LMFDB - Embedded newform 275.4.a.k.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [275,4,Mod(1,275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(275, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("275.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 275 = 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	275.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:	$16.2255252516$
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} - 72x^{8} + 1771x^{6} - 17056x^{4} + 52892x^{2} - 3136 $ x^10 - 72x^8 + 1771x^6 - 17056x^4 + 52892x^2 - 3136
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$5.44091$ of defining polynomial
Character	$\chi$	$=$	275.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+5.44091 q^{2} -5.19344 q^{3} +21.6035 q^{4} -28.2570 q^{6} +18.8252 q^{7} +74.0156 q^{8} -0.0282092 q^{9} +11.0000 q^{11} -112.197 q^{12} +1.72915 q^{13} +102.426 q^{14} +229.884 q^{16} +3.07229 q^{17} -0.153484 q^{18} +123.822 q^{19} -97.7673 q^{21} +59.8500 q^{22} -92.3179 q^{23} -384.395 q^{24} +9.40815 q^{26} +140.369 q^{27} +406.690 q^{28} -119.388 q^{29} +292.648 q^{31} +658.654 q^{32} -57.1278 q^{33} +16.7161 q^{34} -0.609419 q^{36} -235.769 q^{37} +673.704 q^{38} -8.98023 q^{39} -51.2730 q^{41} -531.943 q^{42} -229.869 q^{43} +237.639 q^{44} -502.293 q^{46} -356.494 q^{47} -1193.89 q^{48} +11.3869 q^{49} -15.9557 q^{51} +37.3557 q^{52} -117.848 q^{53} +763.737 q^{54} +1393.36 q^{56} -643.062 q^{57} -649.579 q^{58} -205.210 q^{59} +490.921 q^{61} +1592.27 q^{62} -0.531043 q^{63} +1744.61 q^{64} -310.827 q^{66} -890.724 q^{67} +66.3723 q^{68} +479.447 q^{69} -655.118 q^{71} -2.08792 q^{72} +80.8916 q^{73} -1282.80 q^{74} +2674.99 q^{76} +207.077 q^{77} -48.8606 q^{78} -335.730 q^{79} -728.238 q^{81} -278.972 q^{82} -432.146 q^{83} -2112.12 q^{84} -1250.69 q^{86} +620.033 q^{87} +814.171 q^{88} +455.024 q^{89} +32.5515 q^{91} -1994.39 q^{92} -1519.85 q^{93} -1939.65 q^{94} -3420.68 q^{96} -842.417 q^{97} +61.9550 q^{98} -0.310301 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 64 q^{4} + 26 q^{6} + 152 q^{9} + 110 q^{11} + 34 q^{14} + 468 q^{16} + 90 q^{19} + 302 q^{21} + 206 q^{24} + 392 q^{26} - 58 q^{29} + 1242 q^{31} - 66 q^{34} + 1786 q^{36} - 384 q^{39} + 416 q^{41}+ \cdots + 1672 q^{99}+O(q^{100}) $ 10 * q + 64 * q^4 + 26 * q^6 + 152 * q^9 + 110 * q^11 + 34 * q^14 + 468 * q^16 + 90 * q^19 + 302 * q^21 + 206 * q^24 + 392 * q^26 - 58 * q^29 + 1242 * q^31 - 66 * q^34 + 1786 * q^36 - 384 * q^39 + 416 * q^41 + 704 * q^44 + 1816 * q^46 + 1980 * q^49 + 510 * q^51 + 1882 * q^54 + 2626 * q^56 + 476 * q^59 + 1650 * q^61 + 2576 * q^64 + 286 * q^66 + 4492 * q^69 - 498 * q^71 - 5374 * q^74 + 1410 * q^76 - 416 * q^79 - 710 * q^81 - 9018 * q^84 - 6872 * q^86 + 1918 * q^89 + 1384 * q^91 - 2860 * q^94 - 10114 * q^96 + 1672 * q^99

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 $

$2$	5.44091	1.92365	0.961826	−	0.273660i	$-0.0882344\pi$
$2$	5.44091	1.92365	0.961826	+	0.273660i	$0.0882344\pi$
$3$	−5.19344	−0.999477	−0.499739	−	0.866176i	$-0.666571\pi$
$3$	−5.19344	−0.999477	−0.499739	+	0.866176i	$0.666571\pi$
$4$	21.6035	2.70044
$5$	0	0
$5$	0	0
$6$	−28.2570	−1.92265
$7$	18.8252	1.01646	0.508232	−	0.861220i	$-0.330299\pi$
$7$	18.8252	1.01646	0.508232	+	0.861220i	$0.330299\pi$
$8$	74.0156	3.27106
$9$	−0.0282092	−0.00104479
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−112.197	−2.69903
$13$	1.72915	0.0368907	0.0184454	−	0.999830i	$-0.494128\pi$
$13$	1.72915	0.0368907	0.0184454	+	0.999830i	$0.494128\pi$
$14$	102.426	1.95532
$15$	0	0
$16$	229.884	3.59194
$17$	3.07229	0.0438317	0.0219159	−	0.999760i	$-0.493023\pi$
$17$	3.07229	0.0438317	0.0219159	+	0.999760i	$0.493023\pi$
$18$	−0.153484	−0.00200981
$19$	123.822	1.49509	0.747545	−	0.664211i	$-0.231232\pi$
$19$	123.822	1.49509	0.747545	+	0.664211i	$0.231232\pi$
$20$	0	0
$21$	−97.7673	−1.01593
$22$	59.8500	0.580003
$23$	−92.3179	−0.836939	−0.418470	−	0.908231i	$-0.637433\pi$
$23$	−92.3179	−0.836939	−0.418470	+	0.908231i	$0.637433\pi$
$24$	−384.395	−3.26935
$25$	0	0
$26$	9.40815	0.0709650
$27$	140.369	1.00052
$28$	406.690	2.74490
$29$	−119.388	−0.764475	−0.382237	−	0.924064i	$-0.624846\pi$
$29$	−119.388	−0.764475	−0.382237	+	0.924064i	$0.624846\pi$
$30$	0	0
$31$	292.648	1.69552	0.847761	−	0.530379i	$-0.177950\pi$
$31$	292.648	1.69552	0.847761	+	0.530379i	$0.177950\pi$
$32$	658.654	3.63858
$33$	−57.1278	−0.301354
$34$	16.7161	0.0843170
$35$	0	0
$36$	−0.609419	−0.00282138
$37$	−235.769	−1.04757	−0.523787	−	0.851849i	$-0.675481\pi$
$37$	−235.769	−1.04757	−0.523787	+	0.851849i	$0.675481\pi$
$38$	673.704	2.87603
$39$	−8.98023	−0.0368715
$40$	0	0
$41$	−51.2730	−0.195305	−0.0976524	−	0.995221i	$-0.531133\pi$
$41$	−51.2730	−0.195305	−0.0976524	+	0.995221i	$0.531133\pi$
$42$	−531.943	−1.95430
$43$	−229.869	−0.815224	−0.407612	−	0.913155i	$-0.633638\pi$
$43$	−229.869	−0.815224	−0.407612	+	0.913155i	$0.633638\pi$
$44$	237.639	0.814213
$45$	0	0
$46$	−502.293	−1.60998
$47$	−356.494	−1.10638	−0.553191	−	0.833055i	$-0.686590\pi$
$47$	−356.494	−1.10638	−0.553191	+	0.833055i	$0.686590\pi$
$48$	−1193.89	−3.59006
$49$	11.3869	0.0331979
$50$	0	0
$51$	−15.9557	−0.0438088
$52$	37.3557	0.0996212
$53$	−117.848	−0.305427	−0.152713	−	0.988271i	$-0.548801\pi$
$53$	−117.848	−0.305427	−0.152713	+	0.988271i	$0.548801\pi$
$54$	763.737	1.92466
$55$	0	0
$56$	1393.36	3.32491
$57$	−643.062	−1.49431
$58$	−649.579	−1.47058
$59$	−205.210	−0.452815	−0.226407	−	0.974033i	$-0.572698\pi$
$59$	−205.210	−0.452815	−0.226407	+	0.974033i	$0.572698\pi$
$60$	0	0
$61$	490.921	1.03043	0.515213	−	0.857062i	$-0.327713\pi$
$61$	490.921	1.03043	0.515213	+	0.857062i	$0.327713\pi$
$62$	1592.27	3.26159
$63$	−0.531043	−0.00106199
$64$	1744.61	3.40744
$65$	0	0
$66$	−310.827	−0.579700
$67$	−890.724	−1.62417	−0.812084	−	0.583540i	$-0.801667\pi$
$67$	−890.724	−1.62417	−0.812084	+	0.583540i	$0.801667\pi$
$68$	66.3723	0.118365
$69$	479.447	0.836502
$70$	0	0
$71$	−655.118	−1.09504	−0.547522	−	0.836791i	$-0.684429\pi$
$71$	−655.118	−1.09504	−0.547522	+	0.836791i	$0.684429\pi$
$72$	−2.08792	−0.00341756
$73$	80.8916	0.129694	0.0648469	−	0.997895i	$-0.479344\pi$
$73$	80.8916	0.129694	0.0648469	+	0.997895i	$0.479344\pi$
$74$	−1282.80	−2.01517
$75$	0	0
$76$	2674.99	4.03740
$77$	207.077	0.306475
$78$	−48.8606	−0.0709279
$79$	−335.730	−0.478134	−0.239067	−	0.971003i	$-0.576842\pi$
$79$	−335.730	−0.478134	−0.239067	+	0.971003i	$0.576842\pi$
$80$	0	0
$81$	−728.238	−0.998954
$82$	−278.972	−0.375699
$83$	−432.146	−0.571497	−0.285748	−	0.958305i	$-0.592242\pi$
$83$	−432.146	−0.571497	−0.285748	+	0.958305i	$0.592242\pi$
$84$	−2112.12	−2.74346
$85$	0	0
$86$	−1250.69	−1.56821
$87$	620.033	0.764075
$88$	814.171	0.986261
$89$	455.024	0.541937	0.270969	−	0.962588i	$-0.412656\pi$
$89$	455.024	0.541937	0.270969	+	0.962588i	$0.412656\pi$
$90$	0	0
$91$	32.5515	0.0374981
$92$	−1994.39	−2.26010
$93$	−1519.85	−1.69464
$94$	−1939.65	−2.12829
$95$	0	0
$96$	−3420.68	−3.63668
$97$	−842.417	−0.881798	−0.440899	−	0.897557i	$-0.645340\pi$
$97$	−842.417	−0.881798	−0.440899	+	0.897557i	$0.645340\pi$
$98$	61.9550	0.0638612
$99$	−0.310301	−0.000315015 0
$100$	0	0
$101$	−524.725	−0.516952	−0.258476	−	0.966018i	$-0.583220\pi$
$101$	−524.725	−0.516952	−0.258476	+	0.966018i	$0.583220\pi$
$102$	−86.8138	−0.0842730
$103$	239.172	0.228799	0.114400	−	0.993435i	$-0.463506\pi$
$103$	239.172	0.228799	0.114400	+	0.993435i	$0.463506\pi$
$104$	127.984	0.120672
$105$	0	0
$106$	−641.199	−0.587535
$107$	−891.407	−0.805379	−0.402690	−	0.915337i	$-0.631925\pi$
$107$	−891.407	−0.805379	−0.402690	+	0.915337i	$0.631925\pi$
$108$	3032.47	2.70185
$109$	1423.96	1.25129	0.625645	−	0.780108i	$-0.284836\pi$
$109$	1423.96	1.25129	0.625645	+	0.780108i	$0.284836\pi$
$110$	0	0
$111$	1224.45	1.04703
$112$	4327.61	3.65107
$113$	102.586	0.0854022	0.0427011	−	0.999088i	$-0.486404\pi$
$113$	102.586	0.0854022	0.0427011	+	0.999088i	$0.486404\pi$
$114$	−3498.84	−2.87453
$115$	0	0
$116$	−2579.20	−2.06442
$117$	−0.0487780	−3.85429e−5 0
$118$	−1116.53	−0.871058
$119$	57.8363	0.0445534
$120$	0	0
$121$	121.000	0.0909091
$122$	2671.06	1.98218
$123$	266.283	0.195203
$124$	6322.23	4.57865
$125$	0	0
$126$	−2.88936	−0.00204289
$127$	238.080	0.166348	0.0831740	−	0.996535i	$-0.473494\pi$
$127$	238.080	0.166348	0.0831740	+	0.996535i	$0.473494\pi$
$128$	4223.02	2.91614
$129$	1193.81	0.814798
$130$	0	0
$131$	532.630	0.355238	0.177619	−	0.984099i	$-0.443161\pi$
$131$	532.630	0.355238	0.177619	+	0.984099i	$0.443161\pi$
$132$	−1234.16	−0.813788
$133$	2330.97	1.51970
$134$	−4846.35	−3.12434
$135$	0	0
$136$	227.397	0.143376
$137$	478.506	0.298405	0.149203	−	0.988807i	$-0.452329\pi$
$137$	478.506	0.298405	0.149203	+	0.988807i	$0.452329\pi$
$138$	2608.63	1.60914
$139$	354.237	0.216158	0.108079	−	0.994142i	$-0.465530\pi$
$139$	354.237	0.216158	0.108079	+	0.994142i	$0.465530\pi$
$140$	0	0
$141$	1851.43	1.10580
$142$	−3564.44	−2.10649
$143$	19.0206	0.0111230
$144$	−6.48485	−0.00375281
$145$	0	0
$146$	440.124	0.249486
$147$	−59.1370	−0.0331805
$148$	−5093.45	−2.82891
$149$	3002.16	1.65065	0.825324	−	0.564660i	$-0.190993\pi$
$149$	3002.16	1.65065	0.825324	+	0.564660i	$0.190993\pi$
$150$	0	0
$151$	2231.65	1.20271	0.601356	−	0.798981i	$-0.294627\pi$
$151$	2231.65	1.20271	0.601356	+	0.798981i	$0.294627\pi$
$152$	9164.75	4.89052
$153$	−0.0866669	−4.57948e−5 0
$154$	1126.69	0.589552
$155$	0	0
$156$	−194.005	−0.0995692
$157$	283.657	0.144193	0.0720965	−	0.997398i	$-0.477031\pi$
$157$	283.657	0.144193	0.0720965	+	0.997398i	$0.477031\pi$
$158$	−1826.68	−0.919764
$159$	612.034	0.305267
$160$	0	0
$161$	−1737.90	−0.850718
$162$	−3962.28	−1.92164
$163$	3474.38	1.66953	0.834767	−	0.550603i	$-0.185602\pi$
$163$	3474.38	1.66953	0.834767	+	0.550603i	$0.185602\pi$
$164$	−1107.68	−0.527409
$165$	0	0
$166$	−2351.27	−1.09936
$167$	−1276.60	−0.591536	−0.295768	−	0.955260i	$-0.595576\pi$
$167$	−1276.60	−0.591536	−0.295768	+	0.955260i	$0.595576\pi$
$168$	−7236.30	−3.32317
$169$	−2194.01	−0.998639
$170$	0	0
$171$	−3.49292	−0.00156205
$172$	−4965.97	−2.20146
$173$	−421.138	−0.185078	−0.0925392	−	0.995709i	$-0.529498\pi$
$173$	−421.138	−0.185078	−0.0925392	+	0.995709i	$0.529498\pi$
$174$	3373.55	1.46982
$175$	0	0
$176$	2528.72	1.08301
$177$	1065.75	0.452578
$178$	2475.74	1.04250
$179$	−2631.95	−1.09900	−0.549501	−	0.835493i	$-0.685182\pi$
$179$	−2631.95	−1.09900	−0.549501	+	0.835493i	$0.685182\pi$
$180$	0	0
$181$	−2427.32	−0.996804	−0.498402	−	0.866946i	$-0.666080\pi$
$181$	−2427.32	−0.996804	−0.498402	+	0.866946i	$0.666080\pi$
$182$	177.110	0.0721333
$183$	−2549.57	−1.02989
$184$	−6832.96	−2.73768
$185$	0	0
$186$	−8269.37	−3.25989
$187$	33.7952	0.0132158
$188$	−7701.52	−2.98772
$189$	2642.48	1.01699
$190$	0	0
$191$	927.802	0.351484	0.175742	−	0.984436i	$-0.443768\pi$
$191$	927.802	0.351484	0.175742	+	0.984436i	$0.443768\pi$
$192$	−9060.51	−3.40566
$193$	1353.07	0.504643	0.252322	−	0.967643i	$-0.418806\pi$
$193$	1353.07	0.504643	0.252322	+	0.967643i	$0.418806\pi$
$194$	−4583.51	−1.69627
$195$	0	0
$196$	245.997	0.0896489
$197$	2697.55	0.975596	0.487798	−	0.872957i	$-0.337800\pi$
$197$	2697.55	0.975596	0.487798	+	0.872957i	$0.337800\pi$
$198$	−1.68832	−0.000605979 0
$199$	1796.87	0.640084	0.320042	−	0.947403i	$-0.396303\pi$
$199$	1796.87	0.640084	0.320042	+	0.947403i	$0.396303\pi$
$200$	0	0
$201$	4625.92	1.62332
$202$	−2854.98	−0.994436
$203$	−2247.50	−0.777061
$204$	−344.700	−0.118303
$205$	0	0
$206$	1301.31	0.440130
$207$	2.60422	0.000874423 0
$208$	397.504	0.132509
$209$	1362.04	0.450787
$210$	0	0
$211$	−235.773	−0.0769254	−0.0384627	−	0.999260i	$-0.512246\pi$
$211$	−235.773	−0.0769254	−0.0384627	+	0.999260i	$0.512246\pi$
$212$	−2545.92	−0.824787
$213$	3402.31	1.09447
$214$	−4850.07	−1.54927
$215$	0	0
$216$	10389.5	3.27276
$217$	5509.15	1.72344
$218$	7747.64	2.40705
$219$	−420.105	−0.129626
$220$	0	0
$221$	5.31244	0.00161699
$222$	6662.14	2.01412
$223$	−4355.88	−1.30803	−0.654017	−	0.756480i	$-0.726918\pi$
$223$	−4355.88	−1.30803	−0.654017	+	0.756480i	$0.726918\pi$
$224$	12399.3	3.69849
$225$	0	0
$226$	558.160	0.164284
$227$	3029.90	0.885910	0.442955	−	0.896544i	$-0.353930\pi$
$227$	3029.90	0.885910	0.442955	+	0.896544i	$0.353930\pi$
$228$	−13892.4	−4.03529
$229$	−6092.66	−1.75814	−0.879070	−	0.476693i	$-0.841835\pi$
$229$	−6092.66	−1.75814	−0.879070	+	0.476693i	$0.841835\pi$
$230$	0	0
$231$	−1075.44	−0.306315
$232$	−8836.56	−2.50064
$233$	1632.51	0.459008	0.229504	−	0.973308i	$-0.426290\pi$
$233$	1632.51	0.459008	0.229504	+	0.973308i	$0.426290\pi$
$234$	−0.265397	−7.41432e−5 0
$235$	0	0
$236$	−4433.26	−1.22280
$237$	1743.59	0.477884
$238$	314.682	0.0857052
$239$	5627.21	1.52299	0.761494	−	0.648172i	$-0.224466\pi$
$239$	5627.21	1.52299	0.761494	+	0.648172i	$0.224466\pi$
$240$	0	0
$241$	3486.63	0.931925	0.465962	−	0.884805i	$-0.345708\pi$
$241$	3486.63	0.931925	0.465962	+	0.884805i	$0.345708\pi$
$242$	658.350	0.174878
$243$	−7.91529	−0.00208957
$244$	10605.6	2.78260
$245$	0	0
$246$	1448.82	0.375502
$247$	214.107	0.0551550
$248$	21660.5	5.54615
$249$	2244.33	0.571198
$250$	0	0
$251$	−5704.48	−1.43452	−0.717258	−	0.696808i	$-0.754603\pi$
$251$	−5704.48	−1.43452	−0.717258	+	0.696808i	$0.754603\pi$
$252$	−11.4724	−0.00286783
$253$	−1015.50	−0.252347
$254$	1295.37	0.319996
$255$	0	0
$256$	9020.23	2.20220
$257$	2633.52	0.639200	0.319600	−	0.947553i	$-0.396451\pi$
$257$	2633.52	0.639200	0.319600	+	0.947553i	$0.396451\pi$
$258$	6495.41	1.56739
$259$	−4438.40	−1.06482
$260$	0	0
$261$	3.36784	0.000798713 0
$262$	2898.00	0.683354
$263$	−4713.39	−1.10509	−0.552547	−	0.833482i	$-0.686344\pi$
$263$	−4713.39	−1.10509	−0.552547	+	0.833482i	$0.686344\pi$
$264$	−4228.35	−0.985745
$265$	0	0
$266$	12682.6	2.92338
$267$	−2363.14	−0.541654
$268$	−19242.8	−4.38597
$269$	4145.66	0.939649	0.469825	−	0.882760i	$-0.344317\pi$
$269$	4145.66	0.939649	0.469825	+	0.882760i	$0.344317\pi$
$270$	0	0
$271$	−7155.58	−1.60395	−0.801975	−	0.597357i	$-0.796217\pi$
$271$	−7155.58	−1.60395	−0.801975	+	0.597357i	$0.796217\pi$
$272$	706.270	0.157441
$273$	−169.054	−0.0374785
$274$	2603.51	0.574028
$275$	0	0
$276$	10357.7	2.25892
$277$	−8076.45	−1.75187	−0.875933	−	0.482433i	$-0.839753\pi$
$277$	−8076.45	−1.75187	−0.875933	+	0.482433i	$0.839753\pi$
$278$	1927.37	0.415813
$279$	−8.25538	−0.00177146
$280$	0	0
$281$	1445.70	0.306915	0.153458	−	0.988155i	$-0.450959\pi$
$281$	1445.70	0.306915	0.153458	+	0.988155i	$0.450959\pi$
$282$	10073.5	2.12718
$283$	8260.14	1.73503	0.867517	−	0.497408i	$-0.165715\pi$
$283$	8260.14	1.73503	0.867517	+	0.497408i	$0.165715\pi$
$284$	−14152.8	−2.95710
$285$	0	0
$286$	103.490	0.0213967
$287$	−965.222	−0.198520
$288$	−18.5801	−0.00380154
$289$	−4903.56	−0.998079
$290$	0	0
$291$	4375.04	0.881338
$292$	1747.54	0.350230
$293$	5784.63	1.15338	0.576692	−	0.816961i	$-0.304343\pi$
$293$	5784.63	1.15338	0.576692	+	0.816961i	$0.304343\pi$
$294$	−321.759	−0.0638278
$295$	0	0
$296$	−17450.6	−3.42667
$297$	1544.06	0.301669
$298$	16334.5	3.17527
$299$	−159.631	−0.0308753
$300$	0	0
$301$	−4327.31	−0.828645
$302$	12142.2	2.31360
$303$	2725.13	0.516682
$304$	28464.7	5.37027
$305$	0	0
$306$	−0.471547	−8.80933e−5 0
$307$	9458.25	1.75834	0.879171	−	0.476506i	$-0.158097\pi$
$307$	9458.25	1.75834	0.879171	+	0.476506i	$0.158097\pi$
$308$	4473.59	0.827618
$309$	−1242.12	−0.228680
$310$	0	0
$311$	−5984.54	−1.09116	−0.545582	−	0.838057i	$-0.683691\pi$
$311$	−5984.54	−1.09116	−0.545582	+	0.838057i	$0.683691\pi$
$312$	−664.677	−0.120609
$313$	−1282.31	−0.231568	−0.115784	−	0.993274i	$-0.536938\pi$
$313$	−1282.31	−0.231568	−0.115784	+	0.993274i	$0.536938\pi$
$314$	1543.35	0.277377
$315$	0	0
$316$	−7252.95	−1.29117
$317$	1727.90	0.306147	0.153073	−	0.988215i	$-0.451083\pi$
$317$	1727.90	0.306147	0.153073	+	0.988215i	$0.451083\pi$
$318$	3330.03	0.587228
$319$	−1313.27	−0.230498
$320$	0	0
$321$	4629.47	0.804959
$322$	−9455.76	−1.63649
$323$	380.417	0.0655324
$324$	−15732.5	−2.69762
$325$	0	0
$326$	18903.8	3.21160
$327$	−7395.25	−1.25064
$328$	−3795.00	−0.638853
$329$	−6711.05	−1.12460
$330$	0	0
$331$	−2047.24	−0.339959	−0.169980	−	0.985448i	$-0.554370\pi$
$331$	−2047.24	−0.339959	−0.169980	+	0.985448i	$0.554370\pi$
$332$	−9335.89	−1.54329
$333$	6.65087	0.00109449
$334$	−6945.89	−1.13791
$335$	0	0
$336$	−22475.1	−3.64917
$337$	7259.71	1.17348	0.586738	−	0.809777i	$-0.300412\pi$
$337$	7259.71	1.17348	0.586738	+	0.809777i	$0.300412\pi$
$338$	−11937.4	−1.92103
$339$	−532.772	−0.0853576
$340$	0	0
$341$	3219.13	0.511219
$342$	−19.0047	−0.00300484
$343$	−6242.67	−0.982719
$344$	−17013.9	−2.66664
$345$	0	0
$346$	−2291.38	−0.356027
$347$	−6537.39	−1.01137	−0.505685	−	0.862718i	$-0.668760\pi$
$347$	−6537.39	−1.01137	−0.505685	+	0.862718i	$0.668760\pi$
$348$	13394.9	2.06334
$349$	10911.5	1.67357	0.836787	−	0.547529i	$-0.184431\pi$
$349$	10911.5	1.67357	0.836787	+	0.547529i	$0.184431\pi$
$350$	0	0
$351$	242.719	0.0369100
$352$	7245.20	1.09707
$353$	−4974.07	−0.749980	−0.374990	−	0.927029i	$-0.622354\pi$
$353$	−4974.07	−0.749980	−0.374990	+	0.927029i	$0.622354\pi$
$354$	5798.63	0.870603
$355$	0	0
$356$	9830.11	1.46347
$357$	−300.369	−0.0445301
$358$	−14320.2	−2.11410
$359$	2468.77	0.362943	0.181471	−	0.983396i	$-0.441914\pi$
$359$	2468.77	0.362943	0.181471	+	0.983396i	$0.441914\pi$
$360$	0	0
$361$	8472.88	1.23529
$362$	−13206.9	−1.91750
$363$	−628.406	−0.0908616
$364$	703.227	0.101261
$365$	0	0
$366$	−13872.0	−1.98115
$367$	−7504.87	−1.06744	−0.533721	−	0.845661i	$-0.679207\pi$
$367$	−7504.87	−1.06744	−0.533721	+	0.845661i	$0.679207\pi$
$368$	−21222.4	−3.00623
$369$	1.44637	0.000204052 0
$370$	0	0
$371$	−2218.50	−0.310455
$372$	−32834.1	−4.57626
$373$	7696.58	1.06840	0.534201	−	0.845357i	$-0.320613\pi$
$373$	7696.58	1.06840	0.534201	+	0.845357i	$0.320613\pi$
$374$	183.877	0.0254225
$375$	0	0
$376$	−26386.1	−3.61904
$377$	−206.439	−0.0282020
$378$	14377.5	1.95634
$379$	6248.82	0.846913	0.423457	−	0.905916i	$-0.360817\pi$
$379$	6248.82	0.846913	0.423457	+	0.905916i	$0.360817\pi$
$380$	0	0
$381$	−1236.45	−0.166261
$382$	5048.09	0.676133
$383$	−4505.13	−0.601048	−0.300524	−	0.953774i	$-0.597162\pi$
$383$	−4505.13	−0.601048	−0.300524	+	0.953774i	$0.597162\pi$
$384$	−21932.0	−2.91462
$385$	0	0
$386$	7361.94	0.970759
$387$	6.48442	0.000851735 0
$388$	−18199.2	−2.38124
$389$	5426.65	0.707305	0.353653	−	0.935377i	$-0.384939\pi$
$389$	5426.65	0.707305	0.353653	+	0.935377i	$0.384939\pi$
$390$	0	0
$391$	−283.627	−0.0366845
$392$	842.806	0.108592
$393$	−2766.18	−0.355052
$394$	14677.1	1.87671
$395$	0	0
$396$	−6.70361	−0.000850679 0
$397$	11783.1	1.48961	0.744805	−	0.667282i	$-0.232542\pi$
$397$	11783.1	1.48961	0.744805	+	0.667282i	$0.232542\pi$
$398$	9776.60	1.23130
$399$	−12105.7	−1.51891
$400$	0	0
$401$	10743.2	1.33788	0.668941	−	0.743315i	$-0.266748\pi$
$401$	10743.2	1.33788	0.668941	+	0.743315i	$0.266748\pi$
$402$	25169.2	3.12270
$403$	506.032	0.0625490
$404$	−11335.9	−1.39600
$405$	0	0
$406$	−12228.4	−1.49479
$407$	−2593.46	−0.315855
$408$	−1180.97	−0.143301
$409$	5156.54	0.623409	0.311705	−	0.950179i	$-0.399100\pi$
$409$	5156.54	0.623409	0.311705	+	0.950179i	$0.399100\pi$
$410$	0	0
$411$	−2485.09	−0.298249
$412$	5166.95	0.617858
$413$	−3863.11	−0.460270
$414$	14.1693	0.00168209
$415$	0	0
$416$	1138.91	0.134230
$417$	−1839.71	−0.216045
$418$	7410.75	0.867157
$419$	9993.25	1.16516	0.582580	−	0.812773i	$-0.302043\pi$
$419$	9993.25	1.16516	0.582580	+	0.812773i	$0.302043\pi$
$420$	0	0
$421$	−1823.62	−0.211111	−0.105555	−	0.994413i	$-0.533662\pi$
$421$	−1823.62	−0.211111	−0.105555	+	0.994413i	$0.533662\pi$
$422$	−1282.82	−0.147978
$423$	10.0564	0.00115593
$424$	−8722.56	−0.999069
$425$	0	0
$426$	18511.7	2.10538
$427$	9241.66	1.04739
$428$	−19257.5	−2.17488
$429$	−98.7825	−0.0111172
$430$	0	0
$431$	−6639.30	−0.742005	−0.371002	−	0.928632i	$-0.620986\pi$
$431$	−6639.30	−0.742005	−0.371002	+	0.928632i	$0.620986\pi$
$432$	32268.7	3.59381
$433$	−17034.9	−1.89063	−0.945316	−	0.326156i	$-0.894247\pi$
$433$	−17034.9	−1.89063	−0.945316	+	0.326156i	$0.894247\pi$
$434$	29974.8	3.31529
$435$	0	0
$436$	30762.6	3.37904
$437$	−11431.0	−1.25130
$438$	−2285.76	−0.249355
$439$	−8175.20	−0.888795	−0.444397	−	0.895830i	$-0.646582\pi$
$439$	−8175.20	−0.888795	−0.444397	+	0.895830i	$0.646582\pi$
$440$	0	0
$441$	−0.321215	−3.46847e−5 0
$442$	28.9045	0.00311052
$443$	3336.90	0.357880	0.178940	−	0.983860i	$-0.442733\pi$
$443$	3336.90	0.357880	0.178940	+	0.983860i	$0.442733\pi$
$444$	26452.5	2.82743
$445$	0	0
$446$	−23700.0	−2.51620
$447$	−15591.5	−1.64979
$448$	32842.5	3.46353
$449$	−8068.30	−0.848033	−0.424016	−	0.905654i	$-0.639380\pi$
$449$	−8068.30	−0.848033	−0.424016	+	0.905654i	$0.639380\pi$
$450$	0	0
$451$	−564.003	−0.0588866
$452$	2216.21	0.230624
$453$	−11590.0	−1.20208
$454$	16485.4	1.70418
$455$	0	0
$456$	−47596.6	−4.88797
$457$	−6937.49	−0.710114	−0.355057	−	0.934845i	$-0.615538\pi$
$457$	−6937.49	−0.710114	−0.355057	+	0.934845i	$0.615538\pi$
$458$	−33149.6	−3.38205
$459$	431.255	0.0438546
$460$	0	0
$461$	7068.73	0.714151	0.357075	−	0.934076i	$-0.383774\pi$
$461$	7068.73	0.714151	0.357075	+	0.934076i	$0.383774\pi$
$462$	−5851.38	−0.589244
$463$	5832.99	0.585490	0.292745	−	0.956191i	$-0.405431\pi$
$463$	5832.99	0.585490	0.292745	+	0.956191i	$0.405431\pi$
$464$	−27445.4	−2.74595
$465$	0	0
$466$	8882.32	0.882973
$467$	−6157.77	−0.610166	−0.305083	−	0.952326i	$-0.598684\pi$
$467$	−6157.77	−0.610166	−0.305083	+	0.952326i	$0.598684\pi$
$468$	−1.05378	−0.000104083 0
$469$	−16768.0	−1.65091
$470$	0	0
$471$	−1473.16	−0.144118
$472$	−15188.7	−1.48118
$473$	−2528.55	−0.245799
$474$	9486.74	0.919283
$475$	0	0
$476$	1249.47	0.120314
$477$	3.32439	0.000319106 0
$478$	30617.2	2.92970
$479$	−7042.80	−0.671803	−0.335901	−	0.941897i	$-0.609041\pi$
$479$	−7042.80	−0.671803	−0.335901	+	0.941897i	$0.609041\pi$
$480$	0	0
$481$	−407.680	−0.0386458
$482$	18970.5	1.79270
$483$	9025.67	0.850274
$484$	2614.03	0.245495
$485$	0	0
$486$	−43.0664	−0.00401961
$487$	−4698.83	−0.437216	−0.218608	−	0.975813i	$-0.570152\pi$
$487$	−4698.83	−0.437216	−0.218608	+	0.975813i	$0.570152\pi$
$488$	36335.8	3.37058
$489$	−18044.0	−1.66866
$490$	0	0
$491$	−18714.1	−1.72007	−0.860037	−	0.510232i	$-0.829560\pi$
$491$	−18714.1	−1.72007	−0.860037	+	0.510232i	$0.829560\pi$
$492$	5752.65	0.527133
$493$	−366.794	−0.0335082
$494$	1164.94	0.106099
$495$	0	0
$496$	67275.1	6.09021
$497$	−12332.7	−1.11307
$498$	12211.2	1.09879
$499$	−6019.34	−0.540005	−0.270003	−	0.962860i	$-0.587025\pi$
$499$	−6019.34	−0.540005	−0.270003	+	0.962860i	$0.587025\pi$
$500$	0	0
$501$	6629.96	0.591227
$502$	−31037.6	−2.75951
$503$	171.433	0.0151965	0.00759823	−	0.999971i	$-0.497581\pi$
$503$	171.433	0.0151965	0.00759823	+	0.999971i	$0.497581\pi$
$504$	−39.3055	−0.00347382
$505$	0	0
$506$	−5525.23	−0.485427
$507$	11394.5	0.998117
$508$	5143.37	0.449213
$509$	−5719.31	−0.498043	−0.249022	−	0.968498i	$-0.580109\pi$
$509$	−5719.31	−0.498043	−0.249022	+	0.968498i	$0.580109\pi$
$510$	0	0
$511$	1522.80	0.131829
$512$	15294.1	1.32014
$513$	17380.8	1.49587
$514$	14328.7	1.22960
$515$	0	0
$516$	25790.5	2.20031
$517$	−3921.43	−0.333587
$518$	−24148.9	−2.04834
$519$	2187.16	0.184982
$520$	0	0
$521$	3500.52	0.294358	0.147179	−	0.989110i	$-0.452981\pi$
$521$	3500.52	0.294358	0.147179	+	0.989110i	$0.452981\pi$
$522$	18.3241	0.00153645
$523$	7103.24	0.593887	0.296943	−	0.954895i	$-0.404033\pi$
$523$	7103.24	0.593887	0.296943	+	0.954895i	$0.404033\pi$
$524$	11506.7	0.959298
$525$	0	0
$526$	−25645.1	−2.12582
$527$	899.100	0.0743176
$528$	−13132.8	−1.08244
$529$	−3644.41	−0.299533
$530$	0	0
$531$	5.78882	0.000473095 0
$532$	50357.1	4.10387
$533$	−88.6586	−0.00720494
$534$	−12857.6	−1.04195
$535$	0	0
$536$	−65927.5	−5.31275
$537$	13668.9	1.09843
$538$	22556.2	1.80756
$539$	125.256	0.0100095
$540$	0	0
$541$	2035.89	0.161792	0.0808961	−	0.996723i	$-0.474222\pi$
$541$	2035.89	0.161792	0.0808961	+	0.996723i	$0.474222\pi$
$542$	−38932.9	−3.08544
$543$	12606.2	0.996283
$544$	2023.58	0.159485
$545$	0	0
$546$	−919.809	−0.0720956
$547$	24546.4	1.91870	0.959348	−	0.282226i	$-0.0910728\pi$
$547$	24546.4	1.91870	0.959348	+	0.282226i	$0.0910728\pi$
$548$	10337.4	0.805825
$549$	−13.8485	−0.00107657
$550$	0	0
$551$	−14782.8	−1.14296
$552$	35486.5	2.73625
$553$	−6320.17	−0.486006
$554$	−43943.3	−3.36998
$555$	0	0
$556$	7652.76	0.583722
$557$	−17398.0	−1.32348	−0.661740	−	0.749733i	$-0.730182\pi$
$557$	−17398.0	−1.32348	−0.661740	+	0.749733i	$0.730182\pi$
$558$	−44.9168	−0.00340767
$559$	−397.477	−0.0300742
$560$	0	0
$561$	−175.513	−0.0132089
$562$	7865.93	0.590399
$563$	−13198.2	−0.987992	−0.493996	−	0.869464i	$-0.664464\pi$
$563$	−13198.2	−0.987992	−0.493996	+	0.869464i	$0.664464\pi$
$564$	39997.3	2.98616
$565$	0	0
$566$	44942.7	3.33760
$567$	−13709.2	−1.01540
$568$	−48488.9	−3.58195
$569$	−15124.2	−1.11431	−0.557153	−	0.830410i	$-0.688106\pi$
$569$	−15124.2	−1.11431	−0.557153	+	0.830410i	$0.688106\pi$
$570$	0	0
$571$	8485.10	0.621875	0.310937	−	0.950430i	$-0.399357\pi$
$571$	8485.10	0.621875	0.310937	+	0.950430i	$0.399357\pi$
$572$	410.913	0.0300369
$573$	−4818.48	−0.351300
$574$	−5251.69	−0.381884
$575$	0	0
$576$	−49.2140	−0.00356004
$577$	−14482.9	−1.04494	−0.522471	−	0.852657i	$-0.674990\pi$
$577$	−14482.9	−1.04494	−0.522471	+	0.852657i	$0.674990\pi$
$578$	−26679.8	−1.91996
$579$	−7027.09	−0.504380
$580$	0	0
$581$	−8135.23	−0.580906
$582$	23804.2	1.69539
$583$	−1296.32	−0.0920896
$584$	5987.24	0.424236
$585$	0	0
$586$	31473.7	2.21871
$587$	6985.26	0.491163	0.245581	−	0.969376i	$-0.421021\pi$
$587$	6985.26	0.491163	0.245581	+	0.969376i	$0.421021\pi$
$588$	−1277.57	−0.0896021
$589$	36236.3	2.53496
$590$	0	0
$591$	−14009.5	−0.975086
$592$	−54199.6	−3.76282
$593$	−585.572	−0.0405507	−0.0202753	−	0.999794i	$-0.506454\pi$
$593$	−585.572	−0.0405507	−0.0202753	+	0.999794i	$0.506454\pi$
$594$	8401.11	0.580306
$595$	0	0
$596$	64857.2	4.45748
$597$	−9331.92	−0.639749
$598$	−868.540	−0.0593934
$599$	−19170.0	−1.30762	−0.653809	−	0.756659i	$-0.726830\pi$
$599$	−19170.0	−1.30762	−0.653809	+	0.756659i	$0.726830\pi$
$600$	0	0
$601$	22098.9	1.49989	0.749943	−	0.661502i	$-0.230081\pi$
$601$	22098.9	1.49989	0.749943	+	0.661502i	$0.230081\pi$
$602$	−23544.5	−1.59403
$603$	25.1266	0.00169691
$604$	48211.6	3.24785
$605$	0	0
$606$	14827.2	0.993916
$607$	−3717.96	−0.248612	−0.124306	−	0.992244i	$-0.539670\pi$
$607$	−3717.96	−0.248612	−0.124306	+	0.992244i	$0.539670\pi$
$608$	81555.9	5.44001
$609$	11672.2	0.776655
$610$	0	0
$611$	−616.430	−0.0408152
$612$	−1.87231	−0.000123666 0
$613$	25728.4	1.69520	0.847602	−	0.530632i	$-0.178045\pi$
$613$	25728.4	1.69520	0.847602	+	0.530632i	$0.178045\pi$
$614$	51461.5	3.38244
$615$	0	0
$616$	15326.9	1.00250
$617$	5085.97	0.331853	0.165927	−	0.986138i	$-0.446939\pi$
$617$	5085.97	0.331853	0.165927	+	0.986138i	$0.446939\pi$
$618$	−6758.29	−0.439900
$619$	770.194	0.0500108	0.0250054	−	0.999687i	$-0.492040\pi$
$619$	770.194	0.0500108	0.0250054	+	0.999687i	$0.492040\pi$
$620$	0	0
$621$	−12958.6	−0.837376
$622$	−32561.4	−2.09902
$623$	8565.90	0.550859
$624$	−2064.41	−0.132440
$625$	0	0
$626$	−6976.96	−0.445456
$627$	−7073.68	−0.450551
$628$	6127.99	0.389385
$629$	−724.351	−0.0459170
$630$	0	0
$631$	−12396.1	−0.782063	−0.391031	−	0.920377i	$-0.627882\pi$
$631$	−12396.1	−0.782063	−0.391031	+	0.920377i	$0.627882\pi$
$632$	−24849.3	−1.56400
$633$	1224.47	0.0768852
$634$	9401.35	0.588920
$635$	0	0
$636$	13222.1	0.824356
$637$	19.6896	0.00122469
$638$	−7145.37	−0.443398
$639$	18.4804	0.00114409
$640$	0	0
$641$	5728.50	0.352983	0.176491	−	0.984302i	$-0.443525\pi$
$641$	5728.50	0.352983	0.176491	+	0.984302i	$0.443525\pi$
$642$	25188.5	1.54846
$643$	−1536.24	−0.0942202	−0.0471101	−	0.998890i	$-0.515001\pi$
$643$	−1536.24	−0.0942202	−0.0471101	+	0.998890i	$0.515001\pi$
$644$	−37544.7	−2.29731
$645$	0	0
$646$	2069.81	0.126062
$647$	26647.1	1.61917	0.809587	−	0.586999i	$-0.199691\pi$
$647$	26647.1	1.61917	0.809587	+	0.586999i	$0.199691\pi$
$648$	−53900.9	−3.26764
$649$	−2257.31	−0.136529
$650$	0	0
$651$	−28611.4	−1.72254
$652$	75058.8	4.50848
$653$	−28405.1	−1.70226	−0.851131	−	0.524954i	$-0.824083\pi$
$653$	−28405.1	−1.70226	−0.851131	+	0.524954i	$0.824083\pi$
$654$	−40236.9	−2.40579
$655$	0	0
$656$	−11786.8	−0.701522
$657$	−2.28189	−0.000135502 0
$658$	−36514.2	−2.16333
$659$	12108.5	0.715752	0.357876	−	0.933769i	$-0.383501\pi$
$659$	12108.5	0.715752	0.357876	+	0.933769i	$0.383501\pi$
$660$	0	0
$661$	9862.98	0.580371	0.290186	−	0.956970i	$-0.406283\pi$
$661$	9862.98	0.580371	0.290186	+	0.956970i	$0.406283\pi$
$662$	−11138.9	−0.653964
$663$	−27.5898	−0.00161614
$664$	−31985.6	−1.86940
$665$	0	0
$666$	36.1868	0.00210542
$667$	11021.6	0.639819
$668$	−27579.1	−1.59741
$669$	22622.0	1.30735
$670$	0	0
$671$	5400.13	0.310685
$672$	−64394.9	−3.69656
$673$	2282.66	0.130743	0.0653714	−	0.997861i	$-0.479177\pi$
$673$	2282.66	0.130743	0.0653714	+	0.997861i	$0.479177\pi$
$674$	39499.4	2.25736
$675$	0	0
$676$	−47398.3	−2.69677
$677$	10499.2	0.596038	0.298019	−	0.954560i	$-0.403674\pi$
$677$	10499.2	0.596038	0.298019	+	0.954560i	$0.403674\pi$
$678$	−2898.77	−0.164198
$679$	−15858.6	−0.896316
$680$	0	0
$681$	−15735.6	−0.885447
$682$	17515.0	0.983408
$683$	26681.2	1.49477	0.747386	−	0.664391i	$-0.231309\pi$
$683$	26681.2	1.49477	0.747386	+	0.664391i	$0.231309\pi$
$684$	−75.4594	−0.00421822
$685$	0	0
$686$	−33965.8	−1.89041
$687$	31641.8	1.75722
$688$	−52843.1	−2.92823
$689$	−203.776	−0.0112674
$690$	0	0
$691$	−1530.22	−0.0842434	−0.0421217	−	0.999112i	$-0.513412\pi$
$691$	−1530.22	−0.0842434	−0.0421217	+	0.999112i	$0.513412\pi$
$692$	−9098.07	−0.499793
$693$	−5.84148	−0.000320201 0
$694$	−35569.4	−1.94553
$695$	0	0
$696$	45892.1	2.49933
$697$	−157.525	−0.00856054
$698$	59368.3	3.21937
$699$	−8478.31	−0.458769
$700$	0	0
$701$	25470.4	1.37233	0.686165	−	0.727446i	$-0.259293\pi$
$701$	25470.4	1.37233	0.686165	+	0.727446i	$0.259293\pi$
$702$	1320.62	0.0710020
$703$	−29193.4	−1.56622
$704$	19190.7	1.02738
$705$	0	0
$706$	−27063.5	−1.44270
$707$	−9878.04	−0.525463
$708$	23023.9	1.22216
$709$	36906.7	1.95495	0.977475	−	0.211052i	$-0.0676891\pi$
$709$	36906.7	1.95495	0.977475	+	0.211052i	$0.0676891\pi$
$710$	0	0
$711$	9.47069	0.000499548 0
$712$	33678.8	1.77271
$713$	−27016.7	−1.41905
$714$	−1634.28	−0.0856604
$715$	0	0
$716$	−56859.4	−2.96779
$717$	−29224.6	−1.52219
$718$	13432.3	0.698176
$719$	13606.1	0.705734	0.352867	−	0.935673i	$-0.385207\pi$
$719$	13606.1	0.705734	0.352867	+	0.935673i	$0.385207\pi$
$720$	0	0
$721$	4502.45	0.232566
$722$	46100.2	2.37628
$723$	−18107.6	−0.931438
$724$	−52438.7	−2.69181
$725$	0	0
$726$	−3419.10	−0.174786
$727$	25132.2	1.28212	0.641060	−	0.767491i	$-0.278495\pi$
$727$	25132.2	1.28212	0.641060	+	0.767491i	$0.278495\pi$
$728$	2409.32	0.122658
$729$	19703.5	1.00104
$730$	0	0
$731$	−706.223	−0.0357327
$732$	−55079.6	−2.78115
$733$	10789.5	0.543681	0.271841	−	0.962342i	$-0.412368\pi$
$733$	10789.5	0.543681	0.271841	+	0.962342i	$0.412368\pi$
$734$	−40833.3	−2.05339
$735$	0	0
$736$	−60805.5	−3.04527
$737$	−9797.97	−0.489705
$738$	7.86958	0.000392525 0
$739$	−34785.9	−1.73156	−0.865778	−	0.500428i	$-0.833176\pi$
$739$	−34785.9	−1.73156	−0.865778	+	0.500428i	$0.833176\pi$
$740$	0	0
$741$	−1111.95	−0.0551262
$742$	−12070.7	−0.597208
$743$	−19790.8	−0.977195	−0.488597	−	0.872509i	$-0.662491\pi$
$743$	−19790.8	−0.977195	−0.488597	+	0.872509i	$0.662491\pi$
$744$	−112493.	−5.54325
$745$	0	0
$746$	41876.4	2.05523
$747$	12.1905	0.000597092 0
$748$	730.095	0.0356884
$749$	−16780.9	−0.818639
$750$	0	0
$751$	17836.9	0.866681	0.433341	−	0.901230i	$-0.357335\pi$
$751$	17836.9	0.866681	0.433341	+	0.901230i	$0.357335\pi$
$752$	−81952.2	−3.97405
$753$	29625.8	1.43377
$754$	−1123.22	−0.0542509
$755$	0	0
$756$	57086.8	2.74633
$757$	−26599.8	−1.27713	−0.638564	−	0.769569i	$-0.720471\pi$
$757$	−26599.8	−1.27713	−0.638564	+	0.769569i	$0.720471\pi$
$758$	33999.3	1.62917
$759$	5273.92	0.252215
$760$	0	0
$761$	8248.59	0.392919	0.196459	−	0.980512i	$-0.437056\pi$
$761$	8248.59	0.392919	0.196459	+	0.980512i	$0.437056\pi$
$762$	−6727.44	−0.319828
$763$	26806.3	1.27189
$764$	20043.8	0.949161
$765$	0	0
$766$	−24512.0	−1.15621
$767$	−354.839	−0.0167047
$768$	−46846.0	−2.20105
$769$	1496.76	0.0701880	0.0350940	−	0.999384i	$-0.488827\pi$
$769$	1496.76	0.0701880	0.0350940	+	0.999384i	$0.488827\pi$
$770$	0	0
$771$	−13677.0	−0.638866
$772$	29231.1	1.36276
$773$	−459.376	−0.0213746	−0.0106873	−	0.999943i	$-0.503402\pi$
$773$	−459.376	−0.0213746	−0.0106873	+	0.999943i	$0.503402\pi$
$774$	35.2811	0.00163844
$775$	0	0
$776$	−62351.9	−2.88441
$777$	23050.5	1.06426
$778$	29525.9	1.36061
$779$	−6348.72	−0.291998
$780$	0	0
$781$	−7206.29	−0.330168
$782$	−1543.19	−0.0705682
$783$	−16758.4	−0.764874
$784$	2617.66	0.119245
$785$	0	0
$786$	−15050.6	−0.682997
$787$	21110.4	0.956170	0.478085	−	0.878314i	$-0.341331\pi$
$787$	21110.4	0.956170	0.478085	+	0.878314i	$0.341331\pi$
$788$	58276.5	2.63454
$789$	24478.7	1.10452
$790$	0	0
$791$	1931.19	0.0868082
$792$	−22.9671	−0.00103043
$793$	848.875	0.0380132
$794$	64110.7	2.86549
$795$	0	0
$796$	38818.7	1.72851
$797$	44355.5	1.97133	0.985666	−	0.168711i	$-0.0539605\pi$
$797$	44355.5	1.97133	0.985666	+	0.168711i	$0.0539605\pi$
$798$	−65866.3	−2.92186
$799$	−1095.25	−0.0484946
$800$	0	0
$801$	−12.8359	−0.000566208 0
$802$	58452.9	2.57362
$803$	889.807	0.0391041
$804$	99936.2	4.38368
$805$	0	0
$806$	2753.28	0.120323
$807$	−21530.2	−0.939158
$808$	−38837.8	−1.69098
$809$	−15021.8	−0.652830	−0.326415	−	0.945227i	$-0.605841\pi$
$809$	−15021.8	−0.652830	−0.326415	+	0.945227i	$0.605841\pi$
$810$	0	0
$811$	20178.0	0.873667	0.436833	−	0.899542i	$-0.356100\pi$
$811$	20178.0	0.873667	0.436833	+	0.899542i	$0.356100\pi$
$812$	−48553.8	−2.09841
$813$	37162.1	1.60311
$814$	−14110.8	−0.607596
$815$	0	0
$816$	−3667.97	−0.157359
$817$	−28462.8	−1.21883
$818$	28056.3	1.19922
$819$	−0.918253	−3.91775e−5 0
$820$	0	0
$821$	8507.06	0.361630	0.180815	−	0.983517i	$-0.442126\pi$
$821$	8507.06	0.361630	0.180815	+	0.983517i	$0.442126\pi$
$822$	−13521.2	−0.573728
$823$	13562.9	0.574451	0.287225	−	0.957863i	$-0.407267\pi$
$823$	13562.9	0.574451	0.287225	+	0.957863i	$0.407267\pi$
$824$	17702.4	0.748415
$825$	0	0
$826$	−21018.9	−0.885399
$827$	−42090.2	−1.76979	−0.884897	−	0.465786i	$-0.845771\pi$
$827$	−42090.2	−1.76979	−0.884897	+	0.465786i	$0.845771\pi$
$828$	56.2602	0.00236133
$829$	19524.7	0.817998	0.408999	−	0.912535i	$-0.365878\pi$
$829$	19524.7	0.817998	0.408999	+	0.912535i	$0.365878\pi$
$830$	0	0
$831$	41944.5	1.75095
$832$	3016.69	0.125703
$833$	34.9838	0.00145512
$834$	−10009.7	−0.415596
$835$	0	0
$836$	29424.9	1.21732
$837$	41078.8	1.69641
$838$	54372.4	2.24136
$839$	40733.8	1.67615	0.838073	−	0.545559i	$-0.183683\pi$
$839$	40733.8	1.67615	0.838073	+	0.545559i	$0.183683\pi$
$840$	0	0
$841$	−10135.5	−0.415578
$842$	−9922.13	−0.406104
$843$	−7508.15	−0.306755
$844$	−5093.52	−0.207732
$845$	0	0
$846$	54.7160	0.00222361
$847$	2277.85	0.0924058
$848$	−27091.3	−1.09707
$849$	−42898.5	−1.73413
$850$	0	0
$851$	21765.7	0.876756
$852$	73501.9	2.95556
$853$	7476.59	0.300110	0.150055	−	0.988678i	$-0.452055\pi$
$853$	7476.59	0.300110	0.150055	+	0.988678i	$0.452055\pi$
$854$	50283.1	2.01481
$855$	0	0
$856$	−65978.0	−2.63444
$857$	46353.7	1.84762	0.923812	−	0.382847i	$-0.125056\pi$
$857$	46353.7	1.84762	0.923812	+	0.382847i	$0.125056\pi$
$858$	−537.467	−0.0213856
$859$	−3254.04	−0.129251	−0.0646254	−	0.997910i	$-0.520585\pi$
$859$	−3254.04	−0.129251	−0.0646254	+	0.997910i	$0.520585\pi$
$860$	0	0
$861$	5012.82	0.198416
$862$	−36123.9	−1.42736
$863$	−38864.6	−1.53299	−0.766493	−	0.642252i	$-0.778000\pi$
$863$	−38864.6	−1.53299	−0.766493	+	0.642252i	$0.778000\pi$
$864$	92454.8	3.64048
$865$	0	0
$866$	−92685.2	−3.63692
$867$	25466.3	0.997557
$868$	119017.	4.65404
$869$	−3693.03	−0.144163
$870$	0	0
$871$	−1540.19	−0.0599168
$872$	105395.	4.09304
$873$	23.7639	0.000921291 0
$874$	−62195.0	−2.40707
$875$	0	0
$876$	−9075.75	−0.350047
$877$	−35144.3	−1.35318	−0.676589	−	0.736361i	$-0.736543\pi$
$877$	−35144.3	−1.35318	−0.676589	+	0.736361i	$0.736543\pi$
$878$	−44480.5	−1.70973
$879$	−30042.1	−1.15278
$880$	0	0
$881$	31530.7	1.20578	0.602892	−	0.797823i	$-0.294015\pi$
$881$	31530.7	1.20578	0.602892	+	0.797823i	$0.294015\pi$
$882$	−1.74770	−6.67213e−5 0
$883$	35263.1	1.34394	0.671970	−	0.740579i	$-0.265448\pi$
$883$	35263.1	1.34394	0.671970	+	0.740579i	$0.265448\pi$
$884$	114.768	0.00436657
$885$	0	0
$886$	18155.8	0.688436
$887$	10912.5	0.413086	0.206543	−	0.978438i	$-0.433779\pi$
$887$	10912.5	0.413086	0.206543	+	0.978438i	$0.433779\pi$
$888$	90628.6	3.42488
$889$	4481.90	0.169087
$890$	0	0
$891$	−8010.61	−0.301196
$892$	−94102.4	−3.53227
$893$	−44141.7	−1.65414
$894$	−84832.1	−3.17361
$895$	0	0
$896$	79499.1	2.96415
$897$	829.035	0.0308592
$898$	−43898.9	−1.63132
$899$	−34938.6	−1.29618
$900$	0	0
$901$	−362.062	−0.0133874
$902$	−3068.69	−0.113277
$903$	22473.6	0.828212
$904$	7592.94	0.279356
$905$	0	0
$906$	−63059.9	−2.31239
$907$	6918.62	0.253285	0.126642	−	0.991948i	$-0.459580\pi$
$907$	6918.62	0.253285	0.126642	+	0.991948i	$0.459580\pi$
$908$	65456.5	2.39235
$909$	14.8021	0.000540104 0
$910$	0	0
$911$	9101.44	0.331003	0.165502	−	0.986210i	$-0.447076\pi$
$911$	9101.44	0.331003	0.165502	+	0.986210i	$0.447076\pi$
$912$	−147830.	−5.36746
$913$	−4753.61	−0.172313
$914$	−37746.3	−1.36601
$915$	0	0
$916$	−131623.	−4.74775
$917$	10026.9	0.361086
$918$	2346.42	0.0843610
$919$	34392.5	1.23450	0.617248	−	0.786768i	$-0.288247\pi$
$919$	34392.5	1.23450	0.617248	+	0.786768i	$0.288247\pi$
$920$	0	0
$921$	−49120.8	−1.75742
$922$	38460.3	1.37378
$923$	−1132.80	−0.0403970
$924$	−23233.3	−0.827186
$925$	0	0
$926$	31736.8	1.12628
$927$	−6.74685	−0.000239046 0
$928$	−78635.3	−2.78161
$929$	4364.57	0.154141	0.0770704	−	0.997026i	$-0.475443\pi$
$929$	4364.57	0.154141	0.0770704	+	0.997026i	$0.475443\pi$
$930$	0	0
$931$	1409.95	0.0496338
$932$	35267.9	1.23952
$933$	31080.3	1.09059
$934$	−33503.9	−1.17375
$935$	0	0
$936$	−3.61033	−0.000126076 0
$937$	1816.07	0.0633174	0.0316587	−	0.999499i	$-0.489921\pi$
$937$	1816.07	0.0633174	0.0316587	+	0.999499i	$0.489921\pi$
$938$	−91233.4	−3.17577
$939$	6659.62	0.231447
$940$	0	0
$941$	−35113.5	−1.21644	−0.608218	−	0.793770i	$-0.708115\pi$
$941$	−35113.5	−1.21644	−0.608218	+	0.793770i	$0.708115\pi$
$942$	−8015.31	−0.277232
$943$	4733.41	0.163458
$944$	−47174.5	−1.62648
$945$	0	0
$946$	−13757.6	−0.472833
$947$	−10104.6	−0.346731	−0.173365	−	0.984858i	$-0.555464\pi$
$947$	−10104.6	−0.346731	−0.173365	+	0.984858i	$0.555464\pi$
$948$	37667.8	1.29050
$949$	139.874	0.00478450
$950$	0	0
$951$	−8973.74	−0.305987
$952$	4280.79	0.145737
$953$	−46422.4	−1.57793	−0.788965	−	0.614438i	$-0.789383\pi$
$953$	−46422.4	−1.57793	−0.788965	+	0.614438i	$0.789383\pi$
$954$	18.0877	0.000613849 0
$955$	0	0
$956$	121568.	4.11274
$957$	6820.37	0.230377
$958$	−38319.2	−1.29232
$959$	9007.95	0.303318
$960$	0	0
$961$	55852.0	1.87479
$962$	−2218.15	−0.0743410
$963$	25.1459	0.000841449 0
$964$	75323.6	2.51661
$965$	0	0
$966$	49107.9	1.63563
$967$	928.679	0.0308835	0.0154417	−	0.999881i	$-0.495085\pi$
$967$	928.679	0.0308835	0.0154417	+	0.999881i	$0.495085\pi$
$968$	8955.88	0.297369
$969$	−1975.67	−0.0654981
$970$	0	0
$971$	−51195.8	−1.69202	−0.846011	−	0.533166i	$-0.821002\pi$
$971$	−51195.8	−1.69202	−0.846011	+	0.533166i	$0.821002\pi$
$972$	−170.998	−0.00564276
$973$	6668.56	0.219717
$974$	−25565.9	−0.841052
$975$	0	0
$976$	112855.	3.70122
$977$	−31543.9	−1.03294	−0.516469	−	0.856306i	$-0.672754\pi$
$977$	−31543.9	−1.03294	−0.516469	+	0.856306i	$0.672754\pi$
$978$	−98175.6	−3.20993
$979$	5005.26	0.163400
$980$	0	0
$981$	−40.1688	−0.00130733
$982$	−101822.	−3.30883
$983$	−30743.8	−0.997532	−0.498766	−	0.866737i	$-0.666213\pi$
$983$	−30743.8	−0.997532	−0.498766	+	0.866737i	$0.666213\pi$
$984$	19709.1	0.638519
$985$	0	0
$986$	−1995.69	−0.0644582
$987$	34853.4	1.12401
$988$	4625.46	0.148943
$989$	21221.0	0.682293
$990$	0	0
$991$	−31389.1	−1.00616	−0.503082	−	0.864239i	$-0.667801\pi$
$991$	−31389.1	−1.00616	−0.503082	+	0.864239i	$0.667801\pi$
$992$	192754.	6.16930
$993$	10632.2	0.339782
$994$	−67101.1	−2.14117
$995$	0	0
$996$	48485.3	1.54249
$997$	10343.4	0.328565	0.164282	−	0.986413i	$-0.447469\pi$
$997$	10343.4	0.328565	0.164282	+	0.986413i	$0.447469\pi$
$998$	−32750.7	−1.03878
$999$	−33094.8	−1.04812

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	275.4.a.k.1.10		10
3.2	odd	2		2475.4.a.bw.1.1		10
5.2	odd	4		55.4.b.b.34.10	yes	10
5.3	odd	4		55.4.b.b.34.1	✓	10
5.4	even	2	inner	275.4.a.k.1.1		10
15.2	even	4		495.4.c.b.199.1		10
15.8	even	4		495.4.c.b.199.10		10
15.14	odd	2		2475.4.a.bw.1.10		10
20.3	even	4		880.4.b.i.529.8		10
20.7	even	4		880.4.b.i.529.3		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.4.b.b.34.1	✓	10	5.3	odd	4
55.4.b.b.34.10	yes	10	5.2	odd	4
275.4.a.k.1.1		10	5.4	even	2	inner
275.4.a.k.1.10		10	1.1	even	1	trivial
495.4.c.b.199.1		10	15.2	even	4
495.4.c.b.199.10		10	15.8	even	4
880.4.b.i.529.3		10	20.7	even	4
880.4.b.i.529.8		10	20.3	even	4
2475.4.a.bw.1.1		10	3.2	odd	2
2475.4.a.bw.1.10		10	15.14	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:	\( N \)	\(=\)	\( 275 = 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	275.a (trivial)

\(f(q)\)	\(=\)	\(q+5.44091 q^{2} -5.19344 q^{3} +21.6035 q^{4} -28.2570 q^{6} +18.8252 q^{7} +74.0156 q^{8} -0.0282092 q^{9} +11.0000 q^{11} -112.197 q^{12} +1.72915 q^{13} +102.426 q^{14} +229.884 q^{16} +3.07229 q^{17} -0.153484 q^{18} +123.822 q^{19} -97.7673 q^{21} +59.8500 q^{22} -92.3179 q^{23} -384.395 q^{24} +9.40815 q^{26} +140.369 q^{27} +406.690 q^{28} -119.388 q^{29} +292.648 q^{31} +658.654 q^{32} -57.1278 q^{33} +16.7161 q^{34} -0.609419 q^{36} -235.769 q^{37} +673.704 q^{38} -8.98023 q^{39} -51.2730 q^{41} -531.943 q^{42} -229.869 q^{43} +237.639 q^{44} -502.293 q^{46} -356.494 q^{47} -1193.89 q^{48} +11.3869 q^{49} -15.9557 q^{51} +37.3557 q^{52} -117.848 q^{53} +763.737 q^{54} +1393.36 q^{56} -643.062 q^{57} -649.579 q^{58} -205.210 q^{59} +490.921 q^{61} +1592.27 q^{62} -0.531043 q^{63} +1744.61 q^{64} -310.827 q^{66} -890.724 q^{67} +66.3723 q^{68} +479.447 q^{69} -655.118 q^{71} -2.08792 q^{72} +80.8916 q^{73} -1282.80 q^{74} +2674.99 q^{76} +207.077 q^{77} -48.8606 q^{78} -335.730 q^{79} -728.238 q^{81} -278.972 q^{82} -432.146 q^{83} -2112.12 q^{84} -1250.69 q^{86} +620.033 q^{87} +814.171 q^{88} +455.024 q^{89} +32.5515 q^{91} -1994.39 q^{92} -1519.85 q^{93} -1939.65 q^{94} -3420.68 q^{96} -842.417 q^{97} +61.9550 q^{98} -0.310301 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 64 q^{4} + 26 q^{6} + 152 q^{9} + 110 q^{11} + 34 q^{14} + 468 q^{16} + 90 q^{19} + 302 q^{21} + 206 q^{24} + 392 q^{26} - 58 q^{29} + 1242 q^{31} - 66 q^{34} + 1786 q^{36} - 384 q^{39} + 416 q^{41}+ \cdots + 1672 q^{99}+O(q^{100}) \) 10 * q + 64 * q^4 + 26 * q^6 + 152 * q^9 + 110 * q^11 + 34 * q^14 + 468 * q^16 + 90 * q^19 + 302 * q^21 + 206 * q^24 + 392 * q^26 - 58 * q^29 + 1242 * q^31 - 66 * q^34 + 1786 * q^36 - 384 * q^39 + 416 * q^41 + 704 * q^44 + 1816 * q^46 + 1980 * q^49 + 510 * q^51 + 1882 * q^54 + 2626 * q^56 + 476 * q^59 + 1650 * q^61 + 2576 * q^64 + 286 * q^66 + 4492 * q^69 - 498 * q^71 - 5374 * q^74 + 1410 * q^76 - 416 * q^79 - 710 * q^81 - 9018 * q^84 - 6872 * q^86 + 1918 * q^89 + 1384 * q^91 - 2860 * q^94 - 10114 * q^96 + 1672 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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