LMFDB - Embedded newform 2790.2.a.bk.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2790,2,Mod(1,2790)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2790, 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("2790.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2790.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:	$22.2782621639$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.17428.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 4x + 6 $ x^4 - x^3 - 6x^2 + 4x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.36865$ of defining polynomial
Character	$\chi$	$=$	2790.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.81471 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.81471 q^{7} +1.00000 q^{8} +1.00000 q^{10} +3.81471 q^{11} -2.73730 q^{13} -1.81471 q^{14} +1.00000 q^{16} +0.483701 q^{17} +2.92259 q^{19} +1.00000 q^{20} +3.81471 q^{22} -1.40629 q^{23} +1.00000 q^{25} -2.73730 q^{26} -1.81471 q^{28} +9.47460 q^{29} +1.00000 q^{31} +1.00000 q^{32} +0.483701 q^{34} -1.81471 q^{35} -3.70470 q^{37} +2.92259 q^{38} +1.00000 q^{40} +9.47460 q^{41} +11.5194 q^{43} +3.81471 q^{44} -1.40629 q^{46} -10.4420 q^{47} -3.70683 q^{49} +1.00000 q^{50} -2.73730 q^{52} -8.55201 q^{53} +3.81471 q^{55} -1.81471 q^{56} +9.47460 q^{58} +13.3341 q^{59} +10.7751 q^{61} +1.00000 q^{62} +1.00000 q^{64} -2.73730 q^{65} +9.54988 q^{67} +0.483701 q^{68} -1.81471 q^{70} -5.65989 q^{71} +8.70683 q^{73} -3.70470 q^{74} +2.92259 q^{76} -6.92259 q^{77} -9.36459 q^{79} +1.00000 q^{80} +9.47460 q^{82} +2.81258 q^{83} +0.483701 q^{85} +11.5194 q^{86} +3.81471 q^{88} -12.4115 q^{89} +4.96740 q^{91} -1.40629 q^{92} -10.4420 q^{94} +2.92259 q^{95} +0.154821 q^{97} -3.70683 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 5 q^{7} + 4 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + 5 * q^7 + 4 * q^8	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 5 q^{7} + 4 q^{8} + 4 q^{10} + 3 q^{11} + 6 q^{13} + 5 q^{14} + 4 q^{16} + 7 q^{19} + 4 q^{20} + 3 q^{22} + q^{23} + 4 q^{25} + 6 q^{26} + 5 q^{28} + 4 q^{29} + 4 q^{31} + 4 q^{32} + 5 q^{35} + 6 q^{37} + 7 q^{38} + 4 q^{40} + 4 q^{41} + 13 q^{43} + 3 q^{44} + q^{46} - 4 q^{47} + 5 q^{49} + 4 q^{50} + 6 q^{52} - 5 q^{53} + 3 q^{55} + 5 q^{56} + 4 q^{58} + 8 q^{59} - 4 q^{61} + 4 q^{62} + 4 q^{64} + 6 q^{65} + 8 q^{67} + 5 q^{70} - q^{71} + 15 q^{73} + 6 q^{74} + 7 q^{76} - 23 q^{77} + 5 q^{79} + 4 q^{80} + 4 q^{82} - 2 q^{83} + 13 q^{86} + 3 q^{88} - 9 q^{89} + 16 q^{91} + q^{92} - 4 q^{94} + 7 q^{95} + 10 q^{97} + 5 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + 5 * q^7 + 4 * q^8 + 4 * q^10 + 3 * q^11 + 6 * q^13 + 5 * q^14 + 4 * q^16 + 7 * q^19 + 4 * q^20 + 3 * q^22 + q^23 + 4 * q^25 + 6 * q^26 + 5 * q^28 + 4 * q^29 + 4 * q^31 + 4 * q^32 + 5 * q^35 + 6 * q^37 + 7 * q^38 + 4 * q^40 + 4 * q^41 + 13 * q^43 + 3 * q^44 + q^46 - 4 * q^47 + 5 * q^49 + 4 * q^50 + 6 * q^52 - 5 * q^53 + 3 * q^55 + 5 * q^56 + 4 * q^58 + 8 * q^59 - 4 * q^61 + 4 * q^62 + 4 * q^64 + 6 * q^65 + 8 * q^67 + 5 * q^70 - q^71 + 15 * q^73 + 6 * q^74 + 7 * q^76 - 23 * q^77 + 5 * q^79 + 4 * q^80 + 4 * q^82 - 2 * q^83 + 13 * q^86 + 3 * q^88 - 9 * q^89 + 16 * q^91 + q^92 - 4 * q^94 + 7 * q^95 + 10 * q^97 + 5 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.81471	−0.685896	−0.342948	−	0.939354i	$-0.611425\pi$
$7$	−1.81471	−0.685896	−0.342948	+	0.939354i	$0.611425\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	3.81471	1.15018	0.575089	−	0.818091i	$-0.304967\pi$
$11$	3.81471	1.15018	0.575089	+	0.818091i	$0.304967\pi$
$12$	0	0
$13$	−2.73730	−0.759190	−0.379595	−	0.925153i	$-0.623937\pi$
$13$	−2.73730	−0.759190	−0.379595	+	0.925153i	$0.623937\pi$
$14$	−1.81471	−0.485002
$15$	0	0
$16$	1.00000	0.250000
$17$	0.483701	0.117315	0.0586574	−	0.998278i	$-0.481318\pi$
$17$	0.483701	0.117315	0.0586574	+	0.998278i	$0.481318\pi$
$18$	0	0
$19$	2.92259	0.670488	0.335244	−	0.942131i	$-0.391181\pi$
$19$	2.92259	0.670488	0.335244	+	0.942131i	$0.391181\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	3.81471	0.813299
$23$	−1.40629	−0.293232	−0.146616	−	0.989193i	$-0.546838\pi$
$23$	−1.40629	−0.293232	−0.146616	+	0.989193i	$0.546838\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.73730	−0.536828
$27$	0	0
$28$	−1.81471	−0.342948
$29$	9.47460	1.75939	0.879694	−	0.475540i	$-0.157747\pi$
$29$	9.47460	1.75939	0.879694	+	0.475540i	$0.157747\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	1.00000	0.176777
$33$	0	0
$34$	0.483701	0.0829540
$35$	−1.81471	−0.306742
$36$	0	0
$37$	−3.70470	−0.609049	−0.304525	−	0.952504i	$-0.598498\pi$
$37$	−3.70470	−0.609049	−0.304525	+	0.952504i	$0.598498\pi$
$38$	2.92259	0.474107
$39$	0	0
$40$	1.00000	0.158114
$41$	9.47460	1.47968	0.739842	−	0.672781i	$-0.234900\pi$
$41$	9.47460	1.47968	0.739842	+	0.672781i	$0.234900\pi$
$42$	0	0
$43$	11.5194	1.75669	0.878347	−	0.478024i	$-0.158647\pi$
$43$	11.5194	1.75669	0.878347	+	0.478024i	$0.158647\pi$
$44$	3.81471	0.575089
$45$	0	0
$46$	−1.40629	−0.207346
$47$	−10.4420	−1.52312	−0.761561	−	0.648093i	$-0.775567\pi$
$47$	−10.4420	−1.52312	−0.761561	+	0.648093i	$0.775567\pi$
$48$	0	0
$49$	−3.70683	−0.529547
$50$	1.00000	0.141421
$51$	0	0
$52$	−2.73730	−0.379595
$53$	−8.55201	−1.17471	−0.587354	−	0.809330i	$-0.699830\pi$
$53$	−8.55201	−1.17471	−0.587354	+	0.809330i	$0.699830\pi$
$54$	0	0
$55$	3.81471	0.514375
$56$	−1.81471	−0.242501
$57$	0	0
$58$	9.47460	1.24408
$59$	13.3341	1.73595	0.867977	−	0.496604i	$-0.165420\pi$
$59$	13.3341	1.73595	0.867977	+	0.496604i	$0.165420\pi$
$60$	0	0
$61$	10.7751	1.37961	0.689807	−	0.723993i	$-0.257695\pi$
$61$	10.7751	1.37961	0.689807	+	0.723993i	$0.257695\pi$
$62$	1.00000	0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.73730	−0.339520
$66$	0	0
$67$	9.54988	1.16670	0.583352	−	0.812220i	$-0.301741\pi$
$67$	9.54988	1.16670	0.583352	+	0.812220i	$0.301741\pi$
$68$	0.483701	0.0586574
$69$	0	0
$70$	−1.81471	−0.216899
$71$	−5.65989	−0.671705	−0.335853	−	0.941915i	$-0.609024\pi$
$71$	−5.65989	−0.671705	−0.335853	+	0.941915i	$0.609024\pi$
$72$	0	0
$73$	8.70683	1.01906	0.509529	−	0.860454i	$-0.329820\pi$
$73$	8.70683	1.01906	0.509529	+	0.860454i	$0.329820\pi$
$74$	−3.70470	−0.430663
$75$	0	0
$76$	2.92259	0.335244
$77$	−6.92259	−0.788902
$78$	0	0
$79$	−9.36459	−1.05360	−0.526799	−	0.849990i	$-0.676608\pi$
$79$	−9.36459	−1.05360	−0.526799	+	0.849990i	$0.676608\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	9.47460	1.04629
$83$	2.81258	0.308721	0.154360	−	0.988015i	$-0.450668\pi$
$83$	2.81258	0.308721	0.154360	+	0.988015i	$0.450668\pi$
$84$	0	0
$85$	0.483701	0.0524647
$86$	11.5194	1.24217
$87$	0	0
$88$	3.81471	0.406649
$89$	−12.4115	−1.31562	−0.657810	−	0.753184i	$-0.728517\pi$
$89$	−12.4115	−1.31562	−0.657810	+	0.753184i	$0.728517\pi$
$90$	0	0
$91$	4.96740	0.520725
$92$	−1.40629	−0.146616
$93$	0	0
$94$	−10.4420	−1.07701
$95$	2.92259	0.299851
$96$	0	0
$97$	0.154821	0.0157197	0.00785987	−	0.999969i	$-0.497498\pi$
$97$	0.154821	0.0157197	0.00785987	+	0.999969i	$0.497498\pi$
$98$	−3.70683	−0.374446
$99$	0	0
$100$	1.00000	0.100000
$101$	−1.88999	−0.188061	−0.0940306	−	0.995569i	$-0.529975\pi$
$101$	−1.88999	−0.188061	−0.0940306	+	0.995569i	$0.529975\pi$
$102$	0	0
$103$	−5.33412	−0.525586	−0.262793	−	0.964852i	$-0.584644\pi$
$103$	−5.33412	−0.525586	−0.262793	+	0.964852i	$0.584644\pi$
$104$	−2.73730	−0.268414
$105$	0	0
$106$	−8.55201	−0.830644
$107$	3.51941	0.340234	0.170117	−	0.985424i	$-0.445585\pi$
$107$	3.51941	0.340234	0.170117	+	0.985424i	$0.445585\pi$
$108$	0	0
$109$	20.0714	1.92249	0.961247	−	0.275690i	$-0.0889063\pi$
$109$	20.0714	1.92249	0.961247	+	0.275690i	$0.0889063\pi$
$110$	3.81471	0.363718
$111$	0	0
$112$	−1.81471	−0.171474
$113$	3.73517	0.351375	0.175688	−	0.984446i	$-0.443785\pi$
$113$	3.73517	0.351375	0.175688	+	0.984446i	$0.443785\pi$
$114$	0	0
$115$	−1.40629	−0.131137
$116$	9.47460	0.879694
$117$	0	0
$118$	13.3341	1.22751
$119$	−0.877777	−0.0804657
$120$	0	0
$121$	3.55201	0.322910
$122$	10.7751	0.975535
$123$	0	0
$124$	1.00000	0.0898027
$125$	1.00000	0.0894427
$126$	0	0
$127$	−12.2119	−1.08363	−0.541815	−	0.840498i	$-0.682263\pi$
$127$	−12.2119	−1.08363	−0.541815	+	0.840498i	$0.682263\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−2.73730	−0.240077
$131$	9.55414	0.834749	0.417374	−	0.908735i	$-0.362950\pi$
$131$	9.55414	0.834749	0.417374	+	0.908735i	$0.362950\pi$
$132$	0	0
$133$	−5.30365	−0.459885
$134$	9.54988	0.824984
$135$	0	0
$136$	0.483701	0.0414770
$137$	−12.9300	−1.10468	−0.552340	−	0.833619i	$-0.686265\pi$
$137$	−12.9300	−1.10468	−0.552340	+	0.833619i	$0.686265\pi$
$138$	0	0
$139$	10.1131	0.857784	0.428892	−	0.903356i	$-0.358904\pi$
$139$	10.1131	0.857784	0.428892	+	0.903356i	$0.358904\pi$
$140$	−1.81471	−0.153371
$141$	0	0
$142$	−5.65989	−0.474967
$143$	−10.4420	−0.873204
$144$	0	0
$145$	9.47460	0.786822
$146$	8.70683	0.720582
$147$	0	0
$148$	−3.70470	−0.304525
$149$	−18.9940	−1.55605	−0.778025	−	0.628233i	$-0.783778\pi$
$149$	−18.9940	−1.55605	−0.778025	+	0.628233i	$0.783778\pi$
$150$	0	0
$151$	−4.96740	−0.404241	−0.202121	−	0.979361i	$-0.564783\pi$
$151$	−4.96740	−0.404241	−0.202121	+	0.979361i	$0.564783\pi$
$152$	2.92259	0.237053
$153$	0	0
$154$	−6.92259	−0.557838
$155$	1.00000	0.0803219
$156$	0	0
$157$	5.73517	0.457716	0.228858	−	0.973460i	$-0.426501\pi$
$157$	5.73517	0.457716	0.228858	+	0.973460i	$0.426501\pi$
$158$	−9.36459	−0.745007
$159$	0	0
$160$	1.00000	0.0790569
$161$	2.55201	0.201126
$162$	0	0
$163$	−1.17930	−0.0923698	−0.0461849	−	0.998933i	$-0.514706\pi$
$163$	−1.17930	−0.0923698	−0.0461849	+	0.998933i	$0.514706\pi$
$164$	9.47460	0.739842
$165$	0	0
$166$	2.81258	0.218299
$167$	21.7691	1.68455	0.842273	−	0.539050i	$-0.181217\pi$
$167$	21.7691	1.68455	0.842273	+	0.539050i	$0.181217\pi$
$168$	0	0
$169$	−5.50720	−0.423630
$170$	0.483701	0.0370982
$171$	0	0
$172$	11.5194	0.878347
$173$	−18.2872	−1.39035	−0.695174	−	0.718841i	$-0.744673\pi$
$173$	−18.2872	−1.39035	−0.695174	+	0.718841i	$0.744673\pi$
$174$	0	0
$175$	−1.81471	−0.137179
$176$	3.81471	0.287545
$177$	0	0
$178$	−12.4115	−0.930284
$179$	−5.92046	−0.442516	−0.221258	−	0.975215i	$-0.571016\pi$
$179$	−5.92046	−0.442516	−0.221258	+	0.975215i	$0.571016\pi$
$180$	0	0
$181$	12.8809	0.957429	0.478714	−	0.877971i	$-0.341103\pi$
$181$	12.8809	0.957429	0.478714	+	0.877971i	$0.341103\pi$
$182$	4.96740	0.368208
$183$	0	0
$184$	−1.40629	−0.103673
$185$	−3.70470	−0.272375
$186$	0	0
$187$	1.84518	0.134933
$188$	−10.4420	−0.761561
$189$	0	0
$190$	2.92259	0.212027
$191$	−3.55414	−0.257168	−0.128584	−	0.991699i	$-0.541043\pi$
$191$	−3.55414	−0.257168	−0.128584	+	0.991699i	$0.541043\pi$
$192$	0	0
$193$	17.2588	1.24232	0.621159	−	0.783684i	$-0.286662\pi$
$193$	17.2588	1.24232	0.621159	+	0.783684i	$0.286662\pi$
$194$	0.154821	0.0111155
$195$	0	0
$196$	−3.70683	−0.264774
$197$	−8.52540	−0.607410	−0.303705	−	0.952766i	$-0.598224\pi$
$197$	−8.52540	−0.607410	−0.303705	+	0.952766i	$0.598224\pi$
$198$	0	0
$199$	20.0266	1.41965	0.709824	−	0.704379i	$-0.248774\pi$
$199$	20.0266	1.41965	0.709824	+	0.704379i	$0.248774\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−1.88999	−0.132979
$203$	−17.1936	−1.20676
$204$	0	0
$205$	9.47460	0.661735
$206$	−5.33412	−0.371646
$207$	0	0
$208$	−2.73730	−0.189798
$209$	11.1488	0.771181
$210$	0	0
$211$	−26.4686	−1.82217	−0.911087	−	0.412214i	$-0.864756\pi$
$211$	−26.4686	−1.82217	−0.911087	+	0.412214i	$0.864756\pi$
$212$	−8.55201	−0.587354
$213$	0	0
$214$	3.51941	0.240582
$215$	11.5194	0.785617
$216$	0	0
$217$	−1.81471	−0.123191
$218$	20.0714	1.35941
$219$	0	0
$220$	3.81471	0.257188
$221$	−1.32403	−0.0890642
$222$	0	0
$223$	24.3625	1.63143	0.815715	−	0.578453i	$-0.196344\pi$
$223$	24.3625	1.63143	0.815715	+	0.578453i	$0.196344\pi$
$224$	−1.81471	−0.121250
$225$	0	0
$226$	3.73517	0.248460
$227$	−13.6742	−0.907591	−0.453795	−	0.891106i	$-0.649930\pi$
$227$	−13.6742	−0.907591	−0.453795	+	0.891106i	$0.649930\pi$
$228$	0	0
$229$	−8.97051	−0.592788	−0.296394	−	0.955066i	$-0.595784\pi$
$229$	−8.97051	−0.592788	−0.296394	+	0.955066i	$0.595784\pi$
$230$	−1.40629	−0.0927280
$231$	0	0
$232$	9.47460	0.622038
$233$	25.0654	1.64209	0.821045	−	0.570863i	$-0.193391\pi$
$233$	25.0654	1.64209	0.821045	+	0.570863i	$0.193391\pi$
$234$	0	0
$235$	−10.4420	−0.681161
$236$	13.3341	0.867977
$237$	0	0
$238$	−0.877777	−0.0568978
$239$	−12.4463	−0.805081	−0.402541	−	0.915402i	$-0.631873\pi$
$239$	−12.4463	−0.805081	−0.402541	+	0.915402i	$0.631873\pi$
$240$	0	0
$241$	−9.03260	−0.581841	−0.290920	−	0.956747i	$-0.593961\pi$
$241$	−9.03260	−0.581841	−0.290920	+	0.956747i	$0.593961\pi$
$242$	3.55201	0.228332
$243$	0	0
$244$	10.7751	0.689807
$245$	−3.70683	−0.236821
$246$	0	0
$247$	−8.00000	−0.509028
$248$	1.00000	0.0635001
$249$	0	0
$250$	1.00000	0.0632456
$251$	5.92046	0.373696	0.186848	−	0.982389i	$-0.440173\pi$
$251$	5.92046	0.373696	0.186848	+	0.982389i	$0.440173\pi$
$252$	0	0
$253$	−5.36459	−0.337269
$254$	−12.2119	−0.766243
$255$	0	0
$256$	1.00000	0.0625000
$257$	−31.8718	−1.98811	−0.994054	−	0.108892i	$-0.965270\pi$
$257$	−31.8718	−1.98811	−0.994054	+	0.108892i	$0.965270\pi$
$258$	0	0
$259$	6.72296	0.417744
$260$	−2.73730	−0.169760
$261$	0	0
$262$	9.55414	0.590257
$263$	−1.82594	−0.112592	−0.0562962	−	0.998414i	$-0.517929\pi$
$263$	−1.82594	−0.112592	−0.0562962	+	0.998414i	$0.517929\pi$
$264$	0	0
$265$	−8.55201	−0.525346
$266$	−5.30365	−0.325188
$267$	0	0
$268$	9.54988	0.583352
$269$	−13.8452	−0.844155	−0.422078	−	0.906560i	$-0.638699\pi$
$269$	−13.8452	−0.844155	−0.422078	+	0.906560i	$0.638699\pi$
$270$	0	0
$271$	−25.5012	−1.54909	−0.774544	−	0.632520i	$-0.782021\pi$
$271$	−25.5012	−1.54909	−0.774544	+	0.632520i	$0.782021\pi$
$272$	0.483701	0.0293287
$273$	0	0
$274$	−12.9300	−0.781127
$275$	3.81471	0.230036
$276$	0	0
$277$	−17.9961	−1.08128	−0.540642	−	0.841253i	$-0.681818\pi$
$277$	−17.9961	−1.08128	−0.540642	+	0.841253i	$0.681818\pi$
$278$	10.1131	0.606545
$279$	0	0
$280$	−1.81471	−0.108450
$281$	−18.9492	−1.13041	−0.565207	−	0.824949i	$-0.691204\pi$
$281$	−18.9492	−1.13041	−0.565207	+	0.824949i	$0.691204\pi$
$282$	0	0
$283$	19.7047	1.17132	0.585661	−	0.810556i	$-0.300835\pi$
$283$	19.7047	1.17132	0.585661	+	0.810556i	$0.300835\pi$
$284$	−5.65989	−0.335853
$285$	0	0
$286$	−10.4420	−0.617448
$287$	−17.1936	−1.01491
$288$	0	0
$289$	−16.7660	−0.986237
$290$	9.47460	0.556368
$291$	0	0
$292$	8.70683	0.509529
$293$	−12.6620	−0.739723	−0.369861	−	0.929087i	$-0.620595\pi$
$293$	−12.6620	−0.739723	−0.369861	+	0.929087i	$0.620595\pi$
$294$	0	0
$295$	13.3341	0.776342
$296$	−3.70470	−0.215331
$297$	0	0
$298$	−18.9940	−1.10029
$299$	3.84944	0.222619
$300$	0	0
$301$	−20.9044	−1.20491
$302$	−4.96740	−0.285842
$303$	0	0
$304$	2.92259	0.167622
$305$	10.7751	0.616983
$306$	0	0
$307$	14.8130	0.845421	0.422711	−	0.906265i	$-0.361079\pi$
$307$	14.8130	0.845421	0.422711	+	0.906265i	$0.361079\pi$
$308$	−6.92259	−0.394451
$309$	0	0
$310$	1.00000	0.0567962
$311$	25.0287	1.41925	0.709625	−	0.704580i	$-0.248865\pi$
$311$	25.0287	1.41925	0.709625	+	0.704580i	$0.248865\pi$
$312$	0	0
$313$	−20.1366	−1.13819	−0.569094	−	0.822272i	$-0.692706\pi$
$313$	−20.1366	−1.13819	−0.569094	+	0.822272i	$0.692706\pi$
$314$	5.73517	0.323654
$315$	0	0
$316$	−9.36459	−0.526799
$317$	−21.2588	−1.19402	−0.597008	−	0.802236i	$-0.703644\pi$
$317$	−21.2588	−1.19402	−0.597008	+	0.802236i	$0.703644\pi$
$318$	0	0
$319$	36.1428	2.02361
$320$	1.00000	0.0559017
$321$	0	0
$322$	2.55201	0.142218
$323$	1.41366	0.0786581
$324$	0	0
$325$	−2.73730	−0.151838
$326$	−1.17930	−0.0653153
$327$	0	0
$328$	9.47460	0.523147
$329$	18.9492	1.04470
$330$	0	0
$331$	16.0949	0.884656	0.442328	−	0.896853i	$-0.354153\pi$
$331$	16.0949	0.884656	0.442328	+	0.896853i	$0.354153\pi$
$332$	2.81258	0.154360
$333$	0	0
$334$	21.7691	1.19115
$335$	9.54988	0.521766
$336$	0	0
$337$	2.07142	0.112837	0.0564187	−	0.998407i	$-0.482032\pi$
$337$	2.07142	0.112837	0.0564187	+	0.998407i	$0.482032\pi$
$338$	−5.50720	−0.299552
$339$	0	0
$340$	0.483701	0.0262324
$341$	3.81471	0.206578
$342$	0	0
$343$	19.4298	1.04911
$344$	11.5194	0.621085
$345$	0	0
$346$	−18.2872	−0.983125
$347$	4.22624	0.226876	0.113438	−	0.993545i	$-0.463814\pi$
$347$	4.22624	0.226876	0.113438	+	0.993545i	$0.463814\pi$
$348$	0	0
$349$	13.5685	0.726304	0.363152	−	0.931730i	$-0.381701\pi$
$349$	13.5685	0.726304	0.363152	+	0.931730i	$0.381701\pi$
$350$	−1.81471	−0.0970003
$351$	0	0
$352$	3.81471	0.203325
$353$	7.59198	0.404080	0.202040	−	0.979377i	$-0.435243\pi$
$353$	7.59198	0.404080	0.202040	+	0.979377i	$0.435243\pi$
$354$	0	0
$355$	−5.65989	−0.300396
$356$	−12.4115	−0.657810
$357$	0	0
$358$	−5.92046	−0.312906
$359$	−26.7030	−1.40933	−0.704664	−	0.709541i	$-0.748902\pi$
$359$	−26.7030	−1.40933	−0.704664	+	0.709541i	$0.748902\pi$
$360$	0	0
$361$	−10.4585	−0.550446
$362$	12.8809	0.677004
$363$	0	0
$364$	4.96740	0.260363
$365$	8.70683	0.455736
$366$	0	0
$367$	−29.8371	−1.55748	−0.778741	−	0.627346i	$-0.784141\pi$
$367$	−29.8371	−1.55748	−0.778741	+	0.627346i	$0.784141\pi$
$368$	−1.40629	−0.0733079
$369$	0	0
$370$	−3.70470	−0.192598
$371$	15.5194	0.805728
$372$	0	0
$373$	30.4686	1.57760	0.788802	−	0.614647i	$-0.210702\pi$
$373$	30.4686	1.57760	0.788802	+	0.614647i	$0.210702\pi$
$374$	1.84518	0.0954119
$375$	0	0
$376$	−10.4420	−0.538505
$377$	−25.9348	−1.33571
$378$	0	0
$379$	6.86165	0.352459	0.176230	−	0.984349i	$-0.443610\pi$
$379$	6.86165	0.352459	0.176230	+	0.984349i	$0.443610\pi$
$380$	2.92259	0.149926
$381$	0	0
$382$	−3.55414	−0.181845
$383$	−9.20666	−0.470438	−0.235219	−	0.971942i	$-0.575581\pi$
$383$	−9.20666	−0.470438	−0.235219	+	0.971942i	$0.575581\pi$
$384$	0	0
$385$	−6.92259	−0.352808
$386$	17.2588	0.878452
$387$	0	0
$388$	0.154821	0.00785987
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	−0.680224	−0.0344004
$392$	−3.70683	−0.187223
$393$	0	0
$394$	−8.52540	−0.429504
$395$	−9.36459	−0.471184
$396$	0	0
$397$	9.58461	0.481038	0.240519	−	0.970644i	$-0.422682\pi$
$397$	9.58461	0.481038	0.240519	+	0.970644i	$0.422682\pi$
$398$	20.0266	1.00384
$399$	0	0
$400$	1.00000	0.0500000
$401$	−30.7701	−1.53659	−0.768293	−	0.640098i	$-0.778894\pi$
$401$	−30.7701	−1.53659	−0.768293	+	0.640098i	$0.778894\pi$
$402$	0	0
$403$	−2.73730	−0.136355
$404$	−1.88999	−0.0940306
$405$	0	0
$406$	−17.1936	−0.853306
$407$	−14.1324	−0.700515
$408$	0	0
$409$	−35.4669	−1.75372	−0.876862	−	0.480742i	$-0.840367\pi$
$409$	−35.4669	−1.75372	−0.876862	+	0.480742i	$0.840367\pi$
$410$	9.47460	0.467917
$411$	0	0
$412$	−5.33412	−0.262793
$413$	−24.1976	−1.19068
$414$	0	0
$415$	2.81258	0.138064
$416$	−2.73730	−0.134207
$417$	0	0
$418$	11.1488	0.545307
$419$	5.61508	0.274314	0.137157	−	0.990549i	$-0.456203\pi$
$419$	5.61508	0.274314	0.137157	+	0.990549i	$0.456203\pi$
$420$	0	0
$421$	15.9880	0.779208	0.389604	−	0.920982i	$-0.372612\pi$
$421$	15.9880	0.779208	0.389604	+	0.920982i	$0.372612\pi$
$422$	−26.4686	−1.28847
$423$	0	0
$424$	−8.55201	−0.415322
$425$	0.483701	0.0234629
$426$	0	0
$427$	−19.5537	−0.946272
$428$	3.51941	0.170117
$429$	0	0
$430$	11.5194	0.555515
$431$	23.7047	1.14182	0.570908	−	0.821014i	$-0.306591\pi$
$431$	23.7047	1.14182	0.570908	+	0.821014i	$0.306591\pi$
$432$	0	0
$433$	34.6948	1.66733	0.833664	−	0.552272i	$-0.186239\pi$
$433$	34.6948	1.66733	0.833664	+	0.552272i	$0.186239\pi$
$434$	−1.81471	−0.0871088
$435$	0	0
$436$	20.0714	0.961247
$437$	−4.11001	−0.196608
$438$	0	0
$439$	7.31978	0.349354	0.174677	−	0.984626i	$-0.444112\pi$
$439$	7.31978	0.349354	0.174677	+	0.984626i	$0.444112\pi$
$440$	3.81471	0.181859
$441$	0	0
$442$	−1.32403	−0.0629779
$443$	23.8823	1.13468	0.567340	−	0.823483i	$-0.307972\pi$
$443$	23.8823	1.13468	0.567340	+	0.823483i	$0.307972\pi$
$444$	0	0
$445$	−12.4115	−0.588363
$446$	24.3625	1.15360
$447$	0	0
$448$	−1.81471	−0.0857370
$449$	−17.6333	−0.832166	−0.416083	−	0.909327i	$-0.636597\pi$
$449$	−17.6333	−0.832166	−0.416083	+	0.909327i	$0.636597\pi$
$450$	0	0
$451$	36.1428	1.70190
$452$	3.73517	0.175688
$453$	0	0
$454$	−13.6742	−0.641763
$455$	4.96740	0.232875
$456$	0	0
$457$	−28.4463	−1.33066	−0.665330	−	0.746549i	$-0.731709\pi$
$457$	−28.4463	−1.33066	−0.665330	+	0.746549i	$0.731709\pi$
$458$	−8.97051	−0.419165
$459$	0	0
$460$	−1.40629	−0.0655686
$461$	5.62516	0.261990	0.130995	−	0.991383i	$-0.458183\pi$
$461$	5.62516	0.261990	0.130995	+	0.991383i	$0.458183\pi$
$462$	0	0
$463$	−26.8801	−1.24923	−0.624613	−	0.780935i	$-0.714743\pi$
$463$	−26.8801	−1.24923	−0.624613	+	0.780935i	$0.714743\pi$
$464$	9.47460	0.439847
$465$	0	0
$466$	25.0654	1.16113
$467$	33.6174	1.55563	0.777815	−	0.628494i	$-0.216328\pi$
$467$	33.6174	1.55563	0.777815	+	0.628494i	$0.216328\pi$
$468$	0	0
$469$	−17.3303	−0.800237
$470$	−10.4420	−0.481654
$471$	0	0
$472$	13.3341	0.613753
$473$	43.9432	2.02051
$474$	0	0
$475$	2.92259	0.134098
$476$	−0.877777	−0.0402328
$477$	0	0
$478$	−12.4463	−0.569279
$479$	−1.96953	−0.0899902	−0.0449951	−	0.998987i	$-0.514327\pi$
$479$	−1.96953	−0.0899902	−0.0449951	+	0.998987i	$0.514327\pi$
$480$	0	0
$481$	10.1409	0.462384
$482$	−9.03260	−0.411424
$483$	0	0
$484$	3.55201	0.161455
$485$	0.154821	0.00703008
$486$	0	0
$487$	23.1611	1.04953	0.524765	−	0.851247i	$-0.324153\pi$
$487$	23.1611	1.04953	0.524765	+	0.851247i	$0.324153\pi$
$488$	10.7751	0.487768
$489$	0	0
$490$	−3.70683	−0.167457
$491$	22.0305	0.994221	0.497111	−	0.867687i	$-0.334394\pi$
$491$	22.0305	0.994221	0.497111	+	0.867687i	$0.334394\pi$
$492$	0	0
$493$	4.58287	0.206402
$494$	−8.00000	−0.359937
$495$	0	0
$496$	1.00000	0.0449013
$497$	10.2711	0.460720
$498$	0	0
$499$	−20.9257	−0.936763	−0.468382	−	0.883526i	$-0.655163\pi$
$499$	−20.9257	−0.936763	−0.468382	+	0.883526i	$0.655163\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	5.92046	0.264243
$503$	−16.4238	−0.732301	−0.366150	−	0.930556i	$-0.619324\pi$
$503$	−16.4238	−0.732301	−0.366150	+	0.930556i	$0.619324\pi$
$504$	0	0
$505$	−1.88999	−0.0841035
$506$	−5.36459	−0.238485
$507$	0	0
$508$	−12.2119	−0.541815
$509$	−13.4137	−0.594550	−0.297275	−	0.954792i	$-0.596078\pi$
$509$	−13.4137	−0.594550	−0.297275	+	0.954792i	$0.596078\pi$
$510$	0	0
$511$	−15.8004	−0.698967
$512$	1.00000	0.0441942
$513$	0	0
$514$	−31.8718	−1.40580
$515$	−5.33412	−0.235049
$516$	0	0
$517$	−39.8332	−1.75186
$518$	6.72296	0.295390
$519$	0	0
$520$	−2.73730	−0.120038
$521$	−12.7334	−0.557862	−0.278931	−	0.960311i	$-0.589980\pi$
$521$	−12.7334	−0.557862	−0.278931	+	0.960311i	$0.589980\pi$
$522$	0	0
$523$	6.70257	0.293083	0.146541	−	0.989205i	$-0.453186\pi$
$523$	6.70257	0.293083	0.146541	+	0.989205i	$0.453186\pi$
$524$	9.55414	0.417374
$525$	0	0
$526$	−1.82594	−0.0796148
$527$	0.483701	0.0210703
$528$	0	0
$529$	−21.0223	−0.914015
$530$	−8.55201	−0.371475
$531$	0	0
$532$	−5.30365	−0.229942
$533$	−25.9348	−1.12336
$534$	0	0
$535$	3.51941	0.152157
$536$	9.54988	0.412492
$537$	0	0
$538$	−13.8452	−0.596908
$539$	−14.1405	−0.609074
$540$	0	0
$541$	18.1366	0.779754	0.389877	−	0.920867i	$-0.372518\pi$
$541$	18.1366	0.779754	0.389877	+	0.920867i	$0.372518\pi$
$542$	−25.5012	−1.09537
$543$	0	0
$544$	0.483701	0.0207385
$545$	20.0714	0.859765
$546$	0	0
$547$	−2.45012	−0.104760	−0.0523798	−	0.998627i	$-0.516681\pi$
$547$	−2.45012	−0.104760	−0.0523798	+	0.998627i	$0.516681\pi$
$548$	−12.9300	−0.552340
$549$	0	0
$550$	3.81471	0.162660
$551$	27.6904	1.17965
$552$	0	0
$553$	16.9940	0.722659
$554$	−17.9961	−0.764583
$555$	0	0
$556$	10.1131	0.428892
$557$	−21.5194	−0.911807	−0.455903	−	0.890029i	$-0.650684\pi$
$557$	−21.5194	−0.911807	−0.455903	+	0.890029i	$0.650684\pi$
$558$	0	0
$559$	−31.5321	−1.33366
$560$	−1.81471	−0.0766855
$561$	0	0
$562$	−18.9492	−0.799324
$563$	−18.5177	−0.780427	−0.390213	−	0.920724i	$-0.627599\pi$
$563$	−18.5177	−0.780427	−0.390213	+	0.920724i	$0.627599\pi$
$564$	0	0
$565$	3.73517	0.157140
$566$	19.7047	0.828250
$567$	0	0
$568$	−5.65989	−0.237484
$569$	−26.7030	−1.11945	−0.559723	−	0.828680i	$-0.689093\pi$
$569$	−26.7030	−1.11945	−0.559723	+	0.828680i	$0.689093\pi$
$570$	0	0
$571$	−37.5982	−1.57344	−0.786718	−	0.617313i	$-0.788221\pi$
$571$	−37.5982	−1.57344	−0.786718	+	0.617313i	$0.788221\pi$
$572$	−10.4420	−0.436602
$573$	0	0
$574$	−17.1936	−0.717649
$575$	−1.40629	−0.0586464
$576$	0	0
$577$	−21.4789	−0.894176	−0.447088	−	0.894490i	$-0.647539\pi$
$577$	−21.4789	−0.894176	−0.447088	+	0.894490i	$0.647539\pi$
$578$	−16.7660	−0.697375
$579$	0	0
$580$	9.47460	0.393411
$581$	−5.10402	−0.211750
$582$	0	0
$583$	−32.6234	−1.35112
$584$	8.70683	0.360291
$585$	0	0
$586$	−12.6620	−0.523063
$587$	−16.4238	−0.677882	−0.338941	−	0.940808i	$-0.610069\pi$
$587$	−16.4238	−0.677882	−0.338941	+	0.940808i	$0.610069\pi$
$588$	0	0
$589$	2.92259	0.120423
$590$	13.3341	0.548957
$591$	0	0
$592$	−3.70470	−0.152262
$593$	3.09779	0.127211	0.0636056	−	0.997975i	$-0.479740\pi$
$593$	3.09779	0.127211	0.0636056	+	0.997975i	$0.479740\pi$
$594$	0	0
$595$	−0.877777	−0.0359853
$596$	−18.9940	−0.778025
$597$	0	0
$598$	3.84944	0.157415
$599$	−4.97966	−0.203464	−0.101732	−	0.994812i	$-0.532438\pi$
$599$	−4.97966	−0.203464	−0.101732	+	0.994812i	$0.532438\pi$
$600$	0	0
$601$	−43.8514	−1.78874	−0.894368	−	0.447332i	$-0.852374\pi$
$601$	−43.8514	−1.78874	−0.894368	+	0.447332i	$0.852374\pi$
$602$	−20.9044	−0.851999
$603$	0	0
$604$	−4.96740	−0.202121
$605$	3.55201	0.144410
$606$	0	0
$607$	−39.1387	−1.58859	−0.794296	−	0.607531i	$-0.792160\pi$
$607$	−39.1387	−1.58859	−0.794296	+	0.607531i	$0.792160\pi$
$608$	2.92259	0.118527
$609$	0	0
$610$	10.7751	0.436273
$611$	28.5829	1.15634
$612$	0	0
$613$	−29.9065	−1.20791	−0.603956	−	0.797017i	$-0.706410\pi$
$613$	−29.9065	−1.20791	−0.603956	+	0.797017i	$0.706410\pi$
$614$	14.8130	0.597803
$615$	0	0
$616$	−6.92259	−0.278919
$617$	−21.8066	−0.877900	−0.438950	−	0.898511i	$-0.644650\pi$
$617$	−21.8066	−0.877900	−0.438950	+	0.898511i	$0.644650\pi$
$618$	0	0
$619$	−19.5120	−0.784255	−0.392128	−	0.919911i	$-0.628261\pi$
$619$	−19.5120	−0.784255	−0.392128	+	0.919911i	$0.628261\pi$
$620$	1.00000	0.0401610
$621$	0	0
$622$	25.0287	1.00356
$623$	22.5233	0.902378
$624$	0	0
$625$	1.00000	0.0400000
$626$	−20.1366	−0.804821
$627$	0	0
$628$	5.73517	0.228858
$629$	−1.79197	−0.0714504
$630$	0	0
$631$	−5.42553	−0.215987	−0.107993	−	0.994152i	$-0.534443\pi$
$631$	−5.42553	−0.215987	−0.107993	+	0.994152i	$0.534443\pi$
$632$	−9.36459	−0.372503
$633$	0	0
$634$	−21.2588	−0.844296
$635$	−12.2119	−0.484614
$636$	0	0
$637$	10.1467	0.402027
$638$	36.1428	1.43091
$639$	0	0
$640$	1.00000	0.0395285
$641$	8.51532	0.336335	0.168167	−	0.985758i	$-0.446215\pi$
$641$	8.51532	0.336335	0.168167	+	0.985758i	$0.446215\pi$
$642$	0	0
$643$	19.5055	0.769220	0.384610	−	0.923079i	$-0.374336\pi$
$643$	19.5055	0.769220	0.384610	+	0.923079i	$0.374336\pi$
$644$	2.55201	0.100563
$645$	0	0
$646$	1.41366	0.0556197
$647$	−18.0788	−0.710750	−0.355375	−	0.934724i	$-0.615647\pi$
$647$	−18.0788	−0.710750	−0.355375	+	0.934724i	$0.615647\pi$
$648$	0	0
$649$	50.8658	1.99666
$650$	−2.73730	−0.107366
$651$	0	0
$652$	−1.17930	−0.0461849
$653$	−4.44200	−0.173829	−0.0869144	−	0.996216i	$-0.527701\pi$
$653$	−4.44200	−0.173829	−0.0869144	+	0.996216i	$0.527701\pi$
$654$	0	0
$655$	9.55414	0.373311
$656$	9.47460	0.369921
$657$	0	0
$658$	18.9492	0.738717
$659$	5.79433	0.225715	0.112857	−	0.993611i	$-0.464000\pi$
$659$	5.79433	0.225715	0.112857	+	0.993611i	$0.464000\pi$
$660$	0	0
$661$	−9.91037	−0.385469	−0.192734	−	0.981251i	$-0.561736\pi$
$661$	−9.91037	−0.385469	−0.192734	+	0.981251i	$0.561736\pi$
$662$	16.0949	0.625547
$663$	0	0
$664$	2.81258	0.109149
$665$	−5.30365	−0.205667
$666$	0	0
$667$	−13.3240	−0.515909
$668$	21.7691	0.842273
$669$	0	0
$670$	9.54988	0.368944
$671$	41.1040	1.58680
$672$	0	0
$673$	−28.2872	−1.09039	−0.545195	−	0.838309i	$-0.683545\pi$
$673$	−28.2872	−1.09039	−0.545195	+	0.838309i	$0.683545\pi$
$674$	2.07142	0.0797881
$675$	0	0
$676$	−5.50720	−0.211815
$677$	48.2956	1.85615	0.928075	−	0.372394i	$-0.121463\pi$
$677$	48.2956	1.85615	0.928075	+	0.372394i	$0.121463\pi$
$678$	0	0
$679$	−0.280956	−0.0107821
$680$	0.483701	0.0185491
$681$	0	0
$682$	3.81471	0.146073
$683$	−18.2571	−0.698589	−0.349294	−	0.937013i	$-0.613579\pi$
$683$	−18.2571	−0.698589	−0.349294	+	0.937013i	$0.613579\pi$
$684$	0	0
$685$	−12.9300	−0.494028
$686$	19.4298	0.741833
$687$	0	0
$688$	11.5194	0.439173
$689$	23.4094	0.891827
$690$	0	0
$691$	−39.7926	−1.51378	−0.756892	−	0.653540i	$-0.773283\pi$
$691$	−39.7926	−1.51378	−0.756892	+	0.653540i	$0.773283\pi$
$692$	−18.2872	−0.695174
$693$	0	0
$694$	4.22624	0.160426
$695$	10.1131	0.383613
$696$	0	0
$697$	4.58287	0.173589
$698$	13.5685	0.513575
$699$	0	0
$700$	−1.81471	−0.0685896
$701$	37.5117	1.41680	0.708398	−	0.705813i	$-0.249418\pi$
$701$	37.5117	1.41680	0.708398	+	0.705813i	$0.249418\pi$
$702$	0	0
$703$	−10.8273	−0.408360
$704$	3.81471	0.143772
$705$	0	0
$706$	7.59198	0.285728
$707$	3.42978	0.128990
$708$	0	0
$709$	1.16807	0.0438676	0.0219338	−	0.999759i	$-0.493018\pi$
$709$	1.16807	0.0438676	0.0219338	+	0.999759i	$0.493018\pi$
$710$	−5.65989	−0.212412
$711$	0	0
$712$	−12.4115	−0.465142
$713$	−1.40629	−0.0526660
$714$	0	0
$715$	−10.4420	−0.390509
$716$	−5.92046	−0.221258
$717$	0	0
$718$	−26.7030	−0.996546
$719$	26.8658	1.00192	0.500962	−	0.865469i	$-0.332979\pi$
$719$	26.8658	1.00192	0.500962	+	0.865469i	$0.332979\pi$
$720$	0	0
$721$	9.67988	0.360497
$722$	−10.4585	−0.389224
$723$	0	0
$724$	12.8809	0.478714
$725$	9.47460	0.351878
$726$	0	0
$727$	−15.9086	−0.590017	−0.295009	−	0.955495i	$-0.595322\pi$
$727$	−15.9086	−0.590017	−0.295009	+	0.955495i	$0.595322\pi$
$728$	4.96740	0.184104
$729$	0	0
$730$	8.70683	0.322254
$731$	5.57195	0.206086
$732$	0	0
$733$	−3.31978	−0.122619	−0.0613094	−	0.998119i	$-0.519528\pi$
$733$	−3.31978	−0.122619	−0.0613094	+	0.998119i	$0.519528\pi$
$734$	−29.8371	−1.10131
$735$	0	0
$736$	−1.40629	−0.0518365
$737$	36.4300	1.34192
$738$	0	0
$739$	31.8749	1.17254	0.586268	−	0.810117i	$-0.300596\pi$
$739$	31.8749	1.17254	0.586268	+	0.810117i	$0.300596\pi$
$740$	−3.70470	−0.136188
$741$	0	0
$742$	15.5194	0.569735
$743$	−8.80174	−0.322905	−0.161452	−	0.986881i	$-0.551618\pi$
$743$	−8.80174	−0.322905	−0.161452	+	0.986881i	$0.551618\pi$
$744$	0	0
$745$	−18.9940	−0.695887
$746$	30.4686	1.11553
$747$	0	0
$748$	1.84518	0.0674664
$749$	−6.38671	−0.233365
$750$	0	0
$751$	10.6682	0.389290	0.194645	−	0.980874i	$-0.437645\pi$
$751$	10.6682	0.389290	0.194645	+	0.980874i	$0.437645\pi$
$752$	−10.4420	−0.380781
$753$	0	0
$754$	−25.9348	−0.944490
$755$	−4.96740	−0.180782
$756$	0	0
$757$	−31.2507	−1.13583	−0.567913	−	0.823088i	$-0.692249\pi$
$757$	−31.2507	−1.13583	−0.567913	+	0.823088i	$0.692249\pi$
$758$	6.86165	0.249226
$759$	0	0
$760$	2.92259	0.106013
$761$	47.1407	1.70885	0.854425	−	0.519575i	$-0.173910\pi$
$761$	47.1407	1.70885	0.854425	+	0.519575i	$0.173910\pi$
$762$	0	0
$763$	−36.4238	−1.31863
$764$	−3.55414	−0.128584
$765$	0	0
$766$	−9.20666	−0.332650
$767$	−36.4995	−1.31792
$768$	0	0
$769$	44.5295	1.60578	0.802888	−	0.596130i	$-0.203296\pi$
$769$	44.5295	1.60578	0.802888	+	0.596130i	$0.203296\pi$
$770$	−6.92259	−0.249473
$771$	0	0
$772$	17.2588	0.621159
$773$	47.5971	1.71195	0.855973	−	0.517020i	$-0.172959\pi$
$773$	47.5971	1.71195	0.855973	+	0.517020i	$0.172959\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0.154821	0.00555776
$777$	0	0
$778$	−12.0000	−0.430221
$779$	27.6904	0.992110
$780$	0	0
$781$	−21.5908	−0.772581
$782$	−0.680224	−0.0243248
$783$	0	0
$784$	−3.70683	−0.132387
$785$	5.73517	0.204697
$786$	0	0
$787$	−27.7394	−0.988804	−0.494402	−	0.869233i	$-0.664613\pi$
$787$	−27.7394	−0.988804	−0.494402	+	0.869233i	$0.664613\pi$
$788$	−8.52540	−0.303705
$789$	0	0
$790$	−9.36459	−0.333177
$791$	−6.77825	−0.241007
$792$	0	0
$793$	−29.4948	−1.04739
$794$	9.58461	0.340145
$795$	0	0
$796$	20.0266	0.709824
$797$	−17.8409	−0.631958	−0.315979	−	0.948766i	$-0.602333\pi$
$797$	−17.8409	−0.631958	−0.315979	+	0.948766i	$0.602333\pi$
$798$	0	0
$799$	−5.05081	−0.178685
$800$	1.00000	0.0353553
$801$	0	0
$802$	−30.7701	−1.08653
$803$	33.2140	1.17210
$804$	0	0
$805$	2.55201	0.0899465
$806$	−2.73730	−0.0964172
$807$	0	0
$808$	−1.88999	−0.0664897
$809$	−35.0511	−1.23233	−0.616165	−	0.787617i	$-0.711315\pi$
$809$	−35.0511	−1.23233	−0.616165	+	0.787617i	$0.711315\pi$
$810$	0	0
$811$	41.5908	1.46045	0.730226	−	0.683206i	$-0.239415\pi$
$811$	41.5908	1.46045	0.730226	+	0.683206i	$0.239415\pi$
$812$	−17.1936	−0.603379
$813$	0	0
$814$	−14.1324	−0.495339
$815$	−1.17930	−0.0413090
$816$	0	0
$817$	33.6665	1.17784
$818$	−35.4669	−1.24007
$819$	0	0
$820$	9.47460	0.330867
$821$	5.10402	0.178131	0.0890657	−	0.996026i	$-0.471612\pi$
$821$	5.10402	0.178131	0.0890657	+	0.996026i	$0.471612\pi$
$822$	0	0
$823$	31.3811	1.09388	0.546938	−	0.837173i	$-0.315793\pi$
$823$	31.3811	1.09388	0.546938	+	0.837173i	$0.315793\pi$
$824$	−5.33412	−0.185823
$825$	0	0
$826$	−24.1976	−0.841941
$827$	−11.3198	−0.393627	−0.196814	−	0.980441i	$-0.563059\pi$
$827$	−11.3198	−0.393627	−0.196814	+	0.980441i	$0.563059\pi$
$828$	0	0
$829$	−14.7548	−0.512454	−0.256227	−	0.966617i	$-0.582479\pi$
$829$	−14.7548	−0.512454	−0.256227	+	0.966617i	$0.582479\pi$
$830$	2.81258	0.0976261
$831$	0	0
$832$	−2.73730	−0.0948988
$833$	−1.79300	−0.0621237
$834$	0	0
$835$	21.7691	0.753352
$836$	11.1488	0.385590
$837$	0	0
$838$	5.61508	0.193970
$839$	−53.2121	−1.83709	−0.918543	−	0.395320i	$-0.870634\pi$
$839$	−53.2121	−1.83709	−0.918543	+	0.395320i	$0.870634\pi$
$840$	0	0
$841$	60.7680	2.09545
$842$	15.9880	0.550983
$843$	0	0
$844$	−26.4686	−0.911087
$845$	−5.50720	−0.189453
$846$	0	0
$847$	−6.44586	−0.221482
$848$	−8.55201	−0.293677
$849$	0	0
$850$	0.483701	0.0165908
$851$	5.20988	0.178593
$852$	0	0
$853$	3.97962	0.136259	0.0681297	−	0.997676i	$-0.478297\pi$
$853$	3.97962	0.136259	0.0681297	+	0.997676i	$0.478297\pi$
$854$	−19.5537	−0.669115
$855$	0	0
$856$	3.51941	0.120291
$857$	−1.38071	−0.0471643	−0.0235822	−	0.999722i	$-0.507507\pi$
$857$	−1.38071	−0.0471643	−0.0235822	+	0.999722i	$0.507507\pi$
$858$	0	0
$859$	48.0262	1.63863	0.819317	−	0.573340i	$-0.194353\pi$
$859$	48.0262	1.63863	0.819317	+	0.573340i	$0.194353\pi$
$860$	11.5194	0.392809
$861$	0	0
$862$	23.7047	0.807385
$863$	40.6651	1.38426	0.692129	−	0.721774i	$-0.256673\pi$
$863$	40.6651	1.38426	0.692129	+	0.721774i	$0.256673\pi$
$864$	0	0
$865$	−18.2872	−0.621783
$866$	34.6948	1.17898
$867$	0	0
$868$	−1.81471	−0.0615953
$869$	−35.7232	−1.21183
$870$	0	0
$871$	−26.1409	−0.885750
$872$	20.0714	0.679704
$873$	0	0
$874$	−4.11001	−0.139023
$875$	−1.81471	−0.0613484
$876$	0	0
$877$	36.0819	1.21840	0.609200	−	0.793017i	$-0.291491\pi$
$877$	36.0819	1.21840	0.609200	+	0.793017i	$0.291491\pi$
$878$	7.31978	0.247030
$879$	0	0
$880$	3.81471	0.128594
$881$	−16.3772	−0.551762	−0.275881	−	0.961192i	$-0.588970\pi$
$881$	−16.3772	−0.551762	−0.275881	+	0.961192i	$0.588970\pi$
$882$	0	0
$883$	20.2753	0.682319	0.341159	−	0.940006i	$-0.389180\pi$
$883$	20.2753	0.682319	0.341159	+	0.940006i	$0.389180\pi$
$884$	−1.32403	−0.0445321
$885$	0	0
$886$	23.8823	0.802340
$887$	56.7397	1.90513	0.952566	−	0.304333i	$-0.0984336\pi$
$887$	56.7397	1.90513	0.952566	+	0.304333i	$0.0984336\pi$
$888$	0	0
$889$	22.1610	0.743258
$890$	−12.4115	−0.416035
$891$	0	0
$892$	24.3625	0.815715
$893$	−30.5177	−1.02124
$894$	0	0
$895$	−5.92046	−0.197899
$896$	−1.81471	−0.0606252
$897$	0	0
$898$	−17.6333	−0.588430
$899$	9.47460	0.315996
$900$	0	0
$901$	−4.13661	−0.137811
$902$	36.1428	1.20342
$903$	0	0
$904$	3.73517	0.124230
$905$	12.8809	0.428175
$906$	0	0
$907$	28.6582	0.951578	0.475789	−	0.879559i	$-0.342163\pi$
$907$	28.6582	0.951578	0.475789	+	0.879559i	$0.342163\pi$
$908$	−13.6742	−0.453795
$909$	0	0
$910$	4.96740	0.164668
$911$	−40.3543	−1.33700	−0.668499	−	0.743713i	$-0.733063\pi$
$911$	−40.3543	−1.33700	−0.668499	+	0.743713i	$0.733063\pi$
$912$	0	0
$913$	10.7292	0.355084
$914$	−28.4463	−0.940919
$915$	0	0
$916$	−8.97051	−0.296394
$917$	−17.3380	−0.572551
$918$	0	0
$919$	−1.31552	−0.0433949	−0.0216975	−	0.999765i	$-0.506907\pi$
$919$	−1.31552	−0.0433949	−0.0216975	+	0.999765i	$0.506907\pi$
$920$	−1.40629	−0.0463640
$921$	0	0
$922$	5.62516	0.185255
$923$	15.4928	0.509952
$924$	0	0
$925$	−3.70470	−0.121810
$926$	−26.8801	−0.883336
$927$	0	0
$928$	9.47460	0.311019
$929$	24.4500	0.802179	0.401089	−	0.916039i	$-0.368632\pi$
$929$	24.4500	0.802179	0.401089	+	0.916039i	$0.368632\pi$
$930$	0	0
$931$	−10.8335	−0.355055
$932$	25.0654	0.821045
$933$	0	0
$934$	33.6174	1.10000
$935$	1.84518	0.0603438
$936$	0	0
$937$	10.3096	0.336801	0.168401	−	0.985719i	$-0.446140\pi$
$937$	10.3096	0.336801	0.168401	+	0.985719i	$0.446140\pi$
$938$	−17.3303	−0.565853
$939$	0	0
$940$	−10.4420	−0.340580
$941$	−13.4137	−0.437273	−0.218636	−	0.975806i	$-0.570161\pi$
$941$	−13.4137	−0.437273	−0.218636	+	0.975806i	$0.570161\pi$
$942$	0	0
$943$	−13.3240	−0.433890
$944$	13.3341	0.433989
$945$	0	0
$946$	43.9432	1.42872
$947$	−9.92085	−0.322384	−0.161192	−	0.986923i	$-0.551534\pi$
$947$	−9.92085	−0.322384	−0.161192	+	0.986923i	$0.551534\pi$
$948$	0	0
$949$	−23.8332	−0.773658
$950$	2.92259	0.0948213
$951$	0	0
$952$	−0.877777	−0.0284489
$953$	14.5264	0.470557	0.235279	−	0.971928i	$-0.424400\pi$
$953$	14.5264	0.470557	0.235279	+	0.971928i	$0.424400\pi$
$954$	0	0
$955$	−3.55414	−0.115009
$956$	−12.4463	−0.402541
$957$	0	0
$958$	−1.96953	−0.0636327
$959$	23.4641	0.757696
$960$	0	0
$961$	1.00000	0.0322581
$962$	10.1409	0.326955
$963$	0	0
$964$	−9.03260	−0.290920
$965$	17.2588	0.555582
$966$	0	0
$967$	−29.4952	−0.948501	−0.474250	−	0.880390i	$-0.657281\pi$
$967$	−29.4952	−0.948501	−0.474250	+	0.880390i	$0.657281\pi$
$968$	3.55201	0.114166
$969$	0	0
$970$	0.154821	0.00497102
$971$	17.8638	0.573276	0.286638	−	0.958039i	$-0.407462\pi$
$971$	17.8638	0.573276	0.286638	+	0.958039i	$0.407462\pi$
$972$	0	0
$973$	−18.3524	−0.588350
$974$	23.1611	0.742129
$975$	0	0
$976$	10.7751	0.344904
$977$	−20.2914	−0.649181	−0.324590	−	0.945855i	$-0.605226\pi$
$977$	−20.2914	−0.649181	−0.324590	+	0.945855i	$0.605226\pi$
$978$	0	0
$979$	−47.3464	−1.51320
$980$	−3.70683	−0.118410
$981$	0	0
$982$	22.0305	0.703021
$983$	−23.0666	−0.735709	−0.367855	−	0.929883i	$-0.619908\pi$
$983$	−23.0666	−0.735709	−0.367855	+	0.929883i	$0.619908\pi$
$984$	0	0
$985$	−8.52540	−0.271642
$986$	4.58287	0.145948
$987$	0	0
$988$	−8.00000	−0.254514
$989$	−16.1996	−0.515118
$990$	0	0
$991$	3.95945	0.125776	0.0628880	−	0.998021i	$-0.479969\pi$
$991$	3.95945	0.125776	0.0628880	+	0.998021i	$0.479969\pi$
$992$	1.00000	0.0317500
$993$	0	0
$994$	10.2711	0.325778
$995$	20.0266	0.634886
$996$	0	0
$997$	−5.25458	−0.166414	−0.0832071	−	0.996532i	$-0.526516\pi$
$997$	−5.25458	−0.166414	−0.0832071	+	0.996532i	$0.526516\pi$
$998$	−20.9257	−0.662391
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2790.2.a.bk.1.1	yes	4
3.2	odd	2		2790.2.a.bj.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2790.2.a.bj.1.1	✓	4	3.2	odd	2
2790.2.a.bk.1.1	yes	4	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:	\( N \)	\(=\)	\( 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2790.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.81471 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.81471 q^{7} +1.00000 q^{8} +1.00000 q^{10} +3.81471 q^{11} -2.73730 q^{13} -1.81471 q^{14} +1.00000 q^{16} +0.483701 q^{17} +2.92259 q^{19} +1.00000 q^{20} +3.81471 q^{22} -1.40629 q^{23} +1.00000 q^{25} -2.73730 q^{26} -1.81471 q^{28} +9.47460 q^{29} +1.00000 q^{31} +1.00000 q^{32} +0.483701 q^{34} -1.81471 q^{35} -3.70470 q^{37} +2.92259 q^{38} +1.00000 q^{40} +9.47460 q^{41} +11.5194 q^{43} +3.81471 q^{44} -1.40629 q^{46} -10.4420 q^{47} -3.70683 q^{49} +1.00000 q^{50} -2.73730 q^{52} -8.55201 q^{53} +3.81471 q^{55} -1.81471 q^{56} +9.47460 q^{58} +13.3341 q^{59} +10.7751 q^{61} +1.00000 q^{62} +1.00000 q^{64} -2.73730 q^{65} +9.54988 q^{67} +0.483701 q^{68} -1.81471 q^{70} -5.65989 q^{71} +8.70683 q^{73} -3.70470 q^{74} +2.92259 q^{76} -6.92259 q^{77} -9.36459 q^{79} +1.00000 q^{80} +9.47460 q^{82} +2.81258 q^{83} +0.483701 q^{85} +11.5194 q^{86} +3.81471 q^{88} -12.4115 q^{89} +4.96740 q^{91} -1.40629 q^{92} -10.4420 q^{94} +2.92259 q^{95} +0.154821 q^{97} -3.70683 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 5 q^{7} + 4 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + 5 * q^7 + 4 * q^8	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 5 q^{7} + 4 q^{8} + 4 q^{10} + 3 q^{11} + 6 q^{13} + 5 q^{14} + 4 q^{16} + 7 q^{19} + 4 q^{20} + 3 q^{22} + q^{23} + 4 q^{25} + 6 q^{26} + 5 q^{28} + 4 q^{29} + 4 q^{31} + 4 q^{32} + 5 q^{35} + 6 q^{37} + 7 q^{38} + 4 q^{40} + 4 q^{41} + 13 q^{43} + 3 q^{44} + q^{46} - 4 q^{47} + 5 q^{49} + 4 q^{50} + 6 q^{52} - 5 q^{53} + 3 q^{55} + 5 q^{56} + 4 q^{58} + 8 q^{59} - 4 q^{61} + 4 q^{62} + 4 q^{64} + 6 q^{65} + 8 q^{67} + 5 q^{70} - q^{71} + 15 q^{73} + 6 q^{74} + 7 q^{76} - 23 q^{77} + 5 q^{79} + 4 q^{80} + 4 q^{82} - 2 q^{83} + 13 q^{86} + 3 q^{88} - 9 q^{89} + 16 q^{91} + q^{92} - 4 q^{94} + 7 q^{95} + 10 q^{97} + 5 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + 5 * q^7 + 4 * q^8 + 4 * q^10 + 3 * q^11 + 6 * q^13 + 5 * q^14 + 4 * q^16 + 7 * q^19 + 4 * q^20 + 3 * q^22 + q^23 + 4 * q^25 + 6 * q^26 + 5 * q^28 + 4 * q^29 + 4 * q^31 + 4 * q^32 + 5 * q^35 + 6 * q^37 + 7 * q^38 + 4 * q^40 + 4 * q^41 + 13 * q^43 + 3 * q^44 + q^46 - 4 * q^47 + 5 * q^49 + 4 * q^50 + 6 * q^52 - 5 * q^53 + 3 * q^55 + 5 * q^56 + 4 * q^58 + 8 * q^59 - 4 * q^61 + 4 * q^62 + 4 * q^64 + 6 * q^65 + 8 * q^67 + 5 * q^70 - q^71 + 15 * q^73 + 6 * q^74 + 7 * q^76 - 23 * q^77 + 5 * q^79 + 4 * q^80 + 4 * q^82 - 2 * q^83 + 13 * q^86 + 3 * q^88 - 9 * q^89 + 16 * q^91 + q^92 - 4 * q^94 + 7 * q^95 + 10 * q^97 + 5 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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