LMFDB - Embedded newform 2790.2.a.bj.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	$-0.695033\pi$
$11$	−3.81471	−1.15018	−0.575089	+	0.818091i	$0.695033\pi$
$12$	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	$-0.518682\pi$
$17$	−0.483701	−0.117315	−0.0586574	+	0.998278i	$0.518682\pi$
$18$	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	$-0.453162\pi$
$23$	1.40629	0.293232	0.146616	+	0.989193i	$0.453162\pi$
$24$	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	$-0.842253\pi$
$29$	−9.47460	−1.75939	−0.879694	+	0.475540i	$0.842253\pi$
$30$	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	$-0.765100\pi$
$41$	−9.47460	−1.47968	−0.739842	+	0.672781i	$0.765100\pi$
$42$	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	$-0.224433\pi$
$47$	10.4420	1.52312	0.761561	+	0.648093i	$0.224433\pi$
$48$	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	$-0.300170\pi$
$53$	8.55201	1.17471	0.587354	+	0.809330i	$0.300170\pi$
$54$	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	$-0.834580\pi$
$59$	−13.3341	−1.73595	−0.867977	+	0.496604i	$0.834580\pi$
$60$	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	$-0.390976\pi$
$71$	5.65989	0.671705	0.335853	+	0.941915i	$0.390976\pi$
$72$	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	$-0.549332\pi$
$83$	−2.81258	−0.308721	−0.154360	+	0.988015i	$0.549332\pi$
$84$	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	$-0.271483\pi$
$89$	12.4115	1.31562	0.657810	+	0.753184i	$0.271483\pi$
$90$	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	$-0.470025\pi$
$101$	1.88999	0.188061	0.0940306	+	0.995569i	$0.470025\pi$
$102$	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	$-0.554415\pi$
$107$	−3.51941	−0.340234	−0.170117	+	0.985424i	$0.554415\pi$
$108$	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	$-0.556215\pi$
$113$	−3.73517	−0.351375	−0.175688	+	0.984446i	$0.556215\pi$
$114$	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	$-0.637050\pi$
$131$	−9.55414	−0.834749	−0.417374	+	0.908735i	$0.637050\pi$
$132$	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	$-0.313735\pi$
$137$	12.9300	1.10468	0.552340	+	0.833619i	$0.313735\pi$
$138$	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	$-0.216222\pi$
$149$	18.9940	1.55605	0.778025	+	0.628233i	$0.216222\pi$
$150$	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	$-0.818783\pi$
$167$	−21.7691	−1.68455	−0.842273	+	0.539050i	$0.818783\pi$
$168$	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	$-0.255327\pi$
$173$	18.2872	1.39035	0.695174	+	0.718841i	$0.255327\pi$
$174$	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	$-0.428984\pi$
$179$	5.92046	0.442516	0.221258	+	0.975215i	$0.428984\pi$
$180$	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	$-0.458957\pi$
$191$	3.55414	0.257168	0.128584	+	0.991699i	$0.458957\pi$
$192$	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	$-0.401776\pi$
$197$	8.52540	0.607410	0.303705	+	0.952766i	$0.401776\pi$
$198$	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	$-0.350070\pi$
$227$	13.6742	0.907591	0.453795	+	0.891106i	$0.350070\pi$
$228$	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	$-0.806609\pi$
$233$	−25.0654	−1.64209	−0.821045	+	0.570863i	$0.806609\pi$
$234$	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	$-0.368127\pi$
$239$	12.4463	0.805081	0.402541	+	0.915402i	$0.368127\pi$
$240$	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	$-0.559827\pi$
$251$	−5.92046	−0.373696	−0.186848	+	0.982389i	$0.559827\pi$
$252$	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	$-0.0347301\pi$
$257$	31.8718	1.98811	0.994054	+	0.108892i	$0.0347301\pi$
$258$	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	$-0.482071\pi$
$263$	1.82594	0.112592	0.0562962	+	0.998414i	$0.482071\pi$
$264$	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	$-0.361301\pi$
$269$	13.8452	0.844155	0.422078	+	0.906560i	$0.361301\pi$
$270$	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	$-0.308796\pi$
$281$	18.9492	1.13041	0.565207	+	0.824949i	$0.308796\pi$
$282$	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	$-0.379405\pi$
$293$	12.6620	0.739723	0.369861	+	0.929087i	$0.379405\pi$
$294$	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	$-0.751135\pi$
$311$	−25.0287	−1.41925	−0.709625	+	0.704580i	$0.751135\pi$
$312$	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	$-0.296356\pi$
$317$	21.2588	1.19402	0.597008	+	0.802236i	$0.296356\pi$
$318$	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	$-0.536186\pi$
$347$	−4.22624	−0.226876	−0.113438	+	0.993545i	$0.536186\pi$
$348$	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	$-0.564757\pi$
$353$	−7.59198	−0.404080	−0.202040	+	0.979377i	$0.564757\pi$
$354$	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	$-0.251098\pi$
$359$	26.7030	1.40933	0.704664	+	0.709541i	$0.251098\pi$
$360$	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	$-0.424419\pi$
$383$	9.20666	0.470438	0.235219	+	0.971942i	$0.424419\pi$
$384$	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	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	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	$-0.221106\pi$
$401$	30.7701	1.53659	0.768293	+	0.640098i	$0.221106\pi$
$402$	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	$-0.543797\pi$
$419$	−5.61508	−0.274314	−0.137157	+	0.990549i	$0.543797\pi$
$420$	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	$-0.693409\pi$
$431$	−23.7047	−1.14182	−0.570908	+	0.821014i	$0.693409\pi$
$432$	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	$-0.692028\pi$
$443$	−23.8823	−1.13468	−0.567340	+	0.823483i	$0.692028\pi$
$444$	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	$-0.363403\pi$
$449$	17.6333	0.832166	0.416083	+	0.909327i	$0.363403\pi$
$450$	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	$-0.541817\pi$
$461$	−5.62516	−0.261990	−0.130995	+	0.991383i	$0.541817\pi$
$462$	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	$-0.783672\pi$
$467$	−33.6174	−1.55563	−0.777815	+	0.628494i	$0.783672\pi$
$468$	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	$-0.485673\pi$
$479$	1.96953	0.0899902	0.0449951	+	0.998987i	$0.485673\pi$
$480$	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	$-0.665606\pi$
$491$	−22.0305	−0.994221	−0.497111	+	0.867687i	$0.665606\pi$
$492$	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	$-0.380676\pi$
$503$	16.4238	0.732301	0.366150	+	0.930556i	$0.380676\pi$
$504$	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	$-0.403922\pi$
$509$	13.4137	0.594550	0.297275	+	0.954792i	$0.403922\pi$
$510$	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	$-0.410020\pi$
$521$	12.7334	0.557862	0.278931	+	0.960311i	$0.410020\pi$
$522$	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	$-0.349316\pi$
$557$	21.5194	0.911807	0.455903	+	0.890029i	$0.349316\pi$
$558$	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	$-0.372401\pi$
$563$	18.5177	0.780427	0.390213	+	0.920724i	$0.372401\pi$
$564$	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	$-0.310907\pi$
$569$	26.7030	1.11945	0.559723	+	0.828680i	$0.310907\pi$
$570$	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	$-0.389931\pi$
$587$	16.4238	0.677882	0.338941	+	0.940808i	$0.389931\pi$
$588$	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	$-0.520260\pi$
$593$	−3.09779	−0.127211	−0.0636056	+	0.997975i	$0.520260\pi$
$594$	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	$-0.467562\pi$
$599$	4.97966	0.203464	0.101732	+	0.994812i	$0.467562\pi$
$600$	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	$-0.355350\pi$
$617$	21.8066	0.877900	0.438950	+	0.898511i	$0.355350\pi$
$618$	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	$-0.553785\pi$
$641$	−8.51532	−0.336335	−0.168167	+	0.985758i	$0.553785\pi$
$642$	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	$-0.384353\pi$
$647$	18.0788	0.710750	0.355375	+	0.934724i	$0.384353\pi$
$648$	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	$-0.472299\pi$
$653$	4.44200	0.173829	0.0869144	+	0.996216i	$0.472299\pi$
$654$	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	$-0.536000\pi$
$659$	−5.79433	−0.225715	−0.112857	+	0.993611i	$0.536000\pi$
$660$	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	$-0.878537\pi$
$677$	−48.2956	−1.85615	−0.928075	+	0.372394i	$0.878537\pi$
$678$	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	$-0.386421\pi$
$683$	18.2571	0.698589	0.349294	+	0.937013i	$0.386421\pi$
$684$	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	$-0.750582\pi$
$701$	−37.5117	−1.41680	−0.708398	+	0.705813i	$0.750582\pi$
$702$	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	$-0.667021\pi$
$719$	−26.8658	−1.00192	−0.500962	+	0.865469i	$0.667021\pi$
$720$	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	$-0.448382\pi$
$743$	8.80174	0.322905	0.161452	+	0.986881i	$0.448382\pi$
$744$	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	$-0.826090\pi$
$761$	−47.1407	−1.70885	−0.854425	+	0.519575i	$0.826090\pi$
$762$	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	$-0.827041\pi$
$773$	−47.5971	−1.71195	−0.855973	+	0.517020i	$0.827041\pi$
$774$	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	$-0.397667\pi$
$797$	17.8409	0.631958	0.315979	+	0.948766i	$0.397667\pi$
$798$	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	$-0.288685\pi$
$809$	35.0511	1.23233	0.616165	+	0.787617i	$0.288685\pi$
$810$	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	$-0.528388\pi$
$821$	−5.10402	−0.178131	−0.0890657	+	0.996026i	$0.528388\pi$
$822$	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	$-0.436941\pi$
$827$	11.3198	0.393627	0.196814	+	0.980441i	$0.436941\pi$
$828$	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	$-0.129366\pi$
$839$	53.2121	1.83709	0.918543	+	0.395320i	$0.129366\pi$
$840$	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	$-0.492493\pi$
$857$	1.38071	0.0471643	0.0235822	+	0.999722i	$0.492493\pi$
$858$	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	$-0.743327\pi$
$863$	−40.6651	−1.38426	−0.692129	+	0.721774i	$0.743327\pi$
$864$	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	$-0.411030\pi$
$881$	16.3772	0.551762	0.275881	+	0.961192i	$0.411030\pi$
$882$	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	$-0.901566\pi$
$887$	−56.7397	−1.90513	−0.952566	+	0.304333i	$0.901566\pi$
$888$	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	$-0.266937\pi$
$911$	40.3543	1.33700	0.668499	+	0.743713i	$0.266937\pi$
$912$	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	$-0.631368\pi$
$929$	−24.4500	−0.802179	−0.401089	+	0.916039i	$0.631368\pi$
$930$	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	$-0.429839\pi$
$941$	13.4137	0.437273	0.218636	+	0.975806i	$0.429839\pi$
$942$	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	$-0.448466\pi$
$947$	9.92085	0.322384	0.161192	+	0.986923i	$0.448466\pi$
$948$	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	$-0.575600\pi$
$953$	−14.5264	−0.470557	−0.235279	+	0.971928i	$0.575600\pi$
$954$	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	$-0.592538\pi$
$971$	−17.8638	−0.573276	−0.286638	+	0.958039i	$0.592538\pi$
$972$	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	$-0.394774\pi$
$977$	20.2914	0.649181	0.324590	+	0.945855i	$0.394774\pi$
$978$	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	$-0.380092\pi$
$983$	23.0666	0.735709	0.367855	+	0.929883i	$0.380092\pi$
$984$	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.bj.1.1	✓	4
3.2	odd	2		2790.2.a.bk.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2790.2.a.bj.1.1	✓	4	1.1	even	1	trivial
2790.2.a.bk.1.1	yes	4	3.2	odd	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:	\( N \)	\(=\)	\( 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 \)
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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