LMFDB - Embedded newform 7744.2.a.dg.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7744,2,Mod(1,7744)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7744, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7744.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7744 = 2^{6} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7744.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:	$61.8361513253$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3872)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.65544$ of defining polynomial
Character	$\chi$	$=$	7744.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.05137 q^{3} +4.05137 q^{5} +2.25951 q^{7} +6.31088 q^{9} +3.25951 q^{13} -12.3623 q^{15} -1.79186 q^{17} -7.05137 q^{19} -6.89461 q^{21} +2.25951 q^{23} +11.4136 q^{25} -10.1027 q^{27} -7.36226 q^{29} -4.36226 q^{31} +9.15412 q^{35} -1.25951 q^{37} -9.94599 q^{39} +5.10275 q^{41} +2.10275 q^{43} +25.5678 q^{45} -1.74049 q^{47} -1.89461 q^{49} +5.46765 q^{51} +1.94863 q^{53} +21.5164 q^{57} +3.48098 q^{59} -6.00000 q^{61} +14.2595 q^{63} +13.2055 q^{65} +13.6731 q^{67} -6.89461 q^{69} +8.00000 q^{71} +12.8946 q^{73} -34.8273 q^{75} +13.9460 q^{79} +11.8946 q^{81} -8.63510 q^{83} -7.25951 q^{85} +22.4650 q^{87} +12.3109 q^{89} +7.36490 q^{91} +13.3109 q^{93} -28.5678 q^{95} +9.37559 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} + 4 q^{7} + 7 q^{9} + 7 q^{13} - 16 q^{15} + q^{17} - 12 q^{19} + 4 q^{21} + 4 q^{23} + 4 q^{25} - 12 q^{27} - q^{29} + 8 q^{31} - q^{37} + 4 q^{39} - 3 q^{41} - 12 q^{43} + 19 q^{45} - 8 q^{47}+ \cdots + 9 q^{97}+O(q^{100}) $ 3 * q + 3 * q^5 + 4 * q^7 + 7 * q^9 + 7 * q^13 - 16 * q^15 + q^17 - 12 * q^19 + 4 * q^21 + 4 * q^23 + 4 * q^25 - 12 * q^27 - q^29 + 8 * q^31 - q^37 + 4 * q^39 - 3 * q^41 - 12 * q^43 + 19 * q^45 - 8 * q^47 + 19 * q^49 + 20 * q^51 + 15 * q^53 + 16 * q^57 + 16 * q^59 - 18 * q^61 + 40 * q^63 + 3 * q^65 + 8 * q^67 + 4 * q^69 + 24 * q^71 + 14 * q^73 - 44 * q^75 + 8 * q^79 + 11 * q^81 - 4 * q^83 - 19 * q^85 + 28 * q^87 + 25 * q^89 + 44 * q^91 + 28 * q^93 - 28 * q^95 + 9 * q^97

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
$3$	−3.05137	−1.76171	−0.880856	−	0.473385i	$-0.843032\pi$
$3$	−3.05137	−1.76171	−0.880856	+	0.473385i	$0.843032\pi$
$4$	0	0
$5$	4.05137	1.81183	0.905915	−	0.423460i	$-0.139185\pi$
$5$	4.05137	1.81183	0.905915	+	0.423460i	$0.139185\pi$
$6$	0	0
$7$	2.25951	0.854015	0.427007	−	0.904248i	$-0.359568\pi$
$7$	2.25951	0.854015	0.427007	+	0.904248i	$0.359568\pi$
$8$	0	0
$9$	6.31088	2.10363
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	3.25951	0.904026	0.452013	−	0.892011i	$-0.350706\pi$
$13$	3.25951	0.904026	0.452013	+	0.892011i	$0.350706\pi$
$14$	0	0
$15$	−12.3623	−3.19192
$16$	0	0
$17$	−1.79186	−0.434591	−0.217295	−	0.976106i	$-0.569723\pi$
$17$	−1.79186	−0.434591	−0.217295	+	0.976106i	$0.569723\pi$
$18$	0	0
$19$	−7.05137	−1.61770	−0.808848	−	0.588018i	$-0.799909\pi$
$19$	−7.05137	−1.61770	−0.808848	+	0.588018i	$0.799909\pi$
$20$	0	0
$21$	−6.89461	−1.50453
$22$	0	0
$23$	2.25951	0.471141	0.235570	−	0.971857i	$-0.424304\pi$
$23$	2.25951	0.471141	0.235570	+	0.971857i	$0.424304\pi$
$24$	0	0
$25$	11.4136	2.28273
$26$	0	0
$27$	−10.1027	−1.94427
$28$	0	0
$29$	−7.36226	−1.36714	−0.683569	−	0.729886i	$-0.739573\pi$
$29$	−7.36226	−1.36714	−0.683569	+	0.729886i	$0.739573\pi$
$30$	0	0
$31$	−4.36226	−0.783485	−0.391742	−	0.920075i	$-0.628128\pi$
$31$	−4.36226	−0.783485	−0.391742	+	0.920075i	$0.628128\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	9.15412	1.54733
$36$	0	0
$37$	−1.25951	−0.207062	−0.103531	−	0.994626i	$-0.533014\pi$
$37$	−1.25951	−0.207062	−0.103531	+	0.994626i	$0.533014\pi$
$38$	0	0
$39$	−9.94599	−1.59263
$40$	0	0
$41$	5.10275	0.796915	0.398458	−	0.917187i	$-0.369546\pi$
$41$	5.10275	0.796915	0.398458	+	0.917187i	$0.369546\pi$
$42$	0	0
$43$	2.10275	0.320666	0.160333	−	0.987063i	$-0.448743\pi$
$43$	2.10275	0.320666	0.160333	+	0.987063i	$0.448743\pi$
$44$	0	0
$45$	25.5678	3.81142
$46$	0	0
$47$	−1.74049	−0.253876	−0.126938	−	0.991911i	$-0.540515\pi$
$47$	−1.74049	−0.253876	−0.126938	+	0.991911i	$0.540515\pi$
$48$	0	0
$49$	−1.89461	−0.270659
$50$	0	0
$51$	5.46765	0.765624
$52$	0	0
$53$	1.94863	0.267664	0.133832	−	0.991004i	$-0.457272\pi$
$53$	1.94863	0.267664	0.133832	+	0.991004i	$0.457272\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	21.5164	2.84991
$58$	0	0
$59$	3.48098	0.453185	0.226592	−	0.973990i	$-0.427241\pi$
$59$	3.48098	0.453185	0.226592	+	0.973990i	$0.427241\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	0	0
$63$	14.2595	1.79653
$64$	0	0
$65$	13.2055	1.63794
$66$	0	0
$67$	13.6731	1.67044	0.835220	−	0.549916i	$-0.185340\pi$
$67$	13.6731	1.67044	0.835220	+	0.549916i	$0.185340\pi$
$68$	0	0
$69$	−6.89461	−0.830014
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	12.8946	1.50920	0.754600	−	0.656185i	$-0.227831\pi$
$73$	12.8946	1.50920	0.754600	+	0.656185i	$0.227831\pi$
$74$	0	0
$75$	−34.8273	−4.02151
$76$	0	0
$77$	0	0
$78$	0	0
$79$	13.9460	1.56905	0.784523	−	0.620100i	$-0.212908\pi$
$79$	13.9460	1.56905	0.784523	+	0.620100i	$0.212908\pi$
$80$	0	0
$81$	11.8946	1.32162
$82$	0	0
$83$	−8.63510	−0.947826	−0.473913	−	0.880572i	$-0.657159\pi$
$83$	−8.63510	−0.947826	−0.473913	+	0.880572i	$0.657159\pi$
$84$	0	0
$85$	−7.25951	−0.787404
$86$	0	0
$87$	22.4650	2.40850
$88$	0	0
$89$	12.3109	1.30495	0.652476	−	0.757810i	$-0.273730\pi$
$89$	12.3109	1.30495	0.652476	+	0.757810i	$0.273730\pi$
$90$	0	0
$91$	7.36490	0.772051
$92$	0	0
$93$	13.3109	1.38027
$94$	0	0
$95$	−28.5678	−2.93099
$96$	0	0
$97$	9.37559	0.951947	0.475974	−	0.879460i	$-0.342096\pi$
$97$	9.37559	0.951947	0.475974	+	0.879460i	$0.342096\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.58373	0.356594	0.178297	−	0.983977i	$-0.442941\pi$
$101$	3.58373	0.356594	0.178297	+	0.983977i	$0.442941\pi$
$102$	0	0
$103$	6.62177	0.652462	0.326231	−	0.945290i	$-0.394221\pi$
$103$	6.62177	0.652462	0.326231	+	0.945290i	$0.394221\pi$
$104$	0	0
$105$	−27.9327	−2.72595
$106$	0	0
$107$	1.67314	0.161749	0.0808745	−	0.996724i	$-0.474229\pi$
$107$	1.67314	0.161749	0.0808745	+	0.996724i	$0.474229\pi$
$108$	0	0
$109$	−13.1922	−1.26358	−0.631790	−	0.775140i	$-0.717680\pi$
$109$	−13.1922	−1.26358	−0.631790	+	0.775140i	$0.717680\pi$
$110$	0	0
$111$	3.84324	0.364784
$112$	0	0
$113$	0.583727	0.0549125	0.0274562	−	0.999623i	$-0.491259\pi$
$113$	0.583727	0.0549125	0.0274562	+	0.999623i	$0.491259\pi$
$114$	0	0
$115$	9.15412	0.853626
$116$	0	0
$117$	20.5704	1.90173
$118$	0	0
$119$	−4.04873	−0.371147
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−15.5704	−1.40394
$124$	0	0
$125$	25.9840	2.32408
$126$	0	0
$127$	7.48098	0.663830	0.331915	−	0.943309i	$-0.392305\pi$
$127$	7.48098	0.663830	0.331915	+	0.943309i	$0.392305\pi$
$128$	0	0
$129$	−6.41627	−0.564921
$130$	0	0
$131$	18.8273	1.64495	0.822473	−	0.568804i	$-0.192594\pi$
$131$	18.8273	1.64495	0.822473	+	0.568804i	$0.192594\pi$
$132$	0	0
$133$	−15.9327	−1.38154
$134$	0	0
$135$	−40.9300	−3.52269
$136$	0	0
$137$	−9.93265	−0.848604	−0.424302	−	0.905521i	$-0.639480\pi$
$137$	−9.93265	−0.848604	−0.424302	+	0.905521i	$0.639480\pi$
$138$	0	0
$139$	−13.6731	−1.15974	−0.579870	−	0.814709i	$-0.696897\pi$
$139$	−13.6731	−1.15974	−0.579870	+	0.814709i	$0.696897\pi$
$140$	0	0
$141$	5.31088	0.447257
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−29.8273	−2.47702
$146$	0	0
$147$	5.78117	0.476823
$148$	0	0
$149$	−5.53235	−0.453228	−0.226614	−	0.973985i	$-0.572766\pi$
$149$	−5.53235	−0.453228	−0.226614	+	0.973985i	$0.572766\pi$
$150$	0	0
$151$	7.48098	0.608793	0.304397	−	0.952545i	$-0.401545\pi$
$151$	7.48098	0.608793	0.304397	+	0.952545i	$0.401545\pi$
$152$	0	0
$153$	−11.3082	−0.914217
$154$	0	0
$155$	−17.6731	−1.41954
$156$	0	0
$157$	0.894612	0.0713978	0.0356989	−	0.999363i	$-0.488634\pi$
$157$	0.894612	0.0713978	0.0356989	+	0.999363i	$0.488634\pi$
$158$	0	0
$159$	−5.94599	−0.471547
$160$	0	0
$161$	5.10539	0.402361
$162$	0	0
$163$	−16.6351	−1.30296	−0.651481	−	0.758665i	$-0.725852\pi$
$163$	−16.6351	−1.30296	−0.651481	+	0.758665i	$0.725852\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.51902	0.659222	0.329611	−	0.944117i	$-0.393082\pi$
$167$	8.51902	0.659222	0.329611	+	0.944117i	$0.393082\pi$
$168$	0	0
$169$	−2.37559	−0.182738
$170$	0	0
$171$	−44.5004	−3.40303
$172$	0	0
$173$	16.8273	1.27935	0.639677	−	0.768644i	$-0.279068\pi$
$173$	16.8273	1.27935	0.639677	+	0.768644i	$0.279068\pi$
$174$	0	0
$175$	25.7892	1.94948
$176$	0	0
$177$	−10.6218	−0.798381
$178$	0	0
$179$	21.2702	1.58981	0.794905	−	0.606734i	$-0.207521\pi$
$179$	21.2702	1.58981	0.794905	+	0.606734i	$0.207521\pi$
$180$	0	0
$181$	−0.673144	−0.0500344	−0.0250172	−	0.999687i	$-0.507964\pi$
$181$	−0.673144	−0.0500344	−0.0250172	+	0.999687i	$0.507964\pi$
$182$	0	0
$183$	18.3082	1.35338
$184$	0	0
$185$	−5.10275	−0.375162
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−22.8273	−1.66044
$190$	0	0
$191$	−16.5190	−1.19527	−0.597637	−	0.801767i	$-0.703894\pi$
$191$	−16.5190	−1.19527	−0.597637	+	0.801767i	$0.703894\pi$
$192$	0	0
$193$	10.4136	0.749590	0.374795	−	0.927108i	$-0.377713\pi$
$193$	10.4136	0.749590	0.374795	+	0.927108i	$0.377713\pi$
$194$	0	0
$195$	−40.2949	−2.88558
$196$	0	0
$197$	19.9840	1.42380	0.711902	−	0.702279i	$-0.247834\pi$
$197$	19.9840	1.42380	0.711902	+	0.702279i	$0.247834\pi$
$198$	0	0
$199$	14.1027	0.999717	0.499859	−	0.866107i	$-0.333385\pi$
$199$	14.1027	0.999717	0.499859	+	0.866107i	$0.333385\pi$
$200$	0	0
$201$	−41.7219	−2.94283
$202$	0	0
$203$	−16.6351	−1.16756
$204$	0	0
$205$	20.6731	1.44387
$206$	0	0
$207$	14.2595	0.991104
$208$	0	0
$209$	0	0
$210$	0	0
$211$	13.2435	0.911723	0.455861	−	0.890051i	$-0.349331\pi$
$211$	13.2435	0.911723	0.455861	+	0.890051i	$0.349331\pi$
$212$	0	0
$213$	−24.4110	−1.67261
$214$	0	0
$215$	8.51902	0.580992
$216$	0	0
$217$	−9.85657	−0.669108
$218$	0	0
$219$	−39.3463	−2.65878
$220$	0	0
$221$	−5.84060	−0.392881
$222$	0	0
$223$	10.8273	0.725047	0.362524	−	0.931975i	$-0.381915\pi$
$223$	10.8273	0.725047	0.362524	+	0.931975i	$0.381915\pi$
$224$	0	0
$225$	72.0301	4.80201
$226$	0	0
$227$	−23.3463	−1.54955	−0.774774	−	0.632239i	$-0.782136\pi$
$227$	−23.3463	−1.54955	−0.774774	+	0.632239i	$0.782136\pi$
$228$	0	0
$229$	21.6351	1.42969	0.714844	−	0.699284i	$-0.246498\pi$
$229$	21.6351	1.42969	0.714844	+	0.699284i	$0.246498\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.20550	−0.0789747	−0.0394874	−	0.999220i	$-0.512572\pi$
$233$	−1.20550	−0.0789747	−0.0394874	+	0.999220i	$0.512572\pi$
$234$	0	0
$235$	−7.05137	−0.459981
$236$	0	0
$237$	−42.5544	−2.76421
$238$	0	0
$239$	−8.04873	−0.520629	−0.260315	−	0.965524i	$-0.583826\pi$
$239$	−8.04873	−0.520629	−0.260315	+	0.965524i	$0.583826\pi$
$240$	0	0
$241$	−0.894612	−0.0576270	−0.0288135	−	0.999585i	$-0.509173\pi$
$241$	−0.894612	−0.0576270	−0.0288135	+	0.999585i	$0.509173\pi$
$242$	0	0
$243$	−5.98667	−0.384045
$244$	0	0
$245$	−7.67578	−0.490388
$246$	0	0
$247$	−22.9840	−1.46244
$248$	0	0
$249$	26.3489	1.66980
$250$	0	0
$251$	22.7112	1.43352	0.716759	−	0.697321i	$-0.245625\pi$
$251$	22.7112	1.43352	0.716759	+	0.697321i	$0.245625\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	22.1515	1.38718
$256$	0	0
$257$	5.68912	0.354877	0.177439	−	0.984132i	$-0.443219\pi$
$257$	5.68912	0.354877	0.177439	+	0.984132i	$0.443219\pi$
$258$	0	0
$259$	−2.84588	−0.176834
$260$	0	0
$261$	−46.4624	−2.87595
$262$	0	0
$263$	−20.0487	−1.23626	−0.618129	−	0.786077i	$-0.712109\pi$
$263$	−20.0487	−1.23626	−0.618129	+	0.786077i	$0.712109\pi$
$264$	0	0
$265$	7.89461	0.484962
$266$	0	0
$267$	−37.5651	−2.29895
$268$	0	0
$269$	0.118720	0.00723848	0.00361924	−	0.999993i	$-0.498848\pi$
$269$	0.118720	0.00723848	0.00361924	+	0.999993i	$0.498848\pi$
$270$	0	0
$271$	−6.62177	−0.402244	−0.201122	−	0.979566i	$-0.564459\pi$
$271$	−6.62177	−0.402244	−0.201122	+	0.979566i	$0.564459\pi$
$272$	0	0
$273$	−22.4731	−1.36013
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7.19216	0.432135	0.216068	−	0.976378i	$-0.430677\pi$
$277$	7.19216	0.432135	0.216068	+	0.976378i	$0.430677\pi$
$278$	0	0
$279$	−27.5297	−1.64816
$280$	0	0
$281$	−2.06735	−0.123328	−0.0616638	−	0.998097i	$-0.519641\pi$
$281$	−2.06735	−0.123328	−0.0616638	+	0.998097i	$0.519641\pi$
$282$	0	0
$283$	−0.313524	−0.0186371	−0.00931854	−	0.999957i	$-0.502966\pi$
$283$	−0.313524	−0.0186371	−0.00931854	+	0.999957i	$0.502966\pi$
$284$	0	0
$285$	87.1709	5.16356
$286$	0	0
$287$	11.5297	0.680577
$288$	0	0
$289$	−13.7892	−0.811131
$290$	0	0
$291$	−28.6084	−1.67706
$292$	0	0
$293$	23.2188	1.35646	0.678229	−	0.734850i	$-0.262748\pi$
$293$	23.2188	1.35646	0.678229	+	0.734850i	$0.262748\pi$
$294$	0	0
$295$	14.1027	0.821094
$296$	0	0
$297$	0	0
$298$	0	0
$299$	7.36490	0.425923
$300$	0	0
$301$	4.75118	0.273854
$302$	0	0
$303$	−10.9353	−0.628216
$304$	0	0
$305$	−24.3082	−1.39189
$306$	0	0
$307$	−25.6731	−1.46524	−0.732622	−	0.680636i	$-0.761704\pi$
$307$	−25.6731	−1.46524	−0.732622	+	0.680636i	$0.761704\pi$
$308$	0	0
$309$	−20.2055	−1.14945
$310$	0	0
$311$	−27.3463	−1.55067	−0.775333	−	0.631553i	$-0.782418\pi$
$311$	−27.3463	−1.55067	−0.775333	+	0.631553i	$0.782418\pi$
$312$	0	0
$313$	8.62441	0.487480	0.243740	−	0.969841i	$-0.421626\pi$
$313$	8.62441	0.487480	0.243740	+	0.969841i	$0.421626\pi$
$314$	0	0
$315$	57.7706	3.25501
$316$	0	0
$317$	−28.8273	−1.61910	−0.809550	−	0.587051i	$-0.800289\pi$
$317$	−28.8273	−1.61910	−0.809550	+	0.587051i	$0.800289\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−5.10539	−0.284955
$322$	0	0
$323$	12.6351	0.703036
$324$	0	0
$325$	37.2029	2.06364
$326$	0	0
$327$	40.2542	2.22606
$328$	0	0
$329$	−3.93265	−0.216814
$330$	0	0
$331$	−4.51902	−0.248388	−0.124194	−	0.992258i	$-0.539634\pi$
$331$	−4.51902	−0.248388	−0.124194	+	0.992258i	$0.539634\pi$
$332$	0	0
$333$	−7.94863	−0.435582
$334$	0	0
$335$	55.3950	3.02655
$336$	0	0
$337$	−27.7599	−1.51218	−0.756090	−	0.654468i	$-0.772893\pi$
$337$	−27.7599	−1.51218	−0.756090	+	0.654468i	$0.772893\pi$
$338$	0	0
$339$	−1.78117	−0.0967399
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−20.0975	−1.08516
$344$	0	0
$345$	−27.9327	−1.50384
$346$	0	0
$347$	30.3082	1.62703	0.813516	−	0.581543i	$-0.197551\pi$
$347$	30.3082	1.62703	0.813516	+	0.581543i	$0.197551\pi$
$348$	0	0
$349$	2.67314	0.143090	0.0715451	−	0.997437i	$-0.477207\pi$
$349$	2.67314	0.143090	0.0715451	+	0.997437i	$0.477207\pi$
$350$	0	0
$351$	−32.9300	−1.75767
$352$	0	0
$353$	19.2055	1.02221	0.511103	−	0.859520i	$-0.329237\pi$
$353$	19.2055	1.02221	0.511103	+	0.859520i	$0.329237\pi$
$354$	0	0
$355$	32.4110	1.72020
$356$	0	0
$357$	12.3542	0.653854
$358$	0	0
$359$	−18.7785	−0.991093	−0.495546	−	0.868581i	$-0.665032\pi$
$359$	−18.7785	−0.991093	−0.495546	+	0.868581i	$0.665032\pi$
$360$	0	0
$361$	30.7219	1.61694
$362$	0	0
$363$	0	0
$364$	0	0
$365$	52.2409	2.73441
$366$	0	0
$367$	14.2595	0.744340	0.372170	−	0.928165i	$-0.378614\pi$
$367$	14.2595	0.744340	0.372170	+	0.928165i	$0.378614\pi$
$368$	0	0
$369$	32.2029	1.67641
$370$	0	0
$371$	4.40294	0.228589
$372$	0	0
$373$	−15.0380	−0.778640	−0.389320	−	0.921103i	$-0.627290\pi$
$373$	−15.0380	−0.778640	−0.389320	+	0.921103i	$0.627290\pi$
$374$	0	0
$375$	−79.2870	−4.09436
$376$	0	0
$377$	−23.9974	−1.23593
$378$	0	0
$379$	30.6218	1.57293	0.786467	−	0.617632i	$-0.211908\pi$
$379$	30.6218	1.57293	0.786467	+	0.617632i	$0.211908\pi$
$380$	0	0
$381$	−22.8273	−1.16948
$382$	0	0
$383$	19.3463	0.988549	0.494275	−	0.869306i	$-0.335434\pi$
$383$	19.3463	0.988549	0.494275	+	0.869306i	$0.335434\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	13.2702	0.674562
$388$	0	0
$389$	−4.94599	−0.250771	−0.125386	−	0.992108i	$-0.540017\pi$
$389$	−4.94599	−0.250771	−0.125386	+	0.992108i	$0.540017\pi$
$390$	0	0
$391$	−4.04873	−0.204753
$392$	0	0
$393$	−57.4490	−2.89792
$394$	0	0
$395$	56.5004	2.84284
$396$	0	0
$397$	−26.4624	−1.32811	−0.664054	−	0.747685i	$-0.731166\pi$
$397$	−26.4624	−1.32811	−0.664054	+	0.747685i	$0.731166\pi$
$398$	0	0
$399$	48.6165	2.43387
$400$	0	0
$401$	−5.72452	−0.285869	−0.142934	−	0.989732i	$-0.545654\pi$
$401$	−5.72452	−0.285869	−0.142934	+	0.989732i	$0.545654\pi$
$402$	0	0
$403$	−14.2188	−0.708290
$404$	0	0
$405$	48.1895	2.39456
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−11.3082	−0.559157	−0.279578	−	0.960123i	$-0.590195\pi$
$409$	−11.3082	−0.559157	−0.279578	+	0.960123i	$0.590195\pi$
$410$	0	0
$411$	30.3082	1.49500
$412$	0	0
$413$	7.86531	0.387027
$414$	0	0
$415$	−34.9840	−1.71730
$416$	0	0
$417$	41.7219	2.04313
$418$	0	0
$419$	27.2302	1.33028	0.665141	−	0.746717i	$-0.268371\pi$
$419$	27.2302	1.33028	0.665141	+	0.746717i	$0.268371\pi$
$420$	0	0
$421$	38.0868	1.85624	0.928118	−	0.372286i	$-0.121426\pi$
$421$	38.0868	1.85624	0.928118	+	0.372286i	$0.121426\pi$
$422$	0	0
$423$	−10.9840	−0.534062
$424$	0	0
$425$	−20.4517	−0.992052
$426$	0	0
$427$	−13.5571	−0.656072
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−26.3082	−1.26722	−0.633612	−	0.773651i	$-0.718428\pi$
$431$	−26.3082	−1.26722	−0.633612	+	0.773651i	$0.718428\pi$
$432$	0	0
$433$	18.4136	0.884903	0.442451	−	0.896792i	$-0.354109\pi$
$433$	18.4136	0.884903	0.442451	+	0.896792i	$0.354109\pi$
$434$	0	0
$435$	91.0142	4.36379
$436$	0	0
$437$	−15.9327	−0.762162
$438$	0	0
$439$	11.8432	0.565247	0.282623	−	0.959231i	$-0.408795\pi$
$439$	11.8432	0.565247	0.282623	+	0.959231i	$0.408795\pi$
$440$	0	0
$441$	−11.9567	−0.569366
$442$	0	0
$443$	−3.48098	−0.165386	−0.0826932	−	0.996575i	$-0.526352\pi$
$443$	−3.48098	−0.165386	−0.0826932	+	0.996575i	$0.526352\pi$
$444$	0	0
$445$	49.8760	2.36435
$446$	0	0
$447$	16.8813	0.798457
$448$	0	0
$449$	35.9974	1.69882	0.849410	−	0.527733i	$-0.176958\pi$
$449$	35.9974	1.69882	0.849410	+	0.527733i	$0.176958\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−22.8273	−1.07252
$454$	0	0
$455$	29.8380	1.39883
$456$	0	0
$457$	−1.58637	−0.0742071	−0.0371036	−	0.999311i	$-0.511813\pi$
$457$	−1.58637	−0.0742071	−0.0371036	+	0.999311i	$0.511813\pi$
$458$	0	0
$459$	18.1027	0.844964
$460$	0	0
$461$	−1.98403	−0.0924054	−0.0462027	−	0.998932i	$-0.514712\pi$
$461$	−1.98403	−0.0924054	−0.0462027	+	0.998932i	$0.514712\pi$
$462$	0	0
$463$	2.41627	0.112294	0.0561469	−	0.998423i	$-0.482118\pi$
$463$	2.41627	0.112294	0.0561469	+	0.998423i	$0.482118\pi$
$464$	0	0
$465$	53.9274	2.50082
$466$	0	0
$467$	5.55706	0.257150	0.128575	−	0.991700i	$-0.458960\pi$
$467$	5.55706	0.257150	0.128575	+	0.991700i	$0.458960\pi$
$468$	0	0
$469$	30.8946	1.42658
$470$	0	0
$471$	−2.72980	−0.125782
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−80.4818	−3.69276
$476$	0	0
$477$	12.2976	0.563066
$478$	0	0
$479$	2.82727	0.129181	0.0645906	−	0.997912i	$-0.479426\pi$
$479$	2.82727	0.129181	0.0645906	+	0.997912i	$0.479426\pi$
$480$	0	0
$481$	−4.10539	−0.187190
$482$	0	0
$483$	−15.5784	−0.708844
$484$	0	0
$485$	37.9840	1.72477
$486$	0	0
$487$	29.6058	1.34157	0.670783	−	0.741653i	$-0.265958\pi$
$487$	29.6058	1.34157	0.670783	+	0.741653i	$0.265958\pi$
$488$	0	0
$489$	50.7599	2.29544
$490$	0	0
$491$	32.5004	1.46672	0.733361	−	0.679839i	$-0.237950\pi$
$491$	32.5004	1.46672	0.733361	+	0.679839i	$0.237950\pi$
$492$	0	0
$493$	13.1922	0.594145
$494$	0	0
$495$	0	0
$496$	0	0
$497$	18.0761	0.810823
$498$	0	0
$499$	12.9300	0.578827	0.289413	−	0.957204i	$-0.406540\pi$
$499$	12.9300	0.578827	0.289413	+	0.957204i	$0.406540\pi$
$500$	0	0
$501$	−25.9947	−1.16136
$502$	0	0
$503$	34.1248	1.52155	0.760775	−	0.649015i	$-0.224819\pi$
$503$	34.1248	1.52155	0.760775	+	0.649015i	$0.224819\pi$
$504$	0	0
$505$	14.5190	0.646088
$506$	0	0
$507$	7.24882	0.321931
$508$	0	0
$509$	12.0761	0.535263	0.267632	−	0.963521i	$-0.413759\pi$
$509$	12.0761	0.535263	0.267632	+	0.963521i	$0.413759\pi$
$510$	0	0
$511$	29.1355	1.28888
$512$	0	0
$513$	71.2383	3.14525
$514$	0	0
$515$	26.8273	1.18215
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−51.3463	−2.25385
$520$	0	0
$521$	−23.7219	−1.03927	−0.519637	−	0.854387i	$-0.673933\pi$
$521$	−23.7219	−1.03927	−0.519637	+	0.854387i	$0.673933\pi$
$522$	0	0
$523$	−24.0000	−1.04945	−0.524723	−	0.851273i	$-0.675831\pi$
$523$	−24.0000	−1.04945	−0.524723	+	0.851273i	$0.675831\pi$
$524$	0	0
$525$	−78.6926	−3.43443
$526$	0	0
$527$	7.81657	0.340495
$528$	0	0
$529$	−17.8946	−0.778027
$530$	0	0
$531$	21.9681	0.953332
$532$	0	0
$533$	16.6325	0.720432
$534$	0	0
$535$	6.77853	0.293061
$536$	0	0
$537$	−64.9034	−2.80079
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−44.7599	−1.92438	−0.962190	−	0.272380i	$-0.912189\pi$
$541$	−44.7599	−1.92438	−0.962190	+	0.272380i	$0.912189\pi$
$542$	0	0
$543$	2.05401	0.0881462
$544$	0	0
$545$	−53.4464	−2.28939
$546$	0	0
$547$	19.4810	0.832947	0.416473	−	0.909148i	$-0.363266\pi$
$547$	19.4810	0.832947	0.416473	+	0.909148i	$0.363266\pi$
$548$	0	0
$549$	−37.8653	−1.61605
$550$	0	0
$551$	51.9140	2.21161
$552$	0	0
$553$	31.5111	1.33999
$554$	0	0
$555$	15.5704	0.660926
$556$	0	0
$557$	−10.7512	−0.455542	−0.227771	−	0.973715i	$-0.573144\pi$
$557$	−10.7512	−0.455542	−0.227771	+	0.973715i	$0.573144\pi$
$558$	0	0
$559$	6.85393	0.289890
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−0.116081	−0.00489222	−0.00244611	−	0.999997i	$-0.500779\pi$
$563$	−0.116081	−0.00489222	−0.00244611	+	0.999997i	$0.500779\pi$
$564$	0	0
$565$	2.36490	0.0994920
$566$	0	0
$567$	26.8760	1.12869
$568$	0	0
$569$	3.85657	0.161676	0.0808379	−	0.996727i	$-0.474240\pi$
$569$	3.85657	0.161676	0.0808379	+	0.996727i	$0.474240\pi$
$570$	0	0
$571$	−4.63510	−0.193973	−0.0969865	−	0.995286i	$-0.530920\pi$
$571$	−4.63510	−0.193973	−0.0969865	+	0.995286i	$0.530920\pi$
$572$	0	0
$573$	50.4057	2.10573
$574$	0	0
$575$	25.7892	1.07549
$576$	0	0
$577$	−5.97070	−0.248563	−0.124282	−	0.992247i	$-0.539663\pi$
$577$	−5.97070	−0.248563	−0.124282	+	0.992247i	$0.539663\pi$
$578$	0	0
$579$	−31.7759	−1.32056
$580$	0	0
$581$	−19.5111	−0.809457
$582$	0	0
$583$	0	0
$584$	0	0
$585$	83.3384	3.44562
$586$	0	0
$587$	18.9433	0.781876	0.390938	−	0.920417i	$-0.372151\pi$
$587$	18.9433	0.781876	0.390938	+	0.920417i	$0.372151\pi$
$588$	0	0
$589$	30.7599	1.26744
$590$	0	0
$591$	−60.9787	−2.50833
$592$	0	0
$593$	−23.6891	−0.972795	−0.486398	−	0.873738i	$-0.661689\pi$
$593$	−23.6891	−0.972795	−0.486398	+	0.873738i	$0.661689\pi$
$594$	0	0
$595$	−16.4029	−0.672455
$596$	0	0
$597$	−43.0328	−1.76121
$598$	0	0
$599$	−4.70245	−0.192137	−0.0960684	−	0.995375i	$-0.530627\pi$
$599$	−4.70245	−0.192137	−0.0960684	+	0.995375i	$0.530627\pi$
$600$	0	0
$601$	−5.00000	−0.203954	−0.101977	−	0.994787i	$-0.532517\pi$
$601$	−5.00000	−0.203954	−0.101977	+	0.994787i	$0.532517\pi$
$602$	0	0
$603$	86.2896	3.51399
$604$	0	0
$605$	0	0
$606$	0	0
$607$	11.8432	0.480702	0.240351	−	0.970686i	$-0.422737\pi$
$607$	11.8432	0.480702	0.240351	+	0.970686i	$0.422737\pi$
$608$	0	0
$609$	50.7599	2.05690
$610$	0	0
$611$	−5.67314	−0.229511
$612$	0	0
$613$	43.8087	1.76941	0.884707	−	0.466147i	$-0.154358\pi$
$613$	43.8087	1.76941	0.884707	+	0.466147i	$0.154358\pi$
$614$	0	0
$615$	−63.0815	−2.54369
$616$	0	0
$617$	8.65107	0.348279	0.174140	−	0.984721i	$-0.444286\pi$
$617$	8.65107	0.348279	0.174140	+	0.984721i	$0.444286\pi$
$618$	0	0
$619$	−18.1922	−0.731205	−0.365602	−	0.930771i	$-0.619137\pi$
$619$	−18.1922	−0.731205	−0.365602	+	0.930771i	$0.619137\pi$
$620$	0	0
$621$	−22.8273	−0.916027
$622$	0	0
$623$	27.8166	1.11445
$624$	0	0
$625$	48.2029	1.92811
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.25687	0.0899873
$630$	0	0
$631$	43.0107	1.71223	0.856114	−	0.516786i	$-0.172872\pi$
$631$	43.0107	1.71223	0.856114	+	0.516786i	$0.172872\pi$
$632$	0	0
$633$	−40.4110	−1.60619
$634$	0	0
$635$	30.3082	1.20275
$636$	0	0
$637$	−6.17551	−0.244683
$638$	0	0
$639$	50.4871	1.99724
$640$	0	0
$641$	3.31352	0.130876	0.0654382	−	0.997857i	$-0.479155\pi$
$641$	3.31352	0.130876	0.0654382	+	0.997857i	$0.479155\pi$
$642$	0	0
$643$	19.2835	0.760468	0.380234	−	0.924890i	$-0.375843\pi$
$643$	19.2835	0.760468	0.380234	+	0.924890i	$0.375843\pi$
$644$	0	0
$645$	−25.9947	−1.02354
$646$	0	0
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	30.0761	1.17877
$652$	0	0
$653$	−44.7599	−1.75159	−0.875796	−	0.482682i	$-0.839663\pi$
$653$	−44.7599	−1.75159	−0.875796	+	0.482682i	$0.839663\pi$
$654$	0	0
$655$	76.2763	2.98036
$656$	0	0
$657$	81.3764	3.17480
$658$	0	0
$659$	−23.2302	−0.904920	−0.452460	−	0.891785i	$-0.649454\pi$
$659$	−23.2302	−0.904920	−0.452460	+	0.891785i	$0.649454\pi$
$660$	0	0
$661$	14.0868	0.547912	0.273956	−	0.961742i	$-0.411668\pi$
$661$	14.0868	0.547912	0.273956	+	0.961742i	$0.411668\pi$
$662$	0	0
$663$	17.8219	0.692143
$664$	0	0
$665$	−64.5491	−2.50311
$666$	0	0
$667$	−16.6351	−0.644114
$668$	0	0
$669$	−33.0380	−1.27732
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−47.7219	−1.83954	−0.919772	−	0.392454i	$-0.871626\pi$
$673$	−47.7219	−1.83954	−0.919772	+	0.392454i	$0.871626\pi$
$674$	0	0
$675$	−115.309	−4.43825
$676$	0	0
$677$	−43.7466	−1.68132	−0.840659	−	0.541565i	$-0.817832\pi$
$677$	−43.7466	−1.68132	−0.840659	+	0.541565i	$0.817832\pi$
$678$	0	0
$679$	21.1842	0.812977
$680$	0	0
$681$	71.2383	2.72986
$682$	0	0
$683$	−19.9814	−0.764567	−0.382283	−	0.924045i	$-0.624862\pi$
$683$	−19.9814	−0.764567	−0.382283	+	0.924045i	$0.624862\pi$
$684$	0	0
$685$	−40.2409	−1.53753
$686$	0	0
$687$	−66.0168	−2.51870
$688$	0	0
$689$	6.35157	0.241975
$690$	0	0
$691$	49.4757	1.88214	0.941072	−	0.338206i	$-0.109820\pi$
$691$	49.4757	1.88214	0.941072	+	0.338206i	$0.109820\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−55.3950	−2.10125
$696$	0	0
$697$	−9.14343	−0.346332
$698$	0	0
$699$	3.67842	0.139131
$700$	0	0
$701$	25.6758	0.969761	0.484880	−	0.874580i	$-0.338863\pi$
$701$	25.6758	0.969761	0.484880	+	0.874580i	$0.338863\pi$
$702$	0	0
$703$	8.88128	0.334964
$704$	0	0
$705$	21.5164	0.810354
$706$	0	0
$707$	8.09747	0.304537
$708$	0	0
$709$	45.5784	1.71173	0.855867	−	0.517196i	$-0.173024\pi$
$709$	45.5784	1.71173	0.855867	+	0.517196i	$0.173024\pi$
$710$	0	0
$711$	88.0115	3.30069
$712$	0	0
$713$	−9.85657	−0.369131
$714$	0	0
$715$	0	0
$716$	0	0
$717$	24.5597	0.917199
$718$	0	0
$719$	−18.7785	−0.700321	−0.350161	−	0.936690i	$-0.613873\pi$
$719$	−18.7785	−0.700321	−0.350161	+	0.936690i	$0.613873\pi$
$720$	0	0
$721$	14.9620	0.557212
$722$	0	0
$723$	2.72980	0.101522
$724$	0	0
$725$	−84.0301	−3.12080
$726$	0	0
$727$	−31.3950	−1.16438	−0.582188	−	0.813054i	$-0.697803\pi$
$727$	−31.3950	−1.16438	−0.582188	+	0.813054i	$0.697803\pi$
$728$	0	0
$729$	−17.4163	−0.645047
$730$	0	0
$731$	−3.76784	−0.139359
$732$	0	0
$733$	−6.29755	−0.232605	−0.116303	−	0.993214i	$-0.537104\pi$
$733$	−6.29755	−0.232605	−0.116303	+	0.993214i	$0.537104\pi$
$734$	0	0
$735$	23.4217	0.863922
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−17.6465	−0.649136	−0.324568	−	0.945862i	$-0.605219\pi$
$739$	−17.6465	−0.649136	−0.324568	+	0.945862i	$0.605219\pi$
$740$	0	0
$741$	70.1329	2.57640
$742$	0	0
$743$	−18.7785	−0.688918	−0.344459	−	0.938801i	$-0.611938\pi$
$743$	−18.7785	−0.688918	−0.344459	+	0.938801i	$0.611938\pi$
$744$	0	0
$745$	−22.4136	−0.821172
$746$	0	0
$747$	−54.4951	−1.99387
$748$	0	0
$749$	3.78049	0.138136
$750$	0	0
$751$	−37.2435	−1.35904	−0.679518	−	0.733659i	$-0.737811\pi$
$751$	−37.2435	−1.35904	−0.679518	+	0.733659i	$0.737811\pi$
$752$	0	0
$753$	−69.3003	−2.52544
$754$	0	0
$755$	30.3082	1.10303
$756$	0	0
$757$	−16.6058	−0.603548	−0.301774	−	0.953379i	$-0.597579\pi$
$757$	−16.6058	−0.603548	−0.301774	+	0.953379i	$0.597579\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−26.2169	−0.950361	−0.475180	−	0.879888i	$-0.657617\pi$
$761$	−26.2169	−0.950361	−0.475180	+	0.879888i	$0.657617\pi$
$762$	0	0
$763$	−29.8078	−1.07912
$764$	0	0
$765$	−45.8139	−1.65641
$766$	0	0
$767$	11.3463	0.409691
$768$	0	0
$769$	−37.4517	−1.35054	−0.675271	−	0.737570i	$-0.735973\pi$
$769$	−37.4517	−1.35054	−0.675271	+	0.737570i	$0.735973\pi$
$770$	0	0
$771$	−17.3596	−0.625191
$772$	0	0
$773$	−31.8600	−1.14593	−0.572963	−	0.819581i	$-0.694206\pi$
$773$	−31.8600	−1.14593	−0.572963	+	0.819581i	$0.694206\pi$
$774$	0	0
$775$	−49.7892	−1.78848
$776$	0	0
$777$	8.68384	0.311531
$778$	0	0
$779$	−35.9814	−1.28917
$780$	0	0
$781$	0	0
$782$	0	0
$783$	74.3791	2.65809
$784$	0	0
$785$	3.62441	0.129361
$786$	0	0
$787$	−33.0380	−1.17768	−0.588840	−	0.808250i	$-0.700415\pi$
$787$	−33.0380	−1.17768	−0.588840	+	0.808250i	$0.700415\pi$
$788$	0	0
$789$	61.1762	2.17793
$790$	0	0
$791$	1.31894	0.0468960
$792$	0	0
$793$	−19.5571	−0.694492
$794$	0	0
$795$	−24.0894	−0.854364
$796$	0	0
$797$	−6.54569	−0.231860	−0.115930	−	0.993257i	$-0.536985\pi$
$797$	−6.54569	−0.231860	−0.115930	+	0.993257i	$0.536985\pi$
$798$	0	0
$799$	3.11872	0.110332
$800$	0	0
$801$	77.6926	2.74513
$802$	0	0
$803$	0	0
$804$	0	0
$805$	20.6838	0.729009
$806$	0	0
$807$	−0.362259	−0.0127521
$808$	0	0
$809$	−30.2728	−1.06434	−0.532168	−	0.846639i	$-0.678623\pi$
$809$	−30.2728	−1.06434	−0.532168	+	0.846639i	$0.678623\pi$
$810$	0	0
$811$	19.4810	0.684070	0.342035	−	0.939687i	$-0.388884\pi$
$811$	19.4810	0.684070	0.342035	+	0.939687i	$0.388884\pi$
$812$	0	0
$813$	20.2055	0.708638
$814$	0	0
$815$	−67.3950	−2.36074
$816$	0	0
$817$	−14.8273	−0.518740
$818$	0	0
$819$	46.4790	1.62411
$820$	0	0
$821$	40.8273	1.42488	0.712441	−	0.701732i	$-0.247590\pi$
$821$	40.8273	1.42488	0.712441	+	0.701732i	$0.247590\pi$
$822$	0	0
$823$	−30.8539	−1.07550	−0.537750	−	0.843104i	$-0.680726\pi$
$823$	−30.8539	−1.07550	−0.537750	+	0.843104i	$0.680726\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−15.4624	−0.537679	−0.268840	−	0.963185i	$-0.586640\pi$
$827$	−15.4624	−0.537679	−0.268840	+	0.963185i	$0.586640\pi$
$828$	0	0
$829$	40.9140	1.42100	0.710502	−	0.703695i	$-0.248468\pi$
$829$	40.9140	1.42100	0.710502	+	0.703695i	$0.248468\pi$
$830$	0	0
$831$	−21.9460	−0.761298
$832$	0	0
$833$	3.39489	0.117626
$834$	0	0
$835$	34.5137	1.19440
$836$	0	0
$837$	44.0708	1.52331
$838$	0	0
$839$	4.18343	0.144428	0.0722140	−	0.997389i	$-0.476994\pi$
$839$	4.18343	0.144428	0.0722140	+	0.997389i	$0.476994\pi$
$840$	0	0
$841$	25.2029	0.869064
$842$	0	0
$843$	6.30825	0.217268
$844$	0	0
$845$	−9.62441	−0.331090
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0.956679	0.0328331
$850$	0	0
$851$	−2.84588	−0.0975554
$852$	0	0
$853$	35.4243	1.21291	0.606453	−	0.795119i	$-0.292592\pi$
$853$	35.4243	1.21291	0.606453	+	0.795119i	$0.292592\pi$
$854$	0	0
$855$	−180.288	−6.16571
$856$	0	0
$857$	22.2055	0.758525	0.379263	−	0.925289i	$-0.376178\pi$
$857$	22.2055	0.758525	0.379263	+	0.925289i	$0.376178\pi$
$858$	0	0
$859$	−38.6484	−1.31867	−0.659334	−	0.751850i	$-0.729162\pi$
$859$	−38.6484	−1.31867	−0.659334	+	0.751850i	$0.729162\pi$
$860$	0	0
$861$	−35.1815	−1.19898
$862$	0	0
$863$	−8.33559	−0.283747	−0.141873	−	0.989885i	$-0.545313\pi$
$863$	−8.33559	−0.283747	−0.141873	+	0.989885i	$0.545313\pi$
$864$	0	0
$865$	68.1736	2.31797
$866$	0	0
$867$	42.0761	1.42898
$868$	0	0
$869$	0	0
$870$	0	0
$871$	44.5678	1.51012
$872$	0	0
$873$	59.1683	2.00254
$874$	0	0
$875$	58.7112	1.98480
$876$	0	0
$877$	−20.6731	−0.698082	−0.349041	−	0.937107i	$-0.613493\pi$
$877$	−20.6731	−0.698082	−0.349041	+	0.937107i	$0.613493\pi$
$878$	0	0
$879$	−70.8493	−2.38969
$880$	0	0
$881$	28.5164	0.960741	0.480371	−	0.877066i	$-0.340502\pi$
$881$	28.5164	0.960741	0.480371	+	0.877066i	$0.340502\pi$
$882$	0	0
$883$	18.0761	0.608309	0.304154	−	0.952623i	$-0.401626\pi$
$883$	18.0761	0.608309	0.304154	+	0.952623i	$0.401626\pi$
$884$	0	0
$885$	−43.0328	−1.44653
$886$	0	0
$887$	−10.1248	−0.339958	−0.169979	−	0.985448i	$-0.554370\pi$
$887$	−10.1248	−0.339958	−0.169979	+	0.985448i	$0.554370\pi$
$888$	0	0
$889$	16.9034	0.566920
$890$	0	0
$891$	0	0
$892$	0	0
$893$	12.2728	0.410695
$894$	0	0
$895$	86.1736	2.88046
$896$	0	0
$897$	−22.4731	−0.750354
$898$	0	0
$899$	32.1161	1.07113
$900$	0	0
$901$	−3.49167	−0.116324
$902$	0	0
$903$	−14.4976	−0.482451
$904$	0	0
$905$	−2.72716	−0.0906538
$906$	0	0
$907$	−15.0328	−0.499155	−0.249577	−	0.968355i	$-0.580292\pi$
$907$	−15.0328	−0.499155	−0.249577	+	0.968355i	$0.580292\pi$
$908$	0	0
$909$	22.6165	0.750142
$910$	0	0
$911$	−23.7131	−0.785651	−0.392826	−	0.919613i	$-0.628502\pi$
$911$	−23.7131	−0.785651	−0.392826	+	0.919613i	$0.628502\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	74.1736	2.45210
$916$	0	0
$917$	42.5404	1.40481
$918$	0	0
$919$	−6.38961	−0.210774	−0.105387	−	0.994431i	$-0.533608\pi$
$919$	−6.38961	−0.210774	−0.105387	+	0.994431i	$0.533608\pi$
$920$	0	0
$921$	78.3384	2.58134
$922$	0	0
$923$	26.0761	0.858305
$924$	0	0
$925$	−14.3756	−0.472667
$926$	0	0
$927$	41.7892	1.37254
$928$	0	0
$929$	39.1115	1.28321	0.641603	−	0.767037i	$-0.278270\pi$
$929$	39.1115	1.28321	0.641603	+	0.767037i	$0.278270\pi$
$930$	0	0
$931$	13.3596	0.437844
$932$	0	0
$933$	83.4438	2.73183
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−13.6838	−0.447031	−0.223516	−	0.974700i	$-0.571753\pi$
$937$	−13.6838	−0.447031	−0.223516	+	0.974700i	$0.571753\pi$
$938$	0	0
$939$	−26.3163	−0.858800
$940$	0	0
$941$	−38.2569	−1.24714	−0.623569	−	0.781768i	$-0.714318\pi$
$941$	−38.2569	−1.24714	−0.623569	+	0.781768i	$0.714318\pi$
$942$	0	0
$943$	11.5297	0.375459
$944$	0	0
$945$	−92.4818	−3.00843
$946$	0	0
$947$	37.9053	1.23176	0.615878	−	0.787841i	$-0.288801\pi$
$947$	37.9053	1.23176	0.615878	+	0.787841i	$0.288801\pi$
$948$	0	0
$949$	42.0301	1.36436
$950$	0	0
$951$	87.9628	2.85239
$952$	0	0
$953$	−13.7919	−0.446762	−0.223381	−	0.974731i	$-0.571709\pi$
$953$	−13.7919	−0.446762	−0.223381	+	0.974731i	$0.571709\pi$
$954$	0	0
$955$	−66.9247	−2.16563
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−22.4429	−0.724720
$960$	0	0
$961$	−11.9707	−0.386151
$962$	0	0
$963$	10.5590	0.340260
$964$	0	0
$965$	42.1895	1.35813
$966$	0	0
$967$	−16.0487	−0.516093	−0.258046	−	0.966133i	$-0.583079\pi$
$967$	−16.0487	−0.516093	−0.258046	+	0.966133i	$0.583079\pi$
$968$	0	0
$969$	−38.5544	−1.23855
$970$	0	0
$971$	−33.1541	−1.06397	−0.531983	−	0.846755i	$-0.678553\pi$
$971$	−33.1541	−1.06397	−0.531983	+	0.846755i	$0.678553\pi$
$972$	0	0
$973$	−30.8946	−0.990436
$974$	0	0
$975$	−113.520	−3.63554
$976$	0	0
$977$	−23.8006	−0.761449	−0.380724	−	0.924689i	$-0.624325\pi$
$977$	−23.8006	−0.761449	−0.380724	+	0.924689i	$0.624325\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−83.2542	−2.65810
$982$	0	0
$983$	−1.78922	−0.0570674	−0.0285337	−	0.999593i	$-0.509084\pi$
$983$	−1.78922	−0.0570674	−0.0285337	+	0.999593i	$0.509084\pi$
$984$	0	0
$985$	80.9628	2.57969
$986$	0	0
$987$	12.0000	0.381964
$988$	0	0
$989$	4.75118	0.151079
$990$	0	0
$991$	27.5784	0.876058	0.438029	−	0.898961i	$-0.355677\pi$
$991$	27.5784	0.876058	0.438029	+	0.898961i	$0.355677\pi$
$992$	0	0
$993$	13.7892	0.437588
$994$	0	0
$995$	57.1355	1.81132
$996$	0	0
$997$	−33.2595	−1.05334	−0.526670	−	0.850070i	$-0.676560\pi$
$997$	−33.2595	−1.05334	−0.526670	+	0.850070i	$0.676560\pi$
$998$	0	0
$999$	12.7245	0.402586

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7744.2.a.dg.1.1		3
4.3	odd	2		7744.2.a.de.1.3		3
8.3	odd	2		3872.2.a.bb.1.1	✓	3
8.5	even	2		3872.2.a.bd.1.3	yes	3
11.10	odd	2		7744.2.a.dd.1.1		3
44.43	even	2		7744.2.a.df.1.3		3
88.21	odd	2		3872.2.a.bc.1.3	yes	3
88.43	even	2		3872.2.a.be.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3872.2.a.bb.1.1	✓	3	8.3	odd	2
3872.2.a.bc.1.3	yes	3	88.21	odd	2
3872.2.a.bd.1.3	yes	3	8.5	even	2
3872.2.a.be.1.1	yes	3	88.43	even	2
7744.2.a.dd.1.1		3	11.10	odd	2
7744.2.a.de.1.3		3	4.3	odd	2
7744.2.a.df.1.3		3	44.43	even	2
7744.2.a.dg.1.1		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7744 = 2^{6} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7744.a (trivial)

\(f(q)\)	\(=\)	\(q-3.05137 q^{3} +4.05137 q^{5} +2.25951 q^{7} +6.31088 q^{9} +3.25951 q^{13} -12.3623 q^{15} -1.79186 q^{17} -7.05137 q^{19} -6.89461 q^{21} +2.25951 q^{23} +11.4136 q^{25} -10.1027 q^{27} -7.36226 q^{29} -4.36226 q^{31} +9.15412 q^{35} -1.25951 q^{37} -9.94599 q^{39} +5.10275 q^{41} +2.10275 q^{43} +25.5678 q^{45} -1.74049 q^{47} -1.89461 q^{49} +5.46765 q^{51} +1.94863 q^{53} +21.5164 q^{57} +3.48098 q^{59} -6.00000 q^{61} +14.2595 q^{63} +13.2055 q^{65} +13.6731 q^{67} -6.89461 q^{69} +8.00000 q^{71} +12.8946 q^{73} -34.8273 q^{75} +13.9460 q^{79} +11.8946 q^{81} -8.63510 q^{83} -7.25951 q^{85} +22.4650 q^{87} +12.3109 q^{89} +7.36490 q^{91} +13.3109 q^{93} -28.5678 q^{95} +9.37559 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} + 4 q^{7} + 7 q^{9} + 7 q^{13} - 16 q^{15} + q^{17} - 12 q^{19} + 4 q^{21} + 4 q^{23} + 4 q^{25} - 12 q^{27} - q^{29} + 8 q^{31} - q^{37} + 4 q^{39} - 3 q^{41} - 12 q^{43} + 19 q^{45} - 8 q^{47}+ \cdots + 9 q^{97}+O(q^{100}) \) 3 * q + 3 * q^5 + 4 * q^7 + 7 * q^9 + 7 * q^13 - 16 * q^15 + q^17 - 12 * q^19 + 4 * q^21 + 4 * q^23 + 4 * q^25 - 12 * q^27 - q^29 + 8 * q^31 - q^37 + 4 * q^39 - 3 * q^41 - 12 * q^43 + 19 * q^45 - 8 * q^47 + 19 * q^49 + 20 * q^51 + 15 * q^53 + 16 * q^57 + 16 * q^59 - 18 * q^61 + 40 * q^63 + 3 * q^65 + 8 * q^67 + 4 * q^69 + 24 * q^71 + 14 * q^73 - 44 * q^75 + 8 * q^79 + 11 * q^81 - 4 * q^83 - 19 * q^85 + 28 * q^87 + 25 * q^89 + 44 * q^91 + 28 * q^93 - 28 * q^95 + 9 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more