LMFDB - Embedded newform 3344.2.a.w.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:	$6$
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			1.5
Root			$-2.05861$ 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)

$f(q)$	$=$	$q+2.05861 q^{3} +0.680036 q^{5} -1.59123 q^{7} +1.23787 q^{9} +O(q^{10})$	$q+2.05861 q^{3} +0.680036 q^{5} -1.59123 q^{7} +1.23787 q^{9} +1.00000 q^{11} -0.546391 q^{13} +1.39993 q^{15} +6.61704 q^{17} +1.00000 q^{19} -3.27573 q^{21} +6.44615 q^{23} -4.53755 q^{25} -3.62754 q^{27} +7.32229 q^{29} -3.39993 q^{31} +2.05861 q^{33} -1.08210 q^{35} +8.83065 q^{37} -1.12481 q^{39} +3.18632 q^{41} -10.4060 q^{43} +0.841795 q^{45} +7.25697 q^{47} -4.46798 q^{49} +13.6219 q^{51} +7.47729 q^{53} +0.680036 q^{55} +2.05861 q^{57} -11.1288 q^{59} +8.75715 q^{61} -1.96974 q^{63} -0.371565 q^{65} -11.1326 q^{67} +13.2701 q^{69} +7.88100 q^{71} +3.00155 q^{73} -9.34104 q^{75} -1.59123 q^{77} -8.73426 q^{79} -11.1813 q^{81} +7.76611 q^{83} +4.49983 q^{85} +15.0737 q^{87} +15.7031 q^{89} +0.869435 q^{91} -6.99912 q^{93} +0.680036 q^{95} +9.99329 q^{97} +1.23787 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{3} + 5 q^{5} + 2 q^{7} + 17 q^{9}+O(q^{10}) $ 6 * q - q^3 + 5 * q^5 + 2 * q^7 + 17 * q^9	$ 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}+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

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
$3$	2.05861	1.18854	0.594269	−	0.804266i	$-0.297441\pi$
$3$	2.05861	1.18854	0.594269	+	0.804266i	$0.297441\pi$
$4$	0	0
$5$	0.680036	0.304121	0.152061	−	0.988371i	$-0.451409\pi$
$5$	0.680036	0.304121	0.152061	+	0.988371i	$0.451409\pi$
$6$	0	0
$7$	−1.59123	−0.601429	−0.300715	−	0.953714i	$-0.597225\pi$
$7$	−1.59123	−0.601429	−0.300715	+	0.953714i	$0.597225\pi$
$8$	0	0
$9$	1.23787	0.412623
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−0.546391	−0.151542	−0.0757708	−	0.997125i	$-0.524142\pi$
$13$	−0.546391	−0.151542	−0.0757708	+	0.997125i	$0.524142\pi$
$14$	0	0
$15$	1.39993	0.361460
$16$	0	0
$17$	6.61704	1.60487	0.802434	−	0.596740i	$-0.203538\pi$
$17$	6.61704	1.60487	0.802434	+	0.596740i	$0.203538\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−3.27573	−0.714822
$22$	0	0
$23$	6.44615	1.34411	0.672057	−	0.740499i	$-0.265411\pi$
$23$	6.44615	1.34411	0.672057	+	0.740499i	$0.265411\pi$
$24$	0	0
$25$	−4.53755	−0.907510
$26$	0	0
$27$	−3.62754	−0.698120
$28$	0	0
$29$	7.32229	1.35971	0.679857	−	0.733344i	$-0.262042\pi$
$29$	7.32229	1.35971	0.679857	+	0.733344i	$0.262042\pi$
$30$	0	0
$31$	−3.39993	−0.610645	−0.305323	−	0.952249i	$-0.598764\pi$
$31$	−3.39993	−0.610645	−0.305323	+	0.952249i	$0.598764\pi$
$32$	0	0
$33$	2.05861	0.358358
$34$	0	0
$35$	−1.08210	−0.182907
$36$	0	0
$37$	8.83065	1.45175	0.725875	−	0.687826i	$-0.241435\pi$
$37$	8.83065	1.45175	0.725875	+	0.687826i	$0.241435\pi$
$38$	0	0
$39$	−1.12481	−0.180113
$40$	0	0
$41$	3.18632	0.497619	0.248810	−	0.968552i	$-0.419961\pi$
$41$	3.18632	0.497619	0.248810	+	0.968552i	$0.419961\pi$
$42$	0	0
$43$	−10.4060	−1.58691	−0.793453	−	0.608632i	$-0.791719\pi$
$43$	−10.4060	−1.58691	−0.793453	+	0.608632i	$0.791719\pi$
$44$	0	0
$45$	0.841795	0.125487
$46$	0	0
$47$	7.25697	1.05854	0.529269	−	0.848454i	$-0.322466\pi$
$47$	7.25697	1.05854	0.529269	+	0.848454i	$0.322466\pi$
$48$	0	0
$49$	−4.46798	−0.638283
$50$	0	0
$51$	13.6219	1.90745
$52$	0	0
$53$	7.47729	1.02708	0.513542	−	0.858064i	$-0.328333\pi$
$53$	7.47729	1.02708	0.513542	+	0.858064i	$0.328333\pi$
$54$	0	0
$55$	0.680036	0.0916960
$56$	0	0
$57$	2.05861	0.272669
$58$	0	0
$59$	−11.1288	−1.44884	−0.724422	−	0.689357i	$-0.757893\pi$
$59$	−11.1288	−1.44884	−0.724422	+	0.689357i	$0.757893\pi$
$60$	0	0
$61$	8.75715	1.12124	0.560619	−	0.828074i	$-0.310563\pi$
$61$	8.75715	1.12124	0.560619	+	0.828074i	$0.310563\pi$
$62$	0	0
$63$	−1.96974	−0.248164
$64$	0	0
$65$	−0.371565	−0.0460870
$66$	0	0
$67$	−11.1326	−1.36007	−0.680034	−	0.733180i	$-0.738035\pi$
$67$	−11.1326	−1.36007	−0.680034	+	0.733180i	$0.738035\pi$
$68$	0	0
$69$	13.2701	1.59753
$70$	0	0
$71$	7.88100	0.935303	0.467651	−	0.883913i	$-0.345100\pi$
$71$	7.88100	0.935303	0.467651	+	0.883913i	$0.345100\pi$
$72$	0	0
$73$	3.00155	0.351305	0.175652	−	0.984452i	$-0.443796\pi$
$73$	3.00155	0.351305	0.175652	+	0.984452i	$0.443796\pi$
$74$	0	0
$75$	−9.34104	−1.07861
$76$	0	0
$77$	−1.59123	−0.181338
$78$	0	0
$79$	−8.73426	−0.982681	−0.491341	−	0.870968i	$-0.663493\pi$
$79$	−8.73426	−0.982681	−0.491341	+	0.870968i	$0.663493\pi$
$80$	0	0
$81$	−11.1813	−1.24237
$82$	0	0
$83$	7.76611	0.852441	0.426221	−	0.904619i	$-0.359845\pi$
$83$	7.76611	0.852441	0.426221	+	0.904619i	$0.359845\pi$
$84$	0	0
$85$	4.49983	0.488075
$86$	0	0
$87$	15.0737	1.61607
$88$	0	0
$89$	15.7031	1.66453	0.832264	−	0.554380i	$-0.187045\pi$
$89$	15.7031	1.66453	0.832264	+	0.554380i	$0.187045\pi$
$90$	0	0
$91$	0.869435	0.0911416
$92$	0	0
$93$	−6.99912	−0.725775
$94$	0	0
$95$	0.680036	0.0697702
$96$	0	0
$97$	9.99329	1.01467	0.507333	−	0.861750i	$-0.330632\pi$
$97$	9.99329	1.01467	0.507333	+	0.861750i	$0.330632\pi$
$98$	0	0
$99$	1.23787	0.124411
$100$	0	0
$101$	5.13976	0.511425	0.255712	−	0.966753i	$-0.417690\pi$
$101$	5.13976	0.511425	0.255712	+	0.966753i	$0.417690\pi$
$102$	0	0
$103$	18.5189	1.82472	0.912359	−	0.409390i	$-0.134259\pi$
$103$	18.5189	1.82472	0.912359	+	0.409390i	$0.134259\pi$
$104$	0	0
$105$	−2.22761	−0.217393
$106$	0	0
$107$	20.4593	1.97787	0.988936	−	0.148344i	$-0.0473942\pi$
$107$	20.4593	1.97787	0.988936	+	0.148344i	$0.0473942\pi$
$108$	0	0
$109$	3.07296	0.294336	0.147168	−	0.989112i	$-0.452984\pi$
$109$	3.07296	0.294336	0.147168	+	0.989112i	$0.452984\pi$
$110$	0	0
$111$	18.1789	1.72546
$112$	0	0
$113$	−15.8630	−1.49227	−0.746135	−	0.665795i	$-0.768093\pi$
$113$	−15.8630	−1.49227	−0.746135	+	0.665795i	$0.768093\pi$
$114$	0	0
$115$	4.38361	0.408774
$116$	0	0
$117$	−0.676360	−0.0625295
$118$	0	0
$119$	−10.5293	−0.965215
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	6.55938	0.591440
$124$	0	0
$125$	−6.48588	−0.580114
$126$	0	0
$127$	−17.1267	−1.51975	−0.759876	−	0.650069i	$-0.774740\pi$
$127$	−17.1267	−1.51975	−0.759876	+	0.650069i	$0.774740\pi$
$128$	0	0
$129$	−21.4220	−1.88610
$130$	0	0
$131$	1.47071	0.128496	0.0642482	−	0.997934i	$-0.479535\pi$
$131$	1.47071	0.128496	0.0642482	+	0.997934i	$0.479535\pi$
$132$	0	0
$133$	−1.59123	−0.137977
$134$	0	0
$135$	−2.46686	−0.212313
$136$	0	0
$137$	−8.59949	−0.734704	−0.367352	−	0.930082i	$-0.619736\pi$
$137$	−8.59949	−0.734704	−0.367352	+	0.930082i	$0.619736\pi$
$138$	0	0
$139$	18.5155	1.57047	0.785234	−	0.619199i	$-0.212543\pi$
$139$	18.5155	1.57047	0.785234	+	0.619199i	$0.212543\pi$
$140$	0	0
$141$	14.9393	1.25811
$142$	0	0
$143$	−0.546391	−0.0456915
$144$	0	0
$145$	4.97942	0.413518
$146$	0	0
$147$	−9.19782	−0.758623
$148$	0	0
$149$	11.3272	0.927962	0.463981	−	0.885845i	$-0.346421\pi$
$149$	11.3272	0.927962	0.463981	+	0.885845i	$0.346421\pi$
$150$	0	0
$151$	−13.1169	−1.06744	−0.533718	−	0.845663i	$-0.679206\pi$
$151$	−13.1169	−1.06744	−0.533718	+	0.845663i	$0.679206\pi$
$152$	0	0
$153$	8.19103	0.662206
$154$	0	0
$155$	−2.31207	−0.185710
$156$	0	0
$157$	1.06454	0.0849597	0.0424799	−	0.999097i	$-0.486474\pi$
$157$	1.06454	0.0849597	0.0424799	+	0.999097i	$0.486474\pi$
$158$	0	0
$159$	15.3928	1.22073
$160$	0	0
$161$	−10.2573	−0.808390
$162$	0	0
$163$	11.4544	0.897178	0.448589	−	0.893738i	$-0.351927\pi$
$163$	11.4544	0.897178	0.448589	+	0.893738i	$0.351927\pi$
$164$	0	0
$165$	1.39993	0.108984
$166$	0	0
$167$	18.9897	1.46947	0.734733	−	0.678357i	$-0.237307\pi$
$167$	18.9897	1.46947	0.734733	+	0.678357i	$0.237307\pi$
$168$	0	0
$169$	−12.7015	−0.977035
$170$	0	0
$171$	1.23787	0.0946622
$172$	0	0
$173$	−9.14468	−0.695257	−0.347629	−	0.937632i	$-0.613013\pi$
$173$	−9.14468	−0.695257	−0.347629	+	0.937632i	$0.613013\pi$
$174$	0	0
$175$	7.22030	0.545803
$176$	0	0
$177$	−22.9098	−1.72201
$178$	0	0
$179$	2.19458	0.164030	0.0820152	−	0.996631i	$-0.473864\pi$
$179$	2.19458	0.164030	0.0820152	+	0.996631i	$0.473864\pi$
$180$	0	0
$181$	−10.6906	−0.794622	−0.397311	−	0.917684i	$-0.630057\pi$
$181$	−10.6906	−0.794622	−0.397311	+	0.917684i	$0.630057\pi$
$182$	0	0
$183$	18.0275	1.33263
$184$	0	0
$185$	6.00516	0.441508
$186$	0	0
$187$	6.61704	0.483886
$188$	0	0
$189$	5.77226	0.419870
$190$	0	0
$191$	−18.8704	−1.36541	−0.682707	−	0.730692i	$-0.739198\pi$
$191$	−18.8704	−1.36541	−0.682707	+	0.730692i	$0.739198\pi$
$192$	0	0
$193$	5.89542	0.424361	0.212181	−	0.977230i	$-0.431943\pi$
$193$	5.89542	0.424361	0.212181	+	0.977230i	$0.431943\pi$
$194$	0	0
$195$	−0.764908	−0.0547762
$196$	0	0
$197$	−16.1688	−1.15198	−0.575991	−	0.817456i	$-0.695384\pi$
$197$	−16.1688	−1.15198	−0.575991	+	0.817456i	$0.695384\pi$
$198$	0	0
$199$	−20.6510	−1.46391	−0.731955	−	0.681353i	$-0.761392\pi$
$199$	−20.6510	−1.46391	−0.731955	+	0.681353i	$0.761392\pi$
$200$	0	0
$201$	−22.9177	−1.61649
$202$	0	0
$203$	−11.6515	−0.817773
$204$	0	0
$205$	2.16681	0.151337
$206$	0	0
$207$	7.97948	0.554612
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	−0.700318	−0.0482119	−0.0241059	−	0.999709i	$-0.507674\pi$
$211$	−0.700318	−0.0482119	−0.0241059	+	0.999709i	$0.507674\pi$
$212$	0	0
$213$	16.2239	1.11164
$214$	0	0
$215$	−7.07648	−0.482612
$216$	0	0
$217$	5.41008	0.367260
$218$	0	0
$219$	6.17902	0.417539
$220$	0	0
$221$	−3.61549	−0.243204
$222$	0	0
$223$	−3.32157	−0.222429	−0.111214	−	0.993796i	$-0.535474\pi$
$223$	−3.32157	−0.222429	−0.111214	+	0.993796i	$0.535474\pi$
$224$	0	0
$225$	−5.61689	−0.374460
$226$	0	0
$227$	−16.0767	−1.06704	−0.533522	−	0.845786i	$-0.679132\pi$
$227$	−16.0767	−1.06704	−0.533522	+	0.845786i	$0.679132\pi$
$228$	0	0
$229$	3.00688	0.198700	0.0993502	−	0.995053i	$-0.468324\pi$
$229$	3.00688	0.198700	0.0993502	+	0.995053i	$0.468324\pi$
$230$	0	0
$231$	−3.27573	−0.215527
$232$	0	0
$233$	29.3076	1.92001	0.960004	−	0.279987i	$-0.0903303\pi$
$233$	29.3076	1.92001	0.960004	+	0.279987i	$0.0903303\pi$
$234$	0	0
$235$	4.93500	0.321924
$236$	0	0
$237$	−17.9804	−1.16795
$238$	0	0
$239$	19.8088	1.28133	0.640663	−	0.767822i	$-0.278660\pi$
$239$	19.8088	1.28133	0.640663	+	0.767822i	$0.278660\pi$
$240$	0	0
$241$	−19.6586	−1.26632	−0.633162	−	0.774019i	$-0.718243\pi$
$241$	−19.6586	−1.26632	−0.633162	+	0.774019i	$0.718243\pi$
$242$	0	0
$243$	−12.1353	−0.778478
$244$	0	0
$245$	−3.03839	−0.194115
$246$	0	0
$247$	−0.546391	−0.0347660
$248$	0	0
$249$	15.9874	1.01316
$250$	0	0
$251$	−15.8005	−0.997319	−0.498660	−	0.866798i	$-0.666174\pi$
$251$	−15.8005	−0.997319	−0.498660	+	0.866798i	$0.666174\pi$
$252$	0	0
$253$	6.44615	0.405266
$254$	0	0
$255$	9.26338	0.580096
$256$	0	0
$257$	3.15958	0.197089	0.0985446	−	0.995133i	$-0.468581\pi$
$257$	3.15958	0.197089	0.0985446	+	0.995133i	$0.468581\pi$
$258$	0	0
$259$	−14.0516	−0.873125
$260$	0	0
$261$	9.06403	0.561050
$262$	0	0
$263$	−1.02188	−0.0630116	−0.0315058	−	0.999504i	$-0.510030\pi$
$263$	−1.02188	−0.0630116	−0.0315058	+	0.999504i	$0.510030\pi$
$264$	0	0
$265$	5.08482	0.312358
$266$	0	0
$267$	32.3266	1.97835
$268$	0	0
$269$	−19.3680	−1.18089	−0.590445	−	0.807078i	$-0.701048\pi$
$269$	−19.3680	−1.18089	−0.590445	+	0.807078i	$0.701048\pi$
$270$	0	0
$271$	−16.9326	−1.02858	−0.514290	−	0.857616i	$-0.671945\pi$
$271$	−16.9326	−1.02858	−0.514290	+	0.857616i	$0.671945\pi$
$272$	0	0
$273$	1.78983	0.108325
$274$	0	0
$275$	−4.53755	−0.273625
$276$	0	0
$277$	16.1370	0.969581	0.484790	−	0.874630i	$-0.338896\pi$
$277$	16.1370	0.969581	0.484790	+	0.874630i	$0.338896\pi$
$278$	0	0
$279$	−4.20866	−0.251966
$280$	0	0
$281$	−21.9217	−1.30774	−0.653868	−	0.756609i	$-0.726855\pi$
$281$	−21.9217	−1.30774	−0.653868	+	0.756609i	$0.726855\pi$
$282$	0	0
$283$	−16.4286	−0.976578	−0.488289	−	0.872682i	$-0.662379\pi$
$283$	−16.4286	−0.976578	−0.488289	+	0.872682i	$0.662379\pi$
$284$	0	0
$285$	1.39993	0.0829246
$286$	0	0
$287$	−5.07018	−0.299283
$288$	0	0
$289$	26.7853	1.57560
$290$	0	0
$291$	20.5723	1.20597
$292$	0	0
$293$	−29.9404	−1.74914	−0.874569	−	0.484902i	$-0.838855\pi$
$293$	−29.9404	−1.74914	−0.874569	+	0.484902i	$0.838855\pi$
$294$	0	0
$295$	−7.56797	−0.440624
$296$	0	0
$297$	−3.62754	−0.210491
$298$	0	0
$299$	−3.52212	−0.203689
$300$	0	0
$301$	16.5584	0.954412
$302$	0	0
$303$	10.5807	0.607848
$304$	0	0
$305$	5.95517	0.340992
$306$	0	0
$307$	6.84067	0.390418	0.195209	−	0.980762i	$-0.437462\pi$
$307$	6.84067	0.390418	0.195209	+	0.980762i	$0.437462\pi$
$308$	0	0
$309$	38.1231	2.16875
$310$	0	0
$311$	−6.20500	−0.351853	−0.175927	−	0.984403i	$-0.556292\pi$
$311$	−6.20500	−0.351853	−0.175927	+	0.984403i	$0.556292\pi$
$312$	0	0
$313$	−29.7265	−1.68024	−0.840122	−	0.542398i	$-0.817517\pi$
$313$	−29.7265	−1.68024	−0.840122	+	0.542398i	$0.817517\pi$
$314$	0	0
$315$	−1.33949	−0.0754718
$316$	0	0
$317$	−9.41590	−0.528850	−0.264425	−	0.964406i	$-0.585182\pi$
$317$	−9.41590	−0.528850	−0.264425	+	0.964406i	$0.585182\pi$
$318$	0	0
$319$	7.32229	0.409969
$320$	0	0
$321$	42.1176	2.35078
$322$	0	0
$323$	6.61704	0.368182
$324$	0	0
$325$	2.47928	0.137526
$326$	0	0
$327$	6.32601	0.349829
$328$	0	0
$329$	−11.5475	−0.636636
$330$	0	0
$331$	−17.7803	−0.977294	−0.488647	−	0.872482i	$-0.662509\pi$
$331$	−17.7803	−0.977294	−0.488647	+	0.872482i	$0.662509\pi$
$332$	0	0
$333$	10.9312	0.599026
$334$	0	0
$335$	−7.57059	−0.413626
$336$	0	0
$337$	−7.23368	−0.394044	−0.197022	−	0.980399i	$-0.563127\pi$
$337$	−7.23368	−0.394044	−0.197022	+	0.980399i	$0.563127\pi$
$338$	0	0
$339$	−32.6558	−1.77362
$340$	0	0
$341$	−3.39993	−0.184116
$342$	0	0
$343$	18.2482	0.985311
$344$	0	0
$345$	9.02414	0.485843
$346$	0	0
$347$	7.65820	0.411114	0.205557	−	0.978645i	$-0.434099\pi$
$347$	7.65820	0.411114	0.205557	+	0.978645i	$0.434099\pi$
$348$	0	0
$349$	25.5749	1.36899	0.684497	−	0.729015i	$-0.260022\pi$
$349$	25.5749	1.36899	0.684497	+	0.729015i	$0.260022\pi$
$350$	0	0
$351$	1.98205	0.105794
$352$	0	0
$353$	−14.3449	−0.763504	−0.381752	−	0.924265i	$-0.624679\pi$
$353$	−14.3449	−0.763504	−0.381752	+	0.924265i	$0.624679\pi$
$354$	0	0
$355$	5.35936	0.284446
$356$	0	0
$357$	−21.6756	−1.14720
$358$	0	0
$359$	8.99828	0.474911	0.237455	−	0.971398i	$-0.423687\pi$
$359$	8.99828	0.474911	0.237455	+	0.971398i	$0.423687\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	2.05861	0.108049
$364$	0	0
$365$	2.04116	0.106839
$366$	0	0
$367$	−5.83681	−0.304679	−0.152339	−	0.988328i	$-0.548681\pi$
$367$	−5.83681	−0.304679	−0.152339	+	0.988328i	$0.548681\pi$
$368$	0	0
$369$	3.94425	0.205329
$370$	0	0
$371$	−11.8981	−0.617719
$372$	0	0
$373$	−2.76634	−0.143235	−0.0716177	−	0.997432i	$-0.522816\pi$
$373$	−2.76634	−0.143235	−0.0716177	+	0.997432i	$0.522816\pi$
$374$	0	0
$375$	−13.3519	−0.689488
$376$	0	0
$377$	−4.00083	−0.206053
$378$	0	0
$379$	−37.9993	−1.95189	−0.975946	−	0.218011i	$-0.930043\pi$
$379$	−37.9993	−1.95189	−0.975946	+	0.218011i	$0.930043\pi$
$380$	0	0
$381$	−35.2572	−1.80628
$382$	0	0
$383$	−0.183768	−0.00939013	−0.00469507	−	0.999989i	$-0.501494\pi$
$383$	−0.183768	−0.00939013	−0.00469507	+	0.999989i	$0.501494\pi$
$384$	0	0
$385$	−1.08210	−0.0551487
$386$	0	0
$387$	−12.8813	−0.654794
$388$	0	0
$389$	−1.95683	−0.0992154	−0.0496077	−	0.998769i	$-0.515797\pi$
$389$	−1.95683	−0.0992154	−0.0496077	+	0.998769i	$0.515797\pi$
$390$	0	0
$391$	42.6544	2.15713
$392$	0	0
$393$	3.02761	0.152723
$394$	0	0
$395$	−5.93961	−0.298854
$396$	0	0
$397$	−30.3125	−1.52134	−0.760669	−	0.649139i	$-0.775129\pi$
$397$	−30.3125	−1.52134	−0.760669	+	0.649139i	$0.775129\pi$
$398$	0	0
$399$	−3.27573	−0.163991
$400$	0	0
$401$	10.0474	0.501741	0.250871	−	0.968021i	$-0.419283\pi$
$401$	10.0474	0.501741	0.250871	+	0.968021i	$0.419283\pi$
$402$	0	0
$403$	1.85769	0.0925381
$404$	0	0
$405$	−7.60368	−0.377830
$406$	0	0
$407$	8.83065	0.437719
$408$	0	0
$409$	18.8432	0.931738	0.465869	−	0.884854i	$-0.345742\pi$
$409$	18.8432	0.931738	0.465869	+	0.884854i	$0.345742\pi$
$410$	0	0
$411$	−17.7030	−0.873224
$412$	0	0
$413$	17.7085	0.871378
$414$	0	0
$415$	5.28123	0.259246
$416$	0	0
$417$	38.1163	1.86656
$418$	0	0
$419$	−9.05528	−0.442379	−0.221190	−	0.975231i	$-0.570994\pi$
$419$	−9.05528	−0.442379	−0.221190	+	0.975231i	$0.570994\pi$
$420$	0	0
$421$	15.4995	0.755399	0.377699	−	0.925928i	$-0.376715\pi$
$421$	15.4995	0.755399	0.377699	+	0.925928i	$0.376715\pi$
$422$	0	0
$423$	8.98318	0.436777
$424$	0	0
$425$	−30.0252	−1.45644
$426$	0	0
$427$	−13.9347	−0.674345
$428$	0	0
$429$	−1.12481	−0.0543061
$430$	0	0
$431$	0.871260	0.0419671	0.0209836	−	0.999780i	$-0.493320\pi$
$431$	0.871260	0.0419671	0.0209836	+	0.999780i	$0.493320\pi$
$432$	0	0
$433$	−17.0358	−0.818688	−0.409344	−	0.912380i	$-0.634242\pi$
$433$	−17.0358	−0.818688	−0.409344	+	0.912380i	$0.634242\pi$
$434$	0	0
$435$	10.2507	0.491482
$436$	0	0
$437$	6.44615	0.308361
$438$	0	0
$439$	30.5207	1.45668	0.728338	−	0.685218i	$-0.240293\pi$
$439$	30.5207	1.45668	0.728338	+	0.685218i	$0.240293\pi$
$440$	0	0
$441$	−5.53077	−0.263370
$442$	0	0
$443$	−29.2482	−1.38962	−0.694811	−	0.719192i	$-0.744512\pi$
$443$	−29.2482	−1.38962	−0.694811	+	0.719192i	$0.744512\pi$
$444$	0	0
$445$	10.6787	0.506218
$446$	0	0
$447$	23.3183	1.10292
$448$	0	0
$449$	2.08177	0.0982448	0.0491224	−	0.998793i	$-0.484358\pi$
$449$	2.08177	0.0982448	0.0491224	+	0.998793i	$0.484358\pi$
$450$	0	0
$451$	3.18632	0.150038
$452$	0	0
$453$	−27.0025	−1.26869
$454$	0	0
$455$	0.591247	0.0277181
$456$	0	0
$457$	10.6300	0.497248	0.248624	−	0.968600i	$-0.420022\pi$
$457$	10.6300	0.497248	0.248624	+	0.968600i	$0.420022\pi$
$458$	0	0
$459$	−24.0036	−1.12039
$460$	0	0
$461$	−24.3691	−1.13498	−0.567491	−	0.823380i	$-0.692086\pi$
$461$	−24.3691	−1.13498	−0.567491	+	0.823380i	$0.692086\pi$
$462$	0	0
$463$	−28.2627	−1.31348	−0.656740	−	0.754117i	$-0.728065\pi$
$463$	−28.2627	−1.31348	−0.656740	+	0.754117i	$0.728065\pi$
$464$	0	0
$465$	−4.75965	−0.220724
$466$	0	0
$467$	30.9960	1.43433	0.717163	−	0.696905i	$-0.245440\pi$
$467$	30.9960	1.43433	0.717163	+	0.696905i	$0.245440\pi$
$468$	0	0
$469$	17.7146	0.817985
$470$	0	0
$471$	2.19148	0.100978
$472$	0	0
$473$	−10.4060	−0.478470
$474$	0	0
$475$	−4.53755	−0.208197
$476$	0	0
$477$	9.25590	0.423799
$478$	0	0
$479$	8.49550	0.388169	0.194085	−	0.980985i	$-0.437826\pi$
$479$	8.49550	0.388169	0.194085	+	0.980985i	$0.437826\pi$
$480$	0	0
$481$	−4.82499	−0.220001
$482$	0	0
$483$	−21.1158	−0.960802
$484$	0	0
$485$	6.79580	0.308581
$486$	0	0
$487$	11.4239	0.517666	0.258833	−	0.965922i	$-0.416662\pi$
$487$	11.4239	0.517666	0.258833	+	0.965922i	$0.416662\pi$
$488$	0	0
$489$	23.5801	1.06633
$490$	0	0
$491$	27.2553	1.23001	0.615007	−	0.788522i	$-0.289153\pi$
$491$	27.2553	1.23001	0.615007	+	0.788522i	$0.289153\pi$
$492$	0	0
$493$	48.4519	2.18216
$494$	0	0
$495$	0.841795	0.0378359
$496$	0	0
$497$	−12.5405	−0.562519
$498$	0	0
$499$	−38.6475	−1.73010	−0.865051	−	0.501684i	$-0.832714\pi$
$499$	−38.6475	−1.73010	−0.865051	+	0.501684i	$0.832714\pi$
$500$	0	0
$501$	39.0923	1.74652
$502$	0	0
$503$	43.0702	1.92040	0.960202	−	0.279305i	$-0.0901041\pi$
$503$	43.0702	1.92040	0.960202	+	0.279305i	$0.0901041\pi$
$504$	0	0
$505$	3.49522	0.155535
$506$	0	0
$507$	−26.1473	−1.16124
$508$	0	0
$509$	2.52410	0.111879	0.0559393	−	0.998434i	$-0.482185\pi$
$509$	2.52410	0.111879	0.0559393	+	0.998434i	$0.482185\pi$
$510$	0	0
$511$	−4.77617	−0.211285
$512$	0	0
$513$	−3.62754	−0.160160
$514$	0	0
$515$	12.5935	0.554936
$516$	0	0
$517$	7.25697	0.319161
$518$	0	0
$519$	−18.8253	−0.826340
$520$	0	0
$521$	−0.528257	−0.0231434	−0.0115717	−	0.999933i	$-0.503683\pi$
$521$	−0.528257	−0.0231434	−0.0115717	+	0.999933i	$0.503683\pi$
$522$	0	0
$523$	27.3788	1.19719	0.598597	−	0.801051i	$-0.295725\pi$
$523$	27.3788	1.19719	0.598597	+	0.801051i	$0.295725\pi$
$524$	0	0
$525$	14.8638	0.648708
$526$	0	0
$527$	−22.4975	−0.980005
$528$	0	0
$529$	18.5528	0.806643
$530$	0	0
$531$	−13.7760	−0.597826
$532$	0	0
$533$	−1.74098	−0.0754100
$534$	0	0
$535$	13.9130	0.601513
$536$	0	0
$537$	4.51778	0.194956
$538$	0	0
$539$	−4.46798	−0.192449
$540$	0	0
$541$	−33.1725	−1.42620	−0.713099	−	0.701063i	$-0.752709\pi$
$541$	−33.1725	−1.42620	−0.713099	+	0.701063i	$0.752709\pi$
$542$	0	0
$543$	−22.0077	−0.944439
$544$	0	0
$545$	2.08972	0.0895138
$546$	0	0
$547$	−17.2954	−0.739499	−0.369749	−	0.929132i	$-0.620556\pi$
$547$	−17.2954	−0.739499	−0.369749	+	0.929132i	$0.620556\pi$
$548$	0	0
$549$	10.8402	0.462648
$550$	0	0
$551$	7.32229	0.311940
$552$	0	0
$553$	13.8982	0.591013
$554$	0	0
$555$	12.3623	0.524749
$556$	0	0
$557$	9.53222	0.403893	0.201947	−	0.979397i	$-0.435273\pi$
$557$	9.53222	0.403893	0.201947	+	0.979397i	$0.435273\pi$
$558$	0	0
$559$	5.68577	0.240482
$560$	0	0
$561$	13.6219	0.575117
$562$	0	0
$563$	6.88207	0.290045	0.145022	−	0.989428i	$-0.453675\pi$
$563$	6.88207	0.290045	0.145022	+	0.989428i	$0.453675\pi$
$564$	0	0
$565$	−10.7874	−0.453831
$566$	0	0
$567$	17.7920	0.747195
$568$	0	0
$569$	−16.0041	−0.670927	−0.335463	−	0.942053i	$-0.608893\pi$
$569$	−16.0041	−0.670927	−0.335463	+	0.942053i	$0.608893\pi$
$570$	0	0
$571$	−23.2586	−0.973342	−0.486671	−	0.873585i	$-0.661789\pi$
$571$	−23.2586	−0.973342	−0.486671	+	0.873585i	$0.661789\pi$
$572$	0	0
$573$	−38.8468	−1.62285
$574$	0	0
$575$	−29.2497	−1.21980
$576$	0	0
$577$	9.90262	0.412251	0.206126	−	0.978526i	$-0.433914\pi$
$577$	9.90262	0.412251	0.206126	+	0.978526i	$0.433914\pi$
$578$	0	0
$579$	12.1364	0.504370
$580$	0	0
$581$	−12.3577	−0.512683
$582$	0	0
$583$	7.47729	0.309678
$584$	0	0
$585$	−0.459949	−0.0190166
$586$	0	0
$587$	22.0985	0.912102	0.456051	−	0.889954i	$-0.349263\pi$
$587$	22.0985	0.912102	0.456051	+	0.889954i	$0.349263\pi$
$588$	0	0
$589$	−3.39993	−0.140092
$590$	0	0
$591$	−33.2853	−1.36917
$592$	0	0
$593$	15.4666	0.635138	0.317569	−	0.948235i	$-0.397133\pi$
$593$	15.4666	0.635138	0.317569	+	0.948235i	$0.397133\pi$
$594$	0	0
$595$	−7.16027	−0.293543
$596$	0	0
$597$	−42.5123	−1.73991
$598$	0	0
$599$	3.25975	0.133190	0.0665950	−	0.997780i	$-0.478786\pi$
$599$	3.25975	0.133190	0.0665950	+	0.997780i	$0.478786\pi$
$600$	0	0
$601$	8.34590	0.340436	0.170218	−	0.985406i	$-0.445553\pi$
$601$	8.34590	0.340436	0.170218	+	0.985406i	$0.445553\pi$
$602$	0	0
$603$	−13.7807	−0.561195
$604$	0	0
$605$	0.680036	0.0276474
$606$	0	0
$607$	6.57007	0.266671	0.133335	−	0.991071i	$-0.457431\pi$
$607$	6.57007	0.266671	0.133335	+	0.991071i	$0.457431\pi$
$608$	0	0
$609$	−23.9858	−0.971954
$610$	0	0
$611$	−3.96514	−0.160413
$612$	0	0
$613$	−15.5765	−0.629128	−0.314564	−	0.949236i	$-0.601858\pi$
$613$	−15.5765	−0.629128	−0.314564	+	0.949236i	$0.601858\pi$
$614$	0	0
$615$	4.46062	0.179869
$616$	0	0
$617$	−18.1508	−0.730725	−0.365363	−	0.930865i	$-0.619055\pi$
$617$	−18.1508	−0.730725	−0.365363	+	0.930865i	$0.619055\pi$
$618$	0	0
$619$	−6.94456	−0.279125	−0.139563	−	0.990213i	$-0.544570\pi$
$619$	−6.94456	−0.279125	−0.139563	+	0.990213i	$0.544570\pi$
$620$	0	0
$621$	−23.3836	−0.938353
$622$	0	0
$623$	−24.9873	−1.00110
$624$	0	0
$625$	18.2771	0.731085
$626$	0	0
$627$	2.05861	0.0822129
$628$	0	0
$629$	58.4328	2.32987
$630$	0	0
$631$	0.103144	0.00410608	0.00205304	−	0.999998i	$-0.499346\pi$
$631$	0.103144	0.00410608	0.00205304	+	0.999998i	$0.499346\pi$
$632$	0	0
$633$	−1.44168	−0.0573016
$634$	0	0
$635$	−11.6468	−0.462189
$636$	0	0
$637$	2.44126	0.0967264
$638$	0	0
$639$	9.75565	0.385927
$640$	0	0
$641$	30.4033	1.20086	0.600429	−	0.799678i	$-0.294996\pi$
$641$	30.4033	1.20086	0.600429	+	0.799678i	$0.294996\pi$
$642$	0	0
$643$	−34.7980	−1.37230	−0.686150	−	0.727460i	$-0.740701\pi$
$643$	−34.7980	−1.37230	−0.686150	+	0.727460i	$0.740701\pi$
$644$	0	0
$645$	−14.5677	−0.573603
$646$	0	0
$647$	25.6437	1.00816	0.504078	−	0.863658i	$-0.331832\pi$
$647$	25.6437	1.00816	0.504078	+	0.863658i	$0.331832\pi$
$648$	0	0
$649$	−11.1288	−0.436843
$650$	0	0
$651$	11.1372	0.436502
$652$	0	0
$653$	42.5402	1.66473	0.832363	−	0.554231i	$-0.186987\pi$
$653$	42.5402	1.66473	0.832363	+	0.554231i	$0.186987\pi$
$654$	0	0
$655$	1.00013	0.0390785
$656$	0	0
$657$	3.71553	0.144956
$658$	0	0
$659$	−3.33064	−0.129743	−0.0648717	−	0.997894i	$-0.520664\pi$
$659$	−3.33064	−0.129743	−0.0648717	+	0.997894i	$0.520664\pi$
$660$	0	0
$661$	8.14466	0.316791	0.158395	−	0.987376i	$-0.449368\pi$
$661$	8.14466	0.316791	0.158395	+	0.987376i	$0.449368\pi$
$662$	0	0
$663$	−7.44289	−0.289058
$664$	0	0
$665$	−1.08210	−0.0419619
$666$	0	0
$667$	47.2005	1.82761
$668$	0	0
$669$	−6.83781	−0.264365
$670$	0	0
$671$	8.75715	0.338066
$672$	0	0
$673$	3.62445	0.139712	0.0698561	−	0.997557i	$-0.477746\pi$
$673$	3.62445	0.139712	0.0698561	+	0.997557i	$0.477746\pi$
$674$	0	0
$675$	16.4601	0.633551
$676$	0	0
$677$	−4.21027	−0.161814	−0.0809070	−	0.996722i	$-0.525782\pi$
$677$	−4.21027	−0.161814	−0.0809070	+	0.996722i	$0.525782\pi$
$678$	0	0
$679$	−15.9017	−0.610249
$680$	0	0
$681$	−33.0955	−1.26822
$682$	0	0
$683$	−39.3883	−1.50715	−0.753576	−	0.657361i	$-0.771673\pi$
$683$	−39.3883	−1.50715	−0.753576	+	0.657361i	$0.771673\pi$
$684$	0	0
$685$	−5.84796	−0.223439
$686$	0	0
$687$	6.18999	0.236163
$688$	0	0
$689$	−4.08552	−0.155646
$690$	0	0
$691$	37.5947	1.43017	0.715085	−	0.699038i	$-0.246388\pi$
$691$	37.5947	1.43017	0.715085	+	0.699038i	$0.246388\pi$
$692$	0	0
$693$	−1.96974	−0.0748241
$694$	0	0
$695$	12.5912	0.477613
$696$	0	0
$697$	21.0840	0.798614
$698$	0	0
$699$	60.3330	2.28200
$700$	0	0
$701$	−18.7978	−0.709981	−0.354991	−	0.934870i	$-0.615516\pi$
$701$	−18.7978	−0.709981	−0.354991	+	0.934870i	$0.615516\pi$
$702$	0	0
$703$	8.83065	0.333054
$704$	0	0
$705$	10.1592	0.382619
$706$	0	0
$707$	−8.17855	−0.307586
$708$	0	0
$709$	10.4624	0.392926	0.196463	−	0.980511i	$-0.437055\pi$
$709$	10.4624	0.392926	0.196463	+	0.980511i	$0.437055\pi$
$710$	0	0
$711$	−10.8119	−0.405477
$712$	0	0
$713$	−21.9164	−0.820777
$714$	0	0
$715$	−0.371565	−0.0138958
$716$	0	0
$717$	40.7786	1.52290
$718$	0	0
$719$	−14.2534	−0.531561	−0.265780	−	0.964034i	$-0.585630\pi$
$719$	−14.2534	−0.531561	−0.265780	+	0.964034i	$0.585630\pi$
$720$	0	0
$721$	−29.4678	−1.09744
$722$	0	0
$723$	−40.4694	−1.50507
$724$	0	0
$725$	−33.2253	−1.23396
$726$	0	0
$727$	25.0455	0.928887	0.464443	−	0.885603i	$-0.346254\pi$
$727$	25.0455	0.928887	0.464443	+	0.885603i	$0.346254\pi$
$728$	0	0
$729$	8.56207	0.317114
$730$	0	0
$731$	−68.8572	−2.54678
$732$	0	0
$733$	−39.5761	−1.46178	−0.730889	−	0.682496i	$-0.760894\pi$
$733$	−39.5761	−1.46178	−0.730889	+	0.682496i	$0.760894\pi$
$734$	0	0
$735$	−6.25485	−0.230713
$736$	0	0
$737$	−11.1326	−0.410076
$738$	0	0
$739$	6.93030	0.254935	0.127468	−	0.991843i	$-0.459315\pi$
$739$	6.93030	0.254935	0.127468	+	0.991843i	$0.459315\pi$
$740$	0	0
$741$	−1.12481	−0.0413207
$742$	0	0
$743$	−27.3372	−1.00290	−0.501452	−	0.865185i	$-0.667201\pi$
$743$	−27.3372	−1.00290	−0.501452	+	0.865185i	$0.667201\pi$
$744$	0	0
$745$	7.70291	0.282213
$746$	0	0
$747$	9.61343	0.351737
$748$	0	0
$749$	−32.5554	−1.18955
$750$	0	0
$751$	0.430584	0.0157122	0.00785611	−	0.999969i	$-0.497499\pi$
$751$	0.430584	0.0157122	0.00785611	+	0.999969i	$0.497499\pi$
$752$	0	0
$753$	−32.5271	−1.18535
$754$	0	0
$755$	−8.91994	−0.324630
$756$	0	0
$757$	−44.5951	−1.62083	−0.810417	−	0.585853i	$-0.800760\pi$
$757$	−44.5951	−1.62083	−0.810417	+	0.585853i	$0.800760\pi$
$758$	0	0
$759$	13.2701	0.481674
$760$	0	0
$761$	−48.8363	−1.77031	−0.885157	−	0.465292i	$-0.845949\pi$
$761$	−48.8363	−1.77031	−0.885157	+	0.465292i	$0.845949\pi$
$762$	0	0
$763$	−4.88979	−0.177022
$764$	0	0
$765$	5.57020	0.201391
$766$	0	0
$767$	6.08067	0.219560
$768$	0	0
$769$	−52.3956	−1.88943	−0.944716	−	0.327889i	$-0.893663\pi$
$769$	−52.3956	−1.88943	−0.944716	+	0.327889i	$0.893663\pi$
$770$	0	0
$771$	6.50434	0.234248
$772$	0	0
$773$	−25.3076	−0.910253	−0.455126	−	0.890427i	$-0.650406\pi$
$773$	−25.3076	−0.910253	−0.455126	+	0.890427i	$0.650406\pi$
$774$	0	0
$775$	15.4273	0.554167
$776$	0	0
$777$	−28.9268	−1.03774
$778$	0	0
$779$	3.18632	0.114162
$780$	0	0
$781$	7.88100	0.282004
$782$	0	0
$783$	−26.5619	−0.949244
$784$	0	0
$785$	0.723927	0.0258381
$786$	0	0
$787$	0.354709	0.0126440	0.00632201	−	0.999980i	$-0.497988\pi$
$787$	0.354709	0.0126440	0.00632201	+	0.999980i	$0.497988\pi$
$788$	0	0
$789$	−2.10364	−0.0748917
$790$	0	0
$791$	25.2418	0.897495
$792$	0	0
$793$	−4.78483	−0.169914
$794$	0	0
$795$	10.4677	0.371250
$796$	0	0
$797$	−51.0269	−1.80747	−0.903733	−	0.428097i	$-0.859184\pi$
$797$	−51.0269	−1.80747	−0.903733	+	0.428097i	$0.859184\pi$
$798$	0	0
$799$	48.0197	1.69882
$800$	0	0
$801$	19.4384	0.686822
$802$	0	0
$803$	3.00155	0.105922
$804$	0	0
$805$	−6.97534	−0.245849
$806$	0	0
$807$	−39.8712	−1.40353
$808$	0	0
$809$	23.3721	0.821718	0.410859	−	0.911699i	$-0.365229\pi$
$809$	23.3721	0.821718	0.410859	+	0.911699i	$0.365229\pi$
$810$	0	0
$811$	−15.3578	−0.539286	−0.269643	−	0.962960i	$-0.586906\pi$
$811$	−15.3578	−0.539286	−0.269643	+	0.962960i	$0.586906\pi$
$812$	0	0
$813$	−34.8575	−1.22251
$814$	0	0
$815$	7.78941	0.272851
$816$	0	0
$817$	−10.4060	−0.364061
$818$	0	0
$819$	1.07625	0.0376071
$820$	0	0
$821$	14.4538	0.504441	0.252221	−	0.967670i	$-0.418839\pi$
$821$	14.4538	0.504441	0.252221	+	0.967670i	$0.418839\pi$
$822$	0	0
$823$	−14.2522	−0.496801	−0.248400	−	0.968657i	$-0.579905\pi$
$823$	−14.2522	−0.496801	−0.248400	+	0.968657i	$0.579905\pi$
$824$	0	0
$825$	−9.34104	−0.325213
$826$	0	0
$827$	−6.68323	−0.232399	−0.116199	−	0.993226i	$-0.537071\pi$
$827$	−6.68323	−0.232399	−0.116199	+	0.993226i	$0.537071\pi$
$828$	0	0
$829$	25.9015	0.899596	0.449798	−	0.893130i	$-0.351496\pi$
$829$	25.9015	0.899596	0.449798	+	0.893130i	$0.351496\pi$
$830$	0	0
$831$	33.2199	1.15238
$832$	0	0
$833$	−29.5648	−1.02436
$834$	0	0
$835$	12.9137	0.446896
$836$	0	0
$837$	12.3334	0.426304
$838$	0	0
$839$	−11.9779	−0.413524	−0.206762	−	0.978391i	$-0.566293\pi$
$839$	−11.9779	−0.413524	−0.206762	+	0.978391i	$0.566293\pi$
$840$	0	0
$841$	24.6159	0.848825
$842$	0	0
$843$	−45.1281	−1.55429
$844$	0	0
$845$	−8.63745	−0.297137
$846$	0	0
$847$	−1.59123	−0.0546754
$848$	0	0
$849$	−33.8200	−1.16070
$850$	0	0
$851$	56.9237	1.95132
$852$	0	0
$853$	−3.22318	−0.110359	−0.0551797	−	0.998476i	$-0.517573\pi$
$853$	−3.22318	−0.110359	−0.0551797	+	0.998476i	$0.517573\pi$
$854$	0	0
$855$	0.841795	0.0287888
$856$	0	0
$857$	43.5297	1.48695	0.743473	−	0.668766i	$-0.233177\pi$
$857$	43.5297	1.48695	0.743473	+	0.668766i	$0.233177\pi$
$858$	0	0
$859$	−18.1218	−0.618309	−0.309154	−	0.951012i	$-0.600046\pi$
$859$	−18.1218	−0.618309	−0.309154	+	0.951012i	$0.600046\pi$
$860$	0	0
$861$	−10.4375	−0.355709
$862$	0	0
$863$	−18.8526	−0.641749	−0.320875	−	0.947122i	$-0.603977\pi$
$863$	−18.8526	−0.641749	−0.320875	+	0.947122i	$0.603977\pi$
$864$	0	0
$865$	−6.21871	−0.211443
$866$	0	0
$867$	55.1404	1.87267
$868$	0	0
$869$	−8.73426	−0.296290
$870$	0	0
$871$	6.08277	0.206107
$872$	0	0
$873$	12.3704	0.418674
$874$	0	0
$875$	10.3205	0.348898
$876$	0	0
$877$	−48.4436	−1.63582	−0.817911	−	0.575344i	$-0.804868\pi$
$877$	−48.4436	−1.63582	−0.817911	+	0.575344i	$0.804868\pi$
$878$	0	0
$879$	−61.6356	−2.07892
$880$	0	0
$881$	38.2825	1.28977	0.644885	−	0.764280i	$-0.276905\pi$
$881$	38.2825	1.28977	0.644885	+	0.764280i	$0.276905\pi$
$882$	0	0
$883$	−5.98538	−0.201424	−0.100712	−	0.994916i	$-0.532112\pi$
$883$	−5.98538	−0.201424	−0.100712	+	0.994916i	$0.532112\pi$
$884$	0	0
$885$	−15.5795	−0.523699
$886$	0	0
$887$	13.5203	0.453966	0.226983	−	0.973899i	$-0.427114\pi$
$887$	13.5203	0.453966	0.226983	+	0.973899i	$0.427114\pi$
$888$	0	0
$889$	27.2526	0.914023
$890$	0	0
$891$	−11.1813	−0.374587
$892$	0	0
$893$	7.25697	0.242845
$894$	0	0
$895$	1.49239	0.0498852
$896$	0	0
$897$	−7.25066	−0.242092
$898$	0	0
$899$	−24.8953	−0.830303
$900$	0	0
$901$	49.4775	1.64834
$902$	0	0
$903$	34.0873	1.13436
$904$	0	0
$905$	−7.26996	−0.241662
$906$	0	0
$907$	0.321956	0.0106904	0.00534519	−	0.999986i	$-0.498299\pi$
$907$	0.321956	0.0106904	0.00534519	+	0.999986i	$0.498299\pi$
$908$	0	0
$909$	6.36234	0.211026
$910$	0	0
$911$	−2.89500	−0.0959155	−0.0479578	−	0.998849i	$-0.515271\pi$
$911$	−2.89500	−0.0959155	−0.0479578	+	0.998849i	$0.515271\pi$
$912$	0	0
$913$	7.76611	0.257021
$914$	0	0
$915$	12.2594	0.405282
$916$	0	0
$917$	−2.34024	−0.0772815
$918$	0	0
$919$	22.5001	0.742211	0.371105	−	0.928591i	$-0.378979\pi$
$919$	22.5001	0.742211	0.371105	+	0.928591i	$0.378979\pi$
$920$	0	0
$921$	14.0823	0.464026
$922$	0	0
$923$	−4.30611	−0.141737
$924$	0	0
$925$	−40.0695	−1.31748
$926$	0	0
$927$	22.9239	0.752921
$928$	0	0
$929$	−20.8990	−0.685674	−0.342837	−	0.939395i	$-0.611388\pi$
$929$	−20.8990	−0.685674	−0.342837	+	0.939395i	$0.611388\pi$
$930$	0	0
$931$	−4.46798	−0.146432
$932$	0	0
$933$	−12.7737	−0.418191
$934$	0	0
$935$	4.49983	0.147160
$936$	0	0
$937$	34.5162	1.12760	0.563798	−	0.825913i	$-0.309340\pi$
$937$	34.5162	1.12760	0.563798	+	0.825913i	$0.309340\pi$
$938$	0	0
$939$	−61.1953	−1.99703
$940$	0	0
$941$	−25.8848	−0.843821	−0.421910	−	0.906638i	$-0.638640\pi$
$941$	−25.8848	−0.843821	−0.421910	+	0.906638i	$0.638640\pi$
$942$	0	0
$943$	20.5395	0.668857
$944$	0	0
$945$	3.92534	0.127691
$946$	0	0
$947$	11.2617	0.365955	0.182978	−	0.983117i	$-0.441426\pi$
$947$	11.2617	0.365955	0.182978	+	0.983117i	$0.441426\pi$
$948$	0	0
$949$	−1.64002	−0.0532373
$950$	0	0
$951$	−19.3837	−0.628558
$952$	0	0
$953$	34.3302	1.11206	0.556032	−	0.831161i	$-0.312323\pi$
$953$	34.3302	1.11206	0.556032	+	0.831161i	$0.312323\pi$
$954$	0	0
$955$	−12.8326	−0.415252
$956$	0	0
$957$	15.0737	0.487264
$958$	0	0
$959$	13.6838	0.441873
$960$	0	0
$961$	−19.4405	−0.627113
$962$	0	0
$963$	25.3259	0.816115
$964$	0	0
$965$	4.00910	0.129057
$966$	0	0
$967$	−24.4450	−0.786100	−0.393050	−	0.919517i	$-0.628580\pi$
$967$	−24.4450	−0.786100	−0.393050	+	0.919517i	$0.628580\pi$
$968$	0	0
$969$	13.6219	0.437599
$970$	0	0
$971$	−9.32129	−0.299134	−0.149567	−	0.988752i	$-0.547788\pi$
$971$	−9.32129	−0.299134	−0.149567	+	0.988752i	$0.547788\pi$
$972$	0	0
$973$	−29.4625	−0.944526
$974$	0	0
$975$	5.10386	0.163454
$976$	0	0
$977$	20.0702	0.642102	0.321051	−	0.947062i	$-0.395964\pi$
$977$	20.0702	0.642102	0.321051	+	0.947062i	$0.395964\pi$
$978$	0	0
$979$	15.7031	0.501874
$980$	0	0
$981$	3.80392	0.121450
$982$	0	0
$983$	−0.585569	−0.0186767	−0.00933837	−	0.999956i	$-0.502973\pi$
$983$	−0.585569	−0.0186767	−0.00933837	+	0.999956i	$0.502973\pi$
$984$	0	0
$985$	−10.9954	−0.350342
$986$	0	0
$987$	−23.7718	−0.756666
$988$	0	0
$989$	−67.0788	−2.13298
$990$	0	0
$991$	52.8562	1.67903	0.839516	−	0.543335i	$-0.182839\pi$
$991$	52.8562	1.67903	0.839516	+	0.543335i	$0.182839\pi$
$992$	0	0
$993$	−36.6027	−1.16155
$994$	0	0
$995$	−14.0434	−0.445206
$996$	0	0
$997$	−35.9223	−1.13767	−0.568836	−	0.822451i	$-0.692606\pi$
$997$	−35.9223	−1.13767	−0.568836	+	0.822451i	$0.692606\pi$
$998$	0	0
$999$	−32.0335	−1.01350

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.w.1.5		6
4.3	odd	2		836.2.a.e.1.2	✓	6
12.11	even	2		7524.2.a.q.1.4		6
44.43	even	2		9196.2.a.n.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
836.2.a.e.1.2	✓	6	4.3	odd	2
3344.2.a.w.1.5		6	1.1	even	1	trivial
7524.2.a.q.1.4		6	12.11	even	2
9196.2.a.n.1.2		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+2.05861 q^{3} +0.680036 q^{5} -1.59123 q^{7} +1.23787 q^{9} +O(q^{10})\)	\(q+2.05861 q^{3} +0.680036 q^{5} -1.59123 q^{7} +1.23787 q^{9} +1.00000 q^{11} -0.546391 q^{13} +1.39993 q^{15} +6.61704 q^{17} +1.00000 q^{19} -3.27573 q^{21} +6.44615 q^{23} -4.53755 q^{25} -3.62754 q^{27} +7.32229 q^{29} -3.39993 q^{31} +2.05861 q^{33} -1.08210 q^{35} +8.83065 q^{37} -1.12481 q^{39} +3.18632 q^{41} -10.4060 q^{43} +0.841795 q^{45} +7.25697 q^{47} -4.46798 q^{49} +13.6219 q^{51} +7.47729 q^{53} +0.680036 q^{55} +2.05861 q^{57} -11.1288 q^{59} +8.75715 q^{61} -1.96974 q^{63} -0.371565 q^{65} -11.1326 q^{67} +13.2701 q^{69} +7.88100 q^{71} +3.00155 q^{73} -9.34104 q^{75} -1.59123 q^{77} -8.73426 q^{79} -11.1813 q^{81} +7.76611 q^{83} +4.49983 q^{85} +15.0737 q^{87} +15.7031 q^{89} +0.869435 q^{91} -6.99912 q^{93} +0.680036 q^{95} +9.99329 q^{97} +1.23787 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{3} + 5 q^{5} + 2 q^{7} + 17 q^{9}+O(q^{10}) \) 6 * q - q^3 + 5 * q^5 + 2 * q^7 + 17 * q^9	\( 6 q - q^{3} + 5 q^{5} + 2 q^{7} + 17 q^{9} + 6 q^{11} + 8 q^{13} - 9 q^{15} - 2 q^{17} + 6 q^{19} + 18 q^{21} - 5 q^{23} + 13 q^{25} - 7 q^{27} + 10 q^{29} - 3 q^{31} - q^{33} + 4 q^{35} + 7 q^{37} + 10 q^{39} + 6 q^{41} - 16 q^{43} + 42 q^{45} + 12 q^{49} - 8 q^{51} + 20 q^{53} + 5 q^{55} - q^{57} - 15 q^{59} + 24 q^{61} + 20 q^{63} - 28 q^{65} - 25 q^{67} - 33 q^{69} + 9 q^{71} - 26 q^{73} - 28 q^{75} + 2 q^{77} + 16 q^{79} + 58 q^{81} + 2 q^{83} + 12 q^{85} + 36 q^{87} + 7 q^{89} - 8 q^{91} - 55 q^{93} + 5 q^{95} + 37 q^{97} + 17 q^{99}+O(q^{100}) \) 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

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