LMFDB - Embedded newform 3344.2.a.x.1.6

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:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 12x^{4} + 28x^{3} + 16x^{2} - 60x + 27 $ x^6 - 2x^5 - 12x^4 + 28x^3 + 16x^2 - 60*x + 27
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			$2.81471$ 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)

$f(q)$	$=$	$q+2.81471 q^{3} +4.18951 q^{5} -4.08740 q^{7} +4.92257 q^{9} +O(q^{10})$	$q+2.81471 q^{3} +4.18951 q^{5} -4.08740 q^{7} +4.92257 q^{9} -1.00000 q^{11} +2.68016 q^{13} +11.7922 q^{15} -2.52730 q^{17} -1.00000 q^{19} -11.5048 q^{21} +4.36247 q^{23} +12.5520 q^{25} +5.41146 q^{27} -4.29737 q^{29} +8.43192 q^{31} -2.81471 q^{33} -17.1242 q^{35} +7.26036 q^{37} +7.54385 q^{39} +7.41167 q^{41} +6.55198 q^{43} +20.6231 q^{45} -7.36031 q^{47} +9.70684 q^{49} -7.11361 q^{51} -9.47454 q^{53} -4.18951 q^{55} -2.81471 q^{57} +4.38056 q^{59} +11.7617 q^{61} -20.1205 q^{63} +11.2285 q^{65} -10.6067 q^{67} +12.2791 q^{69} -2.44412 q^{71} -15.8536 q^{73} +35.3301 q^{75} +4.08740 q^{77} +7.62283 q^{79} +0.463960 q^{81} +8.83916 q^{83} -10.5882 q^{85} -12.0958 q^{87} +6.54385 q^{89} -10.9549 q^{91} +23.7334 q^{93} -4.18951 q^{95} -8.16483 q^{97} -4.92257 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} - 2 q^{7} + 10 q^{9}+O(q^{10}) $ 6 * q + 2 * q^3 - 2 * q^7 + 10 * q^9	$ 6 q + 2 q^{3} - 2 q^{7} + 10 q^{9} - 6 q^{11} + 2 q^{13} + 12 q^{15} + 12 q^{17} - 6 q^{19} + 2 q^{21} + 2 q^{23} + 26 q^{25} - 16 q^{27} + 4 q^{29} + 20 q^{31} - 2 q^{33} - 20 q^{35} + 22 q^{37} - 8 q^{39} - 2 q^{41} - 10 q^{43} - 6 q^{45} - 16 q^{47} + 48 q^{49} - 36 q^{51} + 12 q^{53} - 2 q^{57} + 14 q^{59} + 12 q^{61} + 4 q^{63} + 10 q^{65} + 12 q^{67} + 30 q^{69} + 30 q^{71} + 24 q^{73} + 50 q^{75} + 2 q^{77} + 10 q^{81} + 14 q^{83} + 12 q^{85} + 30 q^{87} - 14 q^{89} - 20 q^{91} + 14 q^{93} - 46 q^{97} - 10 q^{99}+O(q^{100}) $ 6 * q + 2 * q^3 - 2 * q^7 + 10 * q^9 - 6 * q^11 + 2 * q^13 + 12 * q^15 + 12 * q^17 - 6 * q^19 + 2 * q^21 + 2 * q^23 + 26 * q^25 - 16 * q^27 + 4 * q^29 + 20 * q^31 - 2 * q^33 - 20 * q^35 + 22 * q^37 - 8 * q^39 - 2 * q^41 - 10 * q^43 - 6 * q^45 - 16 * q^47 + 48 * q^49 - 36 * q^51 + 12 * q^53 - 2 * q^57 + 14 * q^59 + 12 * q^61 + 4 * q^63 + 10 * q^65 + 12 * q^67 + 30 * q^69 + 30 * q^71 + 24 * q^73 + 50 * q^75 + 2 * q^77 + 10 * q^81 + 14 * q^83 + 12 * q^85 + 30 * q^87 - 14 * q^89 - 20 * q^91 + 14 * q^93 - 46 * q^97 - 10 * 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
$3$	2.81471	1.62507	0.812535	−	0.582912i	$-0.198087\pi$
$3$	2.81471	1.62507	0.812535	+	0.582912i	$0.198087\pi$
$4$	0	0
$5$	4.18951	1.87360	0.936802	−	0.349859i	$-0.113770\pi$
$5$	4.18951	1.87360	0.936802	+	0.349859i	$0.113770\pi$
$6$	0	0
$7$	−4.08740	−1.54489	−0.772446	−	0.635080i	$-0.780967\pi$
$7$	−4.08740	−1.54489	−0.772446	+	0.635080i	$0.780967\pi$
$8$	0	0
$9$	4.92257	1.64086
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	2.68016	0.743342	0.371671	−	0.928365i	$-0.378785\pi$
$13$	2.68016	0.743342	0.371671	+	0.928365i	$0.378785\pi$
$14$	0	0
$15$	11.7922	3.04474
$16$	0	0
$17$	−2.52730	−0.612961	−0.306481	−	0.951877i	$-0.599151\pi$
$17$	−2.52730	−0.612961	−0.306481	+	0.951877i	$0.599151\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−11.5048	−2.51056
$22$	0	0
$23$	4.36247	0.909638	0.454819	−	0.890584i	$-0.349704\pi$
$23$	4.36247	0.909638	0.454819	+	0.890584i	$0.349704\pi$
$24$	0	0
$25$	12.5520	2.51040
$26$	0	0
$27$	5.41146	1.04144
$28$	0	0
$29$	−4.29737	−0.798001	−0.399001	−	0.916951i	$-0.630643\pi$
$29$	−4.29737	−0.798001	−0.399001	+	0.916951i	$0.630643\pi$
$30$	0	0
$31$	8.43192	1.51442	0.757209	−	0.653173i	$-0.226563\pi$
$31$	8.43192	1.51442	0.757209	+	0.653173i	$0.226563\pi$
$32$	0	0
$33$	−2.81471	−0.489977
$34$	0	0
$35$	−17.1242	−2.89452
$36$	0	0
$37$	7.26036	1.19360	0.596798	−	0.802391i	$-0.296439\pi$
$37$	7.26036	1.19360	0.596798	+	0.802391i	$0.296439\pi$
$38$	0	0
$39$	7.54385	1.20798
$40$	0	0
$41$	7.41167	1.15751	0.578755	−	0.815502i	$-0.303539\pi$
$41$	7.41167	1.15751	0.578755	+	0.815502i	$0.303539\pi$
$42$	0	0
$43$	6.55198	0.999167	0.499584	−	0.866266i	$-0.333486\pi$
$43$	6.55198	0.999167	0.499584	+	0.866266i	$0.333486\pi$
$44$	0	0
$45$	20.6231	3.07431
$46$	0	0
$47$	−7.36031	−1.07361	−0.536806	−	0.843706i	$-0.680369\pi$
$47$	−7.36031	−1.07361	−0.536806	+	0.843706i	$0.680369\pi$
$48$	0	0
$49$	9.70684	1.38669
$50$	0	0
$51$	−7.11361	−0.996105
$52$	0	0
$53$	−9.47454	−1.30143	−0.650714	−	0.759323i	$-0.725530\pi$
$53$	−9.47454	−1.30143	−0.650714	+	0.759323i	$0.725530\pi$
$54$	0	0
$55$	−4.18951	−0.564913
$56$	0	0
$57$	−2.81471	−0.372817
$58$	0	0
$59$	4.38056	0.570300	0.285150	−	0.958483i	$-0.407957\pi$
$59$	4.38056	0.570300	0.285150	+	0.958483i	$0.407957\pi$
$60$	0	0
$61$	11.7617	1.50594	0.752968	−	0.658058i	$-0.228622\pi$
$61$	11.7617	1.50594	0.752968	+	0.658058i	$0.228622\pi$
$62$	0	0
$63$	−20.1205	−2.53494
$64$	0	0
$65$	11.2285	1.39273
$66$	0	0
$67$	−10.6067	−1.29582	−0.647909	−	0.761718i	$-0.724356\pi$
$67$	−10.6067	−1.29582	−0.647909	+	0.761718i	$0.724356\pi$
$68$	0	0
$69$	12.2791	1.47823
$70$	0	0
$71$	−2.44412	−0.290063	−0.145032	−	0.989427i	$-0.546328\pi$
$71$	−2.44412	−0.290063	−0.145032	+	0.989427i	$0.546328\pi$
$72$	0	0
$73$	−15.8536	−1.85552	−0.927759	−	0.373179i	$-0.878268\pi$
$73$	−15.8536	−1.85552	−0.927759	+	0.373179i	$0.878268\pi$
$74$	0	0
$75$	35.3301	4.07957
$76$	0	0
$77$	4.08740	0.465803
$78$	0	0
$79$	7.62283	0.857635	0.428818	−	0.903391i	$-0.358930\pi$
$79$	7.62283	0.857635	0.428818	+	0.903391i	$0.358930\pi$
$80$	0	0
$81$	0.463960	0.0515511
$82$	0	0
$83$	8.83916	0.970224	0.485112	−	0.874452i	$-0.338779\pi$
$83$	8.83916	0.970224	0.485112	+	0.874452i	$0.338779\pi$
$84$	0	0
$85$	−10.5882	−1.14845
$86$	0	0
$87$	−12.0958	−1.29681
$88$	0	0
$89$	6.54385	0.693647	0.346823	−	0.937930i	$-0.387260\pi$
$89$	6.54385	0.693647	0.346823	+	0.937930i	$0.387260\pi$
$90$	0	0
$91$	−10.9549	−1.14838
$92$	0	0
$93$	23.7334	2.46104
$94$	0	0
$95$	−4.18951	−0.429834
$96$	0	0
$97$	−8.16483	−0.829013	−0.414507	−	0.910046i	$-0.636046\pi$
$97$	−8.16483	−0.829013	−0.414507	+	0.910046i	$0.636046\pi$
$98$	0	0
$99$	−4.92257	−0.494737
$100$	0	0
$101$	7.48112	0.744400	0.372200	−	0.928153i	$-0.378604\pi$
$101$	7.48112	0.744400	0.372200	+	0.928153i	$0.378604\pi$
$102$	0	0
$103$	1.35013	0.133032	0.0665161	−	0.997785i	$-0.478812\pi$
$103$	1.35013	0.133032	0.0665161	+	0.997785i	$0.478812\pi$
$104$	0	0
$105$	−48.1996	−4.70380
$106$	0	0
$107$	−16.4932	−1.59446	−0.797231	−	0.603674i	$-0.793703\pi$
$107$	−16.4932	−1.59446	−0.797231	+	0.603674i	$0.793703\pi$
$108$	0	0
$109$	8.61071	0.824756	0.412378	−	0.911013i	$-0.364698\pi$
$109$	8.61071	0.824756	0.412378	+	0.911013i	$0.364698\pi$
$110$	0	0
$111$	20.4358	1.93968
$112$	0	0
$113$	−9.72936	−0.915261	−0.457631	−	0.889142i	$-0.651302\pi$
$113$	−9.72936	−0.915261	−0.457631	+	0.889142i	$0.651302\pi$
$114$	0	0
$115$	18.2766	1.70430
$116$	0	0
$117$	13.1932	1.21972
$118$	0	0
$119$	10.3301	0.946959
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	20.8617	1.88103
$124$	0	0
$125$	31.6391	2.82988
$126$	0	0
$127$	4.24670	0.376834	0.188417	−	0.982089i	$-0.439664\pi$
$127$	4.24670	0.376834	0.188417	+	0.982089i	$0.439664\pi$
$128$	0	0
$129$	18.4419	1.62372
$130$	0	0
$131$	3.25439	0.284338	0.142169	−	0.989842i	$-0.454592\pi$
$131$	3.25439	0.284338	0.142169	+	0.989842i	$0.454592\pi$
$132$	0	0
$133$	4.08740	0.354423
$134$	0	0
$135$	22.6713	1.95124
$136$	0	0
$137$	−19.3971	−1.65721	−0.828603	−	0.559836i	$-0.810864\pi$
$137$	−19.3971	−1.65721	−0.828603	+	0.559836i	$0.810864\pi$
$138$	0	0
$139$	−7.80895	−0.662347	−0.331173	−	0.943570i	$-0.607444\pi$
$139$	−7.80895	−0.662347	−0.331173	+	0.943570i	$0.607444\pi$
$140$	0	0
$141$	−20.7171	−1.74470
$142$	0	0
$143$	−2.68016	−0.224126
$144$	0	0
$145$	−18.0039	−1.49514
$146$	0	0
$147$	27.3219	2.25347
$148$	0	0
$149$	−2.84390	−0.232982	−0.116491	−	0.993192i	$-0.537165\pi$
$149$	−2.84390	−0.232982	−0.116491	+	0.993192i	$0.537165\pi$
$150$	0	0
$151$	−13.3197	−1.08394	−0.541970	−	0.840398i	$-0.682321\pi$
$151$	−13.3197	−1.08394	−0.541970	+	0.840398i	$0.682321\pi$
$152$	0	0
$153$	−12.4408	−1.00578
$154$	0	0
$155$	35.3256	2.83742
$156$	0	0
$157$	−15.4239	−1.23096	−0.615482	−	0.788151i	$-0.711039\pi$
$157$	−15.4239	−1.23096	−0.615482	+	0.788151i	$0.711039\pi$
$158$	0	0
$159$	−26.6680	−2.11491
$160$	0	0
$161$	−17.8312	−1.40529
$162$	0	0
$163$	−12.4439	−0.974681	−0.487341	−	0.873212i	$-0.662033\pi$
$163$	−12.4439	−0.974681	−0.487341	+	0.873212i	$0.662033\pi$
$164$	0	0
$165$	−11.7922	−0.918024
$166$	0	0
$167$	12.3365	0.954630	0.477315	−	0.878732i	$-0.341610\pi$
$167$	12.3365	0.954630	0.477315	+	0.878732i	$0.341610\pi$
$168$	0	0
$169$	−5.81676	−0.447443
$170$	0	0
$171$	−4.92257	−0.376438
$172$	0	0
$173$	−4.96145	−0.377212	−0.188606	−	0.982053i	$-0.560397\pi$
$173$	−4.96145	−0.377212	−0.188606	+	0.982053i	$0.560397\pi$
$174$	0	0
$175$	−51.3050	−3.87829
$176$	0	0
$177$	12.3300	0.926778
$178$	0	0
$179$	−9.00170	−0.672819	−0.336410	−	0.941716i	$-0.609213\pi$
$179$	−9.00170	−0.672819	−0.336410	+	0.941716i	$0.609213\pi$
$180$	0	0
$181$	−6.78397	−0.504248	−0.252124	−	0.967695i	$-0.581129\pi$
$181$	−6.78397	−0.504248	−0.252124	+	0.967695i	$0.581129\pi$
$182$	0	0
$183$	33.1058	2.44725
$184$	0	0
$185$	30.4173	2.23633
$186$	0	0
$187$	2.52730	0.184815
$188$	0	0
$189$	−22.1188	−1.60891
$190$	0	0
$191$	3.91814	0.283506	0.141753	−	0.989902i	$-0.454726\pi$
$191$	3.91814	0.283506	0.141753	+	0.989902i	$0.454726\pi$
$192$	0	0
$193$	8.21396	0.591254	0.295627	−	0.955303i	$-0.404471\pi$
$193$	8.21396	0.591254	0.295627	+	0.955303i	$0.404471\pi$
$194$	0	0
$195$	31.6050	2.26328
$196$	0	0
$197$	−5.92854	−0.422391	−0.211195	−	0.977444i	$-0.567736\pi$
$197$	−5.92854	−0.422391	−0.211195	+	0.977444i	$0.567736\pi$
$198$	0	0
$199$	−0.550137	−0.0389982	−0.0194991	−	0.999810i	$-0.506207\pi$
$199$	−0.550137	−0.0389982	−0.0194991	+	0.999810i	$0.506207\pi$
$200$	0	0
$201$	−29.8548	−2.10579
$202$	0	0
$203$	17.5651	1.23283
$204$	0	0
$205$	31.0513	2.16871
$206$	0	0
$207$	21.4745	1.49258
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	−21.0162	−1.44682	−0.723409	−	0.690420i	$-0.757426\pi$
$211$	−21.0162	−1.44682	−0.723409	+	0.690420i	$0.757426\pi$
$212$	0	0
$213$	−6.87947	−0.471373
$214$	0	0
$215$	27.4496	1.87204
$216$	0	0
$217$	−34.4646	−2.33961
$218$	0	0
$219$	−44.6231	−3.01535
$220$	0	0
$221$	−6.77357	−0.455640
$222$	0	0
$223$	−25.8635	−1.73195	−0.865975	−	0.500087i	$-0.833301\pi$
$223$	−25.8635	−1.73195	−0.865975	+	0.500087i	$0.833301\pi$
$224$	0	0
$225$	61.7879	4.11920
$226$	0	0
$227$	2.79640	0.185604	0.0928018	−	0.995685i	$-0.470418\pi$
$227$	2.79640	0.185604	0.0928018	+	0.995685i	$0.470418\pi$
$228$	0	0
$229$	4.12678	0.272705	0.136353	−	0.990660i	$-0.456462\pi$
$229$	4.12678	0.272705	0.136353	+	0.990660i	$0.456462\pi$
$230$	0	0
$231$	11.5048	0.756962
$232$	0	0
$233$	−18.0509	−1.18255	−0.591277	−	0.806468i	$-0.701376\pi$
$233$	−18.0509	−1.18255	−0.591277	+	0.806468i	$0.701376\pi$
$234$	0	0
$235$	−30.8361	−2.01152
$236$	0	0
$237$	21.4560	1.39372
$238$	0	0
$239$	−22.5764	−1.46034	−0.730172	−	0.683263i	$-0.760560\pi$
$239$	−22.5764	−1.46034	−0.730172	+	0.683263i	$0.760560\pi$
$240$	0	0
$241$	−23.4553	−1.51089	−0.755445	−	0.655212i	$-0.772579\pi$
$241$	−23.4553	−1.51089	−0.755445	+	0.655212i	$0.772579\pi$
$242$	0	0
$243$	−14.9285	−0.957661
$244$	0	0
$245$	40.6669	2.59811
$246$	0	0
$247$	−2.68016	−0.170534
$248$	0	0
$249$	24.8796	1.57668
$250$	0	0
$251$	−9.36370	−0.591031	−0.295516	−	0.955338i	$-0.595491\pi$
$251$	−9.36370	−0.591031	−0.295516	+	0.955338i	$0.595491\pi$
$252$	0	0
$253$	−4.36247	−0.274266
$254$	0	0
$255$	−29.8025	−1.86631
$256$	0	0
$257$	−20.7740	−1.29585	−0.647923	−	0.761706i	$-0.724362\pi$
$257$	−20.7740	−1.29585	−0.647923	+	0.761706i	$0.724362\pi$
$258$	0	0
$259$	−29.6760	−1.84398
$260$	0	0
$261$	−21.1541	−1.30940
$262$	0	0
$263$	12.4577	0.768173	0.384087	−	0.923297i	$-0.374516\pi$
$263$	12.4577	0.768173	0.384087	+	0.923297i	$0.374516\pi$
$264$	0	0
$265$	−39.6937	−2.43836
$266$	0	0
$267$	18.4190	1.12723
$268$	0	0
$269$	4.68875	0.285878	0.142939	−	0.989731i	$-0.454345\pi$
$269$	4.68875	0.285878	0.142939	+	0.989731i	$0.454345\pi$
$270$	0	0
$271$	4.17080	0.253358	0.126679	−	0.991944i	$-0.459568\pi$
$271$	4.17080	0.253358	0.126679	+	0.991944i	$0.459568\pi$
$272$	0	0
$273$	−30.8347	−1.86620
$274$	0	0
$275$	−12.5520	−0.756913
$276$	0	0
$277$	28.7743	1.72888	0.864440	−	0.502737i	$-0.167673\pi$
$277$	28.7743	1.72888	0.864440	+	0.502737i	$0.167673\pi$
$278$	0	0
$279$	41.5067	2.48494
$280$	0	0
$281$	−3.45285	−0.205980	−0.102990	−	0.994682i	$-0.532841\pi$
$281$	−3.45285	−0.205980	−0.102990	+	0.994682i	$0.532841\pi$
$282$	0	0
$283$	−4.79197	−0.284853	−0.142427	−	0.989805i	$-0.545490\pi$
$283$	−4.79197	−0.284853	−0.142427	+	0.989805i	$0.545490\pi$
$284$	0	0
$285$	−11.7922	−0.698511
$286$	0	0
$287$	−30.2945	−1.78823
$288$	0	0
$289$	−10.6127	−0.624279
$290$	0	0
$291$	−22.9816	−1.34721
$292$	0	0
$293$	−10.3211	−0.602964	−0.301482	−	0.953472i	$-0.597481\pi$
$293$	−10.3211	−0.602964	−0.301482	+	0.953472i	$0.597481\pi$
$294$	0	0
$295$	18.3524	1.06852
$296$	0	0
$297$	−5.41146	−0.314005
$298$	0	0
$299$	11.6921	0.676172
$300$	0	0
$301$	−26.7806	−1.54361
$302$	0	0
$303$	21.0572	1.20970
$304$	0	0
$305$	49.2759	2.82153
$306$	0	0
$307$	17.2614	0.985159	0.492579	−	0.870268i	$-0.336054\pi$
$307$	17.2614	0.985159	0.492579	+	0.870268i	$0.336054\pi$
$308$	0	0
$309$	3.80022	0.216187
$310$	0	0
$311$	−7.41397	−0.420408	−0.210204	−	0.977658i	$-0.567413\pi$
$311$	−7.41397	−0.420408	−0.210204	+	0.977658i	$0.567413\pi$
$312$	0	0
$313$	26.9101	1.52105	0.760525	−	0.649308i	$-0.224941\pi$
$313$	26.9101	1.52105	0.760525	+	0.649308i	$0.224941\pi$
$314$	0	0
$315$	−84.2950	−4.74949
$316$	0	0
$317$	−19.9042	−1.11793	−0.558965	−	0.829192i	$-0.688801\pi$
$317$	−19.9042	−1.11793	−0.558965	+	0.829192i	$0.688801\pi$
$318$	0	0
$319$	4.29737	0.240606
$320$	0	0
$321$	−46.4236	−2.59111
$322$	0	0
$323$	2.52730	0.140623
$324$	0	0
$325$	33.6413	1.86608
$326$	0	0
$327$	24.2366	1.34029
$328$	0	0
$329$	30.0845	1.65861
$330$	0	0
$331$	19.0058	1.04465	0.522326	−	0.852746i	$-0.325065\pi$
$331$	19.0058	1.04465	0.522326	+	0.852746i	$0.325065\pi$
$332$	0	0
$333$	35.7396	1.95852
$334$	0	0
$335$	−44.4369	−2.42785
$336$	0	0
$337$	9.69886	0.528331	0.264165	−	0.964477i	$-0.414904\pi$
$337$	9.69886	0.528331	0.264165	+	0.964477i	$0.414904\pi$
$338$	0	0
$339$	−27.3853	−1.48736
$340$	0	0
$341$	−8.43192	−0.456614
$342$	0	0
$343$	−11.0640	−0.597398
$344$	0	0
$345$	51.4432	2.76961
$346$	0	0
$347$	27.5255	1.47764	0.738822	−	0.673901i	$-0.235382\pi$
$347$	27.5255	1.47764	0.738822	+	0.673901i	$0.235382\pi$
$348$	0	0
$349$	9.79763	0.524455	0.262228	−	0.965006i	$-0.415543\pi$
$349$	9.79763	0.524455	0.262228	+	0.965006i	$0.415543\pi$
$350$	0	0
$351$	14.5036	0.774142
$352$	0	0
$353$	−20.2881	−1.07983	−0.539913	−	0.841721i	$-0.681543\pi$
$353$	−20.2881	−1.07983	−0.539913	+	0.841721i	$0.681543\pi$
$354$	0	0
$355$	−10.2396	−0.543464
$356$	0	0
$357$	29.0762	1.53888
$358$	0	0
$359$	18.2401	0.962675	0.481337	−	0.876535i	$-0.340151\pi$
$359$	18.2401	0.962675	0.481337	+	0.876535i	$0.340151\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	2.81471	0.147734
$364$	0	0
$365$	−66.4186	−3.47651
$366$	0	0
$367$	9.81665	0.512425	0.256212	−	0.966621i	$-0.417525\pi$
$367$	9.81665	0.512425	0.256212	+	0.966621i	$0.417525\pi$
$368$	0	0
$369$	36.4845	1.89931
$370$	0	0
$371$	38.7263	2.01057
$372$	0	0
$373$	21.3095	1.10336	0.551682	−	0.834055i	$-0.313986\pi$
$373$	21.3095	1.10336	0.551682	+	0.834055i	$0.313986\pi$
$374$	0	0
$375$	89.0547	4.59876
$376$	0	0
$377$	−11.5176	−0.593188
$378$	0	0
$379$	32.0403	1.64580	0.822901	−	0.568185i	$-0.192354\pi$
$379$	32.0403	1.64580	0.822901	+	0.568185i	$0.192354\pi$
$380$	0	0
$381$	11.9532	0.612381
$382$	0	0
$383$	22.4408	1.14667	0.573335	−	0.819321i	$-0.305649\pi$
$383$	22.4408	1.14667	0.573335	+	0.819321i	$0.305649\pi$
$384$	0	0
$385$	17.1242	0.872730
$386$	0	0
$387$	32.2525	1.63949
$388$	0	0
$389$	10.7147	0.543257	0.271628	−	0.962402i	$-0.412438\pi$
$389$	10.7147	0.543257	0.271628	+	0.962402i	$0.412438\pi$
$390$	0	0
$391$	−11.0253	−0.557573
$392$	0	0
$393$	9.16015	0.462069
$394$	0	0
$395$	31.9359	1.60687
$396$	0	0
$397$	3.77080	0.189251	0.0946256	−	0.995513i	$-0.469835\pi$
$397$	3.77080	0.189251	0.0946256	+	0.995513i	$0.469835\pi$
$398$	0	0
$399$	11.5048	0.575962
$400$	0	0
$401$	−6.10272	−0.304755	−0.152378	−	0.988322i	$-0.548693\pi$
$401$	−6.10272	−0.304755	−0.152378	+	0.988322i	$0.548693\pi$
$402$	0	0
$403$	22.5989	1.12573
$404$	0	0
$405$	1.94376	0.0965865
$406$	0	0
$407$	−7.26036	−0.359883
$408$	0	0
$409$	24.8555	1.22903	0.614513	−	0.788907i	$-0.289353\pi$
$409$	24.8555	1.22903	0.614513	+	0.788907i	$0.289353\pi$
$410$	0	0
$411$	−54.5971	−2.69308
$412$	0	0
$413$	−17.9051	−0.881052
$414$	0	0
$415$	37.0317	1.81782
$416$	0	0
$417$	−21.9799	−1.07636
$418$	0	0
$419$	35.8373	1.75077	0.875384	−	0.483429i	$-0.160609\pi$
$419$	35.8373	1.75077	0.875384	+	0.483429i	$0.160609\pi$
$420$	0	0
$421$	−22.2979	−1.08673	−0.543366	−	0.839496i	$-0.682851\pi$
$421$	−22.2979	−1.08673	−0.543366	+	0.839496i	$0.682851\pi$
$422$	0	0
$423$	−36.2316	−1.76164
$424$	0	0
$425$	−31.7227	−1.53877
$426$	0	0
$427$	−48.0749	−2.32651
$428$	0	0
$429$	−7.54385	−0.364221
$430$	0	0
$431$	−13.4336	−0.647075	−0.323537	−	0.946215i	$-0.604872\pi$
$431$	−13.4336	−0.647075	−0.323537	+	0.946215i	$0.604872\pi$
$432$	0	0
$433$	−19.6837	−0.945939	−0.472969	−	0.881079i	$-0.656818\pi$
$433$	−19.6837	−0.945939	−0.472969	+	0.881079i	$0.656818\pi$
$434$	0	0
$435$	−50.6756	−2.42971
$436$	0	0
$437$	−4.36247	−0.208685
$438$	0	0
$439$	35.1424	1.67726	0.838628	−	0.544705i	$-0.183358\pi$
$439$	35.1424	1.67726	0.838628	+	0.544705i	$0.183358\pi$
$440$	0	0
$441$	47.7826	2.27536
$442$	0	0
$443$	27.4370	1.30357	0.651786	−	0.758403i	$-0.274020\pi$
$443$	27.4370	1.30357	0.651786	+	0.758403i	$0.274020\pi$
$444$	0	0
$445$	27.4155	1.29962
$446$	0	0
$447$	−8.00475	−0.378612
$448$	0	0
$449$	−18.3578	−0.866359	−0.433180	−	0.901308i	$-0.642608\pi$
$449$	−18.3578	−0.866359	−0.433180	+	0.901308i	$0.642608\pi$
$450$	0	0
$451$	−7.41167	−0.349002
$452$	0	0
$453$	−37.4910	−1.76148
$454$	0	0
$455$	−45.8955	−2.15162
$456$	0	0
$457$	−1.97745	−0.0925012	−0.0462506	−	0.998930i	$-0.514727\pi$
$457$	−1.97745	−0.0925012	−0.0462506	+	0.998930i	$0.514727\pi$
$458$	0	0
$459$	−13.6764	−0.638359
$460$	0	0
$461$	−26.6187	−1.23976	−0.619878	−	0.784698i	$-0.712818\pi$
$461$	−26.6187	−1.23976	−0.619878	+	0.784698i	$0.712818\pi$
$462$	0	0
$463$	−10.1827	−0.473228	−0.236614	−	0.971604i	$-0.576038\pi$
$463$	−10.1827	−0.473228	−0.236614	+	0.971604i	$0.576038\pi$
$464$	0	0
$465$	99.4311	4.61101
$466$	0	0
$467$	25.6903	1.18880	0.594402	−	0.804168i	$-0.297389\pi$
$467$	25.6903	1.18880	0.594402	+	0.804168i	$0.297389\pi$
$468$	0	0
$469$	43.3539	2.00190
$470$	0	0
$471$	−43.4138	−2.00040
$472$	0	0
$473$	−6.55198	−0.301260
$474$	0	0
$475$	−12.5520	−0.575924
$476$	0	0
$477$	−46.6391	−2.13546
$478$	0	0
$479$	−6.66695	−0.304621	−0.152310	−	0.988333i	$-0.548671\pi$
$479$	−6.66695	−0.304621	−0.152310	+	0.988333i	$0.548671\pi$
$480$	0	0
$481$	19.4589	0.887250
$482$	0	0
$483$	−50.1895	−2.28370
$484$	0	0
$485$	−34.2066	−1.55324
$486$	0	0
$487$	0.257807	0.0116824	0.00584118	−	0.999983i	$-0.498141\pi$
$487$	0.257807	0.0116824	0.00584118	+	0.999983i	$0.498141\pi$
$488$	0	0
$489$	−35.0259	−1.58393
$490$	0	0
$491$	−33.1751	−1.49717	−0.748586	−	0.663038i	$-0.769267\pi$
$491$	−33.1751	−1.49717	−0.748586	+	0.663038i	$0.769267\pi$
$492$	0	0
$493$	10.8608	0.489144
$494$	0	0
$495$	−20.6231	−0.926941
$496$	0	0
$497$	9.99008	0.448116
$498$	0	0
$499$	−14.6326	−0.655044	−0.327522	−	0.944844i	$-0.606214\pi$
$499$	−14.6326	−0.655044	−0.327522	+	0.944844i	$0.606214\pi$
$500$	0	0
$501$	34.7237	1.55134
$502$	0	0
$503$	2.50263	0.111587	0.0557934	−	0.998442i	$-0.482231\pi$
$503$	2.50263	0.111587	0.0557934	+	0.998442i	$0.482231\pi$
$504$	0	0
$505$	31.3422	1.39471
$506$	0	0
$507$	−16.3725	−0.727127
$508$	0	0
$509$	31.5334	1.39769	0.698847	−	0.715271i	$-0.253697\pi$
$509$	31.5334	1.39769	0.698847	+	0.715271i	$0.253697\pi$
$510$	0	0
$511$	64.7998	2.86658
$512$	0	0
$513$	−5.41146	−0.238922
$514$	0	0
$515$	5.65638	0.249250
$516$	0	0
$517$	7.36031	0.323706
$518$	0	0
$519$	−13.9650	−0.612997
$520$	0	0
$521$	−8.79180	−0.385176	−0.192588	−	0.981280i	$-0.561688\pi$
$521$	−8.79180	−0.385176	−0.192588	+	0.981280i	$0.561688\pi$
$522$	0	0
$523$	−30.8313	−1.34816	−0.674080	−	0.738658i	$-0.735460\pi$
$523$	−30.8313	−1.34816	−0.674080	+	0.738658i	$0.735460\pi$
$524$	0	0
$525$	−144.408	−6.30250
$526$	0	0
$527$	−21.3100	−0.928279
$528$	0	0
$529$	−3.96886	−0.172559
$530$	0	0
$531$	21.5636	0.935780
$532$	0	0
$533$	19.8644	0.860425
$534$	0	0
$535$	−69.0986	−2.98739
$536$	0	0
$537$	−25.3371	−1.09338
$538$	0	0
$539$	−9.70684	−0.418103
$540$	0	0
$541$	−23.0785	−0.992221	−0.496111	−	0.868259i	$-0.665239\pi$
$541$	−23.0785	−0.992221	−0.496111	+	0.868259i	$0.665239\pi$
$542$	0	0
$543$	−19.0949	−0.819439
$544$	0	0
$545$	36.0746	1.54527
$546$	0	0
$547$	−14.1168	−0.603590	−0.301795	−	0.953373i	$-0.597586\pi$
$547$	−14.1168	−0.603590	−0.301795	+	0.953373i	$0.597586\pi$
$548$	0	0
$549$	57.8979	2.47102
$550$	0	0
$551$	4.29737	0.183074
$552$	0	0
$553$	−31.1576	−1.32495
$554$	0	0
$555$	85.6159	3.63419
$556$	0	0
$557$	27.8620	1.18055	0.590275	−	0.807202i	$-0.299019\pi$
$557$	27.8620	1.18055	0.590275	+	0.807202i	$0.299019\pi$
$558$	0	0
$559$	17.5603	0.742723
$560$	0	0
$561$	7.11361	0.300337
$562$	0	0
$563$	−7.67691	−0.323543	−0.161772	−	0.986828i	$-0.551721\pi$
$563$	−7.67691	−0.323543	−0.161772	+	0.986828i	$0.551721\pi$
$564$	0	0
$565$	−40.7612	−1.71484
$566$	0	0
$567$	−1.89639	−0.0796409
$568$	0	0
$569$	36.7985	1.54267	0.771337	−	0.636427i	$-0.219588\pi$
$569$	36.7985	1.54267	0.771337	+	0.636427i	$0.219588\pi$
$570$	0	0
$571$	−36.2345	−1.51636	−0.758182	−	0.652043i	$-0.773912\pi$
$571$	−36.2345	−1.51636	−0.758182	+	0.652043i	$0.773912\pi$
$572$	0	0
$573$	11.0284	0.460718
$574$	0	0
$575$	54.7576	2.28355
$576$	0	0
$577$	−39.4417	−1.64198	−0.820990	−	0.570943i	$-0.806578\pi$
$577$	−39.4417	−1.64198	−0.820990	+	0.570943i	$0.806578\pi$
$578$	0	0
$579$	23.1199	0.960830
$580$	0	0
$581$	−36.1292	−1.49889
$582$	0	0
$583$	9.47454	0.392395
$584$	0	0
$585$	55.2732	2.28527
$586$	0	0
$587$	42.0399	1.73517	0.867586	−	0.497288i	$-0.165671\pi$
$587$	42.0399	1.73517	0.867586	+	0.497288i	$0.165671\pi$
$588$	0	0
$589$	−8.43192	−0.347431
$590$	0	0
$591$	−16.6871	−0.686415
$592$	0	0
$593$	9.51958	0.390922	0.195461	−	0.980711i	$-0.437380\pi$
$593$	9.51958	0.390922	0.195461	+	0.980711i	$0.437380\pi$
$594$	0	0
$595$	43.2780	1.77423
$596$	0	0
$597$	−1.54847	−0.0633748
$598$	0	0
$599$	−3.35013	−0.136883	−0.0684413	−	0.997655i	$-0.521803\pi$
$599$	−3.35013	−0.136883	−0.0684413	+	0.997655i	$0.521803\pi$
$600$	0	0
$601$	40.1043	1.63589	0.817944	−	0.575297i	$-0.195114\pi$
$601$	40.1043	1.63589	0.817944	+	0.575297i	$0.195114\pi$
$602$	0	0
$603$	−52.2123	−2.12625
$604$	0	0
$605$	4.18951	0.170328
$606$	0	0
$607$	−7.10580	−0.288416	−0.144208	−	0.989547i	$-0.546063\pi$
$607$	−7.10580	−0.288416	−0.144208	+	0.989547i	$0.546063\pi$
$608$	0	0
$609$	49.4405	2.00343
$610$	0	0
$611$	−19.7268	−0.798060
$612$	0	0
$613$	1.26848	0.0512335	0.0256168	−	0.999672i	$-0.491845\pi$
$613$	1.26848	0.0512335	0.0256168	+	0.999672i	$0.491845\pi$
$614$	0	0
$615$	87.4002	3.52432
$616$	0	0
$617$	27.4296	1.10428	0.552138	−	0.833753i	$-0.313812\pi$
$617$	27.4296	1.10428	0.552138	+	0.833753i	$0.313812\pi$
$618$	0	0
$619$	−4.73709	−0.190400	−0.0951998	−	0.995458i	$-0.530349\pi$
$619$	−4.73709	−0.190400	−0.0951998	+	0.995458i	$0.530349\pi$
$620$	0	0
$621$	23.6073	0.947329
$622$	0	0
$623$	−26.7473	−1.07161
$624$	0	0
$625$	69.7922	2.79169
$626$	0	0
$627$	2.81471	0.112409
$628$	0	0
$629$	−18.3491	−0.731628
$630$	0	0
$631$	5.13841	0.204557	0.102278	−	0.994756i	$-0.467387\pi$
$631$	5.13841	0.204557	0.102278	+	0.994756i	$0.467387\pi$
$632$	0	0
$633$	−59.1545	−2.35118
$634$	0	0
$635$	17.7916	0.706037
$636$	0	0
$637$	26.0159	1.03079
$638$	0	0
$639$	−12.0313	−0.475952
$640$	0	0
$641$	−11.9738	−0.472936	−0.236468	−	0.971639i	$-0.575990\pi$
$641$	−11.9738	−0.472936	−0.236468	+	0.971639i	$0.575990\pi$
$642$	0	0
$643$	33.9953	1.34064	0.670322	−	0.742071i	$-0.266156\pi$
$643$	33.9953	1.34064	0.670322	+	0.742071i	$0.266156\pi$
$644$	0	0
$645$	77.2624	3.04220
$646$	0	0
$647$	5.75457	0.226236	0.113118	−	0.993582i	$-0.463916\pi$
$647$	5.75457	0.226236	0.113118	+	0.993582i	$0.463916\pi$
$648$	0	0
$649$	−4.38056	−0.171952
$650$	0	0
$651$	−97.0078	−3.80203
$652$	0	0
$653$	−13.7021	−0.536205	−0.268102	−	0.963390i	$-0.586397\pi$
$653$	−13.7021	−0.536205	−0.268102	+	0.963390i	$0.586397\pi$
$654$	0	0
$655$	13.6343	0.532736
$656$	0	0
$657$	−78.0402	−3.04464
$658$	0	0
$659$	−22.3630	−0.871141	−0.435570	−	0.900155i	$-0.643453\pi$
$659$	−22.3630	−0.871141	−0.435570	+	0.900155i	$0.643453\pi$
$660$	0	0
$661$	−1.70857	−0.0664558	−0.0332279	−	0.999448i	$-0.510579\pi$
$661$	−1.70857	−0.0664558	−0.0332279	+	0.999448i	$0.510579\pi$
$662$	0	0
$663$	−19.0656	−0.740447
$664$	0	0
$665$	17.1242	0.664048
$666$	0	0
$667$	−18.7471	−0.725892
$668$	0	0
$669$	−72.7982	−2.81454
$670$	0	0
$671$	−11.7617	−0.454057
$672$	0	0
$673$	26.3724	1.01658	0.508291	−	0.861185i	$-0.330277\pi$
$673$	26.3724	1.01658	0.508291	+	0.861185i	$0.330277\pi$
$674$	0	0
$675$	67.9245	2.61441
$676$	0	0
$677$	−32.6895	−1.25636	−0.628179	−	0.778069i	$-0.716199\pi$
$677$	−32.6895	−1.25636	−0.628179	+	0.778069i	$0.716199\pi$
$678$	0	0
$679$	33.3729	1.28074
$680$	0	0
$681$	7.87105	0.301619
$682$	0	0
$683$	14.6756	0.561546	0.280773	−	0.959774i	$-0.409409\pi$
$683$	14.6756	0.561546	0.280773	+	0.959774i	$0.409409\pi$
$684$	0	0
$685$	−81.2643	−3.10495
$686$	0	0
$687$	11.6157	0.443166
$688$	0	0
$689$	−25.3933	−0.967406
$690$	0	0
$691$	4.47210	0.170127	0.0850634	−	0.996376i	$-0.472891\pi$
$691$	4.47210	0.170127	0.0850634	+	0.996376i	$0.472891\pi$
$692$	0	0
$693$	20.1205	0.764315
$694$	0	0
$695$	−32.7157	−1.24098
$696$	0	0
$697$	−18.7316	−0.709508
$698$	0	0
$699$	−50.8080	−1.92173
$700$	0	0
$701$	41.9503	1.58444	0.792221	−	0.610235i	$-0.208925\pi$
$701$	41.9503	1.58444	0.792221	+	0.610235i	$0.208925\pi$
$702$	0	0
$703$	−7.26036	−0.273830
$704$	0	0
$705$	−86.7945	−3.26887
$706$	0	0
$707$	−30.5783	−1.15002
$708$	0	0
$709$	43.5918	1.63713	0.818563	−	0.574417i	$-0.194771\pi$
$709$	43.5918	1.63713	0.818563	+	0.574417i	$0.194771\pi$
$710$	0	0
$711$	37.5239	1.40726
$712$	0	0
$713$	36.7840	1.37757
$714$	0	0
$715$	−11.2285	−0.419923
$716$	0	0
$717$	−63.5458	−2.37316
$718$	0	0
$719$	−11.4907	−0.428529	−0.214265	−	0.976776i	$-0.568735\pi$
$719$	−11.4907	−0.428529	−0.214265	+	0.976776i	$0.568735\pi$
$720$	0	0
$721$	−5.51852	−0.205520
$722$	0	0
$723$	−66.0198	−2.45530
$724$	0	0
$725$	−53.9405	−2.00330
$726$	0	0
$727$	19.3567	0.717899	0.358949	−	0.933357i	$-0.383135\pi$
$727$	19.3567	0.717899	0.358949	+	0.933357i	$0.383135\pi$
$728$	0	0
$729$	−43.4111	−1.60782
$730$	0	0
$731$	−16.5588	−0.612451
$732$	0	0
$733$	−27.6969	−1.02301	−0.511503	−	0.859281i	$-0.670911\pi$
$733$	−27.6969	−1.02301	−0.511503	+	0.859281i	$0.670911\pi$
$734$	0	0
$735$	114.465	4.22212
$736$	0	0
$737$	10.6067	0.390704
$738$	0	0
$739$	3.94188	0.145005	0.0725023	−	0.997368i	$-0.476902\pi$
$739$	3.94188	0.145005	0.0725023	+	0.997368i	$0.476902\pi$
$740$	0	0
$741$	−7.54385	−0.277130
$742$	0	0
$743$	36.5153	1.33962	0.669808	−	0.742534i	$-0.266376\pi$
$743$	36.5153	1.33962	0.669808	+	0.742534i	$0.266376\pi$
$744$	0	0
$745$	−11.9145	−0.436515
$746$	0	0
$747$	43.5114	1.59200
$748$	0	0
$749$	67.4145	2.46327
$750$	0	0
$751$	20.6971	0.755247	0.377623	−	0.925959i	$-0.376741\pi$
$751$	20.6971	0.755247	0.377623	+	0.925959i	$0.376741\pi$
$752$	0	0
$753$	−26.3561	−0.960468
$754$	0	0
$755$	−55.8029	−2.03088
$756$	0	0
$757$	33.9811	1.23507	0.617533	−	0.786545i	$-0.288132\pi$
$757$	33.9811	1.23507	0.617533	+	0.786545i	$0.288132\pi$
$758$	0	0
$759$	−12.2791	−0.445702
$760$	0	0
$761$	−26.3388	−0.954781	−0.477391	−	0.878691i	$-0.658417\pi$
$761$	−26.3388	−0.954781	−0.477391	+	0.878691i	$0.658417\pi$
$762$	0	0
$763$	−35.1954	−1.27416
$764$	0	0
$765$	−52.1209	−1.88444
$766$	0	0
$767$	11.7406	0.423928
$768$	0	0
$769$	27.0060	0.973860	0.486930	−	0.873441i	$-0.338117\pi$
$769$	27.0060	0.973860	0.486930	+	0.873441i	$0.338117\pi$
$770$	0	0
$771$	−58.4727	−2.10584
$772$	0	0
$773$	2.54539	0.0915513	0.0457757	−	0.998952i	$-0.485424\pi$
$773$	2.54539	0.0915513	0.0457757	+	0.998952i	$0.485424\pi$
$774$	0	0
$775$	105.837	3.80179
$776$	0	0
$777$	−83.5292	−2.99659
$778$	0	0
$779$	−7.41167	−0.265551
$780$	0	0
$781$	2.44412	0.0874574
$782$	0	0
$783$	−23.2550	−0.831067
$784$	0	0
$785$	−64.6187	−2.30634
$786$	0	0
$787$	−23.4471	−0.835798	−0.417899	−	0.908494i	$-0.637233\pi$
$787$	−23.4471	−0.835798	−0.417899	+	0.908494i	$0.637233\pi$
$788$	0	0
$789$	35.0647	1.24834
$790$	0	0
$791$	39.7678	1.41398
$792$	0	0
$793$	31.5233	1.11942
$794$	0	0
$795$	−111.726	−3.96251
$796$	0	0
$797$	13.1436	0.465570	0.232785	−	0.972528i	$-0.425216\pi$
$797$	13.1436	0.465570	0.232785	+	0.972528i	$0.425216\pi$
$798$	0	0
$799$	18.6017	0.658082
$800$	0	0
$801$	32.2125	1.13817
$802$	0	0
$803$	15.8536	0.559460
$804$	0	0
$805$	−74.7038	−2.63296
$806$	0	0
$807$	13.1975	0.464572
$808$	0	0
$809$	−31.9678	−1.12393	−0.561964	−	0.827162i	$-0.689954\pi$
$809$	−31.9678	−1.12393	−0.561964	+	0.827162i	$0.689954\pi$
$810$	0	0
$811$	8.55364	0.300359	0.150179	−	0.988659i	$-0.452015\pi$
$811$	8.55364	0.300359	0.150179	+	0.988659i	$0.452015\pi$
$812$	0	0
$813$	11.7396	0.411725
$814$	0	0
$815$	−52.1338	−1.82617
$816$	0	0
$817$	−6.55198	−0.229225
$818$	0	0
$819$	−53.9261	−1.88433
$820$	0	0
$821$	−1.51457	−0.0528587	−0.0264293	−	0.999651i	$-0.508414\pi$
$821$	−1.51457	−0.0528587	−0.0264293	+	0.999651i	$0.508414\pi$
$822$	0	0
$823$	9.13955	0.318585	0.159292	−	0.987231i	$-0.449079\pi$
$823$	9.13955	0.318585	0.159292	+	0.987231i	$0.449079\pi$
$824$	0	0
$825$	−35.3301	−1.23004
$826$	0	0
$827$	−1.80683	−0.0628295	−0.0314148	−	0.999506i	$-0.510001\pi$
$827$	−1.80683	−0.0628295	−0.0314148	+	0.999506i	$0.510001\pi$
$828$	0	0
$829$	14.5824	0.506468	0.253234	−	0.967405i	$-0.418506\pi$
$829$	14.5824	0.506468	0.253234	+	0.967405i	$0.418506\pi$
$830$	0	0
$831$	80.9911	2.80955
$832$	0	0
$833$	−24.5321	−0.849988
$834$	0	0
$835$	51.6840	1.78860
$836$	0	0
$837$	45.6290	1.57717
$838$	0	0
$839$	23.9842	0.828028	0.414014	−	0.910270i	$-0.364126\pi$
$839$	23.9842	0.828028	0.414014	+	0.910270i	$0.364126\pi$
$840$	0	0
$841$	−10.5326	−0.363194
$842$	0	0
$843$	−9.71876	−0.334732
$844$	0	0
$845$	−24.3694	−0.838332
$846$	0	0
$847$	−4.08740	−0.140445
$848$	0	0
$849$	−13.4880	−0.462907
$850$	0	0
$851$	31.6731	1.08574
$852$	0	0
$853$	−17.2990	−0.592308	−0.296154	−	0.955140i	$-0.595704\pi$
$853$	−17.2990	−0.592308	−0.296154	+	0.955140i	$0.595704\pi$
$854$	0	0
$855$	−20.6231	−0.705296
$856$	0	0
$857$	−39.1818	−1.33843	−0.669213	−	0.743071i	$-0.733369\pi$
$857$	−39.1818	−1.33843	−0.669213	+	0.743071i	$0.733369\pi$
$858$	0	0
$859$	33.2096	1.13310	0.566549	−	0.824028i	$-0.308278\pi$
$859$	33.2096	1.13310	0.566549	+	0.824028i	$0.308278\pi$
$860$	0	0
$861$	−85.2700	−2.90600
$862$	0	0
$863$	−32.3270	−1.10043	−0.550213	−	0.835025i	$-0.685453\pi$
$863$	−32.3270	−1.10043	−0.550213	+	0.835025i	$0.685453\pi$
$864$	0	0
$865$	−20.7860	−0.706747
$866$	0	0
$867$	−29.8717	−1.01450
$868$	0	0
$869$	−7.62283	−0.258587
$870$	0	0
$871$	−28.4277	−0.963235
$872$	0	0
$873$	−40.1919	−1.36029
$874$	0	0
$875$	−129.322	−4.37187
$876$	0	0
$877$	6.12943	0.206976	0.103488	−	0.994631i	$-0.467000\pi$
$877$	6.12943	0.206976	0.103488	+	0.994631i	$0.467000\pi$
$878$	0	0
$879$	−29.0508	−0.979859
$880$	0	0
$881$	0.834433	0.0281127	0.0140564	−	0.999901i	$-0.495526\pi$
$881$	0.834433	0.0281127	0.0140564	+	0.999901i	$0.495526\pi$
$882$	0	0
$883$	8.89255	0.299258	0.149629	−	0.988742i	$-0.452192\pi$
$883$	8.89255	0.299258	0.149629	+	0.988742i	$0.452192\pi$
$884$	0	0
$885$	51.6565	1.73642
$886$	0	0
$887$	13.2341	0.444359	0.222179	−	0.975006i	$-0.428683\pi$
$887$	13.2341	0.444359	0.222179	+	0.975006i	$0.428683\pi$
$888$	0	0
$889$	−17.3580	−0.582167
$890$	0	0
$891$	−0.463960	−0.0155433
$892$	0	0
$893$	7.36031	0.246303
$894$	0	0
$895$	−37.7127	−1.26060
$896$	0	0
$897$	32.9098	1.09883
$898$	0	0
$899$	−36.2351	−1.20851
$900$	0	0
$901$	23.9450	0.797725
$902$	0	0
$903$	−75.3794	−2.50847
$904$	0	0
$905$	−28.4215	−0.944762
$906$	0	0
$907$	−28.8347	−0.957441	−0.478720	−	0.877967i	$-0.658899\pi$
$907$	−28.8347	−0.957441	−0.478720	+	0.877967i	$0.658899\pi$
$908$	0	0
$909$	36.8263	1.22145
$910$	0	0
$911$	−33.4710	−1.10894	−0.554472	−	0.832202i	$-0.687080\pi$
$911$	−33.4710	−1.10894	−0.554472	+	0.832202i	$0.687080\pi$
$912$	0	0
$913$	−8.83916	−0.292534
$914$	0	0
$915$	138.697	4.58518
$916$	0	0
$917$	−13.3020	−0.439271
$918$	0	0
$919$	−41.6615	−1.37428	−0.687142	−	0.726523i	$-0.741135\pi$
$919$	−41.6615	−1.37428	−0.687142	+	0.726523i	$0.741135\pi$
$920$	0	0
$921$	48.5857	1.60095
$922$	0	0
$923$	−6.55061	−0.215616
$924$	0	0
$925$	91.1319	2.99640
$926$	0	0
$927$	6.64610	0.218287
$928$	0	0
$929$	23.6828	0.777006	0.388503	−	0.921447i	$-0.372992\pi$
$929$	23.6828	0.777006	0.388503	+	0.921447i	$0.372992\pi$
$930$	0	0
$931$	−9.70684	−0.318129
$932$	0	0
$933$	−20.8681	−0.683193
$934$	0	0
$935$	10.5882	0.346270
$936$	0	0
$937$	−20.5898	−0.672640	−0.336320	−	0.941748i	$-0.609182\pi$
$937$	−20.5898	−0.672640	−0.336320	+	0.941748i	$0.609182\pi$
$938$	0	0
$939$	75.7441	2.47182
$940$	0	0
$941$	−0.462136	−0.0150652	−0.00753260	−	0.999972i	$-0.502398\pi$
$941$	−0.462136	−0.0150652	−0.00753260	+	0.999972i	$0.502398\pi$
$942$	0	0
$943$	32.3332	1.05291
$944$	0	0
$945$	−92.6669	−3.01445
$946$	0	0
$947$	−20.8678	−0.678113	−0.339056	−	0.940766i	$-0.610108\pi$
$947$	−20.8678	−0.678113	−0.339056	+	0.940766i	$0.610108\pi$
$948$	0	0
$949$	−42.4900	−1.37928
$950$	0	0
$951$	−56.0244	−1.81671
$952$	0	0
$953$	44.9589	1.45636	0.728181	−	0.685385i	$-0.240366\pi$
$953$	44.9589	1.45636	0.728181	+	0.685385i	$0.240366\pi$
$954$	0	0
$955$	16.4151	0.531179
$956$	0	0
$957$	12.0958	0.391003
$958$	0	0
$959$	79.2838	2.56021
$960$	0	0
$961$	40.0972	1.29346
$962$	0	0
$963$	−81.1891	−2.61628
$964$	0	0
$965$	34.4125	1.10778
$966$	0	0
$967$	−37.3660	−1.20161	−0.600805	−	0.799395i	$-0.705153\pi$
$967$	−37.3660	−1.20161	−0.600805	+	0.799395i	$0.705153\pi$
$968$	0	0
$969$	7.11361	0.228522
$970$	0	0
$971$	44.6226	1.43201	0.716004	−	0.698097i	$-0.245969\pi$
$971$	44.6226	1.43201	0.716004	+	0.698097i	$0.245969\pi$
$972$	0	0
$973$	31.9183	1.02325
$974$	0	0
$975$	94.6902	3.03251
$976$	0	0
$977$	3.97336	0.127119	0.0635595	−	0.997978i	$-0.479755\pi$
$977$	3.97336	0.127119	0.0635595	+	0.997978i	$0.479755\pi$
$978$	0	0
$979$	−6.54385	−0.209142
$980$	0	0
$981$	42.3868	1.35331
$982$	0	0
$983$	−3.47767	−0.110920	−0.0554601	−	0.998461i	$-0.517663\pi$
$983$	−3.47767	−0.110920	−0.0554601	+	0.998461i	$0.517663\pi$
$984$	0	0
$985$	−24.8377	−0.791393
$986$	0	0
$987$	84.6791	2.69537
$988$	0	0
$989$	28.5828	0.908880
$990$	0	0
$991$	21.9392	0.696920	0.348460	−	0.937324i	$-0.386705\pi$
$991$	21.9392	0.696920	0.348460	+	0.937324i	$0.386705\pi$
$992$	0	0
$993$	53.4956	1.69763
$994$	0	0
$995$	−2.30480	−0.0730671
$996$	0	0
$997$	−45.7123	−1.44772	−0.723862	−	0.689945i	$-0.757635\pi$
$997$	−45.7123	−1.44772	−0.723862	+	0.689945i	$0.757635\pi$
$998$	0	0
$999$	39.2891	1.24305

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.x.1.6		6
4.3	odd	2		836.2.a.d.1.1	✓	6
12.11	even	2		7524.2.a.r.1.1		6
44.43	even	2		9196.2.a.k.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
836.2.a.d.1.1	✓	6	4.3	odd	2
3344.2.a.x.1.6		6	1.1	even	1	trivial
7524.2.a.r.1.1		6	12.11	even	2
9196.2.a.k.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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+2.81471 q^{3} +4.18951 q^{5} -4.08740 q^{7} +4.92257 q^{9} +O(q^{10})\)	\(q+2.81471 q^{3} +4.18951 q^{5} -4.08740 q^{7} +4.92257 q^{9} -1.00000 q^{11} +2.68016 q^{13} +11.7922 q^{15} -2.52730 q^{17} -1.00000 q^{19} -11.5048 q^{21} +4.36247 q^{23} +12.5520 q^{25} +5.41146 q^{27} -4.29737 q^{29} +8.43192 q^{31} -2.81471 q^{33} -17.1242 q^{35} +7.26036 q^{37} +7.54385 q^{39} +7.41167 q^{41} +6.55198 q^{43} +20.6231 q^{45} -7.36031 q^{47} +9.70684 q^{49} -7.11361 q^{51} -9.47454 q^{53} -4.18951 q^{55} -2.81471 q^{57} +4.38056 q^{59} +11.7617 q^{61} -20.1205 q^{63} +11.2285 q^{65} -10.6067 q^{67} +12.2791 q^{69} -2.44412 q^{71} -15.8536 q^{73} +35.3301 q^{75} +4.08740 q^{77} +7.62283 q^{79} +0.463960 q^{81} +8.83916 q^{83} -10.5882 q^{85} -12.0958 q^{87} +6.54385 q^{89} -10.9549 q^{91} +23.7334 q^{93} -4.18951 q^{95} -8.16483 q^{97} -4.92257 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} - 2 q^{7} + 10 q^{9}+O(q^{10}) \) 6 * q + 2 * q^3 - 2 * q^7 + 10 * q^9	\( 6 q + 2 q^{3} - 2 q^{7} + 10 q^{9} - 6 q^{11} + 2 q^{13} + 12 q^{15} + 12 q^{17} - 6 q^{19} + 2 q^{21} + 2 q^{23} + 26 q^{25} - 16 q^{27} + 4 q^{29} + 20 q^{31} - 2 q^{33} - 20 q^{35} + 22 q^{37} - 8 q^{39} - 2 q^{41} - 10 q^{43} - 6 q^{45} - 16 q^{47} + 48 q^{49} - 36 q^{51} + 12 q^{53} - 2 q^{57} + 14 q^{59} + 12 q^{61} + 4 q^{63} + 10 q^{65} + 12 q^{67} + 30 q^{69} + 30 q^{71} + 24 q^{73} + 50 q^{75} + 2 q^{77} + 10 q^{81} + 14 q^{83} + 12 q^{85} + 30 q^{87} - 14 q^{89} - 20 q^{91} + 14 q^{93} - 46 q^{97} - 10 q^{99}+O(q^{100}) \) 6 * q + 2 * q^3 - 2 * q^7 + 10 * q^9 - 6 * q^11 + 2 * q^13 + 12 * q^15 + 12 * q^17 - 6 * q^19 + 2 * q^21 + 2 * q^23 + 26 * q^25 - 16 * q^27 + 4 * q^29 + 20 * q^31 - 2 * q^33 - 20 * q^35 + 22 * q^37 - 8 * q^39 - 2 * q^41 - 10 * q^43 - 6 * q^45 - 16 * q^47 + 48 * q^49 - 36 * q^51 + 12 * q^53 - 2 * q^57 + 14 * q^59 + 12 * q^61 + 4 * q^63 + 10 * q^65 + 12 * q^67 + 30 * q^69 + 30 * q^71 + 24 * q^73 + 50 * q^75 + 2 * q^77 + 10 * q^81 + 14 * q^83 + 12 * q^85 + 30 * q^87 - 14 * q^89 - 20 * q^91 + 14 * q^93 - 46 * q^97 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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