LMFDB - Embedded newform 7254.2.a.bk.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7254,2,Mod(1,7254)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7254, 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("7254.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7254 = 2 \cdot 3^{2} \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7254.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:	$57.9234816262$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.58446133.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 7x^{4} + 12x^{3} + 6x^{2} - 3x - 1 $ x^6 - 2x^5 - 7x^4 + 12x^3 + 6x^2 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 806)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.42705$ of defining polynomial
Character	$\chi$	$=$	7254.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -3.28061 q^{5} +3.80073 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -3.28061 q^{5} +3.80073 q^{7} -1.00000 q^{8} +3.28061 q^{10} -1.17595 q^{11} -1.00000 q^{13} -3.80073 q^{14} +1.00000 q^{16} +1.50784 q^{17} -2.81950 q^{19} -3.28061 q^{20} +1.17595 q^{22} -5.32669 q^{23} +5.76238 q^{25} +1.00000 q^{26} +3.80073 q^{28} +8.71517 q^{29} -1.00000 q^{31} -1.00000 q^{32} -1.50784 q^{34} -12.4687 q^{35} -1.41921 q^{37} +2.81950 q^{38} +3.28061 q^{40} +0.248905 q^{41} +5.84149 q^{43} -1.17595 q^{44} +5.32669 q^{46} -10.3034 q^{47} +7.44552 q^{49} -5.76238 q^{50} -1.00000 q^{52} +9.55392 q^{53} +3.85784 q^{55} -3.80073 q^{56} -8.71517 q^{58} -9.72646 q^{59} +3.35920 q^{61} +1.00000 q^{62} +1.00000 q^{64} +3.28061 q^{65} -8.28701 q^{67} +1.50784 q^{68} +12.4687 q^{70} -4.28790 q^{71} +7.52495 q^{73} +1.41921 q^{74} -2.81950 q^{76} -4.46948 q^{77} -6.10675 q^{79} -3.28061 q^{80} -0.248905 q^{82} +6.71839 q^{83} -4.94663 q^{85} -5.84149 q^{86} +1.17595 q^{88} +9.00805 q^{89} -3.80073 q^{91} -5.32669 q^{92} +10.3034 q^{94} +9.24966 q^{95} +5.07778 q^{97} -7.44552 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} + 6 q^{4} - 3 q^{5} + 4 q^{7} - 6 q^{8}+O(q^{10}) $ 6 * q - 6 * q^2 + 6 * q^4 - 3 * q^5 + 4 * q^7 - 6 * q^8	$ 6 q - 6 q^{2} + 6 q^{4} - 3 q^{5} + 4 q^{7} - 6 q^{8} + 3 q^{10} - 6 q^{11} - 6 q^{13} - 4 q^{14} + 6 q^{16} - 10 q^{17} + 15 q^{19} - 3 q^{20} + 6 q^{22} - 2 q^{23} + 13 q^{25} + 6 q^{26} + 4 q^{28} + 3 q^{29} - 6 q^{31} - 6 q^{32} + 10 q^{34} + 6 q^{35} - 26 q^{37} - 15 q^{38} + 3 q^{40} + 2 q^{41} + 3 q^{43} - 6 q^{44} + 2 q^{46} - 12 q^{47} + 24 q^{49} - 13 q^{50} - 6 q^{52} + 25 q^{53} - 6 q^{55} - 4 q^{56} - 3 q^{58} - 3 q^{59} + 17 q^{61} + 6 q^{62} + 6 q^{64} + 3 q^{65} + 3 q^{67} - 10 q^{68} - 6 q^{70} - 8 q^{71} - 19 q^{73} + 26 q^{74} + 15 q^{76} + 10 q^{77} - 8 q^{79} - 3 q^{80} - 2 q^{82} + 4 q^{83} - 44 q^{85} - 3 q^{86} + 6 q^{88} + 5 q^{89} - 4 q^{91} - 2 q^{92} + 12 q^{94} + 38 q^{95} - 24 q^{98}+O(q^{100}) $ 6 * q - 6 * q^2 + 6 * q^4 - 3 * q^5 + 4 * q^7 - 6 * q^8 + 3 * q^10 - 6 * q^11 - 6 * q^13 - 4 * q^14 + 6 * q^16 - 10 * q^17 + 15 * q^19 - 3 * q^20 + 6 * q^22 - 2 * q^23 + 13 * q^25 + 6 * q^26 + 4 * q^28 + 3 * q^29 - 6 * q^31 - 6 * q^32 + 10 * q^34 + 6 * q^35 - 26 * q^37 - 15 * q^38 + 3 * q^40 + 2 * q^41 + 3 * q^43 - 6 * q^44 + 2 * q^46 - 12 * q^47 + 24 * q^49 - 13 * q^50 - 6 * q^52 + 25 * q^53 - 6 * q^55 - 4 * q^56 - 3 * q^58 - 3 * q^59 + 17 * q^61 + 6 * q^62 + 6 * q^64 + 3 * q^65 + 3 * q^67 - 10 * q^68 - 6 * q^70 - 8 * q^71 - 19 * q^73 + 26 * q^74 + 15 * q^76 + 10 * q^77 - 8 * q^79 - 3 * q^80 - 2 * q^82 + 4 * q^83 - 44 * q^85 - 3 * q^86 + 6 * q^88 + 5 * q^89 - 4 * q^91 - 2 * q^92 + 12 * q^94 + 38 * q^95 - 24 * 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
$5$	−3.28061	−1.46713	−0.733566	−	0.679618i	$-0.762145\pi$
$5$	−3.28061	−1.46713	−0.733566	+	0.679618i	$0.762145\pi$
$6$	0	0
$7$	3.80073	1.43654	0.718270	−	0.695765i	$-0.244934\pi$
$7$	3.80073	1.43654	0.718270	+	0.695765i	$0.244934\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	3.28061	1.03742
$11$	−1.17595	−0.354564	−0.177282	−	0.984160i	$-0.556730\pi$
$11$	−1.17595	−0.354564	−0.177282	+	0.984160i	$0.556730\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−3.80073	−1.01579
$15$	0	0
$16$	1.00000	0.250000
$17$	1.50784	0.365705	0.182852	−	0.983140i	$-0.441467\pi$
$17$	1.50784	0.365705	0.182852	+	0.983140i	$0.441467\pi$
$18$	0	0
$19$	−2.81950	−0.646837	−0.323419	−	0.946256i	$-0.604832\pi$
$19$	−2.81950	−0.646837	−0.323419	+	0.946256i	$0.604832\pi$
$20$	−3.28061	−0.733566
$21$	0	0
$22$	1.17595	0.250714
$23$	−5.32669	−1.11069	−0.555346	−	0.831620i	$-0.687414\pi$
$23$	−5.32669	−1.11069	−0.555346	+	0.831620i	$0.687414\pi$
$24$	0	0
$25$	5.76238	1.15248
$26$	1.00000	0.196116
$27$	0	0
$28$	3.80073	0.718270
$29$	8.71517	1.61837	0.809183	−	0.587557i	$-0.199910\pi$
$29$	8.71517	1.61837	0.809183	+	0.587557i	$0.199910\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−1.50784	−0.258592
$35$	−12.4687	−2.10759
$36$	0	0
$37$	−1.41921	−0.233317	−0.116658	−	0.993172i	$-0.537218\pi$
$37$	−1.41921	−0.233317	−0.116658	+	0.993172i	$0.537218\pi$
$38$	2.81950	0.457383
$39$	0	0
$40$	3.28061	0.518709
$41$	0.248905	0.0388724	0.0194362	−	0.999811i	$-0.493813\pi$
$41$	0.248905	0.0388724	0.0194362	+	0.999811i	$0.493813\pi$
$42$	0	0
$43$	5.84149	0.890819	0.445409	−	0.895327i	$-0.353058\pi$
$43$	5.84149	0.890819	0.445409	+	0.895327i	$0.353058\pi$
$44$	−1.17595	−0.177282
$45$	0	0
$46$	5.32669	0.785377
$47$	−10.3034	−1.50290	−0.751450	−	0.659790i	$-0.770645\pi$
$47$	−10.3034	−1.50290	−0.751450	+	0.659790i	$0.770645\pi$
$48$	0	0
$49$	7.44552	1.06365
$50$	−5.76238	−0.814924
$51$	0	0
$52$	−1.00000	−0.138675
$53$	9.55392	1.31233	0.656166	−	0.754617i	$-0.272177\pi$
$53$	9.55392	1.31233	0.656166	+	0.754617i	$0.272177\pi$
$54$	0	0
$55$	3.85784	0.520192
$56$	−3.80073	−0.507893
$57$	0	0
$58$	−8.71517	−1.14436
$59$	−9.72646	−1.26628	−0.633139	−	0.774038i	$-0.718234\pi$
$59$	−9.72646	−1.26628	−0.633139	+	0.774038i	$0.718234\pi$
$60$	0	0
$61$	3.35920	0.430102	0.215051	−	0.976603i	$-0.431008\pi$
$61$	3.35920	0.430102	0.215051	+	0.976603i	$0.431008\pi$
$62$	1.00000	0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	3.28061	0.406909
$66$	0	0
$67$	−8.28701	−1.01242	−0.506210	−	0.862410i	$-0.668954\pi$
$67$	−8.28701	−1.01242	−0.506210	+	0.862410i	$0.668954\pi$
$68$	1.50784	0.182852
$69$	0	0
$70$	12.4687	1.49029
$71$	−4.28790	−0.508880	−0.254440	−	0.967089i	$-0.581891\pi$
$71$	−4.28790	−0.508880	−0.254440	+	0.967089i	$0.581891\pi$
$72$	0	0
$73$	7.52495	0.880729	0.440365	−	0.897819i	$-0.354849\pi$
$73$	7.52495	0.880729	0.440365	+	0.897819i	$0.354849\pi$
$74$	1.41921	0.164980
$75$	0	0
$76$	−2.81950	−0.323419
$77$	−4.46948	−0.509345
$78$	0	0
$79$	−6.10675	−0.687063	−0.343532	−	0.939141i	$-0.611623\pi$
$79$	−6.10675	−0.687063	−0.343532	+	0.939141i	$0.611623\pi$
$80$	−3.28061	−0.366783
$81$	0	0
$82$	−0.248905	−0.0274870
$83$	6.71839	0.737439	0.368719	−	0.929541i	$-0.379796\pi$
$83$	6.71839	0.737439	0.368719	+	0.929541i	$0.379796\pi$
$84$	0	0
$85$	−4.94663	−0.536537
$86$	−5.84149	−0.629904
$87$	0	0
$88$	1.17595	0.125357
$89$	9.00805	0.954852	0.477426	−	0.878672i	$-0.341570\pi$
$89$	9.00805	0.954852	0.477426	+	0.878672i	$0.341570\pi$
$90$	0	0
$91$	−3.80073	−0.398424
$92$	−5.32669	−0.555346
$93$	0	0
$94$	10.3034	1.06271
$95$	9.24966	0.948995
$96$	0	0
$97$	5.07778	0.515571	0.257785	−	0.966202i	$-0.417007\pi$
$97$	5.07778	0.515571	0.257785	+	0.966202i	$0.417007\pi$
$98$	−7.44552	−0.752112
$99$	0	0
$100$	5.76238	0.576238
$101$	6.44362	0.641164	0.320582	−	0.947221i	$-0.396121\pi$
$101$	6.44362	0.641164	0.320582	+	0.947221i	$0.396121\pi$
$102$	0	0
$103$	−17.7023	−1.74426	−0.872131	−	0.489272i	$-0.837262\pi$
$103$	−17.7023	−1.74426	−0.872131	+	0.489272i	$0.837262\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−9.55392	−0.927959
$107$	5.55747	0.537261	0.268630	−	0.963243i	$-0.413429\pi$
$107$	5.55747	0.537261	0.268630	+	0.963243i	$0.413429\pi$
$108$	0	0
$109$	15.2814	1.46369	0.731845	−	0.681471i	$-0.238660\pi$
$109$	15.2814	1.46369	0.731845	+	0.681471i	$0.238660\pi$
$110$	−3.85784	−0.367831
$111$	0	0
$112$	3.80073	0.359135
$113$	10.3313	0.971890	0.485945	−	0.873990i	$-0.338476\pi$
$113$	10.3313	0.971890	0.485945	+	0.873990i	$0.338476\pi$
$114$	0	0
$115$	17.4748	1.62953
$116$	8.71517	0.809183
$117$	0	0
$118$	9.72646	0.895393
$119$	5.73088	0.525349
$120$	0	0
$121$	−9.61713	−0.874285
$122$	−3.35920	−0.304128
$123$	0	0
$124$	−1.00000	−0.0898027
$125$	−2.50107	−0.223702
$126$	0	0
$127$	−9.66047	−0.857229	−0.428614	−	0.903488i	$-0.640998\pi$
$127$	−9.66047	−0.857229	−0.428614	+	0.903488i	$0.640998\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−3.28061	−0.287728
$131$	7.76868	0.678753	0.339376	−	0.940651i	$-0.389784\pi$
$131$	7.76868	0.678753	0.339376	+	0.940651i	$0.389784\pi$
$132$	0	0
$133$	−10.7161	−0.929207
$134$	8.28701	0.715889
$135$	0	0
$136$	−1.50784	−0.129296
$137$	−1.52595	−0.130371	−0.0651854	−	0.997873i	$-0.520764\pi$
$137$	−1.52595	−0.130371	−0.0651854	+	0.997873i	$0.520764\pi$
$138$	0	0
$139$	−2.32942	−0.197579	−0.0987896	−	0.995108i	$-0.531497\pi$
$139$	−2.32942	−0.197579	−0.0987896	+	0.995108i	$0.531497\pi$
$140$	−12.4687	−1.05380
$141$	0	0
$142$	4.28790	0.359833
$143$	1.17595	0.0983383
$144$	0	0
$145$	−28.5910	−2.37436
$146$	−7.52495	−0.622770
$147$	0	0
$148$	−1.41921	−0.116658
$149$	−15.5706	−1.27559	−0.637796	−	0.770205i	$-0.720154\pi$
$149$	−15.5706	−1.27559	−0.637796	+	0.770205i	$0.720154\pi$
$150$	0	0
$151$	10.2243	0.832044	0.416022	−	0.909354i	$-0.363424\pi$
$151$	10.2243	0.832044	0.416022	+	0.909354i	$0.363424\pi$
$152$	2.81950	0.228691
$153$	0	0
$154$	4.46948	0.360161
$155$	3.28061	0.263505
$156$	0	0
$157$	15.5634	1.24209	0.621047	−	0.783773i	$-0.286708\pi$
$157$	15.5634	1.24209	0.621047	+	0.783773i	$0.286708\pi$
$158$	6.10675	0.485827
$159$	0	0
$160$	3.28061	0.259355
$161$	−20.2453	−1.59555
$162$	0	0
$163$	−12.5877	−0.985946	−0.492973	−	0.870045i	$-0.664090\pi$
$163$	−12.5877	−0.985946	−0.492973	+	0.870045i	$0.664090\pi$
$164$	0.248905	0.0194362
$165$	0	0
$166$	−6.71839	−0.521448
$167$	7.70256	0.596042	0.298021	−	0.954559i	$-0.403673\pi$
$167$	7.70256	0.596042	0.298021	+	0.954559i	$0.403673\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	4.94663	0.379389
$171$	0	0
$172$	5.84149	0.445409
$173$	14.9308	1.13517	0.567584	−	0.823316i	$-0.307878\pi$
$173$	14.9308	1.13517	0.567584	+	0.823316i	$0.307878\pi$
$174$	0	0
$175$	21.9012	1.65558
$176$	−1.17595	−0.0886409
$177$	0	0
$178$	−9.00805	−0.675182
$179$	1.81885	0.135947	0.0679736	−	0.997687i	$-0.478347\pi$
$179$	1.81885	0.135947	0.0679736	+	0.997687i	$0.478347\pi$
$180$	0	0
$181$	−17.5553	−1.30488	−0.652439	−	0.757841i	$-0.726254\pi$
$181$	−17.5553	−1.30488	−0.652439	+	0.757841i	$0.726254\pi$
$182$	3.80073	0.281729
$183$	0	0
$184$	5.32669	0.392689
$185$	4.65587	0.342307
$186$	0	0
$187$	−1.77315	−0.129666
$188$	−10.3034	−0.751450
$189$	0	0
$190$	−9.24966	−0.671041
$191$	−0.219133	−0.0158559	−0.00792797	−	0.999969i	$-0.502524\pi$
$191$	−0.219133	−0.0158559	−0.00792797	+	0.999969i	$0.502524\pi$
$192$	0	0
$193$	−23.0468	−1.65895	−0.829473	−	0.558546i	$-0.811359\pi$
$193$	−23.0468	−1.65895	−0.829473	+	0.558546i	$0.811359\pi$
$194$	−5.07778	−0.364564
$195$	0	0
$196$	7.44552	0.531823
$197$	19.6740	1.40172	0.700859	−	0.713300i	$-0.252800\pi$
$197$	19.6740	1.40172	0.700859	+	0.713300i	$0.252800\pi$
$198$	0	0
$199$	22.8733	1.62145	0.810723	−	0.585430i	$-0.199074\pi$
$199$	22.8733	1.62145	0.810723	+	0.585430i	$0.199074\pi$
$200$	−5.76238	−0.407462
$201$	0	0
$202$	−6.44362	−0.453372
$203$	33.1240	2.32485
$204$	0	0
$205$	−0.816559	−0.0570310
$206$	17.7023	1.23338
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	3.31560	0.229345
$210$	0	0
$211$	−2.85909	−0.196828	−0.0984139	−	0.995146i	$-0.531377\pi$
$211$	−2.85909	−0.196828	−0.0984139	+	0.995146i	$0.531377\pi$
$212$	9.55392	0.656166
$213$	0	0
$214$	−5.55747	−0.379901
$215$	−19.1636	−1.30695
$216$	0	0
$217$	−3.80073	−0.258010
$218$	−15.2814	−1.03498
$219$	0	0
$220$	3.85784	0.260096
$221$	−1.50784	−0.101428
$222$	0	0
$223$	11.8268	0.791980	0.395990	−	0.918255i	$-0.370402\pi$
$223$	11.8268	0.791980	0.395990	+	0.918255i	$0.370402\pi$
$224$	−3.80073	−0.253947
$225$	0	0
$226$	−10.3313	−0.687230
$227$	−28.8792	−1.91678	−0.958389	−	0.285467i	$-0.907851\pi$
$227$	−28.8792	−1.91678	−0.958389	+	0.285467i	$0.907851\pi$
$228$	0	0
$229$	9.93379	0.656443	0.328222	−	0.944601i	$-0.393551\pi$
$229$	9.93379	0.656443	0.328222	+	0.944601i	$0.393551\pi$
$230$	−17.4748	−1.15225
$231$	0	0
$232$	−8.71517	−0.572179
$233$	−6.56766	−0.430262	−0.215131	−	0.976585i	$-0.569018\pi$
$233$	−6.56766	−0.430262	−0.215131	+	0.976585i	$0.569018\pi$
$234$	0	0
$235$	33.8013	2.20495
$236$	−9.72646	−0.633139
$237$	0	0
$238$	−5.73088	−0.371478
$239$	−23.6497	−1.52977	−0.764887	−	0.644164i	$-0.777205\pi$
$239$	−23.6497	−1.52977	−0.764887	+	0.644164i	$0.777205\pi$
$240$	0	0
$241$	13.1223	0.845281	0.422641	−	0.906297i	$-0.361103\pi$
$241$	13.1223	0.845281	0.422641	+	0.906297i	$0.361103\pi$
$242$	9.61713	0.618213
$243$	0	0
$244$	3.35920	0.215051
$245$	−24.4258	−1.56051
$246$	0	0
$247$	2.81950	0.179400
$248$	1.00000	0.0635001
$249$	0	0
$250$	2.50107	0.158182
$251$	30.4094	1.91943	0.959714	−	0.280980i	$-0.0906596\pi$
$251$	30.4094	1.91943	0.959714	+	0.280980i	$0.0906596\pi$
$252$	0	0
$253$	6.26394	0.393811
$254$	9.66047	0.606152
$255$	0	0
$256$	1.00000	0.0625000
$257$	−14.6922	−0.916473	−0.458236	−	0.888830i	$-0.651519\pi$
$257$	−14.6922	−0.916473	−0.458236	+	0.888830i	$0.651519\pi$
$258$	0	0
$259$	−5.39403	−0.335169
$260$	3.28061	0.203455
$261$	0	0
$262$	−7.76868	−0.479951
$263$	−9.33107	−0.575378	−0.287689	−	0.957724i	$-0.592887\pi$
$263$	−9.33107	−0.575378	−0.287689	+	0.957724i	$0.592887\pi$
$264$	0	0
$265$	−31.3427	−1.92536
$266$	10.7161	0.657049
$267$	0	0
$268$	−8.28701	−0.506210
$269$	−2.87677	−0.175400	−0.0876999	−	0.996147i	$-0.527952\pi$
$269$	−2.87677	−0.175400	−0.0876999	+	0.996147i	$0.527952\pi$
$270$	0	0
$271$	3.92263	0.238283	0.119141	−	0.992877i	$-0.461986\pi$
$271$	3.92263	0.238283	0.119141	+	0.992877i	$0.461986\pi$
$272$	1.50784	0.0914261
$273$	0	0
$274$	1.52595	0.0921861
$275$	−6.77630	−0.408626
$276$	0	0
$277$	−0.104195	−0.00626047	−0.00313024	−	0.999995i	$-0.500996\pi$
$277$	−0.104195	−0.00626047	−0.00313024	+	0.999995i	$0.500996\pi$
$278$	2.32942	0.139710
$279$	0	0
$280$	12.4687	0.745147
$281$	−11.3347	−0.676170	−0.338085	−	0.941116i	$-0.609779\pi$
$281$	−11.3347	−0.676170	−0.338085	+	0.941116i	$0.609779\pi$
$282$	0	0
$283$	−13.2511	−0.787695	−0.393848	−	0.919176i	$-0.628856\pi$
$283$	−13.2511	−0.787695	−0.393848	+	0.919176i	$0.628856\pi$
$284$	−4.28790	−0.254440
$285$	0	0
$286$	−1.17595	−0.0695357
$287$	0.946020	0.0558418
$288$	0	0
$289$	−14.7264	−0.866260
$290$	28.5910	1.67892
$291$	0	0
$292$	7.52495	0.440365
$293$	20.2988	1.18587	0.592935	−	0.805250i	$-0.297969\pi$
$293$	20.2988	1.18587	0.592935	+	0.805250i	$0.297969\pi$
$294$	0	0
$295$	31.9087	1.85780
$296$	1.41921	0.0824900
$297$	0	0
$298$	15.5706	0.901980
$299$	5.32669	0.308050
$300$	0	0
$301$	22.2019	1.27970
$302$	−10.2243	−0.588344
$303$	0	0
$304$	−2.81950	−0.161709
$305$	−11.0202	−0.631016
$306$	0	0
$307$	22.7053	1.29586	0.647930	−	0.761700i	$-0.275635\pi$
$307$	22.7053	1.29586	0.647930	+	0.761700i	$0.275635\pi$
$308$	−4.46948	−0.254672
$309$	0	0
$310$	−3.28061	−0.186326
$311$	10.7079	0.607192	0.303596	−	0.952801i	$-0.401813\pi$
$311$	10.7079	0.607192	0.303596	+	0.952801i	$0.401813\pi$
$312$	0	0
$313$	−12.5598	−0.709920	−0.354960	−	0.934882i	$-0.615505\pi$
$313$	−12.5598	−0.709920	−0.354960	+	0.934882i	$0.615505\pi$
$314$	−15.5634	−0.878293
$315$	0	0
$316$	−6.10675	−0.343532
$317$	20.1294	1.13058	0.565290	−	0.824893i	$-0.308764\pi$
$317$	20.1294	1.13058	0.565290	+	0.824893i	$0.308764\pi$
$318$	0	0
$319$	−10.2486	−0.573814
$320$	−3.28061	−0.183391
$321$	0	0
$322$	20.2453	1.12823
$323$	−4.25135	−0.236551
$324$	0	0
$325$	−5.76238	−0.319639
$326$	12.5877	0.697169
$327$	0	0
$328$	−0.248905	−0.0137435
$329$	−39.1603	−2.15898
$330$	0	0
$331$	1.22389	0.0672711	0.0336355	−	0.999434i	$-0.489291\pi$
$331$	1.22389	0.0672711	0.0336355	+	0.999434i	$0.489291\pi$
$332$	6.71839	0.368719
$333$	0	0
$334$	−7.70256	−0.421466
$335$	27.1864	1.48535
$336$	0	0
$337$	14.6924	0.800344	0.400172	−	0.916440i	$-0.368950\pi$
$337$	14.6924	0.800344	0.400172	+	0.916440i	$0.368950\pi$
$338$	−1.00000	−0.0543928
$339$	0	0
$340$	−4.94663	−0.268268
$341$	1.17595	0.0636815
$342$	0	0
$343$	1.69332	0.0914305
$344$	−5.84149	−0.314952
$345$	0	0
$346$	−14.9308	−0.802685
$347$	23.3045	1.25105	0.625526	−	0.780203i	$-0.284884\pi$
$347$	23.3045	1.25105	0.625526	+	0.780203i	$0.284884\pi$
$348$	0	0
$349$	30.1334	1.61300	0.806502	−	0.591231i	$-0.201358\pi$
$349$	30.1334	1.61300	0.806502	+	0.591231i	$0.201358\pi$
$350$	−21.9012	−1.17067
$351$	0	0
$352$	1.17595	0.0626786
$353$	30.2593	1.61054	0.805271	−	0.592907i	$-0.202020\pi$
$353$	30.2593	1.61054	0.805271	+	0.592907i	$0.202020\pi$
$354$	0	0
$355$	14.0669	0.746594
$356$	9.00805	0.477426
$357$	0	0
$358$	−1.81885	−0.0961292
$359$	27.4164	1.44698	0.723491	−	0.690334i	$-0.242536\pi$
$359$	27.4164	1.44698	0.723491	+	0.690334i	$0.242536\pi$
$360$	0	0
$361$	−11.0504	−0.581602
$362$	17.5553	0.922688
$363$	0	0
$364$	−3.80073	−0.199212
$365$	−24.6864	−1.29215
$366$	0	0
$367$	−1.11736	−0.0583259	−0.0291630	−	0.999575i	$-0.509284\pi$
$367$	−1.11736	−0.0583259	−0.0291630	+	0.999575i	$0.509284\pi$
$368$	−5.32669	−0.277673
$369$	0	0
$370$	−4.65587	−0.242047
$371$	36.3118	1.88522
$372$	0	0
$373$	−8.07205	−0.417955	−0.208977	−	0.977920i	$-0.567013\pi$
$373$	−8.07205	−0.417955	−0.208977	+	0.977920i	$0.567013\pi$
$374$	1.77315	0.0916874
$375$	0	0
$376$	10.3034	0.531356
$377$	−8.71517	−0.448854
$378$	0	0
$379$	−34.1167	−1.75245	−0.876227	−	0.481898i	$-0.839948\pi$
$379$	−34.1167	−1.75245	−0.876227	+	0.481898i	$0.839948\pi$
$380$	9.24966	0.474498
$381$	0	0
$382$	0.219133	0.0112118
$383$	−12.8136	−0.654743	−0.327371	−	0.944896i	$-0.606163\pi$
$383$	−12.8136	−0.654743	−0.327371	+	0.944896i	$0.606163\pi$
$384$	0	0
$385$	14.6626	0.747276
$386$	23.0468	1.17305
$387$	0	0
$388$	5.07778	0.257785
$389$	30.8506	1.56419	0.782095	−	0.623159i	$-0.214151\pi$
$389$	30.8506	1.56419	0.782095	+	0.623159i	$0.214151\pi$
$390$	0	0
$391$	−8.03178	−0.406185
$392$	−7.44552	−0.376056
$393$	0	0
$394$	−19.6740	−0.991164
$395$	20.0338	1.00801
$396$	0	0
$397$	−11.9150	−0.597998	−0.298999	−	0.954253i	$-0.596653\pi$
$397$	−11.9150	−0.597998	−0.298999	+	0.954253i	$0.596653\pi$
$398$	−22.8733	−1.14654
$399$	0	0
$400$	5.76238	0.288119
$401$	36.2094	1.80821	0.904105	−	0.427310i	$-0.140539\pi$
$401$	36.2094	1.80821	0.904105	+	0.427310i	$0.140539\pi$
$402$	0	0
$403$	1.00000	0.0498135
$404$	6.44362	0.320582
$405$	0	0
$406$	−33.1240	−1.64391
$407$	1.66893	0.0827257
$408$	0	0
$409$	29.1377	1.44077	0.720383	−	0.693576i	$-0.243966\pi$
$409$	29.1377	1.44077	0.720383	+	0.693576i	$0.243966\pi$
$410$	0.816559	0.0403270
$411$	0	0
$412$	−17.7023	−0.872131
$413$	−36.9676	−1.81906
$414$	0	0
$415$	−22.0404	−1.08192
$416$	1.00000	0.0490290
$417$	0	0
$418$	−3.31560	−0.162171
$419$	10.4605	0.511031	0.255515	−	0.966805i	$-0.417755\pi$
$419$	10.4605	0.511031	0.255515	+	0.966805i	$0.417755\pi$
$420$	0	0
$421$	26.1738	1.27563	0.637816	−	0.770189i	$-0.279838\pi$
$421$	26.1738	1.27563	0.637816	+	0.770189i	$0.279838\pi$
$422$	2.85909	0.139178
$423$	0	0
$424$	−9.55392	−0.463979
$425$	8.68874	0.421466
$426$	0	0
$427$	12.7674	0.617858
$428$	5.55747	0.268630
$429$	0	0
$430$	19.1636	0.924152
$431$	−21.9570	−1.05763	−0.528815	−	0.848737i	$-0.677363\pi$
$431$	−21.9570	−1.05763	−0.528815	+	0.848737i	$0.677363\pi$
$432$	0	0
$433$	29.9124	1.43750	0.718748	−	0.695270i	$-0.244715\pi$
$433$	29.9124	1.43750	0.718748	+	0.695270i	$0.244715\pi$
$434$	3.80073	0.182441
$435$	0	0
$436$	15.2814	0.731845
$437$	15.0186	0.718436
$438$	0	0
$439$	32.5127	1.55174	0.775872	−	0.630890i	$-0.217310\pi$
$439$	32.5127	1.55174	0.775872	+	0.630890i	$0.217310\pi$
$440$	−3.85784	−0.183916
$441$	0	0
$442$	1.50784	0.0717206
$443$	29.5405	1.40351	0.701756	−	0.712417i	$-0.252399\pi$
$443$	29.5405	1.40351	0.701756	+	0.712417i	$0.252399\pi$
$444$	0	0
$445$	−29.5519	−1.40089
$446$	−11.8268	−0.560014
$447$	0	0
$448$	3.80073	0.179567
$449$	37.6243	1.77560	0.887801	−	0.460227i	$-0.152232\pi$
$449$	37.6243	1.77560	0.887801	+	0.460227i	$0.152232\pi$
$450$	0	0
$451$	−0.292701	−0.0137828
$452$	10.3313	0.485945
$453$	0	0
$454$	28.8792	1.35537
$455$	12.4687	0.584541
$456$	0	0
$457$	28.2803	1.32290	0.661449	−	0.749990i	$-0.269942\pi$
$457$	28.2803	1.32290	0.661449	+	0.749990i	$0.269942\pi$
$458$	−9.93379	−0.464175
$459$	0	0
$460$	17.4748	0.814765
$461$	11.2717	0.524976	0.262488	−	0.964935i	$-0.415457\pi$
$461$	11.2717	0.524976	0.262488	+	0.964935i	$0.415457\pi$
$462$	0	0
$463$	18.7255	0.870247	0.435124	−	0.900371i	$-0.356705\pi$
$463$	18.7255	0.870247	0.435124	+	0.900371i	$0.356705\pi$
$464$	8.71517	0.404591
$465$	0	0
$466$	6.56766	0.304241
$467$	36.6054	1.69390	0.846948	−	0.531676i	$-0.178438\pi$
$467$	36.6054	1.69390	0.846948	+	0.531676i	$0.178438\pi$
$468$	0	0
$469$	−31.4967	−1.45438
$470$	−33.8013	−1.55914
$471$	0	0
$472$	9.72646	0.447697
$473$	−6.86933	−0.315852
$474$	0	0
$475$	−16.2470	−0.745464
$476$	5.73088	0.262675
$477$	0	0
$478$	23.6497	1.08171
$479$	−39.1426	−1.78847	−0.894236	−	0.447596i	$-0.852280\pi$
$479$	−39.1426	−1.78847	−0.894236	+	0.447596i	$0.852280\pi$
$480$	0	0
$481$	1.41921	0.0647105
$482$	−13.1223	−0.597704
$483$	0	0
$484$	−9.61713	−0.437142
$485$	−16.6582	−0.756410
$486$	0	0
$487$	24.3747	1.10452	0.552262	−	0.833671i	$-0.313765\pi$
$487$	24.3747	1.10452	0.552262	+	0.833671i	$0.313765\pi$
$488$	−3.35920	−0.152064
$489$	0	0
$490$	24.4258	1.10345
$491$	−6.17844	−0.278829	−0.139415	−	0.990234i	$-0.544522\pi$
$491$	−6.17844	−0.278829	−0.139415	+	0.990234i	$0.544522\pi$
$492$	0	0
$493$	13.1411	0.591844
$494$	−2.81950	−0.126855
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	−16.2971	−0.731027
$498$	0	0
$499$	−17.3220	−0.775438	−0.387719	−	0.921778i	$-0.626737\pi$
$499$	−17.3220	−0.775438	−0.387719	+	0.921778i	$0.626737\pi$
$500$	−2.50107	−0.111851
$501$	0	0
$502$	−30.4094	−1.35724
$503$	22.8248	1.01771	0.508853	−	0.860854i	$-0.330070\pi$
$503$	22.8248	1.01771	0.508853	+	0.860854i	$0.330070\pi$
$504$	0	0
$505$	−21.1390	−0.940673
$506$	−6.26394	−0.278466
$507$	0	0
$508$	−9.66047	−0.428614
$509$	−22.2747	−0.987311	−0.493655	−	0.869658i	$-0.664340\pi$
$509$	−22.2747	−0.987311	−0.493655	+	0.869658i	$0.664340\pi$
$510$	0	0
$511$	28.6003	1.26520
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	14.6922	0.648044
$515$	58.0744	2.55906
$516$	0	0
$517$	12.1163	0.532874
$518$	5.39403	0.237000
$519$	0	0
$520$	−3.28061	−0.143864
$521$	−25.0939	−1.09938	−0.549691	−	0.835368i	$-0.685255\pi$
$521$	−25.0939	−1.09938	−0.549691	+	0.835368i	$0.685255\pi$
$522$	0	0
$523$	41.9416	1.83398	0.916988	−	0.398914i	$-0.130613\pi$
$523$	41.9416	1.83398	0.916988	+	0.398914i	$0.130613\pi$
$524$	7.76868	0.339376
$525$	0	0
$526$	9.33107	0.406854
$527$	−1.50784	−0.0656825
$528$	0	0
$529$	5.37360	0.233635
$530$	31.3427	1.36144
$531$	0	0
$532$	−10.7161	−0.464604
$533$	−0.248905	−0.0107813
$534$	0	0
$535$	−18.2319	−0.788233
$536$	8.28701	0.357944
$537$	0	0
$538$	2.87677	0.124026
$539$	−8.75560	−0.377130
$540$	0	0
$541$	31.3775	1.34902	0.674512	−	0.738263i	$-0.264354\pi$
$541$	31.3775	1.34902	0.674512	+	0.738263i	$0.264354\pi$
$542$	−3.92263	−0.168491
$543$	0	0
$544$	−1.50784	−0.0646480
$545$	−50.1322	−2.14743
$546$	0	0
$547$	22.7905	0.974453	0.487227	−	0.873276i	$-0.338009\pi$
$547$	22.7905	0.974453	0.487227	+	0.873276i	$0.338009\pi$
$548$	−1.52595	−0.0651854
$549$	0	0
$550$	6.77630	0.288942
$551$	−24.5724	−1.04682
$552$	0	0
$553$	−23.2101	−0.986993
$554$	0.104195	0.00442682
$555$	0	0
$556$	−2.32942	−0.0987896
$557$	−45.0853	−1.91033	−0.955163	−	0.296081i	$-0.904320\pi$
$557$	−45.0853	−1.91033	−0.955163	+	0.296081i	$0.904320\pi$
$558$	0	0
$559$	−5.84149	−0.247069
$560$	−12.4687	−0.526898
$561$	0	0
$562$	11.3347	0.478124
$563$	22.7229	0.957654	0.478827	−	0.877909i	$-0.341062\pi$
$563$	22.7229	0.957654	0.478827	+	0.877909i	$0.341062\pi$
$564$	0	0
$565$	−33.8930	−1.42589
$566$	13.2511	0.556985
$567$	0	0
$568$	4.28790	0.179916
$569$	−39.9092	−1.67308	−0.836540	−	0.547905i	$-0.815425\pi$
$569$	−39.9092	−1.67308	−0.836540	+	0.547905i	$0.815425\pi$
$570$	0	0
$571$	11.0863	0.463946	0.231973	−	0.972722i	$-0.425482\pi$
$571$	11.0863	0.463946	0.231973	+	0.972722i	$0.425482\pi$
$572$	1.17595	0.0491691
$573$	0	0
$574$	−0.946020	−0.0394861
$575$	−30.6944	−1.28004
$576$	0	0
$577$	12.3064	0.512321	0.256160	−	0.966634i	$-0.417543\pi$
$577$	12.3064	0.512321	0.256160	+	0.966634i	$0.417543\pi$
$578$	14.7264	0.612538
$579$	0	0
$580$	−28.5910	−1.18718
$581$	25.5348	1.05936
$582$	0	0
$583$	−11.2350	−0.465305
$584$	−7.52495	−0.311385
$585$	0	0
$586$	−20.2988	−0.838537
$587$	−33.6583	−1.38923	−0.694613	−	0.719384i	$-0.744424\pi$
$587$	−33.6583	−1.38923	−0.694613	+	0.719384i	$0.744424\pi$
$588$	0	0
$589$	2.81950	0.116175
$590$	−31.9087	−1.31366
$591$	0	0
$592$	−1.41921	−0.0583292
$593$	24.7650	1.01698	0.508489	−	0.861069i	$-0.330204\pi$
$593$	24.7650	1.01698	0.508489	+	0.861069i	$0.330204\pi$
$594$	0	0
$595$	−18.8008	−0.770756
$596$	−15.5706	−0.637796
$597$	0	0
$598$	−5.32669	−0.217824
$599$	15.3751	0.628208	0.314104	−	0.949389i	$-0.398296\pi$
$599$	15.3751	0.628208	0.314104	+	0.949389i	$0.398296\pi$
$600$	0	0
$601$	24.9822	1.01904	0.509522	−	0.860458i	$-0.329822\pi$
$601$	24.9822	1.01904	0.509522	+	0.860458i	$0.329822\pi$
$602$	−22.2019	−0.904882
$603$	0	0
$604$	10.2243	0.416022
$605$	31.5500	1.28269
$606$	0	0
$607$	−5.15928	−0.209409	−0.104704	−	0.994503i	$-0.533390\pi$
$607$	−5.15928	−0.209409	−0.104704	+	0.994503i	$0.533390\pi$
$608$	2.81950	0.114346
$609$	0	0
$610$	11.0202	0.446196
$611$	10.3034	0.416830
$612$	0	0
$613$	−44.1907	−1.78485	−0.892423	−	0.451200i	$-0.850996\pi$
$613$	−44.1907	−1.78485	−0.892423	+	0.451200i	$0.850996\pi$
$614$	−22.7053	−0.916311
$615$	0	0
$616$	4.46948	0.180081
$617$	−3.95272	−0.159131	−0.0795653	−	0.996830i	$-0.525353\pi$
$617$	−3.95272	−0.159131	−0.0795653	+	0.996830i	$0.525353\pi$
$618$	0	0
$619$	−5.02296	−0.201890	−0.100945	−	0.994892i	$-0.532187\pi$
$619$	−5.02296	−0.201890	−0.100945	+	0.994892i	$0.532187\pi$
$620$	3.28061	0.131752
$621$	0	0
$622$	−10.7079	−0.429349
$623$	34.2372	1.37168
$624$	0	0
$625$	−20.6069	−0.824275
$626$	12.5598	0.501989
$627$	0	0
$628$	15.5634	0.621047
$629$	−2.13994	−0.0853250
$630$	0	0
$631$	19.9470	0.794078	0.397039	−	0.917802i	$-0.370038\pi$
$631$	19.9470	0.794078	0.397039	+	0.917802i	$0.370038\pi$
$632$	6.10675	0.242913
$633$	0	0
$634$	−20.1294	−0.799440
$635$	31.6922	1.25767
$636$	0	0
$637$	−7.44552	−0.295002
$638$	10.2486	0.405748
$639$	0	0
$640$	3.28061	0.129677
$641$	8.33747	0.329310	0.164655	−	0.986351i	$-0.447349\pi$
$641$	8.33747	0.329310	0.164655	+	0.986351i	$0.447349\pi$
$642$	0	0
$643$	24.7789	0.977186	0.488593	−	0.872512i	$-0.337510\pi$
$643$	24.7789	0.977186	0.488593	+	0.872512i	$0.337510\pi$
$644$	−20.2453	−0.797776
$645$	0	0
$646$	4.25135	0.167267
$647$	3.83664	0.150834	0.0754169	−	0.997152i	$-0.475971\pi$
$647$	3.83664	0.150834	0.0754169	+	0.997152i	$0.475971\pi$
$648$	0	0
$649$	11.4379	0.448976
$650$	5.76238	0.226019
$651$	0	0
$652$	−12.5877	−0.492973
$653$	15.8504	0.620273	0.310137	−	0.950692i	$-0.399625\pi$
$653$	15.8504	0.620273	0.310137	+	0.950692i	$0.399625\pi$
$654$	0	0
$655$	−25.4860	−0.995819
$656$	0.248905	0.00971811
$657$	0	0
$658$	39.1603	1.52663
$659$	−13.5484	−0.527770	−0.263885	−	0.964554i	$-0.585004\pi$
$659$	−13.5484	−0.527770	−0.263885	+	0.964554i	$0.585004\pi$
$660$	0	0
$661$	8.20135	0.318996	0.159498	−	0.987198i	$-0.449013\pi$
$661$	8.20135	0.318996	0.159498	+	0.987198i	$0.449013\pi$
$662$	−1.22389	−0.0475678
$663$	0	0
$664$	−6.71839	−0.260724
$665$	35.1554	1.36327
$666$	0	0
$667$	−46.4230	−1.79750
$668$	7.70256	0.298021
$669$	0	0
$670$	−27.1864	−1.05030
$671$	−3.95027	−0.152499
$672$	0	0
$673$	−22.6341	−0.872479	−0.436239	−	0.899831i	$-0.643690\pi$
$673$	−22.6341	−0.872479	−0.436239	+	0.899831i	$0.643690\pi$
$674$	−14.6924	−0.565929
$675$	0	0
$676$	1.00000	0.0384615
$677$	−8.50958	−0.327050	−0.163525	−	0.986539i	$-0.552286\pi$
$677$	−8.50958	−0.327050	−0.163525	+	0.986539i	$0.552286\pi$
$678$	0	0
$679$	19.2993	0.740638
$680$	4.94663	0.189694
$681$	0	0
$682$	−1.17595	−0.0450296
$683$	−23.2349	−0.889057	−0.444529	−	0.895765i	$-0.646629\pi$
$683$	−23.2349	−0.889057	−0.444529	+	0.895765i	$0.646629\pi$
$684$	0	0
$685$	5.00605	0.191271
$686$	−1.69332	−0.0646511
$687$	0	0
$688$	5.84149	0.222705
$689$	−9.55392	−0.363975
$690$	0	0
$691$	21.7161	0.826120	0.413060	−	0.910704i	$-0.364460\pi$
$691$	21.7161	0.826120	0.413060	+	0.910704i	$0.364460\pi$
$692$	14.9308	0.567584
$693$	0	0
$694$	−23.3045	−0.884628
$695$	7.64193	0.289875
$696$	0	0
$697$	0.375309	0.0142158
$698$	−30.1334	−1.14057
$699$	0	0
$700$	21.9012	0.827789
$701$	18.4073	0.695235	0.347617	−	0.937637i	$-0.386991\pi$
$701$	18.4073	0.695235	0.347617	+	0.937637i	$0.386991\pi$
$702$	0	0
$703$	4.00146	0.150918
$704$	−1.17595	−0.0443205
$705$	0	0
$706$	−30.2593	−1.13883
$707$	24.4904	0.921058
$708$	0	0
$709$	−12.1127	−0.454903	−0.227452	−	0.973789i	$-0.573039\pi$
$709$	−12.1127	−0.454903	−0.227452	+	0.973789i	$0.573039\pi$
$710$	−14.0669	−0.527922
$711$	0	0
$712$	−9.00805	−0.337591
$713$	5.32669	0.199486
$714$	0	0
$715$	−3.85784	−0.144275
$716$	1.81885	0.0679736
$717$	0	0
$718$	−27.4164	−1.02317
$719$	−33.0590	−1.23289	−0.616446	−	0.787397i	$-0.711428\pi$
$719$	−33.0590	−1.23289	−0.616446	+	0.787397i	$0.711428\pi$
$720$	0	0
$721$	−67.2817	−2.50570
$722$	11.0504	0.411254
$723$	0	0
$724$	−17.5553	−0.652439
$725$	50.2201	1.86513
$726$	0	0
$727$	−13.1750	−0.488635	−0.244318	−	0.969695i	$-0.578564\pi$
$727$	−13.1750	−0.488635	−0.244318	+	0.969695i	$0.578564\pi$
$728$	3.80073	0.140864
$729$	0	0
$730$	24.6864	0.913685
$731$	8.80802	0.325777
$732$	0	0
$733$	22.4664	0.829817	0.414908	−	0.909863i	$-0.363814\pi$
$733$	22.4664	0.829817	0.414908	+	0.909863i	$0.363814\pi$
$734$	1.11736	0.0412427
$735$	0	0
$736$	5.32669	0.196344
$737$	9.74515	0.358967
$738$	0	0
$739$	−30.3685	−1.11712	−0.558561	−	0.829463i	$-0.688646\pi$
$739$	−30.3685	−1.11712	−0.558561	+	0.829463i	$0.688646\pi$
$740$	4.65587	0.171153
$741$	0	0
$742$	−36.3118	−1.33305
$743$	−23.4031	−0.858578	−0.429289	−	0.903167i	$-0.641236\pi$
$743$	−23.4031	−0.858578	−0.429289	+	0.903167i	$0.641236\pi$
$744$	0	0
$745$	51.0810	1.87146
$746$	8.07205	0.295539
$747$	0	0
$748$	−1.77315	−0.0648328
$749$	21.1224	0.771797
$750$	0	0
$751$	27.6332	1.00835	0.504175	−	0.863601i	$-0.331797\pi$
$751$	27.6332	1.00835	0.504175	+	0.863601i	$0.331797\pi$
$752$	−10.3034	−0.375725
$753$	0	0
$754$	8.71517	0.317388
$755$	−33.5420	−1.22072
$756$	0	0
$757$	−0.475687	−0.0172891	−0.00864457	−	0.999963i	$-0.502752\pi$
$757$	−0.475687	−0.0172891	−0.00864457	+	0.999963i	$0.502752\pi$
$758$	34.1167	1.23917
$759$	0	0
$760$	−9.24966	−0.335521
$761$	−0.549865	−0.0199326	−0.00996630	−	0.999950i	$-0.503172\pi$
$761$	−0.549865	−0.0199326	−0.00996630	+	0.999950i	$0.503172\pi$
$762$	0	0
$763$	58.0803	2.10265
$764$	−0.219133	−0.00792797
$765$	0	0
$766$	12.8136	0.462973
$767$	9.72646	0.351202
$768$	0	0
$769$	33.1770	1.19639	0.598197	−	0.801349i	$-0.295884\pi$
$769$	33.1770	1.19639	0.598197	+	0.801349i	$0.295884\pi$
$770$	−14.6626	−0.528404
$771$	0	0
$772$	−23.0468	−0.829473
$773$	8.70307	0.313028	0.156514	−	0.987676i	$-0.449974\pi$
$773$	8.70307	0.313028	0.156514	+	0.987676i	$0.449974\pi$
$774$	0	0
$775$	−5.76238	−0.206991
$776$	−5.07778	−0.182282
$777$	0	0
$778$	−30.8506	−1.10605
$779$	−0.701787	−0.0251441
$780$	0	0
$781$	5.04238	0.180430
$782$	8.03178	0.287216
$783$	0	0
$784$	7.44552	0.265912
$785$	−51.0574	−1.82232
$786$	0	0
$787$	28.2636	1.00749	0.503744	−	0.863853i	$-0.331955\pi$
$787$	28.2636	1.00749	0.503744	+	0.863853i	$0.331955\pi$
$788$	19.6740	0.700859
$789$	0	0
$790$	−20.0338	−0.712772
$791$	39.2666	1.39616
$792$	0	0
$793$	−3.35920	−0.119289
$794$	11.9150	0.422848
$795$	0	0
$796$	22.8733	0.810723
$797$	13.8260	0.489743	0.244872	−	0.969556i	$-0.421254\pi$
$797$	13.8260	0.489743	0.244872	+	0.969556i	$0.421254\pi$
$798$	0	0
$799$	−15.5358	−0.549618
$800$	−5.76238	−0.203731
$801$	0	0
$802$	−36.2094	−1.27860
$803$	−8.84901	−0.312275
$804$	0	0
$805$	66.4168	2.34089
$806$	−1.00000	−0.0352235
$807$	0	0
$808$	−6.44362	−0.226686
$809$	14.3023	0.502841	0.251420	−	0.967878i	$-0.419102\pi$
$809$	14.3023	0.502841	0.251420	+	0.967878i	$0.419102\pi$
$810$	0	0
$811$	35.3622	1.24173	0.620867	−	0.783916i	$-0.286781\pi$
$811$	35.3622	1.24173	0.620867	+	0.783916i	$0.286781\pi$
$812$	33.1240	1.16242
$813$	0	0
$814$	−1.66893	−0.0584959
$815$	41.2954	1.44651
$816$	0	0
$817$	−16.4701	−0.576215
$818$	−29.1377	−1.01878
$819$	0	0
$820$	−0.816559	−0.0285155
$821$	−26.4525	−0.923199	−0.461600	−	0.887088i	$-0.652724\pi$
$821$	−26.4525	−0.923199	−0.461600	+	0.887088i	$0.652724\pi$
$822$	0	0
$823$	43.7152	1.52382	0.761908	−	0.647685i	$-0.224263\pi$
$823$	43.7152	1.52382	0.761908	+	0.647685i	$0.224263\pi$
$824$	17.7023	0.616690
$825$	0	0
$826$	36.9676	1.28627
$827$	33.3321	1.15907	0.579536	−	0.814947i	$-0.303234\pi$
$827$	33.3321	1.15907	0.579536	+	0.814947i	$0.303234\pi$
$828$	0	0
$829$	43.0789	1.49619	0.748096	−	0.663590i	$-0.230968\pi$
$829$	43.0789	1.49619	0.748096	+	0.663590i	$0.230968\pi$
$830$	22.0404	0.765033
$831$	0	0
$832$	−1.00000	−0.0346688
$833$	11.2266	0.388980
$834$	0	0
$835$	−25.2691	−0.874473
$836$	3.31560	0.114672
$837$	0	0
$838$	−10.4605	−0.361353
$839$	−15.6465	−0.540177	−0.270088	−	0.962836i	$-0.587053\pi$
$839$	−15.6465	−0.540177	−0.270088	+	0.962836i	$0.587053\pi$
$840$	0	0
$841$	46.9541	1.61911
$842$	−26.1738	−0.902008
$843$	0	0
$844$	−2.85909	−0.0984139
$845$	−3.28061	−0.112856
$846$	0	0
$847$	−36.5521	−1.25594
$848$	9.55392	0.328083
$849$	0	0
$850$	−8.68874	−0.298021
$851$	7.55969	0.259143
$852$	0	0
$853$	−16.9731	−0.581147	−0.290573	−	0.956853i	$-0.593846\pi$
$853$	−16.9731	−0.581147	−0.290573	+	0.956853i	$0.593846\pi$
$854$	−12.7674	−0.436892
$855$	0	0
$856$	−5.55747	−0.189950
$857$	27.1793	0.928427	0.464214	−	0.885723i	$-0.346337\pi$
$857$	27.1793	0.928427	0.464214	+	0.885723i	$0.346337\pi$
$858$	0	0
$859$	−29.2226	−0.997064	−0.498532	−	0.866871i	$-0.666127\pi$
$859$	−29.2226	−0.997064	−0.498532	+	0.866871i	$0.666127\pi$
$860$	−19.1636	−0.653474
$861$	0	0
$862$	21.9570	0.747857
$863$	−55.4993	−1.88922	−0.944610	−	0.328196i	$-0.893559\pi$
$863$	−55.4993	−1.88922	−0.944610	+	0.328196i	$0.893559\pi$
$864$	0	0
$865$	−48.9821	−1.66544
$866$	−29.9124	−1.01646
$867$	0	0
$868$	−3.80073	−0.129005
$869$	7.18126	0.243608
$870$	0	0
$871$	8.28701	0.280795
$872$	−15.2814	−0.517492
$873$	0	0
$874$	−15.0186	−0.508011
$875$	−9.50588	−0.321357
$876$	0	0
$877$	−34.9732	−1.18096	−0.590480	−	0.807052i	$-0.701061\pi$
$877$	−34.9732	−1.18096	−0.590480	+	0.807052i	$0.701061\pi$
$878$	−32.5127	−1.09725
$879$	0	0
$880$	3.85784	0.130048
$881$	−43.8379	−1.47694	−0.738468	−	0.674289i	$-0.764450\pi$
$881$	−43.8379	−1.47694	−0.738468	+	0.674289i	$0.764450\pi$
$882$	0	0
$883$	26.4608	0.890475	0.445238	−	0.895412i	$-0.353119\pi$
$883$	26.4608	0.890475	0.445238	+	0.895412i	$0.353119\pi$
$884$	−1.50784	−0.0507141
$885$	0	0
$886$	−29.5405	−0.992433
$887$	4.62958	0.155446	0.0777231	−	0.996975i	$-0.475235\pi$
$887$	4.62958	0.155446	0.0777231	+	0.996975i	$0.475235\pi$
$888$	0	0
$889$	−36.7168	−1.23144
$890$	29.5519	0.990581
$891$	0	0
$892$	11.8268	0.395990
$893$	29.0503	0.972132
$894$	0	0
$895$	−5.96693	−0.199452
$896$	−3.80073	−0.126973
$897$	0	0
$898$	−37.6243	−1.25554
$899$	−8.71517	−0.290667
$900$	0	0
$901$	14.4058	0.479926
$902$	0.292701	0.00974588
$903$	0	0
$904$	−10.3313	−0.343615
$905$	57.5922	1.91443
$906$	0	0
$907$	−14.4895	−0.481115	−0.240557	−	0.970635i	$-0.577330\pi$
$907$	−14.4895	−0.481115	−0.240557	+	0.970635i	$0.577330\pi$
$908$	−28.8792	−0.958389
$909$	0	0
$910$	−12.4687	−0.413333
$911$	18.2105	0.603340	0.301670	−	0.953412i	$-0.402456\pi$
$911$	18.2105	0.603340	0.301670	+	0.953412i	$0.402456\pi$
$912$	0	0
$913$	−7.90052	−0.261469
$914$	−28.2803	−0.935430
$915$	0	0
$916$	9.93379	0.328222
$917$	29.5266	0.975055
$918$	0	0
$919$	−2.40156	−0.0792202	−0.0396101	−	0.999215i	$-0.512612\pi$
$919$	−2.40156	−0.0792202	−0.0396101	+	0.999215i	$0.512612\pi$
$920$	−17.4748	−0.576126
$921$	0	0
$922$	−11.2717	−0.371214
$923$	4.28790	0.141138
$924$	0	0
$925$	−8.17803	−0.268892
$926$	−18.7255	−0.615358
$927$	0	0
$928$	−8.71517	−0.286089
$929$	−42.0501	−1.37962	−0.689810	−	0.723990i	$-0.742306\pi$
$929$	−42.0501	−1.37962	−0.689810	+	0.723990i	$0.742306\pi$
$930$	0	0
$931$	−20.9926	−0.688006
$932$	−6.56766	−0.215131
$933$	0	0
$934$	−36.6054	−1.19776
$935$	5.81701	0.190236
$936$	0	0
$937$	42.6350	1.39282	0.696412	−	0.717642i	$-0.254779\pi$
$937$	42.6350	1.39282	0.696412	+	0.717642i	$0.254779\pi$
$938$	31.4967	1.02840
$939$	0	0
$940$	33.8013	1.10248
$941$	−20.3235	−0.662527	−0.331264	−	0.943538i	$-0.607475\pi$
$941$	−20.3235	−0.662527	−0.331264	+	0.943538i	$0.607475\pi$
$942$	0	0
$943$	−1.32584	−0.0431753
$944$	−9.72646	−0.316569
$945$	0	0
$946$	6.86933	0.223341
$947$	60.5428	1.96738	0.983689	−	0.179879i	$-0.0575706\pi$
$947$	60.5428	1.96738	0.983689	+	0.179879i	$0.0575706\pi$
$948$	0	0
$949$	−7.52495	−0.244270
$950$	16.2470	0.527123
$951$	0	0
$952$	−5.73088	−0.185739
$953$	−19.1024	−0.618789	−0.309394	−	0.950934i	$-0.600126\pi$
$953$	−19.1024	−0.618789	−0.309394	+	0.950934i	$0.600126\pi$
$954$	0	0
$955$	0.718890	0.0232627
$956$	−23.6497	−0.764887
$957$	0	0
$958$	39.1426	1.26464
$959$	−5.79972	−0.187283
$960$	0	0
$961$	1.00000	0.0322581
$962$	−1.41921	−0.0457572
$963$	0	0
$964$	13.1223	0.422641
$965$	75.6076	2.43389
$966$	0	0
$967$	−39.6930	−1.27644	−0.638221	−	0.769854i	$-0.720329\pi$
$967$	−39.6930	−1.27644	−0.638221	+	0.769854i	$0.720329\pi$
$968$	9.61713	0.309106
$969$	0	0
$970$	16.6582	0.534863
$971$	36.4222	1.16884	0.584422	−	0.811450i	$-0.301321\pi$
$971$	36.4222	1.16884	0.584422	+	0.811450i	$0.301321\pi$
$972$	0	0
$973$	−8.85351	−0.283830
$974$	−24.3747	−0.781016
$975$	0	0
$976$	3.35920	0.107525
$977$	−1.30792	−0.0418442	−0.0209221	−	0.999781i	$-0.506660\pi$
$977$	−1.30792	−0.0418442	−0.0209221	+	0.999781i	$0.506660\pi$
$978$	0	0
$979$	−10.5931	−0.338556
$980$	−24.4258	−0.780255
$981$	0	0
$982$	6.17844	0.197162
$983$	−29.9914	−0.956576	−0.478288	−	0.878203i	$-0.658742\pi$
$983$	−29.9914	−0.956576	−0.478288	+	0.878203i	$0.658742\pi$
$984$	0	0
$985$	−64.5428	−2.05650
$986$	−13.1411	−0.418497
$987$	0	0
$988$	2.81950	0.0897002
$989$	−31.1158	−0.989425
$990$	0	0
$991$	−37.8792	−1.20327	−0.601636	−	0.798770i	$-0.705484\pi$
$991$	−37.8792	−1.20327	−0.601636	+	0.798770i	$0.705484\pi$
$992$	1.00000	0.0317500
$993$	0	0
$994$	16.2971	0.516914
$995$	−75.0384	−2.37888
$996$	0	0
$997$	13.5245	0.428326	0.214163	−	0.976798i	$-0.431298\pi$
$997$	13.5245	0.428326	0.214163	+	0.976798i	$0.431298\pi$
$998$	17.3220	0.548318
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7254.2.a.bk.1.1		6
3.2	odd	2		806.2.a.l.1.1	✓	6
12.11	even	2		6448.2.a.v.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
806.2.a.l.1.1	✓	6	3.2	odd	2
6448.2.a.v.1.6		6	12.11	even	2
7254.2.a.bk.1.1		6	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:	\( N \)	\(=\)	\( 7254 = 2 \cdot 3^{2} \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7254.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.28061 q^{5} +3.80073 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.28061 q^{5} +3.80073 q^{7} -1.00000 q^{8} +3.28061 q^{10} -1.17595 q^{11} -1.00000 q^{13} -3.80073 q^{14} +1.00000 q^{16} +1.50784 q^{17} -2.81950 q^{19} -3.28061 q^{20} +1.17595 q^{22} -5.32669 q^{23} +5.76238 q^{25} +1.00000 q^{26} +3.80073 q^{28} +8.71517 q^{29} -1.00000 q^{31} -1.00000 q^{32} -1.50784 q^{34} -12.4687 q^{35} -1.41921 q^{37} +2.81950 q^{38} +3.28061 q^{40} +0.248905 q^{41} +5.84149 q^{43} -1.17595 q^{44} +5.32669 q^{46} -10.3034 q^{47} +7.44552 q^{49} -5.76238 q^{50} -1.00000 q^{52} +9.55392 q^{53} +3.85784 q^{55} -3.80073 q^{56} -8.71517 q^{58} -9.72646 q^{59} +3.35920 q^{61} +1.00000 q^{62} +1.00000 q^{64} +3.28061 q^{65} -8.28701 q^{67} +1.50784 q^{68} +12.4687 q^{70} -4.28790 q^{71} +7.52495 q^{73} +1.41921 q^{74} -2.81950 q^{76} -4.46948 q^{77} -6.10675 q^{79} -3.28061 q^{80} -0.248905 q^{82} +6.71839 q^{83} -4.94663 q^{85} -5.84149 q^{86} +1.17595 q^{88} +9.00805 q^{89} -3.80073 q^{91} -5.32669 q^{92} +10.3034 q^{94} +9.24966 q^{95} +5.07778 q^{97} -7.44552 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} + 6 q^{4} - 3 q^{5} + 4 q^{7} - 6 q^{8}+O(q^{10}) \) 6 * q - 6 * q^2 + 6 * q^4 - 3 * q^5 + 4 * q^7 - 6 * q^8	\( 6 q - 6 q^{2} + 6 q^{4} - 3 q^{5} + 4 q^{7} - 6 q^{8} + 3 q^{10} - 6 q^{11} - 6 q^{13} - 4 q^{14} + 6 q^{16} - 10 q^{17} + 15 q^{19} - 3 q^{20} + 6 q^{22} - 2 q^{23} + 13 q^{25} + 6 q^{26} + 4 q^{28} + 3 q^{29} - 6 q^{31} - 6 q^{32} + 10 q^{34} + 6 q^{35} - 26 q^{37} - 15 q^{38} + 3 q^{40} + 2 q^{41} + 3 q^{43} - 6 q^{44} + 2 q^{46} - 12 q^{47} + 24 q^{49} - 13 q^{50} - 6 q^{52} + 25 q^{53} - 6 q^{55} - 4 q^{56} - 3 q^{58} - 3 q^{59} + 17 q^{61} + 6 q^{62} + 6 q^{64} + 3 q^{65} + 3 q^{67} - 10 q^{68} - 6 q^{70} - 8 q^{71} - 19 q^{73} + 26 q^{74} + 15 q^{76} + 10 q^{77} - 8 q^{79} - 3 q^{80} - 2 q^{82} + 4 q^{83} - 44 q^{85} - 3 q^{86} + 6 q^{88} + 5 q^{89} - 4 q^{91} - 2 q^{92} + 12 q^{94} + 38 q^{95} - 24 q^{98}+O(q^{100}) \) 6 * q - 6 * q^2 + 6 * q^4 - 3 * q^5 + 4 * q^7 - 6 * q^8 + 3 * q^10 - 6 * q^11 - 6 * q^13 - 4 * q^14 + 6 * q^16 - 10 * q^17 + 15 * q^19 - 3 * q^20 + 6 * q^22 - 2 * q^23 + 13 * q^25 + 6 * q^26 + 4 * q^28 + 3 * q^29 - 6 * q^31 - 6 * q^32 + 10 * q^34 + 6 * q^35 - 26 * q^37 - 15 * q^38 + 3 * q^40 + 2 * q^41 + 3 * q^43 - 6 * q^44 + 2 * q^46 - 12 * q^47 + 24 * q^49 - 13 * q^50 - 6 * q^52 + 25 * q^53 - 6 * q^55 - 4 * q^56 - 3 * q^58 - 3 * q^59 + 17 * q^61 + 6 * q^62 + 6 * q^64 + 3 * q^65 + 3 * q^67 - 10 * q^68 - 6 * q^70 - 8 * q^71 - 19 * q^73 + 26 * q^74 + 15 * q^76 + 10 * q^77 - 8 * q^79 - 3 * q^80 - 2 * q^82 + 4 * q^83 - 44 * q^85 - 3 * q^86 + 6 * q^88 + 5 * q^89 - 4 * q^91 - 2 * q^92 + 12 * q^94 + 38 * q^95 - 24 * q^98

Introduction

L-functions

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