LMFDB - Embedded newform 1075.6.a.a.1.2

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:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 173x^{6} + 462x^{5} + 9118x^{4} - 14192x^{3} - 167688x^{2} + 106368x + 681984 $ x^8 - 4x^7 - 173x^6 + 462x^5 + 9118x^4 - 14192x^3 - 167688x^2 + 106368*x + 681984
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$7.21373$ 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)

$f(q)$	$=$	$q-5.21373 q^{2} +11.2683 q^{3} -4.81697 q^{4} -58.7500 q^{6} +11.1041 q^{7} +191.954 q^{8} -116.025 q^{9} +O(q^{10})$	\(q-5.21373 q^{2} +11.2683 q^{3} -4.81697 q^{4} -58.7500 q^{6} +11.1041 q^{7} +191.954 q^{8} -116.025 q^{9} -557.274 q^{11} -54.2791 q^{12} -107.663 q^{13} -57.8938 q^{14} -846.654 q^{16} -329.005 q^{17} +604.924 q^{18} +2938.10 q^{19} +125.124 q^{21} +2905.48 q^{22} +385.537 q^{23} +2163.00 q^{24} +561.326 q^{26} -4045.61 q^{27} -53.4881 q^{28} -3309.15 q^{29} -5471.71 q^{31} -1728.30 q^{32} -6279.54 q^{33} +1715.34 q^{34} +558.890 q^{36} -4832.22 q^{37} -15318.5 q^{38} -1213.18 q^{39} +1065.16 q^{41} -652.365 q^{42} +1849.00 q^{43} +2684.37 q^{44} -2010.09 q^{46} +8991.95 q^{47} -9540.36 q^{48} -16683.7 q^{49} -3707.33 q^{51} +518.609 q^{52} +10216.7 q^{53} +21092.7 q^{54} +2131.47 q^{56} +33107.4 q^{57} +17253.0 q^{58} -27457.0 q^{59} -36692.9 q^{61} +28528.1 q^{62} -1288.35 q^{63} +36103.8 q^{64} +32739.9 q^{66} +26272.2 q^{67} +1584.81 q^{68} +4344.35 q^{69} +55130.3 q^{71} -22271.5 q^{72} +9315.14 q^{73} +25193.9 q^{74} -14152.7 q^{76} -6188.02 q^{77} +6325.20 q^{78} +56826.5 q^{79} -17393.1 q^{81} -5553.44 q^{82} +5858.19 q^{83} -602.721 q^{84} -9640.20 q^{86} -37288.5 q^{87} -106971. q^{88} -42815.7 q^{89} -1195.50 q^{91} -1857.12 q^{92} -61657.0 q^{93} -46881.7 q^{94} -19475.0 q^{96} +231.671 q^{97} +86984.4 q^{98} +64657.8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{2} + 26 q^{3} + 122 q^{4} - 69 q^{6} + 136 q^{7} + 666 q^{8} + 546 q^{9}+O(q^{10}) $ 8 * q + 12 * q^2 + 26 * q^3 + 122 * q^4 - 69 * q^6 + 136 * q^7 + 666 * q^8 + 546 * q^9	\( 8 q + 12 q^{2} + 26 q^{3} + 122 q^{4} - 69 q^{6} + 136 q^{7} + 666 q^{8} + 546 q^{9} - 532 q^{11} + 4195 q^{12} + 2492 q^{13} - 4240 q^{14} + 1882 q^{16} + 2534 q^{17} + 3711 q^{18} - 1678 q^{19} - 2256 q^{21} - 11502 q^{22} + 2488 q^{23} + 19953 q^{24} + 4586 q^{26} + 8960 q^{27} - 18640 q^{28} - 4360 q^{29} + 5704 q^{31} + 18294 q^{32} + 12852 q^{33} + 30007 q^{34} + 67969 q^{36} + 3772 q^{37} + 6559 q^{38} + 11120 q^{39} - 10698 q^{41} - 78698 q^{42} + 14792 q^{43} - 356 q^{44} - 19389 q^{46} + 77864 q^{47} + 118727 q^{48} + 7188 q^{49} - 80246 q^{51} + 60736 q^{52} + 62352 q^{53} + 61026 q^{54} - 144528 q^{56} + 808 q^{57} - 52951 q^{58} - 26224 q^{59} - 82540 q^{61} + 9023 q^{62} + 61768 q^{63} + 153858 q^{64} - 48516 q^{66} - 27784 q^{67} - 40507 q^{68} - 93776 q^{69} - 9504 q^{71} + 186687 q^{72} - 14260 q^{73} - 15239 q^{74} + 1279 q^{76} + 218140 q^{77} - 264170 q^{78} + 160248 q^{79} + 161076 q^{81} + 47781 q^{82} + 77176 q^{83} + 16382 q^{84} + 22188 q^{86} - 268136 q^{87} - 129544 q^{88} - 265692 q^{89} + 401148 q^{91} - 190391 q^{92} + 123860 q^{93} + 248737 q^{94} + 950817 q^{96} - 144742 q^{97} + 292244 q^{98} + 239516 q^{99}+O(q^{100}) \) 8 * q + 12 * q^2 + 26 * q^3 + 122 * q^4 - 69 * q^6 + 136 * q^7 + 666 * q^8 + 546 * q^9 - 532 * q^11 + 4195 * q^12 + 2492 * q^13 - 4240 * q^14 + 1882 * q^16 + 2534 * q^17 + 3711 * q^18 - 1678 * q^19 - 2256 * q^21 - 11502 * q^22 + 2488 * q^23 + 19953 * q^24 + 4586 * q^26 + 8960 * q^27 - 18640 * q^28 - 4360 * q^29 + 5704 * q^31 + 18294 * q^32 + 12852 * q^33 + 30007 * q^34 + 67969 * q^36 + 3772 * q^37 + 6559 * q^38 + 11120 * q^39 - 10698 * q^41 - 78698 * q^42 + 14792 * q^43 - 356 * q^44 - 19389 * q^46 + 77864 * q^47 + 118727 * q^48 + 7188 * q^49 - 80246 * q^51 + 60736 * q^52 + 62352 * q^53 + 61026 * q^54 - 144528 * q^56 + 808 * q^57 - 52951 * q^58 - 26224 * q^59 - 82540 * q^61 + 9023 * q^62 + 61768 * q^63 + 153858 * q^64 - 48516 * q^66 - 27784 * q^67 - 40507 * q^68 - 93776 * q^69 - 9504 * q^71 + 186687 * q^72 - 14260 * q^73 - 15239 * q^74 + 1279 * q^76 + 218140 * q^77 - 264170 * q^78 + 160248 * q^79 + 161076 * q^81 + 47781 * q^82 + 77176 * q^83 + 16382 * q^84 + 22188 * q^86 - 268136 * q^87 - 129544 * q^88 - 265692 * q^89 + 401148 * q^91 - 190391 * q^92 + 123860 * q^93 + 248737 * q^94 + 950817 * q^96 - 144742 * q^97 + 292244 * q^98 + 239516 * 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 $

$2$	−5.21373	−0.921667	−0.460833	−	0.887487i	$-0.652449\pi$
$2$	−5.21373	−0.921667	−0.460833	+	0.887487i	$0.652449\pi$
$3$	11.2683	0.722863	0.361431	−	0.932399i	$-0.382288\pi$
$3$	11.2683	0.722863	0.361431	+	0.932399i	$0.382288\pi$
$4$	−4.81697	−0.150530
$5$	0	0
$5$	0	0
$6$	−58.7500	−0.666239
$7$	11.1041	0.0856521	0.0428260	−	0.999083i	$-0.486364\pi$
$7$	11.1041	0.0856521	0.0428260	+	0.999083i	$0.486364\pi$
$8$	191.954	1.06041
$9$	−116.025	−0.477470
$10$	0	0
$11$	−557.274	−1.38863	−0.694316	−	0.719670i	$-0.744293\pi$
$11$	−557.274	−1.38863	−0.694316	+	0.719670i	$0.744293\pi$
$12$	−54.2791	−0.108813
$13$	−107.663	−0.176688	−0.0883441	−	0.996090i	$-0.528158\pi$
$13$	−107.663	−0.176688	−0.0883441	+	0.996090i	$0.528158\pi$
$14$	−57.8938	−0.0789427
$15$	0	0
$16$	−846.654	−0.826810
$17$	−329.005	−0.276109	−0.138054	−	0.990425i	$-0.544085\pi$
$17$	−329.005	−0.276109	−0.138054	+	0.990425i	$0.544085\pi$
$18$	604.924	0.440068
$19$	2938.10	1.86716	0.933582	−	0.358365i	$-0.116666\pi$
$19$	2938.10	1.86716	0.933582	+	0.358365i	$0.116666\pi$
$20$	0	0
$21$	125.124	0.0619147
$22$	2905.48	1.27986
$23$	385.537	0.151966	0.0759830	−	0.997109i	$-0.475791\pi$
$23$	385.537	0.151966	0.0759830	+	0.997109i	$0.475791\pi$
$24$	2163.00	0.766528
$25$	0	0
$26$	561.326	0.162848
$27$	−4045.61	−1.06801
$28$	−53.4881	−0.0128932
$29$	−3309.15	−0.730670	−0.365335	−	0.930876i	$-0.619046\pi$
$29$	−3309.15	−0.730670	−0.365335	+	0.930876i	$0.619046\pi$
$30$	0	0
$31$	−5471.71	−1.02263	−0.511316	−	0.859393i	$-0.670842\pi$
$31$	−5471.71	−1.02263	−0.511316	+	0.859393i	$0.670842\pi$
$32$	−1728.30	−0.298362
$33$	−6279.54	−1.00379
$34$	1715.34	0.254480
$35$	0	0
$36$	558.890	0.0718737
$37$	−4832.22	−0.580286	−0.290143	−	0.956983i	$-0.593703\pi$
$37$	−4832.22	−0.580286	−0.290143	+	0.956983i	$0.593703\pi$
$38$	−15318.5	−1.72090
$39$	−1213.18	−0.127721
$40$	0	0
$41$	1065.16	0.0989587	0.0494793	−	0.998775i	$-0.484244\pi$
$41$	1065.16	0.0989587	0.0494793	+	0.998775i	$0.484244\pi$
$42$	−652.365	−0.0570647
$43$	1849.00	0.152499
$43$	1849.00	0.152499
$44$	2684.37	0.209031
$45$	0	0
$46$	−2010.09	−0.140062
$47$	8991.95	0.593758	0.296879	−	0.954915i	$-0.404054\pi$
$47$	8991.95	0.593758	0.296879	+	0.954915i	$0.404054\pi$
$48$	−9540.36	−0.597670
$49$	−16683.7	−0.992664
$50$	0	0
$51$	−3707.33	−0.199589
$52$	518.609	0.0265969
$53$	10216.7	0.499598	0.249799	−	0.968298i	$-0.419635\pi$
$53$	10216.7	0.499598	0.249799	+	0.968298i	$0.419635\pi$
$54$	21092.7	0.984347
$55$	0	0
$56$	2131.47	0.0908260
$57$	33107.4	1.34970
$58$	17253.0	0.673434
$59$	−27457.0	−1.02689	−0.513444	−	0.858123i	$-0.671631\pi$
$59$	−27457.0	−1.02689	−0.513444	+	0.858123i	$0.671631\pi$
$60$	0	0
$61$	−36692.9	−1.26258	−0.631288	−	0.775548i	$-0.717473\pi$
$61$	−36692.9	−1.26258	−0.631288	+	0.775548i	$0.717473\pi$
$62$	28528.1	0.942525
$63$	−1288.35	−0.0408963
$64$	36103.8	1.10180
$65$	0	0
$66$	32739.9	0.925160
$67$	26272.2	0.715004	0.357502	−	0.933912i	$-0.383628\pi$
$67$	26272.2	0.715004	0.357502	+	0.933912i	$0.383628\pi$
$68$	1584.81	0.0415627
$69$	4344.35	0.109851
$70$	0	0
$71$	55130.3	1.29791	0.648955	−	0.760827i	$-0.275206\pi$
$71$	55130.3	1.29791	0.648955	+	0.760827i	$0.275206\pi$
$72$	−22271.5	−0.506311
$73$	9315.14	0.204589	0.102294	−	0.994754i	$-0.467382\pi$
$73$	9315.14	0.204589	0.102294	+	0.994754i	$0.467382\pi$
$74$	25193.9	0.534830
$75$	0	0
$76$	−14152.7	−0.281065
$77$	−6188.02	−0.118939
$78$	6325.20	0.117717
$79$	56826.5	1.02443	0.512216	−	0.858856i	$-0.328825\pi$
$79$	56826.5	1.02443	0.512216	+	0.858856i	$0.328825\pi$
$80$	0	0
$81$	−17393.1	−0.294553
$82$	−5553.44	−0.0912069
$83$	5858.19	0.0933401	0.0466700	−	0.998910i	$-0.485139\pi$
$83$	5858.19	0.0933401	0.0466700	+	0.998910i	$0.485139\pi$
$84$	−602.721	−0.00932004
$85$	0	0
$86$	−9640.20	−0.140553
$87$	−37288.5	−0.528174
$88$	−106971.	−1.47251
$89$	−42815.7	−0.572965	−0.286483	−	0.958085i	$-0.592486\pi$
$89$	−42815.7	−0.572965	−0.286483	+	0.958085i	$0.592486\pi$
$90$	0	0
$91$	−1195.50	−0.0151337
$92$	−1857.12	−0.0228755
$93$	−61657.0	−0.739222
$94$	−46881.7	−0.547247
$95$	0	0
$96$	−19475.0	−0.215675
$97$	231.671	0.00250002	0.00125001	−	0.999999i	$-0.499602\pi$
$97$	231.671	0.00250002	0.00125001	+	0.999999i	$0.499602\pi$
$98$	86984.4	0.914905
$99$	64657.8	0.663029
$100$	0	0
$101$	−65329.5	−0.637245	−0.318622	−	0.947882i	$-0.603220\pi$
$101$	−65329.5	−0.637245	−0.318622	+	0.947882i	$0.603220\pi$
$102$	19329.0	0.183954
$103$	−210856.	−1.95837	−0.979183	−	0.202980i	$-0.934937\pi$
$103$	−210856.	−1.95837	−0.979183	+	0.202980i	$0.934937\pi$
$104$	−20666.3	−0.187361
$105$	0	0
$106$	−53267.2	−0.460463
$107$	158342.	1.33702	0.668508	−	0.743705i	$-0.266933\pi$
$107$	158342.	1.33702	0.668508	+	0.743705i	$0.266933\pi$
$108$	19487.6	0.160768
$109$	−85822.7	−0.691888	−0.345944	−	0.938255i	$-0.612441\pi$
$109$	−85822.7	−0.691888	−0.345944	+	0.938255i	$0.612441\pi$
$110$	0	0
$111$	−54450.9	−0.419467
$112$	−9401.32	−0.0708180
$113$	−101903.	−0.750745	−0.375372	−	0.926874i	$-0.622485\pi$
$113$	−101903.	−0.750745	−0.375372	+	0.926874i	$0.622485\pi$
$114$	−172613.	−1.24398
$115$	0	0
$116$	15940.1	0.109988
$117$	12491.6	0.0843633
$118$	143153.	0.946448
$119$	−3653.30	−0.0236493
$120$	0	0
$121$	149503.	0.928299
$122$	191307.	1.16367
$123$	12002.5	0.0715335
$124$	26357.1	0.153937
$125$	0	0
$126$	6717.13	0.0376927
$127$	−109901.	−0.604635	−0.302318	−	0.953207i	$-0.597760\pi$
$127$	−109901.	−0.604635	−0.302318	+	0.953207i	$0.597760\pi$
$128$	−132930.	−0.717131
$129$	20835.1	0.110236
$130$	0	0
$131$	−79310.4	−0.403787	−0.201893	−	0.979408i	$-0.564709\pi$
$131$	−79310.4	−0.403787	−0.201893	+	0.979408i	$0.564709\pi$
$132$	30248.4	0.151101
$133$	32624.9	0.159926
$134$	−136976.	−0.658996
$135$	0	0
$136$	−63153.8	−0.292787
$137$	360169.	1.63948	0.819738	−	0.572738i	$-0.194119\pi$
$137$	360169.	1.63948	0.819738	+	0.572738i	$0.194119\pi$
$138$	−22650.3	−0.101246
$139$	330826.	1.45232	0.726160	−	0.687525i	$-0.241303\pi$
$139$	330826.	1.45232	0.726160	+	0.687525i	$0.241303\pi$
$140$	0	0
$141$	101324.	0.429205
$142$	−287435.	−1.19624
$143$	59997.8	0.245355
$144$	98233.1	0.394777
$145$	0	0
$146$	−48566.7	−0.188563
$147$	−187997.	−0.717560
$148$	23276.6	0.0873506
$149$	24179.7	0.0892249	0.0446125	−	0.999004i	$-0.485795\pi$
$149$	24179.7	0.0892249	0.0446125	+	0.999004i	$0.485795\pi$
$150$	0	0
$151$	278667.	0.994589	0.497295	−	0.867582i	$-0.334327\pi$
$151$	278667.	0.994589	0.497295	+	0.867582i	$0.334327\pi$
$152$	563979.	1.97995
$153$	38172.8	0.131833
$154$	32262.7	0.109622
$155$	0	0
$156$	5843.85	0.0192259
$157$	102228.	0.330995	0.165497	−	0.986210i	$-0.447077\pi$
$157$	102228.	0.330995	0.165497	+	0.986210i	$0.447077\pi$
$158$	−296279.	−0.944186
$159$	115125.	0.361141
$160$	0	0
$161$	4281.04	0.0130162
$162$	90682.9	0.271480
$163$	−444216.	−1.30956	−0.654780	−	0.755819i	$-0.727239\pi$
$163$	−444216.	−1.30956	−0.654780	+	0.755819i	$0.727239\pi$
$164$	−5130.83	−0.0148963
$165$	0	0
$166$	−30543.0	−0.0860285
$167$	467231.	1.29640	0.648202	−	0.761469i	$-0.275521\pi$
$167$	467231.	1.29640	0.648202	+	0.761469i	$0.275521\pi$
$168$	24018.1	0.0656547
$169$	−359702.	−0.968781
$170$	0	0
$171$	−340893.	−0.891514
$172$	−8906.58	−0.0229557
$173$	−296435.	−0.753033	−0.376516	−	0.926410i	$-0.622878\pi$
$173$	−296435.	−0.753033	−0.376516	+	0.926410i	$0.622878\pi$
$174$	194412.	0.486801
$175$	0	0
$176$	471818.	1.14814
$177$	−309394.	−0.742299
$178$	223230.	0.528083
$179$	211576.	0.493553	0.246777	−	0.969072i	$-0.420629\pi$
$179$	211576.	0.493553	0.246777	+	0.969072i	$0.420629\pi$
$180$	0	0
$181$	−187769.	−0.426018	−0.213009	−	0.977050i	$-0.568326\pi$
$181$	−187769.	−0.426018	−0.213009	+	0.977050i	$0.568326\pi$
$182$	6233.01	0.0139482
$183$	−413467.	−0.912669
$184$	74005.3	0.161146
$185$	0	0
$186$	321463.	0.681316
$187$	183346.	0.383413
$188$	−43314.0	−0.0893786
$189$	−44922.8	−0.0914771
$190$	0	0
$191$	−655959.	−1.30105	−0.650524	−	0.759486i	$-0.725450\pi$
$191$	−655959.	−1.30105	−0.650524	+	0.759486i	$0.725450\pi$
$192$	406829.	0.796451
$193$	−450394.	−0.870361	−0.435180	−	0.900343i	$-0.643315\pi$
$193$	−450394.	−0.870361	−0.435180	+	0.900343i	$0.643315\pi$
$194$	−1207.87	−0.00230418
$195$	0	0
$196$	80364.9	0.149426
$197$	973133.	1.78652	0.893258	−	0.449545i	$-0.148414\pi$
$197$	973133.	1.78652	0.893258	+	0.449545i	$0.148414\pi$
$198$	−337109.	−0.611092
$199$	259840.	0.465129	0.232564	−	0.972581i	$-0.425288\pi$
$199$	259840.	0.465129	0.232564	+	0.972581i	$0.425288\pi$
$200$	0	0
$201$	296043.	0.516850
$202$	340611.	0.587327
$203$	−36745.1	−0.0625834
$204$	17858.1	0.0300441
$205$	0	0
$206$	1.09935e6	1.80496
$207$	−44732.0	−0.0725591
$208$	91153.2	0.146088
$209$	−1.63733e6	−2.59280
$210$	0	0
$211$	−621161.	−0.960502	−0.480251	−	0.877131i	$-0.659454\pi$
$211$	−621161.	−0.960502	−0.480251	+	0.877131i	$0.659454\pi$
$212$	−49213.5	−0.0752047
$213$	621226.	0.938211
$214$	−825553.	−1.23228
$215$	0	0
$216$	−776570.	−1.13252
$217$	−60758.4	−0.0875905
$218$	447457.	0.637690
$219$	104966.	0.147890
$220$	0	0
$221$	35421.6	0.0487852
$222$	283893.	0.386609
$223$	779614.	1.04983	0.524913	−	0.851156i	$-0.324098\pi$
$223$	779614.	1.04983	0.524913	+	0.851156i	$0.324098\pi$
$224$	−19191.2	−0.0255553
$225$	0	0
$226$	531297.	0.691937
$227$	−795424.	−1.02455	−0.512276	−	0.858821i	$-0.671197\pi$
$227$	−795424.	−1.02455	−0.512276	+	0.858821i	$0.671197\pi$
$228$	−159477.	−0.203171
$229$	−702980.	−0.885838	−0.442919	−	0.896562i	$-0.646057\pi$
$229$	−702980.	−0.885838	−0.442919	+	0.896562i	$0.646057\pi$
$230$	0	0
$231$	−69728.6	−0.0859767
$232$	−635204.	−0.774807
$233$	219327.	0.264669	0.132334	−	0.991205i	$-0.457753\pi$
$233$	219327.	0.264669	0.132334	+	0.991205i	$0.457753\pi$
$234$	−65127.9	−0.0777548
$235$	0	0
$236$	132260.	0.154578
$237$	640339.	0.740524
$238$	19047.3	0.0217968
$239$	1.75148e6	1.98340	0.991701	−	0.128565i	$-0.0410372\pi$
$239$	1.75148e6	1.98340	0.991701	+	0.128565i	$0.0410372\pi$
$240$	0	0
$241$	985600.	1.09310	0.546548	−	0.837428i	$-0.315942\pi$
$241$	985600.	1.09310	0.546548	+	0.837428i	$0.315942\pi$
$242$	−779471.	−0.855582
$243$	787092.	0.855086
$244$	176749.	0.190056
$245$	0	0
$246$	−62578.0	−0.0659301
$247$	−316324.	−0.329906
$248$	−1.05032e6	−1.08440
$249$	66011.9	0.0674721
$250$	0	0
$251$	126142.	0.126379	0.0631894	−	0.998002i	$-0.479873\pi$
$251$	126142.	0.126379	0.0631894	+	0.998002i	$0.479873\pi$
$252$	6205.96	0.00615613
$253$	−214850.	−0.211025
$254$	572996.	0.557272
$255$	0	0
$256$	−462259.	−0.440845
$257$	−1.55933e6	−1.47267	−0.736335	−	0.676617i	$-0.763445\pi$
$257$	−1.55933e6	−1.47267	−0.736335	+	0.676617i	$0.763445\pi$
$258$	−108629.	−0.101600
$259$	−53657.4	−0.0497027
$260$	0	0
$261$	383944.	0.348873
$262$	413503.	0.372157
$263$	107289.	0.0956460	0.0478230	−	0.998856i	$-0.484772\pi$
$263$	107289.	0.0956460	0.0478230	+	0.998856i	$0.484772\pi$
$264$	−1.20538e6	−1.06442
$265$	0	0
$266$	−170098.	−0.147399
$267$	−482461.	−0.414175
$268$	−126552.	−0.107630
$269$	829468.	0.698906	0.349453	−	0.936954i	$-0.386367\pi$
$269$	829468.	0.698906	0.349453	+	0.936954i	$0.386367\pi$
$270$	0	0
$271$	2.22835e6	1.84315	0.921575	−	0.388200i	$-0.126903\pi$
$271$	2.22835e6	1.84315	0.921575	+	0.388200i	$0.126903\pi$
$272$	278553.	0.228289
$273$	−13471.3	−0.0109396
$274$	−1.87783e6	−1.51105
$275$	0	0
$276$	−20926.6	−0.0165358
$277$	1.57467e6	1.23308	0.616538	−	0.787325i	$-0.288535\pi$
$277$	1.57467e6	1.23308	0.616538	+	0.787325i	$0.288535\pi$
$278$	−1.72484e6	−1.33856
$279$	634856.	0.488275
$280$	0	0
$281$	−2.26025e6	−1.70762	−0.853809	−	0.520586i	$-0.825713\pi$
$281$	−2.26025e6	−1.70762	−0.853809	+	0.520586i	$0.825713\pi$
$282$	−528277.	−0.395584
$283$	1.81290e6	1.34558	0.672789	−	0.739835i	$-0.265096\pi$
$283$	1.81290e6	1.34558	0.672789	+	0.739835i	$0.265096\pi$
$284$	−265561.	−0.195375
$285$	0	0
$286$	−312812.	−0.226136
$287$	11827.6	0.00847602
$288$	200526.	0.142459
$289$	−1.31161e6	−0.923764
$290$	0	0
$291$	2610.55	0.00180717
$292$	−44870.7	−0.0307968
$293$	1.92999e6	1.31336	0.656682	−	0.754168i	$-0.271959\pi$
$293$	1.92999e6	1.31336	0.656682	+	0.754168i	$0.271959\pi$
$294$	980167.	0.661351
$295$	0	0
$296$	−927563.	−0.615338
$297$	2.25451e6	1.48307
$298$	−126067.	−0.0822356
$299$	−41508.0	−0.0268506
$300$	0	0
$301$	20531.5	0.0130618
$302$	−1.45290e6	−0.916680
$303$	−736154.	−0.460640
$304$	−2.48755e6	−1.54379
$305$	0	0
$306$	−199023.	−0.121507
$307$	2.35372e6	1.42531	0.712654	−	0.701515i	$-0.247493\pi$
$307$	2.35372e6	1.42531	0.712654	+	0.701515i	$0.247493\pi$
$308$	29807.5	0.0179040
$309$	−2.37600e6	−1.41563
$310$	0	0
$311$	1.25933e6	0.738312	0.369156	−	0.929367i	$-0.379647\pi$
$311$	1.25933e6	0.738312	0.369156	+	0.929367i	$0.379647\pi$
$312$	−232875.	−0.135436
$313$	1.21791e6	0.702676	0.351338	−	0.936249i	$-0.385727\pi$
$313$	1.21791e6	0.702676	0.351338	+	0.936249i	$0.385727\pi$
$314$	−532990.	−0.305067
$315$	0	0
$316$	−273732.	−0.154208
$317$	−689890.	−0.385595	−0.192798	−	0.981239i	$-0.561756\pi$
$317$	−689890.	−0.385595	−0.192798	+	0.981239i	$0.561756\pi$
$318$	−600231.	−0.332852
$319$	1.84410e6	1.01463
$320$	0	0
$321$	1.78425e6	0.966480
$322$	−22320.2	−0.0119966
$323$	−966649.	−0.515540
$324$	83782.0	0.0443392
$325$	0	0
$326$	2.31603e6	1.20698
$327$	−967077.	−0.500140
$328$	204461.	0.104936
$329$	99847.5	0.0508566
$330$	0	0
$331$	1.93331e6	0.969912	0.484956	−	0.874538i	$-0.338836\pi$
$331$	1.93331e6	0.969912	0.484956	+	0.874538i	$0.338836\pi$
$332$	−28218.7	−0.0140505
$333$	560658.	0.277069
$334$	−2.43602e6	−1.19485
$335$	0	0
$336$	−105937.	−0.0511917
$337$	−347418.	−0.166639	−0.0833196	−	0.996523i	$-0.526552\pi$
$337$	−347418.	−0.166639	−0.0833196	+	0.996523i	$0.526552\pi$
$338$	1.87539e6	0.892893
$339$	−1.14828e6	−0.542686
$340$	0	0
$341$	3.04924e6	1.42006
$342$	1.77733e6	0.821678
$343$	−371884.	−0.170676
$344$	354923.	0.161710
$345$	0	0
$346$	1.54553e6	0.694045
$347$	4.28700e6	1.91130	0.955652	−	0.294498i	$-0.0951526\pi$
$347$	4.28700e6	1.91130	0.955652	+	0.294498i	$0.0951526\pi$
$348$	179618.	0.0795062
$349$	2.58234e6	1.13488	0.567439	−	0.823415i	$-0.307934\pi$
$349$	2.58234e6	1.13488	0.567439	+	0.823415i	$0.307934\pi$
$350$	0	0
$351$	435562.	0.188704
$352$	963136.	0.414315
$353$	2.31876e6	0.990419	0.495210	−	0.868774i	$-0.335091\pi$
$353$	2.31876e6	0.990419	0.495210	+	0.868774i	$0.335091\pi$
$354$	1.61310e6	0.684152
$355$	0	0
$356$	206242.	0.0862486
$357$	−41166.5	−0.0170952
$358$	−1.10310e6	−0.454892
$359$	−1.41572e6	−0.579752	−0.289876	−	0.957064i	$-0.593614\pi$
$359$	−1.41572e6	−0.579752	−0.289876	+	0.957064i	$0.593614\pi$
$360$	0	0
$361$	6.15632e6	2.48630
$362$	978979.	0.392647
$363$	1.68465e6	0.671033
$364$	5758.68	0.00227808
$365$	0	0
$366$	2.15571e6	0.841177
$367$	−2.64620e6	−1.02555	−0.512776	−	0.858523i	$-0.671383\pi$
$367$	−2.64620e6	−1.02555	−0.512776	+	0.858523i	$0.671383\pi$
$368$	−326416.	−0.125647
$369$	−123585.	−0.0472498
$370$	0	0
$371$	113447.	0.0427916
$372$	297000.	0.111275
$373$	−3.49265e6	−1.29982	−0.649910	−	0.760011i	$-0.725194\pi$
$373$	−3.49265e6	−1.29982	−0.649910	+	0.760011i	$0.725194\pi$
$374$	−955917.	−0.353379
$375$	0	0
$376$	1.72604e6	0.629624
$377$	356273.	0.129101
$378$	234216.	0.0843114
$379$	1.42204e6	0.508526	0.254263	−	0.967135i	$-0.418167\pi$
$379$	1.42204e6	0.508526	0.254263	+	0.967135i	$0.418167\pi$
$380$	0	0
$381$	−1.23840e6	−0.437068
$382$	3.41999e6	1.19913
$383$	2.81442e6	0.980373	0.490187	−	0.871618i	$-0.336929\pi$
$383$	2.81442e6	0.980373	0.490187	+	0.871618i	$0.336929\pi$
$384$	−1.49790e6	−0.518387
$385$	0	0
$386$	2.34823e6	0.802182
$387$	−214530.	−0.0728134
$388$	−1115.95	−0.000376329 0
$389$	−165325.	−0.0553943	−0.0276971	−	0.999616i	$-0.508817\pi$
$389$	−165325.	−0.0553943	−0.0276971	+	0.999616i	$0.508817\pi$
$390$	0	0
$391$	−126844.	−0.0419591
$392$	−3.20250e6	−1.05263
$393$	−893695.	−0.291882
$394$	−5.07366e6	−1.64657
$395$	0	0
$396$	−311455.	−0.0998061
$397$	3.79445e6	1.20830	0.604148	−	0.796872i	$-0.293514\pi$
$397$	3.79445e6	1.20830	0.604148	+	0.796872i	$0.293514\pi$
$398$	−1.35474e6	−0.428694
$399$	367628.	0.115605
$400$	0	0
$401$	4.59396e6	1.42668	0.713339	−	0.700819i	$-0.247182\pi$
$401$	4.59396e6	1.42668	0.713339	+	0.700819i	$0.247182\pi$
$402$	−1.54349e6	−0.476363
$403$	589100.	0.180687
$404$	314690.	0.0959246
$405$	0	0
$406$	191579.	0.0576811
$407$	2.69287e6	0.805804
$408$	−711637.	−0.211645
$409$	3.83956e6	1.13494	0.567471	−	0.823393i	$-0.307922\pi$
$409$	3.83956e6	1.13494	0.567471	+	0.823393i	$0.307922\pi$
$410$	0	0
$411$	4.05850e6	1.18512
$412$	1.01569e6	0.294793
$413$	−304885.	−0.0879551
$414$	233221.	0.0668753
$415$	0	0
$416$	186074.	0.0527171
$417$	3.72785e6	1.04983
$418$	8.53658e6	2.38970
$419$	2.87922e6	0.801199	0.400600	−	0.916253i	$-0.368802\pi$
$419$	2.87922e6	0.801199	0.400600	+	0.916253i	$0.368802\pi$
$420$	0	0
$421$	1.41982e6	0.390417	0.195208	−	0.980762i	$-0.437462\pi$
$421$	1.41982e6	0.390417	0.195208	+	0.980762i	$0.437462\pi$
$422$	3.23857e6	0.885262
$423$	−1.04329e6	−0.283501
$424$	1.96114e6	0.529777
$425$	0	0
$426$	−3.23891e6	−0.864718
$427$	−407441.	−0.108142
$428$	−762729.	−0.201262
$429$	676074.	0.177358
$430$	0	0
$431$	−4.77968e6	−1.23938	−0.619691	−	0.784846i	$-0.712742\pi$
$431$	−4.77968e6	−1.23938	−0.619691	+	0.784846i	$0.712742\pi$
$432$	3.42523e6	0.883040
$433$	−213516.	−0.0547282	−0.0273641	−	0.999626i	$-0.508711\pi$
$433$	−213516.	−0.0547282	−0.0273641	+	0.999626i	$0.508711\pi$
$434$	316778.	0.0807293
$435$	0	0
$436$	413405.	0.104150
$437$	1.13275e6	0.283745
$438$	−547264.	−0.136305
$439$	555761.	0.137634	0.0688172	−	0.997629i	$-0.478077\pi$
$439$	555761.	0.137634	0.0688172	+	0.997629i	$0.478077\pi$
$440$	0	0
$441$	1.93573e6	0.473967
$442$	−184679.	−0.0449637
$443$	1.23542e6	0.299092	0.149546	−	0.988755i	$-0.452219\pi$
$443$	1.23542e6	0.299092	0.149546	+	0.988755i	$0.452219\pi$
$444$	262289.	0.0631425
$445$	0	0
$446$	−4.06470e6	−0.967591
$447$	272465.	0.0644974
$448$	400900.	0.0943715
$449$	1.61043e6	0.376986	0.188493	−	0.982074i	$-0.439640\pi$
$449$	1.61043e6	0.376986	0.188493	+	0.982074i	$0.439640\pi$
$450$	0	0
$451$	−593584.	−0.137417
$452$	490866.	0.113010
$453$	3.14011e6	0.718951
$454$	4.14713e6	0.944295
$455$	0	0
$456$	6.35510e6	1.43123
$457$	−4.44189e6	−0.994894	−0.497447	−	0.867494i	$-0.665729\pi$
$457$	−4.44189e6	−0.994894	−0.497447	+	0.867494i	$0.665729\pi$
$458$	3.66515e6	0.816447
$459$	1.33102e6	0.294886
$460$	0	0
$461$	−8.65627e6	−1.89705	−0.948524	−	0.316705i	$-0.897423\pi$
$461$	−8.65627e6	−1.89705	−0.948524	+	0.316705i	$0.897423\pi$
$462$	363546.	0.0792419
$463$	8.50880e6	1.84466	0.922329	−	0.386404i	$-0.126283\pi$
$463$	8.50880e6	1.84466	0.922329	+	0.386404i	$0.126283\pi$
$464$	2.80170e6	0.604126
$465$	0	0
$466$	−1.14351e6	−0.243936
$467$	692449.	0.146925	0.0734625	−	0.997298i	$-0.476595\pi$
$467$	692449.	0.146925	0.0734625	+	0.997298i	$0.476595\pi$
$468$	−60171.7	−0.0126992
$469$	291728.	0.0612416
$470$	0	0
$471$	1.15194e6	0.239264
$472$	−5.27048e6	−1.08892
$473$	−1.03040e6	−0.211764
$474$	−3.33856e6	−0.682517
$475$	0	0
$476$	17597.8	0.00355993
$477$	−1.18539e6	−0.238543
$478$	−9.13176e6	−1.82804
$479$	−7.63881e6	−1.52120	−0.760601	−	0.649220i	$-0.775095\pi$
$479$	−7.63881e6	−1.52120	−0.760601	+	0.649220i	$0.775095\pi$
$480$	0	0
$481$	520250.	0.102530
$482$	−5.13865e6	−1.00747
$483$	48240.1	0.00940893
$484$	−720154.	−0.139737
$485$	0	0
$486$	−4.10369e6	−0.788104
$487$	−4.82019e6	−0.920962	−0.460481	−	0.887670i	$-0.652323\pi$
$487$	−4.82019e6	−0.920962	−0.460481	+	0.887670i	$0.652323\pi$
$488$	−7.04335e6	−1.33884
$489$	−5.00557e6	−0.946633
$490$	0	0
$491$	66664.0	0.0124792	0.00623961	−	0.999981i	$-0.498014\pi$
$491$	66664.0	0.0124792	0.00623961	+	0.999981i	$0.498014\pi$
$492$	−57815.8	−0.0107680
$493$	1.08873e6	0.201744
$494$	1.64923e6	0.304063
$495$	0	0
$496$	4.63265e6	0.845522
$497$	612172.	0.111169
$498$	−344169.	−0.0621868
$499$	−2.93211e6	−0.527144	−0.263572	−	0.964640i	$-0.584901\pi$
$499$	−2.93211e6	−0.527144	−0.263572	+	0.964640i	$0.584901\pi$
$500$	0	0
$501$	5.26490e6	0.937122
$502$	−657669.	−0.116479
$503$	3.90093e6	0.687462	0.343731	−	0.939068i	$-0.388309\pi$
$503$	3.90093e6	0.687462	0.343731	+	0.939068i	$0.388309\pi$
$504$	−247305.	−0.0433666
$505$	0	0
$506$	1.12017e6	0.194495
$507$	−4.05323e6	−0.700296
$508$	529391.	0.0910159
$509$	2.09591e6	0.358573	0.179287	−	0.983797i	$-0.442621\pi$
$509$	2.09591e6	0.358573	0.179287	+	0.983797i	$0.442621\pi$
$510$	0	0
$511$	103436.	0.0175235
$512$	6.66386e6	1.12344
$513$	−1.18864e7	−1.99414
$514$	8.12994e6	1.35731
$515$	0	0
$516$	−100362.	−0.0165938
$517$	−5.01098e6	−0.824511
$518$	279755.	0.0458093
$519$	−3.34032e6	−0.544339
$520$	0	0
$521$	3.34502e6	0.539889	0.269944	−	0.962876i	$-0.412995\pi$
$521$	3.34502e6	0.539889	0.269944	+	0.962876i	$0.412995\pi$
$522$	−2.00178e6	−0.321544
$523$	1.04450e7	1.66976	0.834878	−	0.550434i	$-0.185538\pi$
$523$	1.04450e7	1.66976	0.834878	+	0.550434i	$0.185538\pi$
$524$	382036.	0.0607821
$525$	0	0
$526$	−559378.	−0.0881538
$527$	1.80022e6	0.282357
$528$	5.31660e6	0.829944
$529$	−6.28770e6	−0.976906
$530$	0	0
$531$	3.18570e6	0.490308
$532$	−157153.	−0.0240738
$533$	−114678.	−0.0174848
$534$	2.51542e6	0.381731
$535$	0	0
$536$	5.04304e6	0.758195
$537$	2.38411e6	0.356771
$538$	−4.32463e6	−0.644159
$539$	9.29739e6	1.37844
$540$	0	0
$541$	5.88573e6	0.864584	0.432292	−	0.901734i	$-0.357705\pi$
$541$	5.88573e6	0.864584	0.432292	+	0.901734i	$0.357705\pi$
$542$	−1.16180e7	−1.69877
$543$	−2.11584e6	−0.307952
$544$	568618.	0.0823803
$545$	0	0
$546$	70235.6	0.0100827
$547$	2.94922e6	0.421444	0.210722	−	0.977546i	$-0.432419\pi$
$547$	2.94922e6	0.421444	0.210722	+	0.977546i	$0.432419\pi$
$548$	−1.73492e6	−0.246791
$549$	4.25730e6	0.602842
$550$	0	0
$551$	−9.72260e6	−1.36428
$552$	833915.	0.116486
$553$	631007.	0.0877448
$554$	−8.20990e6	−1.13648
$555$	0	0
$556$	−1.59358e6	−0.218618
$557$	1.44836e7	1.97806	0.989029	−	0.147723i	$-0.0471945\pi$
$557$	1.44836e7	1.97806	0.989029	+	0.147723i	$0.0471945\pi$
$558$	−3.30997e6	−0.450027
$559$	−199069.	−0.0269447
$560$	0	0
$561$	2.06600e6	0.277155
$562$	1.17843e7	1.57385
$563$	−6.77740e6	−0.901139	−0.450570	−	0.892741i	$-0.648779\pi$
$563$	−6.77740e6	−0.901139	−0.450570	+	0.892741i	$0.648779\pi$
$564$	−488076.	−0.0646084
$565$	0	0
$566$	−9.45200e6	−1.24017
$567$	−193134.	−0.0252291
$568$	1.05825e7	1.37631
$569$	1.03288e7	1.33742	0.668710	−	0.743523i	$-0.266847\pi$
$569$	1.03288e7	1.33742	0.668710	+	0.743523i	$0.266847\pi$
$570$	0	0
$571$	−9.52868e6	−1.22304	−0.611522	−	0.791227i	$-0.709443\pi$
$571$	−9.52868e6	−1.22304	−0.611522	+	0.791227i	$0.709443\pi$
$572$	−289007.	−0.0369334
$573$	−7.39155e6	−0.940478
$574$	−61666.0	−0.00781206
$575$	0	0
$576$	−4.18895e6	−0.526076
$577$	−5.58573e6	−0.698458	−0.349229	−	0.937037i	$-0.613556\pi$
$577$	−5.58573e6	−0.698458	−0.349229	+	0.937037i	$0.613556\pi$
$578$	6.83840e6	0.851403
$579$	−5.07518e6	−0.629151
$580$	0	0
$581$	65049.9	0.00799478
$582$	−13610.7	−0.00166561
$583$	−5.69350e6	−0.693758
$584$	1.78808e6	0.216947
$585$	0	0
$586$	−1.00624e7	−1.21048
$587$	1.34089e7	1.60619	0.803096	−	0.595849i	$-0.203184\pi$
$587$	1.34089e7	1.60619	0.803096	+	0.595849i	$0.203184\pi$
$588$	905577.	0.108014
$589$	−1.60764e7	−1.90942
$590$	0	0
$591$	1.09656e7	1.29141
$592$	4.09121e6	0.479786
$593$	1.46278e7	1.70821	0.854106	−	0.520099i	$-0.174105\pi$
$593$	1.46278e7	1.70821	0.854106	+	0.520099i	$0.174105\pi$
$594$	−1.17544e7	−1.36690
$595$	0	0
$596$	−116473.	−0.0134311
$597$	2.92796e6	0.336224
$598$	216412.	0.0247473
$599$	1.02632e7	1.16873	0.584367	−	0.811489i	$-0.301343\pi$
$599$	1.02632e7	1.16873	0.584367	+	0.811489i	$0.301343\pi$
$600$	0	0
$601$	9.07707e6	1.02508	0.512542	−	0.858662i	$-0.328704\pi$
$601$	9.07707e6	1.02508	0.512542	+	0.858662i	$0.328704\pi$
$602$	−107046.	−0.0120386
$603$	−3.04823e6	−0.341393
$604$	−1.34233e6	−0.149716
$605$	0	0
$606$	3.83811e6	0.424557
$607$	1.20707e7	1.32972	0.664862	−	0.746966i	$-0.268490\pi$
$607$	1.20707e7	1.32972	0.664862	+	0.746966i	$0.268490\pi$
$608$	−5.07791e6	−0.557091
$609$	−414055.	−0.0452392
$610$	0	0
$611$	−968100.	−0.104910
$612$	−183877.	−0.0198449
$613$	−1.40958e6	−0.151509	−0.0757544	−	0.997127i	$-0.524136\pi$
$613$	−1.40958e6	−0.151509	−0.0757544	+	0.997127i	$0.524136\pi$
$614$	−1.22717e7	−1.31366
$615$	0	0
$616$	−1.18782e6	−0.126124
$617$	−1.51488e7	−1.60201	−0.801005	−	0.598658i	$-0.795701\pi$
$617$	−1.51488e7	−1.60201	−0.801005	+	0.598658i	$0.795701\pi$
$618$	1.23878e7	1.30474
$619$	1.58021e7	1.65763	0.828816	−	0.559521i	$-0.189015\pi$
$619$	1.58021e7	1.65763	0.828816	+	0.559521i	$0.189015\pi$
$620$	0	0
$621$	−1.55973e6	−0.162301
$622$	−6.56584e6	−0.680478
$623$	−475430.	−0.0490757
$624$	1.02714e6	0.105601
$625$	0	0
$626$	−6.34988e6	−0.647634
$627$	−1.84499e7	−1.87424
$628$	−492430.	−0.0498247
$629$	1.58982e6	0.160222
$630$	0	0
$631$	−1.08922e7	−1.08904	−0.544519	−	0.838749i	$-0.683288\pi$
$631$	−1.08922e7	−1.08904	−0.544519	+	0.838749i	$0.683288\pi$
$632$	1.09081e7	1.08631
$633$	−6.99944e6	−0.694311
$634$	3.59690e6	0.355390
$635$	0	0
$636$	−554554.	−0.0543627
$637$	1.79622e6	0.175392
$638$	−9.61466e6	−0.935152
$639$	−6.39650e6	−0.619712
$640$	0	0
$641$	2.41942e6	0.232577	0.116289	−	0.993215i	$-0.462900\pi$
$641$	2.41942e6	0.232577	0.116289	+	0.993215i	$0.462900\pi$
$642$	−9.30260e6	−0.890772
$643$	−1.38769e7	−1.32363	−0.661814	−	0.749668i	$-0.730213\pi$
$643$	−1.38769e7	−1.32363	−0.661814	+	0.749668i	$0.730213\pi$
$644$	−20621.6	−0.00195933
$645$	0	0
$646$	5.03985e6	0.475156
$647$	7.58249e6	0.712116	0.356058	−	0.934464i	$-0.384120\pi$
$647$	7.58249e6	0.712116	0.356058	+	0.934464i	$0.384120\pi$
$648$	−3.33867e6	−0.312346
$649$	1.53011e7	1.42597
$650$	0	0
$651$	−684645.	−0.0633159
$652$	2.13978e6	0.197129
$653$	−3.26919e6	−0.300025	−0.150012	−	0.988684i	$-0.547931\pi$
$653$	−3.26919e6	−0.300025	−0.150012	+	0.988684i	$0.547931\pi$
$654$	5.04208e6	0.460962
$655$	0	0
$656$	−901819.	−0.0818201
$657$	−1.08079e6	−0.0976850
$658$	−520578.	−0.0468728
$659$	1.94787e6	0.174721	0.0873606	−	0.996177i	$-0.472157\pi$
$659$	1.94787e6	0.174721	0.0873606	+	0.996177i	$0.472157\pi$
$660$	0	0
$661$	−1.04897e7	−0.933810	−0.466905	−	0.884307i	$-0.654631\pi$
$661$	−1.04897e7	−0.933810	−0.466905	+	0.884307i	$0.654631\pi$
$662$	−1.00798e7	−0.893936
$663$	399142.	0.0352650
$664$	1.12450e6	0.0989784
$665$	0	0
$666$	−2.92312e6	−0.255365
$667$	−1.27580e6	−0.111037
$668$	−2.25064e6	−0.195148
$669$	8.78494e6	0.758881
$670$	0	0
$671$	2.04480e7	1.75325
$672$	−216252.	−0.0184730
$673$	1.81198e6	0.154211	0.0771054	−	0.997023i	$-0.475432\pi$
$673$	1.81198e6	0.154211	0.0771054	+	0.997023i	$0.475432\pi$
$674$	1.81134e6	0.153586
$675$	0	0
$676$	1.73267e6	0.145831
$677$	1.71538e7	1.43843	0.719216	−	0.694786i	$-0.244501\pi$
$677$	1.71538e7	1.43843	0.719216	+	0.694786i	$0.244501\pi$
$678$	5.98682e6	0.500175
$679$	2572.50	0.000214132 0
$680$	0	0
$681$	−8.96308e6	−0.740610
$682$	−1.58979e7	−1.30882
$683$	−1.80610e7	−1.48146	−0.740732	−	0.671801i	$-0.765521\pi$
$683$	−1.80610e7	−1.48146	−0.740732	+	0.671801i	$0.765521\pi$
$684$	1.64207e6	0.134200
$685$	0	0
$686$	1.93890e6	0.157306
$687$	−7.92140e6	−0.640339
$688$	−1.56546e6	−0.126087
$689$	−1.09996e6	−0.0882732
$690$	0	0
$691$	−1.44368e7	−1.15021	−0.575105	−	0.818080i	$-0.695039\pi$
$691$	−1.44368e7	−1.15021	−0.575105	+	0.818080i	$0.695039\pi$
$692$	1.42792e6	0.113354
$693$	717966.	0.0567899
$694$	−2.23513e7	−1.76159
$695$	0	0
$696$	−7.15768e6	−0.560079
$697$	−350442.	−0.0273233
$698$	−1.34636e7	−1.04598
$699$	2.47145e6	0.191319
$700$	0	0
$701$	−1.39926e7	−1.07549	−0.537743	−	0.843109i	$-0.680723\pi$
$701$	−1.39926e7	−1.07549	−0.537743	+	0.843109i	$0.680723\pi$
$702$	−2.27090e6	−0.173923
$703$	−1.41975e7	−1.08349
$704$	−2.01197e7	−1.53000
$705$	0	0
$706$	−1.20894e7	−0.912836
$707$	−725425.	−0.0545813
$708$	1.49034e6	0.111738
$709$	−1.37678e7	−1.02860	−0.514302	−	0.857609i	$-0.671949\pi$
$709$	−1.37678e7	−1.02860	−0.514302	+	0.857609i	$0.671949\pi$
$710$	0	0
$711$	−6.59331e6	−0.489136
$712$	−8.21864e6	−0.607575
$713$	−2.10955e6	−0.155405
$714$	214631.	0.0157561
$715$	0	0
$716$	−1.01916e6	−0.0742947
$717$	1.97362e7	1.43373
$718$	7.38120e6	0.534338
$719$	−8.26505e6	−0.596243	−0.298121	−	0.954528i	$-0.596360\pi$
$719$	−8.26505e6	−0.596243	−0.298121	+	0.954528i	$0.596360\pi$
$720$	0	0
$721$	−2.34137e6	−0.167738
$722$	−3.20974e7	−2.29154
$723$	1.11060e7	0.790158
$724$	904479.	0.0641286
$725$	0	0
$726$	−8.78333e6	−0.618469
$727$	−7.18688e6	−0.504318	−0.252159	−	0.967686i	$-0.581141\pi$
$727$	−7.18688e6	−0.504318	−0.252159	+	0.967686i	$0.581141\pi$
$728$	−229481.	−0.0160479
$729$	1.30957e7	0.912663
$730$	0	0
$731$	−608330.	−0.0421062
$732$	1.99166e6	0.137384
$733$	1.10728e6	0.0761196	0.0380598	−	0.999275i	$-0.487882\pi$
$733$	1.10728e6	0.0761196	0.0380598	+	0.999275i	$0.487882\pi$
$734$	1.37966e7	0.945217
$735$	0	0
$736$	−666322.	−0.0453409
$737$	−1.46408e7	−0.992878
$738$	644339.	0.0435485
$739$	1.98449e7	1.33671	0.668356	−	0.743841i	$-0.266998\pi$
$739$	1.98449e7	1.33671	0.668356	+	0.743841i	$0.266998\pi$
$740$	0	0
$741$	−3.56444e6	−0.238477
$742$	−591484.	−0.0394396
$743$	2.19204e7	1.45672	0.728360	−	0.685194i	$-0.240283\pi$
$743$	2.19204e7	1.45672	0.728360	+	0.685194i	$0.240283\pi$
$744$	−1.18353e7	−0.783875
$745$	0	0
$746$	1.82098e7	1.19800
$747$	−679697.	−0.0445671
$748$	−883172.	−0.0577153
$749$	1.75825e6	0.114518
$750$	0	0
$751$	−1.62135e7	−1.04900	−0.524502	−	0.851409i	$-0.675748\pi$
$751$	−1.62135e7	−1.04900	−0.524502	+	0.851409i	$0.675748\pi$
$752$	−7.61307e6	−0.490925
$753$	1.42140e6	0.0913545
$754$	−1.85751e6	−0.118988
$755$	0	0
$756$	216392.	0.0137701
$757$	−1.31863e7	−0.836342	−0.418171	−	0.908368i	$-0.637329\pi$
$757$	−1.31863e7	−0.836342	−0.418171	+	0.908368i	$0.637329\pi$
$758$	−7.41412e6	−0.468691
$759$	−2.42099e6	−0.152542
$760$	0	0
$761$	−1.92620e7	−1.20570	−0.602851	−	0.797854i	$-0.705969\pi$
$761$	−1.92620e7	−1.20570	−0.602851	+	0.797854i	$0.705969\pi$
$762$	6.45670e6	0.402831
$763$	−952983.	−0.0592617
$764$	3.15973e6	0.195847
$765$	0	0
$766$	−1.46736e7	−0.903577
$767$	2.95610e6	0.181439
$768$	−5.20888e6	−0.318670
$769$	−456438.	−0.0278334	−0.0139167	−	0.999903i	$-0.504430\pi$
$769$	−456438.	−0.0278334	−0.0139167	+	0.999903i	$0.504430\pi$
$770$	0	0
$771$	−1.75710e7	−1.06454
$772$	2.16953e6	0.131016
$773$	6.14461e6	0.369867	0.184933	−	0.982751i	$-0.440793\pi$
$773$	6.14461e6	0.369867	0.184933	+	0.982751i	$0.440793\pi$
$774$	1.11850e6	0.0671097
$775$	0	0
$776$	44470.2	0.00265103
$777$	−604628.	−0.0359282
$778$	861961.	0.0510551
$779$	3.12953e6	0.184772
$780$	0	0
$781$	−3.07227e7	−1.80232
$782$	661328.	0.0386723
$783$	1.33875e7	0.780361
$784$	1.41253e7	0.820745
$785$	0	0
$786$	4.65949e6	0.269018
$787$	−8.50558e6	−0.489516	−0.244758	−	0.969584i	$-0.578709\pi$
$787$	−8.50558e6	−0.489516	−0.244758	+	0.969584i	$0.578709\pi$
$788$	−4.68755e6	−0.268925
$789$	1.20897e6	0.0691389
$790$	0	0
$791$	−1.13154e6	−0.0643029
$792$	1.24113e7	0.703080
$793$	3.95047e6	0.223082
$794$	−1.97833e7	−1.11365
$795$	0	0
$796$	−1.25164e6	−0.0700160
$797$	−1.28439e7	−0.716227	−0.358113	−	0.933678i	$-0.616580\pi$
$797$	−1.28439e7	−0.716227	−0.358113	+	0.933678i	$0.616580\pi$
$798$	−1.91671e6	−0.106549
$799$	−2.95840e6	−0.163942
$800$	0	0
$801$	4.96770e6	0.273573
$802$	−2.39517e7	−1.31492
$803$	−5.19108e6	−0.284099
$804$	−1.42603e6	−0.0778016
$805$	0	0
$806$	−3.07141e6	−0.166533
$807$	9.34670e6	0.505213
$808$	−1.25403e7	−0.675738
$809$	2.98801e7	1.60513	0.802567	−	0.596562i	$-0.203467\pi$
$809$	2.98801e7	1.60513	0.802567	+	0.596562i	$0.203467\pi$
$810$	0	0
$811$	−2.05001e7	−1.09447	−0.547234	−	0.836980i	$-0.684319\pi$
$811$	−2.05001e7	−1.09447	−0.547234	+	0.836980i	$0.684319\pi$
$812$	177000.	0.00942070
$813$	2.51098e7	1.33234
$814$	−1.40399e7	−0.742682
$815$	0	0
$816$	3.13882e6	0.165022
$817$	5.43254e6	0.284740
$818$	−2.00185e7	−1.04604
$819$	138708.	0.00722589
$820$	0	0
$821$	2.20698e7	1.14272	0.571360	−	0.820699i	$-0.306416\pi$
$821$	2.20698e7	1.14272	0.571360	+	0.820699i	$0.306416\pi$
$822$	−2.11599e7	−1.09228
$823$	−1.59817e6	−0.0822476	−0.0411238	−	0.999154i	$-0.513094\pi$
$823$	−1.59817e6	−0.0822476	−0.0411238	+	0.999154i	$0.513094\pi$
$824$	−4.04747e7	−2.07666
$825$	0	0
$826$	1.58959e6	0.0810653
$827$	1.49218e7	0.758680	0.379340	−	0.925257i	$-0.376151\pi$
$827$	1.49218e7	0.758680	0.379340	+	0.925257i	$0.376151\pi$
$828$	215473.	0.0109223
$829$	7.71986e6	0.390142	0.195071	−	0.980789i	$-0.437506\pi$
$829$	7.71986e6	0.390142	0.195071	+	0.980789i	$0.437506\pi$
$830$	0	0
$831$	1.77439e7	0.891344
$832$	−3.88704e6	−0.194675
$833$	5.48902e6	0.274083
$834$	−1.94360e7	−0.967592
$835$	0	0
$836$	7.88695e6	0.390295
$837$	2.21364e7	1.09218
$838$	−1.50115e7	−0.738439
$839$	3.50556e7	1.71930	0.859652	−	0.510880i	$-0.170680\pi$
$839$	3.50556e7	1.71930	0.859652	+	0.510880i	$0.170680\pi$
$840$	0	0
$841$	−9.56068e6	−0.466121
$842$	−7.40258e6	−0.359834
$843$	−2.54692e7	−1.23437
$844$	2.99211e6	0.144585
$845$	0	0
$846$	5.43945e6	0.261294
$847$	1.66010e6	0.0795108
$848$	−8.65001e6	−0.413073
$849$	2.04284e7	0.972668
$850$	0	0
$851$	−1.86300e6	−0.0881837
$852$	−2.99243e6	−0.141229
$853$	2.21072e7	1.04031	0.520153	−	0.854073i	$-0.325875\pi$
$853$	2.21072e7	1.04031	0.520153	+	0.854073i	$0.325875\pi$
$854$	2.12429e6	0.0996712
$855$	0	0
$856$	3.03944e7	1.41778
$857$	−1.60053e7	−0.744410	−0.372205	−	0.928151i	$-0.621398\pi$
$857$	−1.60053e7	−0.744410	−0.372205	+	0.928151i	$0.621398\pi$
$858$	−3.52487e6	−0.163465
$859$	−2.09057e7	−0.966677	−0.483338	−	0.875434i	$-0.660576\pi$
$859$	−2.09057e7	−0.966677	−0.483338	+	0.875434i	$0.660576\pi$
$860$	0	0
$861$	133277.	0.00612700
$862$	2.49200e7	1.14230
$863$	−1.41874e7	−0.648449	−0.324224	−	0.945980i	$-0.605103\pi$
$863$	−1.41874e7	−0.648449	−0.324224	+	0.945980i	$0.605103\pi$
$864$	6.99201e6	0.318653
$865$	0	0
$866$	1.11322e6	0.0504412
$867$	−1.47797e7	−0.667755
$868$	292671.	0.0131850
$869$	−3.16680e7	−1.42256
$870$	0	0
$871$	−2.82854e6	−0.126333
$872$	−1.64740e7	−0.733682
$873$	−26879.7	−0.00119368
$874$	−5.90583e6	−0.261519
$875$	0	0
$876$	−505618.	−0.0222619
$877$	−2.06566e7	−0.906901	−0.453450	−	0.891282i	$-0.649807\pi$
$877$	−2.06566e7	−0.906901	−0.453450	+	0.891282i	$0.649807\pi$
$878$	−2.89759e6	−0.126853
$879$	2.17477e7	0.949382
$880$	0	0
$881$	−1.35902e7	−0.589909	−0.294955	−	0.955511i	$-0.595305\pi$
$881$	−1.35902e7	−0.589909	−0.294955	+	0.955511i	$0.595305\pi$
$882$	−1.00924e7	−0.436839
$883$	−1.20900e7	−0.521824	−0.260912	−	0.965363i	$-0.584023\pi$
$883$	−1.20900e7	−0.521824	−0.260912	+	0.965363i	$0.584023\pi$
$884$	−170625.	−0.00734365
$885$	0	0
$886$	−6.44115e6	−0.275663
$887$	−1.83559e7	−0.783371	−0.391685	−	0.920099i	$-0.628108\pi$
$887$	−1.83559e7	−0.783371	−0.391685	+	0.920099i	$0.628108\pi$
$888$	−1.04521e7	−0.444805
$889$	−1.22035e6	−0.0517883
$890$	0	0
$891$	9.69271e6	0.409026
$892$	−3.75538e6	−0.158031
$893$	2.64192e7	1.10864
$894$	−1.42056e6	−0.0594451
$895$	0	0
$896$	−1.47607e6	−0.0614238
$897$	−467725.	−0.0194093
$898$	−8.39635e6	−0.347456
$899$	1.81067e7	0.747206
$900$	0	0
$901$	−3.36134e6	−0.137943
$902$	3.09479e6	0.126653
$903$	231355.	0.00944190
$904$	−1.95608e7	−0.796094
$905$	0	0
$906$	−1.63717e7	−0.662634
$907$	−1.78761e7	−0.721531	−0.360765	−	0.932657i	$-0.617485\pi$
$907$	−1.78761e7	−0.721531	−0.360765	+	0.932657i	$0.617485\pi$
$908$	3.83153e6	0.154226
$909$	7.57987e6	0.304265
$910$	0	0
$911$	−1.47994e7	−0.590809	−0.295405	−	0.955372i	$-0.595454\pi$
$911$	−1.47994e7	−0.590809	−0.295405	+	0.955372i	$0.595454\pi$
$912$	−2.80305e7	−1.11595
$913$	−3.26462e6	−0.129615
$914$	2.31588e7	0.916961
$915$	0	0
$916$	3.38623e6	0.133345
$917$	−880670.	−0.0345852
$918$	−6.93961e6	−0.271787
$919$	−2.50852e6	−0.0979780	−0.0489890	−	0.998799i	$-0.515600\pi$
$919$	−2.50852e6	−0.0979780	−0.0489890	+	0.998799i	$0.515600\pi$
$920$	0	0
$921$	2.65225e7	1.03030
$922$	4.51315e7	1.74845
$923$	−5.93549e6	−0.229325
$924$	335881.	0.0129421
$925$	0	0
$926$	−4.43626e7	−1.70016
$927$	2.44646e7	0.935060
$928$	5.71919e6	0.218004
$929$	−9.51733e6	−0.361806	−0.180903	−	0.983501i	$-0.557902\pi$
$929$	−9.51733e6	−0.361806	−0.180903	+	0.983501i	$0.557902\pi$
$930$	0	0
$931$	−4.90183e7	−1.85347
$932$	−1.05649e6	−0.0398407
$933$	1.41906e7	0.533698
$934$	−3.61025e6	−0.135416
$935$	0	0
$936$	2.39781e6	0.0894593
$937$	−4.27838e6	−0.159195	−0.0795977	−	0.996827i	$-0.525364\pi$
$937$	−4.27838e6	−0.159195	−0.0795977	+	0.996827i	$0.525364\pi$
$938$	−1.52099e6	−0.0564444
$939$	1.37238e7	0.507939
$940$	0	0
$941$	−3.20489e7	−1.17988	−0.589941	−	0.807446i	$-0.700849\pi$
$941$	−3.20489e7	−1.17988	−0.589941	+	0.807446i	$0.700849\pi$
$942$	−6.00590e6	−0.220521
$943$	410657.	0.0150384
$944$	2.32466e7	0.849041
$945$	0	0
$946$	5.37223e6	0.195176
$947$	−1.34081e7	−0.485841	−0.242920	−	0.970046i	$-0.578105\pi$
$947$	−1.34081e7	−0.485841	−0.242920	+	0.970046i	$0.578105\pi$
$948$	−3.08450e6	−0.111471
$949$	−1.00289e6	−0.0361485
$950$	0	0
$951$	−7.77389e6	−0.278732
$952$	−701265.	−0.0250778
$953$	−4.23589e7	−1.51082	−0.755409	−	0.655254i	$-0.772562\pi$
$953$	−4.23589e7	−1.51082	−0.755409	+	0.655254i	$0.772562\pi$
$954$	6.18033e6	0.219857
$955$	0	0
$956$	−8.43683e6	−0.298562
$957$	2.07799e7	0.733440
$958$	3.98267e7	1.40204
$959$	3.99935e6	0.140425
$960$	0	0
$961$	1.31048e6	0.0457745
$962$	−2.71245e6	−0.0944982
$963$	−1.83717e7	−0.638385
$964$	−4.74760e6	−0.164544
$965$	0	0
$966$	−251511.	−0.00867190
$967$	4.12202e7	1.41757	0.708785	−	0.705425i	$-0.249244\pi$
$967$	4.12202e7	1.41757	0.708785	+	0.705425i	$0.249244\pi$
$968$	2.86978e7	0.984373
$969$	−1.08925e7	−0.372665
$970$	0	0
$971$	−1.13026e7	−0.384708	−0.192354	−	0.981326i	$-0.561612\pi$
$971$	−1.13026e7	−0.384708	−0.192354	+	0.981326i	$0.561612\pi$
$972$	−3.79140e6	−0.128716
$973$	3.67352e6	0.124394
$974$	2.51312e7	0.848820
$975$	0	0
$976$	3.10662e7	1.04391
$977$	2.79073e7	0.935367	0.467684	−	0.883896i	$-0.345089\pi$
$977$	2.79073e7	0.935367	0.467684	+	0.883896i	$0.345089\pi$
$978$	2.60977e7	0.872480
$979$	2.38601e7	0.795638
$980$	0	0
$981$	9.95758e6	0.330355
$982$	−347568.	−0.0115017
$983$	1.22161e7	0.403226	0.201613	−	0.979465i	$-0.435382\pi$
$983$	1.22161e7	0.403226	0.201613	+	0.979465i	$0.435382\pi$
$984$	2.30393e6	0.0758546
$985$	0	0
$986$	−5.67633e6	−0.185941
$987$	1.12511e6	0.0367623
$988$	1.52372e6	0.0496608
$989$	712858.	0.0231746
$990$	0	0
$991$	2.47723e7	0.801277	0.400638	−	0.916236i	$-0.368788\pi$
$991$	2.47723e7	0.801277	0.400638	+	0.916236i	$0.368788\pi$
$992$	9.45675e6	0.305114
$993$	2.17852e7	0.701113
$994$	−3.19170e6	−0.102461
$995$	0	0
$996$	−317978.	−0.0101566
$997$	−2.12381e7	−0.676670	−0.338335	−	0.941026i	$-0.609864\pi$
$997$	−2.12381e7	−0.676670	−0.338335	+	0.941026i	$0.609864\pi$
$998$	1.52873e7	0.485851
$999$	1.95493e7	0.619750

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1075.6.a.a.1.2		8
5.4	even	2		43.6.a.a.1.7	✓	8
15.14	odd	2		387.6.a.c.1.2		8
20.19	odd	2		688.6.a.e.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.a.a.1.7	✓	8	5.4	even	2
387.6.a.c.1.2		8	15.14	odd	2
688.6.a.e.1.6		8	20.19	odd	2
1075.6.a.a.1.2		8	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)

\(f(q)\)	\(=\)	\(q-5.21373 q^{2} +11.2683 q^{3} -4.81697 q^{4} -58.7500 q^{6} +11.1041 q^{7} +191.954 q^{8} -116.025 q^{9} +O(q^{10})\)	\(q-5.21373 q^{2} +11.2683 q^{3} -4.81697 q^{4} -58.7500 q^{6} +11.1041 q^{7} +191.954 q^{8} -116.025 q^{9} -557.274 q^{11} -54.2791 q^{12} -107.663 q^{13} -57.8938 q^{14} -846.654 q^{16} -329.005 q^{17} +604.924 q^{18} +2938.10 q^{19} +125.124 q^{21} +2905.48 q^{22} +385.537 q^{23} +2163.00 q^{24} +561.326 q^{26} -4045.61 q^{27} -53.4881 q^{28} -3309.15 q^{29} -5471.71 q^{31} -1728.30 q^{32} -6279.54 q^{33} +1715.34 q^{34} +558.890 q^{36} -4832.22 q^{37} -15318.5 q^{38} -1213.18 q^{39} +1065.16 q^{41} -652.365 q^{42} +1849.00 q^{43} +2684.37 q^{44} -2010.09 q^{46} +8991.95 q^{47} -9540.36 q^{48} -16683.7 q^{49} -3707.33 q^{51} +518.609 q^{52} +10216.7 q^{53} +21092.7 q^{54} +2131.47 q^{56} +33107.4 q^{57} +17253.0 q^{58} -27457.0 q^{59} -36692.9 q^{61} +28528.1 q^{62} -1288.35 q^{63} +36103.8 q^{64} +32739.9 q^{66} +26272.2 q^{67} +1584.81 q^{68} +4344.35 q^{69} +55130.3 q^{71} -22271.5 q^{72} +9315.14 q^{73} +25193.9 q^{74} -14152.7 q^{76} -6188.02 q^{77} +6325.20 q^{78} +56826.5 q^{79} -17393.1 q^{81} -5553.44 q^{82} +5858.19 q^{83} -602.721 q^{84} -9640.20 q^{86} -37288.5 q^{87} -106971. q^{88} -42815.7 q^{89} -1195.50 q^{91} -1857.12 q^{92} -61657.0 q^{93} -46881.7 q^{94} -19475.0 q^{96} +231.671 q^{97} +86984.4 q^{98} +64657.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{2} + 26 q^{3} + 122 q^{4} - 69 q^{6} + 136 q^{7} + 666 q^{8} + 546 q^{9}+O(q^{10}) \) 8 * q + 12 * q^2 + 26 * q^3 + 122 * q^4 - 69 * q^6 + 136 * q^7 + 666 * q^8 + 546 * q^9	\( 8 q + 12 q^{2} + 26 q^{3} + 122 q^{4} - 69 q^{6} + 136 q^{7} + 666 q^{8} + 546 q^{9} - 532 q^{11} + 4195 q^{12} + 2492 q^{13} - 4240 q^{14} + 1882 q^{16} + 2534 q^{17} + 3711 q^{18} - 1678 q^{19} - 2256 q^{21} - 11502 q^{22} + 2488 q^{23} + 19953 q^{24} + 4586 q^{26} + 8960 q^{27} - 18640 q^{28} - 4360 q^{29} + 5704 q^{31} + 18294 q^{32} + 12852 q^{33} + 30007 q^{34} + 67969 q^{36} + 3772 q^{37} + 6559 q^{38} + 11120 q^{39} - 10698 q^{41} - 78698 q^{42} + 14792 q^{43} - 356 q^{44} - 19389 q^{46} + 77864 q^{47} + 118727 q^{48} + 7188 q^{49} - 80246 q^{51} + 60736 q^{52} + 62352 q^{53} + 61026 q^{54} - 144528 q^{56} + 808 q^{57} - 52951 q^{58} - 26224 q^{59} - 82540 q^{61} + 9023 q^{62} + 61768 q^{63} + 153858 q^{64} - 48516 q^{66} - 27784 q^{67} - 40507 q^{68} - 93776 q^{69} - 9504 q^{71} + 186687 q^{72} - 14260 q^{73} - 15239 q^{74} + 1279 q^{76} + 218140 q^{77} - 264170 q^{78} + 160248 q^{79} + 161076 q^{81} + 47781 q^{82} + 77176 q^{83} + 16382 q^{84} + 22188 q^{86} - 268136 q^{87} - 129544 q^{88} - 265692 q^{89} + 401148 q^{91} - 190391 q^{92} + 123860 q^{93} + 248737 q^{94} + 950817 q^{96} - 144742 q^{97} + 292244 q^{98} + 239516 q^{99}+O(q^{100}) \) 8 * q + 12 * q^2 + 26 * q^3 + 122 * q^4 - 69 * q^6 + 136 * q^7 + 666 * q^8 + 546 * q^9 - 532 * q^11 + 4195 * q^12 + 2492 * q^13 - 4240 * q^14 + 1882 * q^16 + 2534 * q^17 + 3711 * q^18 - 1678 * q^19 - 2256 * q^21 - 11502 * q^22 + 2488 * q^23 + 19953 * q^24 + 4586 * q^26 + 8960 * q^27 - 18640 * q^28 - 4360 * q^29 + 5704 * q^31 + 18294 * q^32 + 12852 * q^33 + 30007 * q^34 + 67969 * q^36 + 3772 * q^37 + 6559 * q^38 + 11120 * q^39 - 10698 * q^41 - 78698 * q^42 + 14792 * q^43 - 356 * q^44 - 19389 * q^46 + 77864 * q^47 + 118727 * q^48 + 7188 * q^49 - 80246 * q^51 + 60736 * q^52 + 62352 * q^53 + 61026 * q^54 - 144528 * q^56 + 808 * q^57 - 52951 * q^58 - 26224 * q^59 - 82540 * q^61 + 9023 * q^62 + 61768 * q^63 + 153858 * q^64 - 48516 * q^66 - 27784 * q^67 - 40507 * q^68 - 93776 * q^69 - 9504 * q^71 + 186687 * q^72 - 14260 * q^73 - 15239 * q^74 + 1279 * q^76 + 218140 * q^77 - 264170 * q^78 + 160248 * q^79 + 161076 * q^81 + 47781 * q^82 + 77176 * q^83 + 16382 * q^84 + 22188 * q^86 - 268136 * q^87 - 129544 * q^88 - 265692 * q^89 + 401148 * q^91 - 190391 * q^92 + 123860 * q^93 + 248737 * q^94 + 950817 * q^96 - 144742 * q^97 + 292244 * q^98 + 239516 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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