LMFDB - Embedded newform 1075.6.a.b.1.3

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			1.3
Root			$-6.91219$ 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)

$f(q)$	$=$	$q-7.91219 q^{2} -12.8799 q^{3} +30.6028 q^{4} +101.908 q^{6} +172.354 q^{7} +11.0549 q^{8} -77.1083 q^{9} +O(q^{10})$	\(q-7.91219 q^{2} -12.8799 q^{3} +30.6028 q^{4} +101.908 q^{6} +172.354 q^{7} +11.0549 q^{8} -77.1083 q^{9} +452.247 q^{11} -394.161 q^{12} +22.7429 q^{13} -1363.70 q^{14} -1066.76 q^{16} +521.824 q^{17} +610.096 q^{18} +1558.56 q^{19} -2219.90 q^{21} -3578.27 q^{22} +3464.36 q^{23} -142.385 q^{24} -179.946 q^{26} +4122.96 q^{27} +5274.51 q^{28} +4324.79 q^{29} -3987.66 q^{31} +8086.64 q^{32} -5824.90 q^{33} -4128.77 q^{34} -2359.73 q^{36} -10080.4 q^{37} -12331.6 q^{38} -292.926 q^{39} -16408.5 q^{41} +17564.3 q^{42} -1849.00 q^{43} +13840.0 q^{44} -27410.7 q^{46} -24153.7 q^{47} +13739.7 q^{48} +12898.8 q^{49} -6721.04 q^{51} +695.997 q^{52} -21214.2 q^{53} -32621.7 q^{54} +1905.34 q^{56} -20074.0 q^{57} -34218.6 q^{58} -25849.7 q^{59} +28577.7 q^{61} +31551.2 q^{62} -13289.9 q^{63} -29846.8 q^{64} +46087.7 q^{66} -66762.2 q^{67} +15969.3 q^{68} -44620.6 q^{69} -10031.8 q^{71} -852.421 q^{72} -32145.0 q^{73} +79758.3 q^{74} +47696.2 q^{76} +77946.5 q^{77} +2317.69 q^{78} -21913.4 q^{79} -34366.0 q^{81} +129827. q^{82} +66782.4 q^{83} -67935.1 q^{84} +14629.6 q^{86} -55702.8 q^{87} +4999.53 q^{88} +48984.7 q^{89} +3919.82 q^{91} +106019. q^{92} +51360.7 q^{93} +191108. q^{94} -104155. q^{96} -93075.8 q^{97} -102058. q^{98} -34872.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−7.91219	−1.39869	−0.699346	−	0.714784i	$-0.746525\pi$
$2$	−7.91219	−1.39869	−0.699346	+	0.714784i	$0.746525\pi$
$3$	−12.8799	−0.826246	−0.413123	−	0.910675i	$-0.635562\pi$
$3$	−12.8799	−0.826246	−0.413123	+	0.910675i	$0.635562\pi$
$4$	30.6028	0.956338
$5$	0	0
$5$	0	0
$6$	101.908	1.15566
$7$	172.354	1.32946	0.664730	−	0.747083i	$-0.268546\pi$
$7$	172.354	1.32946	0.664730	+	0.747083i	$0.268546\pi$
$8$	11.0549	0.0610700
$9$	−77.1083	−0.317318
$10$	0	0
$11$	452.247	1.12692	0.563461	−	0.826142i	$-0.309469\pi$
$11$	452.247	1.12692	0.563461	+	0.826142i	$0.309469\pi$
$12$	−394.161	−0.790170
$13$	22.7429	0.0373240	0.0186620	−	0.999826i	$-0.494059\pi$
$13$	22.7429	0.0373240	0.0186620	+	0.999826i	$0.494059\pi$
$14$	−1363.70	−1.85950
$15$	0	0
$16$	−1066.76	−1.04176
$17$	521.824	0.437927	0.218964	−	0.975733i	$-0.429732\pi$
$17$	521.824	0.437927	0.218964	+	0.975733i	$0.429732\pi$
$18$	610.096	0.443830
$19$	1558.56	0.990463	0.495232	−	0.868761i	$-0.335083\pi$
$19$	1558.56	0.990463	0.495232	+	0.868761i	$0.335083\pi$
$20$	0	0
$21$	−2219.90	−1.09846
$22$	−3578.27	−1.57622
$23$	3464.36	1.36554	0.682769	−	0.730635i	$-0.260776\pi$
$23$	3464.36	1.36554	0.682769	+	0.730635i	$0.260776\pi$
$24$	−142.385	−0.0504588
$25$	0	0
$26$	−179.946	−0.0522047
$27$	4122.96	1.08843
$28$	5274.51	1.27141
$29$	4324.79	0.954926	0.477463	−	0.878652i	$-0.341556\pi$
$29$	4324.79	0.954926	0.477463	+	0.878652i	$0.341556\pi$
$30$	0	0
$31$	−3987.66	−0.745271	−0.372635	−	0.927978i	$-0.621546\pi$
$31$	−3987.66	−0.745271	−0.372635	+	0.927978i	$0.621546\pi$
$32$	8086.64	1.39602
$33$	−5824.90	−0.931115
$34$	−4128.77	−0.612525
$35$	0	0
$36$	−2359.73	−0.303463
$37$	−10080.4	−1.21053	−0.605264	−	0.796025i	$-0.706932\pi$
$37$	−10080.4	−1.21053	−0.605264	+	0.796025i	$0.706932\pi$
$38$	−12331.6	−1.38535
$39$	−292.926	−0.0308388
$40$	0	0
$41$	−16408.5	−1.52444	−0.762219	−	0.647319i	$-0.775890\pi$
$41$	−16408.5	−1.52444	−0.762219	+	0.647319i	$0.775890\pi$
$42$	17564.3	1.53641
$43$	−1849.00	−0.152499
$43$	−1849.00	−0.152499
$44$	13840.0	1.07772
$45$	0	0
$46$	−27410.7	−1.90997
$47$	−24153.7	−1.59492	−0.797459	−	0.603373i	$-0.793823\pi$
$47$	−24153.7	−1.59492	−0.797459	+	0.603373i	$0.793823\pi$
$48$	13739.7	0.860746
$49$	12898.8	0.767465
$50$	0	0
$51$	−6721.04	−0.361835
$52$	695.997	0.0356943
$53$	−21214.2	−1.03738	−0.518688	−	0.854963i	$-0.673580\pi$
$53$	−21214.2	−1.03738	−0.518688	+	0.854963i	$0.673580\pi$
$54$	−32621.7	−1.52238
$55$	0	0
$56$	1905.34	0.0811902
$57$	−20074.0	−0.818366
$58$	−34218.6	−1.33565
$59$	−25849.7	−0.966775	−0.483387	−	0.875407i	$-0.660594\pi$
$59$	−25849.7	−0.966775	−0.483387	+	0.875407i	$0.660594\pi$
$60$	0	0
$61$	28577.7	0.983338	0.491669	−	0.870782i	$-0.336387\pi$
$61$	28577.7	0.983338	0.491669	+	0.870782i	$0.336387\pi$
$62$	31551.2	1.04240
$63$	−13289.9	−0.421862
$64$	−29846.8	−0.910852
$65$	0	0
$66$	46087.7	1.30234
$67$	−66762.2	−1.81695	−0.908477	−	0.417935i	$-0.862754\pi$
$67$	−66762.2	−1.81695	−0.908477	+	0.417935i	$0.862754\pi$
$68$	15969.3	0.418806
$69$	−44620.6	−1.12827
$70$	0	0
$71$	−10031.8	−0.236173	−0.118087	−	0.993003i	$-0.537676\pi$
$71$	−10031.8	−0.236173	−0.118087	+	0.993003i	$0.537676\pi$
$72$	−852.421	−0.0193786
$73$	−32145.0	−0.706003	−0.353001	−	0.935623i	$-0.614839\pi$
$73$	−32145.0	−0.706003	−0.353001	+	0.935623i	$0.614839\pi$
$74$	79758.3	1.69315
$75$	0	0
$76$	47696.2	0.947217
$77$	77946.5	1.49820
$78$	2317.69	0.0431339
$79$	−21913.4	−0.395040	−0.197520	−	0.980299i	$-0.563289\pi$
$79$	−21913.4	−0.395040	−0.197520	+	0.980299i	$0.563289\pi$
$80$	0	0
$81$	−34366.0	−0.581991
$82$	129827.	2.13222
$83$	66782.4	1.06406	0.532031	−	0.846725i	$-0.321429\pi$
$83$	66782.4	1.06406	0.532031	+	0.846725i	$0.321429\pi$
$84$	−67935.1	−1.05050
$85$	0	0
$86$	14629.6	0.213298
$87$	−55702.8	−0.789004
$88$	4999.53	0.0688212
$89$	48984.7	0.655520	0.327760	−	0.944761i	$-0.393706\pi$
$89$	48984.7	0.655520	0.327760	+	0.944761i	$0.393706\pi$
$90$	0	0
$91$	3919.82	0.0496207
$92$	106019.	1.30591
$93$	51360.7	0.615777
$94$	191108.	2.23080
$95$	0	0
$96$	−104155.	−1.15346
$97$	−93075.8	−1.00440	−0.502201	−	0.864751i	$-0.667476\pi$
$97$	−93075.8	−1.00440	−0.502201	+	0.864751i	$0.667476\pi$
$98$	−102058.	−1.07345
$99$	−34872.0	−0.357593
$100$	0	0
$101$	47004.9	0.458500	0.229250	−	0.973368i	$-0.426373\pi$
$101$	47004.9	0.458500	0.229250	+	0.973368i	$0.426373\pi$
$102$	53178.2	0.506096
$103$	51489.8	0.478220	0.239110	−	0.970992i	$-0.423144\pi$
$103$	51489.8	0.478220	0.239110	+	0.970992i	$0.423144\pi$
$104$	251.420	0.00227938
$105$	0	0
$106$	167851.	1.45097
$107$	−74331.3	−0.627642	−0.313821	−	0.949482i	$-0.601609\pi$
$107$	−74331.3	−0.627642	−0.313821	+	0.949482i	$0.601609\pi$
$108$	126174.	1.04091
$109$	−204775.	−1.65086	−0.825430	−	0.564505i	$-0.809067\pi$
$109$	−204775.	−1.65086	−0.825430	+	0.564505i	$0.809067\pi$
$110$	0	0
$111$	129835.	1.00019
$112$	−183860.	−1.38497
$113$	−125866.	−0.927281	−0.463640	−	0.886023i	$-0.653457\pi$
$113$	−125866.	−0.927281	−0.463640	+	0.886023i	$0.653457\pi$
$114$	158830.	1.14464
$115$	0	0
$116$	132351.	0.913232
$117$	−1753.67	−0.0118436
$118$	204528.	1.35222
$119$	89938.3	0.582207
$120$	0	0
$121$	43476.5	0.269955
$122$	−226112.	−1.37539
$123$	211340.	1.25956
$124$	−122034.	−0.712731
$125$	0	0
$126$	105152.	0.590054
$127$	318214.	1.75069	0.875347	−	0.483495i	$-0.160633\pi$
$127$	318214.	1.75069	0.875347	+	0.483495i	$0.160633\pi$
$128$	−22618.8	−0.122024
$129$	23814.9	0.126001
$130$	0	0
$131$	−261871.	−1.33324	−0.666620	−	0.745398i	$-0.732260\pi$
$131$	−261871.	−1.33324	−0.666620	+	0.745398i	$0.732260\pi$
$132$	−178258.	−0.890460
$133$	268623.	1.31678
$134$	528236.	2.54136
$135$	0	0
$136$	5768.69	0.0267442
$137$	20837.0	0.0948492	0.0474246	−	0.998875i	$-0.484899\pi$
$137$	20837.0	0.0948492	0.0474246	+	0.998875i	$0.484899\pi$
$138$	353047.	1.57810
$139$	−182520.	−0.801260	−0.400630	−	0.916240i	$-0.631209\pi$
$139$	−182520.	−0.801260	−0.400630	+	0.916240i	$0.631209\pi$
$140$	0	0
$141$	311097.	1.31779
$142$	79373.1	0.330334
$143$	10285.4	0.0420612
$144$	82255.9	0.330568
$145$	0	0
$146$	254337.	0.987480
$147$	−166135.	−0.634115
$148$	−308489.	−1.15767
$149$	478460.	1.76555	0.882775	−	0.469796i	$-0.155673\pi$
$149$	478460.	1.76555	0.882775	+	0.469796i	$0.155673\pi$
$150$	0	0
$151$	244853.	0.873901	0.436951	−	0.899485i	$-0.356058\pi$
$151$	244853.	0.873901	0.436951	+	0.899485i	$0.356058\pi$
$152$	17229.6	0.0604876
$153$	−40237.0	−0.138962
$154$	−616727.	−2.09552
$155$	0	0
$156$	−8964.37	−0.0294923
$157$	−12990.1	−0.0420594	−0.0210297	−	0.999779i	$-0.506694\pi$
$157$	−12990.1	−0.0420594	−0.0210297	+	0.999779i	$0.506694\pi$
$158$	173383.	0.552539
$159$	273236.	0.857128
$160$	0	0
$161$	597095.	1.81543
$162$	271910.	0.814026
$163$	−43866.4	−0.129319	−0.0646596	−	0.997907i	$-0.520596\pi$
$163$	−43866.4	−0.129319	−0.0646596	+	0.997907i	$0.520596\pi$
$164$	−502147.	−1.45788
$165$	0	0
$166$	−528395.	−1.48829
$167$	−454884.	−1.26215	−0.631073	−	0.775724i	$-0.717385\pi$
$167$	−454884.	−1.26215	−0.631073	+	0.775724i	$0.717385\pi$
$168$	−24540.6	−0.0670830
$169$	−370776.	−0.998607
$170$	0	0
$171$	−120178.	−0.314292
$172$	−56584.6	−0.145840
$173$	649198.	1.64916	0.824579	−	0.565747i	$-0.191412\pi$
$173$	649198.	1.64916	0.824579	+	0.565747i	$0.191412\pi$
$174$	440731.	1.10357
$175$	0	0
$176$	−482438.	−1.17398
$177$	332941.	0.798793
$178$	−387577.	−0.916870
$179$	−570828.	−1.33160	−0.665798	−	0.746132i	$-0.731909\pi$
$179$	−570828.	−1.33160	−0.665798	+	0.746132i	$0.731909\pi$
$180$	0	0
$181$	−90744.3	−0.205884	−0.102942	−	0.994687i	$-0.532826\pi$
$181$	−90744.3	−0.205884	−0.102942	+	0.994687i	$0.532826\pi$
$182$	−31014.4	−0.0694041
$183$	−368078.	−0.812479
$184$	38298.0	0.0833934
$185$	0	0
$186$	−406376.	−0.861282
$187$	235994.	0.493510
$188$	−739170.	−1.52528
$189$	710607.	1.44702
$190$	0	0
$191$	−546924.	−1.08478	−0.542392	−	0.840126i	$-0.682481\pi$
$191$	−546924.	−1.08478	−0.542392	+	0.840126i	$0.682481\pi$
$192$	384424.	0.752588
$193$	−25064.4	−0.0484356	−0.0242178	−	0.999707i	$-0.507710\pi$
$193$	−25064.4	−0.0484356	−0.0242178	+	0.999707i	$0.507710\pi$
$194$	736434.	1.40485
$195$	0	0
$196$	394739.	0.733956
$197$	−579350.	−1.06359	−0.531797	−	0.846872i	$-0.678483\pi$
$197$	−579350.	−1.06359	−0.531797	+	0.846872i	$0.678483\pi$
$198$	275914.	0.500162
$199$	411324.	0.736295	0.368147	−	0.929767i	$-0.379992\pi$
$199$	411324.	0.736295	0.368147	+	0.929767i	$0.379992\pi$
$200$	0	0
$201$	859891.	1.50125
$202$	−371912.	−0.641301
$203$	745393.	1.26954
$204$	−205683.	−0.346037
$205$	0	0
$206$	−407397.	−0.668883
$207$	−267131.	−0.433310
$208$	−24261.2	−0.0388825
$209$	704852.	1.11618
$210$	0	0
$211$	−385582.	−0.596225	−0.298113	−	0.954531i	$-0.596357\pi$
$211$	−385582.	−0.596225	−0.298113	+	0.954531i	$0.596357\pi$
$212$	−649213.	−0.992083
$213$	129208.	0.195137
$214$	588123.	0.877878
$215$	0	0
$216$	45578.7	0.0664703
$217$	−687288.	−0.990808
$218$	1.62022e6	2.30904
$219$	414024.	0.583332
$220$	0	0
$221$	11867.8	0.0163452
$222$	−1.02728e6	−1.39896
$223$	1.03807e6	1.39786	0.698930	−	0.715190i	$-0.253660\pi$
$223$	1.03807e6	1.39786	0.698930	+	0.715190i	$0.253660\pi$
$224$	1.39376e6	1.85596
$225$	0	0
$226$	995874.	1.29698
$227$	−1.14006e6	−1.46846	−0.734229	−	0.678902i	$-0.762456\pi$
$227$	−1.14006e6	−1.46846	−0.734229	+	0.678902i	$0.762456\pi$
$228$	−614322.	−0.782634
$229$	548999.	0.691804	0.345902	−	0.938271i	$-0.387573\pi$
$229$	548999.	0.691804	0.345902	+	0.938271i	$0.387573\pi$
$230$	0	0
$231$	−1.00394e6	−1.23788
$232$	47809.9	0.0583174
$233$	1.11203e6	1.34191	0.670957	−	0.741496i	$-0.265883\pi$
$233$	1.11203e6	1.34191	0.670957	+	0.741496i	$0.265883\pi$
$234$	13875.4	0.0165655
$235$	0	0
$236$	−791073.	−0.924563
$237$	282242.	0.326400
$238$	−711609.	−0.814328
$239$	−1.28067e6	−1.45025	−0.725123	−	0.688620i	$-0.758217\pi$
$239$	−1.28067e6	−1.45025	−0.725123	+	0.688620i	$0.758217\pi$
$240$	0	0
$241$	725022.	0.804097	0.402049	−	0.915618i	$-0.368298\pi$
$241$	725022.	0.804097	0.402049	+	0.915618i	$0.368298\pi$
$242$	−343995.	−0.377584
$243$	−559249.	−0.607561
$244$	874558.	0.940403
$245$	0	0
$246$	−1.67216e6	−1.76174
$247$	35446.1	0.0369680
$248$	−44083.0	−0.0455137
$249$	−860150.	−0.879176
$250$	0	0
$251$	−1.43637e6	−1.43907	−0.719533	−	0.694458i	$-0.755644\pi$
$251$	−1.43637e6	−1.43907	−0.719533	+	0.694458i	$0.755644\pi$
$252$	−406708.	−0.403442
$253$	1.56675e6	1.53886
$254$	−2.51777e6	−2.44868
$255$	0	0
$256$	1.13406e6	1.08153
$257$	−607070.	−0.573331	−0.286666	−	0.958031i	$-0.592547\pi$
$257$	−607070.	−0.573331	−0.286666	+	0.958031i	$0.592547\pi$
$258$	−188428.	−0.176237
$259$	−1.73740e6	−1.60935
$260$	0	0
$261$	−333477.	−0.303015
$262$	2.07197e6	1.86479
$263$	−169221.	−0.150856	−0.0754282	−	0.997151i	$-0.524032\pi$
$263$	−169221.	−0.150856	−0.0754282	+	0.997151i	$0.524032\pi$
$264$	−64393.4	−0.0568632
$265$	0	0
$266$	−2.12540e6	−1.84177
$267$	−630918.	−0.541620
$268$	−2.04311e6	−1.73762
$269$	1.41683e6	1.19382	0.596908	−	0.802310i	$-0.296396\pi$
$269$	1.41683e6	1.19382	0.596908	+	0.802310i	$0.296396\pi$
$270$	0	0
$271$	947582.	0.783779	0.391889	−	0.920012i	$-0.371822\pi$
$271$	947582.	0.783779	0.391889	+	0.920012i	$0.371822\pi$
$272$	−556660.	−0.456213
$273$	−50486.9	−0.0409989
$274$	−164866.	−0.132665
$275$	0	0
$276$	−1.36552e6	−1.07901
$277$	−37913.3	−0.0296888	−0.0148444	−	0.999890i	$-0.504725\pi$
$277$	−37913.3	−0.0296888	−0.0148444	+	0.999890i	$0.504725\pi$
$278$	1.44413e6	1.12072
$279$	307482.	0.236488
$280$	0	0
$281$	−2.05268e6	−1.55080	−0.775400	−	0.631471i	$-0.782452\pi$
$281$	−2.05268e6	−1.55080	−0.775400	+	0.631471i	$0.782452\pi$
$282$	−2.46146e6	−1.84319
$283$	2.35101e6	1.74497	0.872487	−	0.488636i	$-0.162506\pi$
$283$	2.35101e6	1.74497	0.872487	+	0.488636i	$0.162506\pi$
$284$	−307000.	−0.225861
$285$	0	0
$286$	−81380.2	−0.0588307
$287$	−2.82807e6	−2.02668
$288$	−623547.	−0.442984
$289$	−1.14756e6	−0.808220
$290$	0	0
$291$	1.19881e6	0.829882
$292$	−983727.	−0.675177
$293$	−1.11432e6	−0.758300	−0.379150	−	0.925335i	$-0.623784\pi$
$293$	−1.11432e6	−0.758300	−0.379150	+	0.925335i	$0.623784\pi$
$294$	1.31449e6	0.886931
$295$	0	0
$296$	−111438.	−0.0739269
$297$	1.86460e6	1.22657
$298$	−3.78567e6	−2.46946
$299$	78789.7	0.0509673
$300$	0	0
$301$	−318682.	−0.202741
$302$	−1.93732e6	−1.22232
$303$	−605418.	−0.378834
$304$	−1.66260e6	−1.03182
$305$	0	0
$306$	318363.	0.194365
$307$	1.15903e6	0.701854	0.350927	−	0.936403i	$-0.385866\pi$
$307$	1.15903e6	0.701854	0.350927	+	0.936403i	$0.385866\pi$
$308$	2.38538e6	1.43278
$309$	−663183.	−0.395127
$310$	0	0
$311$	−1.18671e6	−0.695736	−0.347868	−	0.937543i	$-0.613094\pi$
$311$	−1.18671e6	−0.695736	−0.347868	+	0.937543i	$0.613094\pi$
$312$	−3238.26	−0.00188332
$313$	−571287.	−0.329605	−0.164802	−	0.986327i	$-0.552699\pi$
$313$	−571287.	−0.329605	−0.164802	+	0.986327i	$0.552699\pi$
$314$	102780.	0.0588282
$315$	0	0
$316$	−670610.	−0.377792
$317$	796360.	0.445104	0.222552	−	0.974921i	$-0.428561\pi$
$317$	796360.	0.445104	0.222552	+	0.974921i	$0.428561\pi$
$318$	−2.16190e6	−1.19886
$319$	1.95587e6	1.07613
$320$	0	0
$321$	957379.	0.518587
$322$	−4.72433e6	−2.53922
$323$	813292.	0.433751
$324$	−1.05170e6	−0.556580
$325$	0	0
$326$	347080.	0.180878
$327$	2.63748e6	1.36402
$328$	−181394.	−0.0930975
$329$	−4.16297e6	−2.12038
$330$	0	0
$331$	−390295.	−0.195805	−0.0979023	−	0.995196i	$-0.531213\pi$
$331$	−390295.	−0.195805	−0.0979023	+	0.995196i	$0.531213\pi$
$332$	2.04373e6	1.01760
$333$	777285.	0.384122
$334$	3.59913e6	1.76535
$335$	0	0
$336$	2.36809e6	1.14433
$337$	−765533.	−0.367189	−0.183594	−	0.983002i	$-0.558773\pi$
$337$	−765533.	−0.367189	−0.183594	+	0.983002i	$0.558773\pi$
$338$	2.93365e6	1.39674
$339$	1.62114e6	0.766162
$340$	0	0
$341$	−1.80341e6	−0.839863
$342$	950868.	0.439597
$343$	−673595.	−0.309146
$344$	−20440.4	−0.00931309
$345$	0	0
$346$	−5.13658e6	−2.30666
$347$	1.47170e6	0.656140	0.328070	−	0.944653i	$-0.393602\pi$
$347$	1.47170e6	0.656140	0.328070	+	0.944653i	$0.393602\pi$
$348$	−1.70466e6	−0.754554
$349$	3.61738e6	1.58975	0.794877	−	0.606770i	$-0.207535\pi$
$349$	3.61738e6	1.58975	0.794877	+	0.606770i	$0.207535\pi$
$350$	0	0
$351$	93768.2	0.0406245
$352$	3.65716e6	1.57321
$353$	−500946.	−0.213971	−0.106985	−	0.994261i	$-0.534120\pi$
$353$	−500946.	−0.213971	−0.106985	+	0.994261i	$0.534120\pi$
$354$	−2.63430e6	−1.11727
$355$	0	0
$356$	1.49907e6	0.626898
$357$	−1.15840e6	−0.481046
$358$	4.51650e6	1.86249
$359$	−3.84739e6	−1.57554	−0.787770	−	0.615969i	$-0.788765\pi$
$359$	−3.84739e6	−1.57554	−0.787770	+	0.615969i	$0.788765\pi$
$360$	0	0
$361$	−47002.6	−0.0189825
$362$	717987.	0.287968
$363$	−559973.	−0.223049
$364$	119958.	0.0474542
$365$	0	0
$366$	2.91230e6	1.13641
$367$	−2.28572e6	−0.885844	−0.442922	−	0.896560i	$-0.646058\pi$
$367$	−2.28572e6	−0.885844	−0.442922	+	0.896560i	$0.646058\pi$
$368$	−3.69563e6	−1.42256
$369$	1.26523e6	0.483732
$370$	0	0
$371$	−3.65634e6	−1.37915
$372$	1.57178e6	0.588891
$373$	753658.	0.280480	0.140240	−	0.990118i	$-0.455213\pi$
$373$	753658.	0.280480	0.140240	+	0.990118i	$0.455213\pi$
$374$	−1.86723e6	−0.690268
$375$	0	0
$376$	−267015.	−0.0974017
$377$	98358.3	0.0356416
$378$	−5.62246e6	−2.02394
$379$	−3.54911e6	−1.26917	−0.634587	−	0.772851i	$-0.718830\pi$
$379$	−3.54911e6	−1.26917	−0.634587	+	0.772851i	$0.718830\pi$
$380$	0	0
$381$	−4.09857e6	−1.44650
$382$	4.32737e6	1.51728
$383$	−2.29101e6	−0.798049	−0.399025	−	0.916940i	$-0.630651\pi$
$383$	−2.29101e6	−0.798049	−0.399025	+	0.916940i	$0.630651\pi$
$384$	291327.	0.100821
$385$	0	0
$386$	198314.	0.0677464
$387$	142573.	0.0483906
$388$	−2.84838e6	−0.960547
$389$	3.75998e6	1.25983	0.629914	−	0.776665i	$-0.283090\pi$
$389$	3.75998e6	1.25983	0.629914	+	0.776665i	$0.283090\pi$
$390$	0	0
$391$	1.80779e6	0.598006
$392$	142594.	0.0468691
$393$	3.37287e6	1.10158
$394$	4.58393e6	1.48764
$395$	0	0
$396$	−1.06718e6	−0.341980
$397$	−5.66320e6	−1.80337	−0.901686	−	0.432392i	$-0.857670\pi$
$397$	−5.66320e6	−1.80337	−0.901686	+	0.432392i	$0.857670\pi$
$398$	−3.25448e6	−1.02985
$399$	−3.45983e6	−1.08799
$400$	0	0
$401$	3.53734e6	1.09854	0.549270	−	0.835645i	$-0.314906\pi$
$401$	3.53734e6	1.09854	0.549270	+	0.835645i	$0.314906\pi$
$402$	−6.80362e6	−2.09979
$403$	−90691.1	−0.0278165
$404$	1.43848e6	0.438481
$405$	0	0
$406$	−5.89769e6	−1.77569
$407$	−4.55885e6	−1.36417
$408$	−74300.1	−0.0220973
$409$	−2.11034e6	−0.623798	−0.311899	−	0.950115i	$-0.600965\pi$
$409$	−2.11034e6	−0.623798	−0.311899	+	0.950115i	$0.600965\pi$
$410$	0	0
$411$	−268378.	−0.0783688
$412$	1.57573e6	0.457340
$413$	−4.45529e6	−1.28529
$414$	2.11359e6	0.606066
$415$	0	0
$416$	183914.	0.0521052
$417$	2.35084e6	0.662038
$418$	−5.57693e6	−1.56119
$419$	896357.	0.249428	0.124714	−	0.992193i	$-0.460199\pi$
$419$	896357.	0.249428	0.124714	+	0.992193i	$0.460199\pi$
$420$	0	0
$421$	4.09292e6	1.12545	0.562727	−	0.826643i	$-0.309752\pi$
$421$	4.09292e6	1.12545	0.562727	+	0.826643i	$0.309752\pi$
$422$	3.05080e6	0.833935
$423$	1.86245e6	0.506096
$424$	−234520.	−0.0633526
$425$	0	0
$426$	−1.02232e6	−0.272937
$427$	4.92547e6	1.30731
$428$	−2.27475e6	−0.600238
$429$	−132475.	−0.0347529
$430$	0	0
$431$	−1.18319e6	−0.306804	−0.153402	−	0.988164i	$-0.549023\pi$
$431$	−1.18319e6	−0.306804	−0.153402	+	0.988164i	$0.549023\pi$
$432$	−4.39820e6	−1.13388
$433$	−1.63492e6	−0.419061	−0.209530	−	0.977802i	$-0.567194\pi$
$433$	−1.63492e6	−0.419061	−0.209530	+	0.977802i	$0.567194\pi$
$434$	5.43796e6	1.38583
$435$	0	0
$436$	−6.26668e6	−1.57878
$437$	5.39940e6	1.35251
$438$	−3.27584e6	−0.815901
$439$	−2.23900e6	−0.554488	−0.277244	−	0.960800i	$-0.589421\pi$
$439$	−2.23900e6	−0.554488	−0.277244	+	0.960800i	$0.589421\pi$
$440$	0	0
$441$	−994603.	−0.243531
$442$	−93900.4	−0.0228619
$443$	−4.42567e6	−1.07144	−0.535722	−	0.844394i	$-0.679961\pi$
$443$	−4.42567e6	−1.07144	−0.535722	+	0.844394i	$0.679961\pi$
$444$	3.97331e6	0.956522
$445$	0	0
$446$	−8.21339e6	−1.95517
$447$	−6.16252e6	−1.45878
$448$	−5.14421e6	−1.21094
$449$	824623.	0.193037	0.0965183	−	0.995331i	$-0.469229\pi$
$449$	824623.	0.193037	0.0965183	+	0.995331i	$0.469229\pi$
$450$	0	0
$451$	−7.42071e6	−1.71792
$452$	−3.85184e6	−0.886794
$453$	−3.15368e6	−0.722057
$454$	9.02034e6	2.05392
$455$	0	0
$456$	−221916.	−0.0499776
$457$	−4.78062e6	−1.07076	−0.535382	−	0.844610i	$-0.679832\pi$
$457$	−4.78062e6	−1.07076	−0.535382	+	0.844610i	$0.679832\pi$
$458$	−4.34379e6	−0.967621
$459$	2.15146e6	0.476652
$460$	0	0
$461$	177573.	0.0389157	0.0194578	−	0.999811i	$-0.493806\pi$
$461$	177573.	0.0389157	0.0194578	+	0.999811i	$0.493806\pi$
$462$	7.94339e6	1.73141
$463$	−8.24149e6	−1.78671	−0.893353	−	0.449355i	$-0.851654\pi$
$463$	−8.24149e6	−1.78671	−0.893353	+	0.449355i	$0.851654\pi$
$464$	−4.61350e6	−0.994800
$465$	0	0
$466$	−8.79856e6	−1.87692
$467$	−2.93483e6	−0.622718	−0.311359	−	0.950292i	$-0.600784\pi$
$467$	−2.93483e6	−0.622718	−0.311359	+	0.950292i	$0.600784\pi$
$468$	−53667.1	−0.0113265
$469$	−1.15067e7	−2.41557
$470$	0	0
$471$	167311.	0.0347514
$472$	−285765.	−0.0590410
$473$	−836205.	−0.171854
$474$	−2.23315e6	−0.456533
$475$	0	0
$476$	2.75236e6	0.556786
$477$	1.63579e6	0.329178
$478$	1.01329e7	2.02845
$479$	7.53163e6	1.49986	0.749929	−	0.661518i	$-0.230088\pi$
$479$	7.53163e6	1.49986	0.749929	+	0.661518i	$0.230088\pi$
$480$	0	0
$481$	−229258.	−0.0451817
$482$	−5.73651e6	−1.12468
$483$	−7.69053e6	−1.49999
$484$	1.33050e6	0.258168
$485$	0	0
$486$	4.42489e6	0.849790
$487$	2.30284e6	0.439988	0.219994	−	0.975501i	$-0.429396\pi$
$487$	2.30284e6	0.439988	0.219994	+	0.975501i	$0.429396\pi$
$488$	315922.	0.0600525
$489$	564995.	0.106849
$490$	0	0
$491$	4.43974e6	0.831100	0.415550	−	0.909570i	$-0.363589\pi$
$491$	4.43974e6	0.831100	0.415550	+	0.909570i	$0.363589\pi$
$492$	6.46760e6	1.20457
$493$	2.25678e6	0.418188
$494$	−280456.	−0.0517068
$495$	0	0
$496$	4.25387e6	0.776390
$497$	−1.72901e6	−0.313983
$498$	6.80567e6	1.22970
$499$	7.95785e6	1.43069	0.715343	−	0.698774i	$-0.246271\pi$
$499$	7.95785e6	1.43069	0.715343	+	0.698774i	$0.246271\pi$
$500$	0	0
$501$	5.85886e6	1.04284
$502$	1.13648e7	2.01281
$503$	4.75804e6	0.838510	0.419255	−	0.907868i	$-0.362291\pi$
$503$	4.75804e6	0.838510	0.419255	+	0.907868i	$0.362291\pi$
$504$	−146918.	−0.0257631
$505$	0	0
$506$	−1.23964e7	−2.15238
$507$	4.77555e6	0.825095
$508$	9.73825e6	1.67425
$509$	−5.42576e6	−0.928252	−0.464126	−	0.885769i	$-0.653632\pi$
$509$	−5.42576e6	−0.928252	−0.464126	+	0.885769i	$0.653632\pi$
$510$	0	0
$511$	−5.54031e6	−0.938603
$512$	−8.24912e6	−1.39070
$513$	6.42587e6	1.07805
$514$	4.80325e6	0.801914
$515$	0	0
$516$	728804.	0.120500
$517$	−1.09234e7	−1.79735
$518$	1.37466e7	2.25098
$519$	−8.36161e6	−1.36261
$520$	0	0
$521$	−8.88444e6	−1.43396	−0.716978	−	0.697096i	$-0.754475\pi$
$521$	−8.88444e6	−1.43396	−0.716978	+	0.697096i	$0.754475\pi$
$522$	2.63853e6	0.423825
$523$	5.03938e6	0.805606	0.402803	−	0.915287i	$-0.368036\pi$
$523$	5.03938e6	0.805606	0.402803	+	0.915287i	$0.368036\pi$
$524$	−8.01397e6	−1.27503
$525$	0	0
$526$	1.33891e6	0.211002
$527$	−2.08086e6	−0.326374
$528$	6.21375e6	0.969995
$529$	5.56545e6	0.864692
$530$	0	0
$531$	1.99323e6	0.306775
$532$	8.22061e6	1.25929
$533$	−373178.	−0.0568981
$534$	4.99195e6	0.757560
$535$	0	0
$536$	−738047.	−0.110961
$537$	7.35221e6	1.10023
$538$	−1.12102e7	−1.66978
$539$	5.83344e6	0.864874
$540$	0	0
$541$	3.49430e6	0.513295	0.256648	−	0.966505i	$-0.417382\pi$
$541$	3.49430e6	0.513295	0.256648	+	0.966505i	$0.417382\pi$
$542$	−7.49745e6	−1.09626
$543$	1.16878e6	0.170111
$544$	4.21980e6	0.611357
$545$	0	0
$546$	399462.	0.0573448
$547$	2.50382e6	0.357796	0.178898	−	0.983868i	$-0.442747\pi$
$547$	2.50382e6	0.357796	0.178898	+	0.983868i	$0.442747\pi$
$548$	637671.	0.0907079
$549$	−2.20358e6	−0.312031
$550$	0	0
$551$	6.74042e6	0.945819
$552$	−493274.	−0.0689034
$553$	−3.77685e6	−0.525190
$554$	299977.	0.0415254
$555$	0	0
$556$	−5.58563e6	−0.766276
$557$	1.53305e6	0.209371	0.104686	−	0.994505i	$-0.466616\pi$
$557$	1.53305e6	0.209371	0.104686	+	0.994505i	$0.466616\pi$
$558$	−2.43286e6	−0.330774
$559$	−42051.7	−0.00569185
$560$	0	0
$561$	−3.03957e6	−0.407761
$562$	1.62412e7	2.16909
$563$	−8.52078e6	−1.13294	−0.566472	−	0.824081i	$-0.691692\pi$
$563$	−8.52078e6	−1.13294	−0.566472	+	0.824081i	$0.691692\pi$
$564$	9.52043e6	1.26026
$565$	0	0
$566$	−1.86017e7	−2.44068
$567$	−5.92311e6	−0.773734
$568$	−110900.	−0.0144231
$569$	−429293.	−0.0555870	−0.0277935	−	0.999614i	$-0.508848\pi$
$569$	−429293.	−0.0555870	−0.0277935	+	0.999614i	$0.508848\pi$
$570$	0	0
$571$	−6.30376e6	−0.809114	−0.404557	−	0.914513i	$-0.632574\pi$
$571$	−6.30376e6	−0.809114	−0.404557	+	0.914513i	$0.632574\pi$
$572$	314763.	0.0402247
$573$	7.04432e6	0.896298
$574$	2.23762e7	2.83470
$575$	0	0
$576$	2.30144e6	0.289030
$577$	8.10411e6	1.01337	0.506683	−	0.862133i	$-0.330872\pi$
$577$	8.10411e6	1.01337	0.506683	+	0.862133i	$0.330872\pi$
$578$	9.07969e6	1.13045
$579$	322827.	0.0400197
$580$	0	0
$581$	1.15102e7	1.41463
$582$	−9.48519e6	−1.16075
$583$	−9.59405e6	−1.16904
$584$	−355358.	−0.0431156
$585$	0	0
$586$	8.81672e6	1.06063
$587$	−5.93951e6	−0.711468	−0.355734	−	0.934587i	$-0.615769\pi$
$587$	−5.93951e6	−0.711468	−0.355734	+	0.934587i	$0.615769\pi$
$588$	−5.08420e6	−0.606428
$589$	−6.21499e6	−0.738163
$590$	0	0
$591$	7.46197e6	0.878789
$592$	1.07534e7	1.26107
$593$	1.03054e6	0.120345	0.0601727	−	0.998188i	$-0.480835\pi$
$593$	1.03054e6	0.120345	0.0601727	+	0.998188i	$0.480835\pi$
$594$	−1.47531e7	−1.71560
$595$	0	0
$596$	1.46422e7	1.68846
$597$	−5.29781e6	−0.608360
$598$	−623399.	−0.0712875
$599$	8.75551e6	0.997044	0.498522	−	0.866877i	$-0.333876\pi$
$599$	8.75551e6	0.997044	0.498522	+	0.866877i	$0.333876\pi$
$600$	0	0
$601$	−6.36534e6	−0.718845	−0.359423	−	0.933175i	$-0.617026\pi$
$601$	−6.36534e6	−0.718845	−0.359423	+	0.933175i	$0.617026\pi$
$602$	2.52147e6	0.283572
$603$	5.14792e6	0.576552
$604$	7.49318e6	0.835745
$605$	0	0
$606$	4.79019e6	0.529872
$607$	−1.44657e7	−1.59356	−0.796779	−	0.604270i	$-0.793465\pi$
$607$	−1.44657e7	−1.59356	−0.796779	+	0.604270i	$0.793465\pi$
$608$	1.26035e7	1.38271
$609$	−9.60058e6	−1.04895
$610$	0	0
$611$	−549325.	−0.0595287
$612$	−1.23136e6	−0.132895
$613$	1.78097e7	1.91428	0.957142	−	0.289619i	$-0.0935286\pi$
$613$	1.78097e7	1.91428	0.957142	+	0.289619i	$0.0935286\pi$
$614$	−9.17043e6	−0.981678
$615$	0	0
$616$	861687.	0.0914951
$617$	−4.11393e6	−0.435054	−0.217527	−	0.976054i	$-0.569799\pi$
$617$	−4.11393e6	−0.435054	−0.217527	+	0.976054i	$0.569799\pi$
$618$	5.24723e6	0.552661
$619$	−3.56813e6	−0.374295	−0.187148	−	0.982332i	$-0.559924\pi$
$619$	−3.56813e6	−0.374295	−0.187148	+	0.982332i	$0.559924\pi$
$620$	0	0
$621$	1.42834e7	1.48629
$622$	9.38950e6	0.973120
$623$	8.44270e6	0.871488
$624$	312482.	0.0321265
$625$	0	0
$626$	4.52013e6	0.461015
$627$	−9.07843e6	−0.922235
$628$	−397534.	−0.0402230
$629$	−5.26021e6	−0.530123
$630$	0	0
$631$	8.10764e6	0.810627	0.405314	−	0.914178i	$-0.367162\pi$
$631$	8.10764e6	0.810627	0.405314	+	0.914178i	$0.367162\pi$
$632$	−242249.	−0.0241251
$633$	4.96625e6	0.492629
$634$	−6.30095e6	−0.622562
$635$	0	0
$636$	8.36180e6	0.819704
$637$	293356.	0.0286448
$638$	−1.54752e7	−1.50517
$639$	773531.	0.0749421
$640$	0	0
$641$	1.15748e7	1.11267	0.556335	−	0.830958i	$-0.312207\pi$
$641$	1.15748e7	1.11267	0.556335	+	0.830958i	$0.312207\pi$
$642$	−7.57497e6	−0.725343
$643$	865667.	0.0825702	0.0412851	−	0.999147i	$-0.486855\pi$
$643$	865667.	0.0825702	0.0412851	+	0.999147i	$0.486855\pi$
$644$	1.82728e7	1.73616
$645$	0	0
$646$	−6.43493e6	−0.606684
$647$	3.32273e6	0.312057	0.156029	−	0.987753i	$-0.450131\pi$
$647$	3.32273e6	0.312057	0.156029	+	0.987753i	$0.450131\pi$
$648$	−379911.	−0.0355422
$649$	−1.16904e7	−1.08948
$650$	0	0
$651$	8.85220e6	0.818651
$652$	−1.34244e6	−0.123673
$653$	1.42689e7	1.30951	0.654754	−	0.755842i	$-0.272772\pi$
$653$	1.42689e7	1.30951	0.654754	+	0.755842i	$0.272772\pi$
$654$	−2.08682e7	−1.90784
$655$	0	0
$656$	1.75039e7	1.58809
$657$	2.47865e6	0.224027
$658$	3.29382e7	2.96576
$659$	−1.73298e6	−0.155446	−0.0777230	−	0.996975i	$-0.524765\pi$
$659$	−1.73298e6	−0.155446	−0.0777230	+	0.996975i	$0.524765\pi$
$660$	0	0
$661$	456270.	0.0406180	0.0203090	−	0.999794i	$-0.493535\pi$
$661$	456270.	0.0406180	0.0203090	+	0.999794i	$0.493535\pi$
$662$	3.08809e6	0.273870
$663$	−152856.	−0.0135051
$664$	738269.	0.0649822
$665$	0	0
$666$	−6.15003e6	−0.537269
$667$	1.49826e7	1.30399
$668$	−1.39207e7	−1.20704
$669$	−1.33702e7	−1.15498
$670$	0	0
$671$	1.29242e7	1.10815
$672$	−1.79515e7	−1.53348
$673$	1.34881e7	1.14793	0.573963	−	0.818881i	$-0.305405\pi$
$673$	1.34881e7	1.14793	0.573963	+	0.818881i	$0.305405\pi$
$674$	6.05705e6	0.513584
$675$	0	0
$676$	−1.13468e7	−0.955005
$677$	−1.96810e7	−1.65035	−0.825174	−	0.564879i	$-0.808923\pi$
$677$	−1.96810e7	−1.65035	−0.825174	+	0.564879i	$0.808923\pi$
$678$	−1.28268e7	−1.07162
$679$	−1.60420e7	−1.33531
$680$	0	0
$681$	1.46838e7	1.21331
$682$	1.42689e7	1.17471
$683$	−6.25887e6	−0.513386	−0.256693	−	0.966493i	$-0.582633\pi$
$683$	−6.25887e6	−0.513386	−0.256693	+	0.966493i	$0.582633\pi$
$684$	−3.67777e6	−0.300569
$685$	0	0
$686$	5.32962e6	0.432400
$687$	−7.07106e6	−0.571600
$688$	1.97244e6	0.158866
$689$	−482472.	−0.0387190
$690$	0	0
$691$	3.10391e6	0.247295	0.123647	−	0.992326i	$-0.460541\pi$
$691$	3.10391e6	0.247295	0.123647	+	0.992326i	$0.460541\pi$
$692$	1.98673e7	1.57715
$693$	−6.01032e6	−0.475406
$694$	−1.16444e7	−0.917738
$695$	0	0
$696$	−615786.	−0.0481845
$697$	−8.56236e6	−0.667593
$698$	−2.86214e7	−2.22358
$699$	−1.43228e7	−1.10875
$700$	0	0
$701$	5.25605e6	0.403984	0.201992	−	0.979387i	$-0.435258\pi$
$701$	5.25605e6	0.403984	0.201992	+	0.979387i	$0.435258\pi$
$702$	−741912.	−0.0568211
$703$	−1.57109e7	−1.19898
$704$	−1.34981e7	−1.02646
$705$	0	0
$706$	3.96358e6	0.299279
$707$	8.10147e6	0.609558
$708$	1.01889e7	0.763916
$709$	8.29295e6	0.619574	0.309787	−	0.950806i	$-0.399742\pi$
$709$	8.29295e6	0.619574	0.309787	+	0.950806i	$0.399742\pi$
$710$	0	0
$711$	1.68970e6	0.125353
$712$	541519.	0.0400326
$713$	−1.38147e7	−1.01770
$714$	9.16545e6	0.672835
$715$	0	0
$716$	−1.74689e7	−1.27346
$717$	1.64949e7	1.19826
$718$	3.04413e7	2.20370
$719$	3.71146e6	0.267746	0.133873	−	0.990999i	$-0.457259\pi$
$719$	3.71146e6	0.267746	0.133873	+	0.990999i	$0.457259\pi$
$720$	0	0
$721$	8.87445e6	0.635775
$722$	371894.	0.0265507
$723$	−9.33821e6	−0.664382
$724$	−2.77703e6	−0.196895
$725$	0	0
$726$	4.43061e6	0.311977
$727$	3.10084e6	0.217592	0.108796	−	0.994064i	$-0.465300\pi$
$727$	3.10084e6	0.217592	0.108796	+	0.994064i	$0.465300\pi$
$728$	43333.1	0.00303034
$729$	1.55540e7	1.08399
$730$	0	0
$731$	−964853.	−0.0667833
$732$	−1.12642e7	−0.777004
$733$	1.69858e7	1.16769	0.583844	−	0.811866i	$-0.301548\pi$
$733$	1.69858e7	1.16769	0.583844	+	0.811866i	$0.301548\pi$
$734$	1.80850e7	1.23902
$735$	0	0
$736$	2.80150e7	1.90632
$737$	−3.01930e7	−2.04757
$738$	−1.00108e7	−0.676591
$739$	1.13197e7	0.762473	0.381236	−	0.924478i	$-0.375498\pi$
$739$	1.13197e7	0.762473	0.381236	+	0.924478i	$0.375498\pi$
$740$	0	0
$741$	−456542.	−0.0305447
$742$	2.89297e7	1.92901
$743$	9.42589e6	0.626398	0.313199	−	0.949688i	$-0.398599\pi$
$743$	9.42589e6	0.626398	0.313199	+	0.949688i	$0.398599\pi$
$744$	567785.	0.0376055
$745$	0	0
$746$	−5.96309e6	−0.392305
$747$	−5.14947e6	−0.337646
$748$	7.22206e6	0.471962
$749$	−1.28113e7	−0.834425
$750$	0	0
$751$	−3.73658e6	−0.241754	−0.120877	−	0.992667i	$-0.538571\pi$
$751$	−3.73658e6	−0.241754	−0.120877	+	0.992667i	$0.538571\pi$
$752$	2.57661e7	1.66152
$753$	1.85002e7	1.18902
$754$	−778230.	−0.0498516
$755$	0	0
$756$	2.17466e7	1.38384
$757$	−1.39033e7	−0.881815	−0.440908	−	0.897552i	$-0.645343\pi$
$757$	−1.39033e7	−0.881815	−0.440908	+	0.897552i	$0.645343\pi$
$758$	2.80812e7	1.77518
$759$	−2.01795e7	−1.27147
$760$	0	0
$761$	−2.95291e7	−1.84837	−0.924185	−	0.381946i	$-0.875254\pi$
$761$	−2.95291e7	−1.84837	−0.924185	+	0.381946i	$0.875254\pi$
$762$	3.24286e7	2.02321
$763$	−3.52937e7	−2.19475
$764$	−1.67374e7	−1.03742
$765$	0	0
$766$	1.81269e7	1.11622
$767$	−587897.	−0.0360839
$768$	−1.46066e7	−0.893606
$769$	−9.04293e6	−0.551434	−0.275717	−	0.961239i	$-0.588915\pi$
$769$	−9.04293e6	−0.551434	−0.275717	+	0.961239i	$0.588915\pi$
$770$	0	0
$771$	7.81899e6	0.473713
$772$	−767041.	−0.0463207
$773$	661253.	0.0398033	0.0199016	−	0.999802i	$-0.493665\pi$
$773$	661253.	0.0398033	0.0199016	+	0.999802i	$0.493665\pi$
$774$	−1.12807e6	−0.0676834
$775$	0	0
$776$	−1.02894e6	−0.0613388
$777$	2.23775e7	1.32972
$778$	−2.97497e7	−1.76211
$779$	−2.55736e7	−1.50990
$780$	0	0
$781$	−4.53683e6	−0.266149
$782$	−1.43036e7	−0.836426
$783$	1.78309e7	1.03937
$784$	−1.37599e7	−0.799511
$785$	0	0
$786$	−2.66868e7	−1.54078
$787$	6.46523e6	0.372089	0.186045	−	0.982541i	$-0.440433\pi$
$787$	6.46523e6	0.372089	0.186045	+	0.982541i	$0.440433\pi$
$788$	−1.77297e7	−1.01715
$789$	2.17954e6	0.124644
$790$	0	0
$791$	−2.16934e7	−1.23278
$792$	−385505.	−0.0218382
$793$	649940.	0.0367021
$794$	4.48083e7	2.52236
$795$	0	0
$796$	1.25877e7	0.704146
$797$	−2.62978e7	−1.46647	−0.733234	−	0.679976i	$-0.761990\pi$
$797$	−2.62978e7	−1.46647	−0.733234	+	0.679976i	$0.761990\pi$
$798$	2.73749e7	1.52176
$799$	−1.26040e7	−0.698458
$800$	0	0
$801$	−3.77713e6	−0.208008
$802$	−2.79881e7	−1.53652
$803$	−1.45375e7	−0.795610
$804$	2.63151e7	1.43570
$805$	0	0
$806$	717565.	0.0389066
$807$	−1.82486e7	−0.986385
$808$	519632.	0.0280006
$809$	−9.92533e6	−0.533180	−0.266590	−	0.963810i	$-0.585897\pi$
$809$	−9.92533e6	−0.533180	−0.266590	+	0.963810i	$0.585897\pi$
$810$	0	0
$811$	1.86128e7	0.993712	0.496856	−	0.867833i	$-0.334488\pi$
$811$	1.86128e7	0.993712	0.496856	+	0.867833i	$0.334488\pi$
$812$	2.28111e7	1.21411
$813$	−1.22048e7	−0.647594
$814$	3.60705e7	1.90805
$815$	0	0
$816$	7.16972e6	0.376944
$817$	−2.88177e6	−0.151044
$818$	1.66974e7	0.872501
$819$	−302251.	−0.0157456
$820$	0	0
$821$	1.08487e7	0.561718	0.280859	−	0.959749i	$-0.409381\pi$
$821$	1.08487e7	0.561718	0.280859	+	0.959749i	$0.409381\pi$
$822$	2.12346e6	0.109614
$823$	1.29473e7	0.666316	0.333158	−	0.942871i	$-0.391886\pi$
$823$	1.29473e7	0.666316	0.333158	+	0.942871i	$0.391886\pi$
$824$	569212.	0.0292049
$825$	0	0
$826$	3.52511e7	1.79772
$827$	−2.79589e7	−1.42153	−0.710767	−	0.703428i	$-0.751652\pi$
$827$	−2.79589e7	−1.42153	−0.710767	+	0.703428i	$0.751652\pi$
$828$	−8.17496e6	−0.414390
$829$	−1.95464e7	−0.987825	−0.493913	−	0.869512i	$-0.664434\pi$
$829$	−1.95464e7	−0.987825	−0.493913	+	0.869512i	$0.664434\pi$
$830$	0	0
$831$	488319.	0.0245302
$832$	−678803.	−0.0339966
$833$	6.73090e6	0.336094
$834$	−1.86003e7	−0.925987
$835$	0	0
$836$	2.15705e7	1.06744
$837$	−1.64410e7	−0.811174
$838$	−7.09215e6	−0.348873
$839$	1.66180e7	0.815031	0.407516	−	0.913198i	$-0.366395\pi$
$839$	1.66180e7	0.815031	0.407516	+	0.913198i	$0.366395\pi$
$840$	0	0
$841$	−1.80736e6	−0.0881161
$842$	−3.23840e7	−1.57416
$843$	2.64383e7	1.28134
$844$	−1.17999e7	−0.570193
$845$	0	0
$846$	−1.47360e7	−0.707873
$847$	7.49334e6	0.358894
$848$	2.26304e7	1.08069
$849$	−3.02808e7	−1.44178
$850$	0	0
$851$	−3.49222e7	−1.65302
$852$	3.95412e6	0.186617
$853$	1.69144e7	0.795947	0.397974	−	0.917397i	$-0.369714\pi$
$853$	1.69144e7	0.795947	0.397974	+	0.917397i	$0.369714\pi$
$854$	−3.89713e7	−1.82852
$855$	0	0
$856$	−821721.	−0.0383301
$857$	−767365.	−0.0356903	−0.0178451	−	0.999841i	$-0.505681\pi$
$857$	−767365.	−0.0356903	−0.0178451	+	0.999841i	$0.505681\pi$
$858$	1.04817e6	0.0486086
$859$	1.25709e7	0.581278	0.290639	−	0.956833i	$-0.406132\pi$
$859$	1.25709e7	0.581278	0.290639	+	0.956833i	$0.406132\pi$
$860$	0	0
$861$	3.64252e7	1.67454
$862$	9.36163e6	0.429124
$863$	−4.15225e6	−0.189783	−0.0948913	−	0.995488i	$-0.530250\pi$
$863$	−4.15225e6	−0.189783	−0.0948913	+	0.995488i	$0.530250\pi$
$864$	3.33409e7	1.51947
$865$	0	0
$866$	1.29358e7	0.586137
$867$	1.47804e7	0.667788
$868$	−2.10329e7	−0.947547
$869$	−9.91025e6	−0.445180
$870$	0	0
$871$	−1.51837e6	−0.0678159
$872$	−2.26375e6	−0.100818
$873$	7.17691e6	0.318715
$874$	−4.27211e7	−1.89175
$875$	0	0
$876$	1.26703e7	0.557862
$877$	2.05982e6	0.0904337	0.0452168	−	0.998977i	$-0.485602\pi$
$877$	2.05982e6	0.0904337	0.0452168	+	0.998977i	$0.485602\pi$
$878$	1.77154e7	0.775558
$879$	1.43523e7	0.626542
$880$	0	0
$881$	3.31902e6	0.144069	0.0720345	−	0.997402i	$-0.477051\pi$
$881$	3.31902e6	0.144069	0.0720345	+	0.997402i	$0.477051\pi$
$882$	7.86949e6	0.340624
$883$	−3.88804e6	−0.167814	−0.0839071	−	0.996474i	$-0.526740\pi$
$883$	−3.88804e6	−0.167814	−0.0839071	+	0.996474i	$0.526740\pi$
$884$	363188.	0.0156315
$885$	0	0
$886$	3.50168e7	1.49862
$887$	4.11225e6	0.175497	0.0877486	−	0.996143i	$-0.472033\pi$
$887$	4.11225e6	0.175497	0.0877486	+	0.996143i	$0.472033\pi$
$888$	1.43531e6	0.0610818
$889$	5.48454e7	2.32748
$890$	0	0
$891$	−1.55419e7	−0.655859
$892$	3.17678e7	1.33683
$893$	−3.76448e7	−1.57971
$894$	4.87590e7	2.04038
$895$	0	0
$896$	−3.89843e6	−0.162226
$897$	−1.01480e6	−0.0421115
$898$	−6.52458e6	−0.269999
$899$	−1.72458e7	−0.711679
$900$	0	0
$901$	−1.10701e7	−0.454295
$902$	5.87141e7	2.40285
$903$	4.10459e6	0.167514
$904$	−1.39143e6	−0.0566291
$905$	0	0
$906$	2.49525e7	1.00994
$907$	4.04945e7	1.63447	0.817236	−	0.576303i	$-0.195505\pi$
$907$	4.04945e7	1.63447	0.817236	+	0.576303i	$0.195505\pi$
$908$	−3.48889e7	−1.40434
$909$	−3.62447e6	−0.145490
$910$	0	0
$911$	−3.27575e7	−1.30772	−0.653860	−	0.756615i	$-0.726852\pi$
$911$	−3.27575e7	−1.30772	−0.653860	+	0.756615i	$0.726852\pi$
$912$	2.14141e7	0.852538
$913$	3.02021e7	1.19911
$914$	3.78252e7	1.49767
$915$	0	0
$916$	1.68009e7	0.661599
$917$	−4.51343e7	−1.77249
$918$	−1.70228e7	−0.666690
$919$	−8.68701e6	−0.339298	−0.169649	−	0.985505i	$-0.554263\pi$
$919$	−8.68701e6	−0.339298	−0.169649	+	0.985505i	$0.554263\pi$
$920$	0	0
$921$	−1.49281e7	−0.579904
$922$	−1.40499e6	−0.0544310
$923$	−228151.	−0.00881492
$924$	−3.07235e7	−1.18383
$925$	0	0
$926$	6.52083e7	2.49905
$927$	−3.97029e6	−0.151748
$928$	3.49730e7	1.33310
$929$	−6.29057e6	−0.239139	−0.119570	−	0.992826i	$-0.538151\pi$
$929$	−6.29057e6	−0.239139	−0.119570	+	0.992826i	$0.538151\pi$
$930$	0	0
$931$	2.01035e7	0.760146
$932$	3.40311e7	1.28332
$933$	1.52847e7	0.574849
$934$	2.32210e7	0.870990
$935$	0	0
$936$	−19386.5	−0.000723287 0
$937$	6.97275e6	0.259451	0.129725	−	0.991550i	$-0.458590\pi$
$937$	6.97275e6	0.259451	0.129725	+	0.991550i	$0.458590\pi$
$938$	9.10434e7	3.37863
$939$	7.35812e6	0.272335
$940$	0	0
$941$	1.67294e7	0.615894	0.307947	−	0.951404i	$-0.400358\pi$
$941$	1.67294e7	0.615894	0.307947	+	0.951404i	$0.400358\pi$
$942$	−1.32380e6	−0.0486065
$943$	−5.68450e7	−2.08168
$944$	2.75754e7	1.00714
$945$	0	0
$946$	6.61622e6	0.240371
$947$	4.20327e7	1.52304	0.761522	−	0.648140i	$-0.224453\pi$
$947$	4.20327e7	1.52304	0.761522	+	0.648140i	$0.224453\pi$
$948$	8.63739e6	0.312149
$949$	−731071.	−0.0263508
$950$	0	0
$951$	−1.02570e7	−0.367765
$952$	994255.	0.0355554
$953$	−4.38940e7	−1.56557	−0.782787	−	0.622290i	$-0.786202\pi$
$953$	−4.38940e7	−1.56557	−0.782787	+	0.622290i	$0.786202\pi$
$954$	−1.29427e7	−0.460419
$955$	0	0
$956$	−3.91920e7	−1.38692
$957$	−2.51914e7	−0.889146
$958$	−5.95917e7	−2.09784
$959$	3.59133e6	0.126098
$960$	0	0
$961$	−1.27277e7	−0.444571
$962$	1.81394e6	0.0631952
$963$	5.73156e6	0.199162
$964$	2.21877e7	0.768989
$965$	0	0
$966$	6.08489e7	2.09802
$967$	−1.42055e6	−0.0488530	−0.0244265	−	0.999702i	$-0.507776\pi$
$967$	−1.42055e6	−0.0488530	−0.0244265	+	0.999702i	$0.507776\pi$
$968$	480626.	0.0164862
$969$	−1.04751e7	−0.358385
$970$	0	0
$971$	−4.66877e7	−1.58911	−0.794556	−	0.607191i	$-0.792296\pi$
$971$	−4.66877e7	−1.58911	−0.794556	+	0.607191i	$0.792296\pi$
$972$	−1.71146e7	−0.581033
$973$	−3.14580e7	−1.06524
$974$	−1.82205e7	−0.615407
$975$	0	0
$976$	−3.04855e7	−1.02440
$977$	−5.11806e6	−0.171541	−0.0857707	−	0.996315i	$-0.527335\pi$
$977$	−5.11806e6	−0.171541	−0.0857707	+	0.996315i	$0.527335\pi$
$978$	−4.47035e6	−0.149449
$979$	2.21532e7	0.738720
$980$	0	0
$981$	1.57898e7	0.523848
$982$	−3.51280e7	−1.16245
$983$	1.29865e6	0.0428654	0.0214327	−	0.999770i	$-0.493177\pi$
$983$	1.29865e6	0.0428654	0.0214327	+	0.999770i	$0.493177\pi$
$984$	2.33633e6	0.0769214
$985$	0	0
$986$	−1.78561e7	−0.584916
$987$	5.36187e7	1.75196
$988$	1.08475e6	0.0353539
$989$	−6.40560e6	−0.208242
$990$	0	0
$991$	2.92905e7	0.947422	0.473711	−	0.880680i	$-0.342914\pi$
$991$	2.92905e7	0.947422	0.473711	+	0.880680i	$0.342914\pi$
$992$	−3.22468e7	−1.04042
$993$	5.02696e6	0.161783
$994$	1.36803e7	0.439165
$995$	0	0
$996$	−2.63230e7	−0.840789
$997$	2.66478e7	0.849031	0.424515	−	0.905421i	$-0.360445\pi$
$997$	2.66478e7	0.849031	0.424515	+	0.905421i	$0.360445\pi$
$998$	−6.29640e7	−2.00109
$999$	−4.15612e7	−1.31757

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1075.6.a.b.1.3		10
5.4	even	2		43.6.a.b.1.8	✓	10
15.14	odd	2		387.6.a.e.1.3		10
20.19	odd	2		688.6.a.h.1.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.a.b.1.8	✓	10	5.4	even	2
387.6.a.e.1.3		10	15.14	odd	2
688.6.a.h.1.5		10	20.19	odd	2
1075.6.a.b.1.3		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)

\(f(q)\)	\(=\)	\(q-7.91219 q^{2} -12.8799 q^{3} +30.6028 q^{4} +101.908 q^{6} +172.354 q^{7} +11.0549 q^{8} -77.1083 q^{9} +O(q^{10})\)	\(q-7.91219 q^{2} -12.8799 q^{3} +30.6028 q^{4} +101.908 q^{6} +172.354 q^{7} +11.0549 q^{8} -77.1083 q^{9} +452.247 q^{11} -394.161 q^{12} +22.7429 q^{13} -1363.70 q^{14} -1066.76 q^{16} +521.824 q^{17} +610.096 q^{18} +1558.56 q^{19} -2219.90 q^{21} -3578.27 q^{22} +3464.36 q^{23} -142.385 q^{24} -179.946 q^{26} +4122.96 q^{27} +5274.51 q^{28} +4324.79 q^{29} -3987.66 q^{31} +8086.64 q^{32} -5824.90 q^{33} -4128.77 q^{34} -2359.73 q^{36} -10080.4 q^{37} -12331.6 q^{38} -292.926 q^{39} -16408.5 q^{41} +17564.3 q^{42} -1849.00 q^{43} +13840.0 q^{44} -27410.7 q^{46} -24153.7 q^{47} +13739.7 q^{48} +12898.8 q^{49} -6721.04 q^{51} +695.997 q^{52} -21214.2 q^{53} -32621.7 q^{54} +1905.34 q^{56} -20074.0 q^{57} -34218.6 q^{58} -25849.7 q^{59} +28577.7 q^{61} +31551.2 q^{62} -13289.9 q^{63} -29846.8 q^{64} +46087.7 q^{66} -66762.2 q^{67} +15969.3 q^{68} -44620.6 q^{69} -10031.8 q^{71} -852.421 q^{72} -32145.0 q^{73} +79758.3 q^{74} +47696.2 q^{76} +77946.5 q^{77} +2317.69 q^{78} -21913.4 q^{79} -34366.0 q^{81} +129827. q^{82} +66782.4 q^{83} -67935.1 q^{84} +14629.6 q^{86} -55702.8 q^{87} +4999.53 q^{88} +48984.7 q^{89} +3919.82 q^{91} +106019. q^{92} +51360.7 q^{93} +191108. q^{94} -104155. q^{96} -93075.8 q^{97} -102058. q^{98} -34872.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 8 q^{2} - 28 q^{3} + 202 q^{4} + 75 q^{6} - 60 q^{7} - 294 q^{8} + 1356 q^{9}+O(q^{10}) \) 10 * q - 8 * q^2 - 28 * q^3 + 202 * q^4 + 75 * q^6 - 60 * q^7 - 294 * q^8 + 1356 * q^9	\( 10 q - 8 q^{2} - 28 q^{3} + 202 q^{4} + 75 q^{6} - 60 q^{7} - 294 q^{8} + 1356 q^{9} + 745 q^{11} - 4627 q^{12} - 1917 q^{13} + 1936 q^{14} + 5354 q^{16} - 4017 q^{17} + 2725 q^{18} - 2404 q^{19} - 228 q^{21} + 5836 q^{22} - 1733 q^{23} - 10711 q^{24} - 1484 q^{26} + 2324 q^{27} + 15028 q^{28} + 6996 q^{29} - 4899 q^{31} + 7554 q^{32} + 15734 q^{33} - 27033 q^{34} + 4433 q^{36} - 1466 q^{37} - 13905 q^{38} - 26542 q^{39} + 10297 q^{41} + 37642 q^{42} - 18490 q^{43} - 36140 q^{44} + 17991 q^{46} - 48592 q^{47} - 83607 q^{48} + 29458 q^{49} + 92972 q^{51} - 14232 q^{52} - 127165 q^{53} - 92002 q^{54} - 7780 q^{56} - 34060 q^{57} + 10305 q^{58} + 99372 q^{59} + 17408 q^{61} - 28265 q^{62} - 2244 q^{63} + 47202 q^{64} - 150292 q^{66} + 2021 q^{67} - 192151 q^{68} + 1654 q^{69} + 11286 q^{71} + 298365 q^{72} - 49892 q^{73} - 125431 q^{74} - 249803 q^{76} - 98144 q^{77} + 28494 q^{78} - 91524 q^{79} - 26450 q^{81} + 158909 q^{82} + 105203 q^{83} - 357682 q^{84} + 14792 q^{86} - 181200 q^{87} + 461824 q^{88} - 62682 q^{89} - 295304 q^{91} - 183783 q^{92} + 238430 q^{93} + 7259 q^{94} - 162399 q^{96} - 108383 q^{97} - 354656 q^{98} - 270499 q^{99}+O(q^{100}) \) 10 * q - 8 * q^2 - 28 * q^3 + 202 * q^4 + 75 * q^6 - 60 * q^7 - 294 * q^8 + 1356 * q^9 + 745 * q^11 - 4627 * q^12 - 1917 * q^13 + 1936 * q^14 + 5354 * q^16 - 4017 * q^17 + 2725 * q^18 - 2404 * q^19 - 228 * q^21 + 5836 * q^22 - 1733 * q^23 - 10711 * q^24 - 1484 * q^26 + 2324 * q^27 + 15028 * q^28 + 6996 * q^29 - 4899 * q^31 + 7554 * q^32 + 15734 * q^33 - 27033 * q^34 + 4433 * q^36 - 1466 * q^37 - 13905 * q^38 - 26542 * q^39 + 10297 * q^41 + 37642 * q^42 - 18490 * q^43 - 36140 * q^44 + 17991 * q^46 - 48592 * q^47 - 83607 * q^48 + 29458 * q^49 + 92972 * q^51 - 14232 * q^52 - 127165 * q^53 - 92002 * q^54 - 7780 * q^56 - 34060 * q^57 + 10305 * q^58 + 99372 * q^59 + 17408 * q^61 - 28265 * q^62 - 2244 * q^63 + 47202 * q^64 - 150292 * q^66 + 2021 * q^67 - 192151 * q^68 + 1654 * q^69 + 11286 * q^71 + 298365 * q^72 - 49892 * q^73 - 125431 * q^74 - 249803 * q^76 - 98144 * q^77 + 28494 * q^78 - 91524 * q^79 - 26450 * q^81 + 158909 * q^82 + 105203 * q^83 - 357682 * q^84 + 14792 * q^86 - 181200 * q^87 + 461824 * q^88 - 62682 * q^89 - 295304 * q^91 - 183783 * q^92 + 238430 * q^93 + 7259 * q^94 - 162399 * q^96 - 108383 * q^97 - 354656 * q^98 - 270499 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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