LMFDB - Embedded newform 525.4.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,4,Mod(1,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("525.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	525.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:	$30.9760027530$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 24x^{2} - 3x + 46 $ x^4 - 24x^2 - 3x + 46
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.75345$ of defining polynomial
Character	$\chi$	$=$	525.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.75345 q^{2} -3.00000 q^{3} +14.5953 q^{4} +14.2604 q^{6} -7.00000 q^{7} -31.3504 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-4.75345 q^{2} -3.00000 q^{3} +14.5953 q^{4} +14.2604 q^{6} -7.00000 q^{7} -31.3504 q^{8} +9.00000 q^{9} +7.31799 q^{11} -43.7859 q^{12} +4.15422 q^{13} +33.2742 q^{14} +32.2604 q^{16} -53.5216 q^{17} -42.7811 q^{18} +88.9019 q^{19} +21.0000 q^{21} -34.7857 q^{22} -156.780 q^{23} +94.0513 q^{24} -19.7469 q^{26} -27.0000 q^{27} -102.167 q^{28} +42.2570 q^{29} -14.0248 q^{31} +97.4554 q^{32} -21.9540 q^{33} +254.412 q^{34} +131.358 q^{36} +293.336 q^{37} -422.591 q^{38} -12.4627 q^{39} -127.214 q^{41} -99.8225 q^{42} +210.189 q^{43} +106.808 q^{44} +745.244 q^{46} +468.688 q^{47} -96.7811 q^{48} +49.0000 q^{49} +160.565 q^{51} +60.6321 q^{52} -115.973 q^{53} +128.343 q^{54} +219.453 q^{56} -266.706 q^{57} -200.867 q^{58} -314.090 q^{59} +768.386 q^{61} +66.6662 q^{62} -63.0000 q^{63} -721.332 q^{64} +104.357 q^{66} +717.081 q^{67} -781.164 q^{68} +470.339 q^{69} -737.783 q^{71} -282.154 q^{72} -477.618 q^{73} -1394.36 q^{74} +1297.55 q^{76} -51.2259 q^{77} +59.2407 q^{78} -279.262 q^{79} +81.0000 q^{81} +604.703 q^{82} -776.981 q^{83} +306.501 q^{84} -999.123 q^{86} -126.771 q^{87} -229.422 q^{88} -29.7626 q^{89} -29.0796 q^{91} -2288.24 q^{92} +42.0744 q^{93} -2227.88 q^{94} -292.366 q^{96} +231.793 q^{97} -232.919 q^{98} +65.8619 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3} + 16 q^{4} - 28 q^{7} - 9 q^{8} + 36 q^{9}+O(q^{10}) $ 4 * q - 12 * q^3 + 16 * q^4 - 28 * q^7 - 9 * q^8 + 36 * q^9	$ 4 q - 12 q^{3} + 16 q^{4} - 28 q^{7} - 9 q^{8} + 36 q^{9} + 21 q^{11} - 48 q^{12} - 5 q^{13} + 72 q^{16} - 99 q^{17} + 72 q^{19} + 84 q^{21} - 221 q^{22} - 102 q^{23} + 27 q^{24} + 129 q^{26} - 108 q^{27} - 112 q^{28} - 240 q^{29} + 351 q^{31} - 72 q^{32} - 63 q^{33} - 285 q^{34} + 144 q^{36} - 399 q^{37} - 324 q^{38} + 15 q^{39} + 381 q^{41} - 460 q^{43} - 975 q^{44} + 550 q^{46} - 60 q^{47} - 216 q^{48} + 196 q^{49} + 297 q^{51} - 223 q^{52} - 873 q^{53} + 63 q^{56} - 216 q^{57} - 1408 q^{58} - 855 q^{59} + 687 q^{61} + 477 q^{62} - 252 q^{63} - 1285 q^{64} + 663 q^{66} + 503 q^{67} - 1725 q^{68} + 306 q^{69} - 681 q^{71} - 81 q^{72} - 1228 q^{73} - 3369 q^{74} + 1910 q^{76} - 147 q^{77} - 387 q^{78} + 345 q^{79} + 324 q^{81} + 495 q^{82} - 1509 q^{83} + 336 q^{84} - 540 q^{86} + 720 q^{87} - 2266 q^{88} - 198 q^{89} + 35 q^{91} - 2994 q^{92} - 1053 q^{93} - 1732 q^{94} + 216 q^{96} + 372 q^{97} + 189 q^{99}+O(q^{100}) $ 4 * q - 12 * q^3 + 16 * q^4 - 28 * q^7 - 9 * q^8 + 36 * q^9 + 21 * q^11 - 48 * q^12 - 5 * q^13 + 72 * q^16 - 99 * q^17 + 72 * q^19 + 84 * q^21 - 221 * q^22 - 102 * q^23 + 27 * q^24 + 129 * q^26 - 108 * q^27 - 112 * q^28 - 240 * q^29 + 351 * q^31 - 72 * q^32 - 63 * q^33 - 285 * q^34 + 144 * q^36 - 399 * q^37 - 324 * q^38 + 15 * q^39 + 381 * q^41 - 460 * q^43 - 975 * q^44 + 550 * q^46 - 60 * q^47 - 216 * q^48 + 196 * q^49 + 297 * q^51 - 223 * q^52 - 873 * q^53 + 63 * q^56 - 216 * q^57 - 1408 * q^58 - 855 * q^59 + 687 * q^61 + 477 * q^62 - 252 * q^63 - 1285 * q^64 + 663 * q^66 + 503 * q^67 - 1725 * q^68 + 306 * q^69 - 681 * q^71 - 81 * q^72 - 1228 * q^73 - 3369 * q^74 + 1910 * q^76 - 147 * q^77 - 387 * q^78 + 345 * q^79 + 324 * q^81 + 495 * q^82 - 1509 * q^83 + 336 * q^84 - 540 * q^86 + 720 * q^87 - 2266 * q^88 - 198 * q^89 + 35 * q^91 - 2994 * q^92 - 1053 * q^93 - 1732 * q^94 + 216 * q^96 + 372 * q^97 + 189 * q^99

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 $

$2$	−4.75345	−1.68060	−0.840299	−	0.542123i	$-0.817621\pi$
$2$	−4.75345	−1.68060	−0.840299	+	0.542123i	$0.817621\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	14.5953	1.82441
$5$	0	0
$5$	0	0
$6$	14.2604	0.970294
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−31.3504	−1.38551
$9$	9.00000	0.333333
$10$	0	0
$11$	7.31799	0.200587	0.100293	−	0.994958i	$-0.468022\pi$
$11$	7.31799	0.200587	0.100293	+	0.994958i	$0.468022\pi$
$12$	−43.7859	−1.05332
$13$	4.15422	0.0886288	0.0443144	−	0.999018i	$-0.485890\pi$
$13$	4.15422	0.0886288	0.0443144	+	0.999018i	$0.485890\pi$
$14$	33.2742	0.635207
$15$	0	0
$16$	32.2604	0.504068
$17$	−53.5216	−0.763582	−0.381791	−	0.924249i	$-0.624693\pi$
$17$	−53.5216	−0.763582	−0.381791	+	0.924249i	$0.624693\pi$
$18$	−42.7811	−0.560200
$19$	88.9019	1.07345	0.536724	−	0.843758i	$-0.319662\pi$
$19$	88.9019	1.07345	0.536724	+	0.843758i	$0.319662\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	−34.7857	−0.337106
$23$	−156.780	−1.42134	−0.710670	−	0.703526i	$-0.751608\pi$
$23$	−156.780	−1.42134	−0.710670	+	0.703526i	$0.751608\pi$
$24$	94.0513	0.799922
$25$	0	0
$26$	−19.7469	−0.148949
$27$	−27.0000	−0.192450
$28$	−102.167	−0.689563
$29$	42.2570	0.270584	0.135292	−	0.990806i	$-0.456803\pi$
$29$	42.2570	0.270584	0.135292	+	0.990806i	$0.456803\pi$
$30$	0	0
$31$	−14.0248	−0.0812557	−0.0406279	−	0.999174i	$-0.512936\pi$
$31$	−14.0248	−0.0812557	−0.0406279	+	0.999174i	$0.512936\pi$
$32$	97.4554	0.538370
$33$	−21.9540	−0.115809
$34$	254.412	1.28328
$35$	0	0
$36$	131.358	0.608137
$37$	293.336	1.30335	0.651677	−	0.758496i	$-0.274066\pi$
$37$	293.336	1.30335	0.651677	+	0.758496i	$0.274066\pi$
$38$	−422.591	−1.80403
$39$	−12.4627	−0.0511699
$40$	0	0
$41$	−127.214	−0.484571	−0.242286	−	0.970205i	$-0.577897\pi$
$41$	−127.214	−0.484571	−0.242286	+	0.970205i	$0.577897\pi$
$42$	−99.8225	−0.366737
$43$	210.189	0.745431	0.372715	−	0.927946i	$-0.378427\pi$
$43$	210.189	0.745431	0.372715	+	0.927946i	$0.378427\pi$
$44$	106.808	0.365953
$45$	0	0
$46$	745.244	2.38870
$47$	468.688	1.45458	0.727289	−	0.686332i	$-0.240780\pi$
$47$	468.688	1.45458	0.727289	+	0.686332i	$0.240780\pi$
$48$	−96.7811	−0.291024
$49$	49.0000	0.142857
$50$	0	0
$51$	160.565	0.440854
$52$	60.6321	0.161696
$53$	−115.973	−0.300569	−0.150285	−	0.988643i	$-0.548019\pi$
$53$	−115.973	−0.300569	−0.150285	+	0.988643i	$0.548019\pi$
$54$	128.343	0.323431
$55$	0	0
$56$	219.453	0.523672
$57$	−266.706	−0.619755
$58$	−200.867	−0.454743
$59$	−314.090	−0.693069	−0.346534	−	0.938037i	$-0.612642\pi$
$59$	−314.090	−0.693069	−0.346534	+	0.938037i	$0.612642\pi$
$60$	0	0
$61$	768.386	1.61281	0.806407	−	0.591361i	$-0.201409\pi$
$61$	768.386	1.61281	0.806407	+	0.591361i	$0.201409\pi$
$62$	66.6662	0.136558
$63$	−63.0000	−0.125988
$64$	−721.332	−1.40885
$65$	0	0
$66$	104.357	0.194628
$67$	717.081	1.30754	0.653772	−	0.756692i	$-0.273186\pi$
$67$	717.081	1.30754	0.653772	+	0.756692i	$0.273186\pi$
$68$	−781.164	−1.39309
$69$	470.339	0.820610
$70$	0	0
$71$	−737.783	−1.23322	−0.616611	−	0.787268i	$-0.711495\pi$
$71$	−737.783	−1.23322	−0.616611	+	0.787268i	$0.711495\pi$
$72$	−282.154	−0.461835
$73$	−477.618	−0.765767	−0.382884	−	0.923797i	$-0.625069\pi$
$73$	−477.618	−0.765767	−0.382884	+	0.923797i	$0.625069\pi$
$74$	−1394.36	−2.19042
$75$	0	0
$76$	1297.55	1.95841
$77$	−51.2259	−0.0758147
$78$	59.2407	0.0859960
$79$	−279.262	−0.397714	−0.198857	−	0.980029i	$-0.563723\pi$
$79$	−279.262	−0.397714	−0.198857	+	0.980029i	$0.563723\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	604.703	0.814370
$83$	−776.981	−1.02753	−0.513764	−	0.857932i	$-0.671749\pi$
$83$	−776.981	−1.02753	−0.513764	+	0.857932i	$0.671749\pi$
$84$	306.501	0.398119
$85$	0	0
$86$	−999.123	−1.25277
$87$	−126.771	−0.156222
$88$	−229.422	−0.277914
$89$	−29.7626	−0.0354476	−0.0177238	−	0.999843i	$-0.505642\pi$
$89$	−29.7626	−0.0354476	−0.0177238	+	0.999843i	$0.505642\pi$
$90$	0	0
$91$	−29.0796	−0.0334985
$92$	−2288.24	−2.59311
$93$	42.0744	0.0469130
$94$	−2227.88	−2.44456
$95$	0	0
$96$	−292.366	−0.310828
$97$	231.793	0.242629	0.121314	−	0.992614i	$-0.461289\pi$
$97$	231.793	0.242629	0.121314	+	0.992614i	$0.461289\pi$
$98$	−232.919	−0.240086
$99$	65.8619	0.0668623
$100$	0	0
$101$	−1898.26	−1.87014	−0.935069	−	0.354467i	$-0.884662\pi$
$101$	−1898.26	−1.87014	−0.935069	+	0.354467i	$0.884662\pi$
$102$	−763.237	−0.740899
$103$	−1375.67	−1.31601	−0.658003	−	0.753015i	$-0.728599\pi$
$103$	−1375.67	−1.31601	−0.658003	+	0.753015i	$0.728599\pi$
$104$	−130.237	−0.122796
$105$	0	0
$106$	551.274	0.505136
$107$	−166.359	−0.150304	−0.0751521	−	0.997172i	$-0.523944\pi$
$107$	−166.359	−0.150304	−0.0751521	+	0.997172i	$0.523944\pi$
$108$	−394.073	−0.351108
$109$	−1346.00	−1.18279	−0.591393	−	0.806383i	$-0.701422\pi$
$109$	−1346.00	−1.18279	−0.591393	+	0.806383i	$0.701422\pi$
$110$	0	0
$111$	−880.008	−0.752492
$112$	−225.822	−0.190520
$113$	1322.25	1.10077	0.550386	−	0.834911i	$-0.314481\pi$
$113$	1322.25	1.10077	0.550386	+	0.834911i	$0.314481\pi$
$114$	1267.77	1.04156
$115$	0	0
$116$	616.754	0.493657
$117$	37.3880	0.0295429
$118$	1493.01	1.16477
$119$	374.651	0.288607
$120$	0	0
$121$	−1277.45	−0.959765
$122$	−3652.48	−2.71049
$123$	381.641	0.279767
$124$	−204.696	−0.148244
$125$	0	0
$126$	299.467	0.211736
$127$	−2111.85	−1.47556	−0.737781	−	0.675040i	$-0.764126\pi$
$127$	−2111.85	−1.47556	−0.737781	+	0.675040i	$0.764126\pi$
$128$	2649.18	1.82935
$129$	−630.567	−0.430375
$130$	0	0
$131$	−209.773	−0.139908	−0.0699541	−	0.997550i	$-0.522285\pi$
$131$	−209.773	−0.139908	−0.0699541	+	0.997550i	$0.522285\pi$
$132$	−320.425	−0.211283
$133$	−622.313	−0.405725
$134$	−3408.61	−2.19746
$135$	0	0
$136$	1677.93	1.05795
$137$	−2386.27	−1.48812	−0.744062	−	0.668111i	$-0.767103\pi$
$137$	−2386.27	−1.48812	−0.744062	+	0.668111i	$0.767103\pi$
$138$	−2235.73	−1.37912
$139$	1215.45	0.741675	0.370838	−	0.928698i	$-0.379071\pi$
$139$	1215.45	0.741675	0.370838	+	0.928698i	$0.379071\pi$
$140$	0	0
$141$	−1406.06	−0.839801
$142$	3507.01	2.07255
$143$	30.4006	0.0177778
$144$	290.343	0.168023
$145$	0	0
$146$	2270.34	1.28695
$147$	−147.000	−0.0824786
$148$	4281.33	2.37786
$149$	2849.51	1.56672	0.783359	−	0.621570i	$-0.213505\pi$
$149$	2849.51	1.56672	0.783359	+	0.621570i	$0.213505\pi$
$150$	0	0
$151$	−2643.84	−1.42485	−0.712426	−	0.701747i	$-0.752403\pi$
$151$	−2643.84	−1.42485	−0.712426	+	0.701747i	$0.752403\pi$
$152$	−2787.11	−1.48727
$153$	−481.695	−0.254527
$154$	243.500	0.127414
$155$	0	0
$156$	−181.896	−0.0933549
$157$	2563.09	1.30291	0.651454	−	0.758688i	$-0.274159\pi$
$157$	2563.09	1.30291	0.651454	+	0.758688i	$0.274159\pi$
$158$	1327.46	0.668398
$159$	347.920	0.173534
$160$	0	0
$161$	1097.46	0.537216
$162$	−385.030	−0.186733
$163$	1403.46	0.674400	0.337200	−	0.941433i	$-0.390520\pi$
$163$	1403.46	0.674400	0.337200	+	0.941433i	$0.390520\pi$
$164$	−1856.72	−0.884057
$165$	0	0
$166$	3693.34	1.72686
$167$	2658.97	1.23208	0.616040	−	0.787715i	$-0.288736\pi$
$167$	2658.97	1.23208	0.616040	+	0.787715i	$0.288736\pi$
$168$	−658.359	−0.302342
$169$	−2179.74	−0.992145
$170$	0	0
$171$	800.117	0.357816
$172$	3067.77	1.35997
$173$	−3763.19	−1.65382	−0.826909	−	0.562336i	$-0.809903\pi$
$173$	−3763.19	−1.65382	−0.826909	+	0.562336i	$0.809903\pi$
$174$	602.600	0.262546
$175$	0	0
$176$	236.081	0.101109
$177$	942.270	0.400143
$178$	141.475	0.0595731
$179$	−1056.94	−0.441336	−0.220668	−	0.975349i	$-0.570824\pi$
$179$	−1056.94	−0.441336	−0.220668	+	0.975349i	$0.570824\pi$
$180$	0	0
$181$	537.439	0.220705	0.110352	−	0.993893i	$-0.464802\pi$
$181$	537.439	0.220705	0.110352	+	0.993893i	$0.464802\pi$
$182$	138.228	0.0562976
$183$	−2305.16	−0.931159
$184$	4915.11	1.96927
$185$	0	0
$186$	−199.999	−0.0788420
$187$	−391.670	−0.153165
$188$	6840.64	2.65375
$189$	189.000	0.0727393
$190$	0	0
$191$	−3236.34	−1.22604	−0.613020	−	0.790068i	$-0.710045\pi$
$191$	−3236.34	−1.22604	−0.613020	+	0.790068i	$0.710045\pi$
$192$	2164.00	0.813401
$193$	−4620.71	−1.72335	−0.861673	−	0.507464i	$-0.830583\pi$
$193$	−4620.71	−1.72335	−0.861673	+	0.507464i	$0.830583\pi$
$194$	−1101.82	−0.407762
$195$	0	0
$196$	715.170	0.260630
$197$	2519.57	0.911230	0.455615	−	0.890177i	$-0.349419\pi$
$197$	2519.57	0.911230	0.455615	+	0.890177i	$0.349419\pi$
$198$	−313.071	−0.112369
$199$	−2121.77	−0.755819	−0.377910	−	0.925842i	$-0.623357\pi$
$199$	−2121.77	−0.755819	−0.377910	+	0.925842i	$0.623357\pi$
$200$	0	0
$201$	−2151.24	−0.754911
$202$	9023.28	3.14295
$203$	−295.799	−0.102271
$204$	2343.49	0.804300
$205$	0	0
$206$	6539.17	2.21168
$207$	−1411.02	−0.473780
$208$	134.017	0.0446750
$209$	650.583	0.215319
$210$	0	0
$211$	−1557.91	−0.508297	−0.254149	−	0.967165i	$-0.581795\pi$
$211$	−1557.91	−0.508297	−0.254149	+	0.967165i	$0.581795\pi$
$212$	−1692.67	−0.548362
$213$	2213.35	0.712001
$214$	790.780	0.252601
$215$	0	0
$216$	846.462	0.266641
$217$	98.1736	0.0307118
$218$	6398.16	1.98779
$219$	1432.85	0.442116
$220$	0	0
$221$	−222.341	−0.0676754
$222$	4183.07	1.26464
$223$	−1319.83	−0.396333	−0.198167	−	0.980168i	$-0.563499\pi$
$223$	−1319.83	−0.396333	−0.198167	+	0.980168i	$0.563499\pi$
$224$	−682.188	−0.203485
$225$	0	0
$226$	−6285.27	−1.84995
$227$	−6442.42	−1.88369	−0.941847	−	0.336043i	$-0.890911\pi$
$227$	−6442.42	−1.88369	−0.941847	+	0.336043i	$0.890911\pi$
$228$	−3892.65	−1.13069
$229$	4654.11	1.34302	0.671511	−	0.740995i	$-0.265646\pi$
$229$	4654.11	1.34302	0.671511	+	0.740995i	$0.265646\pi$
$230$	0	0
$231$	153.678	0.0437716
$232$	−1324.78	−0.374896
$233$	2628.61	0.739080	0.369540	−	0.929215i	$-0.379515\pi$
$233$	2628.61	0.739080	0.369540	+	0.929215i	$0.379515\pi$
$234$	−177.722	−0.0496498
$235$	0	0
$236$	−4584.24	−1.26444
$237$	837.786	0.229620
$238$	−1780.89	−0.485033
$239$	586.369	0.158699	0.0793495	−	0.996847i	$-0.474716\pi$
$239$	586.369	0.158699	0.0793495	+	0.996847i	$0.474716\pi$
$240$	0	0
$241$	3141.71	0.839731	0.419865	−	0.907586i	$-0.362077\pi$
$241$	3141.71	0.839731	0.419865	+	0.907586i	$0.362077\pi$
$242$	6072.28	1.61298
$243$	−243.000	−0.0641500
$244$	11214.8	2.94244
$245$	0	0
$246$	−1814.11	−0.470176
$247$	369.318	0.0951383
$248$	439.683	0.112580
$249$	2330.94	0.593243
$250$	0	0
$251$	−2929.38	−0.736656	−0.368328	−	0.929696i	$-0.620070\pi$
$251$	−2929.38	−0.736656	−0.368328	+	0.929696i	$0.620070\pi$
$252$	−919.504	−0.229854
$253$	−1147.31	−0.285102
$254$	10038.6	2.47983
$255$	0	0
$256$	−6822.07	−1.66554
$257$	388.552	0.0943081	0.0471541	−	0.998888i	$-0.484985\pi$
$257$	388.552	0.0943081	0.0471541	+	0.998888i	$0.484985\pi$
$258$	2997.37	0.723287
$259$	−2053.35	−0.492622
$260$	0	0
$261$	380.313	0.0901946
$262$	997.147	0.235130
$263$	−766.349	−0.179677	−0.0898387	−	0.995956i	$-0.528635\pi$
$263$	−766.349	−0.179677	−0.0898387	+	0.995956i	$0.528635\pi$
$264$	688.266	0.160454
$265$	0	0
$266$	2958.14	0.681861
$267$	89.2879	0.0204657
$268$	10466.0	2.38550
$269$	2842.03	0.644170	0.322085	−	0.946711i	$-0.395616\pi$
$269$	2842.03	0.644170	0.322085	+	0.946711i	$0.395616\pi$
$270$	0	0
$271$	3512.72	0.787390	0.393695	−	0.919241i	$-0.371197\pi$
$271$	3512.72	0.787390	0.393695	+	0.919241i	$0.371197\pi$
$272$	−1726.63	−0.384897
$273$	87.2387	0.0193404
$274$	11343.0	2.50094
$275$	0	0
$276$	6864.73	1.49713
$277$	−6388.71	−1.38578	−0.692889	−	0.721044i	$-0.743663\pi$
$277$	−6388.71	−1.38578	−0.692889	+	0.721044i	$0.743663\pi$
$278$	−5777.57	−1.24646
$279$	−126.223	−0.0270852
$280$	0	0
$281$	−2126.77	−0.451503	−0.225752	−	0.974185i	$-0.572484\pi$
$281$	−2126.77	−0.451503	−0.225752	+	0.974185i	$0.572484\pi$
$282$	6683.65	1.41137
$283$	−8332.13	−1.75015	−0.875077	−	0.483983i	$-0.839190\pi$
$283$	−8332.13	−1.75015	−0.875077	+	0.483983i	$0.839190\pi$
$284$	−10768.2	−2.24990
$285$	0	0
$286$	−144.508	−0.0298773
$287$	890.495	0.183151
$288$	877.099	0.179457
$289$	−2048.44	−0.416942
$290$	0	0
$291$	−695.379	−0.140082
$292$	−6970.98	−1.39707
$293$	8654.11	1.72552	0.862762	−	0.505610i	$-0.168733\pi$
$293$	8654.11	1.72552	0.862762	+	0.505610i	$0.168733\pi$
$294$	698.757	0.138613
$295$	0	0
$296$	−9196.21	−1.80581
$297$	−197.586	−0.0386030
$298$	−13545.0	−2.63302
$299$	−651.297	−0.125972
$300$	0	0
$301$	−1471.32	−0.281746
$302$	12567.4	2.39460
$303$	5694.78	1.07972
$304$	2868.01	0.541090
$305$	0	0
$306$	2289.71	0.427758
$307$	−5318.56	−0.988749	−0.494375	−	0.869249i	$-0.664603\pi$
$307$	−5318.56	−0.988749	−0.494375	+	0.869249i	$0.664603\pi$
$308$	−747.657	−0.138317
$309$	4127.00	0.759796
$310$	0	0
$311$	−7097.87	−1.29416	−0.647079	−	0.762423i	$-0.724010\pi$
$311$	−7097.87	−1.29416	−0.647079	+	0.762423i	$0.724010\pi$
$312$	390.710	0.0708962
$313$	2080.78	0.375759	0.187879	−	0.982192i	$-0.439839\pi$
$313$	2080.78	0.375759	0.187879	+	0.982192i	$0.439839\pi$
$314$	−12183.5	−2.18967
$315$	0	0
$316$	−4075.91	−0.725595
$317$	−2644.61	−0.468568	−0.234284	−	0.972168i	$-0.575275\pi$
$317$	−2644.61	−0.468568	−0.234284	+	0.972168i	$0.575275\pi$
$318$	−1653.82	−0.291641
$319$	309.236	0.0542756
$320$	0	0
$321$	499.078	0.0867782
$322$	−5216.71	−0.902844
$323$	−4758.17	−0.819665
$324$	1182.22	0.202712
$325$	0	0
$326$	−6671.26	−1.13340
$327$	4038.01	0.682882
$328$	3988.20	0.671376
$329$	−3280.81	−0.549779
$330$	0	0
$331$	−1831.16	−0.304077	−0.152039	−	0.988375i	$-0.548584\pi$
$331$	−1831.16	−0.304077	−0.152039	+	0.988375i	$0.548584\pi$
$332$	−11340.3	−1.87463
$333$	2640.02	0.434452
$334$	−12639.3	−2.07063
$335$	0	0
$336$	677.467	0.109997
$337$	−9307.67	−1.50451	−0.752257	−	0.658870i	$-0.771035\pi$
$337$	−9307.67	−1.50451	−0.752257	+	0.658870i	$0.771035\pi$
$338$	10361.3	1.66740
$339$	−3966.76	−0.635531
$340$	0	0
$341$	−102.633	−0.0162988
$342$	−3803.32	−0.601345
$343$	−343.000	−0.0539949
$344$	−6589.52	−1.03280
$345$	0	0
$346$	17888.2	2.77940
$347$	10802.7	1.67124	0.835619	−	0.549310i	$-0.185109\pi$
$347$	10802.7	1.67124	0.835619	+	0.549310i	$0.185109\pi$
$348$	−1850.26	−0.285013
$349$	2242.94	0.344017	0.172009	−	0.985095i	$-0.444974\pi$
$349$	2242.94	0.344017	0.172009	+	0.985095i	$0.444974\pi$
$350$	0	0
$351$	−112.164	−0.0170566
$352$	713.177	0.107990
$353$	−3295.68	−0.496916	−0.248458	−	0.968643i	$-0.579924\pi$
$353$	−3295.68	−0.496916	−0.248458	+	0.968643i	$0.579924\pi$
$354$	−4479.04	−0.672480
$355$	0	0
$356$	−434.395	−0.0646710
$357$	−1123.95	−0.166627
$358$	5024.09	0.741709
$359$	−4634.55	−0.681344	−0.340672	−	0.940182i	$-0.610655\pi$
$359$	−4634.55	−0.681344	−0.340672	+	0.940182i	$0.610655\pi$
$360$	0	0
$361$	1044.55	0.152289
$362$	−2554.69	−0.370916
$363$	3832.34	0.554121
$364$	−424.425	−0.0611152
$365$	0	0
$366$	10957.4	1.56490
$367$	−6316.17	−0.898369	−0.449185	−	0.893439i	$-0.648285\pi$
$367$	−6316.17	−0.898369	−0.449185	+	0.893439i	$0.648285\pi$
$368$	−5057.76	−0.716452
$369$	−1144.92	−0.161524
$370$	0	0
$371$	811.814	0.113604
$372$	614.088	0.0855887
$373$	7880.83	1.09398	0.546989	−	0.837140i	$-0.315774\pi$
$373$	7880.83	1.09398	0.546989	+	0.837140i	$0.315774\pi$
$374$	1861.79	0.257408
$375$	0	0
$376$	−14693.6	−2.01533
$377$	175.545	0.0239815
$378$	−898.402	−0.122246
$379$	9853.13	1.33541	0.667706	−	0.744425i	$-0.267276\pi$
$379$	9853.13	1.33541	0.667706	+	0.744425i	$0.267276\pi$
$380$	0	0
$381$	6335.55	0.851916
$382$	15383.8	2.06048
$383$	−1193.49	−0.159228	−0.0796141	−	0.996826i	$-0.525369\pi$
$383$	−1193.49	−0.159228	−0.0796141	+	0.996826i	$0.525369\pi$
$384$	−7947.53	−1.05617
$385$	0	0
$386$	21964.3	2.89625
$387$	1891.70	0.248477
$388$	3383.09	0.442655
$389$	−10425.7	−1.35888	−0.679442	−	0.733729i	$-0.737778\pi$
$389$	−10425.7	−1.35888	−0.679442	+	0.733729i	$0.737778\pi$
$390$	0	0
$391$	8391.09	1.08531
$392$	−1536.17	−0.197929
$393$	629.320	0.0807761
$394$	−11976.7	−1.53141
$395$	0	0
$396$	961.274	0.121984
$397$	−3650.05	−0.461437	−0.230719	−	0.973021i	$-0.574108\pi$
$397$	−3650.05	−0.461437	−0.230719	+	0.973021i	$0.574108\pi$
$398$	10085.7	1.27023
$399$	1866.94	0.234245
$400$	0	0
$401$	−8202.17	−1.02144	−0.510719	−	0.859747i	$-0.670621\pi$
$401$	−8202.17	−1.02144	−0.510719	+	0.859747i	$0.670621\pi$
$402$	10225.8	1.26870
$403$	−58.2622	−0.00720160
$404$	−27705.7	−3.41190
$405$	0	0
$406$	1406.07	0.171877
$407$	2146.63	0.261436
$408$	−5033.78	−0.610807
$409$	9097.72	1.09989	0.549943	−	0.835202i	$-0.314649\pi$
$409$	9097.72	1.09989	0.549943	+	0.835202i	$0.314649\pi$
$410$	0	0
$411$	7158.81	0.859169
$412$	−20078.3	−2.40094
$413$	2198.63	0.261955
$414$	6707.20	0.796234
$415$	0	0
$416$	404.852	0.0477151
$417$	−3646.34	−0.428206
$418$	−3092.51	−0.361866
$419$	−4704.01	−0.548462	−0.274231	−	0.961664i	$-0.588423\pi$
$419$	−4704.01	−0.548462	−0.274231	+	0.961664i	$0.588423\pi$
$420$	0	0
$421$	1596.95	0.184871	0.0924355	−	0.995719i	$-0.470535\pi$
$421$	1596.95	0.184871	0.0924355	+	0.995719i	$0.470535\pi$
$422$	7405.44	0.854244
$423$	4218.19	0.484859
$424$	3635.82	0.416441
$425$	0	0
$426$	−10521.0	−1.19659
$427$	−5378.70	−0.609587
$428$	−2428.06	−0.274217
$429$	−91.2017	−0.0102640
$430$	0	0
$431$	6235.23	0.696845	0.348423	−	0.937338i	$-0.386717\pi$
$431$	6235.23	0.696845	0.348423	+	0.937338i	$0.386717\pi$
$432$	−871.030	−0.0970079
$433$	−2363.94	−0.262364	−0.131182	−	0.991358i	$-0.541877\pi$
$433$	−2363.94	−0.262364	−0.131182	+	0.991358i	$0.541877\pi$
$434$	−466.663	−0.0516142
$435$	0	0
$436$	−19645.3	−2.15789
$437$	−13938.0	−1.52573
$438$	−6811.01	−0.743019
$439$	−17537.7	−1.90667	−0.953335	−	0.301916i	$-0.902374\pi$
$439$	−17537.7	−1.90667	−0.953335	+	0.301916i	$0.902374\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	1056.89	0.113735
$443$	4488.83	0.481424	0.240712	−	0.970597i	$-0.422619\pi$
$443$	4488.83	0.481424	0.240712	+	0.970597i	$0.422619\pi$
$444$	−12844.0	−1.37286
$445$	0	0
$446$	6273.74	0.666077
$447$	−8548.53	−0.904544
$448$	5049.33	0.532496
$449$	13188.3	1.38618	0.693090	−	0.720851i	$-0.256249\pi$
$449$	13188.3	1.38618	0.693090	+	0.720851i	$0.256249\pi$
$450$	0	0
$451$	−930.947	−0.0971986
$452$	19298.7	2.00826
$453$	7931.52	0.822638
$454$	30623.7	3.16573
$455$	0	0
$456$	8361.34	0.858674
$457$	−8085.04	−0.827576	−0.413788	−	0.910373i	$-0.635794\pi$
$457$	−8085.04	−0.827576	−0.413788	+	0.910373i	$0.635794\pi$
$458$	−22123.1	−2.25708
$459$	1445.08	0.146951
$460$	0	0
$461$	−825.258	−0.0833754	−0.0416877	−	0.999131i	$-0.513273\pi$
$461$	−825.258	−0.0833754	−0.0416877	+	0.999131i	$0.513273\pi$
$462$	−730.499	−0.0735626
$463$	−16607.8	−1.66702	−0.833509	−	0.552505i	$-0.813672\pi$
$463$	−16607.8	−1.66702	−0.833509	+	0.552505i	$0.813672\pi$
$464$	1363.23	0.136393
$465$	0	0
$466$	−12495.0	−1.24210
$467$	−6003.84	−0.594914	−0.297457	−	0.954735i	$-0.596138\pi$
$467$	−6003.84	−0.594914	−0.297457	+	0.954735i	$0.596138\pi$
$468$	545.689	0.0538985
$469$	−5019.57	−0.494205
$470$	0	0
$471$	−7689.26	−0.752235
$472$	9846.86	0.960251
$473$	1538.16	0.149524
$474$	−3982.37	−0.385900
$475$	0	0
$476$	5468.15	0.526538
$477$	−1043.76	−0.100190
$478$	−2787.28	−0.266709
$479$	20386.0	1.94459	0.972295	−	0.233759i	$-0.0751025\pi$
$479$	20386.0	1.94459	0.972295	+	0.233759i	$0.0751025\pi$
$480$	0	0
$481$	1218.58	0.115515
$482$	−14933.9	−1.41125
$483$	−3292.37	−0.310162
$484$	−18644.7	−1.75101
$485$	0	0
$486$	1155.09	0.107810
$487$	−9422.61	−0.876754	−0.438377	−	0.898791i	$-0.644447\pi$
$487$	−9422.61	−0.876754	−0.438377	+	0.898791i	$0.644447\pi$
$488$	−24089.2	−2.23456
$489$	−4210.37	−0.389365
$490$	0	0
$491$	5908.30	0.543051	0.271525	−	0.962431i	$-0.412472\pi$
$491$	5908.30	0.543051	0.271525	+	0.962431i	$0.412472\pi$
$492$	5570.16	0.510411
$493$	−2261.66	−0.206613
$494$	−1755.54	−0.159889
$495$	0	0
$496$	−452.445	−0.0409584
$497$	5164.48	0.466114
$498$	−11080.0	−0.997003
$499$	−3237.65	−0.290455	−0.145227	−	0.989398i	$-0.546391\pi$
$499$	−3237.65	−0.290455	−0.145227	+	0.989398i	$0.546391\pi$
$500$	0	0
$501$	−7976.92	−0.711342
$502$	13924.7	1.23802
$503$	−1112.05	−0.0985760	−0.0492880	−	0.998785i	$-0.515695\pi$
$503$	−1112.05	−0.0985760	−0.0492880	+	0.998785i	$0.515695\pi$
$504$	1975.08	0.174557
$505$	0	0
$506$	5453.68	0.479142
$507$	6539.23	0.572815
$508$	−30823.1	−2.69203
$509$	15737.5	1.37044	0.685220	−	0.728336i	$-0.259706\pi$
$509$	15737.5	1.37044	0.685220	+	0.728336i	$0.259706\pi$
$510$	0	0
$511$	3343.33	0.289433
$512$	11235.0	0.969765
$513$	−2400.35	−0.206585
$514$	−1846.96	−0.158494
$515$	0	0
$516$	−9203.32	−0.785181
$517$	3429.85	0.291769
$518$	9760.51	0.827900
$519$	11289.6	0.954832
$520$	0	0
$521$	−5009.73	−0.421267	−0.210633	−	0.977565i	$-0.567553\pi$
$521$	−5009.73	−0.421267	−0.210633	+	0.977565i	$0.567553\pi$
$522$	−1807.80	−0.151581
$523$	−14162.2	−1.18407	−0.592035	−	0.805912i	$-0.701675\pi$
$523$	−14162.2	−1.18407	−0.592035	+	0.805912i	$0.701675\pi$
$524$	−3061.70	−0.255250
$525$	0	0
$526$	3642.80	0.301965
$527$	750.630	0.0620454
$528$	−708.242	−0.0583756
$529$	12412.8	1.02020
$530$	0	0
$531$	−2826.81	−0.231023
$532$	−9082.85	−0.740209
$533$	−528.474	−0.0429470
$534$	−424.426	−0.0343946
$535$	0	0
$536$	−22480.8	−1.81161
$537$	3170.81	0.254805
$538$	−13509.5	−1.08259
$539$	358.581	0.0286553
$540$	0	0
$541$	14815.3	1.17738	0.588689	−	0.808360i	$-0.299644\pi$
$541$	14815.3	1.17738	0.588689	+	0.808360i	$0.299644\pi$
$542$	−16697.5	−1.32329
$543$	−1612.32	−0.127424
$544$	−5215.97	−0.411090
$545$	0	0
$546$	−414.685	−0.0325034
$547$	−14248.3	−1.11373	−0.556867	−	0.830601i	$-0.687997\pi$
$547$	−14248.3	−1.11373	−0.556867	+	0.830601i	$0.687997\pi$
$548$	−34828.3	−2.71495
$549$	6915.47	0.537605
$550$	0	0
$551$	3756.73	0.290458
$552$	−14745.3	−1.13696
$553$	1954.83	0.150322
$554$	30368.4	2.32894
$555$	0	0
$556$	17739.8	1.35312
$557$	−9394.37	−0.714636	−0.357318	−	0.933983i	$-0.616309\pi$
$557$	−9394.37	−0.714636	−0.357318	+	0.933983i	$0.616309\pi$
$558$	599.996	0.0455194
$559$	873.173	0.0660667
$560$	0	0
$561$	1175.01	0.0884296
$562$	10109.5	0.758796
$563$	8572.89	0.641748	0.320874	−	0.947122i	$-0.396023\pi$
$563$	8572.89	0.641748	0.320874	+	0.947122i	$0.396023\pi$
$564$	−20521.9	−1.53214
$565$	0	0
$566$	39606.4	2.94131
$567$	−567.000	−0.0419961
$568$	23129.8	1.70864
$569$	8954.65	0.659751	0.329876	−	0.944024i	$-0.392993\pi$
$569$	8954.65	0.659751	0.329876	+	0.944024i	$0.392993\pi$
$570$	0	0
$571$	21592.5	1.58252	0.791260	−	0.611479i	$-0.209425\pi$
$571$	21592.5	1.58252	0.791260	+	0.611479i	$0.209425\pi$
$572$	443.705	0.0324340
$573$	9709.03	0.707854
$574$	−4232.92	−0.307803
$575$	0	0
$576$	−6491.99	−0.469617
$577$	17587.5	1.26894	0.634470	−	0.772947i	$-0.281218\pi$
$577$	17587.5	1.26894	0.634470	+	0.772947i	$0.281218\pi$
$578$	9737.14	0.700712
$579$	13862.1	0.994974
$580$	0	0
$581$	5438.87	0.388369
$582$	3305.45	0.235421
$583$	−848.692	−0.0602903
$584$	14973.5	1.06098
$585$	0	0
$586$	−41136.9	−2.89991
$587$	4613.88	0.324421	0.162210	−	0.986756i	$-0.448138\pi$
$587$	4613.88	0.324421	0.162210	+	0.986756i	$0.448138\pi$
$588$	−2145.51	−0.150475
$589$	−1246.83	−0.0872237
$590$	0	0
$591$	−7558.72	−0.526099
$592$	9463.12	0.656980
$593$	8688.07	0.601647	0.300823	−	0.953680i	$-0.402739\pi$
$593$	8688.07	0.601647	0.300823	+	0.953680i	$0.402739\pi$
$594$	939.214	0.0648761
$595$	0	0
$596$	41589.4	2.85834
$597$	6365.30	0.436372
$598$	3095.91	0.211708
$599$	7361.43	0.502137	0.251068	−	0.967969i	$-0.419218\pi$
$599$	7361.43	0.502137	0.251068	+	0.967969i	$0.419218\pi$
$600$	0	0
$601$	16441.9	1.11594	0.557971	−	0.829861i	$-0.311580\pi$
$601$	16441.9	1.11594	0.557971	+	0.829861i	$0.311580\pi$
$602$	6993.86	0.473503
$603$	6453.73	0.435848
$604$	−38587.6	−2.59952
$605$	0	0
$606$	−27069.9	−1.81458
$607$	21024.5	1.40586	0.702931	−	0.711258i	$-0.251874\pi$
$607$	21024.5	1.40586	0.702931	+	0.711258i	$0.251874\pi$
$608$	8663.97	0.577912
$609$	887.398	0.0590463
$610$	0	0
$611$	1947.03	0.128917
$612$	−7030.48	−0.464363
$613$	−3278.79	−0.216034	−0.108017	−	0.994149i	$-0.534450\pi$
$613$	−3278.79	−0.216034	−0.108017	+	0.994149i	$0.534450\pi$
$614$	25281.5	1.66169
$615$	0	0
$616$	1605.95	0.105042
$617$	−136.641	−0.00891567	−0.00445783	−	0.999990i	$-0.501419\pi$
$617$	−136.641	−0.00891567	−0.00445783	+	0.999990i	$0.501419\pi$
$618$	−19617.5	−1.27691
$619$	−13109.0	−0.851205	−0.425603	−	0.904910i	$-0.639938\pi$
$619$	−13109.0	−0.851205	−0.425603	+	0.904910i	$0.639938\pi$
$620$	0	0
$621$	4233.05	0.273537
$622$	33739.4	2.17496
$623$	208.338	0.0133979
$624$	−402.050	−0.0257931
$625$	0	0
$626$	−9890.87	−0.631500
$627$	−1951.75	−0.124315
$628$	37409.0	2.37704
$629$	−15699.8	−0.995219
$630$	0	0
$631$	−351.608	−0.0221827	−0.0110913	−	0.999938i	$-0.503531\pi$
$631$	−351.608	−0.0221827	−0.0110913	+	0.999938i	$0.503531\pi$
$632$	8754.98	0.551035
$633$	4673.72	0.293466
$634$	12571.0	0.787474
$635$	0	0
$636$	5078.00	0.316597
$637$	203.557	0.0126613
$638$	−1469.94	−0.0912155
$639$	−6640.04	−0.411074
$640$	0	0
$641$	−17551.0	−1.08147	−0.540736	−	0.841192i	$-0.681854\pi$
$641$	−17551.0	−1.08147	−0.540736	+	0.841192i	$0.681854\pi$
$642$	−2372.34	−0.145839
$643$	−20683.7	−1.26856	−0.634281	−	0.773103i	$-0.718704\pi$
$643$	−20683.7	−1.26856	−0.634281	+	0.773103i	$0.718704\pi$
$644$	16017.7	0.980103
$645$	0	0
$646$	22617.7	1.37753
$647$	−1782.40	−0.108305	−0.0541526	−	0.998533i	$-0.517246\pi$
$647$	−1782.40	−0.108305	−0.0541526	+	0.998533i	$0.517246\pi$
$648$	−2539.38	−0.153945
$649$	−2298.51	−0.139020
$650$	0	0
$651$	−294.521	−0.0177315
$652$	20483.9	1.23038
$653$	3642.28	0.218275	0.109137	−	0.994027i	$-0.465191\pi$
$653$	3642.28	0.218275	0.109137	+	0.994027i	$0.465191\pi$
$654$	−19194.5	−1.14765
$655$	0	0
$656$	−4103.95	−0.244257
$657$	−4298.56	−0.255256
$658$	15595.2	0.923957
$659$	−14883.2	−0.879768	−0.439884	−	0.898055i	$-0.644980\pi$
$659$	−14883.2	−0.879768	−0.439884	+	0.898055i	$0.644980\pi$
$660$	0	0
$661$	−1871.82	−0.110145	−0.0550723	−	0.998482i	$-0.517539\pi$
$661$	−1871.82	−0.110145	−0.0550723	+	0.998482i	$0.517539\pi$
$662$	8704.33	0.511032
$663$	667.022	0.0390724
$664$	24358.7	1.42365
$665$	0	0
$666$	−12549.2	−0.730139
$667$	−6625.04	−0.384592
$668$	38808.5	2.24782
$669$	3959.49	0.228823
$670$	0	0
$671$	5623.03	0.323509
$672$	2046.56	0.117482
$673$	22208.8	1.27205	0.636023	−	0.771670i	$-0.280578\pi$
$673$	22208.8	1.27205	0.636023	+	0.771670i	$0.280578\pi$
$674$	44243.6	2.52848
$675$	0	0
$676$	−31814.0	−1.81008
$677$	31340.2	1.77918	0.889589	−	0.456762i	$-0.150991\pi$
$677$	31340.2	1.77918	0.889589	+	0.456762i	$0.150991\pi$
$678$	18855.8	1.06807
$679$	−1622.55	−0.0917051
$680$	0	0
$681$	19327.3	1.08755
$682$	487.862	0.0273918
$683$	−27666.5	−1.54997	−0.774985	−	0.631979i	$-0.782243\pi$
$683$	−27666.5	−1.54997	−0.774985	+	0.631979i	$0.782243\pi$
$684$	11677.9	0.652803
$685$	0	0
$686$	1630.43	0.0907438
$687$	−13962.3	−0.775394
$688$	6780.77	0.375748
$689$	−481.780	−0.0266391
$690$	0	0
$691$	−7978.64	−0.439250	−0.219625	−	0.975584i	$-0.570483\pi$
$691$	−7978.64	−0.439250	−0.219625	+	0.975584i	$0.570483\pi$
$692$	−54925.0	−3.01724
$693$	−461.033	−0.0252716
$694$	−51350.1	−2.80868
$695$	0	0
$696$	3974.33	0.216446
$697$	6808.67	0.370010
$698$	−10661.7	−0.578155
$699$	−7885.82	−0.426708
$700$	0	0
$701$	−16299.3	−0.878197	−0.439099	−	0.898439i	$-0.644702\pi$
$701$	−16299.3	−0.878197	−0.439099	+	0.898439i	$0.644702\pi$
$702$	533.166	0.0286653
$703$	26078.1	1.39908
$704$	−5278.70	−0.282597
$705$	0	0
$706$	15665.8	0.835116
$707$	13287.8	0.706845
$708$	13752.7	0.730026
$709$	−36223.4	−1.91876	−0.959379	−	0.282122i	$-0.908962\pi$
$709$	−36223.4	−1.91876	−0.959379	+	0.282122i	$0.908962\pi$
$710$	0	0
$711$	−2513.36	−0.132571
$712$	933.071	0.0491128
$713$	2198.80	0.115492
$714$	5342.66	0.280034
$715$	0	0
$716$	−15426.3	−0.805179
$717$	−1759.11	−0.0916249
$718$	22030.1	1.14507
$719$	−636.264	−0.0330023	−0.0165011	−	0.999864i	$-0.505253\pi$
$719$	−636.264	−0.0330023	−0.0165011	+	0.999864i	$0.505253\pi$
$720$	0	0
$721$	9629.67	0.497403
$722$	−4965.21	−0.255936
$723$	−9425.12	−0.484819
$724$	7844.08	0.402656
$725$	0	0
$726$	−18216.8	−0.931254
$727$	33098.8	1.68854	0.844270	−	0.535918i	$-0.180035\pi$
$727$	33098.8	1.68854	0.844270	+	0.535918i	$0.180035\pi$
$728$	911.657	0.0464124
$729$	729.000	0.0370370
$730$	0	0
$731$	−11249.7	−0.569198
$732$	−33644.4	−1.69882
$733$	12019.8	0.605678	0.302839	−	0.953042i	$-0.402066\pi$
$733$	12019.8	0.605678	0.302839	+	0.953042i	$0.402066\pi$
$734$	30023.6	1.50980
$735$	0	0
$736$	−15279.0	−0.765207
$737$	5247.59	0.262276
$738$	5442.33	0.271457
$739$	5332.38	0.265433	0.132716	−	0.991154i	$-0.457630\pi$
$739$	5332.38	0.265433	0.132716	+	0.991154i	$0.457630\pi$
$740$	0	0
$741$	−1107.96	−0.0549281
$742$	−3858.92	−0.190924
$743$	−6547.51	−0.323290	−0.161645	−	0.986849i	$-0.551680\pi$
$743$	−6547.51	−0.323290	−0.161645	+	0.986849i	$0.551680\pi$
$744$	−1319.05	−0.0649983
$745$	0	0
$746$	−37461.1	−1.83854
$747$	−6992.83	−0.342509
$748$	−5716.55	−0.279435
$749$	1164.51	0.0568097
$750$	0	0
$751$	19316.3	0.938565	0.469282	−	0.883048i	$-0.344513\pi$
$751$	19316.3	0.938565	0.469282	+	0.883048i	$0.344513\pi$
$752$	15120.0	0.733206
$753$	8788.14	0.425309
$754$	−834.446	−0.0403033
$755$	0	0
$756$	2758.51	0.132706
$757$	−40483.3	−1.94371	−0.971857	−	0.235573i	$-0.924303\pi$
$757$	−40483.3	−1.94371	−0.971857	+	0.235573i	$0.924303\pi$
$758$	−46836.3	−2.24429
$759$	3441.93	0.164604
$760$	0	0
$761$	−5470.00	−0.260561	−0.130281	−	0.991477i	$-0.541588\pi$
$761$	−5470.00	−0.260561	−0.130281	+	0.991477i	$0.541588\pi$
$762$	−30115.7	−1.43173
$763$	9422.02	0.447051
$764$	−47235.4	−2.23680
$765$	0	0
$766$	5673.19	0.267599
$767$	−1304.80	−0.0614258
$768$	20466.2	0.961602
$769$	17692.4	0.829653	0.414826	−	0.909901i	$-0.363842\pi$
$769$	17692.4	0.829653	0.414826	+	0.909901i	$0.363842\pi$
$770$	0	0
$771$	−1165.65	−0.0544488
$772$	−67440.6	−3.14409
$773$	3338.50	0.155340	0.0776699	−	0.996979i	$-0.475252\pi$
$773$	3338.50	0.155340	0.0776699	+	0.996979i	$0.475252\pi$
$774$	−8992.11	−0.417590
$775$	0	0
$776$	−7266.81	−0.336164
$777$	6160.05	0.284415
$778$	49558.2	2.28374
$779$	−11309.5	−0.520161
$780$	0	0
$781$	−5399.08	−0.247368
$782$	−39886.7	−1.82397
$783$	−1140.94	−0.0520739
$784$	1580.76	0.0720097
$785$	0	0
$786$	−2991.44	−0.135752
$787$	−11738.0	−0.531657	−0.265829	−	0.964020i	$-0.585646\pi$
$787$	−11738.0	−0.531657	−0.265829	+	0.964020i	$0.585646\pi$
$788$	36773.9	1.66246
$789$	2299.05	0.103737
$790$	0	0
$791$	−9255.77	−0.416052
$792$	−2064.80	−0.0926381
$793$	3192.05	0.142942
$794$	17350.3	0.775491
$795$	0	0
$796$	−30967.8	−1.37893
$797$	20150.5	0.895568	0.447784	−	0.894142i	$-0.352213\pi$
$797$	20150.5	0.895568	0.447784	+	0.894142i	$0.352213\pi$
$798$	−8874.41	−0.393672
$799$	−25084.9	−1.11069
$800$	0	0
$801$	−267.864	−0.0118159
$802$	38988.6	1.71663
$803$	−3495.20	−0.153603
$804$	−31398.0	−1.37727
$805$	0	0
$806$	276.946	0.0121030
$807$	−8526.09	−0.371912
$808$	59511.3	2.59109
$809$	−3381.53	−0.146957	−0.0734785	−	0.997297i	$-0.523410\pi$
$809$	−3381.53	−0.146957	−0.0734785	+	0.997297i	$0.523410\pi$
$810$	0	0
$811$	11375.3	0.492528	0.246264	−	0.969203i	$-0.420797\pi$
$811$	11375.3	0.492528	0.246264	+	0.969203i	$0.420797\pi$
$812$	−4317.28	−0.186585
$813$	−10538.2	−0.454600
$814$	−10203.9	−0.439369
$815$	0	0
$816$	5179.88	0.222221
$817$	18686.2	0.800181
$818$	−43245.6	−1.84847
$819$	−261.716	−0.0111662
$820$	0	0
$821$	−9610.94	−0.408556	−0.204278	−	0.978913i	$-0.565485\pi$
$821$	−9610.94	−0.408556	−0.204278	+	0.978913i	$0.565485\pi$
$822$	−34029.1	−1.44392
$823$	−13767.3	−0.583109	−0.291555	−	0.956554i	$-0.594173\pi$
$823$	−13767.3	−0.583109	−0.291555	+	0.956554i	$0.594173\pi$
$824$	43127.8	1.82333
$825$	0	0
$826$	−10451.1	−0.440242
$827$	−31978.6	−1.34462	−0.672312	−	0.740268i	$-0.734699\pi$
$827$	−31978.6	−1.34462	−0.672312	+	0.740268i	$0.734699\pi$
$828$	−20594.2	−0.864369
$829$	18477.2	0.774111	0.387055	−	0.922056i	$-0.373492\pi$
$829$	18477.2	0.774111	0.387055	+	0.922056i	$0.373492\pi$
$830$	0	0
$831$	19166.1	0.800080
$832$	−2996.58	−0.124865
$833$	−2622.56	−0.109083
$834$	17332.7	0.719643
$835$	0	0
$836$	9495.45	0.392831
$837$	378.669	0.0156377
$838$	22360.3	0.921745
$839$	−38552.6	−1.58639	−0.793196	−	0.608967i	$-0.791584\pi$
$839$	−38552.6	−1.58639	−0.793196	+	0.608967i	$0.791584\pi$
$840$	0	0
$841$	−22603.3	−0.926784
$842$	−7591.04	−0.310694
$843$	6380.31	0.260676
$844$	−22738.1	−0.927344
$845$	0	0
$846$	−20051.0	−0.814854
$847$	8942.13	0.362757
$848$	−3741.34	−0.151507
$849$	24996.4	1.01045
$850$	0	0
$851$	−45989.1	−1.85251
$852$	32304.5	1.29898
$853$	−3080.15	−0.123637	−0.0618185	−	0.998087i	$-0.519690\pi$
$853$	−3080.15	−0.123637	−0.0618185	+	0.998087i	$0.519690\pi$
$854$	25567.4	1.02447
$855$	0	0
$856$	5215.43	0.208247
$857$	28362.3	1.13050	0.565249	−	0.824920i	$-0.308780\pi$
$857$	28362.3	1.13050	0.565249	+	0.824920i	$0.308780\pi$
$858$	433.523	0.0172497
$859$	−4928.50	−0.195761	−0.0978803	−	0.995198i	$-0.531206\pi$
$859$	−4928.50	−0.195761	−0.0978803	+	0.995198i	$0.531206\pi$
$860$	0	0
$861$	−2671.48	−0.105742
$862$	−29638.9	−1.17112
$863$	−39910.8	−1.57425	−0.787126	−	0.616792i	$-0.788432\pi$
$863$	−39910.8	−1.57425	−0.787126	+	0.616792i	$0.788432\pi$
$864$	−2631.30	−0.103609
$865$	0	0
$866$	11236.9	0.440928
$867$	6145.31	0.240722
$868$	1432.87	0.0560309
$869$	−2043.63	−0.0797762
$870$	0	0
$871$	2978.92	0.115886
$872$	42197.8	1.63876
$873$	2086.14	0.0808763
$874$	66253.6	2.56414
$875$	0	0
$876$	20912.9	0.806602
$877$	29231.5	1.12552	0.562759	−	0.826621i	$-0.309740\pi$
$877$	29231.5	1.12552	0.562759	+	0.826621i	$0.309740\pi$
$878$	83364.5	3.20435
$879$	−25962.3	−0.996232
$880$	0	0
$881$	−23473.3	−0.897657	−0.448829	−	0.893618i	$-0.648159\pi$
$881$	−23473.3	−0.897657	−0.448829	+	0.893618i	$0.648159\pi$
$882$	−2096.27	−0.0800285
$883$	−21250.7	−0.809902	−0.404951	−	0.914338i	$-0.632711\pi$
$883$	−21250.7	−0.809902	−0.404951	+	0.914338i	$0.632711\pi$
$884$	−3245.13	−0.123468
$885$	0	0
$886$	−21337.4	−0.809080
$887$	48748.6	1.84534	0.922671	−	0.385588i	$-0.126001\pi$
$887$	48748.6	1.84534	0.922671	+	0.385588i	$0.126001\pi$
$888$	27588.6	1.04258
$889$	14783.0	0.557710
$890$	0	0
$891$	592.757	0.0222874
$892$	−19263.3	−0.723075
$893$	41667.2	1.56141
$894$	40635.0	1.52018
$895$	0	0
$896$	−18544.2	−0.691428
$897$	1953.89	0.0727297
$898$	−62690.0	−2.32961
$899$	−592.646	−0.0219865
$900$	0	0
$901$	6207.08	0.229509
$902$	4425.21	0.163352
$903$	4413.97	0.162666
$904$	−41453.2	−1.52513
$905$	0	0
$906$	−37702.1	−1.38253
$907$	13144.5	0.481209	0.240604	−	0.970623i	$-0.422654\pi$
$907$	13144.5	0.481209	0.240604	+	0.970623i	$0.422654\pi$
$908$	−94029.0	−3.43663
$909$	−17084.3	−0.623379
$910$	0	0
$911$	−29731.7	−1.08129	−0.540644	−	0.841251i	$-0.681819\pi$
$911$	−29731.7	−1.08129	−0.540644	+	0.841251i	$0.681819\pi$
$912$	−8604.02	−0.312399
$913$	−5685.94	−0.206108
$914$	38431.8	1.39082
$915$	0	0
$916$	67928.1	2.45022
$917$	1468.41	0.0528803
$918$	−6869.13	−0.246966
$919$	−47715.4	−1.71272	−0.856358	−	0.516383i	$-0.827278\pi$
$919$	−47715.4	−1.71272	−0.856358	+	0.516383i	$0.827278\pi$
$920$	0	0
$921$	15955.7	0.570855
$922$	3922.82	0.140121
$923$	−3064.92	−0.109299
$924$	2242.97	0.0798575
$925$	0	0
$926$	78944.4	2.80159
$927$	−12381.0	−0.438669
$928$	4118.18	0.145674
$929$	−19795.2	−0.699094	−0.349547	−	0.936919i	$-0.613665\pi$
$929$	−19795.2	−0.699094	−0.349547	+	0.936919i	$0.613665\pi$
$930$	0	0
$931$	4356.19	0.153350
$932$	38365.3	1.34839
$933$	21293.6	0.747183
$934$	28539.0	0.999811
$935$	0	0
$936$	−1172.13	−0.0409319
$937$	17993.9	0.627359	0.313680	−	0.949529i	$-0.398438\pi$
$937$	17993.9	0.627359	0.313680	+	0.949529i	$0.398438\pi$
$938$	23860.3	0.830560
$939$	−6242.33	−0.216944
$940$	0	0
$941$	−34667.7	−1.20099	−0.600497	−	0.799627i	$-0.705030\pi$
$941$	−34667.7	−1.20099	−0.600497	+	0.799627i	$0.705030\pi$
$942$	36550.5	1.26420
$943$	19944.5	0.688740
$944$	−10132.7	−0.349354
$945$	0	0
$946$	−7311.57	−0.251289
$947$	56727.3	1.94656	0.973279	−	0.229627i	$-0.0737505\pi$
$947$	56727.3	1.94656	0.973279	+	0.229627i	$0.0737505\pi$
$948$	12227.7	0.418922
$949$	−1984.13	−0.0678690
$950$	0	0
$951$	7933.82	0.270528
$952$	−11745.5	−0.399867
$953$	−46267.5	−1.57267	−0.786333	−	0.617802i	$-0.788023\pi$
$953$	−46267.5	−1.57267	−0.786333	+	0.617802i	$0.788023\pi$
$954$	4961.46	0.168379
$955$	0	0
$956$	8558.23	0.289532
$957$	−927.709	−0.0313360
$958$	−96903.7	−3.26807
$959$	16703.9	0.562458
$960$	0	0
$961$	−29594.3	−0.993398
$962$	−5792.48	−0.194134
$963$	−1497.23	−0.0501014
$964$	45854.1	1.53201
$965$	0	0
$966$	15650.1	0.521257
$967$	−26514.2	−0.881738	−0.440869	−	0.897571i	$-0.645330\pi$
$967$	−26514.2	−0.881738	−0.440869	+	0.897571i	$0.645330\pi$
$968$	40048.5	1.32976
$969$	14274.5	0.473234
$970$	0	0
$971$	38864.6	1.28447	0.642237	−	0.766506i	$-0.278006\pi$
$971$	38864.6	1.28447	0.642237	+	0.766506i	$0.278006\pi$
$972$	−3546.66	−0.117036
$973$	−8508.13	−0.280327
$974$	44789.9	1.47347
$975$	0	0
$976$	24788.4	0.812968
$977$	33774.5	1.10598	0.552990	−	0.833188i	$-0.313487\pi$
$977$	33774.5	1.10598	0.552990	+	0.833188i	$0.313487\pi$
$978$	20013.8	0.654367
$979$	−217.803	−0.00711032
$980$	0	0
$981$	−12114.0	−0.394262
$982$	−28084.8	−0.912651
$983$	−23536.8	−0.763691	−0.381846	−	0.924226i	$-0.624711\pi$
$983$	−23536.8	−0.763691	−0.381846	+	0.924226i	$0.624711\pi$
$984$	−11964.6	−0.387619
$985$	0	0
$986$	10750.7	0.347234
$987$	9842.44	0.317415
$988$	5390.31	0.173572
$989$	−32953.3	−1.05951
$990$	0	0
$991$	29012.6	0.929984	0.464992	−	0.885315i	$-0.346057\pi$
$991$	29012.6	0.929984	0.464992	+	0.885315i	$0.346057\pi$
$992$	−1366.79	−0.0437457
$993$	5493.48	0.175559
$994$	−24549.1	−0.783350
$995$	0	0
$996$	34020.8	1.08232
$997$	−14027.1	−0.445579	−0.222790	−	0.974867i	$-0.571516\pi$
$997$	−14027.1	−0.445579	−0.222790	+	0.974867i	$0.571516\pi$
$998$	15390.0	0.488138
$999$	−7920.07	−0.250831

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.4.a.t.1.1	✓	4
3.2	odd	2		1575.4.a.bj.1.4		4
5.2	odd	4		525.4.d.n.274.1		8
5.3	odd	4		525.4.d.n.274.8		8
5.4	even	2		525.4.a.u.1.4	yes	4
15.14	odd	2		1575.4.a.bk.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.4.a.t.1.1	✓	4	1.1	even	1	trivial
525.4.a.u.1.4	yes	4	5.4	even	2
525.4.d.n.274.1		8	5.2	odd	4
525.4.d.n.274.8		8	5.3	odd	4
1575.4.a.bj.1.4		4	3.2	odd	2
1575.4.a.bk.1.1		4	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:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	525.a (trivial)

\(f(q)\)	\(=\)	\(q-4.75345 q^{2} -3.00000 q^{3} +14.5953 q^{4} +14.2604 q^{6} -7.00000 q^{7} -31.3504 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-4.75345 q^{2} -3.00000 q^{3} +14.5953 q^{4} +14.2604 q^{6} -7.00000 q^{7} -31.3504 q^{8} +9.00000 q^{9} +7.31799 q^{11} -43.7859 q^{12} +4.15422 q^{13} +33.2742 q^{14} +32.2604 q^{16} -53.5216 q^{17} -42.7811 q^{18} +88.9019 q^{19} +21.0000 q^{21} -34.7857 q^{22} -156.780 q^{23} +94.0513 q^{24} -19.7469 q^{26} -27.0000 q^{27} -102.167 q^{28} +42.2570 q^{29} -14.0248 q^{31} +97.4554 q^{32} -21.9540 q^{33} +254.412 q^{34} +131.358 q^{36} +293.336 q^{37} -422.591 q^{38} -12.4627 q^{39} -127.214 q^{41} -99.8225 q^{42} +210.189 q^{43} +106.808 q^{44} +745.244 q^{46} +468.688 q^{47} -96.7811 q^{48} +49.0000 q^{49} +160.565 q^{51} +60.6321 q^{52} -115.973 q^{53} +128.343 q^{54} +219.453 q^{56} -266.706 q^{57} -200.867 q^{58} -314.090 q^{59} +768.386 q^{61} +66.6662 q^{62} -63.0000 q^{63} -721.332 q^{64} +104.357 q^{66} +717.081 q^{67} -781.164 q^{68} +470.339 q^{69} -737.783 q^{71} -282.154 q^{72} -477.618 q^{73} -1394.36 q^{74} +1297.55 q^{76} -51.2259 q^{77} +59.2407 q^{78} -279.262 q^{79} +81.0000 q^{81} +604.703 q^{82} -776.981 q^{83} +306.501 q^{84} -999.123 q^{86} -126.771 q^{87} -229.422 q^{88} -29.7626 q^{89} -29.0796 q^{91} -2288.24 q^{92} +42.0744 q^{93} -2227.88 q^{94} -292.366 q^{96} +231.793 q^{97} -232.919 q^{98} +65.8619 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} + 16 q^{4} - 28 q^{7} - 9 q^{8} + 36 q^{9}+O(q^{10}) \) 4 * q - 12 * q^3 + 16 * q^4 - 28 * q^7 - 9 * q^8 + 36 * q^9	\( 4 q - 12 q^{3} + 16 q^{4} - 28 q^{7} - 9 q^{8} + 36 q^{9} + 21 q^{11} - 48 q^{12} - 5 q^{13} + 72 q^{16} - 99 q^{17} + 72 q^{19} + 84 q^{21} - 221 q^{22} - 102 q^{23} + 27 q^{24} + 129 q^{26} - 108 q^{27} - 112 q^{28} - 240 q^{29} + 351 q^{31} - 72 q^{32} - 63 q^{33} - 285 q^{34} + 144 q^{36} - 399 q^{37} - 324 q^{38} + 15 q^{39} + 381 q^{41} - 460 q^{43} - 975 q^{44} + 550 q^{46} - 60 q^{47} - 216 q^{48} + 196 q^{49} + 297 q^{51} - 223 q^{52} - 873 q^{53} + 63 q^{56} - 216 q^{57} - 1408 q^{58} - 855 q^{59} + 687 q^{61} + 477 q^{62} - 252 q^{63} - 1285 q^{64} + 663 q^{66} + 503 q^{67} - 1725 q^{68} + 306 q^{69} - 681 q^{71} - 81 q^{72} - 1228 q^{73} - 3369 q^{74} + 1910 q^{76} - 147 q^{77} - 387 q^{78} + 345 q^{79} + 324 q^{81} + 495 q^{82} - 1509 q^{83} + 336 q^{84} - 540 q^{86} + 720 q^{87} - 2266 q^{88} - 198 q^{89} + 35 q^{91} - 2994 q^{92} - 1053 q^{93} - 1732 q^{94} + 216 q^{96} + 372 q^{97} + 189 q^{99}+O(q^{100}) \) 4 * q - 12 * q^3 + 16 * q^4 - 28 * q^7 - 9 * q^8 + 36 * q^9 + 21 * q^11 - 48 * q^12 - 5 * q^13 + 72 * q^16 - 99 * q^17 + 72 * q^19 + 84 * q^21 - 221 * q^22 - 102 * q^23 + 27 * q^24 + 129 * q^26 - 108 * q^27 - 112 * q^28 - 240 * q^29 + 351 * q^31 - 72 * q^32 - 63 * q^33 - 285 * q^34 + 144 * q^36 - 399 * q^37 - 324 * q^38 + 15 * q^39 + 381 * q^41 - 460 * q^43 - 975 * q^44 + 550 * q^46 - 60 * q^47 - 216 * q^48 + 196 * q^49 + 297 * q^51 - 223 * q^52 - 873 * q^53 + 63 * q^56 - 216 * q^57 - 1408 * q^58 - 855 * q^59 + 687 * q^61 + 477 * q^62 - 252 * q^63 - 1285 * q^64 + 663 * q^66 + 503 * q^67 - 1725 * q^68 + 306 * q^69 - 681 * q^71 - 81 * q^72 - 1228 * q^73 - 3369 * q^74 + 1910 * q^76 - 147 * q^77 - 387 * q^78 + 345 * q^79 + 324 * q^81 + 495 * q^82 - 1509 * q^83 + 336 * q^84 - 540 * q^86 + 720 * q^87 - 2266 * q^88 - 198 * q^89 + 35 * q^91 - 2994 * q^92 - 1053 * q^93 - 1732 * q^94 + 216 * q^96 + 372 * q^97 + 189 * q^99

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