649	−	0.999644i	$-0.508489\pi$
$13$	−0.192283	−0.0533298	−0.0266649	+	0.999644i	$0.508489\pi$
$14$	0	0
$15$	5.33391	1.37721
$16$	0	0
$17$	−7.62265	−1.84876	−0.924382	−	0.381469i	$-0.875418\pi$
$17$	−7.62265	−1.84876	−0.924382	+	0.381469i	$0.875418\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	13.6394	2.97636
$22$	0	0
$23$	−6.73080	−1.40347	−0.701734	−	0.712439i	$-0.747590\pi$
$23$	−6.73080	−1.40347	−0.701734	+	0.712439i	$0.747590\pi$
$24$	0	0
$25$	−2.41443	−0.482887
$26$	0	0
$27$	16.5978	3.19425
$28$	0	0
$29$	6.80988	1.26456	0.632282	−	0.774738i	$-0.282118\pi$
$29$	6.80988	1.26456	0.632282	+	0.774738i	$0.282118\pi$
$30$	0	0
$31$	−7.33391	−1.31721	−0.658604	−	0.752490i	$-0.728853\pi$
$31$	−7.33391	−1.31721	−0.658604	+	0.752490i	$0.728853\pi$
$32$	0	0
$33$	3.31717	0.577445
$34$	0	0
$35$	6.61158	1.11756
$36$	0	0
$37$	0.734914	0.120819	0.0604095	−	0.998174i	$-0.480759\pi$
$37$	0.734914	0.120819	0.0604095	+	0.998174i	$0.480759\pi$
$38$	0	0
$39$	−0.637837	−0.102136
$40$	0	0
$41$	0.976345	0.152479	0.0762397	−	0.997090i	$-0.475709\pi$
$41$	0.976345	0.152479	0.0762397	+	0.997090i	$0.475709\pi$
$42$	0	0
$43$	5.55470	0.847084	0.423542	−	0.905876i	$-0.360787\pi$
$43$	5.55470	0.847084	0.423542	+	0.905876i	$0.360787\pi$
$44$	0	0
$45$	12.8696	1.91848
$46$	0	0
$47$	−8.83859	−1.28924	−0.644620	−	0.764503i	$-0.722984\pi$
$47$	−8.83859	−1.28924	−0.644620	+	0.764503i	$0.722984\pi$
$48$	0	0
$49$	9.90653	1.41522
$50$	0	0
$51$	−25.2856	−3.54069
$52$	0	0
$53$	11.8503	1.62776	0.813880	−	0.581033i	$-0.197351\pi$
$53$	11.8503	1.62776	0.813880	+	0.581033i	$0.197351\pi$
$54$	0	0
$55$	1.60797	0.216819
$56$	0	0
$57$	3.31717	0.439370
$58$	0	0
$59$	−3.30268	−0.429972	−0.214986	−	0.976617i	$-0.568971\pi$
$59$	−3.30268	−0.429972	−0.214986	+	0.976617i	$0.568971\pi$
$60$	0	0
$61$	9.41840	1.20590	0.602951	−	0.797778i	$-0.293991\pi$
$61$	9.41840	1.20590	0.602951	+	0.797778i	$0.293991\pi$
$62$	0	0
$63$	32.9089	4.14613
$64$	0	0
$65$	−0.309186	−0.0383498
$66$	0	0
$67$	−12.5025	−1.52743	−0.763714	−	0.645555i	$-0.776626\pi$
$67$	−12.5025	−1.52743	−0.763714	+	0.645555i	$0.776626\pi$
$68$	0	0
$69$	−22.3272	−2.68788
$70$	0	0
$71$	−4.12228	−0.489225	−0.244612	−	0.969621i	$-0.578661\pi$
$71$	−4.12228	−0.489225	−0.244612	+	0.969621i	$0.578661\pi$
$72$	0	0
$73$	−6.15694	−0.720615	−0.360308	−	0.932834i	$-0.617328\pi$
$73$	−6.15694	−0.720615	−0.360308	+	0.932834i	$0.617328\pi$
$74$	0	0
$75$	−8.00909	−0.924809
$76$	0	0
$77$	4.11176	0.468578
$78$	0	0
$79$	2.98831	0.336211	0.168106	−	0.985769i	$-0.446235\pi$
$79$	2.98831	0.336211	0.168106	+	0.985769i	$0.446235\pi$
$80$	0	0
$81$	31.0469	3.44966
$82$	0	0
$83$	−6.33876	−0.695770	−0.347885	−	0.937537i	$-0.613100\pi$
$83$	−6.33876	−0.695770	−0.347885	+	0.937537i	$0.613100\pi$
$84$	0	0
$85$	−12.2570	−1.32946
$86$	0	0
$87$	22.5895	2.42185
$88$	0	0
$89$	−13.5694	−1.43835	−0.719176	−	0.694828i	$-0.755480\pi$
$89$	−13.5694	−1.43835	−0.719176	+	0.694828i	$0.755480\pi$
$90$	0	0
$91$	−0.790623	−0.0828798
$92$	0	0
$93$	−24.3278	−2.52268
$94$	0	0
$95$	1.60797	0.164974
$96$	0	0
$97$	−4.33130	−0.439777	−0.219889	−	0.975525i	$-0.570569\pi$
$97$	−4.33130	−0.439777	−0.219889	+	0.975525i	$0.570569\pi$
$98$	0	0
$99$	8.00361	0.804393
$100$	0	0
$101$	−13.4729	−1.34061	−0.670303	−	0.742088i	$-0.733836\pi$
$101$	−13.4729	−1.34061	−0.670303	+	0.742088i	$0.733836\pi$
$102$	0	0
$103$	−8.83382	−0.870422	−0.435211	−	0.900329i	$-0.643326\pi$
$103$	−8.83382	−0.870422	−0.435211	+	0.900329i	$0.643326\pi$
$104$	0	0
$105$	21.9317	2.14032
$106$	0	0
$107$	−12.2194	−1.18129	−0.590647	−	0.806930i	$-0.701127\pi$
$107$	−12.2194	−1.18129	−0.590647	+	0.806930i	$0.701127\pi$
$108$	0	0
$109$	7.54186	0.722379	0.361190	−	0.932492i	$-0.382371\pi$
$109$	7.54186	0.722379	0.361190	+	0.932492i	$0.382371\pi$
$110$	0	0
$111$	2.43783	0.231389
$112$	0	0
$113$	3.68563	0.346715	0.173358	−	0.984859i	$-0.444538\pi$
$113$	3.68563	0.346715	0.173358	+	0.984859i	$0.444538\pi$
$114$	0	0
$115$	−10.8229	−1.00924
$116$	0	0
$117$	−1.53896	−0.142277
$118$	0	0
$119$	−31.3425	−2.87316
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	3.23870	0.292024
$124$	0	0
$125$	−11.9222	−1.06635
$126$	0	0
$127$	4.19292	0.372061	0.186031	−	0.982544i	$-0.440438\pi$
$127$	4.19292	0.372061	0.186031	+	0.982544i	$0.440438\pi$
$128$	0	0
$129$	18.4259	1.62231
$130$	0	0
$131$	8.75993	0.765359	0.382679	−	0.923881i	$-0.375001\pi$
$131$	8.75993	0.765359	0.382679	+	0.923881i	$0.375001\pi$
$132$	0	0
$133$	4.11176	0.356534
$134$	0	0
$135$	26.6888	2.29700
$136$	0	0
$137$	−8.06261	−0.688835	−0.344418	−	0.938817i	$-0.611924\pi$
$137$	−8.06261	−0.688835	−0.344418	+	0.938817i	$0.611924\pi$
$138$	0	0
$139$	−13.3201	−1.12979	−0.564897	−	0.825161i	$-0.691084\pi$
$139$	−13.3201	−1.12979	−0.564897	+	0.825161i	$0.691084\pi$
$140$	0	0
$141$	−29.3191	−2.46911
$142$	0	0
$143$	−0.192283	−0.0160796
$144$	0	0
$145$	10.9501	0.909355
$146$	0	0
$147$	32.8616	2.71038
$148$	0	0
$149$	15.6532	1.28236	0.641182	−	0.767389i	$-0.278444\pi$
$149$	15.6532	1.28236	0.641182	+	0.767389i	$0.278444\pi$
$150$	0	0
$151$	17.8796	1.45502	0.727512	−	0.686095i	$-0.240677\pi$
$151$	17.8796	1.45502	0.727512	+	0.686095i	$0.240677\pi$
$152$	0	0
$153$	−61.0087	−4.93226
$154$	0	0
$155$	−11.7927	−0.947212
$156$	0	0
$157$	7.07368	0.564541	0.282271	−	0.959335i	$-0.408912\pi$
$157$	7.07368	0.564541	0.282271	+	0.959335i	$0.408912\pi$
$158$	0	0
$159$	39.3094	3.11744
$160$	0	0
$161$	−27.6754	−2.18113
$162$	0	0
$163$	3.44357	0.269721	0.134861	−	0.990865i	$-0.456941\pi$
$163$	3.44357	0.269721	0.134861	+	0.990865i	$0.456941\pi$
$164$	0	0
$165$	5.33391	0.415244
$166$	0	0
$167$	0.330041	0.0255394	0.0127697	−	0.999918i	$-0.495935\pi$
$167$	0.330041	0.0255394	0.0127697	+	0.999918i	$0.495935\pi$
$168$	0	0
$169$	−12.9630	−0.997156
$170$	0	0
$171$	8.00361	0.612051
$172$	0	0
$173$	4.62957	0.351979	0.175990	−	0.984392i	$-0.443687\pi$
$173$	4.62957	0.351979	0.175990	+	0.984392i	$0.443687\pi$
$174$	0	0
$175$	−9.92756	−0.750453
$176$	0	0
$177$	−10.9555	−0.823469
$178$	0	0
$179$	5.15071	0.384982	0.192491	−	0.981299i	$-0.438343\pi$
$179$	5.15071	0.384982	0.192491	+	0.981299i	$0.438343\pi$
$180$	0	0
$181$	12.3061	0.914708	0.457354	−	0.889285i	$-0.348797\pi$
$181$	12.3061	0.914708	0.457354	+	0.889285i	$0.348797\pi$
$182$	0	0
$183$	31.2424	2.30951
$184$	0	0
$185$	1.18172	0.0868817
$186$	0	0
$187$	−7.62265	−0.557423
$188$	0	0
$189$	68.2461	4.96418
$190$	0	0
$191$	15.7999	1.14324	0.571620	−	0.820518i	$-0.306315\pi$
$191$	15.7999	1.14324	0.571620	+	0.820518i	$0.306315\pi$
$192$	0	0
$193$	26.6087	1.91534	0.957669	−	0.287871i	$-0.0929475\pi$
$193$	26.6087	1.91534	0.957669	+	0.287871i	$0.0929475\pi$
$194$	0	0
$195$	−1.02562	−0.0734463
$196$	0	0
$197$	−1.61255	−0.114890	−0.0574448	−	0.998349i	$-0.518295\pi$
$197$	−1.61255	−0.114890	−0.0574448	+	0.998349i	$0.518295\pi$
$198$	0	0
$199$	14.2001	1.00662	0.503310	−	0.864106i	$-0.332115\pi$
$199$	14.2001	1.00662	0.503310	+	0.864106i	$0.332115\pi$
$200$	0	0
$201$	−41.4730	−2.92528
$202$	0	0
$203$	28.0006	1.96525
$204$	0	0
$205$	1.56993	0.109649
$206$	0	0
$207$	−53.8706	−3.74427
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	−9.58917	−0.660146	−0.330073	−	0.943955i	$-0.607073\pi$
$211$	−9.58917	−0.660146	−0.330073	+	0.943955i	$0.607073\pi$
$212$	0	0
$213$	−13.6743	−0.936948
$214$	0	0
$215$	8.93179	0.609143
$216$	0	0
$217$	−30.1552	−2.04707
$218$	0	0
$219$	−20.4236	−1.38010
$220$	0	0
$221$	1.46571	0.0985943
$222$	0	0
$223$	−2.74913	−0.184095	−0.0920476	−	0.995755i	$-0.529341\pi$
$223$	−2.74913	−0.184095	−0.0920476	+	0.995755i	$0.529341\pi$
$224$	0	0
$225$	−19.3242	−1.28828
$226$	0	0
$227$	−2.67193	−0.177342	−0.0886711	−	0.996061i	$-0.528262\pi$
$227$	−2.67193	−0.177342	−0.0886711	+	0.996061i	$0.528262\pi$
$228$	0	0
$229$	19.9350	1.31734	0.658672	−	0.752430i	$-0.271119\pi$
$229$	19.9350	1.31734	0.658672	+	0.752430i	$0.271119\pi$
$230$	0	0
$231$	13.6394	0.897406
$232$	0	0
$233$	5.74051	0.376073	0.188037	−	0.982162i	$-0.439788\pi$
$233$	5.74051	0.376073	0.188037	+	0.982162i	$0.439788\pi$
$234$	0	0
$235$	−14.2122	−0.927100
$236$	0	0
$237$	9.91273	0.643901
$238$	0	0
$239$	12.9106	0.835120	0.417560	−	0.908649i	$-0.362885\pi$
$239$	12.9106	0.835120	0.417560	+	0.908649i	$0.362885\pi$
$240$	0	0
$241$	26.3067	1.69457	0.847283	−	0.531142i	$-0.178237\pi$
$241$	26.3067	1.69457	0.847283	+	0.531142i	$0.178237\pi$
$242$	0	0
$243$	53.1944	3.41242
$244$	0	0
$245$	15.9294	1.01769
$246$	0	0
$247$	−0.192283	−0.0122347
$248$	0	0
$249$	−21.0268	−1.33252
$250$	0	0
$251$	5.77775	0.364688	0.182344	−	0.983235i	$-0.441631\pi$
$251$	5.77775	0.364688	0.182344	+	0.983235i	$0.441631\pi$
$252$	0	0
$253$	−6.73080	−0.423161
$254$	0	0
$255$	−40.6585	−2.54613
$256$	0	0
$257$	−20.6302	−1.28688	−0.643439	−	0.765498i	$-0.722493\pi$
$257$	−20.6302	−1.28688	−0.643439	+	0.765498i	$0.722493\pi$
$258$	0	0
$259$	3.02179	0.187765
$260$	0	0
$261$	54.5036	3.37369
$262$	0	0
$263$	−13.4936	−0.832048	−0.416024	−	0.909354i	$-0.636577\pi$
$263$	−13.4936	−0.832048	−0.416024	+	0.909354i	$0.636577\pi$
$264$	0	0
$265$	19.0549	1.17053
$266$	0	0
$267$	−45.0119	−2.75469
$268$	0	0
$269$	−6.54562	−0.399094	−0.199547	−	0.979888i	$-0.563947\pi$
$269$	−6.54562	−0.399094	−0.199547	+	0.979888i	$0.563947\pi$
$270$	0	0
$271$	−0.121298	−0.00736831	−0.00368415	−	0.999993i	$-0.501173\pi$
$271$	−0.121298	−0.00736831	−0.00368415	+	0.999993i	$0.501173\pi$
$272$	0	0
$273$	−2.62263	−0.158729
$274$	0	0
$275$	−2.41443	−0.145596
$276$	0	0
$277$	13.4770	0.809757	0.404878	−	0.914371i	$-0.367314\pi$
$277$	13.4770	0.809757	0.404878	+	0.914371i	$0.367314\pi$
$278$	0	0
$279$	−58.6977	−3.51414
$280$	0	0
$281$	−3.63154	−0.216639	−0.108320	−	0.994116i	$-0.534547\pi$
$281$	−3.63154	−0.216639	−0.108320	+	0.994116i	$0.534547\pi$
$282$	0	0
$283$	−16.9545	−1.00784	−0.503919	−	0.863751i	$-0.668109\pi$
$283$	−16.9545	−1.00784	−0.503919	+	0.863751i	$0.668109\pi$
$284$	0	0
$285$	5.33391	0.315953
$286$	0	0
$287$	4.01449	0.236968
$288$	0	0
$289$	41.1048	2.41793
$290$	0	0
$291$	−14.3677	−0.842247
$292$	0	0
$293$	−10.8307	−0.632739	−0.316369	−	0.948636i	$-0.602464\pi$
$293$	−10.8307	−0.632739	−0.316369	+	0.948636i	$0.602464\pi$
$294$	0	0
$295$	−5.31061	−0.309195
$296$	0	0
$297$	16.5978	0.963103
$298$	0	0
$299$	1.29422	0.0748467
$300$	0	0
$301$	22.8396	1.31645
$302$	0	0
$303$	−44.6920	−2.56749
$304$	0	0
$305$	15.1445	0.867171
$306$	0	0
$307$	−2.43981	−0.139247	−0.0696235	−	0.997573i	$-0.522180\pi$
$307$	−2.43981	−0.139247	−0.0696235	+	0.997573i	$0.522180\pi$
$308$	0	0
$309$	−29.3033	−1.66700
$310$	0	0
$311$	26.3308	1.49308	0.746541	−	0.665340i	$-0.231713\pi$
$311$	26.3308	1.49308	0.746541	+	0.665340i	$0.231713\pi$
$312$	0	0
$313$	−3.76724	−0.212937	−0.106468	−	0.994316i	$-0.533954\pi$
$313$	−3.76724	−0.212937	−0.106468	+	0.994316i	$0.533954\pi$
$314$	0	0
$315$	52.9165	2.98150
$316$	0	0
$317$	1.02324	0.0574711	0.0287356	−	0.999587i	$-0.490852\pi$
$317$	1.02324	0.0574711	0.0287356	+	0.999587i	$0.490852\pi$
$318$	0	0
$319$	6.80988	0.381280
$320$	0	0
$321$	−40.5338	−2.26238
$322$	0	0
$323$	−7.62265	−0.424135
$324$	0	0
$325$	0.464256	0.0257523
$326$	0	0
$327$	25.0176	1.38348
$328$	0	0
$329$	−36.3421	−2.00361
$330$	0	0
$331$	−13.5921	−0.747090	−0.373545	−	0.927612i	$-0.621858\pi$
$331$	−13.5921	−0.747090	−0.373545	+	0.927612i	$0.621858\pi$
$332$	0	0
$333$	5.88196	0.322329
$334$	0	0
$335$	−20.1037	−1.09838
$336$	0	0
$337$	−13.0615	−0.711503	−0.355752	−	0.934581i	$-0.615775\pi$
$337$	−13.0615	−0.711503	−0.355752	+	0.934581i	$0.615775\pi$
$338$	0	0
$339$	12.2259	0.664018
$340$	0	0
$341$	−7.33391	−0.397153
$342$	0	0
$343$	11.9509	0.645290
$344$	0	0
$345$	−35.9014	−1.93287
$346$	0	0
$347$	9.78370	0.525217	0.262608	−	0.964903i	$-0.415417\pi$
$347$	9.78370	0.525217	0.262608	+	0.964903i	$0.415417\pi$
$348$	0	0
$349$	4.57188	0.244727	0.122364	−	0.992485i	$-0.460953\pi$
$349$	4.57188	0.244727	0.122364	+	0.992485i	$0.460953\pi$
$350$	0	0
$351$	−3.19148	−0.170349
$352$	0	0
$353$	−7.57171	−0.403001	−0.201501	−	0.979488i	$-0.564582\pi$
$353$	−7.57171	−0.403001	−0.201501	+	0.979488i	$0.564582\pi$
$354$	0	0
$355$	−6.62850	−0.351804
$356$	0	0
$357$	−103.968	−5.50258
$358$	0	0
$359$	−12.7533	−0.673094	−0.336547	−	0.941667i	$-0.609259\pi$
$359$	−12.7533	−0.673094	−0.336547	+	0.941667i	$0.609259\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	3.31717	0.174106
$364$	0	0
$365$	−9.90017	−0.518199
$366$	0	0
$367$	−25.2916	−1.32021	−0.660104	−	0.751174i	$-0.729488\pi$
$367$	−25.2916	−1.32021	−0.660104	+	0.751174i	$0.729488\pi$
$368$	0	0
$369$	7.81428	0.406795
$370$	0	0
$371$	48.7254	2.52970
$372$	0	0
$373$	16.8451	0.872209	0.436104	−	0.899896i	$-0.356358\pi$
$373$	16.8451	0.872209	0.436104	+	0.899896i	$0.356358\pi$
$374$	0	0
$375$	−39.5479	−2.04224
$376$	0	0
$377$	−1.30943	−0.0674390
$378$	0	0
$379$	−24.1556	−1.24079	−0.620394	−	0.784290i	$-0.713027\pi$
$379$	−24.1556	−1.24079	−0.620394	+	0.784290i	$0.713027\pi$
$380$	0	0
$381$	13.9086	0.712560
$382$	0	0
$383$	−3.06487	−0.156608	−0.0783039	−	0.996930i	$-0.524950\pi$
$383$	−3.06487	−0.156608	−0.0783039	+	0.996930i	$0.524950\pi$
$384$	0	0
$385$	6.61158	0.336957
$386$	0	0
$387$	44.4577	2.25991
$388$	0	0
$389$	18.3879	0.932304	0.466152	−	0.884705i	$-0.345640\pi$
$389$	18.3879	0.932304	0.466152	+	0.884705i	$0.345640\pi$
$390$	0	0
$391$	51.3065	2.59468
$392$	0	0
$393$	29.0582	1.46579
$394$	0	0
$395$	4.80511	0.241771
$396$	0	0
$397$	−10.1799	−0.510917	−0.255458	−	0.966820i	$-0.582226\pi$
$397$	−10.1799	−0.510917	−0.255458	+	0.966820i	$0.582226\pi$
$398$	0	0
$399$	13.6394	0.682823
$400$	0	0
$401$	18.0851	0.903128	0.451564	−	0.892239i	$-0.350866\pi$
$401$	18.0851	0.903128	0.451564	+	0.892239i	$0.350866\pi$
$402$	0	0
$403$	1.41019	0.0702465
$404$	0	0
$405$	49.9225	2.48067
$406$	0	0
$407$	0.734914	0.0364283
$408$	0	0
$409$	20.3087	1.00420	0.502100	−	0.864810i	$-0.332561\pi$
$409$	20.3087	1.00420	0.502100	+	0.864810i	$0.332561\pi$
$410$	0	0
$411$	−26.7450	−1.31923
$412$	0	0
$413$	−13.5798	−0.668219
$414$	0	0
$415$	−10.1925	−0.500332
$416$	0	0
$417$	−44.1849	−2.16375
$418$	0	0
$419$	−9.98616	−0.487856	−0.243928	−	0.969793i	$-0.578436\pi$
$419$	−9.98616	−0.487856	−0.243928	+	0.969793i	$0.578436\pi$
$420$	0	0
$421$	−34.7710	−1.69463	−0.847316	−	0.531089i	$-0.821783\pi$
$421$	−34.7710	−1.69463	−0.847316	+	0.531089i	$0.821783\pi$
$422$	0	0
$423$	−70.7406	−3.43952
$424$	0	0
$425$	18.4044	0.892744
$426$	0	0
$427$	38.7261	1.87409
$428$	0	0
$429$	−0.637837	−0.0307950
$430$	0	0
$431$	22.3666	1.07736	0.538681	−	0.842510i	$-0.318923\pi$
$431$	22.3666	1.07736	0.538681	+	0.842510i	$0.318923\pi$
$432$	0	0
$433$	−35.4168	−1.70202	−0.851011	−	0.525148i	$-0.824010\pi$
$433$	−35.4168	−1.70202	−0.851011	+	0.525148i	$0.824010\pi$
$434$	0	0
$435$	36.3233	1.74157
$436$	0	0
$437$	−6.73080	−0.321978
$438$	0	0
$439$	−9.55446	−0.456010	−0.228005	−	0.973660i	$-0.573220\pi$
$439$	−9.55446	−0.456010	−0.228005	+	0.973660i	$0.573220\pi$
$440$	0	0
$441$	79.2880	3.77562
$442$	0	0
$443$	28.3348	1.34623	0.673113	−	0.739540i	$-0.264957\pi$
$443$	28.3348	1.34623	0.673113	+	0.739540i	$0.264957\pi$
$444$	0	0
$445$	−21.8192	−1.03433
$446$	0	0
$447$	51.9244	2.45594
$448$	0	0
$449$	19.7315	0.931188	0.465594	−	0.884998i	$-0.345841\pi$
$449$	19.7315	0.931188	0.465594	+	0.884998i	$0.345841\pi$
$450$	0	0
$451$	0.976345	0.0459743
$452$	0	0
$453$	59.3098	2.78662
$454$	0	0
$455$	−1.27130	−0.0595993
$456$	0	0
$457$	23.6281	1.10527	0.552637	−	0.833422i	$-0.313621\pi$
$457$	23.6281	1.10527	0.552637	+	0.833422i	$0.313621\pi$
$458$	0	0
$459$	−126.519	−5.90541
$460$	0	0
$461$	−0.976802	−0.0454942	−0.0227471	−	0.999741i	$-0.507241\pi$
$461$	−0.976802	−0.0454942	−0.0227471	+	0.999741i	$0.507241\pi$
$462$	0	0
$463$	−7.78708	−0.361896	−0.180948	−	0.983493i	$-0.557917\pi$
$463$	−7.78708	−0.361896	−0.180948	+	0.983493i	$0.557917\pi$
$464$	0	0
$465$	−39.1184	−1.81407
$466$	0	0
$467$	−35.1591	−1.62697	−0.813486	−	0.581585i	$-0.802433\pi$
$467$	−35.1591	−1.62697	−0.813486	+	0.581585i	$0.802433\pi$
$468$	0	0
$469$	−51.4074	−2.37377
$470$	0	0
$471$	23.4646	1.08119
$472$	0	0
$473$	5.55470	0.255406
$474$	0	0
$475$	−2.41443	−0.110782
$476$	0	0
$477$	94.8450	4.34265
$478$	0	0
$479$	−30.2805	−1.38355	−0.691777	−	0.722112i	$-0.743172\pi$
$479$	−30.2805	−1.38355	−0.691777	+	0.722112i	$0.743172\pi$
$480$	0	0
$481$	−0.141312	−0.00644326
$482$	0	0
$483$	−91.8039	−4.17722
$484$	0	0
$485$	−6.96460	−0.316246
$486$	0	0
$487$	23.3841	1.05964	0.529818	−	0.848112i	$-0.322260\pi$
$487$	23.3841	1.05964	0.529818	+	0.848112i	$0.322260\pi$
$488$	0	0
$489$	11.4229	0.516561
$490$	0	0
$491$	−18.1633	−0.819696	−0.409848	−	0.912154i	$-0.634418\pi$
$491$	−18.1633	−0.819696	−0.409848	+	0.912154i	$0.634418\pi$
$492$	0	0
$493$	−51.9093	−2.33788
$494$	0	0
$495$	12.8696	0.578444
$496$	0	0
$497$	−16.9498	−0.760303
$498$	0	0
$499$	39.7067	1.77752	0.888759	−	0.458375i	$-0.151568\pi$
$499$	39.7067	1.77752	0.888759	+	0.458375i	$0.151568\pi$
$500$	0	0
$501$	1.09480	0.0489122
$502$	0	0
$503$	−13.5741	−0.605241	−0.302621	−	0.953111i	$-0.597861\pi$
$503$	−13.5741	−0.605241	−0.302621	+	0.953111i	$0.597861\pi$
$504$	0	0
$505$	−21.6641	−0.964038
$506$	0	0
$507$	−43.0005	−1.90972
$508$	0	0
$509$	28.1607	1.24820	0.624101	−	0.781343i	$-0.285465\pi$
$509$	28.1607	1.24820	0.624101	+	0.781343i	$0.285465\pi$
$510$	0	0
$511$	−25.3158	−1.11991
$512$	0	0
$513$	16.5978	0.732811
$514$	0	0
$515$	−14.2045	−0.625925
$516$	0	0
$517$	−8.83859	−0.388721
$518$	0	0
$519$	15.3571	0.674100
$520$	0	0
$521$	−17.2741	−0.756790	−0.378395	−	0.925644i	$-0.623524\pi$
$521$	−17.2741	−0.756790	−0.378395	+	0.925644i	$0.623524\pi$
$522$	0	0
$523$	14.6315	0.639789	0.319894	−	0.947453i	$-0.396353\pi$
$523$	14.6315	0.639789	0.319894	+	0.947453i	$0.396353\pi$
$524$	0	0
$525$	−32.9314	−1.43724
$526$	0	0
$527$	55.9038	2.43521
$528$	0	0
$529$	22.3036	0.969722
$530$	0	0
$531$	−26.4333	−1.14711
$532$	0	0
$533$	−0.187735	−0.00813170
$534$	0	0
$535$	−19.6484	−0.849475
$536$	0	0
$537$	17.0858	0.737305
$538$	0	0
$539$	9.90653	0.426704
$540$	0	0
$541$	−0.742620	−0.0319277	−0.0159639	−	0.999873i	$-0.505082\pi$
$541$	−0.742620	−0.0319277	−0.0159639	+	0.999873i	$0.505082\pi$
$542$	0	0
$543$	40.8215	1.75182
$544$	0	0
$545$	12.1271	0.519467
$546$	0	0
$547$	−33.5177	−1.43311	−0.716557	−	0.697528i	$-0.754283\pi$
$547$	−33.5177	−1.43311	−0.716557	+	0.697528i	$0.754283\pi$
$548$	0	0
$549$	75.3812	3.21719
$550$	0	0
$551$	6.80988	0.290111
$552$	0	0
$553$	12.2872	0.522505
$554$	0	0
$555$	3.91996	0.166393
$556$	0	0
$557$	−18.6775	−0.791392	−0.395696	−	0.918381i	$-0.629497\pi$
$557$	−18.6775	−0.791392	−0.395696	+	0.918381i	$0.629497\pi$
$558$	0	0
$559$	−1.06808	−0.0451749
$560$	0	0
$561$	−25.2856	−1.06756
$562$	0	0
$563$	28.4524	1.19913	0.599564	−	0.800327i	$-0.295341\pi$
$563$	28.4524	1.19913	0.599564	+	0.800327i	$0.295341\pi$
$564$	0	0
$565$	5.92638	0.249325
$566$	0	0
$567$	127.657	5.36110
$568$	0	0
$569$	7.87017	0.329935	0.164967	−	0.986299i	$-0.447248\pi$
$569$	7.87017	0.329935	0.164967	+	0.986299i	$0.447248\pi$
$570$	0	0
$571$	37.6693	1.57641	0.788205	−	0.615413i	$-0.211011\pi$
$571$	37.6693	1.57641	0.788205	+	0.615413i	$0.211011\pi$
$572$	0	0
$573$	52.4109	2.18950
$574$	0	0
$575$	16.2511	0.677716
$576$	0	0
$577$	43.9800	1.83091	0.915457	−	0.402417i	$-0.131830\pi$
$577$	43.9800	1.83091	0.915457	+	0.402417i	$0.131830\pi$
$578$	0	0
$579$	88.2656	3.66819
$580$	0	0
$581$	−26.0634	−1.08129
$582$	0	0
$583$	11.8503	0.490788
$584$	0	0
$585$	−2.47460	−0.102312
$586$	0	0
$587$	11.6515	0.480907	0.240453	−	0.970661i	$-0.422704\pi$
$587$	11.6515	0.480907	0.240453	+	0.970661i	$0.422704\pi$
$588$	0	0
$589$	−7.33391	−0.302188
$590$	0	0
$591$	−5.34911	−0.220033
$592$	0	0
$593$	−32.3337	−1.32778	−0.663892	−	0.747828i	$-0.731097\pi$
$593$	−32.3337	−1.32778	−0.663892	+	0.747828i	$0.731097\pi$
$594$	0	0
$595$	−50.3977	−2.06611
$596$	0	0
$597$	47.1042	1.92785
$598$	0	0
$599$	0.717765	0.0293271	0.0146635	−	0.999892i	$-0.495332\pi$
$599$	0.717765	0.0293271	0.0146635	+	0.999892i	$0.495332\pi$
$600$	0	0
$601$	−17.6539	−0.720117	−0.360058	−	0.932930i	$-0.617243\pi$
$601$	−17.6539	−0.720117	−0.360058	+	0.932930i	$0.617243\pi$
$602$	0	0
$603$	−100.065	−4.07498
$604$	0	0
$605$	1.60797	0.0653733
$606$	0	0
$607$	10.2348	0.415419	0.207710	−	0.978191i	$-0.433399\pi$
$607$	10.2348	0.415419	0.207710	+	0.978191i	$0.433399\pi$
$608$	0	0
$609$	92.8826	3.76379
$610$	0	0
$611$	1.69951	0.0687550
$612$	0	0
$613$	−13.7215	−0.554205	−0.277102	−	0.960840i	$-0.589374\pi$
$613$	−13.7215	−0.554205	−0.277102	+	0.960840i	$0.589374\pi$
$614$	0	0
$615$	5.20773	0.209996
$616$	0	0
$617$	−21.2447	−0.855280	−0.427640	−	0.903949i	$-0.640655\pi$
$617$	−21.2447	−0.855280	−0.427640	+	0.903949i	$0.640655\pi$
$618$	0	0
$619$	−16.8753	−0.678274	−0.339137	−	0.940737i	$-0.610135\pi$
$619$	−16.8753	−0.678274	−0.339137	+	0.940737i	$0.610135\pi$
$620$	0	0
$621$	−111.716	−4.48303
$622$	0	0
$623$	−55.7940	−2.23534
$624$	0	0
$625$	−7.09834	−0.283933
$626$	0	0
$627$	3.31717	0.132475
$628$	0	0
$629$	−5.60199	−0.223366
$630$	0	0
$631$	31.7238	1.26291	0.631453	−	0.775414i	$-0.282459\pi$
$631$	31.7238	1.26291	0.631453	+	0.775414i	$0.282459\pi$
$632$	0	0
$633$	−31.8089	−1.26429
$634$	0	0
$635$	6.74208	0.267551
$636$	0	0
$637$	−1.90486	−0.0754734
$638$	0	0
$639$	−32.9931	−1.30519
$640$	0	0
$641$	34.5505	1.36466	0.682332	−	0.731043i	$-0.260966\pi$
$641$	34.5505	1.36466	0.682332	+	0.731043i	$0.260966\pi$
$642$	0	0
$643$	3.89781	0.153715	0.0768573	−	0.997042i	$-0.475511\pi$
$643$	3.89781	0.153715	0.0768573	+	0.997042i	$0.475511\pi$
$644$	0	0
$645$	29.6283	1.16661
$646$	0	0
$647$	19.5834	0.769904	0.384952	−	0.922937i	$-0.374218\pi$
$647$	19.5834	0.769904	0.384952	+	0.922937i	$0.374218\pi$
$648$	0	0
$649$	−3.30268	−0.129641
$650$	0	0
$651$	−100.030	−3.92048
$652$	0	0
$653$	13.1173	0.513320	0.256660	−	0.966502i	$-0.417378\pi$
$653$	13.1173	0.513320	0.256660	+	0.966502i	$0.417378\pi$
$654$	0	0
$655$	14.0857	0.550374
$656$	0	0
$657$	−49.2777	−1.92251
$658$	0	0
$659$	32.3835	1.26148	0.630741	−	0.775993i	$-0.282751\pi$
$659$	32.3835	1.26148	0.630741	+	0.775993i	$0.282751\pi$
$660$	0	0
$661$	−21.4666	−0.834954	−0.417477	−	0.908688i	$-0.637086\pi$
$661$	−21.4666	−0.834954	−0.417477	+	0.908688i	$0.637086\pi$
$662$	0	0
$663$	4.86200	0.188825
$664$	0	0
$665$	6.61158	0.256386
$666$	0	0
$667$	−45.8359	−1.77477
$668$	0	0
$669$	−9.11932	−0.352573
$670$	0	0
$671$	9.41840	0.363593
$672$	0	0
$673$	32.8054	1.26455	0.632277	−	0.774742i	$-0.282120\pi$
$673$	32.8054	1.26455	0.632277	+	0.774742i	$0.282120\pi$
$674$	0	0
$675$	−40.0743	−1.54246
$676$	0	0
$677$	−38.3331	−1.47326	−0.736630	−	0.676296i	$-0.763584\pi$
$677$	−38.3331	−1.47326	−0.736630	+	0.676296i	$0.763584\pi$
$678$	0	0
$679$	−17.8093	−0.683456
$680$	0	0
$681$	−8.86324	−0.339640
$682$	0	0
$683$	−10.3756	−0.397010	−0.198505	−	0.980100i	$-0.563609\pi$
$683$	−10.3756	−0.397010	−0.198505	+	0.980100i	$0.563609\pi$
$684$	0	0
$685$	−12.9644	−0.495345
$686$	0	0
$687$	66.1278	2.52293
$688$	0	0
$689$	−2.27861	−0.0868082
$690$	0	0
$691$	6.05581	0.230374	0.115187	−	0.993344i	$-0.463253\pi$
$691$	6.05581	0.230374	0.115187	+	0.993344i	$0.463253\pi$
$692$	0	0
$693$	32.9089	1.25010
$694$	0	0
$695$	−21.4183	−0.812442
$696$	0	0
$697$	−7.44233	−0.281898
$698$	0	0
$699$	19.0422	0.720244
$700$	0	0
$701$	0.497075	0.0187743	0.00938714	−	0.999956i	$-0.497012\pi$
$701$	0.497075	0.0187743	0.00938714	+	0.999956i	$0.497012\pi$
$702$	0	0
$703$	0.734914	0.0277178
$704$	0	0
$705$	−47.1442	−1.77555
$706$	0	0
$707$	−55.3974	−2.08343
$708$	0	0
$709$	12.5856	0.472661	0.236330	−	0.971673i	$-0.424055\pi$
$709$	12.5856	0.472661	0.236330	+	0.971673i	$0.424055\pi$
$710$	0	0
$711$	23.9173	0.896967
$712$	0	0
$713$	49.3630	1.84866
$714$	0	0
$715$	−0.309186	−0.0115629
$716$	0	0
$717$	42.8268	1.59940
$718$	0	0
$719$	6.17724	0.230372	0.115186	−	0.993344i	$-0.463254\pi$
$719$	6.17724	0.230372	0.115186	+	0.993344i	$0.463254\pi$
$720$	0	0
$721$	−36.3225	−1.35272
$722$	0	0
$723$	87.2639	3.24538
$724$	0	0
$725$	−16.4420	−0.610641
$726$	0	0
$727$	−5.90914	−0.219158	−0.109579	−	0.993978i	$-0.534950\pi$
$727$	−5.90914	−0.219158	−0.109579	+	0.993978i	$0.534950\pi$
$728$	0	0
$729$	83.3141	3.08571
$730$	0	0
$731$	−42.3416	−1.56606
$732$	0	0
$733$	24.0990	0.890118	0.445059	−	0.895501i	$-0.353183\pi$
$733$	24.0990	0.890118	0.445059	+	0.895501i	$0.353183\pi$
$734$	0	0
$735$	52.8405	1.94905
$736$	0	0
$737$	−12.5025	−0.460537
$738$	0	0
$739$	5.74460	0.211319	0.105659	−	0.994402i	$-0.466305\pi$
$739$	5.74460	0.211319	0.105659	+	0.994402i	$0.466305\pi$
$740$	0	0
$741$	−0.637837	−0.0234315
$742$	0	0
$743$	−16.8092	−0.616671	−0.308335	−	0.951278i	$-0.599772\pi$
$743$	−16.8092	−0.616671	−0.308335	+	0.951278i	$0.599772\pi$
$744$	0	0
$745$	25.1699	0.922155
$746$	0	0
$747$	−50.7330	−1.85622
$748$	0	0
$749$	−50.2431	−1.83584
$750$	0	0
$751$	−32.6804	−1.19252	−0.596262	−	0.802790i	$-0.703348\pi$
$751$	−32.6804	−1.19252	−0.596262	+	0.802790i	$0.703348\pi$
$752$	0	0
$753$	19.1658	0.698439
$754$	0	0
$755$	28.7499	1.04632
$756$	0	0
$757$	40.3897	1.46799	0.733994	−	0.679156i	$-0.237654\pi$
$757$	40.3897	1.46799	0.733994	+	0.679156i	$0.237654\pi$
$758$	0	0
$759$	−22.3272	−0.810425
$760$	0	0
$761$	−45.6390	−1.65441	−0.827207	−	0.561898i	$-0.810071\pi$
$761$	−45.6390	−1.65441	−0.827207	+	0.561898i	$0.810071\pi$
$762$	0	0
$763$	31.0103	1.12265
$764$	0	0
$765$	−98.1001	−3.54682
$766$	0	0
$767$	0.635051	0.0229303
$768$	0	0
$769$	−24.4815	−0.882826	−0.441413	−	0.897304i	$-0.645523\pi$
$769$	−24.4815	−0.882826	−0.441413	+	0.897304i	$0.645523\pi$
$770$	0	0
$771$	−68.4339	−2.46459
$772$	0	0
$773$	−1.74051	−0.0626018	−0.0313009	−	0.999510i	$-0.509965\pi$
$773$	−1.74051	−0.0626018	−0.0313009	+	0.999510i	$0.509965\pi$
$774$	0	0
$775$	17.7072	0.636063
$776$	0	0
$777$	10.0238	0.359601
$778$	0	0
$779$	0.976345	0.0349812
$780$	0	0
$781$	−4.12228	−0.147507
$782$	0	0
$783$	113.029	4.03933
$784$	0	0
$785$	11.3743	0.405965
$786$	0	0
$787$	25.4670	0.907801	0.453901	−	0.891052i	$-0.350032\pi$
$787$	25.4670	0.907801	0.453901	+	0.891052i	$0.350032\pi$
$788$	0	0
$789$	−44.7604	−1.59351
$790$	0	0
$791$	15.1544	0.538829
$792$	0	0
$793$	−1.81100	−0.0643106
$794$	0	0
$795$	63.2083	2.24177
$796$	0	0
$797$	17.4228	0.617146	0.308573	−	0.951201i	$-0.400149\pi$
$797$	17.4228	0.617146	0.308573	+	0.951201i	$0.400149\pi$
$798$	0	0
$799$	67.3734	2.38350
$800$	0	0
$801$	−108.604	−3.83733
$802$	0	0
$803$	−6.15694	−0.217274
$804$	0	0
$805$	−44.5012	−1.56846
$806$	0	0
$807$	−21.7129	−0.764332
$808$	0	0
$809$	26.1897	0.920779	0.460390	−	0.887717i	$-0.347710\pi$
$809$	26.1897	0.920779	0.460390	+	0.887717i	$0.347710\pi$
$810$	0	0
$811$	−50.4597	−1.77188	−0.885939	−	0.463801i	$-0.846485\pi$
$811$	−50.4597	−1.77188	−0.885939	+	0.463801i	$0.846485\pi$
$812$	0	0
$813$	−0.402365	−0.0141115
$814$	0	0
$815$	5.53715	0.193958
$816$	0	0
$817$	5.55470	0.194334
$818$	0	0
$819$	−6.32783	−0.221112
$820$	0	0
$821$	5.99962	0.209388	0.104694	−	0.994504i	$-0.466614\pi$
$821$	5.99962	0.209388	0.104694	+	0.994504i	$0.466614\pi$
$822$	0	0
$823$	−0.615783	−0.0214649	−0.0107324	−	0.999942i	$-0.503416\pi$
$823$	−0.615783	−0.0214649	−0.0107324	+	0.999942i	$0.503416\pi$
$824$	0	0
$825$	−8.00909	−0.278841
$826$	0	0
$827$	−40.7839	−1.41820	−0.709098	−	0.705110i	$-0.750898\pi$
$827$	−40.7839	−1.41820	−0.709098	+	0.705110i	$0.750898\pi$
$828$	0	0
$829$	38.6707	1.34309	0.671544	−	0.740964i	$-0.265631\pi$
$829$	38.6707	1.34309	0.671544	+	0.740964i	$0.265631\pi$
$830$	0	0
$831$	44.7056	1.55082
$832$	0	0
$833$	−75.5140	−2.61640
$834$	0	0
$835$	0.530696	0.0183655
$836$	0	0
$837$	−121.727	−4.20749
$838$	0	0
$839$	−18.6821	−0.644978	−0.322489	−	0.946573i	$-0.604520\pi$
$839$	−18.6821	−0.644978	−0.322489	+	0.946573i	$0.604520\pi$
$840$	0	0
$841$	17.3745	0.599121
$842$	0	0
$843$	−12.0464	−0.414901
$844$	0	0
$845$	−20.8442	−0.717061
$846$	0	0
$847$	4.11176	0.141282
$848$	0	0
$849$	−56.2408	−1.93018
$850$	0	0
$851$	−4.94655	−0.169566
$852$	0	0
$853$	44.3592	1.51883	0.759415	−	0.650607i	$-0.225485\pi$
$853$	44.3592	1.51883	0.759415	+	0.650607i	$0.225485\pi$
$854$	0	0
$855$	12.8696	0.440130
$856$	0	0
$857$	22.4643	0.767367	0.383684	−	0.923465i	$-0.374655\pi$
$857$	22.4643	0.767367	0.383684	+	0.923465i	$0.374655\pi$
$858$	0	0
$859$	−7.80075	−0.266158	−0.133079	−	0.991105i	$-0.542486\pi$
$859$	−7.80075	−0.266158	−0.133079	+	0.991105i	$0.542486\pi$
$860$	0	0
$861$	13.3167	0.453833
$862$	0	0
$863$	−42.3672	−1.44220	−0.721098	−	0.692833i	$-0.756362\pi$
$863$	−42.3672	−1.44220	−0.721098	+	0.692833i	$0.756362\pi$
$864$	0	0
$865$	7.44420	0.253110
$866$	0	0
$867$	136.351	4.63074
$868$	0	0
$869$	2.98831	0.101371
$870$	0	0
$871$	2.40403	0.0814575
$872$	0	0
$873$	−34.6660	−1.17327
$874$	0	0
$875$	−49.0211	−1.65722
$876$	0	0
$877$	16.7160	0.564458	0.282229	−	0.959347i	$-0.408926\pi$
$877$	16.7160	0.564458	0.282229	+	0.959347i	$0.408926\pi$
$878$	0	0
$879$	−35.9274	−1.21180
$880$	0	0
$881$	−34.5007	−1.16236	−0.581179	−	0.813776i	$-0.697408\pi$
$881$	−34.5007	−1.16236	−0.581179	+	0.813776i	$0.697408\pi$
$882$	0	0
$883$	−20.0083	−0.673333	−0.336667	−	0.941624i	$-0.609300\pi$
$883$	−20.0083	−0.673333	−0.336667	+	0.941624i	$0.609300\pi$
$884$	0	0
$885$	−17.6162	−0.592161
$886$	0	0
$887$	−51.5291	−1.73018	−0.865089	−	0.501618i	$-0.832738\pi$
$887$	−51.5291	−1.73018	−0.865089	+	0.501618i	$0.832738\pi$
$888$	0	0
$889$	17.2402	0.578219
$890$	0	0
$891$	31.0469	1.04011
$892$	0	0
$893$	−8.83859	−0.295772
$894$	0	0
$895$	8.28218	0.276843
$896$	0	0
$897$	4.29315	0.143344
$898$	0	0
$899$	−49.9430	−1.66569
$900$	0	0
$901$	−90.3305	−3.00934
$902$	0	0
$903$	75.7627	2.52123
$904$	0	0
$905$	19.7879	0.657772
$906$	0	0
$907$	46.9324	1.55836	0.779182	−	0.626798i	$-0.215635\pi$
$907$	46.9324	1.55836	0.779182	+	0.626798i	$0.215635\pi$
$908$	0	0
$909$	−107.832	−3.57656
$910$	0	0
$911$	−43.3927	−1.43766	−0.718831	−	0.695184i	$-0.755323\pi$
$911$	−43.3927	−1.43766	−0.718831	+	0.695184i	$0.755323\pi$
$912$	0	0
$913$	−6.33876	−0.209783
$914$	0	0
$915$	50.2368	1.66078
$916$	0	0
$917$	36.0187	1.18944
$918$	0	0
$919$	−46.1195	−1.52134	−0.760671	−	0.649138i	$-0.775130\pi$
$919$	−46.1195	−1.52134	−0.760671	+	0.649138i	$0.775130\pi$
$920$	0	0
$921$	−8.09325	−0.266682
$922$	0	0
$923$	0.792646	0.0260903
$924$	0	0
$925$	−1.77440	−0.0583419
$926$	0	0
$927$	−70.7024	−2.32217
$928$	0	0
$929$	−50.6414	−1.66149	−0.830745	−	0.556653i	$-0.812085\pi$
$929$	−50.6414	−1.66149	−0.830745	+	0.556653i	$0.812085\pi$
$930$	0	0
$931$	9.90653	0.324673
$932$	0	0
$933$	87.3436	2.85950
$934$	0	0
$935$	−12.2570	−0.400846
$936$	0	0
$937$	52.6225	1.71910	0.859551	−	0.511051i	$-0.170744\pi$
$937$	52.6225	1.71910	0.859551	+	0.511051i	$0.170744\pi$
$938$	0	0
$939$	−12.4966	−0.407810
$940$	0	0
$941$	−39.1834	−1.27734	−0.638671	−	0.769480i	$-0.720516\pi$
$941$	−39.1834	−1.27734	−0.638671	+	0.769480i	$0.720516\pi$
$942$	0	0
$943$	−6.57158	−0.214000
$944$	0	0
$945$	109.738	3.56977
$946$	0	0
$947$	1.35590	0.0440608	0.0220304	−	0.999757i	$-0.492987\pi$
$947$	1.35590	0.0440608	0.0220304	+	0.999757i	$0.492987\pi$
$948$	0	0
$949$	1.18388	0.0384303
$950$	0	0
$951$	3.39427	0.110067
$952$	0	0
$953$	−10.3667	−0.335812	−0.167906	−	0.985803i	$-0.553701\pi$
$953$	−10.3667	−0.335812	−0.167906	+	0.985803i	$0.553701\pi$
$954$	0	0
$955$	25.4057	0.822111
$956$	0	0
$957$	22.5895	0.730216
$958$	0	0
$959$	−33.1515	−1.07052
$960$	0	0
$961$	22.7862	0.735038
$962$	0	0
$963$	−97.7992	−3.15154
$964$	0	0
$965$	42.7860	1.37733
$966$	0	0
$967$	11.8273	0.380340	0.190170	−	0.981751i	$-0.439096\pi$
$967$	11.8273	0.380340	0.190170	+	0.981751i	$0.439096\pi$
$968$	0	0
$969$	−25.2856	−0.812291
$970$	0	0
$971$	−4.74148	−0.152161	−0.0760806	−	0.997102i	$-0.524241\pi$
$971$	−4.74148	−0.152161	−0.0760806	+	0.997102i	$0.524241\pi$
$972$	0	0
$973$	−54.7689	−1.75581
$974$	0	0
$975$	1.54001	0.0493199
$976$	0	0
$977$	35.1330	1.12401	0.562003	−	0.827135i	$-0.310031\pi$
$977$	35.1330	1.12401	0.562003	+	0.827135i	$0.310031\pi$
$978$	0	0
$979$	−13.5694	−0.433679
$980$	0	0
$981$	60.3621	1.92721
$982$	0	0
$983$	13.8563	0.441948	0.220974	−	0.975280i	$-0.429076\pi$
$983$	13.8563	0.441948	0.220974	+	0.975280i	$0.429076\pi$
$984$	0	0
$985$	−2.59293	−0.0826178
$986$	0	0
$987$	−120.553	−3.83724
$988$	0	0
$989$	−37.3876	−1.18886
$990$	0	0
$991$	−8.08672	−0.256883	−0.128442	−	0.991717i	$-0.540997\pi$
$991$	−8.08672	−0.256883	−0.128442	+	0.991717i	$0.540997\pi$
$992$	0	0
$993$	−45.0873	−1.43080
$994$	0	0
$995$	22.8334	0.723867
$996$	0	0
$997$	−4.02300	−0.127410	−0.0637049	−	0.997969i	$-0.520292\pi$
$997$	−4.02300	−0.127410	−0.0637049	+	0.997969i	$0.520292\pi$
$998$	0	0
$999$	12.1980	0.385926

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.w.1.6		6
4.3	odd	2		836.2.a.e.1.1	✓	6
12.11	even	2		7524.2.a.q.1.3		6
44.43	even	2		9196.2.a.n.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
836.2.a.e.1.1	✓	6	4.3	odd	2
3344.2.a.w.1.6		6	1.1	even	1	trivial
7524.2.a.q.1.3		6	12.11	even	2
9196.2.a.n.1.1		6	44.43	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+3.31717 q^{3} +1.60797 q^{5} +4.11176 q^{7} +8.00361 q^{9} +1.00000 q^{11} -0.192283 q^{13} +5.33391 q^{15} -7.62265 q^{17} +1.00000 q^{19} +13.6394 q^{21} -6.73080 q^{23} -2.41443 q^{25} +16.5978 q^{27} +6.80988 q^{29} -7.33391 q^{31} +3.31717 q^{33} +6.61158 q^{35} +0.734914 q^{37} -0.637837 q^{39} +0.976345 q^{41} +5.55470 q^{43} +12.8696 q^{45} -8.83859 q^{47} +9.90653 q^{49} -25.2856 q^{51} +11.8503 q^{53} +1.60797 q^{55} +3.31717 q^{57} -3.30268 q^{59} +9.41840 q^{61} +32.9089 q^{63} -0.309186 q^{65} -12.5025 q^{67} -22.3272 q^{69} -4.12228 q^{71} -6.15694 q^{73} -8.00909 q^{75} +4.11176 q^{77} +2.98831 q^{79} +31.0469 q^{81} -6.33876 q^{83} -12.2570 q^{85} +22.5895 q^{87} -13.5694 q^{89} -0.790623 q^{91} -24.3278 q^{93} +1.60797 q^{95} -4.33130 q^{97} +8.00361 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{3} + 5 q^{5} + 2 q^{7} + 17 q^{9} + 6 q^{11} + 8 q^{13} - 9 q^{15} - 2 q^{17} + 6 q^{19} + 18 q^{21} - 5 q^{23} + 13 q^{25} - 7 q^{27} + 10 q^{29} - 3 q^{31} - q^{33} + 4 q^{35} + 7 q^{37} + 10 q^{39}+ \cdots + 17 q^{99}+O(q^{100}) \) 6 * q - q^3 + 5 * q^5 + 2 * q^7 + 17 * q^9 + 6 * q^11 + 8 * q^13 - 9 * q^15 - 2 * q^17 + 6 * q^19 + 18 * q^21 - 5 * q^23 + 13 * q^25 - 7 * q^27 + 10 * q^29 - 3 * q^31 - q^33 + 4 * q^35 + 7 * q^37 + 10 * q^39 + 6 * q^41 - 16 * q^43 + 42 * q^45 + 12 * q^49 - 8 * q^51 + 20 * q^53 + 5 * q^55 - q^57 - 15 * q^59 + 24 * q^61 + 20 * q^63 - 28 * q^65 - 25 * q^67 - 33 * q^69 + 9 * q^71 - 26 * q^73 - 28 * q^75 + 2 * q^77 + 16 * q^79 + 58 * q^81 + 2 * q^83 + 12 * q^85 + 36 * q^87 + 7 * q^89 - 8 * q^91 - 55 * q^93 + 5 * q^95 + 37 * q^97 + 17 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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