LMFDB - Embedded newform 1075.6.a.b.1.6

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			1.6
Root			$2.86024$ 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)

$f(q)$	$=$	$q+1.86024 q^{2} +14.8716 q^{3} -28.5395 q^{4} +27.6647 q^{6} -202.971 q^{7} -112.618 q^{8} -21.8357 q^{9} +O(q^{10})$	\(q+1.86024 q^{2} +14.8716 q^{3} -28.5395 q^{4} +27.6647 q^{6} -202.971 q^{7} -112.618 q^{8} -21.8357 q^{9} +436.853 q^{11} -424.428 q^{12} +617.704 q^{13} -377.575 q^{14} +703.768 q^{16} -833.458 q^{17} -40.6196 q^{18} +1356.52 q^{19} -3018.51 q^{21} +812.652 q^{22} +904.822 q^{23} -1674.81 q^{24} +1149.08 q^{26} -3938.53 q^{27} +5792.70 q^{28} +5326.52 q^{29} +919.124 q^{31} +4912.95 q^{32} +6496.71 q^{33} -1550.43 q^{34} +623.180 q^{36} -4962.82 q^{37} +2523.46 q^{38} +9186.24 q^{39} -5931.38 q^{41} -5615.15 q^{42} -1849.00 q^{43} -12467.6 q^{44} +1683.19 q^{46} -17815.9 q^{47} +10466.2 q^{48} +24390.4 q^{49} -12394.9 q^{51} -17629.0 q^{52} -24713.3 q^{53} -7326.61 q^{54} +22858.2 q^{56} +20173.7 q^{57} +9908.61 q^{58} +33834.6 q^{59} +45948.5 q^{61} +1709.79 q^{62} +4432.02 q^{63} -13381.3 q^{64} +12085.4 q^{66} +58523.0 q^{67} +23786.5 q^{68} +13456.1 q^{69} +1792.64 q^{71} +2459.09 q^{72} +46710.3 q^{73} -9232.03 q^{74} -38714.5 q^{76} -88668.8 q^{77} +17088.6 q^{78} -79459.7 q^{79} -53266.1 q^{81} -11033.8 q^{82} -91200.9 q^{83} +86146.8 q^{84} -3439.58 q^{86} +79213.9 q^{87} -49197.6 q^{88} -16470.2 q^{89} -125376. q^{91} -25823.2 q^{92} +13668.8 q^{93} -33141.8 q^{94} +73063.4 q^{96} -145639. q^{97} +45372.0 q^{98} -9539.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.86024	0.328847	0.164423	−	0.986390i	$-0.447424\pi$
$2$	1.86024	0.328847	0.164423	+	0.986390i	$0.447424\pi$
$3$	14.8716	0.954013	0.477007	−	0.878900i	$-0.341722\pi$
$3$	14.8716	0.954013	0.477007	+	0.878900i	$0.341722\pi$
$4$	−28.5395	−0.891860
$5$	0	0
$5$	0	0
$6$	27.6647	0.313724
$7$	−202.971	−1.56563	−0.782816	−	0.622253i	$-0.786218\pi$
$7$	−202.971	−1.56563	−0.782816	+	0.622253i	$0.786218\pi$
$8$	−112.618	−0.622132
$9$	−21.8357	−0.0898588
$10$	0	0
$11$	436.853	1.08856	0.544282	−	0.838902i	$-0.316802\pi$
$11$	436.853	1.08856	0.544282	+	0.838902i	$0.316802\pi$
$12$	−424.428	−0.850846
$13$	617.704	1.01373	0.506864	−	0.862026i	$-0.330804\pi$
$13$	617.704	1.01373	0.506864	+	0.862026i	$0.330804\pi$
$14$	−377.575	−0.514853
$15$	0	0
$16$	703.768	0.687273
$17$	−833.458	−0.699458	−0.349729	−	0.936851i	$-0.613726\pi$
$17$	−833.458	−0.699458	−0.349729	+	0.936851i	$0.613726\pi$
$18$	−40.6196	−0.0295498
$19$	1356.52	0.862071	0.431036	−	0.902335i	$-0.358148\pi$
$19$	1356.52	0.862071	0.431036	+	0.902335i	$0.358148\pi$
$20$	0	0
$21$	−3018.51	−1.49363
$22$	812.652	0.357971
$23$	904.822	0.356651	0.178326	−	0.983972i	$-0.442932\pi$
$23$	904.822	0.356651	0.178326	+	0.983972i	$0.442932\pi$
$24$	−1674.81	−0.593522
$25$	0	0
$26$	1149.08	0.333362
$27$	−3938.53	−1.03974
$28$	5792.70	1.39632
$29$	5326.52	1.17611	0.588056	−	0.808820i	$-0.299893\pi$
$29$	5326.52	1.17611	0.588056	+	0.808820i	$0.299893\pi$
$30$	0	0
$31$	919.124	0.171779	0.0858895	−	0.996305i	$-0.472627\pi$
$31$	919.124	0.171779	0.0858895	+	0.996305i	$0.472627\pi$
$32$	4912.95	0.848140
$33$	6496.71	1.03850
$34$	−1550.43	−0.230015
$35$	0	0
$36$	623.180	0.0801415
$37$	−4962.82	−0.595969	−0.297985	−	0.954571i	$-0.596314\pi$
$37$	−4962.82	−0.595969	−0.297985	+	0.954571i	$0.596314\pi$
$38$	2523.46	0.283490
$39$	9186.24	0.967111
$40$	0	0
$41$	−5931.38	−0.551057	−0.275528	−	0.961293i	$-0.588853\pi$
$41$	−5931.38	−0.551057	−0.275528	+	0.961293i	$0.588853\pi$
$42$	−5615.15	−0.491177
$43$	−1849.00	−0.152499
$43$	−1849.00	−0.152499
$44$	−12467.6	−0.970847
$45$	0	0
$46$	1683.19	0.117284
$47$	−17815.9	−1.17642	−0.588210	−	0.808708i	$-0.700167\pi$
$47$	−17815.9	−1.17642	−0.588210	+	0.808708i	$0.700167\pi$
$48$	10466.2	0.655668
$49$	24390.4	1.45120
$50$	0	0
$51$	−12394.9	−0.667292
$52$	−17629.0	−0.904104
$53$	−24713.3	−1.20848	−0.604241	−	0.796801i	$-0.706524\pi$
$53$	−24713.3	−1.20848	−0.604241	+	0.796801i	$0.706524\pi$
$54$	−7326.61	−0.341915
$55$	0	0
$56$	22858.2	0.974030
$57$	20173.7	0.822428
$58$	9908.61	0.386761
$59$	33834.6	1.26541	0.632706	−	0.774392i	$-0.281944\pi$
$59$	33834.6	1.26541	0.632706	+	0.774392i	$0.281944\pi$
$60$	0	0
$61$	45948.5	1.58105	0.790527	−	0.612427i	$-0.209807\pi$
$61$	45948.5	1.58105	0.790527	+	0.612427i	$0.209807\pi$
$62$	1709.79	0.0564890
$63$	4432.02	0.140686
$64$	−13381.3	−0.408365
$65$	0	0
$66$	12085.4	0.341509
$67$	58523.0	1.59272	0.796361	−	0.604822i	$-0.206756\pi$
$67$	58523.0	1.59272	0.796361	+	0.604822i	$0.206756\pi$
$68$	23786.5	0.623818
$69$	13456.1	0.340250
$70$	0	0
$71$	1792.64	0.0422034	0.0211017	−	0.999777i	$-0.493283\pi$
$71$	1792.64	0.0422034	0.0211017	+	0.999777i	$0.493283\pi$
$72$	2459.09	0.0559041
$73$	46710.3	1.02590	0.512951	−	0.858418i	$-0.328552\pi$
$73$	46710.3	1.02590	0.512951	+	0.858418i	$0.328552\pi$
$74$	−9232.03	−0.195983
$75$	0	0
$76$	−38714.5	−0.768847
$77$	−88668.8	−1.70429
$78$	17088.6	0.318031
$79$	−79459.7	−1.43245	−0.716225	−	0.697870i	$-0.754131\pi$
$79$	−79459.7	−1.43245	−0.716225	+	0.697870i	$0.754131\pi$
$80$	0	0
$81$	−53266.1	−0.902067
$82$	−11033.8	−0.181213
$83$	−91200.9	−1.45313	−0.726564	−	0.687099i	$-0.758884\pi$
$83$	−91200.9	−1.45313	−0.726564	+	0.687099i	$0.758884\pi$
$84$	86146.8	1.33211
$85$	0	0
$86$	−3439.58	−0.0501487
$87$	79213.9	1.12203
$88$	−49197.6	−0.677231
$89$	−16470.2	−0.220406	−0.110203	−	0.993909i	$-0.535150\pi$
$89$	−16470.2	−0.220406	−0.110203	+	0.993909i	$0.535150\pi$
$90$	0	0
$91$	−125376.	−1.58713
$92$	−25823.2	−0.318083
$93$	13668.8	0.163879
$94$	−33141.8	−0.386862
$95$	0	0
$96$	73063.4	0.809137
$97$	−145639.	−1.57163	−0.785813	−	0.618464i	$-0.787755\pi$
$97$	−145639.	−1.57163	−0.785813	+	0.618464i	$0.787755\pi$
$98$	45372.0	0.477224
$99$	−9539.00	−0.0978171
$100$	0	0
$101$	−59937.9	−0.584653	−0.292327	−	0.956319i	$-0.594429\pi$
$101$	−59937.9	−0.584653	−0.292327	+	0.956319i	$0.594429\pi$
$102$	−23057.4	−0.219437
$103$	176381.	1.63817	0.819083	−	0.573675i	$-0.194483\pi$
$103$	176381.	1.63817	0.819083	+	0.573675i	$0.194483\pi$
$104$	−69564.5	−0.630674
$105$	0	0
$106$	−45972.6	−0.397406
$107$	−145711.	−1.23036	−0.615180	−	0.788387i	$-0.710917\pi$
$107$	−145711.	−1.23036	−0.615180	+	0.788387i	$0.710917\pi$
$108$	112404.	0.927302
$109$	−74113.8	−0.597493	−0.298746	−	0.954333i	$-0.596568\pi$
$109$	−74113.8	−0.597493	−0.298746	+	0.954333i	$0.596568\pi$
$110$	0	0
$111$	−73805.0	−0.568563
$112$	−142845.	−1.07602
$113$	−170870.	−1.25884	−0.629419	−	0.777066i	$-0.716707\pi$
$113$	−170870.	−1.25884	−0.629419	+	0.777066i	$0.716707\pi$
$114$	37527.8	0.270453
$115$	0	0
$116$	−152016.	−1.04893
$117$	−13488.0	−0.0910925
$118$	62940.5	0.416127
$119$	169168.	1.09509
$120$	0	0
$121$	29789.9	0.184972
$122$	85475.2	0.519925
$123$	−88209.1	−0.525716
$124$	−26231.3	−0.153203
$125$	0	0
$126$	8244.62	0.0462641
$127$	−117267.	−0.645156	−0.322578	−	0.946543i	$-0.604549\pi$
$127$	−117267.	−0.645156	−0.322578	+	0.946543i	$0.604549\pi$
$128$	−182107.	−0.982430
$129$	−27497.6	−0.145486
$130$	0	0
$131$	−64361.8	−0.327680	−0.163840	−	0.986487i	$-0.552388\pi$
$131$	−64361.8	−0.327680	−0.163840	+	0.986487i	$0.552388\pi$
$132$	−185413.	−0.926200
$133$	−275335.	−1.34969
$134$	108867.	0.523762
$135$	0	0
$136$	93862.4	0.435155
$137$	219763.	1.00035	0.500176	−	0.865924i	$-0.333269\pi$
$137$	219763.	1.00035	0.500176	+	0.865924i	$0.333269\pi$
$138$	25031.7	0.111890
$139$	14710.6	0.0645793	0.0322896	−	0.999479i	$-0.489720\pi$
$139$	14710.6	0.0645793	0.0322896	+	0.999479i	$0.489720\pi$
$140$	0	0
$141$	−264950.	−1.12232
$142$	3334.74	0.0138785
$143$	269846.	1.10351
$144$	−15367.3	−0.0617576
$145$	0	0
$146$	86892.4	0.337365
$147$	362724.	1.38447
$148$	141636.	0.531521
$149$	−396393.	−1.46272	−0.731359	−	0.681993i	$-0.761113\pi$
$149$	−396393.	−1.46272	−0.731359	+	0.681993i	$0.761113\pi$
$150$	0	0
$151$	70722.8	0.252416	0.126208	−	0.992004i	$-0.459719\pi$
$151$	70722.8	0.252416	0.126208	+	0.992004i	$0.459719\pi$
$152$	−152769.	−0.536323
$153$	18199.1	0.0628525
$154$	−164945.	−0.560451
$155$	0	0
$156$	−262171.	−0.862527
$157$	469441.	1.51996	0.759980	−	0.649946i	$-0.225209\pi$
$157$	469441.	1.51996	0.759980	+	0.649946i	$0.225209\pi$
$158$	−147814.	−0.471057
$159$	−367526.	−1.15291
$160$	0	0
$161$	−183653.	−0.558385
$162$	−99087.7	−0.296642
$163$	−660494.	−1.94715	−0.973576	−	0.228362i	$-0.926663\pi$
$163$	−660494.	−1.94715	−0.973576	+	0.228362i	$0.926663\pi$
$164$	169279.	0.491465
$165$	0	0
$166$	−169655.	−0.477857
$167$	−488059.	−1.35419	−0.677097	−	0.735894i	$-0.736762\pi$
$167$	−488059.	−1.35419	−0.677097	+	0.735894i	$0.736762\pi$
$168$	339938.	0.929238
$169$	10264.9	0.0276464
$170$	0	0
$171$	−29620.6	−0.0774647
$172$	52769.6	0.136007
$173$	−648829.	−1.64822	−0.824109	−	0.566431i	$-0.808324\pi$
$173$	−648829.	−1.64822	−0.824109	+	0.566431i	$0.808324\pi$
$174$	147357.	0.368975
$175$	0	0
$176$	307443.	0.748141
$177$	503175.	1.20722
$178$	−30638.4	−0.0724797
$179$	−203377.	−0.474428	−0.237214	−	0.971457i	$-0.576234\pi$
$179$	−203377.	−0.474428	−0.237214	+	0.971457i	$0.576234\pi$
$180$	0	0
$181$	174043.	0.394875	0.197438	−	0.980315i	$-0.436738\pi$
$181$	174043.	0.394875	0.197438	+	0.980315i	$0.436738\pi$
$182$	−233230.	−0.521922
$183$	683327.	1.50835
$184$	−101899.	−0.221884
$185$	0	0
$186$	25427.3	0.0538912
$187$	−364099.	−0.761405
$188$	508456.	1.04920
$189$	799409.	1.62785
$190$	0	0
$191$	−529048.	−1.04933	−0.524664	−	0.851309i	$-0.675809\pi$
$191$	−529048.	−1.04933	−0.524664	+	0.851309i	$0.675809\pi$
$192$	−199001.	−0.389586
$193$	74996.7	0.144927	0.0724634	−	0.997371i	$-0.476914\pi$
$193$	74996.7	0.144927	0.0724634	+	0.997371i	$0.476914\pi$
$194$	−270924.	−0.516824
$195$	0	0
$196$	−696090.	−1.29427
$197$	−741584.	−1.36143	−0.680714	−	0.732549i	$-0.738331\pi$
$197$	−741584.	−1.36143	−0.680714	+	0.732549i	$0.738331\pi$
$198$	−17744.8	−0.0321669
$199$	−46763.5	−0.0837093	−0.0418547	−	0.999124i	$-0.513327\pi$
$199$	−46763.5	−0.0837093	−0.0418547	+	0.999124i	$0.513327\pi$
$200$	0	0
$201$	870331.	1.51948
$202$	−111499.	−0.192261
$203$	−1.08113e6	−1.84136
$204$	353743.	0.595131
$205$	0	0
$206$	328110.	0.538706
$207$	−19757.4	−0.0320483
$208$	434720.	0.696709
$209$	592602.	0.938420
$210$	0	0
$211$	360269.	0.557084	0.278542	−	0.960424i	$-0.410149\pi$
$211$	360269.	0.557084	0.278542	+	0.960424i	$0.410149\pi$
$212$	705304.	1.07780
$213$	26659.4	0.0402626
$214$	−271057.	−0.404600
$215$	0	0
$216$	443549.	0.646856
$217$	−186556.	−0.268943
$218$	−137869.	−0.196484
$219$	694657.	0.978723
$220$	0	0
$221$	−514830.	−0.709061
$222$	−137295.	−0.186970
$223$	184494.	0.248440	0.124220	−	0.992255i	$-0.460357\pi$
$223$	184494.	0.248440	0.124220	+	0.992255i	$0.460357\pi$
$224$	−997189.	−1.32788
$225$	0	0
$226$	−317859.	−0.413965
$227$	404674.	0.521243	0.260621	−	0.965441i	$-0.416073\pi$
$227$	404674.	0.521243	0.260621	+	0.965441i	$0.416073\pi$
$228$	−575746.	−0.733490
$229$	605861.	0.763456	0.381728	−	0.924275i	$-0.375329\pi$
$229$	605861.	0.763456	0.381728	+	0.924275i	$0.375329\pi$
$230$	0	0
$231$	−1.31865e6	−1.62592
$232$	−599862.	−0.731698
$233$	126192.	0.152280	0.0761400	−	0.997097i	$-0.475740\pi$
$233$	126192.	0.152280	0.0761400	+	0.997097i	$0.475740\pi$
$234$	−25090.9	−0.0299555
$235$	0	0
$236$	−965624.	−1.12857
$237$	−1.18169e6	−1.36658
$238$	314693.	0.360118
$239$	1.31948e6	1.49420	0.747098	−	0.664714i	$-0.231446\pi$
$239$	1.31948e6	1.49420	0.747098	+	0.664714i	$0.231446\pi$
$240$	0	0
$241$	1.51669e6	1.68211	0.841055	−	0.540949i	$-0.181935\pi$
$241$	1.51669e6	1.68211	0.841055	+	0.540949i	$0.181935\pi$
$242$	55416.4	0.0608275
$243$	164910.	0.179156
$244$	−1.31135e6	−1.41008
$245$	0	0
$246$	−164090.	−0.172880
$247$	837930.	0.873907
$248$	−103510.	−0.106869
$249$	−1.35630e6	−1.38630
$250$	0	0
$251$	−1.30092e6	−1.30337	−0.651685	−	0.758490i	$-0.725938\pi$
$251$	−1.30092e6	−1.30337	−0.651685	+	0.758490i	$0.725938\pi$
$252$	−126488.	−0.125472
$253$	395275.	0.388238
$254$	−218144.	−0.212158
$255$	0	0
$256$	89439.4	0.0852960
$257$	223343.	0.210930	0.105465	−	0.994423i	$-0.466367\pi$
$257$	223343.	0.210930	0.105465	+	0.994423i	$0.466367\pi$
$258$	−51152.1	−0.0478425
$259$	1.00731e6	0.933069
$260$	0	0
$261$	−116308.	−0.105684
$262$	−119728.	−0.107757
$263$	−216725.	−0.193206	−0.0966028	−	0.995323i	$-0.530798\pi$
$263$	−216725.	−0.193206	−0.0966028	+	0.995323i	$0.530798\pi$
$264$	−731646.	−0.646087
$265$	0	0
$266$	−512190.	−0.443840
$267$	−244938.	−0.210270
$268$	−1.67022e6	−1.42048
$269$	1.12157e6	0.945029	0.472515	−	0.881323i	$-0.343346\pi$
$269$	1.12157e6	0.945029	0.472515	+	0.881323i	$0.343346\pi$
$270$	0	0
$271$	−318880.	−0.263757	−0.131878	−	0.991266i	$-0.542101\pi$
$271$	−318880.	−0.263757	−0.131878	+	0.991266i	$0.542101\pi$
$272$	−586561.	−0.480719
$273$	−1.86454e6	−1.51414
$274$	408812.	0.328963
$275$	0	0
$276$	−384032.	−0.303455
$277$	353813.	0.277061	0.138530	−	0.990358i	$-0.455762\pi$
$277$	353813.	0.277061	0.138530	+	0.990358i	$0.455762\pi$
$278$	27365.2	0.0212367
$279$	−20069.7	−0.0154359
$280$	0	0
$281$	−248237.	−0.187543	−0.0937717	−	0.995594i	$-0.529892\pi$
$281$	−248237.	−0.187543	−0.0937717	+	0.995594i	$0.529892\pi$
$282$	−492871.	−0.369072
$283$	420812.	0.312336	0.156168	−	0.987730i	$-0.450086\pi$
$283$	420812.	0.312336	0.156168	+	0.987730i	$0.450086\pi$
$284$	−51161.1	−0.0376395
$285$	0	0
$286$	501978.	0.362886
$287$	1.20390e6	0.862753
$288$	−107278.	−0.0762129
$289$	−725205.	−0.510759
$290$	0	0
$291$	−2.16589e6	−1.49935
$292$	−1.33309e6	−0.914960
$293$	59022.2	0.0401649	0.0200824	−	0.999798i	$-0.493607\pi$
$293$	59022.2	0.0401649	0.0200824	+	0.999798i	$0.493607\pi$
$294$	674754.	0.455278
$295$	0	0
$296$	558903.	0.370772
$297$	−1.72056e6	−1.13182
$298$	−737386.	−0.481010
$299$	558912.	0.361548
$300$	0	0
$301$	375294.	0.238757
$302$	131561.	0.0830063
$303$	−891373.	−0.557767
$304$	954678.	0.592479
$305$	0	0
$306$	33854.8	0.0206688
$307$	−969345.	−0.586993	−0.293496	−	0.955960i	$-0.594819\pi$
$307$	−969345.	−0.586993	−0.293496	+	0.955960i	$0.594819\pi$
$308$	2.53056e6	1.51999
$309$	2.62306e6	1.56283
$310$	0	0
$311$	−3.12666e6	−1.83307	−0.916537	−	0.399950i	$-0.869028\pi$
$311$	−3.12666e6	−1.83307	−0.916537	+	0.399950i	$0.869028\pi$
$312$	−1.03454e6	−0.601671
$313$	−230443.	−0.132955	−0.0664773	−	0.997788i	$-0.521176\pi$
$313$	−230443.	−0.132955	−0.0664773	+	0.997788i	$0.521176\pi$
$314$	873274.	0.499834
$315$	0	0
$316$	2.26774e6	1.27754
$317$	580982.	0.324724	0.162362	−	0.986731i	$-0.448089\pi$
$317$	580982.	0.324724	0.162362	+	0.986731i	$0.448089\pi$
$318$	−683686.	−0.379130
$319$	2.32691e6	1.28027
$320$	0	0
$321$	−2.16695e6	−1.17378
$322$	−341639.	−0.183623
$323$	−1.13061e6	−0.602983
$324$	1.52019e6	0.804517
$325$	0	0
$326$	−1.22868e6	−0.640315
$327$	−1.10219e6	−0.570016
$328$	667981.	0.342830
$329$	3.61611e6	1.84184
$330$	0	0
$331$	−2.54254e6	−1.27555	−0.637776	−	0.770222i	$-0.720145\pi$
$331$	−2.54254e6	−1.27555	−0.637776	+	0.770222i	$0.720145\pi$
$332$	2.60283e6	1.29599
$333$	108367.	0.0535531
$334$	−907906.	−0.445323
$335$	0	0
$336$	−2.12433e6	−1.02653
$337$	−2.09346e6	−1.00413	−0.502066	−	0.864829i	$-0.667427\pi$
$337$	−2.09346e6	−1.00413	−0.502066	+	0.864829i	$0.667427\pi$
$338$	19095.2	0.00909143
$339$	−2.54111e6	−1.20095
$340$	0	0
$341$	401522.	0.186992
$342$	−55101.5	−0.0254740
$343$	−1.53921e6	−0.706421
$344$	208231.	0.0948743
$345$	0	0
$346$	−1.20698e6	−0.542012
$347$	4.07369e6	1.81620	0.908100	−	0.418753i	$-0.137533\pi$
$347$	4.07369e6	1.81620	0.908100	+	0.418753i	$0.137533\pi$
$348$	−2.26073e6	−1.00069
$349$	−1.01046e6	−0.444073	−0.222037	−	0.975038i	$-0.571270\pi$
$349$	−1.01046e6	−0.444073	−0.222037	+	0.975038i	$0.571270\pi$
$350$	0	0
$351$	−2.43284e6	−1.05401
$352$	2.14624e6	0.923255
$353$	655684.	0.280064	0.140032	−	0.990147i	$-0.455279\pi$
$353$	655684.	0.280064	0.140032	+	0.990147i	$0.455279\pi$
$354$	936026.	0.396990
$355$	0	0
$356$	470050.	0.196571
$357$	2.51580e6	1.04473
$358$	−378330.	−0.156014
$359$	1.78546e6	0.731162	0.365581	−	0.930780i	$-0.380870\pi$
$359$	1.78546e6	0.731162	0.365581	+	0.930780i	$0.380870\pi$
$360$	0	0
$361$	−635943.	−0.256833
$362$	323762.	0.129854
$363$	443024.	0.176466
$364$	3.57818e6	1.41549
$365$	0	0
$366$	1.27115e6	0.496015
$367$	−2.96896e6	−1.15064	−0.575320	−	0.817928i	$-0.695122\pi$
$367$	−2.96896e6	−1.15064	−0.575320	+	0.817928i	$0.695122\pi$
$368$	636785.	0.245117
$369$	129516.	0.0495173
$370$	0	0
$371$	5.01609e6	1.89204
$372$	−390102.	−0.146157
$373$	3.94005e6	1.46632	0.733162	−	0.680055i	$-0.238044\pi$
$373$	3.94005e6	1.46632	0.733162	+	0.680055i	$0.238044\pi$
$374$	−677311.	−0.250386
$375$	0	0
$376$	2.00639e6	0.731889
$377$	3.29021e6	1.19226
$378$	1.48709e6	0.535314
$379$	−4.36353e6	−1.56042	−0.780208	−	0.625520i	$-0.784887\pi$
$379$	−4.36353e6	−1.56042	−0.780208	+	0.625520i	$0.784887\pi$
$380$	0	0
$381$	−1.74394e6	−0.615487
$382$	−984155.	−0.345068
$383$	3.53079e6	1.22991	0.614957	−	0.788561i	$-0.289173\pi$
$383$	3.53079e6	1.22991	0.614957	+	0.788561i	$0.289173\pi$
$384$	−2.70822e6	−0.937251
$385$	0	0
$386$	139512.	0.0476587
$387$	40374.2	0.0137033
$388$	4.15647e6	1.40167
$389$	−5.57419e6	−1.86770	−0.933851	−	0.357663i	$-0.883574\pi$
$389$	−5.57419e6	−1.86770	−0.933851	+	0.357663i	$0.883574\pi$
$390$	0	0
$391$	−754131.	−0.249462
$392$	−2.74680e6	−0.902841
$393$	−957163.	−0.312611
$394$	−1.37952e6	−0.447702
$395$	0	0
$396$	272238.	0.0872391
$397$	−5.87343e6	−1.87032	−0.935159	−	0.354229i	$-0.884743\pi$
$397$	−5.87343e6	−1.87032	−0.935159	+	0.354229i	$0.884743\pi$
$398$	−86991.2	−0.0275276
$399$	−4.09468e6	−1.28762
$400$	0	0
$401$	−46354.8	−0.0143957	−0.00719786	−	0.999974i	$-0.502291\pi$
$401$	−46354.8	−0.0143957	−0.00719786	+	0.999974i	$0.502291\pi$
$402$	1.61902e6	0.499676
$403$	567746.	0.174137
$404$	1.71060e6	0.521429
$405$	0	0
$406$	−2.01116e6	−0.605526
$407$	−2.16802e6	−0.648751
$408$	1.39588e6	0.415144
$409$	2.92397e6	0.864301	0.432150	−	0.901802i	$-0.357755\pi$
$409$	2.92397e6	0.864301	0.432150	+	0.901802i	$0.357755\pi$
$410$	0	0
$411$	3.26822e6	0.954349
$412$	−5.03382e6	−1.46101
$413$	−6.86747e6	−1.98117
$414$	−36753.5	−0.0105390
$415$	0	0
$416$	3.03475e6	0.859784
$417$	218770.	0.0616095
$418$	1.10238e6	0.308597
$419$	3.23273e6	0.899569	0.449785	−	0.893137i	$-0.351501\pi$
$419$	3.23273e6	0.899569	0.449785	+	0.893137i	$0.351501\pi$
$420$	0	0
$421$	−4.62842e6	−1.27271	−0.636353	−	0.771398i	$-0.719558\pi$
$421$	−4.62842e6	−1.27271	−0.636353	+	0.771398i	$0.719558\pi$
$422$	670186.	0.183195
$423$	389022.	0.105712
$424$	2.78316e6	0.751836
$425$	0	0
$426$	49592.9	0.0132402
$427$	−9.32623e6	−2.47535
$428$	4.15851e6	1.09731
$429$	4.01304e6	1.05276
$430$	0	0
$431$	1.87590e6	0.486425	0.243213	−	0.969973i	$-0.421799\pi$
$431$	1.87590e6	0.486425	0.243213	+	0.969973i	$0.421799\pi$
$432$	−2.77181e6	−0.714585
$433$	3.80994e6	0.976559	0.488280	−	0.872687i	$-0.337625\pi$
$433$	3.80994e6	0.976559	0.488280	+	0.872687i	$0.337625\pi$
$434$	−347039.	−0.0884410
$435$	0	0
$436$	2.11517e6	0.532880
$437$	1.22741e6	0.307459
$438$	1.29223e6	0.321850
$439$	2.35939e6	0.584304	0.292152	−	0.956372i	$-0.405629\pi$
$439$	2.35939e6	0.584304	0.292152	+	0.956372i	$0.405629\pi$
$440$	0	0
$441$	−532581.	−0.130404
$442$	−957707.	−0.233172
$443$	2.55633e6	0.618881	0.309440	−	0.950919i	$-0.399858\pi$
$443$	2.55633e6	0.618881	0.309440	+	0.950919i	$0.399858\pi$
$444$	2.10636e6	0.507078
$445$	0	0
$446$	343204.	0.0816987
$447$	−5.89500e6	−1.39545
$448$	2.71602e6	0.639350
$449$	3.71934e6	0.870662	0.435331	−	0.900270i	$-0.356631\pi$
$449$	3.71934e6	0.870662	0.435331	+	0.900270i	$0.356631\pi$
$450$	0	0
$451$	−2.59115e6	−0.599861
$452$	4.87655e6	1.12271
$453$	1.05176e6	0.240808
$454$	752790.	0.171409
$455$	0	0
$456$	−2.27192e6	−0.511659
$457$	2.59594e6	0.581438	0.290719	−	0.956808i	$-0.406105\pi$
$457$	2.59594e6	0.581438	0.290719	+	0.956808i	$0.406105\pi$
$458$	1.12705e6	0.251060
$459$	3.28260e6	0.727254
$460$	0	0
$461$	−6.62260e6	−1.45136	−0.725682	−	0.688031i	$-0.758475\pi$
$461$	−6.62260e6	−1.45136	−0.725682	+	0.688031i	$0.758475\pi$
$462$	−2.45300e6	−0.534678
$463$	−5.09992e6	−1.10563	−0.552817	−	0.833303i	$-0.686447\pi$
$463$	−5.09992e6	−1.10563	−0.552817	+	0.833303i	$0.686447\pi$
$464$	3.74864e6	0.808311
$465$	0	0
$466$	234748.	0.0500768
$467$	−3.91009e6	−0.829649	−0.414825	−	0.909901i	$-0.636157\pi$
$467$	−3.91009e6	−0.829649	−0.414825	+	0.909901i	$0.636157\pi$
$468$	384941.	0.0812417
$469$	−1.18785e7	−2.49362
$470$	0	0
$471$	6.98134e6	1.45006
$472$	−3.81039e6	−0.787253
$473$	−807742.	−0.166004
$474$	−2.19823e6	−0.449394
$475$	0	0
$476$	−4.82798e6	−0.976670
$477$	539631.	0.108593
$478$	2.45455e6	0.491362
$479$	−8.49397e6	−1.69150	−0.845750	−	0.533580i	$-0.820846\pi$
$479$	−8.49397e6	−1.69150	−0.845750	+	0.533580i	$0.820846\pi$
$480$	0	0
$481$	−3.06555e6	−0.604152
$482$	2.82141e6	0.553157
$483$	−2.73121e6	−0.532706
$484$	−850190.	−0.164969
$485$	0	0
$486$	306773.	0.0589150
$487$	−7.46234e6	−1.42578	−0.712890	−	0.701276i	$-0.752614\pi$
$487$	−7.46234e6	−1.42578	−0.712890	+	0.701276i	$0.752614\pi$
$488$	−5.17463e6	−0.983625
$489$	−9.82260e6	−1.85761
$490$	0	0
$491$	2.46640e6	0.461700	0.230850	−	0.972989i	$-0.425849\pi$
$491$	2.46640e6	0.461700	0.230850	+	0.972989i	$0.425849\pi$
$492$	2.51745e6	0.468864
$493$	−4.43943e6	−0.822641
$494$	1.55875e6	0.287382
$495$	0	0
$496$	646850.	0.118059
$497$	−363855.	−0.0660750
$498$	−2.52305e6	−0.455882
$499$	−2.03466e6	−0.365798	−0.182899	−	0.983132i	$-0.558548\pi$
$499$	−2.03466e6	−0.365798	−0.182899	+	0.983132i	$0.558548\pi$
$500$	0	0
$501$	−7.25821e6	−1.29192
$502$	−2.42003e6	−0.428609
$503$	5.11148e6	0.900797	0.450398	−	0.892828i	$-0.351282\pi$
$503$	5.11148e6	0.900797	0.450398	+	0.892828i	$0.351282\pi$
$504$	−499125.	−0.0875253
$505$	0	0
$506$	735305.	0.127671
$507$	152656.	0.0263750
$508$	3.34673e6	0.575389
$509$	5.43324e6	0.929532	0.464766	−	0.885433i	$-0.346138\pi$
$509$	5.43324e6	0.929532	0.464766	+	0.885433i	$0.346138\pi$
$510$	0	0
$511$	−9.48086e6	−1.60618
$512$	5.99380e6	1.01048
$513$	−5.34271e6	−0.896330
$514$	415471.	0.0693638
$515$	0	0
$516$	784767.	0.129753
$517$	−7.78292e6	−1.28061
$518$	1.87384e6	0.306837
$519$	−9.64911e6	−1.57242
$520$	0	0
$521$	8.89569e6	1.43577	0.717886	−	0.696161i	$-0.245110\pi$
$521$	8.89569e6	1.43577	0.717886	+	0.696161i	$0.245110\pi$
$522$	−216361.	−0.0347539
$523$	−5.98910e6	−0.957430	−0.478715	−	0.877970i	$-0.658897\pi$
$523$	−5.98910e6	−0.957430	−0.478715	+	0.877970i	$0.658897\pi$
$524$	1.83686e6	0.292245
$525$	0	0
$526$	−403160.	−0.0635351
$527$	−766051.	−0.120152
$528$	4.57217e6	0.713737
$529$	−5.61764e6	−0.872800
$530$	0	0
$531$	−738803.	−0.113708
$532$	7.85794e6	1.20373
$533$	−3.66384e6	−0.558622
$534$	−455642.	−0.0691466
$535$	0	0
$536$	−6.59075e6	−0.990884
$537$	−3.02454e6	−0.452610
$538$	2.08639e6	0.310770
$539$	1.06550e7	1.57973
$540$	0	0
$541$	3.07873e6	0.452249	0.226125	−	0.974098i	$-0.427394\pi$
$541$	3.07873e6	0.452249	0.226125	+	0.974098i	$0.427394\pi$
$542$	−593193.	−0.0867356
$543$	2.58830e6	0.376716
$544$	−4.09474e6	−0.593238
$545$	0	0
$546$	−3.46850e6	−0.497920
$547$	1.37829e7	1.96958	0.984789	−	0.173755i	$-0.0555902\pi$
$547$	1.37829e7	1.96958	0.984789	+	0.173755i	$0.0555902\pi$
$548$	−6.27192e6	−0.892174
$549$	−1.00332e6	−0.142072
$550$	0	0
$551$	7.22555e6	1.01389
$552$	−1.51540e6	−0.211680
$553$	1.61280e7	2.24269
$554$	658177.	0.0911105
$555$	0	0
$556$	−419833.	−0.0575956
$557$	−3.17087e6	−0.433053	−0.216526	−	0.976277i	$-0.569473\pi$
$557$	−3.17087e6	−0.433053	−0.216526	+	0.976277i	$0.569473\pi$
$558$	−37334.5	−0.00507603
$559$	−1.14213e6	−0.154592
$560$	0	0
$561$	−5.41473e6	−0.726390
$562$	−461781.	−0.0616730
$563$	−1.15175e7	−1.53139	−0.765697	−	0.643201i	$-0.777606\pi$
$563$	−1.15175e7	−1.53139	−0.765697	+	0.643201i	$0.777606\pi$
$564$	7.56155e6	1.00095
$565$	0	0
$566$	782812.	0.102711
$567$	1.08115e7	1.41230
$568$	−201884.	−0.0262561
$569$	−1.09971e7	−1.42397	−0.711983	−	0.702197i	$-0.752203\pi$
$569$	−1.09971e7	−1.42397	−0.711983	+	0.702197i	$0.752203\pi$
$570$	0	0
$571$	−2.17295e6	−0.278907	−0.139454	−	0.990229i	$-0.544535\pi$
$571$	−2.17295e6	−0.278907	−0.139454	+	0.990229i	$0.544535\pi$
$572$	−7.70127e6	−0.984175
$573$	−7.86778e6	−1.00107
$574$	2.23955e6	0.283714
$575$	0	0
$576$	292190.	0.0366952
$577$	1.05167e7	1.31505	0.657524	−	0.753434i	$-0.271604\pi$
$577$	1.05167e7	1.31505	0.657524	+	0.753434i	$0.271604\pi$
$578$	−1.34905e6	−0.167962
$579$	1.11532e6	0.138262
$580$	0	0
$581$	1.85112e7	2.27506
$582$	−4.02907e6	−0.493057
$583$	−1.07961e7	−1.31551
$584$	−5.26042e6	−0.638246
$585$	0	0
$586$	109795.	0.0132081
$587$	−2.96292e6	−0.354915	−0.177457	−	0.984128i	$-0.556787\pi$
$587$	−2.96292e6	−0.354915	−0.177457	+	0.984128i	$0.556787\pi$
$588$	−1.03520e7	−1.23475
$589$	1.24681e6	0.148086
$590$	0	0
$591$	−1.10285e7	−1.29882
$592$	−3.49267e6	−0.409594
$593$	−7.54824e6	−0.881473	−0.440737	−	0.897636i	$-0.645283\pi$
$593$	−7.54824e6	−0.881473	−0.440737	+	0.897636i	$0.645283\pi$
$594$	−3.20065e6	−0.372197
$595$	0	0
$596$	1.13129e7	1.30454
$597$	−695447.	−0.0798598
$598$	1.03971e6	0.118894
$599$	7.70971e6	0.877953	0.438976	−	0.898499i	$-0.355341\pi$
$599$	7.70971e6	0.877953	0.438976	+	0.898499i	$0.355341\pi$
$600$	0	0
$601$	3.40171e6	0.384159	0.192079	−	0.981379i	$-0.438477\pi$
$601$	3.40171e6	0.384159	0.192079	+	0.981379i	$0.438477\pi$
$602$	698137.	0.0785144
$603$	−1.27789e6	−0.143120
$604$	−2.01839e6	−0.225120
$605$	0	0
$606$	−1.65817e6	−0.183420
$607$	−5.29342e6	−0.583129	−0.291564	−	0.956551i	$-0.594176\pi$
$607$	−5.29342e6	−0.583129	−0.291564	+	0.956551i	$0.594176\pi$
$608$	6.66453e6	0.731157
$609$	−1.60782e7	−1.75668
$610$	0	0
$611$	−1.10049e7	−1.19257
$612$	−519395.	−0.0560556
$613$	−1.51319e7	−1.62646	−0.813229	−	0.581944i	$-0.802292\pi$
$613$	−1.51319e7	−1.62646	−0.813229	+	0.581944i	$0.802292\pi$
$614$	−1.80321e6	−0.193031
$615$	0	0
$616$	9.98570e6	1.06029
$617$	−2.54291e6	−0.268917	−0.134458	−	0.990919i	$-0.542929\pi$
$617$	−2.54291e6	−0.268917	−0.134458	+	0.990919i	$0.542929\pi$
$618$	4.87952e6	0.513932
$619$	8.70953e6	0.913626	0.456813	−	0.889563i	$-0.348991\pi$
$619$	8.70953e6	0.913626	0.456813	+	0.889563i	$0.348991\pi$
$620$	0	0
$621$	−3.56367e6	−0.370824
$622$	−5.81634e6	−0.602801
$623$	3.34297e6	0.345074
$624$	6.46498e6	0.664669
$625$	0	0
$626$	−428680.	−0.0437217
$627$	8.81293e6	0.895265
$628$	−1.33976e7	−1.35559
$629$	4.13630e6	0.416855
$630$	0	0
$631$	1.28039e7	1.28018	0.640089	−	0.768301i	$-0.278898\pi$
$631$	1.28039e7	1.28018	0.640089	+	0.768301i	$0.278898\pi$
$632$	8.94859e6	0.891173
$633$	5.35777e6	0.531465
$634$	1.08077e6	0.106785
$635$	0	0
$636$	1.04890e7	1.02823
$637$	1.50660e7	1.47113
$638$	4.32861e6	0.421014
$639$	−39143.6	−0.00379235
$640$	0	0
$641$	−1.14881e7	−1.10434	−0.552170	−	0.833732i	$-0.686200\pi$
$641$	−1.14881e7	−1.10434	−0.552170	+	0.833732i	$0.686200\pi$
$642$	−4.03105e6	−0.385994
$643$	1.13485e7	1.08246	0.541230	−	0.840875i	$-0.317959\pi$
$643$	1.13485e7	1.08246	0.541230	+	0.840875i	$0.317959\pi$
$644$	5.24137e6	0.498001
$645$	0	0
$646$	−2.10320e6	−0.198289
$647$	6.22456e6	0.584585	0.292293	−	0.956329i	$-0.405582\pi$
$647$	6.22456e6	0.584585	0.292293	+	0.956329i	$0.405582\pi$
$648$	5.99872e6	0.561205
$649$	1.47808e7	1.37748
$650$	0	0
$651$	−2.77438e6	−0.256575
$652$	1.88502e7	1.73659
$653$	382501.	0.0351034	0.0175517	−	0.999846i	$-0.494413\pi$
$653$	382501.	0.0351034	0.0175517	+	0.999846i	$0.494413\pi$
$654$	−2.05034e6	−0.187448
$655$	0	0
$656$	−4.17432e6	−0.378727
$657$	−1.01995e6	−0.0921863
$658$	6.72683e6	0.605684
$659$	−2.06346e7	−1.85090	−0.925449	−	0.378872i	$-0.876312\pi$
$659$	−2.06346e7	−1.85090	−0.925449	+	0.378872i	$0.876312\pi$
$660$	0	0
$661$	−3.78028e6	−0.336527	−0.168264	−	0.985742i	$-0.553816\pi$
$661$	−3.78028e6	−0.336527	−0.168264	+	0.985742i	$0.553816\pi$
$662$	−4.72973e6	−0.419461
$663$	−7.65635e6	−0.676453
$664$	1.02709e7	0.904038
$665$	0	0
$666$	201588.	0.0176108
$667$	4.81956e6	0.419462
$668$	1.39290e7	1.20775
$669$	2.74373e6	0.237015
$670$	0	0
$671$	2.00728e7	1.72108
$672$	−1.48298e7	−1.26681
$673$	7.82427e6	0.665895	0.332948	−	0.942945i	$-0.391957\pi$
$673$	7.82427e6	0.665895	0.332948	+	0.942945i	$0.391957\pi$
$674$	−3.89435e6	−0.330206
$675$	0	0
$676$	−292955.	−0.0246567
$677$	−5.22258e6	−0.437939	−0.218969	−	0.975732i	$-0.570270\pi$
$677$	−5.22258e6	−0.437939	−0.218969	+	0.975732i	$0.570270\pi$
$678$	−4.72707e6	−0.394928
$679$	2.95606e7	2.46059
$680$	0	0
$681$	6.01814e6	0.497273
$682$	746928.	0.0614919
$683$	1.18085e7	0.968598	0.484299	−	0.874902i	$-0.339075\pi$
$683$	1.18085e7	0.968598	0.484299	+	0.874902i	$0.339075\pi$
$684$	845358.	0.0690877
$685$	0	0
$686$	−2.86330e6	−0.232304
$687$	9.01012e6	0.728348
$688$	−1.30127e6	−0.104808
$689$	−1.52655e7	−1.22507
$690$	0	0
$691$	1.90341e7	1.51648	0.758241	−	0.651974i	$-0.226059\pi$
$691$	1.90341e7	1.51648	0.758241	+	0.651974i	$0.226059\pi$
$692$	1.85172e7	1.46998
$693$	1.93614e6	0.153146
$694$	7.57803e6	0.597252
$695$	0	0
$696$	−8.92091e6	−0.698049
$697$	4.94356e6	0.385441
$698$	−1.87969e6	−0.146032
$699$	1.87668e6	0.145277
$700$	0	0
$701$	2.48342e7	1.90878	0.954390	−	0.298564i	$-0.0965075\pi$
$701$	2.48342e7	1.90878	0.954390	+	0.298564i	$0.0965075\pi$
$702$	−4.52567e6	−0.346609
$703$	−6.73218e6	−0.513768
$704$	−5.84567e6	−0.444532
$705$	0	0
$706$	1.21973e6	0.0920983
$707$	1.21657e7	0.915352
$708$	−1.43604e7	−1.07667
$709$	−1.83486e7	−1.37084	−0.685422	−	0.728146i	$-0.740382\pi$
$709$	−1.83486e7	−1.37084	−0.685422	+	0.728146i	$0.740382\pi$
$710$	0	0
$711$	1.73506e6	0.128718
$712$	1.85484e6	0.137122
$713$	831644.	0.0612651
$714$	4.67999e6	0.343558
$715$	0	0
$716$	5.80429e6	0.423123
$717$	1.96228e7	1.42548
$718$	3.32138e6	0.240440
$719$	−387175.	−0.0279309	−0.0139655	−	0.999902i	$-0.504445\pi$
$719$	−387175.	−0.0279309	−0.0139655	+	0.999902i	$0.504445\pi$
$720$	0	0
$721$	−3.58002e7	−2.56476
$722$	−1.18301e6	−0.0844587
$723$	2.25556e7	1.60476
$724$	−4.96710e6	−0.352173
$725$	0	0
$726$	824130.	0.0580302
$727$	−5.28478e6	−0.370843	−0.185422	−	0.982659i	$-0.559365\pi$
$727$	−5.28478e6	−0.370843	−0.185422	+	0.982659i	$0.559365\pi$
$728$	1.41196e7	0.987403
$729$	1.53961e7	1.07298
$730$	0	0
$731$	1.54106e6	0.106666
$732$	−1.95018e7	−1.34523
$733$	−2.20879e7	−1.51843	−0.759215	−	0.650840i	$-0.774417\pi$
$733$	−2.20879e7	−1.51843	−0.759215	+	0.650840i	$0.774417\pi$
$734$	−5.52298e6	−0.378385
$735$	0	0
$736$	4.44535e6	0.302490
$737$	2.55660e7	1.73378
$738$	240931.	0.0162836
$739$	−1.90903e7	−1.28588	−0.642941	−	0.765916i	$-0.722286\pi$
$739$	−1.90903e7	−1.28588	−0.642941	+	0.765916i	$0.722286\pi$
$740$	0	0
$741$	1.24613e7	0.833719
$742$	9.33112e6	0.622191
$743$	8.08984e6	0.537610	0.268805	−	0.963195i	$-0.413371\pi$
$743$	8.08984e6	0.537610	0.268805	+	0.963195i	$0.413371\pi$
$744$	−1.53936e6	−0.101955
$745$	0	0
$746$	7.32944e6	0.482196
$747$	1.99144e6	0.130576
$748$	1.03912e7	0.679066
$749$	2.95751e7	1.92629
$750$	0	0
$751$	3.66332e6	0.237015	0.118507	−	0.992953i	$-0.462189\pi$
$751$	3.66332e6	0.237015	0.118507	+	0.992953i	$0.462189\pi$
$752$	−1.25382e7	−0.808522
$753$	−1.93468e7	−1.24343
$754$	6.12059e6	0.392071
$755$	0	0
$756$	−2.28147e7	−1.45181
$757$	−3.20177e6	−0.203072	−0.101536	−	0.994832i	$-0.532376\pi$
$757$	−3.20177e6	−0.203072	−0.101536	+	0.994832i	$0.532376\pi$
$758$	−8.11722e6	−0.513138
$759$	5.87836e6	0.370384
$760$	0	0
$761$	−1.65236e7	−1.03429	−0.517147	−	0.855897i	$-0.673006\pi$
$761$	−1.65236e7	−1.03429	−0.517147	+	0.855897i	$0.673006\pi$
$762$	−3.24415e6	−0.202401
$763$	1.50430e7	0.935454
$764$	1.50988e7	0.935854
$765$	0	0
$766$	6.56811e6	0.404453
$767$	2.08998e7	1.28278
$768$	1.33011e6	0.0813735
$769$	1.98289e7	1.20916	0.604579	−	0.796545i	$-0.293342\pi$
$769$	1.98289e7	1.20916	0.604579	+	0.796545i	$0.293342\pi$
$770$	0	0
$771$	3.32146e6	0.201230
$772$	−2.14037e6	−0.129254
$773$	−2.13192e7	−1.28329	−0.641643	−	0.767004i	$-0.721747\pi$
$773$	−2.13192e7	−1.28329	−0.641643	+	0.767004i	$0.721747\pi$
$774$	75105.7	0.00450630
$775$	0	0
$776$	1.64016e7	0.977759
$777$	1.49803e7	0.890160
$778$	−1.03693e7	−0.614188
$779$	−8.04606e6	−0.475050
$780$	0	0
$781$	783122.	0.0459411
$782$	−1.40286e6	−0.0820350
$783$	−2.09787e7	−1.22285
$784$	1.71652e7	0.997374
$785$	0	0
$786$	−1.78055e6	−0.102801
$787$	−2.08248e7	−1.19852	−0.599258	−	0.800556i	$-0.704538\pi$
$787$	−2.08248e7	−1.19852	−0.599258	+	0.800556i	$0.704538\pi$
$788$	2.11645e7	1.21420
$789$	−3.22305e6	−0.184321
$790$	0	0
$791$	3.46817e7	1.97088
$792$	1.07426e6	0.0608552
$793$	2.83826e7	1.60276
$794$	−1.09260e7	−0.615048
$795$	0	0
$796$	1.33461e6	0.0746570
$797$	1.52522e7	0.850527	0.425263	−	0.905070i	$-0.360181\pi$
$797$	1.52522e7	0.850527	0.425263	+	0.905070i	$0.360181\pi$
$798$	−7.61708e6	−0.423430
$799$	1.48488e7	0.822856
$800$	0	0
$801$	359637.	0.0198054
$802$	−86231.0	−0.00473399
$803$	2.04056e7	1.11676
$804$	−2.48388e7	−1.35516
$805$	0	0
$806$	1.05614e6	0.0572645
$807$	1.66795e7	0.901571
$808$	6.75009e6	0.363732
$809$	1.69851e6	0.0912426	0.0456213	−	0.998959i	$-0.485473\pi$
$809$	1.69851e6	0.0912426	0.0456213	+	0.998959i	$0.485473\pi$
$810$	0	0
$811$	−8.51667e6	−0.454693	−0.227346	−	0.973814i	$-0.573005\pi$
$811$	−8.51667e6	−0.454693	−0.227346	+	0.973814i	$0.573005\pi$
$812$	3.08550e7	1.64223
$813$	−4.74225e6	−0.251627
$814$	−4.03304e6	−0.213340
$815$	0	0
$816$	−8.72310e6	−0.458612
$817$	−2.50821e6	−0.131465
$818$	5.43929e6	0.284223
$819$	2.73768e6	0.142617
$820$	0	0
$821$	1.11108e7	0.575293	0.287647	−	0.957737i	$-0.407127\pi$
$821$	1.11108e7	0.575293	0.287647	+	0.957737i	$0.407127\pi$
$822$	6.07968e6	0.313835
$823$	1.20698e7	0.621157	0.310578	−	0.950548i	$-0.399477\pi$
$823$	1.20698e7	0.621157	0.310578	+	0.950548i	$0.399477\pi$
$824$	−1.98636e7	−1.01916
$825$	0	0
$826$	−1.27751e7	−0.651501
$827$	−1.25034e7	−0.635719	−0.317859	−	0.948138i	$-0.602964\pi$
$827$	−1.25034e7	−0.635719	−0.317859	+	0.948138i	$0.602964\pi$
$828$	563867.	0.0285825
$829$	−4.43044e6	−0.223903	−0.111952	−	0.993714i	$-0.535710\pi$
$829$	−4.43044e6	−0.223903	−0.111952	+	0.993714i	$0.535710\pi$
$830$	0	0
$831$	5.26177e6	0.264319
$832$	−8.26568e6	−0.413971
$833$	−2.03284e7	−1.01506
$834$	406964.	0.0202601
$835$	0	0
$836$	−1.69126e7	−0.836939
$837$	−3.62000e6	−0.178605
$838$	6.01365e6	0.295821
$839$	−2.49502e7	−1.22369	−0.611843	−	0.790979i	$-0.709572\pi$
$839$	−2.49502e7	−1.22369	−0.611843	+	0.790979i	$0.709572\pi$
$840$	0	0
$841$	7.86071e6	0.383241
$842$	−8.60998e6	−0.418525
$843$	−3.69169e6	−0.178919
$844$	−1.02819e7	−0.496840
$845$	0	0
$846$	723674.	0.0347630
$847$	−6.04651e6	−0.289598
$848$	−1.73924e7	−0.830558
$849$	6.25815e6	0.297973
$850$	0	0
$851$	−4.49047e6	−0.212553
$852$	−760847.	−0.0359086
$853$	−1.87125e7	−0.880561	−0.440280	−	0.897860i	$-0.645121\pi$
$853$	−1.87125e7	−0.880561	−0.440280	+	0.897860i	$0.645121\pi$
$854$	−1.73490e7	−0.814011
$855$	0	0
$856$	1.64096e7	0.765446
$857$	9.57836e6	0.445491	0.222746	−	0.974877i	$-0.428498\pi$
$857$	9.57836e6	0.445491	0.222746	+	0.974877i	$0.428498\pi$
$858$	7.46522e6	0.346198
$859$	−2.56964e7	−1.18820	−0.594100	−	0.804391i	$-0.702492\pi$
$859$	−2.56964e7	−1.18820	−0.594100	+	0.804391i	$0.702492\pi$
$860$	0	0
$861$	1.79039e7	0.823077
$862$	3.48962e6	0.159959
$863$	3.20924e7	1.46681	0.733406	−	0.679791i	$-0.237929\pi$
$863$	3.20924e7	1.46681	0.733406	+	0.679791i	$0.237929\pi$
$864$	−1.93498e7	−0.881845
$865$	0	0
$866$	7.08740e6	0.321139
$867$	−1.07849e7	−0.487271
$868$	5.32421e6	0.239859
$869$	−3.47122e7	−1.55931
$870$	0	0
$871$	3.61499e7	1.61459
$872$	8.34654e6	0.371720
$873$	3.18014e6	0.141225
$874$	2.28328e6	0.101107
$875$	0	0
$876$	−1.98252e7	−0.872884
$877$	−5.49507e6	−0.241254	−0.120627	−	0.992698i	$-0.538490\pi$
$877$	−5.49507e6	−0.241254	−0.120627	+	0.992698i	$0.538490\pi$
$878$	4.38903e6	0.192147
$879$	877754.	0.0383178
$880$	0	0
$881$	−2.28313e7	−0.991038	−0.495519	−	0.868597i	$-0.665022\pi$
$881$	−2.28313e7	−0.991038	−0.495519	+	0.868597i	$0.665022\pi$
$882$	−990729.	−0.0428828
$883$	2.80025e7	1.20864	0.604318	−	0.796743i	$-0.293446\pi$
$883$	2.80025e7	1.20864	0.604318	+	0.796743i	$0.293446\pi$
$884$	1.46930e7	0.632383
$885$	0	0
$886$	4.75538e6	0.203517
$887$	4.93689e6	0.210690	0.105345	−	0.994436i	$-0.466405\pi$
$887$	4.93689e6	0.210690	0.105345	+	0.994436i	$0.466405\pi$
$888$	8.31177e6	0.353721
$889$	2.38018e7	1.01008
$890$	0	0
$891$	−2.32695e7	−0.981957
$892$	−5.26538e6	−0.221573
$893$	−2.41676e7	−1.01416
$894$	−1.09661e7	−0.458890
$895$	0	0
$896$	3.69625e7	1.53812
$897$	8.31191e6	0.344921
$898$	6.91886e6	0.286315
$899$	4.89574e6	0.202031
$900$	0	0
$901$	2.05975e7	0.845282
$902$	−4.82015e6	−0.197262
$903$	5.58122e6	0.227777
$904$	1.92430e7	0.783164
$905$	0	0
$906$	1.95653e6	0.0791891
$907$	1.84794e7	0.745880	0.372940	−	0.927856i	$-0.378350\pi$
$907$	1.84794e7	0.745880	0.372940	+	0.927856i	$0.378350\pi$
$908$	−1.15492e7	−0.464876
$909$	1.30879e6	0.0525363
$910$	0	0
$911$	1.37578e6	0.0549229	0.0274615	−	0.999623i	$-0.491258\pi$
$911$	1.37578e6	0.0549229	0.0274615	+	0.999623i	$0.491258\pi$
$912$	1.41976e7	0.565233
$913$	−3.98414e7	−1.58182
$914$	4.82906e6	0.191204
$915$	0	0
$916$	−1.72910e7	−0.680896
$917$	1.30636e7	0.513027
$918$	6.10642e6	0.239155
$919$	−2.38412e7	−0.931192	−0.465596	−	0.884997i	$-0.654160\pi$
$919$	−2.38412e7	−0.931192	−0.465596	+	0.884997i	$0.654160\pi$
$920$	0	0
$921$	−1.44157e7	−0.559999
$922$	−1.23196e7	−0.477276
$923$	1.10732e6	0.0427828
$924$	3.76335e7	1.45009
$925$	0	0
$926$	−9.48708e6	−0.363584
$927$	−3.85139e6	−0.147204
$928$	2.61690e7	0.997508
$929$	4.74191e6	0.180266	0.0901331	−	0.995930i	$-0.471271\pi$
$929$	4.74191e6	0.180266	0.0901331	+	0.995930i	$0.471271\pi$
$930$	0	0
$931$	3.30861e7	1.25104
$932$	−3.60147e6	−0.135812
$933$	−4.64985e7	−1.74878
$934$	−7.27370e6	−0.272828
$935$	0	0
$936$	1.51899e6	0.0566716
$937$	1.07340e7	0.399404	0.199702	−	0.979857i	$-0.436003\pi$
$937$	1.07340e7	0.399404	0.199702	+	0.979857i	$0.436003\pi$
$938$	−2.20969e7	−0.820018
$939$	−3.42706e6	−0.126840
$940$	0	0
$941$	2.93932e7	1.08211	0.541057	−	0.840986i	$-0.318024\pi$
$941$	2.93932e7	1.08211	0.541057	+	0.840986i	$0.318024\pi$
$942$	1.29870e7	0.476849
$943$	−5.36685e6	−0.196535
$944$	2.38117e7	0.869683
$945$	0	0
$946$	−1.50259e6	−0.0545901
$947$	−4.42575e7	−1.60366	−0.801830	−	0.597552i	$-0.796140\pi$
$947$	−4.42575e7	−1.60366	−0.801830	+	0.597552i	$0.796140\pi$
$948$	3.37249e7	1.21879
$949$	2.88531e7	1.03999
$950$	0	0
$951$	8.64013e6	0.309791
$952$	−1.90514e7	−0.681293
$953$	−2.46336e7	−0.878609	−0.439304	−	0.898338i	$-0.644775\pi$
$953$	−2.46336e7	−0.878609	−0.439304	+	0.898338i	$0.644775\pi$
$954$	1.00384e6	0.0357104
$955$	0	0
$956$	−3.76573e7	−1.33261
$957$	3.46049e7	1.22140
$958$	−1.58008e7	−0.556244
$959$	−4.46056e7	−1.56618
$960$	0	0
$961$	−2.77844e7	−0.970492
$962$	−5.70266e6	−0.198673
$963$	3.18170e6	0.110559
$964$	−4.32856e7	−1.50021
$965$	0	0
$966$	−5.08071e6	−0.175179
$967$	1.51031e7	0.519397	0.259698	−	0.965690i	$-0.416377\pi$
$967$	1.51031e7	0.519397	0.259698	+	0.965690i	$0.416377\pi$
$968$	−3.35488e6	−0.115077
$969$	−1.68139e7	−0.575253
$970$	0	0
$971$	2.95351e7	1.00529	0.502643	−	0.864494i	$-0.332361\pi$
$971$	2.95351e7	1.00529	0.502643	+	0.864494i	$0.332361\pi$
$972$	−4.70646e6	−0.159782
$973$	−2.98583e6	−0.101107
$974$	−1.38817e7	−0.468863
$975$	0	0
$976$	3.23371e7	1.08662
$977$	−2.11745e7	−0.709705	−0.354852	−	0.934922i	$-0.615469\pi$
$977$	−2.11745e7	−0.709705	−0.354852	+	0.934922i	$0.615469\pi$
$978$	−1.82724e7	−0.610869
$979$	−7.19505e6	−0.239926
$980$	0	0
$981$	1.61833e6	0.0536900
$982$	4.58810e6	0.151829
$983$	1.08531e7	0.358236	0.179118	−	0.983828i	$-0.442676\pi$
$983$	1.08531e7	0.358236	0.179118	+	0.983828i	$0.442676\pi$
$984$	9.93394e6	0.327065
$985$	0	0
$986$	−8.25841e6	−0.270523
$987$	5.37774e7	1.75714
$988$	−2.39141e7	−0.779402
$989$	−1.67302e6	−0.0543888
$990$	0	0
$991$	−1.68782e7	−0.545936	−0.272968	−	0.962023i	$-0.588005\pi$
$991$	−1.68782e7	−0.545936	−0.272968	+	0.962023i	$0.588005\pi$
$992$	4.51561e6	0.145693
$993$	−3.78116e7	−1.21689
$994$	−676857.	−0.0217286
$995$	0	0
$996$	3.87082e7	1.23639
$997$	1.21396e7	0.386783	0.193392	−	0.981122i	$-0.438051\pi$
$997$	1.21396e7	0.386783	0.193392	+	0.981122i	$0.438051\pi$
$998$	−3.78496e6	−0.120291
$999$	1.95462e7	0.619653

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1075.6.a.b.1.6		10
5.4	even	2		43.6.a.b.1.5	✓	10
15.14	odd	2		387.6.a.e.1.6		10
20.19	odd	2		688.6.a.h.1.8		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.a.b.1.5	✓	10	5.4	even	2
387.6.a.e.1.6		10	15.14	odd	2
688.6.a.h.1.8		10	20.19	odd	2
1075.6.a.b.1.6		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)

\(f(q)\)	\(=\)	\(q+1.86024 q^{2} +14.8716 q^{3} -28.5395 q^{4} +27.6647 q^{6} -202.971 q^{7} -112.618 q^{8} -21.8357 q^{9} +O(q^{10})\)	\(q+1.86024 q^{2} +14.8716 q^{3} -28.5395 q^{4} +27.6647 q^{6} -202.971 q^{7} -112.618 q^{8} -21.8357 q^{9} +436.853 q^{11} -424.428 q^{12} +617.704 q^{13} -377.575 q^{14} +703.768 q^{16} -833.458 q^{17} -40.6196 q^{18} +1356.52 q^{19} -3018.51 q^{21} +812.652 q^{22} +904.822 q^{23} -1674.81 q^{24} +1149.08 q^{26} -3938.53 q^{27} +5792.70 q^{28} +5326.52 q^{29} +919.124 q^{31} +4912.95 q^{32} +6496.71 q^{33} -1550.43 q^{34} +623.180 q^{36} -4962.82 q^{37} +2523.46 q^{38} +9186.24 q^{39} -5931.38 q^{41} -5615.15 q^{42} -1849.00 q^{43} -12467.6 q^{44} +1683.19 q^{46} -17815.9 q^{47} +10466.2 q^{48} +24390.4 q^{49} -12394.9 q^{51} -17629.0 q^{52} -24713.3 q^{53} -7326.61 q^{54} +22858.2 q^{56} +20173.7 q^{57} +9908.61 q^{58} +33834.6 q^{59} +45948.5 q^{61} +1709.79 q^{62} +4432.02 q^{63} -13381.3 q^{64} +12085.4 q^{66} +58523.0 q^{67} +23786.5 q^{68} +13456.1 q^{69} +1792.64 q^{71} +2459.09 q^{72} +46710.3 q^{73} -9232.03 q^{74} -38714.5 q^{76} -88668.8 q^{77} +17088.6 q^{78} -79459.7 q^{79} -53266.1 q^{81} -11033.8 q^{82} -91200.9 q^{83} +86146.8 q^{84} -3439.58 q^{86} +79213.9 q^{87} -49197.6 q^{88} -16470.2 q^{89} -125376. q^{91} -25823.2 q^{92} +13668.8 q^{93} -33141.8 q^{94} +73063.4 q^{96} -145639. q^{97} +45372.0 q^{98} -9539.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 8 q^{2} - 28 q^{3} + 202 q^{4} + 75 q^{6} - 60 q^{7} - 294 q^{8} + 1356 q^{9}+O(q^{10}) \) 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

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