LMFDB - Embedded newform 1075.6.a.b.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1075,6,Mod(1,1075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1075, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("1075.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1075 = 5^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1075.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:	$172.412606299$
Analytic rank:	$1$
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} - 2 x^{9} - 256 x^{8} + 266 x^{7} + 21986 x^{6} - 10450 x^{5} - 719484 x^{4} + 384582 x^{3} + \cdots - 22734604 $ x^10 - 2x^9 - 256x^8 + 266x^7 + 21986x^6 - 10450x^5 - 719484x^4 + 384582x^3 + 8437093x^2 - 5752252*x - 22734604
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-4.38824$ of defining polynomial
Character	$\chi$	$=$	1075.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.38824 q^{2} -25.0462 q^{3} -2.96684 q^{4} +134.955 q^{6} -166.517 q^{7} +188.410 q^{8} +384.312 q^{9} +O(q^{10})$	\(q-5.38824 q^{2} -25.0462 q^{3} -2.96684 q^{4} +134.955 q^{6} -166.517 q^{7} +188.410 q^{8} +384.312 q^{9} -65.6764 q^{11} +74.3080 q^{12} +689.371 q^{13} +897.236 q^{14} -920.259 q^{16} -737.385 q^{17} -2070.77 q^{18} -609.887 q^{19} +4170.63 q^{21} +353.880 q^{22} -1312.45 q^{23} -4718.95 q^{24} -3714.50 q^{26} -3539.34 q^{27} +494.030 q^{28} -8969.74 q^{29} +5858.93 q^{31} -1070.53 q^{32} +1644.94 q^{33} +3973.21 q^{34} -1140.19 q^{36} +55.9069 q^{37} +3286.22 q^{38} -17266.1 q^{39} -10450.3 q^{41} -22472.4 q^{42} -1849.00 q^{43} +194.851 q^{44} +7071.82 q^{46} -1942.82 q^{47} +23049.0 q^{48} +10921.1 q^{49} +18468.7 q^{51} -2045.25 q^{52} -30129.2 q^{53} +19070.8 q^{54} -31373.5 q^{56} +15275.3 q^{57} +48331.2 q^{58} +52167.4 q^{59} -14040.1 q^{61} -31569.4 q^{62} -63994.7 q^{63} +35216.6 q^{64} -8863.36 q^{66} -51437.7 q^{67} +2187.70 q^{68} +32872.0 q^{69} +11268.6 q^{71} +72408.2 q^{72} +61124.5 q^{73} -301.240 q^{74} +1809.44 q^{76} +10936.3 q^{77} +93034.1 q^{78} +96018.3 q^{79} -4740.91 q^{81} +56308.8 q^{82} -30378.0 q^{83} -12373.6 q^{84} +9962.86 q^{86} +224658. q^{87} -12374.1 q^{88} -65040.4 q^{89} -114792. q^{91} +3893.84 q^{92} -146744. q^{93} +10468.4 q^{94} +26812.8 q^{96} -12067.6 q^{97} -58845.3 q^{98} -25240.2 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 8 q^{2} - 28 q^{3} + 202 q^{4} + 75 q^{6} - 60 q^{7} - 294 q^{8} + 1356 q^{9}+O(q^{10}) $ 10 * q - 8 * q^2 - 28 * q^3 + 202 * q^4 + 75 * q^6 - 60 * q^7 - 294 * q^8 + 1356 * q^9	\( 10 q - 8 q^{2} - 28 q^{3} + 202 q^{4} + 75 q^{6} - 60 q^{7} - 294 q^{8} + 1356 q^{9} + 745 q^{11} - 4627 q^{12} - 1917 q^{13} + 1936 q^{14} + 5354 q^{16} - 4017 q^{17} + 2725 q^{18} - 2404 q^{19} - 228 q^{21} + 5836 q^{22} - 1733 q^{23} - 10711 q^{24} - 1484 q^{26} + 2324 q^{27} + 15028 q^{28} + 6996 q^{29} - 4899 q^{31} + 7554 q^{32} + 15734 q^{33} - 27033 q^{34} + 4433 q^{36} - 1466 q^{37} - 13905 q^{38} - 26542 q^{39} + 10297 q^{41} + 37642 q^{42} - 18490 q^{43} - 36140 q^{44} + 17991 q^{46} - 48592 q^{47} - 83607 q^{48} + 29458 q^{49} + 92972 q^{51} - 14232 q^{52} - 127165 q^{53} - 92002 q^{54} - 7780 q^{56} - 34060 q^{57} + 10305 q^{58} + 99372 q^{59} + 17408 q^{61} - 28265 q^{62} - 2244 q^{63} + 47202 q^{64} - 150292 q^{66} + 2021 q^{67} - 192151 q^{68} + 1654 q^{69} + 11286 q^{71} + 298365 q^{72} - 49892 q^{73} - 125431 q^{74} - 249803 q^{76} - 98144 q^{77} + 28494 q^{78} - 91524 q^{79} - 26450 q^{81} + 158909 q^{82} + 105203 q^{83} - 357682 q^{84} + 14792 q^{86} - 181200 q^{87} + 461824 q^{88} - 62682 q^{89} - 295304 q^{91} - 183783 q^{92} + 238430 q^{93} + 7259 q^{94} - 162399 q^{96} - 108383 q^{97} - 354656 q^{98} - 270499 q^{99}+O(q^{100}) \) 10 * q - 8 * q^2 - 28 * q^3 + 202 * q^4 + 75 * q^6 - 60 * q^7 - 294 * q^8 + 1356 * q^9 + 745 * q^11 - 4627 * q^12 - 1917 * q^13 + 1936 * q^14 + 5354 * q^16 - 4017 * q^17 + 2725 * q^18 - 2404 * q^19 - 228 * q^21 + 5836 * q^22 - 1733 * q^23 - 10711 * q^24 - 1484 * q^26 + 2324 * q^27 + 15028 * q^28 + 6996 * q^29 - 4899 * q^31 + 7554 * q^32 + 15734 * q^33 - 27033 * q^34 + 4433 * q^36 - 1466 * q^37 - 13905 * q^38 - 26542 * q^39 + 10297 * q^41 + 37642 * q^42 - 18490 * q^43 - 36140 * q^44 + 17991 * q^46 - 48592 * q^47 - 83607 * q^48 + 29458 * q^49 + 92972 * q^51 - 14232 * q^52 - 127165 * q^53 - 92002 * q^54 - 7780 * q^56 - 34060 * q^57 + 10305 * q^58 + 99372 * q^59 + 17408 * q^61 - 28265 * q^62 - 2244 * q^63 + 47202 * q^64 - 150292 * q^66 + 2021 * q^67 - 192151 * q^68 + 1654 * q^69 + 11286 * q^71 + 298365 * q^72 - 49892 * q^73 - 125431 * q^74 - 249803 * q^76 - 98144 * q^77 + 28494 * q^78 - 91524 * q^79 - 26450 * q^81 + 158909 * q^82 + 105203 * q^83 - 357682 * q^84 + 14792 * q^86 - 181200 * q^87 + 461824 * q^88 - 62682 * q^89 - 295304 * q^91 - 183783 * q^92 + 238430 * q^93 + 7259 * q^94 - 162399 * q^96 - 108383 * q^97 - 354656 * q^98 - 270499 * 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 $

$2$	−5.38824	−0.952516	−0.476258	−	0.879306i	$-0.658007\pi$
$2$	−5.38824	−0.952516	−0.476258	+	0.879306i	$0.658007\pi$
$3$	−25.0462	−1.60671	−0.803357	−	0.595497i	$-0.796955\pi$
$3$	−25.0462	−1.60671	−0.803357	+	0.595497i	$0.796955\pi$
$4$	−2.96684	−0.0927137
$5$	0	0
$5$	0	0
$6$	134.955	1.53042
$7$	−166.517	−1.28444	−0.642221	−	0.766519i	$-0.721987\pi$
$7$	−166.517	−1.28444	−0.642221	+	0.766519i	$0.721987\pi$
$8$	188.410	1.04083
$9$	384.312	1.58153
$10$	0	0
$11$	−65.6764	−0.163654	−0.0818272	−	0.996647i	$-0.526076\pi$
$11$	−65.6764	−0.163654	−0.0818272	+	0.996647i	$0.526076\pi$
$12$	74.3080	0.148964
$13$	689.371	1.13134	0.565672	−	0.824630i	$-0.308617\pi$
$13$	689.371	1.13134	0.565672	+	0.824630i	$0.308617\pi$
$14$	897.236	1.22345
$15$	0	0
$16$	−920.259	−0.898690
$17$	−737.385	−0.618831	−0.309415	−	0.950927i	$-0.600133\pi$
$17$	−737.385	−0.618831	−0.309415	+	0.950927i	$0.600133\pi$
$18$	−2070.77	−1.50643
$19$	−609.887	−0.387583	−0.193792	−	0.981043i	$-0.562079\pi$
$19$	−609.887	−0.387583	−0.193792	+	0.981043i	$0.562079\pi$
$20$	0	0
$21$	4170.63	2.06373
$22$	353.880	0.155883
$23$	−1312.45	−0.517326	−0.258663	−	0.965968i	$-0.583282\pi$
$23$	−1312.45	−0.517326	−0.258663	+	0.965968i	$0.583282\pi$
$24$	−4718.95	−1.67231
$25$	0	0
$26$	−3714.50	−1.07762
$27$	−3539.34	−0.934357
$28$	494.030	0.119085
$29$	−8969.74	−1.98055	−0.990273	−	0.139137i	$-0.955567\pi$
$29$	−8969.74	−1.98055	−0.990273	+	0.139137i	$0.955567\pi$
$30$	0	0
$31$	5858.93	1.09500	0.547500	−	0.836806i	$-0.315580\pi$
$31$	5858.93	1.09500	0.547500	+	0.836806i	$0.315580\pi$
$32$	−1070.53	−0.184810
$33$	1644.94	0.262946
$34$	3973.21	0.589446
$35$	0	0
$36$	−1140.19	−0.146630
$37$	55.9069	0.00671369	0.00335685	−	0.999994i	$-0.498931\pi$
$37$	55.9069	0.00671369	0.00335685	+	0.999994i	$0.498931\pi$
$38$	3286.22	0.369179
$39$	−17266.1	−1.81775
$40$	0	0
$41$	−10450.3	−0.970889	−0.485445	−	0.874267i	$-0.661342\pi$
$41$	−10450.3	−0.970889	−0.485445	+	0.874267i	$0.661342\pi$
$42$	−22472.4	−1.96574
$43$	−1849.00	−0.152499
$43$	−1849.00	−0.152499
$44$	194.851	0.0151730
$45$	0	0
$46$	7071.82	0.492762
$47$	−1942.82	−0.128288	−0.0641442	−	0.997941i	$-0.520432\pi$
$47$	−1942.82	−0.128288	−0.0641442	+	0.997941i	$0.520432\pi$
$48$	23049.0	1.44394
$49$	10921.1	0.649792
$50$	0	0
$51$	18468.7	0.994284
$52$	−2045.25	−0.104891
$53$	−30129.2	−1.47332	−0.736661	−	0.676262i	$-0.763599\pi$
$53$	−30129.2	−1.47332	−0.736661	+	0.676262i	$0.763599\pi$
$54$	19070.8	0.889989
$55$	0	0
$56$	−31373.5	−1.33688
$57$	15275.3	0.622736
$58$	48331.2	1.88650
$59$	52167.4	1.95106	0.975528	−	0.219877i	$-0.0705657\pi$
$59$	52167.4	1.95106	0.975528	+	0.219877i	$0.0705657\pi$
$60$	0	0
$61$	−14040.1	−0.483108	−0.241554	−	0.970387i	$-0.577657\pi$
$61$	−14040.1	−0.483108	−0.241554	+	0.970387i	$0.577657\pi$
$62$	−31569.4	−1.04301
$63$	−63994.7	−2.03139
$64$	35216.6	1.07473
$65$	0	0
$66$	−8863.36	−0.250460
$67$	−51437.7	−1.39989	−0.699946	−	0.714196i	$-0.746792\pi$
$67$	−51437.7	−1.39989	−0.699946	+	0.714196i	$0.746792\pi$
$68$	2187.70	0.0573741
$69$	32872.0	0.831196
$70$	0	0
$71$	11268.6	0.265292	0.132646	−	0.991163i	$-0.457653\pi$
$71$	11268.6	0.265292	0.132646	+	0.991163i	$0.457653\pi$
$72$	72408.2	1.64610
$73$	61124.5	1.34248	0.671241	−	0.741239i	$-0.265762\pi$
$73$	61124.5	1.34248	0.671241	+	0.741239i	$0.265762\pi$
$74$	−301.240	−0.00639490
$75$	0	0
$76$	1809.44	0.0359343
$77$	10936.3	0.210205
$78$	93034.1	1.73143
$79$	96018.3	1.73096	0.865478	−	0.500947i	$-0.167015\pi$
$79$	96018.3	1.73096	0.865478	+	0.500947i	$0.167015\pi$
$80$	0	0
$81$	−4740.91	−0.0802878
$82$	56308.8	0.924787
$83$	−30378.0	−0.484021	−0.242010	−	0.970274i	$-0.577807\pi$
$83$	−30378.0	−0.484021	−0.242010	+	0.970274i	$0.577807\pi$
$84$	−12373.6	−0.191336
$85$	0	0
$86$	9962.86	0.145257
$87$	224658.	3.18217
$88$	−12374.1	−0.170336
$89$	−65040.4	−0.870378	−0.435189	−	0.900339i	$-0.643319\pi$
$89$	−65040.4	−0.870378	−0.435189	+	0.900339i	$0.643319\pi$
$90$	0	0
$91$	−114792.	−1.45315
$92$	3893.84	0.0479632
$93$	−146744.	−1.75935
$94$	10468.4	0.122197
$95$	0	0
$96$	26812.8	0.296937
$97$	−12067.6	−0.130224	−0.0651119	−	0.997878i	$-0.520740\pi$
$97$	−12067.6	−0.130224	−0.0651119	+	0.997878i	$0.520740\pi$
$98$	−58845.3	−0.618938
$99$	−25240.2	−0.258825
$100$	0	0
$101$	107046.	1.04416	0.522079	−	0.852897i	$-0.325157\pi$
$101$	107046.	1.04416	0.522079	+	0.852897i	$0.325157\pi$
$102$	−99513.8	−0.947072
$103$	86209.7	0.800688	0.400344	−	0.916365i	$-0.368891\pi$
$103$	86209.7	0.800688	0.400344	+	0.916365i	$0.368891\pi$
$104$	129884.	1.17753
$105$	0	0
$106$	162343.	1.40336
$107$	−197153.	−1.66473	−0.832365	−	0.554227i	$-0.813014\pi$
$107$	−197153.	−1.66473	−0.832365	+	0.554227i	$0.813014\pi$
$108$	10500.6	0.0866276
$109$	165311.	1.33271	0.666355	−	0.745634i	$-0.267853\pi$
$109$	165311.	1.33271	0.666355	+	0.745634i	$0.267853\pi$
$110$	0	0
$111$	−1400.26	−0.0107870
$112$	153239.	1.15432
$113$	179790.	1.32455	0.662275	−	0.749260i	$-0.269591\pi$
$113$	179790.	1.32455	0.662275	+	0.749260i	$0.269591\pi$
$114$	−82307.3	−0.593166
$115$	0	0
$116$	26611.8	0.183624
$117$	264934.	1.78926
$118$	−281091.	−1.85841
$119$	122787.	0.794852
$120$	0	0
$121$	−156738.	−0.973217
$122$	75651.3	0.460168
$123$	261741.	1.55994
$124$	−17382.5	−0.101522
$125$	0	0
$126$	344819.	1.93493
$127$	286585.	1.57668	0.788341	−	0.615238i	$-0.210940\pi$
$127$	286585.	1.57668	0.788341	+	0.615238i	$0.210940\pi$
$128$	−155498.	−0.838882
$129$	46310.4	0.245022
$130$	0	0
$131$	67619.1	0.344263	0.172132	−	0.985074i	$-0.444935\pi$
$131$	67619.1	0.344263	0.172132	+	0.985074i	$0.444935\pi$
$132$	−4880.28	−0.0243787
$133$	101557.	0.497829
$134$	277159.	1.33342
$135$	0	0
$136$	−138931.	−0.644096
$137$	−133580.	−0.608050	−0.304025	−	0.952664i	$-0.598331\pi$
$137$	−133580.	−0.608050	−0.304025	+	0.952664i	$0.598331\pi$
$138$	−177122.	−0.791727
$139$	−262195.	−1.15103	−0.575516	−	0.817790i	$-0.695199\pi$
$139$	−262195.	−1.15103	−0.575516	+	0.817790i	$0.695199\pi$
$140$	0	0
$141$	48660.2	0.206123
$142$	−60718.0	−0.252695
$143$	−45275.4	−0.185149
$144$	−353667.	−1.42131
$145$	0	0
$146$	−329354.	−1.27873
$147$	−273531.	−1.04403
$148$	−165.867	−0.000622451 0
$149$	187414.	0.691569	0.345785	−	0.938314i	$-0.387613\pi$
$149$	187414.	0.691569	0.345785	+	0.938314i	$0.387613\pi$
$150$	0	0
$151$	−131632.	−0.469807	−0.234904	−	0.972019i	$-0.575477\pi$
$151$	−131632.	−0.469807	−0.234904	+	0.972019i	$0.575477\pi$
$152$	−114909.	−0.403407
$153$	−283386.	−0.978701
$154$	−58927.3	−0.200223
$155$	0	0
$156$	51225.8	0.168530
$157$	252270.	0.816800	0.408400	−	0.912803i	$-0.366087\pi$
$157$	252270.	0.816800	0.408400	+	0.912803i	$0.366087\pi$
$158$	−517370.	−1.64876
$159$	754622.	2.36721
$160$	0	0
$161$	218547.	0.664476
$162$	25545.2	0.0764754
$163$	518121.	1.52743	0.763717	−	0.645552i	$-0.223373\pi$
$163$	518121.	1.52743	0.763717	+	0.645552i	$0.223373\pi$
$164$	31004.4	0.0900147
$165$	0	0
$166$	163684.	0.461038
$167$	681328.	1.89045	0.945224	−	0.326422i	$-0.105843\pi$
$167$	681328.	1.89045	0.945224	+	0.326422i	$0.105843\pi$
$168$	785788.	2.14799
$169$	103939.	0.279939
$170$	0	0
$171$	−234387.	−0.612976
$172$	5485.68	0.0141387
$173$	148596.	0.377478	0.188739	−	0.982027i	$-0.439560\pi$
$173$	148596.	0.377478	0.188739	+	0.982027i	$0.439560\pi$
$174$	−1.21051e6	−3.03107
$175$	0	0
$176$	60439.3	0.147075
$177$	−1.30660e6	−3.13479
$178$	350453.	0.829049
$179$	103068.	0.240432	0.120216	−	0.992748i	$-0.461641\pi$
$179$	103068.	0.240432	0.120216	+	0.992748i	$0.461641\pi$
$180$	0	0
$181$	309316.	0.701789	0.350894	−	0.936415i	$-0.385878\pi$
$181$	309316.	0.701789	0.350894	+	0.936415i	$0.385878\pi$
$182$	618529.	1.38414
$183$	351650.	0.776217
$184$	−247279.	−0.538447
$185$	0	0
$186$	790693.	1.67581
$187$	48428.8	0.101274
$188$	5764.02	0.0118941
$189$	589362.	1.20013
$190$	0	0
$191$	373258.	0.740330	0.370165	−	0.928966i	$-0.379301\pi$
$191$	373258.	0.740330	0.370165	+	0.928966i	$0.379301\pi$
$192$	−882042.	−1.72678
$193$	−85782.8	−0.165770	−0.0828852	−	0.996559i	$-0.526413\pi$
$193$	−85782.8	−0.165770	−0.0828852	+	0.996559i	$0.526413\pi$
$194$	65023.0	0.124040
$195$	0	0
$196$	−32401.0	−0.0602446
$197$	178146.	0.327047	0.163523	−	0.986539i	$-0.447714\pi$
$197$	178146.	0.327047	0.163523	+	0.986539i	$0.447714\pi$
$198$	136001.	0.246535
$199$	154725.	0.276967	0.138484	−	0.990365i	$-0.455777\pi$
$199$	154725.	0.276967	0.138484	+	0.990365i	$0.455777\pi$
$200$	0	0
$201$	1.28832e6	2.24923
$202$	−576789.	−0.994577
$203$	1.49362e6	2.54390
$204$	−54793.6	−0.0921838
$205$	0	0
$206$	−464519.	−0.762668
$207$	−504392.	−0.818168
$208$	−634400.	−1.01673
$209$	40055.2	0.0634297
$210$	0	0
$211$	−682180.	−1.05485	−0.527427	−	0.849600i	$-0.676843\pi$
$211$	−682180.	−1.05485	−0.527427	+	0.849600i	$0.676843\pi$
$212$	89388.4	0.136597
$213$	−282236.	−0.426249
$214$	1.06231e6	1.58568
$215$	0	0
$216$	−666846.	−0.972504
$217$	−975615.	−1.40647
$218$	−890736.	−1.26943
$219$	−1.53094e6	−2.15698
$220$	0	0
$221$	−508332.	−0.700110
$222$	7544.92	0.0102748
$223$	464004.	0.624827	0.312413	−	0.949946i	$-0.398863\pi$
$223$	464004.	0.624827	0.312413	+	0.949946i	$0.398863\pi$
$224$	178263.	0.237378
$225$	0	0
$226$	−968750.	−1.26166
$227$	−1.13876e6	−1.46679	−0.733396	−	0.679801i	$-0.762066\pi$
$227$	−1.13876e6	−1.46679	−0.733396	+	0.679801i	$0.762066\pi$
$228$	−45319.5	−0.0577361
$229$	648718.	0.817462	0.408731	−	0.912655i	$-0.365971\pi$
$229$	648718.	0.817462	0.408731	+	0.912655i	$0.365971\pi$
$230$	0	0
$231$	−273912.	−0.337739
$232$	−1.68999e6	−2.06141
$233$	588684.	0.710383	0.355191	−	0.934794i	$-0.384416\pi$
$233$	588684.	0.710383	0.355191	+	0.934794i	$0.384416\pi$
$234$	−1.42753e6	−1.70430
$235$	0	0
$236$	−154772.	−0.180889
$237$	−2.40489e6	−2.78115
$238$	−661608.	−0.757109
$239$	739129.	0.837000	0.418500	−	0.908217i	$-0.362556\pi$
$239$	739129.	0.837000	0.418500	+	0.908217i	$0.362556\pi$
$240$	0	0
$241$	−1.57360e6	−1.74523	−0.872615	−	0.488409i	$-0.837577\pi$
$241$	−1.57360e6	−1.74523	−0.872615	+	0.488409i	$0.837577\pi$
$242$	844540.	0.927005
$243$	978801.	1.06336
$244$	41654.6	0.0447908
$245$	0	0
$246$	−1.41032e6	−1.48587
$247$	−420438.	−0.438490
$248$	1.10388e6	1.13971
$249$	760854.	0.777684
$250$	0	0
$251$	456789.	0.457648	0.228824	−	0.973468i	$-0.426512\pi$
$251$	456789.	0.457648	0.228824	+	0.973468i	$0.426512\pi$
$252$	189862.	0.188337
$253$	86197.3	0.0846627
$254$	−1.54419e6	−1.50181
$255$	0	0
$256$	−289068.	−0.275676
$257$	−1.12132e6	−1.05900	−0.529499	−	0.848310i	$-0.677620\pi$
$257$	−1.12132e6	−1.05900	−0.529499	+	0.848310i	$0.677620\pi$
$258$	−249532.	−0.233387
$259$	−9309.48	−0.00862335
$260$	0	0
$261$	−3.44718e6	−3.13230
$262$	−364348.	−0.327916
$263$	−1.75620e6	−1.56562	−0.782808	−	0.622263i	$-0.786214\pi$
$263$	−1.75620e6	−1.56562	−0.782808	+	0.622263i	$0.786214\pi$
$264$	309924.	0.273681
$265$	0	0
$266$	−547213.	−0.474190
$267$	1.62901e6	1.39845
$268$	152607.	0.129789
$269$	1.22933e6	1.03583	0.517915	−	0.855432i	$-0.326708\pi$
$269$	1.22933e6	1.03583	0.517915	+	0.855432i	$0.326708\pi$
$270$	0	0
$271$	1.32305e6	1.09435	0.547173	−	0.837020i	$-0.315704\pi$
$271$	1.32305e6	1.09435	0.547173	+	0.837020i	$0.315704\pi$
$272$	678585.	0.556137
$273$	2.87511e6	2.33479
$274$	719760.	0.579177
$275$	0	0
$276$	−97525.9	−0.0770632
$277$	−1.96854e6	−1.54151	−0.770754	−	0.637133i	$-0.780120\pi$
$277$	−1.96854e6	−1.54151	−0.770754	+	0.637133i	$0.780120\pi$
$278$	1.41277e6	1.09638
$279$	2.25166e6	1.73178
$280$	0	0
$281$	−1.11820e6	−0.844801	−0.422400	−	0.906409i	$-0.638812\pi$
$281$	−1.11820e6	−0.844801	−0.422400	+	0.906409i	$0.638812\pi$
$282$	−262193.	−0.196335
$283$	−1.25727e6	−0.933174	−0.466587	−	0.884475i	$-0.654517\pi$
$283$	−1.25727e6	−0.933174	−0.466587	+	0.884475i	$0.654517\pi$
$284$	−33432.2	−0.0245962
$285$	0	0
$286$	243955.	0.176358
$287$	1.74016e6	1.24705
$288$	−411420.	−0.292283
$289$	−876121.	−0.617049
$290$	0	0
$291$	302247.	0.209233
$292$	−181347.	−0.124466
$293$	260406.	0.177207	0.0886037	−	0.996067i	$-0.471760\pi$
$293$	260406.	0.177207	0.0886037	+	0.996067i	$0.471760\pi$
$294$	1.47385e6	0.994456
$295$	0	0
$296$	10533.4	0.00698779
$297$	232451.	0.152912
$298$	−1.00983e6	−0.658731
$299$	−904768.	−0.585274
$300$	0	0
$301$	307891.	0.195876
$302$	709266.	0.447499
$303$	−2.68109e6	−1.67766
$304$	561254.	0.348318
$305$	0	0
$306$	1.52695e6	0.932228
$307$	1.56144e6	0.945536	0.472768	−	0.881187i	$-0.343255\pi$
$307$	1.56144e6	0.945536	0.472768	+	0.881187i	$0.343255\pi$
$308$	−32446.1	−0.0194888
$309$	−2.15923e6	−1.28648
$310$	0	0
$311$	655137.	0.384088	0.192044	−	0.981386i	$-0.438488\pi$
$311$	655137.	0.384088	0.192044	+	0.981386i	$0.438488\pi$
$312$	−3.25311e6	−1.89196
$313$	212890.	0.122827	0.0614136	−	0.998112i	$-0.480439\pi$
$313$	212890.	0.122827	0.0614136	+	0.998112i	$0.480439\pi$
$314$	−1.35929e6	−0.778015
$315$	0	0
$316$	−284871.	−0.160483
$317$	385858.	0.215665	0.107832	−	0.994169i	$-0.465609\pi$
$317$	385858.	0.215665	0.107832	+	0.994169i	$0.465609\pi$
$318$	−4.06608e6	−2.25480
$319$	589100.	0.324125
$320$	0	0
$321$	4.93794e6	2.67475
$322$	−1.17758e6	−0.632924
$323$	449721.	0.239849
$324$	14065.5	0.00744378
$325$	0	0
$326$	−2.79176e6	−1.45490
$327$	−4.14042e6	−2.14129
$328$	−1.96894e6	−1.01053
$329$	323513.	0.164779
$330$	0	0
$331$	−1.62905e6	−0.817269	−0.408634	−	0.912698i	$-0.633995\pi$
$331$	−1.62905e6	−0.817269	−0.408634	+	0.912698i	$0.633995\pi$
$332$	90126.6	0.0448754
$333$	21485.7	0.0106179
$334$	−3.67116e6	−1.80068
$335$	0	0
$336$	−3.83806e6	−1.85466
$337$	2.32344e6	1.11444	0.557220	−	0.830365i	$-0.311868\pi$
$337$	2.32344e6	1.11444	0.557220	+	0.830365i	$0.311868\pi$
$338$	−560050.	−0.266646
$339$	−4.50305e6	−2.12818
$340$	0	0
$341$	−384794.	−0.179202
$342$	1.26293e6	0.583869
$343$	980112.	0.449822
$344$	−348370.	−0.158725
$345$	0	0
$346$	−800670.	−0.359553
$347$	2.45789e6	1.09582	0.547910	−	0.836537i	$-0.315424\pi$
$347$	2.45789e6	1.09582	0.547910	+	0.836537i	$0.315424\pi$
$348$	−666524.	−0.295031
$349$	−4.36489e6	−1.91827	−0.959135	−	0.282950i	$-0.908687\pi$
$349$	−4.36489e6	−1.91827	−0.959135	+	0.282950i	$0.908687\pi$
$350$	0	0
$351$	−2.43992e6	−1.05708
$352$	70308.9	0.0302450
$353$	−898354.	−0.383717	−0.191858	−	0.981423i	$-0.561451\pi$
$353$	−898354.	−0.383717	−0.191858	+	0.981423i	$0.561451\pi$
$354$	7.04026e6	2.98594
$355$	0	0
$356$	192964.	0.0806960
$357$	−3.07536e6	−1.27710
$358$	−555357.	−0.229015
$359$	1.81866e6	0.744759	0.372379	−	0.928081i	$-0.378542\pi$
$359$	1.81866e6	0.744759	0.372379	+	0.928081i	$0.378542\pi$
$360$	0	0
$361$	−2.10414e6	−0.849779
$362$	−1.66667e6	−0.668465
$363$	3.92568e6	1.56368
$364$	340570.	0.134727
$365$	0	0
$366$	−1.89478e6	−0.739359
$367$	2.34762e6	0.909836	0.454918	−	0.890533i	$-0.349669\pi$
$367$	2.34762e6	0.909836	0.454918	+	0.890533i	$0.349669\pi$
$368$	1.20780e6	0.464916
$369$	−4.01619e6	−1.53549
$370$	0	0
$371$	5.01703e6	1.89240
$372$	435366.	0.163116
$373$	−2.97881e6	−1.10859	−0.554295	−	0.832321i	$-0.687012\pi$
$373$	−2.97881e6	−1.10859	−0.554295	+	0.832321i	$0.687012\pi$
$374$	−260946.	−0.0964654
$375$	0	0
$376$	−366046.	−0.133526
$377$	−6.18348e6	−2.24068
$378$	−3.17562e6	−1.14314
$379$	1.77718e6	0.635527	0.317764	−	0.948170i	$-0.397068\pi$
$379$	1.77718e6	0.635527	0.317764	+	0.948170i	$0.397068\pi$
$380$	0	0
$381$	−7.17787e6	−2.53328
$382$	−2.01120e6	−0.705176
$383$	2.08193e6	0.725220	0.362610	−	0.931941i	$-0.381886\pi$
$383$	2.08193e6	0.725220	0.362610	+	0.931941i	$0.381886\pi$
$384$	3.89465e6	1.34784
$385$	0	0
$386$	462219.	0.157899
$387$	−710594.	−0.241181
$388$	35802.5	0.0120735
$389$	−1.81080e6	−0.606730	−0.303365	−	0.952874i	$-0.598110\pi$
$389$	−1.81080e6	−0.606730	−0.303365	+	0.952874i	$0.598110\pi$
$390$	0	0
$391$	967784.	0.320137
$392$	2.05764e6	0.676321
$393$	−1.69360e6	−0.553133
$394$	−959892.	−0.311517
$395$	0	0
$396$	74883.7	0.0239966
$397$	−185900.	−0.0591975	−0.0295988	−	0.999562i	$-0.509423\pi$
$397$	−185900.	−0.0591975	−0.0295988	+	0.999562i	$0.509423\pi$
$398$	−833697.	−0.263816
$399$	−2.54361e6	−0.799869
$400$	0	0
$401$	−3.41892e6	−1.06176	−0.530882	−	0.847446i	$-0.678139\pi$
$401$	−3.41892e6	−1.06176	−0.530882	+	0.847446i	$0.678139\pi$
$402$	−6.94177e6	−2.14242
$403$	4.03898e6	1.23882
$404$	−317587.	−0.0968077
$405$	0	0
$406$	−8.04798e6	−2.42310
$407$	−3671.77	−0.00109873
$408$	3.47968e6	1.03488
$409$	−428179.	−0.126566	−0.0632830	−	0.997996i	$-0.520157\pi$
$409$	−428179.	−0.126566	−0.0632830	+	0.997996i	$0.520157\pi$
$410$	0	0
$411$	3.34567e6	0.976963
$412$	−255770.	−0.0742347
$413$	−8.68679e6	−2.50602
$414$	2.71779e6	0.779318
$415$	0	0
$416$	−737996.	−0.209084
$417$	6.56699e6	1.84938
$418$	−215827.	−0.0604178
$419$	3.84011e6	1.06858	0.534292	−	0.845300i	$-0.320578\pi$
$419$	3.84011e6	1.06858	0.534292	+	0.845300i	$0.320578\pi$
$420$	0	0
$421$	7.05884e6	1.94101	0.970506	−	0.241078i	$-0.0775010\pi$
$421$	7.05884e6	1.94101	0.970506	+	0.241078i	$0.0775010\pi$
$422$	3.67575e6	1.00477
$423$	−746649.	−0.202892
$424$	−5.67663e6	−1.53347
$425$	0	0
$426$	1.52076e6	0.406009
$427$	2.33792e6	0.620525
$428$	584921.	0.154343
$429$	1.13398e6	0.297482
$430$	0	0
$431$	−2.11813e6	−0.549237	−0.274619	−	0.961553i	$-0.588552\pi$
$431$	−2.11813e6	−0.549237	−0.274619	+	0.961553i	$0.588552\pi$
$432$	3.25711e6	0.839697
$433$	1.58858e6	0.407182	0.203591	−	0.979056i	$-0.434739\pi$
$433$	1.58858e6	0.407182	0.203591	+	0.979056i	$0.434739\pi$
$434$	5.25685e6	1.33968
$435$	0	0
$436$	−490451.	−0.123560
$437$	800448.	0.200507
$438$	8.24906e6	2.05456
$439$	7.11302e6	1.76154	0.880771	−	0.473543i	$-0.157025\pi$
$439$	7.11302e6	1.76154	0.880771	+	0.473543i	$0.157025\pi$
$440$	0	0
$441$	4.19710e6	1.02767
$442$	2.73901e6	0.666866
$443$	−2.52423e6	−0.611110	−0.305555	−	0.952174i	$-0.598842\pi$
$443$	−2.52423e6	−0.611110	−0.305555	+	0.952174i	$0.598842\pi$
$444$	4154.33	0.00100010
$445$	0	0
$446$	−2.50017e6	−0.595157
$447$	−4.69400e6	−1.11115
$448$	−5.86418e6	−1.38042
$449$	−471868.	−0.110460	−0.0552300	−	0.998474i	$-0.517589\pi$
$449$	−471868.	−0.110460	−0.0552300	+	0.998474i	$0.517589\pi$
$450$	0	0
$451$	686339.	0.158890
$452$	−533407.	−0.122804
$453$	3.29689e6	0.754846
$454$	6.13593e6	1.39714
$455$	0	0
$456$	2.87803e6	0.648160
$457$	6.11039e6	1.36861	0.684303	−	0.729198i	$-0.260107\pi$
$457$	6.11039e6	1.36861	0.684303	+	0.729198i	$0.260107\pi$
$458$	−3.49545e6	−0.778645
$459$	2.60985e6	0.578209
$460$	0	0
$461$	7.47383e6	1.63791	0.818957	−	0.573855i	$-0.194553\pi$
$461$	7.47383e6	1.63791	0.818957	+	0.573855i	$0.194553\pi$
$462$	1.47590e6	0.321702
$463$	4.14981e6	0.899654	0.449827	−	0.893116i	$-0.351486\pi$
$463$	4.14981e6	0.899654	0.449827	+	0.893116i	$0.351486\pi$
$464$	8.25449e6	1.77990
$465$	0	0
$466$	−3.17197e6	−0.676651
$467$	1.18789e6	0.252048	0.126024	−	0.992027i	$-0.459778\pi$
$467$	1.18789e6	0.252048	0.126024	+	0.992027i	$0.459778\pi$
$468$	−786016.	−0.165889
$469$	8.56527e6	1.79808
$470$	0	0
$471$	−6.31840e6	−1.31237
$472$	9.82886e6	2.03071
$473$	121436.	0.0249571
$474$	1.29581e7	2.64909
$475$	0	0
$476$	−364290.	−0.0736937
$477$	−1.15790e7	−2.33011
$478$	−3.98260e6	−0.797255
$479$	3.74102e6	0.744992	0.372496	−	0.928034i	$-0.378502\pi$
$479$	3.74102e6	0.744992	0.372496	+	0.928034i	$0.378502\pi$
$480$	0	0
$481$	38540.6	0.00759549
$482$	8.47896e6	1.66236
$483$	−5.47376e6	−1.06762
$484$	465015.	0.0902306
$485$	0	0
$486$	−5.27402e6	−1.01286
$487$	−7.18109e6	−1.37204	−0.686021	−	0.727581i	$-0.740644\pi$
$487$	−7.18109e6	−1.37204	−0.686021	+	0.727581i	$0.740644\pi$
$488$	−2.64529e6	−0.502832
$489$	−1.29770e7	−2.45415
$490$	0	0
$491$	−7.37443e6	−1.38046	−0.690231	−	0.723589i	$-0.742491\pi$
$491$	−7.37443e6	−1.38046	−0.690231	+	0.723589i	$0.742491\pi$
$492$	−776542.	−0.144628
$493$	6.61415e6	1.22562
$494$	2.26542e6	0.417669
$495$	0	0
$496$	−5.39174e6	−0.984067
$497$	−1.87642e6	−0.340753
$498$	−4.09967e6	−0.740756
$499$	2.61086e6	0.469388	0.234694	−	0.972069i	$-0.424591\pi$
$499$	2.61086e6	0.469388	0.234694	+	0.972069i	$0.424591\pi$
$500$	0	0
$501$	−1.70647e7	−3.03741
$502$	−2.46129e6	−0.435917
$503$	398438.	0.0702167	0.0351084	−	0.999384i	$-0.488822\pi$
$503$	398438.	0.0702167	0.0351084	+	0.999384i	$0.488822\pi$
$504$	−1.20572e7	−2.11432
$505$	0	0
$506$	−464452.	−0.0806426
$507$	−2.60329e6	−0.449782
$508$	−850251.	−0.146180
$509$	556201.	0.0951563	0.0475781	−	0.998868i	$-0.484850\pi$
$509$	556201.	0.0951563	0.0475781	+	0.998868i	$0.484850\pi$
$510$	0	0
$511$	−1.01783e7	−1.72434
$512$	6.53352e6	1.10147
$513$	2.15860e6	0.362141
$514$	6.04193e6	1.00871
$515$	0	0
$516$	−137396.	−0.0227169
$517$	127597.	0.0209950
$518$	50161.8	0.00821388
$519$	−3.72176e6	−0.606499
$520$	0	0
$521$	3.39465e6	0.547898	0.273949	−	0.961744i	$-0.411670\pi$
$521$	3.39465e6	0.547898	0.273949	+	0.961744i	$0.411670\pi$
$522$	1.85743e7	2.98356
$523$	9.40326e6	1.50323	0.751613	−	0.659605i	$-0.229276\pi$
$523$	9.40326e6	1.50323	0.751613	+	0.659605i	$0.229276\pi$
$524$	−200615.	−0.0319179
$525$	0	0
$526$	9.46285e6	1.49127
$527$	−4.32029e6	−0.677620
$528$	−1.51377e6	−0.236307
$529$	−4.71381e6	−0.732373
$530$	0	0
$531$	2.00486e7	3.08566
$532$	−301303.	−0.0461555
$533$	−7.20414e6	−1.09841
$534$	−8.77753e6	−1.33205
$535$	0	0
$536$	−9.69136e6	−1.45704
$537$	−2.58147e6	−0.386306
$538$	−6.62393e6	−0.986644
$539$	−717256.	−0.106341
$540$	0	0
$541$	4.76159e6	0.699453	0.349727	−	0.936852i	$-0.386274\pi$
$541$	4.76159e6	0.699453	0.349727	+	0.936852i	$0.386274\pi$
$542$	−7.12894e6	−1.04238
$543$	−7.74720e6	−1.12757
$544$	789396.	0.114366
$545$	0	0
$546$	−1.54918e7	−2.22393
$547$	−4.68380e6	−0.669314	−0.334657	−	0.942340i	$-0.608621\pi$
$547$	−4.68380e6	−0.669314	−0.334657	+	0.942340i	$0.608621\pi$
$548$	396309.	0.0563745
$549$	−5.39577e6	−0.764051
$550$	0	0
$551$	5.47053e6	0.767627
$552$	6.19341e6	0.865131
$553$	−1.59887e7	−2.22331
$554$	1.06070e7	1.46831
$555$	0	0
$556$	777890.	0.106716
$557$	−4.37082e6	−0.596931	−0.298466	−	0.954420i	$-0.596475\pi$
$557$	−4.37082e6	−0.596931	−0.298466	+	0.954420i	$0.596475\pi$
$558$	−1.21325e7	−1.64955
$559$	−1.27465e6	−0.172528
$560$	0	0
$561$	−1.21296e6	−0.162719
$562$	6.02514e6	0.804686
$563$	9.98439e6	1.32755	0.663774	−	0.747933i	$-0.268954\pi$
$563$	9.98439e6	1.32755	0.663774	+	0.747933i	$0.268954\pi$
$564$	−144367.	−0.0191104
$565$	0	0
$566$	6.77448e6	0.888863
$567$	789445.	0.103125
$568$	2.12312e6	0.276123
$569$	−3.03616e6	−0.393137	−0.196568	−	0.980490i	$-0.562980\pi$
$569$	−3.03616e6	−0.393137	−0.196568	+	0.980490i	$0.562980\pi$
$570$	0	0
$571$	−4.66426e6	−0.598677	−0.299339	−	0.954147i	$-0.596766\pi$
$571$	−4.66426e6	−0.598677	−0.299339	+	0.954147i	$0.596766\pi$
$572$	134325.	0.0171659
$573$	−9.34869e6	−1.18950
$574$	−9.37640e6	−1.18784
$575$	0	0
$576$	1.35342e7	1.69971
$577$	−2.53940e6	−0.317536	−0.158768	−	0.987316i	$-0.550752\pi$
$577$	−2.53940e6	−0.317536	−0.158768	+	0.987316i	$0.550752\pi$
$578$	4.72075e6	0.587749
$579$	2.14853e6	0.266346
$580$	0	0
$581$	5.05847e6	0.621697
$582$	−1.62858e6	−0.199297
$583$	1.97878e6	0.241116
$584$	1.15165e7	1.39729
$585$	0	0
$586$	−1.40313e6	−0.168793
$587$	−1.18921e7	−1.42450	−0.712248	−	0.701927i	$-0.752323\pi$
$587$	−1.18921e7	−1.42450	−0.712248	+	0.701927i	$0.752323\pi$
$588$	811522.	0.0967960
$589$	−3.57329e6	−0.424404
$590$	0	0
$591$	−4.46187e6	−0.525471
$592$	−51448.9	−0.00603353
$593$	−1.49053e6	−0.174062	−0.0870311	−	0.996206i	$-0.527738\pi$
$593$	−1.49053e6	−0.174062	−0.0870311	+	0.996206i	$0.527738\pi$
$594$	−1.25250e6	−0.145651
$595$	0	0
$596$	−556026.	−0.0641179
$597$	−3.87528e6	−0.445008
$598$	4.87511e6	0.557483
$599$	−1.13339e7	−1.29066	−0.645328	−	0.763905i	$-0.723279\pi$
$599$	−1.13339e7	−1.29066	−0.645328	+	0.763905i	$0.723279\pi$
$600$	0	0
$601$	1.07157e7	1.21014	0.605069	−	0.796173i	$-0.293145\pi$
$601$	1.07157e7	1.21014	0.605069	+	0.796173i	$0.293145\pi$
$602$	−1.65899e6	−0.186575
$603$	−1.97681e7	−2.21397
$604$	390531.	0.0435576
$605$	0	0
$606$	1.44464e7	1.59800
$607$	871321.	0.0959857	0.0479929	−	0.998848i	$-0.484718\pi$
$607$	871321.	0.0959857	0.0479929	+	0.998848i	$0.484718\pi$
$608$	652905.	0.0716293
$609$	−3.74095e7	−4.08732
$610$	0	0
$611$	−1.33932e6	−0.145138
$612$	840760.	0.0907389
$613$	1.62264e7	1.74410	0.872051	−	0.489415i	$-0.162790\pi$
$613$	1.62264e7	1.74410	0.872051	+	0.489415i	$0.162790\pi$
$614$	−8.41339e6	−0.900638
$615$	0	0
$616$	2.06050e6	0.218787
$617$	−1.18192e6	−0.124990	−0.0624950	−	0.998045i	$-0.519906\pi$
$617$	−1.18192e6	−0.124990	−0.0624950	+	0.998045i	$0.519906\pi$
$618$	1.16344e7	1.22539
$619$	1.78346e7	1.87084	0.935418	−	0.353544i	$-0.115023\pi$
$619$	1.78346e7	1.87084	0.935418	+	0.353544i	$0.115023\pi$
$620$	0	0
$621$	4.64522e6	0.483367
$622$	−3.53004e6	−0.365850
$623$	1.08304e7	1.11795
$624$	1.58893e7	1.63359
$625$	0	0
$626$	−1.14710e6	−0.116995
$627$	−1.00323e6	−0.101913
$628$	−748443.	−0.0757286
$629$	−41224.9	−0.00415464
$630$	0	0
$631$	−1.56569e7	−1.56543	−0.782715	−	0.622381i	$-0.786166\pi$
$631$	−1.56569e7	−1.56543	−0.782715	+	0.622381i	$0.786166\pi$
$632$	1.80908e7	1.80163
$633$	1.70860e7	1.69485
$634$	−2.07910e6	−0.205424
$635$	0	0
$636$	−2.23884e6	−0.219473
$637$	7.52866e6	0.735139
$638$	−3.17422e6	−0.308734
$639$	4.33067e6	0.419568
$640$	0	0
$641$	−9.56677e6	−0.919645	−0.459822	−	0.888011i	$-0.652087\pi$
$641$	−9.56677e6	−0.919645	−0.459822	+	0.888011i	$0.652087\pi$
$642$	−2.66068e7	−2.54774
$643$	−6.58449e6	−0.628050	−0.314025	−	0.949415i	$-0.601678\pi$
$643$	−6.58449e6	−0.628050	−0.314025	+	0.949415i	$0.601678\pi$
$644$	−648392.	−0.0616060
$645$	0	0
$646$	−2.42321e6	−0.228459
$647$	5.77942e6	0.542779	0.271390	−	0.962470i	$-0.412517\pi$
$647$	5.77942e6	0.542779	0.271390	+	0.962470i	$0.412517\pi$
$648$	−893235.	−0.0835657
$649$	−3.42617e6	−0.319299
$650$	0	0
$651$	2.44354e7	2.25979
$652$	−1.53718e6	−0.141614
$653$	1.76847e7	1.62298	0.811492	−	0.584364i	$-0.198656\pi$
$653$	1.76847e7	1.62298	0.811492	+	0.584364i	$0.198656\pi$
$654$	2.23096e7	2.03961
$655$	0	0
$656$	9.61700e6	0.872529
$657$	2.34909e7	2.12318
$658$	−1.74317e6	−0.156955
$659$	1.07891e7	0.967769	0.483884	−	0.875132i	$-0.339226\pi$
$659$	1.07891e7	0.967769	0.483884	+	0.875132i	$0.339226\pi$
$660$	0	0
$661$	−1.27678e7	−1.13662	−0.568308	−	0.822816i	$-0.692402\pi$
$661$	−1.27678e7	−1.13662	−0.568308	+	0.822816i	$0.692402\pi$
$662$	8.77773e6	0.778461
$663$	1.27318e7	1.12488
$664$	−5.72352e6	−0.503782
$665$	0	0
$666$	−115770.	−0.0101137
$667$	1.17724e7	1.02459
$668$	−2.02139e6	−0.175270
$669$	−1.16215e7	−1.00392
$670$	0	0
$671$	922101.	0.0790628
$672$	−4.46480e6	−0.381399
$673$	−5.05831e6	−0.430495	−0.215247	−	0.976560i	$-0.569056\pi$
$673$	−5.05831e6	−0.430495	−0.215247	+	0.976560i	$0.569056\pi$
$674$	−1.25193e7	−1.06152
$675$	0	0
$676$	−308371.	−0.0259542
$677$	−1.48127e7	−1.24211	−0.621057	−	0.783766i	$-0.713296\pi$
$677$	−1.48127e7	−1.24211	−0.621057	+	0.783766i	$0.713296\pi$
$678$	2.42635e7	2.02712
$679$	2.00946e6	0.167265
$680$	0	0
$681$	2.85217e7	2.35672
$682$	2.07336e6	0.170692
$683$	9.52658e6	0.781421	0.390711	−	0.920514i	$-0.372229\pi$
$683$	9.52658e6	0.781421	0.390711	+	0.920514i	$0.372229\pi$
$684$	695388.	0.0568312
$685$	0	0
$686$	−5.28108e6	−0.428462
$687$	−1.62479e7	−1.31343
$688$	1.70156e6	0.137049
$689$	−2.07702e7	−1.66683
$690$	0	0
$691$	−4.04497e6	−0.322270	−0.161135	−	0.986932i	$-0.551516\pi$
$691$	−4.04497e6	−0.322270	−0.161135	+	0.986932i	$0.551516\pi$
$692$	−440860.	−0.0349973
$693$	4.20294e6	0.332445
$694$	−1.32437e7	−1.04379
$695$	0	0
$696$	4.23278e7	3.31209
$697$	7.70590e6	0.600816
$698$	2.35191e7	1.82718
$699$	−1.47443e7	−1.14138
$700$	0	0
$701$	1.78401e6	0.137120	0.0685602	−	0.997647i	$-0.478159\pi$
$701$	1.78401e6	0.137120	0.0685602	+	0.997647i	$0.478159\pi$
$702$	1.31469e7	1.00688
$703$	−34096.9	−0.00260212
$704$	−2.31290e6	−0.175883
$705$	0	0
$706$	4.84055e6	0.365496
$707$	−1.78250e7	−1.34116
$708$	3.87646e6	0.290638
$709$	−1.22322e7	−0.913878	−0.456939	−	0.889498i	$-0.651054\pi$
$709$	−1.22322e7	−0.913878	−0.456939	+	0.889498i	$0.651054\pi$
$710$	0	0
$711$	3.69010e7	2.73756
$712$	−1.22542e7	−0.905913
$713$	−7.68958e6	−0.566473
$714$	1.65708e7	1.21646
$715$	0	0
$716$	−305787.	−0.0222913
$717$	−1.85124e7	−1.34482
$718$	−9.79938e6	−0.709394
$719$	8.42924e6	0.608088	0.304044	−	0.952658i	$-0.401663\pi$
$719$	8.42924e6	0.608088	0.304044	+	0.952658i	$0.401663\pi$
$720$	0	0
$721$	−1.43554e7	−1.02844
$722$	1.13376e7	0.809428
$723$	3.94128e7	2.80409
$724$	−917691.	−0.0650654
$725$	0	0
$726$	−2.11525e7	−1.48943
$727$	−1.10329e7	−0.774201	−0.387101	−	0.922037i	$-0.626523\pi$
$727$	−1.10329e7	−0.774201	−0.387101	+	0.922037i	$0.626523\pi$
$728$	−2.16280e7	−1.51247
$729$	−2.33632e7	−1.62822
$730$	0	0
$731$	1.36342e6	0.0943708
$732$	−1.04329e6	−0.0719660
$733$	5.02000e6	0.345099	0.172550	−	0.985001i	$-0.444799\pi$
$733$	5.02000e6	0.345099	0.172550	+	0.985001i	$0.444799\pi$
$734$	−1.26496e7	−0.866633
$735$	0	0
$736$	1.40503e6	0.0956072
$737$	3.37824e6	0.229098
$738$	2.16402e7	1.46258
$739$	2.49061e6	0.167762	0.0838810	−	0.996476i	$-0.473268\pi$
$739$	2.49061e6	0.167762	0.0838810	+	0.996476i	$0.473268\pi$
$740$	0	0
$741$	1.05304e7	0.704529
$742$	−2.70330e7	−1.80254
$743$	1.21111e7	0.804847	0.402423	−	0.915454i	$-0.368168\pi$
$743$	1.21111e7	0.804847	0.402423	+	0.915454i	$0.368168\pi$
$744$	−2.76480e7	−1.83118
$745$	0	0
$746$	1.60506e7	1.05595
$747$	−1.16746e7	−0.765495
$748$	−143680.	−0.00938952
$749$	3.28294e7	2.13825
$750$	0	0
$751$	2.21669e7	1.43418	0.717092	−	0.696978i	$-0.245472\pi$
$751$	2.21669e7	1.43418	0.717092	+	0.696978i	$0.245472\pi$
$752$	1.78790e6	0.115292
$753$	−1.14408e7	−0.735310
$754$	3.33181e7	2.13428
$755$	0	0
$756$	−1.74854e6	−0.111268
$757$	−1.03112e7	−0.653986	−0.326993	−	0.945027i	$-0.606035\pi$
$757$	−1.03112e7	−0.653986	−0.326993	+	0.945027i	$0.606035\pi$
$758$	−9.57590e6	−0.605350
$759$	−2.15891e6	−0.136029
$760$	0	0
$761$	5.23003e6	0.327373	0.163686	−	0.986512i	$-0.447661\pi$
$761$	5.23003e6	0.327373	0.163686	+	0.986512i	$0.447661\pi$
$762$	3.86761e7	2.41299
$763$	−2.75272e7	−1.71179
$764$	−1.10740e6	−0.0686387
$765$	0	0
$766$	−1.12180e7	−0.690783
$767$	3.59627e7	2.20731
$768$	7.24005e6	0.442933
$769$	1.16812e7	0.712315	0.356157	−	0.934426i	$-0.384087\pi$
$769$	1.16812e7	0.712315	0.356157	+	0.934426i	$0.384087\pi$
$770$	0	0
$771$	2.80847e7	1.70151
$772$	254504.	0.0153692
$773$	−1.35127e7	−0.813379	−0.406690	−	0.913566i	$-0.633317\pi$
$773$	−1.35127e7	−0.813379	−0.406690	+	0.913566i	$0.633317\pi$
$774$	3.82885e6	0.229729
$775$	0	0
$776$	−2.27365e6	−0.135540
$777$	233167.	0.0138553
$778$	9.75702e6	0.577920
$779$	6.37351e6	0.376301
$780$	0	0
$781$	−740082.	−0.0434162
$782$	−5.21465e6	−0.304936
$783$	3.17470e7	1.85054
$784$	−1.00502e7	−0.583962
$785$	0	0
$786$	9.12553e6	0.526868
$787$	2.08780e7	1.20158	0.600789	−	0.799407i	$-0.294853\pi$
$787$	2.08780e7	1.20158	0.600789	+	0.799407i	$0.294853\pi$
$788$	−528529.	−0.0303217
$789$	4.39862e7	2.51550
$790$	0	0
$791$	−2.99381e7	−1.70131
$792$	−4.75551e6	−0.269392
$793$	−9.67881e6	−0.546562
$794$	1.00167e6	0.0563866
$795$	0	0
$796$	−459045.	−0.0256787
$797$	1.17024e7	0.652573	0.326286	−	0.945271i	$-0.394203\pi$
$797$	1.17024e7	0.652573	0.326286	+	0.945271i	$0.394203\pi$
$798$	1.37056e7	0.761887
$799$	1.43260e6	0.0793888
$800$	0	0
$801$	−2.49958e7	−1.37653
$802$	1.84220e7	1.01135
$803$	−4.01444e6	−0.219703
$804$	−3.82223e6	−0.208534
$805$	0	0
$806$	−2.17630e7	−1.18000
$807$	−3.07901e7	−1.66428
$808$	2.01685e7	1.08679
$809$	−2.76467e7	−1.48516	−0.742578	−	0.669760i	$-0.766397\pi$
$809$	−2.76467e7	−1.48516	−0.742578	+	0.669760i	$0.766397\pi$
$810$	0	0
$811$	−8.32113e6	−0.444253	−0.222126	−	0.975018i	$-0.571300\pi$
$811$	−8.32113e6	−0.444253	−0.222126	+	0.975018i	$0.571300\pi$
$812$	−4.43132e6	−0.235854
$813$	−3.31375e7	−1.75830
$814$	19784.4	0.00104655
$815$	0	0
$816$	−1.69960e7	−0.893554
$817$	1.12768e6	0.0591059
$818$	2.30713e6	0.120556
$819$	−4.41161e7	−2.29820
$820$	0	0
$821$	8.01626e6	0.415063	0.207531	−	0.978228i	$-0.433457\pi$
$821$	8.01626e6	0.415063	0.207531	+	0.978228i	$0.433457\pi$
$822$	−1.80273e7	−0.930573
$823$	−4.63830e6	−0.238704	−0.119352	−	0.992852i	$-0.538082\pi$
$823$	−4.63830e6	−0.238704	−0.119352	+	0.992852i	$0.538082\pi$
$824$	1.62428e7	0.833378
$825$	0	0
$826$	4.68065e7	2.38702
$827$	−5.55667e6	−0.282521	−0.141261	−	0.989972i	$-0.545116\pi$
$827$	−5.55667e6	−0.282521	−0.141261	+	0.989972i	$0.545116\pi$
$828$	1.49645e6	0.0758554
$829$	2.47536e7	1.25098	0.625491	−	0.780231i	$-0.284899\pi$
$829$	2.47536e7	1.25098	0.625491	+	0.780231i	$0.284899\pi$
$830$	0	0
$831$	4.93046e7	2.47676
$832$	2.42773e7	1.21588
$833$	−8.05302e6	−0.402111
$834$	−3.53846e7	−1.76156
$835$	0	0
$836$	−118837.	−0.00588080
$837$	−2.07367e7	−1.02312
$838$	−2.06914e7	−1.01784
$839$	−3.24512e7	−1.59157	−0.795785	−	0.605579i	$-0.792942\pi$
$839$	−3.24512e7	−1.59157	−0.795785	+	0.605579i	$0.792942\pi$
$840$	0	0
$841$	5.99451e7	2.92256
$842$	−3.80347e7	−1.84884
$843$	2.80067e7	1.35735
$844$	2.02392e6	0.0977994
$845$	0	0
$846$	4.02313e6	0.193258
$847$	2.60995e7	1.25004
$848$	2.77267e7	1.32406
$849$	3.14899e7	1.49935
$850$	0	0
$851$	−73375.3	−0.00347317
$852$	837349.	0.0395191
$853$	−2.47564e7	−1.16497	−0.582485	−	0.812841i	$-0.697920\pi$
$853$	−2.47564e7	−1.16497	−0.582485	+	0.812841i	$0.697920\pi$
$854$	−1.25973e7	−0.591060
$855$	0	0
$856$	−3.71456e7	−1.73270
$857$	−1.28652e7	−0.598363	−0.299182	−	0.954196i	$-0.596714\pi$
$857$	−1.28652e7	−0.598363	−0.299182	+	0.954196i	$0.596714\pi$
$858$	−6.11014e6	−0.283356
$859$	1.98227e6	0.0916601	0.0458300	−	0.998949i	$-0.485407\pi$
$859$	1.98227e6	0.0916601	0.0458300	+	0.998949i	$0.485407\pi$
$860$	0	0
$861$	−4.35844e7	−2.00366
$862$	1.14130e7	0.523157
$863$	3.11840e6	0.142529	0.0712647	−	0.997457i	$-0.477296\pi$
$863$	3.11840e6	0.142529	0.0712647	+	0.997457i	$0.477296\pi$
$864$	3.78898e6	0.172679
$865$	0	0
$866$	−8.55964e6	−0.387847
$867$	2.19435e7	0.991421
$868$	2.89449e6	0.130399
$869$	−6.30613e6	−0.283279
$870$	0	0
$871$	−3.54596e7	−1.58376
$872$	3.11462e7	1.38712
$873$	−4.63772e6	−0.205953
$874$	−4.31301e6	−0.190986
$875$	0	0
$876$	4.54204e6	0.199982
$877$	2.15790e7	0.947398	0.473699	−	0.880687i	$-0.342918\pi$
$877$	2.15790e7	0.947398	0.473699	+	0.880687i	$0.342918\pi$
$878$	−3.83267e7	−1.67790
$879$	−6.52218e6	−0.284722
$880$	0	0
$881$	−1.27720e7	−0.554393	−0.277196	−	0.960813i	$-0.589405\pi$
$881$	−1.27720e7	−0.554393	−0.277196	+	0.960813i	$0.589405\pi$
$882$	−2.26150e7	−0.978870
$883$	−8.80113e6	−0.379871	−0.189936	−	0.981797i	$-0.560828\pi$
$883$	−8.80113e6	−0.379871	−0.189936	+	0.981797i	$0.560828\pi$
$884$	1.50814e6	0.0649098
$885$	0	0
$886$	1.36012e7	0.582092
$887$	3.47818e7	1.48437	0.742187	−	0.670192i	$-0.233788\pi$
$887$	3.47818e7	1.48437	0.742187	+	0.670192i	$0.233788\pi$
$888$	−263822.	−0.0112274
$889$	−4.77214e7	−2.02516
$890$	0	0
$891$	311366.	0.0131394
$892$	−1.37662e6	−0.0579300
$893$	1.18490e6	0.0497225
$894$	2.52924e7	1.05839
$895$	0	0
$896$	2.58932e7	1.07750
$897$	2.26610e7	0.940368
$898$	2.54254e6	0.105215
$899$	−5.25531e7	−2.16870
$900$	0	0
$901$	2.22168e7	0.911737
$902$	−3.69816e6	−0.151345
$903$	−7.71150e6	−0.314716
$904$	3.38741e7	1.37863
$905$	0	0
$906$	−1.77644e7	−0.719003
$907$	−1.91865e7	−0.774420	−0.387210	−	0.921992i	$-0.626561\pi$
$907$	−1.91865e7	−0.774420	−0.387210	+	0.921992i	$0.626561\pi$
$908$	3.37853e6	0.135992
$909$	4.11390e7	1.65137
$910$	0	0
$911$	834307.	0.0333066	0.0166533	−	0.999861i	$-0.494699\pi$
$911$	834307.	0.0333066	0.0166533	+	0.999861i	$0.494699\pi$
$912$	−1.40573e7	−0.559647
$913$	1.99512e6	0.0792121
$914$	−3.29242e7	−1.30362
$915$	0	0
$916$	−1.92464e6	−0.0757899
$917$	−1.12598e7	−0.442187
$918$	−1.40625e7	−0.550753
$919$	1.11037e7	0.433690	0.216845	−	0.976206i	$-0.430423\pi$
$919$	1.11037e7	0.433690	0.216845	+	0.976206i	$0.430423\pi$
$920$	0	0
$921$	−3.91080e7	−1.51921
$922$	−4.02708e7	−1.56014
$923$	7.76826e6	0.300137
$924$	812652.	0.0313130
$925$	0	0
$926$	−2.23602e7	−0.856935
$927$	3.31315e7	1.26631
$928$	9.60242e6	0.366025
$929$	−1.03857e7	−0.394816	−0.197408	−	0.980321i	$-0.563252\pi$
$929$	−1.03857e7	−0.394816	−0.197408	+	0.980321i	$0.563252\pi$
$930$	0	0
$931$	−6.66061e6	−0.251849
$932$	−1.74653e6	−0.0658622
$933$	−1.64087e7	−0.617120
$934$	−6.40063e6	−0.240080
$935$	0	0
$936$	4.99161e7	1.86231
$937$	2.10627e7	0.783729	0.391865	−	0.920023i	$-0.371830\pi$
$937$	2.10627e7	0.783729	0.391865	+	0.920023i	$0.371830\pi$
$938$	−4.61518e7	−1.71270
$939$	−5.33209e6	−0.197348
$940$	0	0
$941$	−3.67199e7	−1.35185	−0.675924	−	0.736971i	$-0.736255\pi$
$941$	−3.67199e7	−1.35185	−0.675924	+	0.736971i	$0.736255\pi$
$942$	3.40451e7	1.25005
$943$	1.37156e7	0.502267
$944$	−4.80076e7	−1.75339
$945$	0	0
$946$	−654325.	−0.0237720
$947$	−1.07332e7	−0.388914	−0.194457	−	0.980911i	$-0.562294\pi$
$947$	−1.07332e7	−0.388914	−0.194457	+	0.980911i	$0.562294\pi$
$948$	7.13493e6	0.257851
$949$	4.21375e7	1.51881
$950$	0	0
$951$	−9.66428e6	−0.346512
$952$	2.31344e7	0.827304
$953$	−4.10240e7	−1.46321	−0.731603	−	0.681731i	$-0.761228\pi$
$953$	−4.10240e7	−1.46321	−0.731603	+	0.681731i	$0.761228\pi$
$954$	6.23906e7	2.21946
$955$	0	0
$956$	−2.19287e6	−0.0776013
$957$	−1.47547e7	−0.520776
$958$	−2.01575e7	−0.709617
$959$	2.22434e7	0.781005
$960$	0	0
$961$	5.69795e6	0.199026
$962$	−207666.	−0.00723483
$963$	−7.57683e7	−2.63283
$964$	4.66862e6	0.161807
$965$	0	0
$966$	2.94940e7	1.01693
$967$	−2.93253e7	−1.00850	−0.504251	−	0.863557i	$-0.668231\pi$
$967$	−2.93253e7	−1.00850	−0.504251	+	0.863557i	$0.668231\pi$
$968$	−2.95309e7	−1.01295
$969$	−1.12638e7	−0.385368
$970$	0	0
$971$	3.44554e6	0.117276	0.0586381	−	0.998279i	$-0.481324\pi$
$971$	3.44554e6	0.117276	0.0586381	+	0.998279i	$0.481324\pi$
$972$	−2.90394e6	−0.0985877
$973$	4.36601e7	1.47843
$974$	3.86934e7	1.30689
$975$	0	0
$976$	1.29205e7	0.434165
$977$	−3.77647e7	−1.26576	−0.632878	−	0.774252i	$-0.718126\pi$
$977$	−3.77647e7	−1.26576	−0.632878	+	0.774252i	$0.718126\pi$
$978$	6.99230e7	2.33762
$979$	4.27162e6	0.142441
$980$	0	0
$981$	6.35311e7	2.10772
$982$	3.97352e7	1.31491
$983$	−1.69058e7	−0.558024	−0.279012	−	0.960288i	$-0.590007\pi$
$983$	−1.69058e7	−0.558024	−0.279012	+	0.960288i	$0.590007\pi$
$984$	4.93145e7	1.62363
$985$	0	0
$986$	−3.56387e7	−1.16743
$987$	−8.10277e6	−0.264753
$988$	1.24737e6	0.0406540
$989$	2.42673e6	0.0788915
$990$	0	0
$991$	1.83818e7	0.594570	0.297285	−	0.954789i	$-0.403919\pi$
$991$	1.83818e7	0.594570	0.297285	+	0.954789i	$0.403919\pi$
$992$	−6.27219e6	−0.202367
$993$	4.08016e7	1.31312
$994$	1.01106e7	0.324572
$995$	0	0
$996$	−2.25733e6	−0.0721019
$997$	2.05937e7	0.656139	0.328069	−	0.944654i	$-0.393602\pi$
$997$	2.05937e7	0.656139	0.328069	+	0.944654i	$0.393602\pi$
$998$	−1.40679e7	−0.447099
$999$	−197874.	−0.00627298

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1075.6.a.b.1.4		10
5.4	even	2		43.6.a.b.1.7	✓	10
15.14	odd	2		387.6.a.e.1.4		10
20.19	odd	2		688.6.a.h.1.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.a.b.1.7	✓	10	5.4	even	2
387.6.a.e.1.4		10	15.14	odd	2
688.6.a.h.1.2		10	20.19	odd	2
1075.6.a.b.1.4		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:	\( N \)	\(=\)	\( 1075 = 5^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1075.a (trivial)

\(f(q)\)	\(=\)	\(q-5.38824 q^{2} -25.0462 q^{3} -2.96684 q^{4} +134.955 q^{6} -166.517 q^{7} +188.410 q^{8} +384.312 q^{9} +O(q^{10})\)	\(q-5.38824 q^{2} -25.0462 q^{3} -2.96684 q^{4} +134.955 q^{6} -166.517 q^{7} +188.410 q^{8} +384.312 q^{9} -65.6764 q^{11} +74.3080 q^{12} +689.371 q^{13} +897.236 q^{14} -920.259 q^{16} -737.385 q^{17} -2070.77 q^{18} -609.887 q^{19} +4170.63 q^{21} +353.880 q^{22} -1312.45 q^{23} -4718.95 q^{24} -3714.50 q^{26} -3539.34 q^{27} +494.030 q^{28} -8969.74 q^{29} +5858.93 q^{31} -1070.53 q^{32} +1644.94 q^{33} +3973.21 q^{34} -1140.19 q^{36} +55.9069 q^{37} +3286.22 q^{38} -17266.1 q^{39} -10450.3 q^{41} -22472.4 q^{42} -1849.00 q^{43} +194.851 q^{44} +7071.82 q^{46} -1942.82 q^{47} +23049.0 q^{48} +10921.1 q^{49} +18468.7 q^{51} -2045.25 q^{52} -30129.2 q^{53} +19070.8 q^{54} -31373.5 q^{56} +15275.3 q^{57} +48331.2 q^{58} +52167.4 q^{59} -14040.1 q^{61} -31569.4 q^{62} -63994.7 q^{63} +35216.6 q^{64} -8863.36 q^{66} -51437.7 q^{67} +2187.70 q^{68} +32872.0 q^{69} +11268.6 q^{71} +72408.2 q^{72} +61124.5 q^{73} -301.240 q^{74} +1809.44 q^{76} +10936.3 q^{77} +93034.1 q^{78} +96018.3 q^{79} -4740.91 q^{81} +56308.8 q^{82} -30378.0 q^{83} -12373.6 q^{84} +9962.86 q^{86} +224658. q^{87} -12374.1 q^{88} -65040.4 q^{89} -114792. q^{91} +3893.84 q^{92} -146744. q^{93} +10468.4 q^{94} +26812.8 q^{96} -12067.6 q^{97} -58845.3 q^{98} -25240.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 8 q^{2} - 28 q^{3} + 202 q^{4} + 75 q^{6} - 60 q^{7} - 294 q^{8} + 1356 q^{9}+O(q^{10}) \) 10 * q - 8 * q^2 - 28 * q^3 + 202 * q^4 + 75 * q^6 - 60 * q^7 - 294 * q^8 + 1356 * q^9	\( 10 q - 8 q^{2} - 28 q^{3} + 202 q^{4} + 75 q^{6} - 60 q^{7} - 294 q^{8} + 1356 q^{9} + 745 q^{11} - 4627 q^{12} - 1917 q^{13} + 1936 q^{14} + 5354 q^{16} - 4017 q^{17} + 2725 q^{18} - 2404 q^{19} - 228 q^{21} + 5836 q^{22} - 1733 q^{23} - 10711 q^{24} - 1484 q^{26} + 2324 q^{27} + 15028 q^{28} + 6996 q^{29} - 4899 q^{31} + 7554 q^{32} + 15734 q^{33} - 27033 q^{34} + 4433 q^{36} - 1466 q^{37} - 13905 q^{38} - 26542 q^{39} + 10297 q^{41} + 37642 q^{42} - 18490 q^{43} - 36140 q^{44} + 17991 q^{46} - 48592 q^{47} - 83607 q^{48} + 29458 q^{49} + 92972 q^{51} - 14232 q^{52} - 127165 q^{53} - 92002 q^{54} - 7780 q^{56} - 34060 q^{57} + 10305 q^{58} + 99372 q^{59} + 17408 q^{61} - 28265 q^{62} - 2244 q^{63} + 47202 q^{64} - 150292 q^{66} + 2021 q^{67} - 192151 q^{68} + 1654 q^{69} + 11286 q^{71} + 298365 q^{72} - 49892 q^{73} - 125431 q^{74} - 249803 q^{76} - 98144 q^{77} + 28494 q^{78} - 91524 q^{79} - 26450 q^{81} + 158909 q^{82} + 105203 q^{83} - 357682 q^{84} + 14792 q^{86} - 181200 q^{87} + 461824 q^{88} - 62682 q^{89} - 295304 q^{91} - 183783 q^{92} + 238430 q^{93} + 7259 q^{94} - 162399 q^{96} - 108383 q^{97} - 354656 q^{98} - 270499 q^{99}+O(q^{100}) \) 10 * q - 8 * q^2 - 28 * q^3 + 202 * q^4 + 75 * q^6 - 60 * q^7 - 294 * q^8 + 1356 * q^9 + 745 * q^11 - 4627 * q^12 - 1917 * q^13 + 1936 * q^14 + 5354 * q^16 - 4017 * q^17 + 2725 * q^18 - 2404 * q^19 - 228 * q^21 + 5836 * q^22 - 1733 * q^23 - 10711 * q^24 - 1484 * q^26 + 2324 * q^27 + 15028 * q^28 + 6996 * q^29 - 4899 * q^31 + 7554 * q^32 + 15734 * q^33 - 27033 * q^34 + 4433 * q^36 - 1466 * q^37 - 13905 * q^38 - 26542 * q^39 + 10297 * q^41 + 37642 * q^42 - 18490 * q^43 - 36140 * q^44 + 17991 * q^46 - 48592 * q^47 - 83607 * q^48 + 29458 * q^49 + 92972 * q^51 - 14232 * q^52 - 127165 * q^53 - 92002 * q^54 - 7780 * q^56 - 34060 * q^57 + 10305 * q^58 + 99372 * q^59 + 17408 * q^61 - 28265 * q^62 - 2244 * q^63 + 47202 * q^64 - 150292 * q^66 + 2021 * q^67 - 192151 * q^68 + 1654 * q^69 + 11286 * q^71 + 298365 * q^72 - 49892 * q^73 - 125431 * q^74 - 249803 * q^76 - 98144 * q^77 + 28494 * q^78 - 91524 * q^79 - 26450 * q^81 + 158909 * q^82 + 105203 * q^83 - 357682 * q^84 + 14792 * q^86 - 181200 * q^87 + 461824 * q^88 - 62682 * q^89 - 295304 * q^91 - 183783 * q^92 + 238430 * q^93 + 7259 * q^94 - 162399 * q^96 - 108383 * q^97 - 354656 * q^98 - 270499 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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