LMFDB - Embedded newform 1075.6.a.b.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:	$1$
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			1.2
Root			$-8.57770$ 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)

$f(q)$	$=$	$q-9.57770 q^{2} +7.84343 q^{3} +59.7324 q^{4} -75.1221 q^{6} -195.604 q^{7} -265.613 q^{8} -181.481 q^{9} +O(q^{10})$	\(q-9.57770 q^{2} +7.84343 q^{3} +59.7324 q^{4} -75.1221 q^{6} -195.604 q^{7} -265.613 q^{8} -181.481 q^{9} +72.8476 q^{11} +468.507 q^{12} -301.666 q^{13} +1873.44 q^{14} +632.525 q^{16} +1207.20 q^{17} +1738.17 q^{18} -2350.21 q^{19} -1534.21 q^{21} -697.713 q^{22} -516.070 q^{23} -2083.32 q^{24} +2889.27 q^{26} -3329.38 q^{27} -11683.9 q^{28} +1531.55 q^{29} +1126.13 q^{31} +2441.48 q^{32} +571.375 q^{33} -11562.2 q^{34} -10840.3 q^{36} +9339.18 q^{37} +22509.6 q^{38} -2366.10 q^{39} +19704.2 q^{41} +14694.2 q^{42} -1849.00 q^{43} +4351.37 q^{44} +4942.77 q^{46} +13797.6 q^{47} +4961.16 q^{48} +21453.9 q^{49} +9468.56 q^{51} -18019.2 q^{52} +3351.03 q^{53} +31887.9 q^{54} +51954.9 q^{56} -18433.7 q^{57} -14668.7 q^{58} +2511.18 q^{59} +49249.3 q^{61} -10785.7 q^{62} +35498.3 q^{63} -43624.6 q^{64} -5472.46 q^{66} -9116.54 q^{67} +72108.8 q^{68} -4047.76 q^{69} +43397.3 q^{71} +48203.6 q^{72} -80067.4 q^{73} -89447.9 q^{74} -140384. q^{76} -14249.3 q^{77} +22661.8 q^{78} -65991.7 q^{79} +17986.0 q^{81} -188721. q^{82} +76880.9 q^{83} -91641.8 q^{84} +17709.2 q^{86} +12012.6 q^{87} -19349.3 q^{88} -75722.6 q^{89} +59007.1 q^{91} -30826.1 q^{92} +8832.73 q^{93} -132150. q^{94} +19149.6 q^{96} +67921.7 q^{97} -205479. q^{98} -13220.4 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−9.57770	−1.69311	−0.846557	−	0.532297i	$-0.821329\pi$
$2$	−9.57770	−1.69311	−0.846557	+	0.532297i	$0.821329\pi$
$3$	7.84343	0.503156	0.251578	−	0.967837i	$-0.419050\pi$
$3$	7.84343	0.503156	0.251578	+	0.967837i	$0.419050\pi$
$4$	59.7324	1.86664
$5$	0	0
$5$	0	0
$6$	−75.1221	−0.851901
$7$	−195.604	−1.50880	−0.754401	−	0.656413i	$-0.772073\pi$
$7$	−195.604	−1.50880	−0.754401	+	0.656413i	$0.772073\pi$
$8$	−265.613	−1.46732
$9$	−181.481	−0.746834
$10$	0	0
$11$	72.8476	0.181524	0.0907619	−	0.995873i	$-0.471070\pi$
$11$	72.8476	0.181524	0.0907619	+	0.995873i	$0.471070\pi$
$12$	468.507	0.939211
$13$	−301.666	−0.495072	−0.247536	−	0.968879i	$-0.579621\pi$
$13$	−301.666	−0.495072	−0.247536	+	0.968879i	$0.579621\pi$
$14$	1873.44	2.55458
$15$	0	0
$16$	632.525	0.617700
$17$	1207.20	1.01311	0.506554	−	0.862208i	$-0.330919\pi$
$17$	1207.20	1.01311	0.506554	+	0.862208i	$0.330919\pi$
$18$	1738.17	1.26448
$19$	−2350.21	−1.49356	−0.746780	−	0.665071i	$-0.768401\pi$
$19$	−2350.21	−1.49356	−0.746780	+	0.665071i	$0.768401\pi$
$20$	0	0
$21$	−1534.21	−0.759164
$22$	−697.713	−0.307341
$23$	−516.070	−0.203418	−0.101709	−	0.994814i	$-0.532431\pi$
$23$	−516.070	−0.203418	−0.101709	+	0.994814i	$0.532431\pi$
$24$	−2083.32	−0.738290
$25$	0	0
$26$	2889.27	0.838213
$27$	−3329.38	−0.878930
$28$	−11683.9	−2.81639
$29$	1531.55	0.338171	0.169085	−	0.985601i	$-0.445919\pi$
$29$	1531.55	0.338171	0.169085	+	0.985601i	$0.445919\pi$
$30$	0	0
$31$	1126.13	0.210467	0.105234	−	0.994448i	$-0.466441\pi$
$31$	1126.13	0.210467	0.105234	+	0.994448i	$0.466441\pi$
$32$	2441.48	0.421481
$33$	571.375	0.0913349
$34$	−11562.2	−1.71531
$35$	0	0
$36$	−10840.3	−1.39407
$37$	9339.18	1.12151	0.560756	−	0.827981i	$-0.310510\pi$
$37$	9339.18	1.12151	0.560756	+	0.827981i	$0.310510\pi$
$38$	22509.6	2.52877
$39$	−2366.10	−0.249098
$40$	0	0
$41$	19704.2	1.83062	0.915310	−	0.402750i	$-0.131946\pi$
$41$	19704.2	1.83062	0.915310	+	0.402750i	$0.131946\pi$
$42$	14694.2	1.28535
$43$	−1849.00	−0.152499
$43$	−1849.00	−0.152499
$44$	4351.37	0.338839
$45$	0	0
$46$	4942.77	0.344410
$47$	13797.6	0.911088	0.455544	−	0.890213i	$-0.349445\pi$
$47$	13797.6	0.911088	0.455544	+	0.890213i	$0.349445\pi$
$48$	4961.16	0.310800
$49$	21453.9	1.27649
$50$	0	0
$51$	9468.56	0.509752
$52$	−18019.2	−0.924120
$53$	3351.03	0.163866	0.0819330	−	0.996638i	$-0.473891\pi$
$53$	3351.03	0.163866	0.0819330	+	0.996638i	$0.473891\pi$
$54$	31887.9	1.48813
$55$	0	0
$56$	51954.9	2.21389
$57$	−18433.7	−0.751494
$58$	−14668.7	−0.572562
$59$	2511.18	0.0939177	0.0469588	−	0.998897i	$-0.485047\pi$
$59$	2511.18	0.0939177	0.0469588	+	0.998897i	$0.485047\pi$
$60$	0	0
$61$	49249.3	1.69463	0.847316	−	0.531088i	$-0.178217\pi$
$61$	49249.3	1.69463	0.847316	+	0.531088i	$0.178217\pi$
$62$	−10785.7	−0.356345
$63$	35498.3	1.12682
$64$	−43624.6	−1.33132
$65$	0	0
$66$	−5472.46	−0.154640
$67$	−9116.54	−0.248109	−0.124055	−	0.992275i	$-0.539590\pi$
$67$	−9116.54	−0.248109	−0.124055	+	0.992275i	$0.539590\pi$
$68$	72108.8	1.89111
$69$	−4047.76	−0.102351
$70$	0	0
$71$	43397.3	1.02168	0.510842	−	0.859675i	$-0.329334\pi$
$71$	43397.3	1.02168	0.510842	+	0.859675i	$0.329334\pi$
$72$	48203.6	1.09584
$73$	−80067.4	−1.75852	−0.879262	−	0.476338i	$-0.841964\pi$
$73$	−80067.4	−1.75852	−0.879262	+	0.476338i	$0.841964\pi$
$74$	−89447.9	−1.89885
$75$	0	0
$76$	−140384.	−2.78794
$77$	−14249.3	−0.273884
$78$	22661.8	0.421752
$79$	−65991.7	−1.18966	−0.594828	−	0.803853i	$-0.702780\pi$
$79$	−65991.7	−1.18966	−0.594828	+	0.803853i	$0.702780\pi$
$80$	0	0
$81$	17986.0	0.304594
$82$	−188721.	−3.09945
$83$	76880.9	1.22496	0.612482	−	0.790484i	$-0.290171\pi$
$83$	76880.9	1.22496	0.612482	+	0.790484i	$0.290171\pi$
$84$	−91641.8	−1.41708
$85$	0	0
$86$	17709.2	0.258198
$87$	12012.6	0.170153
$88$	−19349.3	−0.266353
$89$	−75722.6	−1.01333	−0.506665	−	0.862143i	$-0.669122\pi$
$89$	−75722.6	−1.01333	−0.506665	+	0.862143i	$0.669122\pi$
$90$	0	0
$91$	59007.1	0.746965
$92$	−30826.1	−0.379708
$93$	8832.73	0.105898
$94$	−132150.	−1.54258
$95$	0	0
$96$	19149.6	0.212071
$97$	67921.7	0.732958	0.366479	−	0.930426i	$-0.380563\pi$
$97$	67921.7	0.732958	0.366479	+	0.930426i	$0.380563\pi$
$98$	−205479.	−2.16124
$99$	−13220.4	−0.135568
$100$	0	0
$101$	130824.	1.27610	0.638051	−	0.769994i	$-0.279741\pi$
$101$	130824.	1.27610	0.638051	+	0.769994i	$0.279741\pi$
$102$	−90687.1	−0.863068
$103$	−41945.9	−0.389580	−0.194790	−	0.980845i	$-0.562403\pi$
$103$	−41945.9	−0.389580	−0.194790	+	0.980845i	$0.562403\pi$
$104$	80126.4	0.726428
$105$	0	0
$106$	−32095.2	−0.277444
$107$	5277.45	0.0445620	0.0222810	−	0.999752i	$-0.492907\pi$
$107$	5277.45	0.0445620	0.0222810	+	0.999752i	$0.492907\pi$
$108$	−198872.	−1.64064
$109$	−138680.	−1.11802	−0.559008	−	0.829163i	$-0.688818\pi$
$109$	−138680.	−1.11802	−0.559008	+	0.829163i	$0.688818\pi$
$110$	0	0
$111$	73251.2	0.564296
$112$	−123724.	−0.931988
$113$	−251325.	−1.85157	−0.925783	−	0.378055i	$-0.876593\pi$
$113$	−251325.	−1.85157	−0.925783	+	0.378055i	$0.876593\pi$
$114$	176553.	1.27237
$115$	0	0
$116$	91483.1	0.631242
$117$	54746.5	0.369736
$118$	−24051.3	−0.159013
$119$	−236132.	−1.52858
$120$	0	0
$121$	−155744.	−0.967049
$122$	−471695.	−2.86921
$123$	154548.	0.921088
$124$	67266.5	0.392866
$125$	0	0
$126$	−339992.	−1.90784
$127$	175695.	0.966606	0.483303	−	0.875453i	$-0.339437\pi$
$127$	175695.	0.966606	0.483303	+	0.875453i	$0.339437\pi$
$128$	339696.	1.83259
$129$	−14502.5	−0.0767306
$130$	0	0
$131$	213758.	1.08829	0.544144	−	0.838992i	$-0.316854\pi$
$131$	213758.	1.08829	0.544144	+	0.838992i	$0.316854\pi$
$132$	34129.6	0.170489
$133$	459711.	2.25349
$134$	87315.5	0.420077
$135$	0	0
$136$	−320647.	−1.48655
$137$	−126026.	−0.573664	−0.286832	−	0.957981i	$-0.592602\pi$
$137$	−126026.	−0.573664	−0.286832	+	0.957981i	$0.592602\pi$
$138$	38768.2	0.173292
$139$	181105.	0.795049	0.397524	−	0.917592i	$-0.369869\pi$
$139$	181105.	0.795049	0.397524	+	0.917592i	$0.369869\pi$
$140$	0	0
$141$	108221.	0.458419
$142$	−415646.	−1.72983
$143$	−21975.7	−0.0898673
$144$	−114791.	−0.461319
$145$	0	0
$146$	766862.	2.97738
$147$	168272.	0.642272
$148$	557852.	2.09346
$149$	187084.	0.690354	0.345177	−	0.938538i	$-0.387819\pi$
$149$	187084.	0.690354	0.345177	+	0.938538i	$0.387819\pi$
$150$	0	0
$151$	−396158.	−1.41392	−0.706962	−	0.707252i	$-0.749935\pi$
$151$	−396158.	−1.41392	−0.706962	+	0.707252i	$0.749935\pi$
$152$	624247.	2.19153
$153$	−219083.	−0.756623
$154$	136475.	0.463717
$155$	0	0
$156$	−141333.	−0.464977
$157$	−504286.	−1.63278	−0.816390	−	0.577501i	$-0.804028\pi$
$157$	−504286.	−1.63278	−0.816390	+	0.577501i	$0.804028\pi$
$158$	632049.	2.01423
$159$	26283.6	0.0824502
$160$	0	0
$161$	100945.	0.306917
$162$	−172265.	−0.515713
$163$	466039.	1.37389	0.686946	−	0.726708i	$-0.258951\pi$
$163$	466039.	1.37389	0.686946	+	0.726708i	$0.258951\pi$
$164$	1.17698e6	3.41711
$165$	0	0
$166$	−736343.	−2.07401
$167$	142267.	0.394742	0.197371	−	0.980329i	$-0.436760\pi$
$167$	142267.	0.394742	0.197371	+	0.980329i	$0.436760\pi$
$168$	407505.	1.11393
$169$	−280291.	−0.754904
$170$	0	0
$171$	426518.	1.11544
$172$	−110445.	−0.284660
$173$	−526184.	−1.33666	−0.668332	−	0.743863i	$-0.732991\pi$
$173$	−526184.	−1.33666	−0.668332	+	0.743863i	$0.732991\pi$
$174$	−115053.	−0.288088
$175$	0	0
$176$	46077.9	0.112127
$177$	19696.2	0.0472553
$178$	725249.	1.71568
$179$	692470.	1.61536	0.807678	−	0.589624i	$-0.200724\pi$
$179$	692470.	1.61536	0.807678	+	0.589624i	$0.200724\pi$
$180$	0	0
$181$	286664.	0.650393	0.325197	−	0.945646i	$-0.394569\pi$
$181$	286664.	0.650393	0.325197	+	0.945646i	$0.394569\pi$
$182$	−565152.	−1.26470
$183$	386284.	0.852665
$184$	137075.	0.298479
$185$	0	0
$186$	−84597.3	−0.179297
$187$	87941.4	0.183903
$188$	824167.	1.70067
$189$	651241.	1.32613
$190$	0	0
$191$	−692472.	−1.37347	−0.686734	−	0.726909i	$-0.740956\pi$
$191$	−692472.	−1.37347	−0.686734	+	0.726909i	$0.740956\pi$
$192$	−342166.	−0.669860
$193$	−855425.	−1.65306	−0.826530	−	0.562893i	$-0.809688\pi$
$193$	−855425.	−1.65306	−0.826530	+	0.562893i	$0.809688\pi$
$194$	−650534.	−1.24098
$195$	0	0
$196$	1.28149e6	2.38274
$197$	749649.	1.37623	0.688117	−	0.725600i	$-0.258438\pi$
$197$	749649.	1.37623	0.688117	+	0.725600i	$0.258438\pi$
$198$	126621.	0.229532
$199$	990082.	1.77230	0.886152	−	0.463394i	$-0.153369\pi$
$199$	990082.	1.77230	0.886152	+	0.463394i	$0.153369\pi$
$200$	0	0
$201$	−71504.9	−0.124838
$202$	−1.25300e6	−2.16059
$203$	−299577.	−0.510233
$204$	565580.	0.951522
$205$	0	0
$206$	401746.	0.659604
$207$	93656.7	0.151919
$208$	−190811.	−0.305806
$209$	−171207.	−0.271117
$210$	0	0
$211$	−274555.	−0.424545	−0.212272	−	0.977211i	$-0.568086\pi$
$211$	−274555.	−0.424545	−0.212272	+	0.977211i	$0.568086\pi$
$212$	200165.	0.305879
$213$	340384.	0.514067
$214$	−50545.8	−0.0754486
$215$	0	0
$216$	884327.	1.28967
$217$	−220276.	−0.317554
$218$	1.32824e6	1.89293
$219$	−628003.	−0.884813
$220$	0	0
$221$	−364170.	−0.501561
$222$	−701578.	−0.955418
$223$	568751.	0.765878	0.382939	−	0.923774i	$-0.374912\pi$
$223$	568751.	0.765878	0.382939	+	0.923774i	$0.374912\pi$
$224$	−477563.	−0.635932
$225$	0	0
$226$	2.40712e6	3.13492
$227$	−607473.	−0.782461	−0.391230	−	0.920293i	$-0.627950\pi$
$227$	−607473.	−0.782461	−0.391230	+	0.920293i	$0.627950\pi$
$228$	−1.10109e6	−1.40277
$229$	104213.	0.131320	0.0656600	−	0.997842i	$-0.479085\pi$
$229$	104213.	0.131320	0.0656600	+	0.997842i	$0.479085\pi$
$230$	0	0
$231$	−111763.	−0.137806
$232$	−406799.	−0.496204
$233$	−1.28050e6	−1.54521	−0.772606	−	0.634885i	$-0.781047\pi$
$233$	−1.28050e6	−1.54521	−0.772606	+	0.634885i	$0.781047\pi$
$234$	−524346.	−0.626006
$235$	0	0
$236$	149999.	0.175310
$237$	−517601.	−0.598583
$238$	2.26161e6	2.58806
$239$	221000.	0.250264	0.125132	−	0.992140i	$-0.460065\pi$
$239$	221000.	0.250264	0.125132	+	0.992140i	$0.460065\pi$
$240$	0	0
$241$	−480522.	−0.532931	−0.266465	−	0.963845i	$-0.585856\pi$
$241$	−480522.	−0.532931	−0.266465	+	0.963845i	$0.585856\pi$
$242$	1.49167e6	1.63733
$243$	950112.	1.03219
$244$	2.94178e6	3.16327
$245$	0	0
$246$	−1.48022e6	−1.55951
$247$	708979.	0.739420
$248$	−299115.	−0.308822
$249$	603010.	0.616348
$250$	0	0
$251$	722732.	0.724091	0.362045	−	0.932161i	$-0.382079\pi$
$251$	722732.	0.724091	0.362045	+	0.932161i	$0.382079\pi$
$252$	2.12040e6	2.10337
$253$	−37594.5	−0.0369252
$254$	−1.68275e6	−1.63657
$255$	0	0
$256$	−1.85752e6	−1.77147
$257$	1.34492e6	1.27018	0.635089	−	0.772439i	$-0.280963\pi$
$257$	1.34492e6	1.27018	0.635089	+	0.772439i	$0.280963\pi$
$258$	138901.	0.129914
$259$	−1.82678e6	−1.69214
$260$	0	0
$261$	−277946.	−0.252557
$262$	−2.04731e6	−1.84260
$263$	18066.1	0.0161055	0.00805275	−	0.999968i	$-0.497437\pi$
$263$	18066.1	0.0161055	0.00805275	+	0.999968i	$0.497437\pi$
$264$	−151765.	−0.134017
$265$	0	0
$266$	−4.40297e6	−3.81542
$267$	−593925.	−0.509863
$268$	−544553.	−0.463130
$269$	−549502.	−0.463008	−0.231504	−	0.972834i	$-0.574365\pi$
$269$	−549502.	−0.463008	−0.231504	+	0.972834i	$0.574365\pi$
$270$	0	0
$271$	13732.4	0.0113586	0.00567929	−	0.999984i	$-0.498192\pi$
$271$	13732.4	0.0113586	0.00567929	+	0.999984i	$0.498192\pi$
$272$	763582.	0.625797
$273$	462818.	0.375840
$274$	1.20704e6	0.971278
$275$	0	0
$276$	−241782.	−0.191052
$277$	501657.	0.392833	0.196416	−	0.980521i	$-0.437070\pi$
$277$	501657.	0.392833	0.196416	+	0.980521i	$0.437070\pi$
$278$	−1.73457e6	−1.34611
$279$	−204371.	−0.157184
$280$	0	0
$281$	605657.	0.457574	0.228787	−	0.973477i	$-0.426524\pi$
$281$	605657.	0.457574	0.228787	+	0.973477i	$0.426524\pi$
$282$	−1.03651e6	−0.776157
$283$	1.85484e6	1.37671	0.688353	−	0.725376i	$-0.258334\pi$
$283$	1.85484e6	1.37671	0.688353	+	0.725376i	$0.258334\pi$
$284$	2.59223e6	1.90711
$285$	0	0
$286$	210476.	0.152156
$287$	−3.85421e6	−2.76204
$288$	−443081.	−0.314776
$289$	37467.4	0.0263882
$290$	0	0
$291$	532739.	0.368792
$292$	−4.78262e6	−3.28253
$293$	−950968.	−0.647138	−0.323569	−	0.946205i	$-0.604883\pi$
$293$	−950968.	−0.647138	−0.323569	+	0.946205i	$0.604883\pi$
$294$	−1.61166e6	−1.08744
$295$	0	0
$296$	−2.48061e6	−1.64562
$297$	−242538.	−0.159547
$298$	−1.79184e6	−1.16885
$299$	155681.	0.100706
$300$	0	0
$301$	361672.	0.230090
$302$	3.79428e6	2.39393
$303$	1.02611e6	0.642079
$304$	−1.48657e6	−0.922572
$305$	0	0
$306$	2.09831e6	1.28105
$307$	736219.	0.445822	0.222911	−	0.974839i	$-0.428444\pi$
$307$	736219.	0.445822	0.222911	+	0.974839i	$0.428444\pi$
$308$	−851144.	−0.511242
$309$	−329000.	−0.196020
$310$	0	0
$311$	−1.02220e6	−0.599286	−0.299643	−	0.954051i	$-0.596867\pi$
$311$	−1.02220e6	−0.599286	−0.299643	+	0.954051i	$0.596867\pi$
$312$	628466.	0.365507
$313$	60474.7	0.0348909	0.0174455	−	0.999848i	$-0.494447\pi$
$313$	60474.7	0.0348909	0.0174455	+	0.999848i	$0.494447\pi$
$314$	4.82990e6	2.76448
$315$	0	0
$316$	−3.94184e6	−2.22066
$317$	−842265.	−0.470761	−0.235381	−	0.971903i	$-0.575634\pi$
$317$	−842265.	−0.470761	−0.235381	+	0.971903i	$0.575634\pi$
$318$	−251736.	−0.139598
$319$	111570.	0.0613860
$320$	0	0
$321$	41393.3	0.0224216
$322$	−966825.	−0.519646
$323$	−2.83717e6	−1.51314
$324$	1.07435e6	0.568568
$325$	0	0
$326$	−4.46358e6	−2.32616
$327$	−1.08773e6	−0.562536
$328$	−5.23368e6	−2.68610
$329$	−2.69887e6	−1.37465
$330$	0	0
$331$	−3.50942e6	−1.76062	−0.880308	−	0.474402i	$-0.842664\pi$
$331$	−3.50942e6	−1.76062	−0.880308	+	0.474402i	$0.842664\pi$
$332$	4.59228e6	2.28657
$333$	−1.69488e6	−0.837584
$334$	−1.36259e6	−0.668344
$335$	0	0
$336$	−970423.	−0.468935
$337$	1.49987e6	0.719414	0.359707	−	0.933065i	$-0.382877\pi$
$337$	1.49987e6	0.719414	0.359707	+	0.933065i	$0.382877\pi$
$338$	2.68454e6	1.27814
$339$	−1.97125e6	−0.931627
$340$	0	0
$341$	82036.0	0.0382048
$342$	−4.08506e6	−1.88857
$343$	−908952.	−0.417163
$344$	491118.	0.223764
$345$	0	0
$346$	5.03963e6	2.26313
$347$	−2.16958e6	−0.967280	−0.483640	−	0.875267i	$-0.660686\pi$
$347$	−2.16958e6	−0.967280	−0.483640	+	0.875267i	$0.660686\pi$
$348$	717541.	0.317613
$349$	393576.	0.172968	0.0864838	−	0.996253i	$-0.472437\pi$
$349$	393576.	0.172968	0.0864838	+	0.996253i	$0.472437\pi$
$350$	0	0
$351$	1.00436e6	0.435133
$352$	177856.	0.0765089
$353$	−272744.	−0.116498	−0.0582491	−	0.998302i	$-0.518552\pi$
$353$	−272744.	−0.116498	−0.0582491	+	0.998302i	$0.518552\pi$
$354$	−188645.	−0.0800086
$355$	0	0
$356$	−4.52309e6	−1.89152
$357$	−1.85209e6	−0.769115
$358$	−6.63227e6	−2.73498
$359$	−3.96827e6	−1.62504	−0.812521	−	0.582932i	$-0.801906\pi$
$359$	−3.96827e6	−1.62504	−0.812521	+	0.582932i	$0.801906\pi$
$360$	0	0
$361$	3.04739e6	1.23072
$362$	−2.74558e6	−1.10119
$363$	−1.22157e6	−0.486577
$364$	3.52464e6	1.39431
$365$	0	0
$366$	−3.69971e6	−1.44366
$367$	4.90395e6	1.90056	0.950278	−	0.311404i	$-0.100799\pi$
$367$	4.90395e6	1.90056	0.950278	+	0.311404i	$0.100799\pi$
$368$	−326427.	−0.125651
$369$	−3.57592e6	−1.36717
$370$	0	0
$371$	−655475.	−0.247242
$372$	527600.	0.197673
$373$	−2.52633e6	−0.940196	−0.470098	−	0.882614i	$-0.655781\pi$
$373$	−2.52633e6	−0.940196	−0.470098	+	0.882614i	$0.655781\pi$
$374$	−842277.	−0.311369
$375$	0	0
$376$	−3.66483e6	−1.33686
$377$	−462016.	−0.167419
$378$	−6.23739e6	−2.24529
$379$	−607631.	−0.217291	−0.108646	−	0.994081i	$-0.534651\pi$
$379$	−607631.	−0.217291	−0.108646	+	0.994081i	$0.534651\pi$
$380$	0	0
$381$	1.37805e6	0.486354
$382$	6.63229e6	2.32544
$383$	3.94370e6	1.37375	0.686874	−	0.726777i	$-0.258982\pi$
$383$	3.94370e6	1.37375	0.686874	+	0.726777i	$0.258982\pi$
$384$	2.66438e6	0.922079
$385$	0	0
$386$	8.19301e6	2.79882
$387$	335558.	0.113891
$388$	4.05713e6	1.36817
$389$	−2.42464e6	−0.812405	−0.406203	−	0.913783i	$-0.633147\pi$
$389$	−2.42464e6	−0.812405	−0.406203	+	0.913783i	$0.633147\pi$
$390$	0	0
$391$	−622998.	−0.206084
$392$	−5.69843e6	−1.87301
$393$	1.67660e6	0.547579
$394$	−7.17991e6	−2.33012
$395$	0	0
$396$	−789689.	−0.253057
$397$	2.94322e6	0.937232	0.468616	−	0.883402i	$-0.344753\pi$
$397$	2.94322e6	0.937232	0.468616	+	0.883402i	$0.344753\pi$
$398$	−9.48271e6	−3.00072
$399$	3.60571e6	1.13386
$400$	0	0
$401$	−5.82714e6	−1.80965	−0.904824	−	0.425785i	$-0.859998\pi$
$401$	−5.82714e6	−1.80965	−0.904824	+	0.425785i	$0.859998\pi$
$402$	684853.	0.211365
$403$	−339715.	−0.104196
$404$	7.81446e6	2.38202
$405$	0	0
$406$	2.86926e6	0.863883
$407$	680337.	0.203581
$408$	−2.51497e6	−0.747968
$409$	1.22650e6	0.362542	0.181271	−	0.983433i	$-0.441979\pi$
$409$	1.22650e6	0.362542	0.181271	+	0.983433i	$0.441979\pi$
$410$	0	0
$411$	−988473.	−0.288642
$412$	−2.50553e6	−0.727205
$413$	−491196.	−0.141703
$414$	−897016.	−0.257217
$415$	0	0
$416$	−736511.	−0.208663
$417$	1.42049e6	0.400034
$418$	1.63977e6	0.459032
$419$	1.13490e6	0.315809	0.157904	−	0.987454i	$-0.449526\pi$
$419$	1.13490e6	0.315809	0.157904	+	0.987454i	$0.449526\pi$
$420$	0	0
$421$	−5.21555e6	−1.43415	−0.717075	−	0.696996i	$-0.754520\pi$
$421$	−5.21555e6	−1.43415	−0.717075	+	0.696996i	$0.754520\pi$
$422$	2.62961e6	0.718803
$423$	−2.50400e6	−0.680431
$424$	−890078.	−0.240444
$425$	0	0
$426$	−3.26009e6	−0.870374
$427$	−9.63336e6	−2.55687
$428$	315235.	0.0831811
$429$	−172365.	−0.0452173
$430$	0	0
$431$	4.94938e6	1.28339	0.641693	−	0.766961i	$-0.278232\pi$
$431$	4.94938e6	1.28339	0.641693	+	0.766961i	$0.278232\pi$
$432$	−2.10592e6	−0.542915
$433$	2.02994e6	0.520311	0.260156	−	0.965567i	$-0.416226\pi$
$433$	2.02994e6	0.520311	0.260156	+	0.965567i	$0.416226\pi$
$434$	2.10973e6	0.537655
$435$	0	0
$436$	−8.28369e6	−2.08693
$437$	1.21287e6	0.303817
$438$	6.01482e6	1.49809
$439$	−4.05767e6	−1.00488	−0.502442	−	0.864611i	$-0.667565\pi$
$439$	−4.05767e6	−1.00488	−0.502442	+	0.864611i	$0.667565\pi$
$440$	0	0
$441$	−3.89347e6	−0.953323
$442$	3.48792e6	0.849201
$443$	−3.73717e6	−0.904761	−0.452380	−	0.891825i	$-0.649425\pi$
$443$	−3.73717e6	−0.904761	−0.452380	+	0.891825i	$0.649425\pi$
$444$	4.37547e6	1.05334
$445$	0	0
$446$	−5.44733e6	−1.29672
$447$	1.46738e6	0.347356
$448$	8.53313e6	2.00869
$449$	−2.08643e6	−0.488414	−0.244207	−	0.969723i	$-0.578528\pi$
$449$	−2.08643e6	−0.488414	−0.244207	+	0.969723i	$0.578528\pi$
$450$	0	0
$451$	1.43540e6	0.332301
$452$	−1.50122e7	−3.45620
$453$	−3.10724e6	−0.711424
$454$	5.81820e6	1.32480
$455$	0	0
$456$	4.89623e6	1.10268
$457$	−383012.	−0.0857871	−0.0428936	−	0.999080i	$-0.513658\pi$
$457$	−383012.	−0.0857871	−0.0428936	+	0.999080i	$0.513658\pi$
$458$	−998117.	−0.222340
$459$	−4.01922e6	−0.890452
$460$	0	0
$461$	−767599.	−0.168222	−0.0841109	−	0.996456i	$-0.526805\pi$
$461$	−767599.	−0.168222	−0.0841109	+	0.996456i	$0.526805\pi$
$462$	1.07044e6	0.233322
$463$	−2.87956e6	−0.624272	−0.312136	−	0.950037i	$-0.601045\pi$
$463$	−2.87956e6	−0.624272	−0.312136	+	0.950037i	$0.601045\pi$
$464$	968743.	0.208888
$465$	0	0
$466$	1.22642e7	2.61622
$467$	−4.73971e6	−1.00568	−0.502840	−	0.864379i	$-0.667712\pi$
$467$	−4.73971e6	−1.00568	−0.502840	+	0.864379i	$0.667712\pi$
$468$	3.27014e6	0.690164
$469$	1.78323e6	0.374348
$470$	0	0
$471$	−3.95533e6	−0.821543
$472$	−667001.	−0.137807
$473$	−134695.	−0.0276821
$474$	4.95743e6	1.01347
$475$	0	0
$476$	−1.41048e7	−2.85331
$477$	−608147.	−0.122381
$478$	−2.11668e6	−0.423726
$479$	5.41230e6	1.07781	0.538906	−	0.842366i	$-0.318838\pi$
$479$	5.41230e6	1.07781	0.538906	+	0.842366i	$0.318838\pi$
$480$	0	0
$481$	−2.81731e6	−0.555229
$482$	4.60230e6	0.902313
$483$	791758.	0.154427
$484$	−9.30298e6	−1.80513
$485$	0	0
$486$	−9.09989e6	−1.74761
$487$	−5.16701e6	−0.987227	−0.493614	−	0.869681i	$-0.664324\pi$
$487$	−5.16701e6	−0.987227	−0.493614	+	0.869681i	$0.664324\pi$
$488$	−1.30813e7	−2.48657
$489$	3.65534e6	0.691283
$490$	0	0
$491$	1.68112e6	0.314698	0.157349	−	0.987543i	$-0.449705\pi$
$491$	1.68112e6	0.314698	0.157349	+	0.987543i	$0.449705\pi$
$492$	9.23154e6	1.71934
$493$	1.84888e6	0.342603
$494$	−6.79039e6	−1.25192
$495$	0	0
$496$	712306.	0.130006
$497$	−8.48868e6	−1.54152
$498$	−5.77545e6	−1.04355
$499$	3.57956e6	0.643545	0.321772	−	0.946817i	$-0.395721\pi$
$499$	3.57956e6	0.643545	0.321772	+	0.946817i	$0.395721\pi$
$500$	0	0
$501$	1.11586e6	0.198617
$502$	−6.92211e6	−1.22597
$503$	1.07972e6	0.190279	0.0951397	−	0.995464i	$-0.469670\pi$
$503$	1.07972e6	0.190279	0.0951397	+	0.995464i	$0.469670\pi$
$504$	−9.42881e6	−1.65341
$505$	0	0
$506$	360069.	0.0625186
$507$	−2.19844e6	−0.379835
$508$	1.04947e7	1.80430
$509$	1.11760e6	0.191202	0.0956012	−	0.995420i	$-0.469523\pi$
$509$	1.11760e6	0.191202	0.0956012	+	0.995420i	$0.469523\pi$
$510$	0	0
$511$	1.56615e7	2.65327
$512$	6.92051e6	1.16671
$513$	7.82475e6	1.31274
$514$	−1.28813e7	−2.15056
$515$	0	0
$516$	−866270.	−0.143228
$517$	1.00513e6	0.165384
$518$	1.74964e7	2.86499
$519$	−4.12709e6	−0.672551
$520$	0	0
$521$	−6.50399e6	−1.04975	−0.524874	−	0.851180i	$-0.675888\pi$
$521$	−6.50399e6	−1.04975	−0.524874	+	0.851180i	$0.675888\pi$
$522$	2.66209e6	0.427608
$523$	−437675.	−0.0699677	−0.0349838	−	0.999388i	$-0.511138\pi$
$523$	−437675.	−0.0699677	−0.0349838	+	0.999388i	$0.511138\pi$
$524$	1.27683e7	2.03144
$525$	0	0
$526$	−173031.	−0.0272685
$527$	1.35946e6	0.213226
$528$	361409.	0.0564176
$529$	−6.17001e6	−0.958621
$530$	0	0
$531$	−455730.	−0.0701409
$532$	2.74596e7	4.20645
$533$	−5.94407e6	−0.906288
$534$	5.68844e6	0.863256
$535$	0	0
$536$	2.42147e6	0.364055
$537$	5.43134e6	0.812777
$538$	5.26297e6	0.783926
$539$	1.56287e6	0.231713
$540$	0	0
$541$	6.56956e6	0.965035	0.482518	−	0.875886i	$-0.339722\pi$
$541$	6.56956e6	0.965035	0.482518	+	0.875886i	$0.339722\pi$
$542$	−131525.	−0.0192314
$543$	2.24843e6	0.327249
$544$	2.94735e6	0.427006
$545$	0	0
$546$	−4.43273e6	−0.636341
$547$	2.84947e6	0.407189	0.203595	−	0.979055i	$-0.434738\pi$
$547$	2.84947e6	0.407189	0.203595	+	0.979055i	$0.434738\pi$
$548$	−7.52781e6	−1.07082
$549$	−8.93780e6	−1.26561
$550$	0	0
$551$	−3.59946e6	−0.505078
$552$	1.07514e6	0.150181
$553$	1.29082e7	1.79496
$554$	−4.80472e6	−0.665111
$555$	0	0
$556$	1.08178e7	1.48407
$557$	−1.14478e7	−1.56344	−0.781722	−	0.623627i	$-0.785658\pi$
$557$	−1.14478e7	−1.56344	−0.781722	+	0.623627i	$0.785658\pi$
$558$	1.95740e6	0.266131
$559$	557781.	0.0754977
$560$	0	0
$561$	689763.	0.0925321
$562$	−5.80080e6	−0.774725
$563$	−3.98082e6	−0.529300	−0.264650	−	0.964345i	$-0.585256\pi$
$563$	−3.98082e6	−0.529300	−0.264650	+	0.964345i	$0.585256\pi$
$564$	6.46429e6	0.855703
$565$	0	0
$566$	−1.77652e7	−2.33092
$567$	−3.51813e6	−0.459573
$568$	−1.15269e7	−1.49914
$569$	8.49999e6	1.10062	0.550311	−	0.834960i	$-0.314509\pi$
$569$	8.49999e6	1.10062	0.550311	+	0.834960i	$0.314509\pi$
$570$	0	0
$571$	5.24847e6	0.673663	0.336831	−	0.941565i	$-0.390645\pi$
$571$	5.24847e6	0.673663	0.336831	+	0.941565i	$0.390645\pi$
$572$	−1.31266e6	−0.167750
$573$	−5.43135e6	−0.691069
$574$	3.69145e7	4.67646
$575$	0	0
$576$	7.91701e6	0.994272
$577$	−1.94309e6	−0.242970	−0.121485	−	0.992593i	$-0.538766\pi$
$577$	−1.94309e6	−0.242970	−0.121485	+	0.992593i	$0.538766\pi$
$578$	−358852.	−0.0446782
$579$	−6.70946e6	−0.831747
$580$	0	0
$581$	−1.50382e7	−1.84823
$582$	−5.10242e6	−0.624408
$583$	244115.	0.0297456
$584$	2.12669e7	2.58031
$585$	0	0
$586$	9.10809e6	1.09568
$587$	−924441.	−0.110735	−0.0553674	−	0.998466i	$-0.517633\pi$
$587$	−924441.	−0.110735	−0.0553674	+	0.998466i	$0.517633\pi$
$588$	1.00513e7	1.19889
$589$	−2.64665e6	−0.314346
$590$	0	0
$591$	5.87982e6	0.692461
$592$	5.90726e6	0.692759
$593$	1.09140e7	1.27453	0.637263	−	0.770646i	$-0.280066\pi$
$593$	1.09140e7	1.27453	0.637263	+	0.770646i	$0.280066\pi$
$594$	2.32296e6	0.270131
$595$	0	0
$596$	1.11750e7	1.28864
$597$	7.76564e6	0.891746
$598$	−1.49106e6	−0.170508
$599$	−6.47200e6	−0.737006	−0.368503	−	0.929626i	$-0.620130\pi$
$599$	−6.47200e6	−0.737006	−0.368503	+	0.929626i	$0.620130\pi$
$600$	0	0
$601$	−9.66608e6	−1.09160	−0.545801	−	0.837915i	$-0.683775\pi$
$601$	−9.66608e6	−1.09160	−0.545801	+	0.837915i	$0.683775\pi$
$602$	−3.46398e6	−0.389569
$603$	1.65447e6	0.185296
$604$	−2.36635e7	−2.63928
$605$	0	0
$606$	−9.82780e6	−1.08711
$607$	−1.50825e7	−1.66151	−0.830753	−	0.556641i	$-0.812090\pi$
$607$	−1.50825e7	−1.66151	−0.830753	+	0.556641i	$0.812090\pi$
$608$	−5.73799e6	−0.629507
$609$	−2.34971e6	−0.256727
$610$	0	0
$611$	−4.16228e6	−0.451054
$612$	−1.30863e7	−1.41234
$613$	2.07592e6	0.223131	0.111565	−	0.993757i	$-0.464414\pi$
$613$	2.07592e6	0.223131	0.111565	+	0.993757i	$0.464414\pi$
$614$	−7.05129e6	−0.754827
$615$	0	0
$616$	3.78479e6	0.401875
$617$	−1.05100e7	−1.11145	−0.555727	−	0.831365i	$-0.687560\pi$
$617$	−1.05100e7	−1.11145	−0.555727	+	0.831365i	$0.687560\pi$
$618$	3.15106e6	0.331884
$619$	−6.33902e6	−0.664960	−0.332480	−	0.943110i	$-0.607885\pi$
$619$	−6.33902e6	−0.664960	−0.332480	+	0.943110i	$0.607885\pi$
$620$	0	0
$621$	1.71820e6	0.178790
$622$	9.79030e6	1.01466
$623$	1.48116e7	1.52891
$624$	−1.49661e6	−0.153868
$625$	0	0
$626$	−579208.	−0.0590744
$627$	−1.34285e6	−0.136414
$628$	−3.01222e7	−3.04781
$629$	1.12742e7	1.13621
$630$	0	0
$631$	−1.15541e7	−1.15521	−0.577606	−	0.816316i	$-0.696013\pi$
$631$	−1.15541e7	−1.15521	−0.577606	+	0.816316i	$0.696013\pi$
$632$	1.75283e7	1.74560
$633$	−2.15345e6	−0.213612
$634$	8.06697e6	0.797053
$635$	0	0
$636$	1.56998e6	0.153905
$637$	−6.47191e6	−0.631952
$638$	−1.06858e6	−0.103934
$639$	−7.87577e6	−0.763028
$640$	0	0
$641$	−3.92627e6	−0.377429	−0.188714	−	0.982032i	$-0.560432\pi$
$641$	−3.92627e6	−0.377429	−0.188714	+	0.982032i	$0.560432\pi$
$642$	−396453.	−0.0379624
$643$	1.87570e7	1.78911	0.894553	−	0.446961i	$-0.147494\pi$
$643$	1.87570e7	1.78911	0.894553	+	0.446961i	$0.147494\pi$
$644$	6.02971e6	0.572904
$645$	0	0
$646$	2.71736e7	2.56192
$647$	9.03463e6	0.848496	0.424248	−	0.905546i	$-0.360538\pi$
$647$	9.03463e6	0.848496	0.424248	+	0.905546i	$0.360538\pi$
$648$	−4.77731e6	−0.446937
$649$	182933.	0.0170483
$650$	0	0
$651$	−1.72772e6	−0.159779
$652$	2.78376e7	2.56456
$653$	−1.74423e7	−1.60074	−0.800371	−	0.599505i	$-0.795364\pi$
$653$	−1.74423e7	−1.60074	−0.800371	+	0.599505i	$0.795364\pi$
$654$	1.04179e7	0.952439
$655$	0	0
$656$	1.24634e7	1.13077
$657$	1.45307e7	1.31333
$658$	2.58490e7	2.32744
$659$	1.74636e7	1.56646	0.783230	−	0.621732i	$-0.213571\pi$
$659$	1.74636e7	1.56646	0.783230	+	0.621732i	$0.213571\pi$
$660$	0	0
$661$	5.12626e6	0.456348	0.228174	−	0.973620i	$-0.426724\pi$
$661$	5.12626e6	0.456348	0.228174	+	0.973620i	$0.426724\pi$
$662$	3.36121e7	2.98093
$663$	−2.85634e6	−0.252364
$664$	−2.04206e7	−1.79741
$665$	0	0
$666$	1.62331e7	1.41813
$667$	−790386.	−0.0687899
$668$	8.49796e6	0.736841
$669$	4.46096e6	0.385356
$670$	0	0
$671$	3.58770e6	0.307616
$672$	−3.74573e6	−0.319973
$673$	−4.98529e6	−0.424280	−0.212140	−	0.977239i	$-0.568043\pi$
$673$	−4.98529e6	−0.424280	−0.212140	+	0.977239i	$0.568043\pi$
$674$	−1.43653e7	−1.21805
$675$	0	0
$676$	−1.67424e7	−1.40913
$677$	−8.15170e6	−0.683560	−0.341780	−	0.939780i	$-0.611030\pi$
$677$	−8.15170e6	−0.683560	−0.341780	+	0.939780i	$0.611030\pi$
$678$	1.88800e7	1.57735
$679$	−1.32857e7	−1.10589
$680$	0	0
$681$	−4.76467e6	−0.393700
$682$	−785716.	−0.0646852
$683$	3.28051e6	0.269085	0.134543	−	0.990908i	$-0.457043\pi$
$683$	3.28051e6	0.269085	0.134543	+	0.990908i	$0.457043\pi$
$684$	2.54769e7	2.08213
$685$	0	0
$686$	8.70567e6	0.706305
$687$	817384.	0.0660745
$688$	−1.16954e6	−0.0941984
$689$	−1.01089e6	−0.0811254
$690$	0	0
$691$	1.54825e7	1.23352	0.616758	−	0.787153i	$-0.288446\pi$
$691$	1.54825e7	1.23352	0.616758	+	0.787153i	$0.288446\pi$
$692$	−3.14302e7	−2.49507
$693$	2.58597e6	0.204546
$694$	2.07796e7	1.63772
$695$	0	0
$696$	−3.19070e6	−0.249668
$697$	2.37868e7	1.85462
$698$	−3.76955e6	−0.292854
$699$	−1.00435e7	−0.777483
$700$	0	0
$701$	−4.91809e6	−0.378008	−0.189004	−	0.981976i	$-0.560526\pi$
$701$	−4.91809e6	−0.378008	−0.189004	+	0.981976i	$0.560526\pi$
$702$	−9.61948e6	−0.736731
$703$	−2.19490e7	−1.67505
$704$	−3.17795e6	−0.241666
$705$	0	0
$706$	2.61227e6	0.197245
$707$	−2.55898e7	−1.92539
$708$	1.17650e6	0.0882085
$709$	−8.69925e6	−0.649930	−0.324965	−	0.945726i	$-0.605352\pi$
$709$	−8.69925e6	−0.649930	−0.324965	+	0.945726i	$0.605352\pi$
$710$	0	0
$711$	1.19762e7	0.888476
$712$	2.01129e7	1.48688
$713$	−581162.	−0.0428128
$714$	1.77388e7	1.30220
$715$	0	0
$716$	4.13629e7	3.01529
$717$	1.73340e6	0.125922
$718$	3.80069e7	2.75138
$719$	−8.92512e6	−0.643861	−0.321930	−	0.946763i	$-0.604332\pi$
$719$	−8.92512e6	−0.643861	−0.321930	+	0.946763i	$0.604332\pi$
$720$	0	0
$721$	8.20479e6	0.587799
$722$	−2.91870e7	−2.08376
$723$	−3.76894e6	−0.268147
$724$	1.71231e7	1.21405
$725$	0	0
$726$	1.16998e7	0.823830
$727$	−1.33205e7	−0.934729	−0.467365	−	0.884065i	$-0.654796\pi$
$727$	−1.33205e7	−0.934729	−0.467365	+	0.884065i	$0.654796\pi$
$728$	−1.56730e7	−1.09604
$729$	3.08154e6	0.214758
$730$	0	0
$731$	−2.23211e6	−0.154498
$732$	2.30737e7	1.59162
$733$	−1.41244e7	−0.970981	−0.485490	−	0.874242i	$-0.661359\pi$
$733$	−1.41244e7	−0.970981	−0.485490	+	0.874242i	$0.661359\pi$
$734$	−4.69685e7	−3.21786
$735$	0	0
$736$	−1.25997e6	−0.0857368
$737$	−664118.	−0.0450377
$738$	3.42491e7	2.31477
$739$	−1.65506e7	−1.11482	−0.557408	−	0.830239i	$-0.688204\pi$
$739$	−1.65506e7	−1.11482	−0.557408	+	0.830239i	$0.688204\pi$
$740$	0	0
$741$	5.56083e6	0.372044
$742$	6.27795e6	0.418608
$743$	−1.32085e7	−0.877771	−0.438885	−	0.898543i	$-0.644627\pi$
$743$	−1.32085e7	−0.877771	−0.438885	+	0.898543i	$0.644627\pi$
$744$	−2.34609e6	−0.155386
$745$	0	0
$746$	2.41965e7	1.59186
$747$	−1.39524e7	−0.914845
$748$	5.25296e6	0.343281
$749$	−1.03229e6	−0.0672353
$750$	0	0
$751$	1.62740e6	0.105291	0.0526457	−	0.998613i	$-0.483235\pi$
$751$	1.62740e6	0.105291	0.0526457	+	0.998613i	$0.483235\pi$
$752$	8.72735e6	0.562779
$753$	5.66870e6	0.364331
$754$	4.42506e6	0.283459
$755$	0	0
$756$	3.89002e7	2.47541
$757$	1.41817e7	0.899472	0.449736	−	0.893162i	$-0.351518\pi$
$757$	1.41817e7	0.899472	0.449736	+	0.893162i	$0.351518\pi$
$758$	5.81971e6	0.367899
$759$	−294870.	−0.0185791
$760$	0	0
$761$	−631434.	−0.0395245	−0.0197622	−	0.999805i	$-0.506291\pi$
$761$	−631434.	−0.0395245	−0.0197622	+	0.999805i	$0.506291\pi$
$762$	−1.31985e7	−0.823453
$763$	2.71264e7	1.68686
$764$	−4.13630e7	−2.56377
$765$	0	0
$766$	−3.77716e7	−2.32591
$767$	−757537.	−0.0464960
$768$	−1.45693e7	−0.891326
$769$	2920.58	0.000178096 0	8.90480e−5	−	1.00000i	$-0.499972\pi$
$769$	2920.58	0.000178096 0	8.90480e−5		1.00000i	$0.499972\pi$
$770$	0	0
$771$	1.05488e7	0.639098
$772$	−5.10966e7	−3.08566
$773$	8.64224e6	0.520209	0.260104	−	0.965580i	$-0.416243\pi$
$773$	8.64224e6	0.520209	0.260104	+	0.965580i	$0.416243\pi$
$774$	−3.21387e6	−0.192831
$775$	0	0
$776$	−1.80409e7	−1.07548
$777$	−1.43282e7	−0.851412
$778$	2.32225e7	1.37550
$779$	−4.63089e7	−2.73414
$780$	0	0
$781$	3.16139e6	0.185460
$782$	5.96689e6	0.348924
$783$	−5.09911e6	−0.297228
$784$	1.35701e7	0.788485
$785$	0	0
$786$	−1.60579e7	−0.927115
$787$	−6.88838e6	−0.396443	−0.198221	−	0.980157i	$-0.563516\pi$
$787$	−6.88838e6	−0.396443	−0.198221	+	0.980157i	$0.563516\pi$
$788$	4.47783e7	2.56893
$789$	141700.	0.00810358
$790$	0	0
$791$	4.91601e7	2.79365
$792$	3.51152e6	0.198922
$793$	−1.48568e7	−0.838965
$794$	−2.81893e7	−1.58684
$795$	0	0
$796$	5.91400e7	3.30825
$797$	−2.93617e7	−1.63733	−0.818664	−	0.574273i	$-0.805285\pi$
$797$	−2.93617e7	−1.63733	−0.818664	+	0.574273i	$0.805285\pi$
$798$	−3.45344e7	−1.91975
$799$	1.66565e7	0.923030
$800$	0	0
$801$	1.37422e7	0.756788
$802$	5.58106e7	3.06394
$803$	−5.83272e6	−0.319214
$804$	−4.27116e6	−0.233027
$805$	0	0
$806$	3.25369e6	0.176416
$807$	−4.30998e6	−0.232965
$808$	−3.47487e7	−1.87245
$809$	−2.84195e6	−0.152667	−0.0763334	−	0.997082i	$-0.524321\pi$
$809$	−2.84195e6	−0.152667	−0.0763334	+	0.997082i	$0.524321\pi$
$810$	0	0
$811$	1.67390e6	0.0893670	0.0446835	−	0.999001i	$-0.485772\pi$
$811$	1.67390e6	0.0893670	0.0446835	+	0.999001i	$0.485772\pi$
$812$	−1.78945e7	−0.952420
$813$	107709.	0.00571515
$814$	−6.51607e6	−0.344687
$815$	0	0
$816$	5.98910e6	0.314874
$817$	4.34554e6	0.227766
$818$	−1.17470e7	−0.613825
$819$	−1.07086e7	−0.557859
$820$	0	0
$821$	3.75970e7	1.94668	0.973340	−	0.229365i	$-0.0736650\pi$
$821$	3.75970e7	1.94668	0.973340	+	0.229365i	$0.0736650\pi$
$822$	9.46730e6	0.488705
$823$	−1.67263e7	−0.860797	−0.430398	−	0.902639i	$-0.641627\pi$
$823$	−1.67263e7	−0.860797	−0.430398	+	0.902639i	$0.641627\pi$
$824$	1.11414e7	0.571638
$825$	0	0
$826$	4.70453e6	0.239920
$827$	2.69145e7	1.36843	0.684216	−	0.729279i	$-0.260144\pi$
$827$	2.69145e7	1.36843	0.684216	+	0.729279i	$0.260144\pi$
$828$	5.59434e6	0.283578
$829$	−7.34223e6	−0.371058	−0.185529	−	0.982639i	$-0.559400\pi$
$829$	−7.34223e6	−0.371058	−0.185529	+	0.982639i	$0.559400\pi$
$830$	0	0
$831$	3.93471e6	0.197656
$832$	1.31600e7	0.659097
$833$	2.58991e7	1.29322
$834$	−1.36050e7	−0.677303
$835$	0	0
$836$	−1.02266e7	−0.506077
$837$	−3.74932e6	−0.184986
$838$	−1.08698e7	−0.534701
$839$	−1.44619e7	−0.709284	−0.354642	−	0.935002i	$-0.615397\pi$
$839$	−1.44619e7	−0.709284	−0.354642	+	0.935002i	$0.615397\pi$
$840$	0	0
$841$	−1.81655e7	−0.885641
$842$	4.99530e7	2.42818
$843$	4.75043e6	0.230231
$844$	−1.63998e7	−0.792471
$845$	0	0
$846$	2.39826e7	1.15205
$847$	3.04642e7	1.45909
$848$	2.11961e6	0.101220
$849$	1.45483e7	0.692699
$850$	0	0
$851$	−4.81967e6	−0.228136
$852$	2.03319e7	0.959577
$853$	3.00835e6	0.141565	0.0707825	−	0.997492i	$-0.477450\pi$
$853$	3.00835e6	0.141565	0.0707825	+	0.997492i	$0.477450\pi$
$854$	9.22655e7	4.32907
$855$	0	0
$856$	−1.40176e6	−0.0653866
$857$	−5.35883e6	−0.249240	−0.124620	−	0.992205i	$-0.539771\pi$
$857$	−5.35883e6	−0.249240	−0.124620	+	0.992205i	$0.539771\pi$
$858$	1.65086e6	0.0765581
$859$	2.59898e7	1.20176	0.600882	−	0.799338i	$-0.294816\pi$
$859$	2.59898e7	1.20176	0.600882	+	0.799338i	$0.294816\pi$
$860$	0	0
$861$	−3.02302e7	−1.38974
$862$	−4.74037e7	−2.17292
$863$	4.32743e7	1.97789	0.988947	−	0.148266i	$-0.0473693\pi$
$863$	4.32743e7	1.97789	0.988947	+	0.148266i	$0.0473693\pi$
$864$	−8.12862e6	−0.370452
$865$	0	0
$866$	−1.94421e7	−0.880947
$867$	293873.	0.0132774
$868$	−1.31576e7	−0.592758
$869$	−4.80734e6	−0.215951
$870$	0	0
$871$	2.75015e6	0.122832
$872$	3.68352e7	1.64048
$873$	−1.23265e7	−0.547398
$874$	−1.16165e7	−0.514397
$875$	0	0
$876$	−3.75121e7	−1.65162
$877$	−1.03640e7	−0.455017	−0.227508	−	0.973776i	$-0.573058\pi$
$877$	−1.03640e7	−0.455017	−0.227508	+	0.973776i	$0.573058\pi$
$878$	3.88632e7	1.70138
$879$	−7.45885e6	−0.325611
$880$	0	0
$881$	−2.43175e7	−1.05555	−0.527776	−	0.849384i	$-0.676974\pi$
$881$	−2.43175e7	−1.05555	−0.527776	+	0.849384i	$0.676974\pi$
$882$	3.72905e7	1.61409
$883$	−1.78896e7	−0.772147	−0.386073	−	0.922468i	$-0.626169\pi$
$883$	−1.78896e7	−0.772147	−0.386073	+	0.922468i	$0.626169\pi$
$884$	−2.17528e7	−0.936233
$885$	0	0
$886$	3.57935e7	1.53186
$887$	2.17492e7	0.928185	0.464092	−	0.885787i	$-0.346381\pi$
$887$	2.17492e7	0.928185	0.464092	+	0.885787i	$0.346381\pi$
$888$	−1.94565e7	−0.828002
$889$	−3.43666e7	−1.45842
$890$	0	0
$891$	1.31024e6	0.0552912
$892$	3.39729e7	1.42962
$893$	−3.24274e7	−1.36076
$894$	−1.40542e7	−0.588114
$895$	0	0
$896$	−6.64458e7	−2.76502
$897$	1.22107e6	0.0506711
$898$	1.99832e7	0.826941
$899$	1.72472e6	0.0711738
$900$	0	0
$901$	4.04536e6	0.166014
$902$	−1.37478e7	−0.562624
$903$	2.83675e6	0.115771
$904$	6.67551e7	2.71684
$905$	0	0
$906$	2.97602e7	1.20452
$907$	−2.98499e7	−1.20483	−0.602414	−	0.798184i	$-0.705794\pi$
$907$	−2.98499e7	−1.20483	−0.602414	+	0.798184i	$0.705794\pi$
$908$	−3.62859e7	−1.46057
$909$	−2.37421e7	−0.953036
$910$	0	0
$911$	−1.09193e7	−0.435912	−0.217956	−	0.975959i	$-0.569939\pi$
$911$	−1.09193e7	−0.435912	−0.217956	+	0.975959i	$0.569939\pi$
$912$	−1.16598e7	−0.464198
$913$	5.60060e6	0.222360
$914$	3.66838e6	0.145247
$915$	0	0
$916$	6.22487e6	0.245127
$917$	−4.18119e7	−1.64201
$918$	3.84949e7	1.50764
$919$	9.16828e6	0.358096	0.179048	−	0.983840i	$-0.442698\pi$
$919$	9.16828e6	0.358096	0.179048	+	0.983840i	$0.442698\pi$
$920$	0	0
$921$	5.77448e6	0.224318
$922$	7.35184e6	0.284819
$923$	−1.30915e7	−0.505807
$924$	−6.67589e6	−0.257235
$925$	0	0
$926$	2.75796e7	1.05697
$927$	7.61237e6	0.290952
$928$	3.73924e6	0.142532
$929$	−1.38913e6	−0.0528083	−0.0264041	−	0.999651i	$-0.508406\pi$
$929$	−1.38913e6	−0.0528083	−0.0264041	+	0.999651i	$0.508406\pi$
$930$	0	0
$931$	−5.04212e7	−1.90651
$932$	−7.64871e7	−2.88435
$933$	−8.01753e6	−0.301534
$934$	4.53956e7	1.70273
$935$	0	0
$936$	−1.45414e7	−0.542521
$937$	2.52918e7	0.941090	0.470545	−	0.882376i	$-0.344057\pi$
$937$	2.52918e7	0.941090	0.470545	+	0.882376i	$0.344057\pi$
$938$	−1.70793e7	−0.633814
$939$	474329.	0.0175556
$940$	0	0
$941$	2.69608e7	0.992563	0.496282	−	0.868162i	$-0.334698\pi$
$941$	2.69608e7	0.992563	0.496282	+	0.868162i	$0.334698\pi$
$942$	3.78830e7	1.39097
$943$	−1.01687e7	−0.372381
$944$	1.58838e6	0.0580129
$945$	0	0
$946$	1.29007e6	0.0468690
$947$	3.19275e6	0.115689	0.0578443	−	0.998326i	$-0.481577\pi$
$947$	3.19275e6	0.115689	0.0578443	+	0.998326i	$0.481577\pi$
$948$	−3.09176e7	−1.11734
$949$	2.41536e7	0.870596
$950$	0	0
$951$	−6.60625e6	−0.236866
$952$	6.27198e7	2.24291
$953$	3.59618e7	1.28265	0.641327	−	0.767268i	$-0.278384\pi$
$953$	3.59618e7	1.28265	0.641327	+	0.767268i	$0.278384\pi$
$954$	5.82466e6	0.207205
$955$	0	0
$956$	1.32009e7	0.467152
$957$	875089.	0.0308868
$958$	−5.18374e7	−1.82486
$959$	2.46511e7	0.865545
$960$	0	0
$961$	−2.73610e7	−0.955704
$962$	2.69834e7	0.940067
$963$	−957755.	−0.0332804
$964$	−2.87027e7	−0.994789
$965$	0	0
$966$	−7.58322e6	−0.261463
$967$	1.05454e7	0.362659	0.181330	−	0.983422i	$-0.441960\pi$
$967$	1.05454e7	0.362659	0.181330	+	0.983422i	$0.441960\pi$
$968$	4.13677e7	1.41897
$969$	−2.22531e7	−0.761345
$970$	0	0
$971$	−1.47340e7	−0.501500	−0.250750	−	0.968052i	$-0.580677\pi$
$971$	−1.47340e7	−0.501500	−0.250750	+	0.968052i	$0.580677\pi$
$972$	5.67525e7	1.92672
$973$	−3.54249e7	−1.19957
$974$	4.94881e7	1.67149
$975$	0	0
$976$	3.11514e7	1.04677
$977$	−4.83492e7	−1.62051	−0.810257	−	0.586075i	$-0.800673\pi$
$977$	−4.83492e7	−1.62051	−0.810257	+	0.586075i	$0.800673\pi$
$978$	−3.50098e7	−1.17042
$979$	−5.51621e6	−0.183943
$980$	0	0
$981$	2.51677e7	0.834971
$982$	−1.61012e7	−0.532820
$983$	1.25654e7	0.414754	0.207377	−	0.978261i	$-0.433507\pi$
$983$	1.25654e7	0.414754	0.207377	+	0.978261i	$0.433507\pi$
$984$	−4.10500e7	−1.35153
$985$	0	0
$986$	−1.77080e7	−0.580067
$987$	−2.11684e7	−0.691665
$988$	4.23490e7	1.38023
$989$	954214.	0.0310209
$990$	0	0
$991$	−4.25363e7	−1.37586	−0.687932	−	0.725776i	$-0.741481\pi$
$991$	−4.25363e7	−1.37586	−0.687932	+	0.725776i	$0.741481\pi$
$992$	2.74942e6	0.0887080
$993$	−2.75259e7	−0.885865
$994$	8.13021e7	2.60997
$995$	0	0
$996$	3.60193e7	1.15050
$997$	3.70772e7	1.18132	0.590662	−	0.806919i	$-0.298867\pi$
$997$	3.70772e7	1.18132	0.590662	+	0.806919i	$0.298867\pi$
$998$	−3.42840e7	−1.08960
$999$	−3.10937e7	−0.985732

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1075.6.a.b.1.2		10
5.4	even	2		43.6.a.b.1.9	✓	10
15.14	odd	2		387.6.a.e.1.2		10
20.19	odd	2		688.6.a.h.1.7		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.a.b.1.9	✓	10	5.4	even	2
387.6.a.e.1.2		10	15.14	odd	2
688.6.a.h.1.7		10	20.19	odd	2
1075.6.a.b.1.2		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)

\(f(q)\)	\(=\)	\(q-9.57770 q^{2} +7.84343 q^{3} +59.7324 q^{4} -75.1221 q^{6} -195.604 q^{7} -265.613 q^{8} -181.481 q^{9} +O(q^{10})\)	\(q-9.57770 q^{2} +7.84343 q^{3} +59.7324 q^{4} -75.1221 q^{6} -195.604 q^{7} -265.613 q^{8} -181.481 q^{9} +72.8476 q^{11} +468.507 q^{12} -301.666 q^{13} +1873.44 q^{14} +632.525 q^{16} +1207.20 q^{17} +1738.17 q^{18} -2350.21 q^{19} -1534.21 q^{21} -697.713 q^{22} -516.070 q^{23} -2083.32 q^{24} +2889.27 q^{26} -3329.38 q^{27} -11683.9 q^{28} +1531.55 q^{29} +1126.13 q^{31} +2441.48 q^{32} +571.375 q^{33} -11562.2 q^{34} -10840.3 q^{36} +9339.18 q^{37} +22509.6 q^{38} -2366.10 q^{39} +19704.2 q^{41} +14694.2 q^{42} -1849.00 q^{43} +4351.37 q^{44} +4942.77 q^{46} +13797.6 q^{47} +4961.16 q^{48} +21453.9 q^{49} +9468.56 q^{51} -18019.2 q^{52} +3351.03 q^{53} +31887.9 q^{54} +51954.9 q^{56} -18433.7 q^{57} -14668.7 q^{58} +2511.18 q^{59} +49249.3 q^{61} -10785.7 q^{62} +35498.3 q^{63} -43624.6 q^{64} -5472.46 q^{66} -9116.54 q^{67} +72108.8 q^{68} -4047.76 q^{69} +43397.3 q^{71} +48203.6 q^{72} -80067.4 q^{73} -89447.9 q^{74} -140384. q^{76} -14249.3 q^{77} +22661.8 q^{78} -65991.7 q^{79} +17986.0 q^{81} -188721. q^{82} +76880.9 q^{83} -91641.8 q^{84} +17709.2 q^{86} +12012.6 q^{87} -19349.3 q^{88} -75722.6 q^{89} +59007.1 q^{91} -30826.1 q^{92} +8832.73 q^{93} -132150. q^{94} +19149.6 q^{96} +67921.7 q^{97} -205479. q^{98} -13220.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 8 q^{2} - 28 q^{3} + 202 q^{4} + 75 q^{6} - 60 q^{7} - 294 q^{8} + 1356 q^{9}+O(q^{10}) \) 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

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