LMFDB - Embedded newform 7245.2.a.bv.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7245,2,Mod(1,7245)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7245, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7245.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7245 = 3^{2} \cdot 5 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7245.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:	$57.8516162644$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 16x^{8} + 30x^{7} + 87x^{6} - 143x^{5} - 196x^{4} + 244x^{3} + 160x^{2} - 89x - 12 $ x^10 - 2x^9 - 16x^8 + 30x^7 + 87x^6 - 143x^5 - 196x^4 + 244x^3 + 160x^2 - 89*x - 12
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2415)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-1.78717$ of defining polynomial
Character	$\chi$	$=$	7245.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.78717 q^{2} +1.19399 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.44048 q^{8} +O(q^{10})$	\(q+1.78717 q^{2} +1.19399 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.44048 q^{8} -1.78717 q^{10} +5.76695 q^{11} +4.94069 q^{13} +1.78717 q^{14} -4.96237 q^{16} +1.91492 q^{17} +5.05003 q^{19} -1.19399 q^{20} +10.3066 q^{22} +1.00000 q^{23} +1.00000 q^{25} +8.82988 q^{26} +1.19399 q^{28} -5.57387 q^{29} -9.61929 q^{31} -5.98766 q^{32} +3.42229 q^{34} -1.00000 q^{35} +2.49363 q^{37} +9.02529 q^{38} +1.44048 q^{40} -4.66157 q^{41} +6.84362 q^{43} +6.88570 q^{44} +1.78717 q^{46} -2.41954 q^{47} +1.00000 q^{49} +1.78717 q^{50} +5.89915 q^{52} -5.06154 q^{53} -5.76695 q^{55} -1.44048 q^{56} -9.96147 q^{58} +4.43104 q^{59} +9.83388 q^{61} -17.1913 q^{62} -0.776261 q^{64} -4.94069 q^{65} +3.52693 q^{67} +2.28639 q^{68} -1.78717 q^{70} +8.09959 q^{71} +7.86940 q^{73} +4.45655 q^{74} +6.02970 q^{76} +5.76695 q^{77} -4.46714 q^{79} +4.96237 q^{80} -8.33104 q^{82} +2.52906 q^{83} -1.91492 q^{85} +12.2307 q^{86} -8.30716 q^{88} -0.000952551 q^{89} +4.94069 q^{91} +1.19399 q^{92} -4.32414 q^{94} -5.05003 q^{95} -4.66488 q^{97} +1.78717 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{2} + 16 q^{4} - 10 q^{5} + 10 q^{7} - 6 q^{8}+O(q^{10}) $ 10 * q - 2 * q^2 + 16 * q^4 - 10 * q^5 + 10 * q^7 - 6 * q^8	$ 10 q - 2 q^{2} + 16 q^{4} - 10 q^{5} + 10 q^{7} - 6 q^{8} + 2 q^{10} - 9 q^{11} + 14 q^{13} - 2 q^{14} + 20 q^{16} - 8 q^{17} + 13 q^{19} - 16 q^{20} + 10 q^{23} + 10 q^{25} + 11 q^{26} + 16 q^{28} - 10 q^{29} + 8 q^{31} + 11 q^{32} - 5 q^{34} - 10 q^{35} + 8 q^{37} + 10 q^{38} + 6 q^{40} + 5 q^{41} + 4 q^{43} - 3 q^{44} - 2 q^{46} - q^{47} + 10 q^{49} - 2 q^{50} + 14 q^{52} - 9 q^{53} + 9 q^{55} - 6 q^{56} - 28 q^{58} + 17 q^{59} + 19 q^{61} + 28 q^{62} + 24 q^{64} - 14 q^{65} - 8 q^{68} + 2 q^{70} + 6 q^{73} - 3 q^{74} + 15 q^{76} - 9 q^{77} + 32 q^{79} - 20 q^{80} + 14 q^{82} + 2 q^{83} + 8 q^{85} - 2 q^{86} - 3 q^{88} - 10 q^{89} + 14 q^{91} + 16 q^{92} - 10 q^{94} - 13 q^{95} + 18 q^{97} - 2 q^{98}+O(q^{100}) $ 10 * q - 2 * q^2 + 16 * q^4 - 10 * q^5 + 10 * q^7 - 6 * q^8 + 2 * q^10 - 9 * q^11 + 14 * q^13 - 2 * q^14 + 20 * q^16 - 8 * q^17 + 13 * q^19 - 16 * q^20 + 10 * q^23 + 10 * q^25 + 11 * q^26 + 16 * q^28 - 10 * q^29 + 8 * q^31 + 11 * q^32 - 5 * q^34 - 10 * q^35 + 8 * q^37 + 10 * q^38 + 6 * q^40 + 5 * q^41 + 4 * q^43 - 3 * q^44 - 2 * q^46 - q^47 + 10 * q^49 - 2 * q^50 + 14 * q^52 - 9 * q^53 + 9 * q^55 - 6 * q^56 - 28 * q^58 + 17 * q^59 + 19 * q^61 + 28 * q^62 + 24 * q^64 - 14 * q^65 - 8 * q^68 + 2 * q^70 + 6 * q^73 - 3 * q^74 + 15 * q^76 - 9 * q^77 + 32 * q^79 - 20 * q^80 + 14 * q^82 + 2 * q^83 + 8 * q^85 - 2 * q^86 - 3 * q^88 - 10 * q^89 + 14 * q^91 + 16 * q^92 - 10 * q^94 - 13 * q^95 + 18 * q^97 - 2 * 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 $

$2$	1.78717	1.26372	0.631862	−	0.775081i	$-0.282291\pi$
$2$	1.78717	1.26372	0.631862	+	0.775081i	$0.282291\pi$
$3$	0	0
$3$	0	0
$4$	1.19399	0.596996
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.44048	−0.509285
$9$	0	0
$10$	−1.78717	−0.565154
$11$	5.76695	1.73880	0.869401	−	0.494107i	$-0.164505\pi$
$11$	5.76695	1.73880	0.869401	+	0.494107i	$0.164505\pi$
$12$	0	0
$13$	4.94069	1.37030	0.685151	−	0.728401i	$-0.259736\pi$
$13$	4.94069	1.37030	0.685151	+	0.728401i	$0.259736\pi$
$14$	1.78717	0.477642
$15$	0	0
$16$	−4.96237	−1.24059
$17$	1.91492	0.464435	0.232218	−	0.972664i	$-0.425402\pi$
$17$	1.91492	0.464435	0.232218	+	0.972664i	$0.425402\pi$
$18$	0	0
$19$	5.05003	1.15856	0.579279	−	0.815130i	$-0.303334\pi$
$19$	5.05003	1.15856	0.579279	+	0.815130i	$0.303334\pi$
$20$	−1.19399	−0.266985
$21$	0	0
$22$	10.3066	2.19736
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	8.82988	1.73168
$27$	0	0
$28$	1.19399	0.225643
$29$	−5.57387	−1.03504	−0.517521	−	0.855671i	$-0.673145\pi$
$29$	−5.57387	−1.03504	−0.517521	+	0.855671i	$0.673145\pi$
$30$	0	0
$31$	−9.61929	−1.72768	−0.863838	−	0.503771i	$-0.831946\pi$
$31$	−9.61929	−1.72768	−0.863838	+	0.503771i	$0.831946\pi$
$32$	−5.98766	−1.05848
$33$	0	0
$34$	3.42229	0.586918
$35$	−1.00000	−0.169031
$36$	0	0
$37$	2.49363	0.409950	0.204975	−	0.978767i	$-0.434289\pi$
$37$	2.49363	0.409950	0.204975	+	0.978767i	$0.434289\pi$
$38$	9.02529	1.46410
$39$	0	0
$40$	1.44048	0.227759
$41$	−4.66157	−0.728015	−0.364007	−	0.931396i	$-0.618592\pi$
$41$	−4.66157	−0.728015	−0.364007	+	0.931396i	$0.618592\pi$
$42$	0	0
$43$	6.84362	1.04364	0.521821	−	0.853055i	$-0.325253\pi$
$43$	6.84362	1.04364	0.521821	+	0.853055i	$0.325253\pi$
$44$	6.88570	1.03806
$45$	0	0
$46$	1.78717	0.263504
$47$	−2.41954	−0.352926	−0.176463	−	0.984307i	$-0.556466\pi$
$47$	−2.41954	−0.352926	−0.176463	+	0.984307i	$0.556466\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.78717	0.252745
$51$	0	0
$52$	5.89915	0.818065
$53$	−5.06154	−0.695256	−0.347628	−	0.937633i	$-0.613013\pi$
$53$	−5.06154	−0.695256	−0.347628	+	0.937633i	$0.613013\pi$
$54$	0	0
$55$	−5.76695	−0.777616
$56$	−1.44048	−0.192492
$57$	0	0
$58$	−9.96147	−1.30801
$59$	4.43104	0.576873	0.288436	−	0.957499i	$-0.406865\pi$
$59$	4.43104	0.576873	0.288436	+	0.957499i	$0.406865\pi$
$60$	0	0
$61$	9.83388	1.25910	0.629550	−	0.776960i	$-0.283239\pi$
$61$	9.83388	1.25910	0.629550	+	0.776960i	$0.283239\pi$
$62$	−17.1913	−2.18330
$63$	0	0
$64$	−0.776261	−0.0970326
$65$	−4.94069	−0.612818
$66$	0	0
$67$	3.52693	0.430883	0.215442	−	0.976517i	$-0.430881\pi$
$67$	3.52693	0.430883	0.215442	+	0.976517i	$0.430881\pi$
$68$	2.28639	0.277266
$69$	0	0
$70$	−1.78717	−0.213608
$71$	8.09959	0.961244	0.480622	−	0.876928i	$-0.340411\pi$
$71$	8.09959	0.961244	0.480622	+	0.876928i	$0.340411\pi$
$72$	0	0
$73$	7.86940	0.921044	0.460522	−	0.887648i	$-0.347662\pi$
$73$	7.86940	0.921044	0.460522	+	0.887648i	$0.347662\pi$
$74$	4.45655	0.518063
$75$	0	0
$76$	6.02970	0.691654
$77$	5.76695	0.657205
$78$	0	0
$79$	−4.46714	−0.502593	−0.251296	−	0.967910i	$-0.580857\pi$
$79$	−4.46714	−0.502593	−0.251296	+	0.967910i	$0.580857\pi$
$80$	4.96237	0.554810
$81$	0	0
$82$	−8.33104	−0.920009
$83$	2.52906	0.277600	0.138800	−	0.990320i	$-0.455675\pi$
$83$	2.52906	0.277600	0.138800	+	0.990320i	$0.455675\pi$
$84$	0	0
$85$	−1.91492	−0.207702
$86$	12.2307	1.31888
$87$	0	0
$88$	−8.30716	−0.885547
$89$	−0.000952551 0	−0.000100970 0	−5.04851e−5	−	1.00000i	$-0.500016\pi$
$89$	−0.000952551 0	−0.000100970 0	−5.04851e−5		1.00000i	$0.500016\pi$
$90$	0	0
$91$	4.94069	0.517925
$92$	1.19399	0.124482
$93$	0	0
$94$	−4.32414	−0.446001
$95$	−5.05003	−0.518123
$96$	0	0
$97$	−4.66488	−0.473647	−0.236823	−	0.971553i	$-0.576106\pi$
$97$	−4.66488	−0.473647	−0.236823	+	0.971553i	$0.576106\pi$
$98$	1.78717	0.180532
$99$	0	0
$100$	1.19399	0.119399
$101$	−7.30587	−0.726961	−0.363481	−	0.931602i	$-0.618412\pi$
$101$	−7.30587	−0.726961	−0.363481	+	0.931602i	$0.618412\pi$
$102$	0	0
$103$	15.5664	1.53380	0.766902	−	0.641764i	$-0.221797\pi$
$103$	15.5664	1.53380	0.766902	+	0.641764i	$0.221797\pi$
$104$	−7.11695	−0.697875
$105$	0	0
$106$	−9.04585	−0.878611
$107$	2.54596	0.246127	0.123063	−	0.992399i	$-0.460728\pi$
$107$	2.54596	0.246127	0.123063	+	0.992399i	$0.460728\pi$
$108$	0	0
$109$	−7.03401	−0.673736	−0.336868	−	0.941552i	$-0.609368\pi$
$109$	−7.03401	−0.673736	−0.336868	+	0.941552i	$0.609368\pi$
$110$	−10.3066	−0.982691
$111$	0	0
$112$	−4.96237	−0.468900
$113$	10.9575	1.03079	0.515397	−	0.856952i	$-0.327644\pi$
$113$	10.9575	1.03079	0.515397	+	0.856952i	$0.327644\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	−6.65515	−0.617916
$117$	0	0
$118$	7.91905	0.729008
$119$	1.91492	0.175540
$120$	0	0
$121$	22.2578	2.02343
$122$	17.5749	1.59115
$123$	0	0
$124$	−11.4854	−1.03142
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−3.29967	−0.292798	−0.146399	−	0.989226i	$-0.546768\pi$
$127$	−3.29967	−0.292798	−0.146399	+	0.989226i	$0.546768\pi$
$128$	10.5880	0.935857
$129$	0	0
$130$	−8.82988	−0.774432
$131$	−21.8368	−1.90789	−0.953943	−	0.299988i	$-0.903017\pi$
$131$	−21.8368	−1.90789	−0.953943	+	0.299988i	$0.903017\pi$
$132$	0	0
$133$	5.05003	0.437894
$134$	6.30324	0.544517
$135$	0	0
$136$	−2.75839	−0.236530
$137$	14.4643	1.23577	0.617884	−	0.786270i	$-0.287990\pi$
$137$	14.4643	1.23577	0.617884	+	0.786270i	$0.287990\pi$
$138$	0	0
$139$	−0.321407	−0.0272614	−0.0136307	−	0.999907i	$-0.504339\pi$
$139$	−0.321407	−0.0272614	−0.0136307	+	0.999907i	$0.504339\pi$
$140$	−1.19399	−0.100911
$141$	0	0
$142$	14.4754	1.21475
$143$	28.4927	2.38268
$144$	0	0
$145$	5.57387	0.462885
$146$	14.0640	1.16394
$147$	0	0
$148$	2.97737	0.244738
$149$	−14.7093	−1.20503	−0.602516	−	0.798107i	$-0.705835\pi$
$149$	−14.7093	−1.20503	−0.602516	+	0.798107i	$0.705835\pi$
$150$	0	0
$151$	3.01776	0.245582	0.122791	−	0.992433i	$-0.460815\pi$
$151$	3.01776	0.245582	0.122791	+	0.992433i	$0.460815\pi$
$152$	−7.27446	−0.590036
$153$	0	0
$154$	10.3066	0.830526
$155$	9.61929	0.772640
$156$	0	0
$157$	16.9272	1.35093	0.675467	−	0.737390i	$-0.263942\pi$
$157$	16.9272	1.35093	0.675467	+	0.737390i	$0.263942\pi$
$158$	−7.98356	−0.635138
$159$	0	0
$160$	5.98766	0.473366
$161$	1.00000	0.0788110
$162$	0	0
$163$	−22.5392	−1.76541	−0.882703	−	0.469931i	$-0.844279\pi$
$163$	−22.5392	−1.76541	−0.882703	+	0.469931i	$0.844279\pi$
$164$	−5.56588	−0.434622
$165$	0	0
$166$	4.51987	0.350810
$167$	−0.590948	−0.0457290	−0.0228645	−	0.999739i	$-0.507279\pi$
$167$	−0.590948	−0.0457290	−0.0228645	+	0.999739i	$0.507279\pi$
$168$	0	0
$169$	11.4104	0.877727
$170$	−3.42229	−0.262478
$171$	0	0
$172$	8.17123	0.623051
$173$	9.23360	0.702017	0.351009	−	0.936372i	$-0.385839\pi$
$173$	9.23360	0.702017	0.351009	+	0.936372i	$0.385839\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	−28.6177	−2.15714
$177$	0	0
$178$	−0.00170237	−0.000127598 0
$179$	−10.2776	−0.768184	−0.384092	−	0.923295i	$-0.625486\pi$
$179$	−10.2776	−0.768184	−0.384092	+	0.923295i	$0.625486\pi$
$180$	0	0
$181$	24.9286	1.85293	0.926463	−	0.376387i	$-0.122834\pi$
$181$	24.9286	1.85293	0.926463	+	0.376387i	$0.122834\pi$
$182$	8.82988	0.654514
$183$	0	0
$184$	−1.44048	−0.106193
$185$	−2.49363	−0.183335
$186$	0	0
$187$	11.0432	0.807561
$188$	−2.88891	−0.210696
$189$	0	0
$190$	−9.02529	−0.654764
$191$	10.3252	0.747105	0.373552	−	0.927609i	$-0.378140\pi$
$191$	10.3252	0.747105	0.373552	+	0.927609i	$0.378140\pi$
$192$	0	0
$193$	12.0562	0.867826	0.433913	−	0.900955i	$-0.357133\pi$
$193$	12.0562	0.867826	0.433913	+	0.900955i	$0.357133\pi$
$194$	−8.33696	−0.598559
$195$	0	0
$196$	1.19399	0.0852851
$197$	7.99771	0.569813	0.284907	−	0.958555i	$-0.408037\pi$
$197$	7.99771	0.569813	0.284907	+	0.958555i	$0.408037\pi$
$198$	0	0
$199$	17.3059	1.22678	0.613392	−	0.789779i	$-0.289805\pi$
$199$	17.3059	1.22678	0.613392	+	0.789779i	$0.289805\pi$
$200$	−1.44048	−0.101857
$201$	0	0
$202$	−13.0569	−0.918678
$203$	−5.57387	−0.391209
$204$	0	0
$205$	4.66157	0.325578
$206$	27.8199	1.93830
$207$	0	0
$208$	−24.5175	−1.69998
$209$	29.1233	2.01450
$210$	0	0
$211$	2.35893	0.162396	0.0811978	−	0.996698i	$-0.474125\pi$
$211$	2.35893	0.162396	0.0811978	+	0.996698i	$0.474125\pi$
$212$	−6.04344	−0.415065
$213$	0	0
$214$	4.55007	0.311036
$215$	−6.84362	−0.466731
$216$	0	0
$217$	−9.61929	−0.653000
$218$	−12.5710	−0.851416
$219$	0	0
$220$	−6.88570	−0.464234
$221$	9.46101	0.636416
$222$	0	0
$223$	−27.7755	−1.85998	−0.929992	−	0.367580i	$-0.880186\pi$
$223$	−27.7755	−1.85998	−0.929992	+	0.367580i	$0.880186\pi$
$224$	−5.98766	−0.400067
$225$	0	0
$226$	19.5829	1.30264
$227$	−5.02844	−0.333749	−0.166875	−	0.985978i	$-0.553368\pi$
$227$	−5.02844	−0.333749	−0.166875	+	0.985978i	$0.553368\pi$
$228$	0	0
$229$	28.4382	1.87925	0.939624	−	0.342208i	$-0.111175\pi$
$229$	28.4382	1.87925	0.939624	+	0.342208i	$0.111175\pi$
$230$	−1.78717	−0.117843
$231$	0	0
$232$	8.02903	0.527131
$233$	17.1117	1.12103	0.560513	−	0.828146i	$-0.310604\pi$
$233$	17.1117	1.12103	0.560513	+	0.828146i	$0.310604\pi$
$234$	0	0
$235$	2.41954	0.157833
$236$	5.29063	0.344391
$237$	0	0
$238$	3.42229	0.221834
$239$	−17.3184	−1.12023	−0.560116	−	0.828414i	$-0.689244\pi$
$239$	−17.3184	−1.12023	−0.560116	+	0.828414i	$0.689244\pi$
$240$	0	0
$241$	5.81472	0.374559	0.187280	−	0.982307i	$-0.440033\pi$
$241$	5.81472	0.374559	0.187280	+	0.982307i	$0.440033\pi$
$242$	39.7785	2.55706
$243$	0	0
$244$	11.7416	0.751678
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	24.9507	1.58757
$248$	13.8564	0.879880
$249$	0	0
$250$	−1.78717	−0.113031
$251$	26.4843	1.67167	0.835837	−	0.548977i	$-0.184983\pi$
$251$	26.4843	1.67167	0.835837	+	0.548977i	$0.184983\pi$
$252$	0	0
$253$	5.76695	0.362565
$254$	−5.89708	−0.370016
$255$	0	0
$256$	20.4751	1.27970
$257$	−15.0921	−0.941422	−0.470711	−	0.882287i	$-0.656003\pi$
$257$	−15.0921	−0.941422	−0.470711	+	0.882287i	$0.656003\pi$
$258$	0	0
$259$	2.49363	0.154946
$260$	−5.89915	−0.365850
$261$	0	0
$262$	−39.0261	−2.41104
$263$	−11.7189	−0.722620	−0.361310	−	0.932446i	$-0.617670\pi$
$263$	−11.7189	−0.722620	−0.361310	+	0.932446i	$0.617670\pi$
$264$	0	0
$265$	5.06154	0.310928
$266$	9.02529	0.553376
$267$	0	0
$268$	4.21113	0.257236
$269$	5.71714	0.348580	0.174290	−	0.984694i	$-0.444237\pi$
$269$	5.71714	0.348580	0.174290	+	0.984694i	$0.444237\pi$
$270$	0	0
$271$	23.7785	1.44444	0.722220	−	0.691664i	$-0.243122\pi$
$271$	23.7785	1.44444	0.722220	+	0.691664i	$0.243122\pi$
$272$	−9.50251	−0.576175
$273$	0	0
$274$	25.8502	1.56167
$275$	5.76695	0.347760
$276$	0	0
$277$	−17.4415	−1.04796	−0.523980	−	0.851730i	$-0.675553\pi$
$277$	−17.4415	−1.04796	−0.523980	+	0.851730i	$0.675553\pi$
$278$	−0.574411	−0.0344509
$279$	0	0
$280$	1.44048	0.0860849
$281$	−22.6653	−1.35210	−0.676051	−	0.736855i	$-0.736310\pi$
$281$	−22.6653	−1.35210	−0.676051	+	0.736855i	$0.736310\pi$
$282$	0	0
$283$	31.7559	1.88769	0.943846	−	0.330385i	$-0.107179\pi$
$283$	31.7559	1.88769	0.943846	+	0.330385i	$0.107179\pi$
$284$	9.67084	0.573859
$285$	0	0
$286$	50.9215	3.01105
$287$	−4.66157	−0.275164
$288$	0	0
$289$	−13.3331	−0.784300
$290$	9.96147	0.584958
$291$	0	0
$292$	9.39600	0.549859
$293$	−22.9821	−1.34263	−0.671314	−	0.741173i	$-0.734270\pi$
$293$	−22.9821	−1.34263	−0.671314	+	0.741173i	$0.734270\pi$
$294$	0	0
$295$	−4.43104	−0.257985
$296$	−3.59201	−0.208781
$297$	0	0
$298$	−26.2881	−1.52283
$299$	4.94069	0.285728
$300$	0	0
$301$	6.84362	0.394460
$302$	5.39327	0.310348
$303$	0	0
$304$	−25.0601	−1.43730
$305$	−9.83388	−0.563087
$306$	0	0
$307$	14.3476	0.818858	0.409429	−	0.912342i	$-0.365728\pi$
$307$	14.3476	0.818858	0.409429	+	0.912342i	$0.365728\pi$
$308$	6.88570	0.392349
$309$	0	0
$310$	17.1913	0.976403
$311$	31.2661	1.77294	0.886469	−	0.462789i	$-0.153151\pi$
$311$	31.2661	1.77294	0.886469	+	0.462789i	$0.153151\pi$
$312$	0	0
$313$	−12.4626	−0.704429	−0.352215	−	0.935919i	$-0.614571\pi$
$313$	−12.4626	−0.704429	−0.352215	+	0.935919i	$0.614571\pi$
$314$	30.2518	1.70721
$315$	0	0
$316$	−5.33373	−0.300046
$317$	−33.0448	−1.85598	−0.927992	−	0.372601i	$-0.878466\pi$
$317$	−33.0448	−1.85598	−0.927992	+	0.372601i	$0.878466\pi$
$318$	0	0
$319$	−32.1442	−1.79973
$320$	0.776261	0.0433943
$321$	0	0
$322$	1.78717	0.0995953
$323$	9.67039	0.538075
$324$	0	0
$325$	4.94069	0.274060
$326$	−40.2815	−2.23099
$327$	0	0
$328$	6.71488	0.370767
$329$	−2.41954	−0.133394
$330$	0	0
$331$	12.1192	0.666130	0.333065	−	0.942904i	$-0.391917\pi$
$331$	12.1192	0.666130	0.333065	+	0.942904i	$0.391917\pi$
$332$	3.01968	0.165726
$333$	0	0
$334$	−1.05613	−0.0577887
$335$	−3.52693	−0.192697
$336$	0	0
$337$	7.17203	0.390686	0.195343	−	0.980735i	$-0.437418\pi$
$337$	7.17203	0.390686	0.195343	+	0.980735i	$0.437418\pi$
$338$	20.3925	1.10920
$339$	0	0
$340$	−2.28639	−0.123997
$341$	−55.4740	−3.00408
$342$	0	0
$343$	1.00000	0.0539949
$344$	−9.85808	−0.531512
$345$	0	0
$346$	16.5020	0.887155
$347$	−10.0503	−0.539530	−0.269765	−	0.962926i	$-0.586946\pi$
$347$	−10.0503	−0.539530	−0.269765	+	0.962926i	$0.586946\pi$
$348$	0	0
$349$	−1.66481	−0.0891152	−0.0445576	−	0.999007i	$-0.514188\pi$
$349$	−1.66481	−0.0891152	−0.0445576	+	0.999007i	$0.514188\pi$
$350$	1.78717	0.0955285
$351$	0	0
$352$	−34.5306	−1.84049
$353$	−4.57542	−0.243525	−0.121763	−	0.992559i	$-0.538855\pi$
$353$	−4.57542	−0.243525	−0.121763	+	0.992559i	$0.538855\pi$
$354$	0	0
$355$	−8.09959	−0.429882
$356$	−0.00113734	−6.02788e−5 0
$357$	0	0
$358$	−18.3679	−0.970772
$359$	−27.0943	−1.42998	−0.714992	−	0.699133i	$-0.753569\pi$
$359$	−27.0943	−1.42998	−0.714992	+	0.699133i	$0.753569\pi$
$360$	0	0
$361$	6.50285	0.342255
$362$	44.5517	2.34158
$363$	0	0
$364$	5.89915	0.309199
$365$	−7.86940	−0.411903
$366$	0	0
$367$	−11.2992	−0.589811	−0.294906	−	0.955526i	$-0.595288\pi$
$367$	−11.2992	−0.589811	−0.294906	+	0.955526i	$0.595288\pi$
$368$	−4.96237	−0.258681
$369$	0	0
$370$	−4.45655	−0.231685
$371$	−5.06154	−0.262782
$372$	0	0
$373$	−12.1176	−0.627424	−0.313712	−	0.949518i	$-0.601573\pi$
$373$	−12.1176	−0.627424	−0.313712	+	0.949518i	$0.601573\pi$
$374$	19.7362	1.02053
$375$	0	0
$376$	3.48529	0.179740
$377$	−27.5388	−1.41832
$378$	0	0
$379$	1.84435	0.0947381	0.0473690	−	0.998877i	$-0.484916\pi$
$379$	1.84435	0.0947381	0.0473690	+	0.998877i	$0.484916\pi$
$380$	−6.02970	−0.309317
$381$	0	0
$382$	18.4529	0.944133
$383$	−4.33144	−0.221326	−0.110663	−	0.993858i	$-0.535297\pi$
$383$	−4.33144	−0.221326	−0.110663	+	0.993858i	$0.535297\pi$
$384$	0	0
$385$	−5.76695	−0.293911
$386$	21.5466	1.09669
$387$	0	0
$388$	−5.56983	−0.282765
$389$	0.930716	0.0471892	0.0235946	−	0.999722i	$-0.492489\pi$
$389$	0.930716	0.0471892	0.0235946	+	0.999722i	$0.492489\pi$
$390$	0	0
$391$	1.91492	0.0968415
$392$	−1.44048	−0.0727551
$393$	0	0
$394$	14.2933	0.720086
$395$	4.46714	0.224766
$396$	0	0
$397$	−6.38273	−0.320340	−0.160170	−	0.987089i	$-0.551204\pi$
$397$	−6.38273	−0.320340	−0.160170	+	0.987089i	$0.551204\pi$
$398$	30.9287	1.55031
$399$	0	0
$400$	−4.96237	−0.248118
$401$	−2.12336	−0.106036	−0.0530178	−	0.998594i	$-0.516884\pi$
$401$	−2.12336	−0.106036	−0.0530178	+	0.998594i	$0.516884\pi$
$402$	0	0
$403$	−47.5259	−2.36744
$404$	−8.72315	−0.433993
$405$	0	0
$406$	−9.96147	−0.494380
$407$	14.3806	0.712821
$408$	0	0
$409$	−14.1723	−0.700773	−0.350386	−	0.936605i	$-0.613950\pi$
$409$	−14.1723	−0.700773	−0.350386	+	0.936605i	$0.613950\pi$
$410$	8.33104	0.411441
$411$	0	0
$412$	18.5862	0.915675
$413$	4.43104	0.218037
$414$	0	0
$415$	−2.52906	−0.124147
$416$	−29.5832	−1.45044
$417$	0	0
$418$	52.0484	2.54577
$419$	14.1577	0.691649	0.345825	−	0.938299i	$-0.387599\pi$
$419$	14.1577	0.691649	0.345825	+	0.938299i	$0.387599\pi$
$420$	0	0
$421$	21.7007	1.05763	0.528813	−	0.848738i	$-0.322637\pi$
$421$	21.7007	1.05763	0.528813	+	0.848738i	$0.322637\pi$
$422$	4.21582	0.205223
$423$	0	0
$424$	7.29103	0.354084
$425$	1.91492	0.0928871
$426$	0	0
$427$	9.83388	0.475895
$428$	3.03985	0.146937
$429$	0	0
$430$	−12.2307	−0.589819
$431$	23.7748	1.14519	0.572597	−	0.819837i	$-0.305936\pi$
$431$	23.7748	1.14519	0.572597	+	0.819837i	$0.305936\pi$
$432$	0	0
$433$	8.76012	0.420985	0.210492	−	0.977596i	$-0.432493\pi$
$433$	8.76012	0.420985	0.210492	+	0.977596i	$0.432493\pi$
$434$	−17.1913	−0.825211
$435$	0	0
$436$	−8.39855	−0.402218
$437$	5.05003	0.241576
$438$	0	0
$439$	−26.3016	−1.25531	−0.627654	−	0.778492i	$-0.715985\pi$
$439$	−26.3016	−1.25531	−0.627654	+	0.778492i	$0.715985\pi$
$440$	8.30716	0.396028
$441$	0	0
$442$	16.9085	0.804254
$443$	23.6660	1.12441	0.562203	−	0.826999i	$-0.309954\pi$
$443$	23.6660	1.12441	0.562203	+	0.826999i	$0.309954\pi$
$444$	0	0
$445$	0.000952551 0	4.51552e−5 0
$446$	−49.6396	−2.35050
$447$	0	0
$448$	−0.776261	−0.0366749
$449$	33.5290	1.58233	0.791166	−	0.611602i	$-0.209474\pi$
$449$	33.5290	1.58233	0.791166	+	0.611602i	$0.209474\pi$
$450$	0	0
$451$	−26.8831	−1.26587
$452$	13.0832	0.615380
$453$	0	0
$454$	−8.98670	−0.421767
$455$	−4.94069	−0.231623
$456$	0	0
$457$	−39.6400	−1.85428	−0.927140	−	0.374716i	$-0.877740\pi$
$457$	−39.6400	−1.85428	−0.927140	+	0.374716i	$0.877740\pi$
$458$	50.8240	2.37485
$459$	0	0
$460$	−1.19399	−0.0556702
$461$	−12.0439	−0.560940	−0.280470	−	0.959863i	$-0.590490\pi$
$461$	−12.0439	−0.560940	−0.280470	+	0.959863i	$0.590490\pi$
$462$	0	0
$463$	−26.9415	−1.25208	−0.626038	−	0.779792i	$-0.715325\pi$
$463$	−26.9415	−1.25208	−0.626038	+	0.779792i	$0.715325\pi$
$464$	27.6596	1.28406
$465$	0	0
$466$	30.5816	1.41667
$467$	1.68825	0.0781230	0.0390615	−	0.999237i	$-0.487563\pi$
$467$	1.68825	0.0781230	0.0390615	+	0.999237i	$0.487563\pi$
$468$	0	0
$469$	3.52693	0.162859
$470$	4.32414	0.199458
$471$	0	0
$472$	−6.38282	−0.293793
$473$	39.4669	1.81469
$474$	0	0
$475$	5.05003	0.231711
$476$	2.28639	0.104797
$477$	0	0
$478$	−30.9510	−1.41566
$479$	−33.6192	−1.53610	−0.768051	−	0.640388i	$-0.778773\pi$
$479$	−33.6192	−1.53610	−0.768051	+	0.640388i	$0.778773\pi$
$480$	0	0
$481$	12.3202	0.561755
$482$	10.3919	0.473339
$483$	0	0
$484$	26.5756	1.20798
$485$	4.66488	0.211821
$486$	0	0
$487$	−6.91562	−0.313377	−0.156688	−	0.987648i	$-0.550082\pi$
$487$	−6.91562	−0.313377	−0.156688	+	0.987648i	$0.550082\pi$
$488$	−14.1655	−0.641241
$489$	0	0
$490$	−1.78717	−0.0807363
$491$	−15.2501	−0.688227	−0.344114	−	0.938928i	$-0.611821\pi$
$491$	−15.2501	−0.688227	−0.344114	+	0.938928i	$0.611821\pi$
$492$	0	0
$493$	−10.6735	−0.480710
$494$	44.5912	2.00625
$495$	0	0
$496$	47.7344	2.14334
$497$	8.09959	0.363316
$498$	0	0
$499$	−19.4633	−0.871295	−0.435647	−	0.900117i	$-0.643481\pi$
$499$	−19.4633	−0.871295	−0.435647	+	0.900117i	$0.643481\pi$
$500$	−1.19399	−0.0533969
$501$	0	0
$502$	47.3321	2.11253
$503$	10.9238	0.487069	0.243534	−	0.969892i	$-0.421693\pi$
$503$	10.9238	0.487069	0.243534	+	0.969892i	$0.421693\pi$
$504$	0	0
$505$	7.30587	0.325107
$506$	10.3066	0.458182
$507$	0	0
$508$	−3.93978	−0.174799
$509$	1.04529	0.0463317	0.0231658	−	0.999732i	$-0.492625\pi$
$509$	1.04529	0.0463317	0.0231658	+	0.999732i	$0.492625\pi$
$510$	0	0
$511$	7.86940	0.348122
$512$	15.4166	0.681325
$513$	0	0
$514$	−26.9723	−1.18970
$515$	−15.5664	−0.685938
$516$	0	0
$517$	−13.9534	−0.613669
$518$	4.45655	0.195809
$519$	0	0
$520$	7.11695	0.312099
$521$	38.1598	1.67181	0.835906	−	0.548873i	$-0.184943\pi$
$521$	38.1598	1.67181	0.835906	+	0.548873i	$0.184943\pi$
$522$	0	0
$523$	37.1955	1.62645	0.813223	−	0.581953i	$-0.197711\pi$
$523$	37.1955	1.62645	0.813223	+	0.581953i	$0.197711\pi$
$524$	−26.0729	−1.13900
$525$	0	0
$526$	−20.9438	−0.913192
$527$	−18.4201	−0.802393
$528$	0	0
$529$	1.00000	0.0434783
$530$	9.04585	0.392927
$531$	0	0
$532$	6.02970	0.261421
$533$	−23.0314	−0.997600
$534$	0	0
$535$	−2.54596	−0.110071
$536$	−5.08046	−0.219443
$537$	0	0
$538$	10.2175	0.440508
$539$	5.76695	0.248400
$540$	0	0
$541$	−29.7889	−1.28073	−0.640363	−	0.768072i	$-0.721216\pi$
$541$	−29.7889	−1.28073	−0.640363	+	0.768072i	$0.721216\pi$
$542$	42.4963	1.82537
$543$	0	0
$544$	−11.4659	−0.491595
$545$	7.03401	0.301304
$546$	0	0
$547$	−29.0335	−1.24138	−0.620691	−	0.784055i	$-0.713148\pi$
$547$	−29.0335	−1.24138	−0.620691	+	0.784055i	$0.713148\pi$
$548$	17.2702	0.737748
$549$	0	0
$550$	10.3066	0.439473
$551$	−28.1482	−1.19915
$552$	0	0
$553$	−4.46714	−0.189962
$554$	−31.1711	−1.32433
$555$	0	0
$556$	−0.383758	−0.0162750
$557$	−34.1718	−1.44791	−0.723953	−	0.689850i	$-0.757677\pi$
$557$	−34.1718	−1.44791	−0.723953	+	0.689850i	$0.757677\pi$
$558$	0	0
$559$	33.8122	1.43011
$560$	4.96237	0.209698
$561$	0	0
$562$	−40.5069	−1.70868
$563$	−18.4099	−0.775887	−0.387943	−	0.921683i	$-0.626814\pi$
$563$	−18.4099	−0.775887	−0.387943	+	0.921683i	$0.626814\pi$
$564$	0	0
$565$	−10.9575	−0.460985
$566$	56.7533	2.38552
$567$	0	0
$568$	−11.6673	−0.489548
$569$	30.9396	1.29705	0.648527	−	0.761192i	$-0.275385\pi$
$569$	30.9396	1.29705	0.648527	+	0.761192i	$0.275385\pi$
$570$	0	0
$571$	−5.88560	−0.246305	−0.123152	−	0.992388i	$-0.539300\pi$
$571$	−5.88560	−0.246305	−0.123152	+	0.992388i	$0.539300\pi$
$572$	34.0201	1.42245
$573$	0	0
$574$	−8.33104	−0.347731
$575$	1.00000	0.0417029
$576$	0	0
$577$	23.3157	0.970646	0.485323	−	0.874335i	$-0.338702\pi$
$577$	23.3157	0.970646	0.485323	+	0.874335i	$0.338702\pi$
$578$	−23.8286	−0.991138
$579$	0	0
$580$	6.65515	0.276340
$581$	2.52906	0.104923
$582$	0	0
$583$	−29.1897	−1.20891
$584$	−11.3357	−0.469074
$585$	0	0
$586$	−41.0730	−1.69671
$587$	−26.6706	−1.10082	−0.550408	−	0.834896i	$-0.685528\pi$
$587$	−26.6706	−1.10082	−0.550408	+	0.834896i	$0.685528\pi$
$588$	0	0
$589$	−48.5777	−2.00161
$590$	−7.91905	−0.326022
$591$	0	0
$592$	−12.3743	−0.508580
$593$	48.4665	1.99028	0.995141	−	0.0984620i	$-0.0313923\pi$
$593$	48.4665	1.99028	0.995141	+	0.0984620i	$0.0313923\pi$
$594$	0	0
$595$	−1.91492	−0.0785039
$596$	−17.5628	−0.719399
$597$	0	0
$598$	8.82988	0.361081
$599$	−41.4387	−1.69314	−0.846571	−	0.532276i	$-0.821337\pi$
$599$	−41.4387	−1.69314	−0.846571	+	0.532276i	$0.821337\pi$
$600$	0	0
$601$	−35.4264	−1.44507	−0.722537	−	0.691332i	$-0.757024\pi$
$601$	−35.4264	−1.44507	−0.722537	+	0.691332i	$0.757024\pi$
$602$	12.2307	0.498488
$603$	0	0
$604$	3.60319	0.146612
$605$	−22.2578	−0.904907
$606$	0	0
$607$	−43.9254	−1.78288	−0.891438	−	0.453143i	$-0.850303\pi$
$607$	−43.9254	−1.78288	−0.891438	+	0.453143i	$0.850303\pi$
$608$	−30.2379	−1.22631
$609$	0	0
$610$	−17.5749	−0.711585
$611$	−11.9542	−0.483615
$612$	0	0
$613$	−35.8777	−1.44909	−0.724544	−	0.689228i	$-0.757950\pi$
$613$	−35.8777	−1.44909	−0.724544	+	0.689228i	$0.757950\pi$
$614$	25.6416	1.03481
$615$	0	0
$616$	−8.30716	−0.334705
$617$	−16.3149	−0.656811	−0.328406	−	0.944537i	$-0.606511\pi$
$617$	−16.3149	−0.656811	−0.328406	+	0.944537i	$0.606511\pi$
$618$	0	0
$619$	−36.8025	−1.47922	−0.739610	−	0.673036i	$-0.764990\pi$
$619$	−36.8025	−1.47922	−0.739610	+	0.673036i	$0.764990\pi$
$620$	11.4854	0.461263
$621$	0	0
$622$	55.8779	2.24050
$623$	−0.000952551 0	−3.81631e−5 0
$624$	0	0
$625$	1.00000	0.0400000
$626$	−22.2729	−0.890203
$627$	0	0
$628$	20.2109	0.806502
$629$	4.77508	0.190395
$630$	0	0
$631$	−27.8981	−1.11061	−0.555303	−	0.831648i	$-0.687398\pi$
$631$	−27.8981	−1.11061	−0.555303	+	0.831648i	$0.687398\pi$
$632$	6.43482	0.255963
$633$	0	0
$634$	−59.0569	−2.34545
$635$	3.29967	0.130943
$636$	0	0
$637$	4.94069	0.195757
$638$	−57.4474	−2.27436
$639$	0	0
$640$	−10.5880	−0.418528
$641$	33.9862	1.34237	0.671187	−	0.741288i	$-0.265785\pi$
$641$	33.9862	1.34237	0.671187	+	0.741288i	$0.265785\pi$
$642$	0	0
$643$	−19.7573	−0.779153	−0.389576	−	0.920994i	$-0.627379\pi$
$643$	−19.7573	−0.779153	−0.389576	+	0.920994i	$0.627379\pi$
$644$	1.19399	0.0470499
$645$	0	0
$646$	17.2827	0.679978
$647$	−7.95723	−0.312831	−0.156416	−	0.987691i	$-0.549994\pi$
$647$	−7.95723	−0.312831	−0.156416	+	0.987691i	$0.549994\pi$
$648$	0	0
$649$	25.5536	1.00307
$650$	8.82988	0.346336
$651$	0	0
$652$	−26.9116	−1.05394
$653$	21.0436	0.823498	0.411749	−	0.911297i	$-0.364918\pi$
$653$	21.0436	0.823498	0.411749	+	0.911297i	$0.364918\pi$
$654$	0	0
$655$	21.8368	0.853233
$656$	23.1324	0.903169
$657$	0	0
$658$	−4.32414	−0.168573
$659$	−38.8732	−1.51428	−0.757142	−	0.653250i	$-0.773405\pi$
$659$	−38.8732	−1.51428	−0.757142	+	0.653250i	$0.773405\pi$
$660$	0	0
$661$	−11.4815	−0.446578	−0.223289	−	0.974752i	$-0.571679\pi$
$661$	−11.4815	−0.446578	−0.223289	+	0.974752i	$0.571679\pi$
$662$	21.6591	0.841804
$663$	0	0
$664$	−3.64305	−0.141378
$665$	−5.05003	−0.195832
$666$	0	0
$667$	−5.57387	−0.215821
$668$	−0.705588	−0.0273000
$669$	0	0
$670$	−6.30324	−0.243515
$671$	56.7116	2.18933
$672$	0	0
$673$	28.9694	1.11669	0.558345	−	0.829609i	$-0.311437\pi$
$673$	28.9694	1.11669	0.558345	+	0.829609i	$0.311437\pi$
$674$	12.8177	0.493718
$675$	0	0
$676$	13.6240	0.523999
$677$	45.0231	1.73038	0.865189	−	0.501445i	$-0.167198\pi$
$677$	45.0231	1.73038	0.865189	+	0.501445i	$0.167198\pi$
$678$	0	0
$679$	−4.66488	−0.179022
$680$	2.75839	0.105779
$681$	0	0
$682$	−99.1417	−3.79633
$683$	17.6839	0.676657	0.338329	−	0.941028i	$-0.390138\pi$
$683$	17.6839	0.676657	0.338329	+	0.941028i	$0.390138\pi$
$684$	0	0
$685$	−14.4643	−0.552652
$686$	1.78717	0.0682346
$687$	0	0
$688$	−33.9606	−1.29473
$689$	−25.0075	−0.952710
$690$	0	0
$691$	−4.13911	−0.157459	−0.0787296	−	0.996896i	$-0.525086\pi$
$691$	−4.13911	−0.157459	−0.0787296	+	0.996896i	$0.525086\pi$
$692$	11.0248	0.419102
$693$	0	0
$694$	−17.9617	−0.681816
$695$	0.321407	0.0121917
$696$	0	0
$697$	−8.92651	−0.338116
$698$	−2.97530	−0.112617
$699$	0	0
$700$	1.19399	0.0451287
$701$	−10.1301	−0.382607	−0.191304	−	0.981531i	$-0.561271\pi$
$701$	−10.1301	−0.382607	−0.191304	+	0.981531i	$0.561271\pi$
$702$	0	0
$703$	12.5929	0.474950
$704$	−4.47666	−0.168721
$705$	0	0
$706$	−8.17707	−0.307748
$707$	−7.30587	−0.274766
$708$	0	0
$709$	−32.2470	−1.21106	−0.605531	−	0.795822i	$-0.707039\pi$
$709$	−32.2470	−1.21106	−0.605531	+	0.795822i	$0.707039\pi$
$710$	−14.4754	−0.543251
$711$	0	0
$712$	0.00137213	5.14226e−5 0
$713$	−9.61929	−0.360245
$714$	0	0
$715$	−28.4927	−1.06557
$716$	−12.2714	−0.458603
$717$	0	0
$718$	−48.4223	−1.80710
$719$	45.6042	1.70075	0.850374	−	0.526178i	$-0.176375\pi$
$719$	45.6042	1.70075	0.850374	+	0.526178i	$0.176375\pi$
$720$	0	0
$721$	15.5664	0.579724
$722$	11.6217	0.432516
$723$	0	0
$724$	29.7645	1.10619
$725$	−5.57387	−0.207008
$726$	0	0
$727$	34.1485	1.26650	0.633248	−	0.773949i	$-0.281721\pi$
$727$	34.1485	1.26650	0.633248	+	0.773949i	$0.281721\pi$
$728$	−7.11695	−0.263772
$729$	0	0
$730$	−14.0640	−0.520532
$731$	13.1050	0.484705
$732$	0	0
$733$	−38.7771	−1.43226	−0.716132	−	0.697965i	$-0.754089\pi$
$733$	−38.7771	−1.43226	−0.716132	+	0.697965i	$0.754089\pi$
$734$	−20.1936	−0.745358
$735$	0	0
$736$	−5.98766	−0.220708
$737$	20.3397	0.749221
$738$	0	0
$739$	−16.9026	−0.621773	−0.310887	−	0.950447i	$-0.600626\pi$
$739$	−16.9026	−0.621773	−0.310887	+	0.950447i	$0.600626\pi$
$740$	−2.97737	−0.109450
$741$	0	0
$742$	−9.04585	−0.332084
$743$	−5.83512	−0.214070	−0.107035	−	0.994255i	$-0.534136\pi$
$743$	−5.83512	−0.214070	−0.107035	+	0.994255i	$0.534136\pi$
$744$	0	0
$745$	14.7093	0.538906
$746$	−21.6562	−0.792890
$747$	0	0
$748$	13.1855	0.482111
$749$	2.54596	0.0930272
$750$	0	0
$751$	30.2412	1.10352	0.551759	−	0.834003i	$-0.313957\pi$
$751$	30.2412	1.10352	0.551759	+	0.834003i	$0.313957\pi$
$752$	12.0066	0.437837
$753$	0	0
$754$	−49.2166	−1.79236
$755$	−3.01776	−0.109828
$756$	0	0
$757$	−48.5862	−1.76589	−0.882947	−	0.469473i	$-0.844444\pi$
$757$	−48.5862	−1.76589	−0.882947	+	0.469473i	$0.844444\pi$
$758$	3.29618	0.119723
$759$	0	0
$760$	7.27446	0.263872
$761$	40.0065	1.45024	0.725118	−	0.688625i	$-0.241785\pi$
$761$	40.0065	1.45024	0.725118	+	0.688625i	$0.241785\pi$
$762$	0	0
$763$	−7.03401	−0.254648
$764$	12.3282	0.446018
$765$	0	0
$766$	−7.74103	−0.279695
$767$	21.8924	0.790490
$768$	0	0
$769$	−30.6407	−1.10493	−0.552466	−	0.833535i	$-0.686313\pi$
$769$	−30.6407	−1.10493	−0.552466	+	0.833535i	$0.686313\pi$
$770$	−10.3066	−0.371422
$771$	0	0
$772$	14.3950	0.518089
$773$	−10.5381	−0.379030	−0.189515	−	0.981878i	$-0.560692\pi$
$773$	−10.5381	−0.379030	−0.189515	+	0.981878i	$0.560692\pi$
$774$	0	0
$775$	−9.61929	−0.345535
$776$	6.71965	0.241221
$777$	0	0
$778$	1.66335	0.0596340
$779$	−23.5411	−0.843447
$780$	0	0
$781$	46.7100	1.67141
$782$	3.42229	0.122381
$783$	0	0
$784$	−4.96237	−0.177227
$785$	−16.9272	−0.604156
$786$	0	0
$787$	−23.7561	−0.846814	−0.423407	−	0.905940i	$-0.639166\pi$
$787$	−23.7561	−0.846814	−0.423407	+	0.905940i	$0.639166\pi$
$788$	9.54920	0.340176
$789$	0	0
$790$	7.98356	0.284042
$791$	10.9575	0.389603
$792$	0	0
$793$	48.5862	1.72535
$794$	−11.4071	−0.404821
$795$	0	0
$796$	20.6631	0.732385
$797$	−49.5197	−1.75408	−0.877039	−	0.480419i	$-0.840485\pi$
$797$	−49.5197	−1.75408	−0.877039	+	0.480419i	$0.840485\pi$
$798$	0	0
$799$	−4.63322	−0.163911
$800$	−5.98766	−0.211696
$801$	0	0
$802$	−3.79481	−0.134000
$803$	45.3825	1.60151
$804$	0	0
$805$	−1.00000	−0.0352454
$806$	−84.9371	−2.99178
$807$	0	0
$808$	10.5239	0.370231
$809$	4.98889	0.175400	0.0877000	−	0.996147i	$-0.472048\pi$
$809$	4.98889	0.175400	0.0877000	+	0.996147i	$0.472048\pi$
$810$	0	0
$811$	2.61994	0.0919986	0.0459993	−	0.998941i	$-0.485353\pi$
$811$	2.61994	0.0919986	0.0459993	+	0.998941i	$0.485353\pi$
$812$	−6.65515	−0.233550
$813$	0	0
$814$	25.7007	0.900809
$815$	22.5392	0.789514
$816$	0	0
$817$	34.5605	1.20912
$818$	−25.3283	−0.885583
$819$	0	0
$820$	5.56588	0.194369
$821$	14.0592	0.490670	0.245335	−	0.969438i	$-0.421102\pi$
$821$	14.0592	0.490670	0.245335	+	0.969438i	$0.421102\pi$
$822$	0	0
$823$	46.9108	1.63521	0.817603	−	0.575782i	$-0.195302\pi$
$823$	46.9108	1.63521	0.817603	+	0.575782i	$0.195302\pi$
$824$	−22.4231	−0.781144
$825$	0	0
$826$	7.91905	0.275539
$827$	−9.97661	−0.346921	−0.173460	−	0.984841i	$-0.555495\pi$
$827$	−9.97661	−0.346921	−0.173460	+	0.984841i	$0.555495\pi$
$828$	0	0
$829$	27.6546	0.960485	0.480242	−	0.877136i	$-0.340549\pi$
$829$	27.6546	0.960485	0.480242	+	0.877136i	$0.340549\pi$
$830$	−4.51987	−0.156887
$831$	0	0
$832$	−3.83527	−0.132964
$833$	1.91492	0.0663479
$834$	0	0
$835$	0.590948	0.0204506
$836$	34.7730	1.20265
$837$	0	0
$838$	25.3023	0.874053
$839$	−8.92508	−0.308128	−0.154064	−	0.988061i	$-0.549236\pi$
$839$	−8.92508	−0.308128	−0.154064	+	0.988061i	$0.549236\pi$
$840$	0	0
$841$	2.06800	0.0713104
$842$	38.7829	1.33655
$843$	0	0
$844$	2.81655	0.0969495
$845$	−11.4104	−0.392531
$846$	0	0
$847$	22.2578	0.764786
$848$	25.1172	0.862529
$849$	0	0
$850$	3.42229	0.117384
$851$	2.49363	0.0854804
$852$	0	0
$853$	−8.56231	−0.293168	−0.146584	−	0.989198i	$-0.546828\pi$
$853$	−8.56231	−0.293168	−0.146584	+	0.989198i	$0.546828\pi$
$854$	17.5749	0.601399
$855$	0	0
$856$	−3.66739	−0.125349
$857$	−7.46128	−0.254872	−0.127436	−	0.991847i	$-0.540675\pi$
$857$	−7.46128	−0.254872	−0.127436	+	0.991847i	$0.540675\pi$
$858$	0	0
$859$	28.9288	0.987038	0.493519	−	0.869735i	$-0.335710\pi$
$859$	28.9288	0.987038	0.493519	+	0.869735i	$0.335710\pi$
$860$	−8.17123	−0.278637
$861$	0	0
$862$	42.4898	1.44721
$863$	−23.3137	−0.793609	−0.396804	−	0.917903i	$-0.629881\pi$
$863$	−23.3137	−0.793609	−0.396804	+	0.917903i	$0.629881\pi$
$864$	0	0
$865$	−9.23360	−0.313952
$866$	15.6559	0.532008
$867$	0	0
$868$	−11.4854	−0.389838
$869$	−25.7618	−0.873910
$870$	0	0
$871$	17.4255	0.590440
$872$	10.1323	0.343124
$873$	0	0
$874$	9.02529	0.305285
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	4.79479	0.161908	0.0809542	−	0.996718i	$-0.474203\pi$
$877$	4.79479	0.161908	0.0809542	+	0.996718i	$0.474203\pi$
$878$	−47.0056	−1.58636
$879$	0	0
$880$	28.6177	0.964704
$881$	−12.2290	−0.412005	−0.206003	−	0.978551i	$-0.566046\pi$
$881$	−12.2290	−0.412005	−0.206003	+	0.978551i	$0.566046\pi$
$882$	0	0
$883$	−1.28633	−0.0432885	−0.0216442	−	0.999766i	$-0.506890\pi$
$883$	−1.28633	−0.0432885	−0.0216442	+	0.999766i	$0.506890\pi$
$884$	11.2964	0.379938
$885$	0	0
$886$	42.2953	1.42094
$887$	14.1304	0.474453	0.237227	−	0.971454i	$-0.423762\pi$
$887$	14.1304	0.474453	0.237227	+	0.971454i	$0.423762\pi$
$888$	0	0
$889$	−3.29967	−0.110667
$890$	0.00170237	5.70637e−5 0
$891$	0	0
$892$	−33.1637	−1.11040
$893$	−12.2188	−0.408885
$894$	0	0
$895$	10.2776	0.343542
$896$	10.5880	0.353721
$897$	0	0
$898$	59.9222	1.99963
$899$	53.6166	1.78821
$900$	0	0
$901$	−9.69242	−0.322901
$902$	−48.0447	−1.59971
$903$	0	0
$904$	−15.7840	−0.524968
$905$	−24.9286	−0.828653
$906$	0	0
$907$	18.6996	0.620910	0.310455	−	0.950588i	$-0.399519\pi$
$907$	18.6996	0.620910	0.310455	+	0.950588i	$0.399519\pi$
$908$	−6.00392	−0.199247
$909$	0	0
$910$	−8.82988	−0.292708
$911$	−31.4339	−1.04145	−0.520726	−	0.853724i	$-0.674339\pi$
$911$	−31.4339	−1.04145	−0.520726	+	0.853724i	$0.674339\pi$
$912$	0	0
$913$	14.5850	0.482692
$914$	−70.8435	−2.34330
$915$	0	0
$916$	33.9550	1.12190
$917$	−21.8368	−0.721113
$918$	0	0
$919$	−3.19367	−0.105349	−0.0526747	−	0.998612i	$-0.516775\pi$
$919$	−3.19367	−0.105349	−0.0526747	+	0.998612i	$0.516775\pi$
$920$	1.44048	0.0474911
$921$	0	0
$922$	−21.5245	−0.708873
$923$	40.0176	1.31719
$924$	0	0
$925$	2.49363	0.0819899
$926$	−48.1491	−1.58228
$927$	0	0
$928$	33.3744	1.09557
$929$	−31.0697	−1.01936	−0.509682	−	0.860363i	$-0.670237\pi$
$929$	−31.0697	−1.01936	−0.509682	+	0.860363i	$0.670237\pi$
$930$	0	0
$931$	5.05003	0.165508
$932$	20.4313	0.669248
$933$	0	0
$934$	3.01720	0.0987258
$935$	−11.0432	−0.361152
$936$	0	0
$937$	−21.1432	−0.690717	−0.345358	−	0.938471i	$-0.612243\pi$
$937$	−21.1432	−0.690717	−0.345358	+	0.938471i	$0.612243\pi$
$938$	6.30324	0.205808
$939$	0	0
$940$	2.88891	0.0942259
$941$	19.8996	0.648708	0.324354	−	0.945936i	$-0.394853\pi$
$941$	19.8996	0.648708	0.324354	+	0.945936i	$0.394853\pi$
$942$	0	0
$943$	−4.66157	−0.151802
$944$	−21.9885	−0.715664
$945$	0	0
$946$	70.5342	2.29326
$947$	18.5243	0.601959	0.300979	−	0.953631i	$-0.402686\pi$
$947$	18.5243	0.601959	0.300979	+	0.953631i	$0.402686\pi$
$948$	0	0
$949$	38.8803	1.26211
$950$	9.02529	0.292819
$951$	0	0
$952$	−2.75839	−0.0894000
$953$	5.04401	0.163391	0.0816957	−	0.996657i	$-0.473966\pi$
$953$	5.04401	0.163391	0.0816957	+	0.996657i	$0.473966\pi$
$954$	0	0
$955$	−10.3252	−0.334115
$956$	−20.6780	−0.668774
$957$	0	0
$958$	−60.0835	−1.94121
$959$	14.4643	0.467076
$960$	0	0
$961$	61.5307	1.98486
$962$	22.0184	0.709902
$963$	0	0
$964$	6.94273	0.223610
$965$	−12.0562	−0.388104
$966$	0	0
$967$	−46.3809	−1.49151	−0.745755	−	0.666220i	$-0.767911\pi$
$967$	−46.3809	−1.49151	−0.745755	+	0.666220i	$0.767911\pi$
$968$	−32.0618	−1.03050
$969$	0	0
$970$	8.33696	0.267684
$971$	33.4656	1.07396	0.536981	−	0.843594i	$-0.319565\pi$
$971$	33.4656	1.07396	0.536981	+	0.843594i	$0.319565\pi$
$972$	0	0
$973$	−0.321407	−0.0103038
$974$	−12.3594	−0.396021
$975$	0	0
$976$	−48.7993	−1.56203
$977$	56.3976	1.80432	0.902160	−	0.431401i	$-0.141981\pi$
$977$	56.3976	1.80432	0.902160	+	0.431401i	$0.141981\pi$
$978$	0	0
$979$	−0.00549332	−0.000175567 0
$980$	−1.19399	−0.0381407
$981$	0	0
$982$	−27.2546	−0.869729
$983$	37.0224	1.18083	0.590416	−	0.807099i	$-0.298964\pi$
$983$	37.0224	1.18083	0.590416	+	0.807099i	$0.298964\pi$
$984$	0	0
$985$	−7.99771	−0.254828
$986$	−19.0754	−0.607484
$987$	0	0
$988$	29.7909	0.947775
$989$	6.84362	0.217615
$990$	0	0
$991$	11.4366	0.363294	0.181647	−	0.983364i	$-0.441857\pi$
$991$	11.4366	0.363294	0.181647	+	0.983364i	$0.441857\pi$
$992$	57.5970	1.82871
$993$	0	0
$994$	14.4754	0.459131
$995$	−17.3059	−0.548634
$996$	0	0
$997$	−29.7278	−0.941490	−0.470745	−	0.882269i	$-0.656015\pi$
$997$	−29.7278	−0.941490	−0.470745	+	0.882269i	$0.656015\pi$
$998$	−34.7842	−1.10108
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7245.2.a.bv.1.8		10
3.2	odd	2		2415.2.a.w.1.3	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2415.2.a.w.1.3	✓	10	3.2	odd	2
7245.2.a.bv.1.8		10	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:	\( N \)	\(=\)	\( 7245 = 3^{2} \cdot 5 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7245.a (trivial)

\(f(q)\)	\(=\)	\(q+1.78717 q^{2} +1.19399 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.44048 q^{8} +O(q^{10})\)	\(q+1.78717 q^{2} +1.19399 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.44048 q^{8} -1.78717 q^{10} +5.76695 q^{11} +4.94069 q^{13} +1.78717 q^{14} -4.96237 q^{16} +1.91492 q^{17} +5.05003 q^{19} -1.19399 q^{20} +10.3066 q^{22} +1.00000 q^{23} +1.00000 q^{25} +8.82988 q^{26} +1.19399 q^{28} -5.57387 q^{29} -9.61929 q^{31} -5.98766 q^{32} +3.42229 q^{34} -1.00000 q^{35} +2.49363 q^{37} +9.02529 q^{38} +1.44048 q^{40} -4.66157 q^{41} +6.84362 q^{43} +6.88570 q^{44} +1.78717 q^{46} -2.41954 q^{47} +1.00000 q^{49} +1.78717 q^{50} +5.89915 q^{52} -5.06154 q^{53} -5.76695 q^{55} -1.44048 q^{56} -9.96147 q^{58} +4.43104 q^{59} +9.83388 q^{61} -17.1913 q^{62} -0.776261 q^{64} -4.94069 q^{65} +3.52693 q^{67} +2.28639 q^{68} -1.78717 q^{70} +8.09959 q^{71} +7.86940 q^{73} +4.45655 q^{74} +6.02970 q^{76} +5.76695 q^{77} -4.46714 q^{79} +4.96237 q^{80} -8.33104 q^{82} +2.52906 q^{83} -1.91492 q^{85} +12.2307 q^{86} -8.30716 q^{88} -0.000952551 q^{89} +4.94069 q^{91} +1.19399 q^{92} -4.32414 q^{94} -5.05003 q^{95} -4.66488 q^{97} +1.78717 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{2} + 16 q^{4} - 10 q^{5} + 10 q^{7} - 6 q^{8}+O(q^{10}) \) 10 * q - 2 * q^2 + 16 * q^4 - 10 * q^5 + 10 * q^7 - 6 * q^8	\( 10 q - 2 q^{2} + 16 q^{4} - 10 q^{5} + 10 q^{7} - 6 q^{8} + 2 q^{10} - 9 q^{11} + 14 q^{13} - 2 q^{14} + 20 q^{16} - 8 q^{17} + 13 q^{19} - 16 q^{20} + 10 q^{23} + 10 q^{25} + 11 q^{26} + 16 q^{28} - 10 q^{29} + 8 q^{31} + 11 q^{32} - 5 q^{34} - 10 q^{35} + 8 q^{37} + 10 q^{38} + 6 q^{40} + 5 q^{41} + 4 q^{43} - 3 q^{44} - 2 q^{46} - q^{47} + 10 q^{49} - 2 q^{50} + 14 q^{52} - 9 q^{53} + 9 q^{55} - 6 q^{56} - 28 q^{58} + 17 q^{59} + 19 q^{61} + 28 q^{62} + 24 q^{64} - 14 q^{65} - 8 q^{68} + 2 q^{70} + 6 q^{73} - 3 q^{74} + 15 q^{76} - 9 q^{77} + 32 q^{79} - 20 q^{80} + 14 q^{82} + 2 q^{83} + 8 q^{85} - 2 q^{86} - 3 q^{88} - 10 q^{89} + 14 q^{91} + 16 q^{92} - 10 q^{94} - 13 q^{95} + 18 q^{97} - 2 q^{98}+O(q^{100}) \) 10 * q - 2 * q^2 + 16 * q^4 - 10 * q^5 + 10 * q^7 - 6 * q^8 + 2 * q^10 - 9 * q^11 + 14 * q^13 - 2 * q^14 + 20 * q^16 - 8 * q^17 + 13 * q^19 - 16 * q^20 + 10 * q^23 + 10 * q^25 + 11 * q^26 + 16 * q^28 - 10 * q^29 + 8 * q^31 + 11 * q^32 - 5 * q^34 - 10 * q^35 + 8 * q^37 + 10 * q^38 + 6 * q^40 + 5 * q^41 + 4 * q^43 - 3 * q^44 - 2 * q^46 - q^47 + 10 * q^49 - 2 * q^50 + 14 * q^52 - 9 * q^53 + 9 * q^55 - 6 * q^56 - 28 * q^58 + 17 * q^59 + 19 * q^61 + 28 * q^62 + 24 * q^64 - 14 * q^65 - 8 * q^68 + 2 * q^70 + 6 * q^73 - 3 * q^74 + 15 * q^76 - 9 * q^77 + 32 * q^79 - 20 * q^80 + 14 * q^82 + 2 * q^83 + 8 * q^85 - 2 * q^86 - 3 * q^88 - 10 * q^89 + 14 * q^91 + 16 * q^92 - 10 * q^94 - 13 * q^95 + 18 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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