LMFDB - Embedded newform 825.6.a.o.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,6,Mod(1,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	825.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:	$132.316651346$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 152x^{5} + 358x^{4} + 5771x^{3} - 13444x^{2} - 51316x + 92576 $ x^7 - 2x^6 - 152x^5 + 358x^4 + 5771x^3 - 13444x^2 - 51316x + 92576
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.60892$ of defining polynomial
Character	$\chi$	$=$	825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.60892 q^{2} -9.00000 q^{3} -25.1935 q^{4} -23.4803 q^{6} -174.969 q^{7} -149.214 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+2.60892 q^{2} -9.00000 q^{3} -25.1935 q^{4} -23.4803 q^{6} -174.969 q^{7} -149.214 q^{8} +81.0000 q^{9} +121.000 q^{11} +226.742 q^{12} +815.527 q^{13} -456.481 q^{14} +416.906 q^{16} -462.991 q^{17} +211.323 q^{18} -1996.09 q^{19} +1574.72 q^{21} +315.680 q^{22} -1342.70 q^{23} +1342.92 q^{24} +2127.65 q^{26} -729.000 q^{27} +4408.08 q^{28} +6586.33 q^{29} -4492.74 q^{31} +5862.51 q^{32} -1089.00 q^{33} -1207.91 q^{34} -2040.67 q^{36} +9089.95 q^{37} -5207.64 q^{38} -7339.74 q^{39} -1108.16 q^{41} +4108.32 q^{42} +20231.4 q^{43} -3048.42 q^{44} -3503.01 q^{46} +9098.00 q^{47} -3752.15 q^{48} +13807.1 q^{49} +4166.92 q^{51} -20546.0 q^{52} +38340.2 q^{53} -1901.91 q^{54} +26107.7 q^{56} +17964.8 q^{57} +17183.2 q^{58} +29309.0 q^{59} +32390.1 q^{61} -11721.2 q^{62} -14172.5 q^{63} +1953.85 q^{64} -2841.12 q^{66} -64412.9 q^{67} +11664.4 q^{68} +12084.3 q^{69} -63314.4 q^{71} -12086.3 q^{72} +4121.83 q^{73} +23715.0 q^{74} +50288.4 q^{76} -21171.2 q^{77} -19148.8 q^{78} -53233.3 q^{79} +6561.00 q^{81} -2891.10 q^{82} -119847. q^{83} -39672.7 q^{84} +52782.1 q^{86} -59277.0 q^{87} -18054.8 q^{88} +66306.8 q^{89} -142692. q^{91} +33827.4 q^{92} +40434.6 q^{93} +23736.0 q^{94} -52762.6 q^{96} -180567. q^{97} +36021.7 q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 9 q^{2} - 63 q^{3} + 95 q^{4} - 81 q^{6} + 65 q^{7} + 207 q^{8} + 567 q^{9}+O(q^{10}) $ 7 * q + 9 * q^2 - 63 * q^3 + 95 * q^4 - 81 * q^6 + 65 * q^7 + 207 * q^8 + 567 * q^9	\( 7 q + 9 q^{2} - 63 q^{3} + 95 q^{4} - 81 q^{6} + 65 q^{7} + 207 q^{8} + 567 q^{9} + 847 q^{11} - 855 q^{12} - 113 q^{13} - 1212 q^{14} + 499 q^{16} + 1030 q^{17} + 729 q^{18} - 3803 q^{19} - 585 q^{21} + 1089 q^{22} + 514 q^{23} - 1863 q^{24} - 12111 q^{26} - 5103 q^{27} - 342 q^{28} - 2698 q^{29} - 17233 q^{31} + 9943 q^{32} - 7623 q^{33} + 4090 q^{34} + 7695 q^{36} + 23182 q^{37} - 11943 q^{38} + 1017 q^{39} - 16158 q^{41} + 10908 q^{42} - 4249 q^{43} + 11495 q^{44} - 28769 q^{46} + 7580 q^{47} - 4491 q^{48} + 37140 q^{49} - 9270 q^{51} - 23887 q^{52} + 20574 q^{53} - 6561 q^{54} - 73276 q^{56} + 34227 q^{57} + 8733 q^{58} - 364 q^{59} - 28127 q^{61} - 71917 q^{62} + 5265 q^{63} + 43379 q^{64} - 9801 q^{66} - 21493 q^{67} + 160660 q^{68} - 4626 q^{69} - 177084 q^{71} + 16767 q^{72} + 78670 q^{73} + 196750 q^{74} - 32701 q^{76} + 7865 q^{77} + 108999 q^{78} - 187432 q^{79} + 45927 q^{81} + 179552 q^{82} - 44592 q^{83} + 3078 q^{84} - 110433 q^{86} + 24282 q^{87} + 25047 q^{88} - 151168 q^{89} - 230153 q^{91} - 44767 q^{92} + 155097 q^{93} + 54040 q^{94} - 89487 q^{96} - 55589 q^{97} + 478341 q^{98} + 68607 q^{99}+O(q^{100}) \) 7 * q + 9 * q^2 - 63 * q^3 + 95 * q^4 - 81 * q^6 + 65 * q^7 + 207 * q^8 + 567 * q^9 + 847 * q^11 - 855 * q^12 - 113 * q^13 - 1212 * q^14 + 499 * q^16 + 1030 * q^17 + 729 * q^18 - 3803 * q^19 - 585 * q^21 + 1089 * q^22 + 514 * q^23 - 1863 * q^24 - 12111 * q^26 - 5103 * q^27 - 342 * q^28 - 2698 * q^29 - 17233 * q^31 + 9943 * q^32 - 7623 * q^33 + 4090 * q^34 + 7695 * q^36 + 23182 * q^37 - 11943 * q^38 + 1017 * q^39 - 16158 * q^41 + 10908 * q^42 - 4249 * q^43 + 11495 * q^44 - 28769 * q^46 + 7580 * q^47 - 4491 * q^48 + 37140 * q^49 - 9270 * q^51 - 23887 * q^52 + 20574 * q^53 - 6561 * q^54 - 73276 * q^56 + 34227 * q^57 + 8733 * q^58 - 364 * q^59 - 28127 * q^61 - 71917 * q^62 + 5265 * q^63 + 43379 * q^64 - 9801 * q^66 - 21493 * q^67 + 160660 * q^68 - 4626 * q^69 - 177084 * q^71 + 16767 * q^72 + 78670 * q^73 + 196750 * q^74 - 32701 * q^76 + 7865 * q^77 + 108999 * q^78 - 187432 * q^79 + 45927 * q^81 + 179552 * q^82 - 44592 * q^83 + 3078 * q^84 - 110433 * q^86 + 24282 * q^87 + 25047 * q^88 - 151168 * q^89 - 230153 * q^91 - 44767 * q^92 + 155097 * q^93 + 54040 * q^94 - 89487 * q^96 - 55589 * q^97 + 478341 * q^98 + 68607 * 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 $

$2$	2.60892	0.461197	0.230598	−	0.973049i	$-0.425932\pi$
$2$	2.60892	0.461197	0.230598	+	0.973049i	$0.425932\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	−25.1935	−0.787297
$5$	0	0
$5$	0	0
$6$	−23.4803	−0.266272
$7$	−174.969	−1.34963	−0.674817	−	0.737985i	$-0.735777\pi$
$7$	−174.969	−1.34963	−0.674817	+	0.737985i	$0.735777\pi$
$8$	−149.214	−0.824296
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	226.742	0.454546
$13$	815.527	1.33838	0.669191	−	0.743091i	$-0.266641\pi$
$13$	815.527	1.33838	0.669191	+	0.743091i	$0.266641\pi$
$14$	−456.481	−0.622447
$15$	0	0
$16$	416.906	0.407135
$17$	−462.991	−0.388553	−0.194276	−	0.980947i	$-0.562236\pi$
$17$	−462.991	−0.388553	−0.194276	+	0.980947i	$0.562236\pi$
$18$	211.323	0.153732
$19$	−1996.09	−1.26851	−0.634257	−	0.773122i	$-0.718694\pi$
$19$	−1996.09	−1.26851	−0.634257	+	0.773122i	$0.718694\pi$
$20$	0	0
$21$	1574.72	0.779211
$22$	315.680	0.139056
$23$	−1342.70	−0.529250	−0.264625	−	0.964351i	$-0.585248\pi$
$23$	−1342.70	−0.529250	−0.264625	+	0.964351i	$0.585248\pi$
$24$	1342.92	0.475908
$25$	0	0
$26$	2127.65	0.617257
$27$	−729.000	−0.192450
$28$	4408.08	1.06256
$29$	6586.33	1.45428	0.727141	−	0.686488i	$-0.240849\pi$
$29$	6586.33	1.45428	0.727141	+	0.686488i	$0.240849\pi$
$30$	0	0
$31$	−4492.74	−0.839667	−0.419833	−	0.907601i	$-0.637911\pi$
$31$	−4492.74	−0.839667	−0.419833	+	0.907601i	$0.637911\pi$
$32$	5862.51	1.01207
$33$	−1089.00	−0.174078
$34$	−1207.91	−0.179199
$35$	0	0
$36$	−2040.67	−0.262432
$37$	9089.95	1.09158	0.545792	−	0.837921i	$-0.316229\pi$
$37$	9089.95	1.09158	0.545792	+	0.837921i	$0.316229\pi$
$38$	−5207.64	−0.585035
$39$	−7339.74	−0.772715
$40$	0	0
$41$	−1108.16	−0.102954	−0.0514769	−	0.998674i	$-0.516393\pi$
$41$	−1108.16	−0.102954	−0.0514769	+	0.998674i	$0.516393\pi$
$42$	4108.32	0.359370
$43$	20231.4	1.66861	0.834304	−	0.551304i	$-0.185870\pi$
$43$	20231.4	1.66861	0.834304	+	0.551304i	$0.185870\pi$
$44$	−3048.42	−0.237379
$45$	0	0
$46$	−3503.01	−0.244088
$47$	9098.00	0.600761	0.300380	−	0.953820i	$-0.402886\pi$
$47$	9098.00	0.600761	0.300380	+	0.953820i	$0.402886\pi$
$48$	−3752.15	−0.235059
$49$	13807.1	0.821510
$50$	0	0
$51$	4166.92	0.224331
$52$	−20546.0	−1.05370
$53$	38340.2	1.87484	0.937422	−	0.348196i	$-0.113206\pi$
$53$	38340.2	1.87484	0.937422	+	0.348196i	$0.113206\pi$
$54$	−1901.91	−0.0887574
$55$	0	0
$56$	26107.7	1.11250
$57$	17964.8	0.732377
$58$	17183.2	0.670711
$59$	29309.0	1.09615	0.548076	−	0.836429i	$-0.315361\pi$
$59$	29309.0	1.09615	0.548076	+	0.836429i	$0.315361\pi$
$60$	0	0
$61$	32390.1	1.11452	0.557260	−	0.830338i	$-0.311853\pi$
$61$	32390.1	1.11452	0.557260	+	0.830338i	$0.311853\pi$
$62$	−11721.2	−0.387252
$63$	−14172.5	−0.449878
$64$	1953.85	0.0596267
$65$	0	0
$66$	−2841.12	−0.0802841
$67$	−64412.9	−1.75302	−0.876508	−	0.481387i	$-0.840133\pi$
$67$	−64412.9	−1.75302	−0.876508	+	0.481387i	$0.840133\pi$
$68$	11664.4	0.305907
$69$	12084.3	0.305562
$70$	0	0
$71$	−63314.4	−1.49058	−0.745292	−	0.666738i	$-0.767690\pi$
$71$	−63314.4	−1.49058	−0.745292	+	0.666738i	$0.767690\pi$
$72$	−12086.3	−0.274765
$73$	4121.83	0.0905279	0.0452640	−	0.998975i	$-0.485587\pi$
$73$	4121.83	0.0905279	0.0452640	+	0.998975i	$0.485587\pi$
$74$	23715.0	0.503435
$75$	0	0
$76$	50288.4	0.998698
$77$	−21171.2	−0.406930
$78$	−19148.8	−0.356374
$79$	−53233.3	−0.959656	−0.479828	−	0.877363i	$-0.659301\pi$
$79$	−53233.3	−0.959656	−0.479828	+	0.877363i	$0.659301\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−2891.10	−0.0474820
$83$	−119847.	−1.90956	−0.954780	−	0.297315i	$-0.903909\pi$
$83$	−119847.	−1.90956	−0.954780	+	0.297315i	$0.903909\pi$
$84$	−39672.7	−0.613471
$85$	0	0
$86$	52782.1	0.769557
$87$	−59277.0	−0.839630
$88$	−18054.8	−0.248535
$89$	66306.8	0.887326	0.443663	−	0.896194i	$-0.353679\pi$
$89$	66306.8	0.887326	0.443663	+	0.896194i	$0.353679\pi$
$90$	0	0
$91$	−142692.	−1.80632
$92$	33827.4	0.416677
$93$	40434.6	0.484782
$94$	23736.0	0.277069
$95$	0	0
$96$	−52762.6	−0.584316
$97$	−180567.	−1.94854	−0.974268	−	0.225394i	$-0.927633\pi$
$97$	−180567.	−1.94854	−0.974268	+	0.225394i	$0.927633\pi$
$98$	36021.7	0.378878
$99$	9801.00	0.100504
$100$	0	0
$101$	−113813.	−1.11017	−0.555084	−	0.831794i	$-0.687314\pi$
$101$	−113813.	−1.11017	−0.555084	+	0.831794i	$0.687314\pi$
$102$	10871.2	0.103461
$103$	192758.	1.79027	0.895135	−	0.445795i	$-0.147079\pi$
$103$	192758.	1.79027	0.895135	+	0.445795i	$0.147079\pi$
$104$	−121688.	−1.10322
$105$	0	0
$106$	100027.	0.864672
$107$	−139114.	−1.17465	−0.587327	−	0.809350i	$-0.699820\pi$
$107$	−139114.	−1.17465	−0.587327	+	0.809350i	$0.699820\pi$
$108$	18366.1	0.151515
$109$	97360.0	0.784900	0.392450	−	0.919773i	$-0.371628\pi$
$109$	97360.0	0.784900	0.392450	+	0.919773i	$0.371628\pi$
$110$	0	0
$111$	−81809.6	−0.630227
$112$	−72945.6	−0.549483
$113$	−20952.6	−0.154362	−0.0771811	−	0.997017i	$-0.524592\pi$
$113$	−20952.6	−0.154362	−0.0771811	+	0.997017i	$0.524592\pi$
$114$	46868.7	0.337770
$115$	0	0
$116$	−165933.	−1.14495
$117$	66057.7	0.446127
$118$	76464.9	0.505542
$119$	81009.0	0.524404
$120$	0	0
$121$	14641.0	0.0909091
$122$	84503.4	0.514013
$123$	9973.43	0.0594404
$124$	113188.	0.661067
$125$	0	0
$126$	−36974.9	−0.207482
$127$	347758.	1.91323	0.956616	−	0.291351i	$-0.0941046\pi$
$127$	347758.	1.91323	0.956616	+	0.291351i	$0.0941046\pi$
$128$	−182503.	−0.984566
$129$	−182083.	−0.963372
$130$	0	0
$131$	−160486.	−0.817070	−0.408535	−	0.912743i	$-0.633960\pi$
$131$	−160486.	−0.817070	−0.408535	+	0.912743i	$0.633960\pi$
$132$	27435.7	0.137051
$133$	349253.	1.71203
$134$	−168048.	−0.808486
$135$	0	0
$136$	69084.5	0.320282
$137$	−142596.	−0.649090	−0.324545	−	0.945870i	$-0.605211\pi$
$137$	−142596.	−0.649090	−0.324545	+	0.945870i	$0.605211\pi$
$138$	31527.1	0.140924
$139$	−124192.	−0.545201	−0.272600	−	0.962127i	$-0.587884\pi$
$139$	−124192.	−0.545201	−0.272600	+	0.962127i	$0.587884\pi$
$140$	0	0
$141$	−81882.0	−0.346849
$142$	−165182.	−0.687453
$143$	98678.7	0.403537
$144$	33769.4	0.135712
$145$	0	0
$146$	10753.5	0.0417512
$147$	−124264.	−0.474299
$148$	−229008.	−0.859402
$149$	−332027.	−1.22520	−0.612601	−	0.790393i	$-0.709877\pi$
$149$	−332027.	−1.22520	−0.612601	+	0.790393i	$0.709877\pi$
$150$	0	0
$151$	−221114.	−0.789177	−0.394589	−	0.918858i	$-0.629113\pi$
$151$	−221114.	−0.789177	−0.394589	+	0.918858i	$0.629113\pi$
$152$	297843.	1.04563
$153$	−37502.2	−0.129518
$154$	−55234.1	−0.187675
$155$	0	0
$156$	184914.	0.608356
$157$	178933.	0.579350	0.289675	−	0.957125i	$-0.406453\pi$
$157$	178933.	0.579350	0.289675	+	0.957125i	$0.406453\pi$
$158$	−138882.	−0.442590
$159$	−345062.	−1.08244
$160$	0	0
$161$	234931.	0.714293
$162$	17117.1	0.0512441
$163$	222185.	0.655007	0.327504	−	0.944850i	$-0.393793\pi$
$163$	222185.	0.655007	0.327504	+	0.944850i	$0.393793\pi$
$164$	27918.4	0.0810553
$165$	0	0
$166$	−312673.	−0.880683
$167$	286150.	0.793968	0.396984	−	0.917826i	$-0.370057\pi$
$167$	286150.	0.793968	0.396984	+	0.917826i	$0.370057\pi$
$168$	−234970.	−0.642301
$169$	293791.	0.791265
$170$	0	0
$171$	−161683.	−0.422838
$172$	−509700.	−1.31369
$173$	585626.	1.48767	0.743833	−	0.668366i	$-0.233006\pi$
$173$	585626.	1.48767	0.743833	+	0.668366i	$0.233006\pi$
$174$	−154649.	−0.387235
$175$	0	0
$176$	50445.6	0.122756
$177$	−263781.	−0.632863
$178$	172989.	0.409232
$179$	−511386.	−1.19293	−0.596467	−	0.802638i	$-0.703429\pi$
$179$	−511386.	−1.19293	−0.596467	+	0.802638i	$0.703429\pi$
$180$	0	0
$181$	−16277.3	−0.0369306	−0.0184653	−	0.999830i	$-0.505878\pi$
$181$	−16277.3	−0.0369306	−0.0184653	+	0.999830i	$0.505878\pi$
$182$	−372272.	−0.833071
$183$	−291511.	−0.643469
$184$	200350.	0.436258
$185$	0	0
$186$	105491.	0.223580
$187$	−56021.9	−0.117153
$188$	−229211.	−0.472977
$189$	127552.	0.259737
$190$	0	0
$191$	−219087.	−0.434543	−0.217271	−	0.976111i	$-0.569716\pi$
$191$	−219087.	−0.434543	−0.217271	+	0.976111i	$0.569716\pi$
$192$	−17584.6	−0.0344255
$193$	−704505.	−1.36142	−0.680708	−	0.732555i	$-0.738328\pi$
$193$	−704505.	−1.36142	−0.680708	+	0.732555i	$0.738328\pi$
$194$	−471085.	−0.898658
$195$	0	0
$196$	−347850.	−0.646773
$197$	−670027.	−1.23006	−0.615031	−	0.788503i	$-0.710856\pi$
$197$	−670027.	−1.23006	−0.615031	+	0.788503i	$0.710856\pi$
$198$	25570.1	0.0463520
$199$	−591128.	−1.05815	−0.529077	−	0.848574i	$-0.677462\pi$
$199$	−591128.	−1.05815	−0.529077	+	0.848574i	$0.677462\pi$
$200$	0	0
$201$	579716.	1.01210
$202$	−296930.	−0.512006
$203$	−1.15240e6	−1.96275
$204$	−104979.	−0.176615
$205$	0	0
$206$	502890.	0.825667
$207$	−108759.	−0.176417
$208$	339998.	0.544901
$209$	−241526.	−0.382471
$210$	0	0
$211$	−1070.59	−0.00165545	−0.000827723	−	1.00000i	$-0.500263\pi$
$211$	−1070.59	−0.00165545	−0.000827723		1.00000i	$0.500263\pi$
$212$	−965925.	−1.47606
$213$	569829.	0.860589
$214$	−362937.	−0.541747
$215$	0	0
$216$	108777.	0.158636
$217$	786090.	1.13324
$218$	254005.	0.361993
$219$	−37096.4	−0.0522663
$220$	0	0
$221$	−377581.	−0.520032
$222$	−213435.	−0.290659
$223$	515555.	0.694246	0.347123	−	0.937820i	$-0.387159\pi$
$223$	515555.	0.694246	0.347123	+	0.937820i	$0.387159\pi$
$224$	−1.02576e6	−1.36592
$225$	0	0
$226$	−54663.6	−0.0711914
$227$	69874.2	0.0900020	0.0450010	−	0.998987i	$-0.485671\pi$
$227$	69874.2	0.0900020	0.0450010	+	0.998987i	$0.485671\pi$
$228$	−452596.	−0.576599
$229$	502115.	0.632725	0.316362	−	0.948638i	$-0.397538\pi$
$229$	502115.	0.632725	0.316362	+	0.948638i	$0.397538\pi$
$230$	0	0
$231$	190541.	0.234941
$232$	−982770.	−1.19876
$233$	1.52257e6	1.83733	0.918664	−	0.395041i	$-0.129270\pi$
$233$	1.52257e6	1.83733	0.918664	+	0.395041i	$0.129270\pi$
$234$	172339.	0.205752
$235$	0	0
$236$	−738396.	−0.862997
$237$	479099.	0.554058
$238$	211346.	0.241853
$239$	−601975.	−0.681685	−0.340843	−	0.940120i	$-0.610712\pi$
$239$	−601975.	−0.681685	−0.340843	+	0.940120i	$0.610712\pi$
$240$	0	0
$241$	−960570.	−1.06534	−0.532668	−	0.846324i	$-0.678811\pi$
$241$	−960570.	−1.06534	−0.532668	+	0.846324i	$0.678811\pi$
$242$	38197.3	0.0419270
$243$	−59049.0	−0.0641500
$244$	−816021.	−0.877459
$245$	0	0
$246$	26019.9	0.0274137
$247$	−1.62786e6	−1.69776
$248$	670377.	0.692134
$249$	1.07863e6	1.10248
$250$	0	0
$251$	−999437.	−1.00132	−0.500658	−	0.865645i	$-0.666909\pi$
$251$	−999437.	−1.00132	−0.500658	+	0.865645i	$0.666909\pi$
$252$	357055.	0.354188
$253$	−162467.	−0.159575
$254$	907274.	0.882377
$255$	0	0
$256$	−538659.	−0.513705
$257$	222038.	0.209698	0.104849	−	0.994488i	$-0.466564\pi$
$257$	222038.	0.209698	0.104849	+	0.994488i	$0.466564\pi$
$258$	−475039.	−0.444304
$259$	−1.59046e6	−1.47324
$260$	0	0
$261$	533493.	0.484761
$262$	−418696.	−0.376830
$263$	−501628.	−0.447190	−0.223595	−	0.974682i	$-0.571779\pi$
$263$	−501628.	−0.447190	−0.223595	+	0.974682i	$0.571779\pi$
$264$	162494.	0.143492
$265$	0	0
$266$	911175.	0.789583
$267$	−596761.	−0.512298
$268$	1.62279e6	1.38015
$269$	−1.50544e6	−1.26848	−0.634238	−	0.773138i	$-0.718686\pi$
$269$	−1.50544e6	−1.26848	−0.634238	+	0.773138i	$0.718686\pi$
$270$	0	0
$271$	935698.	0.773949	0.386975	−	0.922090i	$-0.373520\pi$
$271$	935698.	0.773949	0.386975	+	0.922090i	$0.373520\pi$
$272$	−193024.	−0.158193
$273$	1.28423e6	1.04288
$274$	−372021.	−0.299358
$275$	0	0
$276$	−304447.	−0.240569
$277$	1.06811e6	0.836404	0.418202	−	0.908354i	$-0.362660\pi$
$277$	1.06811e6	0.836404	0.418202	+	0.908354i	$0.362660\pi$
$278$	−324007.	−0.251445
$279$	−363912.	−0.279889
$280$	0	0
$281$	683491.	0.516377	0.258189	−	0.966095i	$-0.416874\pi$
$281$	683491.	0.516377	0.258189	+	0.966095i	$0.416874\pi$
$282$	−213624.	−0.159966
$283$	272388.	0.202172	0.101086	−	0.994878i	$-0.467768\pi$
$283$	272388.	0.202172	0.101086	+	0.994878i	$0.467768\pi$
$284$	1.59511e6	1.17353
$285$	0	0
$286$	257445.	0.186110
$287$	193893.	0.138950
$288$	474863.	0.337355
$289$	−1.20550e6	−0.849027
$290$	0	0
$291$	1.62510e6	1.12499
$292$	−103843.	−0.0712724
$293$	1.53914e6	1.04739	0.523697	−	0.851905i	$-0.324553\pi$
$293$	1.53914e6	1.04739	0.523697	+	0.851905i	$0.324553\pi$
$294$	−324196.	−0.218745
$295$	0	0
$296$	−1.35634e6	−0.899789
$297$	−88209.0	−0.0580259
$298$	−866233.	−0.565059
$299$	−1.09501e6	−0.708338
$300$	0	0
$301$	−3.53986e6	−2.25201
$302$	−576870.	−0.363966
$303$	1.02432e6	0.640956
$304$	−832180.	−0.516456
$305$	0	0
$306$	−97840.5	−0.0597331
$307$	−800678.	−0.484855	−0.242428	−	0.970170i	$-0.577944\pi$
$307$	−800678.	−0.484855	−0.242428	+	0.970170i	$0.577944\pi$
$308$	533378.	0.320375
$309$	−1.73482e6	−1.03361
$310$	0	0
$311$	3.17233e6	1.85985	0.929925	−	0.367750i	$-0.119872\pi$
$311$	3.17233e6	1.85985	0.929925	+	0.367750i	$0.119872\pi$
$312$	1.09519e6	0.636946
$313$	996401.	0.574875	0.287437	−	0.957799i	$-0.407197\pi$
$313$	996401.	0.574875	0.287437	+	0.957799i	$0.407197\pi$
$314$	466822.	0.267194
$315$	0	0
$316$	1.34113e6	0.755535
$317$	−1.73853e6	−0.971702	−0.485851	−	0.874042i	$-0.661490\pi$
$317$	−1.73853e6	−0.971702	−0.485851	+	0.874042i	$0.661490\pi$
$318$	−900241.	−0.499219
$319$	796946.	0.438483
$320$	0	0
$321$	1.25202e6	0.678187
$322$	612918.	0.329430
$323$	924170.	0.492885
$324$	−165295.	−0.0874775
$325$	0	0
$326$	579664.	0.302087
$327$	−876240.	−0.453162
$328$	165352.	0.0848644
$329$	−1.59187e6	−0.810807
$330$	0	0
$331$	−530501.	−0.266144	−0.133072	−	0.991106i	$-0.542484\pi$
$331$	−530501.	−0.266144	−0.133072	+	0.991106i	$0.542484\pi$
$332$	3.01938e6	1.50339
$333$	736286.	0.363861
$334$	746544.	0.366176
$335$	0	0
$336$	656510.	0.317244
$337$	−1110.26	−0.000532538 0	−0.000266269	−	1.00000i	$-0.500085\pi$
$337$	−1110.26	−0.000532538 0	−0.000266269		1.00000i	$0.500085\pi$
$338$	766478.	0.364929
$339$	188573.	0.0891211
$340$	0	0
$341$	−543621.	−0.253169
$342$	−421819.	−0.195012
$343$	524885.	0.240896
$344$	−3.01880e6	−1.37543
$345$	0	0
$346$	1.52785e6	0.686107
$347$	1.08849e6	0.485289	0.242644	−	0.970115i	$-0.421985\pi$
$347$	1.08849e6	0.485289	0.242644	+	0.970115i	$0.421985\pi$
$348$	1.49340e6	0.661039
$349$	−591038.	−0.259748	−0.129874	−	0.991531i	$-0.541457\pi$
$349$	−591038.	−0.259748	−0.129874	+	0.991531i	$0.541457\pi$
$350$	0	0
$351$	−594519.	−0.257572
$352$	709363.	0.305149
$353$	980027.	0.418602	0.209301	−	0.977851i	$-0.432881\pi$
$353$	980027.	0.418602	0.209301	+	0.977851i	$0.432881\pi$
$354$	−688184.	−0.291875
$355$	0	0
$356$	−1.67050e6	−0.698589
$357$	−729081.	−0.302765
$358$	−1.33417e6	−0.550177
$359$	−2.87712e6	−1.17821	−0.589104	−	0.808057i	$-0.700519\pi$
$359$	−2.87712e6	−1.17821	−0.589104	+	0.808057i	$0.700519\pi$
$360$	0	0
$361$	1.50826e6	0.609129
$362$	−42466.3	−0.0170323
$363$	−131769.	−0.0524864
$364$	3.59491e6	1.42211
$365$	0	0
$366$	−760530.	−0.296766
$367$	3.28496e6	1.27311	0.636554	−	0.771232i	$-0.280359\pi$
$367$	3.28496e6	1.27311	0.636554	+	0.771232i	$0.280359\pi$
$368$	−559781.	−0.215476
$369$	−89760.9	−0.0343179
$370$	0	0
$371$	−6.70835e6	−2.53035
$372$	−1.01869e6	−0.381667
$373$	1.03382e6	0.384744	0.192372	−	0.981322i	$-0.438382\pi$
$373$	1.03382e6	0.384744	0.192372	+	0.981322i	$0.438382\pi$
$374$	−146157.	−0.0540306
$375$	0	0
$376$	−1.35755e6	−0.495205
$377$	5.37133e6	1.94638
$378$	332774.	0.119790
$379$	3.41421e6	1.22094	0.610468	−	0.792041i	$-0.290982\pi$
$379$	3.41421e6	1.22094	0.610468	+	0.792041i	$0.290982\pi$
$380$	0	0
$381$	−3.12982e6	−1.10461
$382$	−571580.	−0.200410
$383$	−1.82511e6	−0.635758	−0.317879	−	0.948131i	$-0.602971\pi$
$383$	−1.82511e6	−0.635758	−0.317879	+	0.948131i	$0.602971\pi$
$384$	1.64253e6	0.568439
$385$	0	0
$386$	−1.83800e6	−0.627881
$387$	1.63874e6	0.556203
$388$	4.54911e6	1.53408
$389$	−1.31381e6	−0.440210	−0.220105	−	0.975476i	$-0.570640\pi$
$389$	−1.31381e6	−0.440210	−0.220105	+	0.975476i	$0.570640\pi$
$390$	0	0
$391$	621659.	0.205641
$392$	−2.06021e6	−0.677168
$393$	1.44438e6	0.471736
$394$	−1.74805e6	−0.567301
$395$	0	0
$396$	−246922.	−0.0791264
$397$	−4.07499e6	−1.29763	−0.648814	−	0.760947i	$-0.724735\pi$
$397$	−4.07499e6	−1.29763	−0.648814	+	0.760947i	$0.724735\pi$
$398$	−1.54221e6	−0.488017
$399$	−3.14328e6	−0.988441
$400$	0	0
$401$	1.88854e6	0.586495	0.293247	−	0.956037i	$-0.405264\pi$
$401$	1.88854e6	0.586495	0.293247	+	0.956037i	$0.405264\pi$
$402$	1.51244e6	0.466779
$403$	−3.66395e6	−1.12379
$404$	2.86735e6	0.874033
$405$	0	0
$406$	−3.00653e6	−0.905213
$407$	1.09988e6	0.329125
$408$	−621760.	−0.184915
$409$	−2.77474e6	−0.820188	−0.410094	−	0.912043i	$-0.634504\pi$
$409$	−2.77474e6	−0.820188	−0.410094	+	0.912043i	$0.634504\pi$
$410$	0	0
$411$	1.28336e6	0.374752
$412$	−4.85624e6	−1.40948
$413$	−5.12816e6	−1.47940
$414$	−283744.	−0.0813628
$415$	0	0
$416$	4.78103e6	1.35453
$417$	1.11773e6	0.314772
$418$	−630124.	−0.176395
$419$	1.74694e6	0.486119	0.243060	−	0.970011i	$-0.421849\pi$
$419$	1.74694e6	0.486119	0.243060	+	0.970011i	$0.421849\pi$
$420$	0	0
$421$	321171.	0.0883143	0.0441572	−	0.999025i	$-0.485940\pi$
$421$	321171.	0.0883143	0.0441572	+	0.999025i	$0.485940\pi$
$422$	−2793.08	−0.000763487 0
$423$	736938.	0.200254
$424$	−5.72088e6	−1.54543
$425$	0	0
$426$	1.48664e6	0.396901
$427$	−5.66726e6	−1.50419
$428$	3.50476e6	0.924802
$429$	−888109.	−0.232982
$430$	0	0
$431$	1.16098e6	0.301046	0.150523	−	0.988607i	$-0.451904\pi$
$431$	1.16098e6	0.301046	0.150523	+	0.988607i	$0.451904\pi$
$432$	−303924.	−0.0783531
$433$	−1.65899e6	−0.425229	−0.212615	−	0.977136i	$-0.568198\pi$
$433$	−1.65899e6	−0.425229	−0.212615	+	0.977136i	$0.568198\pi$
$434$	2.05085e6	0.522648
$435$	0	0
$436$	−2.45284e6	−0.617950
$437$	2.68015e6	0.671361
$438$	−96781.8	−0.0241051
$439$	7.30846e6	1.80994	0.904971	−	0.425473i	$-0.139892\pi$
$439$	7.30846e6	1.80994	0.904971	+	0.425473i	$0.139892\pi$
$440$	0	0
$441$	1.11838e6	0.273837
$442$	−985081.	−0.239837
$443$	−3.34639e6	−0.810153	−0.405077	−	0.914283i	$-0.632755\pi$
$443$	−3.34639e6	−0.810153	−0.405077	+	0.914283i	$0.632755\pi$
$444$	2.06107e6	0.496176
$445$	0	0
$446$	1.34504e6	0.320184
$447$	2.98824e6	0.707370
$448$	−341863.	−0.0804742
$449$	−2.28621e6	−0.535180	−0.267590	−	0.963533i	$-0.586227\pi$
$449$	−2.28621e6	−0.535180	−0.267590	+	0.963533i	$0.586227\pi$
$450$	0	0
$451$	−134087.	−0.0310417
$452$	527869.	0.121529
$453$	1.99003e6	0.455632
$454$	182297.	0.0415087
$455$	0	0
$456$	−2.68059e6	−0.603696
$457$	−3.57185e6	−0.800024	−0.400012	−	0.916510i	$-0.630994\pi$
$457$	−3.57185e6	−0.800024	−0.400012	+	0.916510i	$0.630994\pi$
$458$	1.30998e6	0.291811
$459$	337520.	0.0747770
$460$	0	0
$461$	−7.95525e6	−1.74342	−0.871709	−	0.490024i	$-0.836988\pi$
$461$	−7.95525e6	−1.74342	−0.871709	+	0.490024i	$0.836988\pi$
$462$	497107.	0.108354
$463$	−5.70155e6	−1.23606	−0.618032	−	0.786153i	$-0.712070\pi$
$463$	−5.70155e6	−1.23606	−0.618032	+	0.786153i	$0.712070\pi$
$464$	2.74588e6	0.592089
$465$	0	0
$466$	3.97226e6	0.847370
$467$	−513186.	−0.108889	−0.0544444	−	0.998517i	$-0.517339\pi$
$467$	−513186.	−0.108889	−0.0544444	+	0.998517i	$0.517339\pi$
$468$	−1.66423e6	−0.351235
$469$	1.12703e7	2.36593
$470$	0	0
$471$	−1.61040e6	−0.334488
$472$	−4.37329e6	−0.903553
$473$	2.44800e6	0.503105
$474$	1.24993e6	0.255530
$475$	0	0
$476$	−2.04090e6	−0.412862
$477$	3.10556e6	0.624948
$478$	−1.57051e6	−0.314391
$479$	5.59628e6	1.11445	0.557225	−	0.830361i	$-0.311866\pi$
$479$	5.59628e6	1.11445	0.557225	+	0.830361i	$0.311866\pi$
$480$	0	0
$481$	7.41310e6	1.46096
$482$	−2.50605e6	−0.491330
$483$	−2.11438e6	−0.412397
$484$	−368858.	−0.0715725
$485$	0	0
$486$	−154054.	−0.0295858
$487$	−425345.	−0.0812678	−0.0406339	−	0.999174i	$-0.512938\pi$
$487$	−425345.	−0.0812678	−0.0406339	+	0.999174i	$0.512938\pi$
$488$	−4.83304e6	−0.918695
$489$	−1.99967e6	−0.378169
$490$	0	0
$491$	−5.56781e6	−1.04227	−0.521136	−	0.853474i	$-0.674492\pi$
$491$	−5.56781e6	−1.04227	−0.521136	+	0.853474i	$0.674492\pi$
$492$	−251266.	−0.0467973
$493$	−3.04941e6	−0.565065
$494$	−4.24697e6	−0.783000
$495$	0	0
$496$	−1.87305e6	−0.341857
$497$	1.10780e7	2.01174
$498$	2.81405e6	0.508462
$499$	−5.31330e6	−0.955241	−0.477621	−	0.878566i	$-0.658501\pi$
$499$	−5.31330e6	−0.955241	−0.477621	+	0.878566i	$0.658501\pi$
$500$	0	0
$501$	−2.57535e6	−0.458398
$502$	−2.60745e6	−0.461804
$503$	1.03261e7	1.81976	0.909882	−	0.414867i	$-0.136172\pi$
$503$	1.03261e7	1.81976	0.909882	+	0.414867i	$0.136172\pi$
$504$	2.11473e6	0.370832
$505$	0	0
$506$	−423864.	−0.0735954
$507$	−2.64412e6	−0.456837
$508$	−8.76125e6	−1.50628
$509$	1.37144e6	0.234629	0.117314	−	0.993095i	$-0.462571\pi$
$509$	1.37144e6	0.234629	0.117314	+	0.993095i	$0.462571\pi$
$510$	0	0
$511$	−721191.	−0.122179
$512$	4.43477e6	0.747646
$513$	1.45515e6	0.244126
$514$	579280.	0.0967121
$515$	0	0
$516$	4.58730e6	0.758460
$517$	1.10086e6	0.181136
$518$	−4.14939e6	−0.679453
$519$	−5.27064e6	−0.858904
$520$	0	0
$521$	524277.	0.0846187	0.0423094	−	0.999105i	$-0.486528\pi$
$521$	524277.	0.0846187	0.0423094	+	0.999105i	$0.486528\pi$
$522$	1.39184e6	0.223570
$523$	−1.42764e6	−0.228225	−0.114113	−	0.993468i	$-0.536402\pi$
$523$	−1.42764e6	−0.228225	−0.114113	+	0.993468i	$0.536402\pi$
$524$	4.04321e6	0.643277
$525$	0	0
$526$	−1.30871e6	−0.206243
$527$	2.08010e6	0.326255
$528$	−454010.	−0.0708730
$529$	−4.63349e6	−0.719895
$530$	0	0
$531$	2.37403e6	0.365384
$532$	−8.79892e6	−1.34788
$533$	−903733.	−0.137791
$534$	−1.55690e6	−0.236270
$535$	0	0
$536$	9.61128e6	1.44500
$537$	4.60248e6	0.688741
$538$	−3.92757e6	−0.585017
$539$	1.67066e6	0.247695
$540$	0	0
$541$	5.39626e6	0.792683	0.396341	−	0.918103i	$-0.370280\pi$
$541$	5.39626e6	0.792683	0.396341	+	0.918103i	$0.370280\pi$
$542$	2.44116e6	0.356943
$543$	146496.	0.0213219
$544$	−2.71429e6	−0.393241
$545$	0	0
$546$	3.35045e6	0.480974
$547$	−7.34437e6	−1.04951	−0.524754	−	0.851254i	$-0.675843\pi$
$547$	−7.34437e6	−1.04951	−0.524754	+	0.851254i	$0.675843\pi$
$548$	3.59249e6	0.511027
$549$	2.62360e6	0.371507
$550$	0	0
$551$	−1.31469e7	−1.84478
$552$	−1.80315e6	−0.251874
$553$	9.31417e6	1.29518
$554$	2.78662e6	0.385747
$555$	0	0
$556$	3.12883e6	0.429235
$557$	7.05449e6	0.963447	0.481723	−	0.876323i	$-0.340011\pi$
$557$	7.05449e6	0.963447	0.481723	+	0.876323i	$0.340011\pi$
$558$	−949418.	−0.129084
$559$	1.64992e7	2.23324
$560$	0	0
$561$	504197.	0.0676383
$562$	1.78318e6	0.238152
$563$	126964.	0.0168814	0.00844072	−	0.999964i	$-0.497313\pi$
$563$	126964.	0.0168814	0.00844072	+	0.999964i	$0.497313\pi$
$564$	2.06290e6	0.273074
$565$	0	0
$566$	710638.	0.0932411
$567$	−1.14797e6	−0.149959
$568$	9.44736e6	1.22868
$569$	−1.36950e7	−1.77330	−0.886648	−	0.462445i	$-0.846972\pi$
$569$	−1.36950e7	−1.77330	−0.886648	+	0.462445i	$0.846972\pi$
$570$	0	0
$571$	−484553.	−0.0621944	−0.0310972	−	0.999516i	$-0.509900\pi$
$571$	−484553.	−0.0621944	−0.0310972	+	0.999516i	$0.509900\pi$
$572$	−2.48606e6	−0.317704
$573$	1.97178e6	0.250883
$574$	505853.	0.0640833
$575$	0	0
$576$	158262.	0.0198756
$577$	−4.31027e6	−0.538970	−0.269485	−	0.963005i	$-0.586853\pi$
$577$	−4.31027e6	−0.538970	−0.269485	+	0.963005i	$0.586853\pi$
$578$	−3.14505e6	−0.391569
$579$	6.34054e6	0.786014
$580$	0	0
$581$	2.09696e7	2.57720
$582$	4.23976e6	0.518841
$583$	4.63917e6	0.565287
$584$	−615032.	−0.0746218
$585$	0	0
$586$	4.01551e6	0.483055
$587$	−4.35136e6	−0.521230	−0.260615	−	0.965443i	$-0.583925\pi$
$587$	−4.35136e6	−0.521230	−0.260615	+	0.965443i	$0.583925\pi$
$588$	3.13065e6	0.373415
$589$	8.96790e6	1.06513
$590$	0	0
$591$	6.03024e6	0.710176
$592$	3.78965e6	0.444422
$593$	−2.85779e6	−0.333729	−0.166865	−	0.985980i	$-0.553364\pi$
$593$	−2.85779e6	−0.333729	−0.166865	+	0.985980i	$0.553364\pi$
$594$	−230131.	−0.0267614
$595$	0	0
$596$	8.36492e6	0.964598
$597$	5.32015e6	0.610926
$598$	−2.85680e6	−0.326683
$599$	1.62076e7	1.84566	0.922831	−	0.385206i	$-0.125870\pi$
$599$	1.62076e7	1.84566	0.922831	+	0.385206i	$0.125870\pi$
$600$	0	0
$601$	−6.74322e6	−0.761520	−0.380760	−	0.924674i	$-0.624338\pi$
$601$	−6.74322e6	−0.761520	−0.380760	+	0.924674i	$0.624338\pi$
$602$	−9.23524e6	−1.03862
$603$	−5.21745e6	−0.584339
$604$	5.57065e6	0.621317
$605$	0	0
$606$	2.67237e6	0.295607
$607$	−1.76156e6	−0.194055	−0.0970276	−	0.995282i	$-0.530933\pi$
$607$	−1.76156e6	−0.194055	−0.0970276	+	0.995282i	$0.530933\pi$
$608$	−1.17021e7	−1.28382
$609$	1.03716e7	1.13319
$610$	0	0
$611$	7.41967e6	0.804047
$612$	944813.	0.101969
$613$	−1.71643e7	−1.84491	−0.922456	−	0.386102i	$-0.873821\pi$
$613$	−1.71643e7	−1.84491	−0.922456	+	0.386102i	$0.873821\pi$
$614$	−2.08891e6	−0.223614
$615$	0	0
$616$	3.15904e6	0.335431
$617$	5.60639e6	0.592885	0.296443	−	0.955051i	$-0.404200\pi$
$617$	5.60639e6	0.592885	0.296443	+	0.955051i	$0.404200\pi$
$618$	−4.52601e6	−0.476699
$619$	679689.	0.0712990	0.0356495	−	0.999364i	$-0.488650\pi$
$619$	679689.	0.0712990	0.0356495	+	0.999364i	$0.488650\pi$
$620$	0	0
$621$	978831.	0.101854
$622$	8.27637e6	0.857757
$623$	−1.16016e7	−1.19756
$624$	−3.05998e6	−0.314599
$625$	0	0
$626$	2.59953e6	0.265131
$627$	2.17374e6	0.220820
$628$	−4.50795e6	−0.456121
$629$	−4.20856e6	−0.424138
$630$	0	0
$631$	−1.79404e6	−0.179374	−0.0896868	−	0.995970i	$-0.528587\pi$
$631$	−1.79404e6	−0.179374	−0.0896868	+	0.995970i	$0.528587\pi$
$632$	7.94312e6	0.791040
$633$	9635.27	0.000955773 0
$634$	−4.53568e6	−0.448146
$635$	0	0
$636$	8.69333e6	0.852203
$637$	1.12601e7	1.09949
$638$	2.07917e6	0.202227
$639$	−5.12846e6	−0.496861
$640$	0	0
$641$	−1.87077e6	−0.179836	−0.0899179	−	0.995949i	$-0.528660\pi$
$641$	−1.87077e6	−0.179836	−0.0899179	+	0.995949i	$0.528660\pi$
$642$	3.26643e6	0.312778
$643$	−1.17131e7	−1.11724	−0.558618	−	0.829425i	$-0.688668\pi$
$643$	−1.17131e7	−1.11724	−0.558618	+	0.829425i	$0.688668\pi$
$644$	−5.91875e6	−0.562361
$645$	0	0
$646$	2.41109e6	0.227317
$647$	−8.80680e6	−0.827099	−0.413550	−	0.910482i	$-0.635711\pi$
$647$	−8.80680e6	−0.827099	−0.413550	+	0.910482i	$0.635711\pi$
$648$	−978990.	−0.0915884
$649$	3.54638e6	0.330502
$650$	0	0
$651$	−7.07481e6	−0.654278
$652$	−5.59763e6	−0.515686
$653$	6.56212e6	0.602229	0.301114	−	0.953588i	$-0.402641\pi$
$653$	6.56212e6	0.602229	0.301114	+	0.953588i	$0.402641\pi$
$654$	−2.28604e6	−0.208997
$655$	0	0
$656$	−461998.	−0.0419161
$657$	333868.	0.0301760
$658$	−4.15306e6	−0.373941
$659$	−1.31437e7	−1.17897	−0.589486	−	0.807778i	$-0.700670\pi$
$659$	−1.31437e7	−1.17897	−0.589486	+	0.807778i	$0.700670\pi$
$660$	0	0
$661$	−1.73548e7	−1.54496	−0.772478	−	0.635041i	$-0.780983\pi$
$661$	−1.73548e7	−1.54496	−0.772478	+	0.635041i	$0.780983\pi$
$662$	−1.38404e6	−0.122745
$663$	3.39823e6	0.300240
$664$	1.78828e7	1.57404
$665$	0	0
$666$	1.92091e6	0.167812
$667$	−8.84350e6	−0.769678
$668$	−7.20913e6	−0.625089
$669$	−4.64000e6	−0.400823
$670$	0	0
$671$	3.91920e6	0.336041
$672$	9.23181e6	0.788613
$673$	−1.02596e7	−0.873159	−0.436580	−	0.899666i	$-0.643810\pi$
$673$	−1.02596e7	−0.873159	−0.436580	+	0.899666i	$0.643810\pi$
$674$	−2896.59	−0.000245605 0
$675$	0	0
$676$	−7.40163e6	−0.622961
$677$	1.22085e6	0.102375	0.0511873	−	0.998689i	$-0.483699\pi$
$677$	1.22085e6	0.102375	0.0511873	+	0.998689i	$0.483699\pi$
$678$	491973.	0.0411024
$679$	3.15936e7	2.62981
$680$	0	0
$681$	−628868.	−0.0519627
$682$	−1.41827e6	−0.116761
$683$	8.61333e6	0.706511	0.353256	−	0.935527i	$-0.385075\pi$
$683$	8.61333e6	0.706511	0.353256	+	0.935527i	$0.385075\pi$
$684$	4.07336e6	0.332899
$685$	0	0
$686$	1.36939e6	0.111100
$687$	−4.51904e6	−0.365304
$688$	8.43458e6	0.679348
$689$	3.12675e7	2.50926
$690$	0	0
$691$	−1.53208e7	−1.22064	−0.610319	−	0.792156i	$-0.708959\pi$
$691$	−1.53208e7	−1.22064	−0.610319	+	0.792156i	$0.708959\pi$
$692$	−1.47540e7	−1.17124
$693$	−1.71487e6	−0.135643
$694$	2.83978e6	0.223814
$695$	0	0
$696$	8.84493e6	0.692104
$697$	513067.	0.0400030
$698$	−1.54197e6	−0.119795
$699$	−1.37031e7	−1.06078
$700$	0	0
$701$	233349.	0.0179354	0.00896768	−	0.999960i	$-0.497145\pi$
$701$	233349.	0.0179354	0.00896768	+	0.999960i	$0.497145\pi$
$702$	−1.55105e6	−0.118791
$703$	−1.81443e7	−1.38469
$704$	236416.	0.0179781
$705$	0	0
$706$	2.55682e6	0.193058
$707$	1.99138e7	1.49832
$708$	6.64556e6	0.498252
$709$	−3.15108e6	−0.235420	−0.117710	−	0.993048i	$-0.537555\pi$
$709$	−3.15108e6	−0.235420	−0.117710	+	0.993048i	$0.537555\pi$
$710$	0	0
$711$	−4.31190e6	−0.319885
$712$	−9.89387e6	−0.731419
$713$	6.03242e6	0.444393
$714$	−1.90212e6	−0.139634
$715$	0	0
$716$	1.28836e7	0.939194
$717$	5.41778e6	0.393571
$718$	−7.50619e6	−0.543386
$719$	−2.38682e7	−1.72186	−0.860930	−	0.508723i	$-0.830118\pi$
$719$	−2.38682e7	−1.72186	−0.860930	+	0.508723i	$0.830118\pi$
$720$	0	0
$721$	−3.37266e7	−2.41621
$722$	3.93494e6	0.280928
$723$	8.64513e6	0.615072
$724$	410083.	0.0290754
$725$	0	0
$726$	−343775.	−0.0242066
$727$	2.60780e7	1.82995	0.914974	−	0.403513i	$-0.132211\pi$
$727$	2.60780e7	1.82995	0.914974	+	0.403513i	$0.132211\pi$
$728$	2.12916e7	1.48895
$729$	531441.	0.0370370
$730$	0	0
$731$	−9.36694e6	−0.648343
$732$	7.34419e6	0.506601
$733$	−822931.	−0.0565723	−0.0282861	−	0.999600i	$-0.509005\pi$
$733$	−822931.	−0.0565723	−0.0282861	+	0.999600i	$0.509005\pi$
$734$	8.57021e6	0.587153
$735$	0	0
$736$	−7.87161e6	−0.535635
$737$	−7.79396e6	−0.528554
$738$	−234179.	−0.0158273
$739$	1.58916e7	1.07043	0.535215	−	0.844716i	$-0.320231\pi$
$739$	1.58916e7	1.07043	0.535215	+	0.844716i	$0.320231\pi$
$740$	0	0
$741$	1.46508e7	0.980200
$742$	−1.75016e7	−1.16699
$743$	−8.78449e6	−0.583774	−0.291887	−	0.956453i	$-0.594283\pi$
$743$	−8.78449e6	−0.583774	−0.291887	+	0.956453i	$0.594283\pi$
$744$	−6.03340e6	−0.399604
$745$	0	0
$746$	2.69715e6	0.177443
$747$	−9.70763e6	−0.636520
$748$	1.41139e6	0.0922343
$749$	2.43406e7	1.58535
$750$	0	0
$751$	−1.46203e7	−0.945926	−0.472963	−	0.881082i	$-0.656816\pi$
$751$	−1.46203e7	−0.945926	−0.472963	+	0.881082i	$0.656816\pi$
$752$	3.79301e6	0.244590
$753$	8.99493e6	0.578110
$754$	1.40134e7	0.897666
$755$	0	0
$756$	−3.21349e6	−0.204490
$757$	7.20669e6	0.457084	0.228542	−	0.973534i	$-0.426604\pi$
$757$	7.20669e6	0.457084	0.228542	+	0.973534i	$0.426604\pi$
$758$	8.90742e6	0.563092
$759$	1.46220e6	0.0921305
$760$	0	0
$761$	−2.92986e7	−1.83394	−0.916969	−	0.398958i	$-0.869372\pi$
$761$	−2.92986e7	−1.83394	−0.916969	+	0.398958i	$0.869372\pi$
$762$	−8.16547e6	−0.509441
$763$	−1.70350e7	−1.05933
$764$	5.51956e6	0.342114
$765$	0	0
$766$	−4.76157e6	−0.293209
$767$	2.39022e7	1.46707
$768$	4.84793e6	0.296588
$769$	−9.38411e6	−0.572239	−0.286119	−	0.958194i	$-0.592365\pi$
$769$	−9.38411e6	−0.572239	−0.286119	+	0.958194i	$0.592365\pi$
$770$	0	0
$771$	−1.99834e6	−0.121069
$772$	1.77490e7	1.07184
$773$	3.43323e6	0.206659	0.103329	−	0.994647i	$-0.467050\pi$
$773$	3.43323e6	0.206659	0.103329	+	0.994647i	$0.467050\pi$
$774$	4.27535e6	0.256519
$775$	0	0
$776$	2.69430e7	1.60617
$777$	1.43141e7	0.850575
$778$	−3.42764e6	−0.203024
$779$	2.21198e6	0.130598
$780$	0	0
$781$	−7.66104e6	−0.449428
$782$	1.62186e6	0.0948412
$783$	−4.80144e6	−0.279877
$784$	5.75627e6	0.334465
$785$	0	0
$786$	3.76827e6	0.217563
$787$	−9.21027e6	−0.530073	−0.265036	−	0.964238i	$-0.585384\pi$
$787$	−9.21027e6	−0.530073	−0.265036	+	0.964238i	$0.585384\pi$
$788$	1.68803e7	0.968424
$789$	4.51465e6	0.258185
$790$	0	0
$791$	3.66605e6	0.208332
$792$	−1.46244e6	−0.0828449
$793$	2.64150e7	1.49165
$794$	−1.06313e7	−0.598462
$795$	0	0
$796$	1.48926e7	0.833082
$797$	1.76659e7	0.985120	0.492560	−	0.870279i	$-0.336061\pi$
$797$	1.76659e7	0.985120	0.492560	+	0.870279i	$0.336061\pi$
$798$	−8.20057e6	−0.455866
$799$	−4.21229e6	−0.233427
$800$	0	0
$801$	5.37085e6	0.295775
$802$	4.92705e6	0.270490
$803$	498741.	0.0272952
$804$	−1.46051e7	−0.796827
$805$	0	0
$806$	−9.55896e6	−0.518290
$807$	1.35489e7	0.732355
$808$	1.69825e7	0.915108
$809$	5.29802e6	0.284605	0.142302	−	0.989823i	$-0.454549\pi$
$809$	5.29802e6	0.284605	0.142302	+	0.989823i	$0.454549\pi$
$810$	0	0
$811$	1.81272e7	0.967786	0.483893	−	0.875127i	$-0.339222\pi$
$811$	1.81272e7	0.967786	0.483893	+	0.875127i	$0.339222\pi$
$812$	2.90331e7	1.54527
$813$	−8.42128e6	−0.446840
$814$	2.86951e6	0.151791
$815$	0	0
$816$	1.73721e6	0.0913329
$817$	−4.03836e7	−2.11665
$818$	−7.23908e6	−0.378268
$819$	−1.15580e7	−0.602108
$820$	0	0
$821$	3.68262e7	1.90677	0.953387	−	0.301751i	$-0.0975710\pi$
$821$	3.68262e7	1.90677	0.953387	+	0.301751i	$0.0975710\pi$
$822$	3.34819e6	0.172835
$823$	3.25146e7	1.67332	0.836660	−	0.547723i	$-0.184505\pi$
$823$	3.25146e7	1.67332	0.836660	+	0.547723i	$0.184505\pi$
$824$	−2.87621e7	−1.47571
$825$	0	0
$826$	−1.33790e7	−0.682296
$827$	−4.14035e6	−0.210510	−0.105255	−	0.994445i	$-0.533566\pi$
$827$	−4.14035e6	−0.210510	−0.105255	+	0.994445i	$0.533566\pi$
$828$	2.74002e6	0.138892
$829$	−2.81888e7	−1.42459	−0.712296	−	0.701879i	$-0.752345\pi$
$829$	−2.81888e7	−1.42459	−0.712296	+	0.701879i	$0.752345\pi$
$830$	0	0
$831$	−9.61298e6	−0.482898
$832$	1.59342e6	0.0798033
$833$	−6.39257e6	−0.319200
$834$	2.91606e6	0.145172
$835$	0	0
$836$	6.08490e6	0.301119
$837$	3.27521e6	0.161594
$838$	4.55763e6	0.224197
$839$	−470091.	−0.0230556	−0.0115278	−	0.999934i	$-0.503669\pi$
$839$	−470091.	−0.0230556	−0.0115278	+	0.999934i	$0.503669\pi$
$840$	0	0
$841$	2.28686e7	1.11494
$842$	837911.	0.0407303
$843$	−6.15142e6	−0.298130
$844$	26971.8	0.00130333
$845$	0	0
$846$	1.92262e6	0.0923563
$847$	−2.56172e6	−0.122694
$848$	1.59843e7	0.763314
$849$	−2.45149e6	−0.116724
$850$	0	0
$851$	−1.22051e7	−0.577721
$852$	−1.43560e7	−0.677540
$853$	−3.80179e7	−1.78902	−0.894510	−	0.447047i	$-0.852476\pi$
$853$	−3.80179e7	−1.78902	−0.894510	+	0.447047i	$0.852476\pi$
$854$	−1.47855e7	−0.693730
$855$	0	0
$856$	2.07576e7	0.968263
$857$	2.72661e7	1.26815	0.634076	−	0.773271i	$-0.281381\pi$
$857$	2.72661e7	1.26815	0.634076	+	0.773271i	$0.281381\pi$
$858$	−2.31701e6	−0.107451
$859$	3.07007e7	1.41960	0.709799	−	0.704404i	$-0.248786\pi$
$859$	3.07007e7	1.41960	0.709799	+	0.704404i	$0.248786\pi$
$860$	0	0
$861$	−1.74504e6	−0.0802228
$862$	3.02892e6	0.138841
$863$	−2.20422e7	−1.00746	−0.503731	−	0.863861i	$-0.668040\pi$
$863$	−2.20422e7	−1.00746	−0.503731	+	0.863861i	$0.668040\pi$
$864$	−4.27377e6	−0.194772
$865$	0	0
$866$	−4.32817e6	−0.196114
$867$	1.08495e7	0.490186
$868$	−1.98044e7	−0.892199
$869$	−6.44123e6	−0.289347
$870$	0	0
$871$	−5.25305e7	−2.34620
$872$	−1.45274e7	−0.646990
$873$	−1.46259e7	−0.649512
$874$	6.99231e6	0.309630
$875$	0	0
$876$	934590.	0.0411491
$877$	6.53105e6	0.286737	0.143369	−	0.989669i	$-0.454207\pi$
$877$	6.53105e6	0.286737	0.143369	+	0.989669i	$0.454207\pi$
$878$	1.90672e7	0.834740
$879$	−1.38523e7	−0.604713
$880$	0	0
$881$	2.27023e7	0.985439	0.492719	−	0.870188i	$-0.336003\pi$
$881$	2.27023e7	0.985439	0.492719	+	0.870188i	$0.336003\pi$
$882$	2.91776e6	0.126293
$883$	2.91594e7	1.25857	0.629285	−	0.777174i	$-0.283348\pi$
$883$	2.91594e7	1.25857	0.629285	+	0.777174i	$0.283348\pi$
$884$	9.51260e6	0.409420
$885$	0	0
$886$	−8.73048e6	−0.373640
$887$	−1.47884e7	−0.631120	−0.315560	−	0.948906i	$-0.602192\pi$
$887$	−1.47884e7	−0.631120	−0.315560	+	0.948906i	$0.602192\pi$
$888$	1.22071e7	0.519493
$889$	−6.08468e7	−2.58216
$890$	0	0
$891$	793881.	0.0335013
$892$	−1.29887e7	−0.546578
$893$	−1.81604e7	−0.762073
$894$	7.79609e6	0.326237
$895$	0	0
$896$	3.19323e7	1.32880
$897$	9.85510e6	0.408959
$898$	−5.96454e6	−0.246823
$899$	−2.95907e7	−1.22111
$900$	0	0
$901$	−1.77512e7	−0.728475
$902$	−349823.	−0.0143164
$903$	3.18588e7	1.30020
$904$	3.12641e6	0.127240
$905$	0	0
$906$	5.19183e6	0.210136
$907$	−6.96226e6	−0.281017	−0.140508	−	0.990080i	$-0.544874\pi$
$907$	−6.96226e6	−0.281017	−0.140508	+	0.990080i	$0.544874\pi$
$908$	−1.76038e6	−0.0708584
$909$	−9.21887e6	−0.370056
$910$	0	0
$911$	3.40287e7	1.35847	0.679234	−	0.733922i	$-0.262312\pi$
$911$	3.40287e7	1.35847	0.679234	+	0.733922i	$0.262312\pi$
$912$	7.48962e6	0.298176
$913$	−1.45015e7	−0.575754
$914$	−9.31869e6	−0.368968
$915$	0	0
$916$	−1.26501e7	−0.498143
$917$	2.80801e7	1.10275
$918$	880564.	0.0344869
$919$	−2.25426e7	−0.880470	−0.440235	−	0.897883i	$-0.645105\pi$
$919$	−2.25426e7	−0.880470	−0.440235	+	0.897883i	$0.645105\pi$
$920$	0	0
$921$	7.20610e6	0.279931
$922$	−2.07546e7	−0.804059
$923$	−5.16346e7	−1.99497
$924$	−4.80040e6	−0.184968
$925$	0	0
$926$	−1.48749e7	−0.570069
$927$	1.56134e7	0.596757
$928$	3.86124e7	1.47183
$929$	3.13230e6	0.119076	0.0595381	−	0.998226i	$-0.481037\pi$
$929$	3.13230e6	0.119076	0.0595381	+	0.998226i	$0.481037\pi$
$930$	0	0
$931$	−2.75602e7	−1.04210
$932$	−3.83588e7	−1.44652
$933$	−2.85510e7	−1.07378
$934$	−1.33886e6	−0.0502191
$935$	0	0
$936$	−9.85670e6	−0.367741
$937$	−1.11644e7	−0.415417	−0.207709	−	0.978191i	$-0.566601\pi$
$937$	−1.11644e7	−0.415417	−0.207709	+	0.978191i	$0.566601\pi$
$938$	2.94032e7	1.09116
$939$	−8.96761e6	−0.331904
$940$	0	0
$941$	2.20914e7	0.813296	0.406648	−	0.913585i	$-0.366698\pi$
$941$	2.20914e7	0.813296	0.406648	+	0.913585i	$0.366698\pi$
$942$	−4.20140e6	−0.154265
$943$	1.48793e6	0.0544883
$944$	1.22191e7	0.446281
$945$	0	0
$946$	6.38664e6	0.232030
$947$	3.88629e7	1.40819	0.704093	−	0.710107i	$-0.251354\pi$
$947$	3.88629e7	1.40819	0.704093	+	0.710107i	$0.251354\pi$
$948$	−1.20702e7	−0.436208
$949$	3.36146e6	0.121161
$950$	0	0
$951$	1.56467e7	0.561012
$952$	−1.20876e7	−0.432264
$953$	−1.65779e6	−0.0591287	−0.0295643	−	0.999563i	$-0.509412\pi$
$953$	−1.65779e6	−0.0591287	−0.0295643	+	0.999563i	$0.509412\pi$
$954$	8.10216e6	0.288224
$955$	0	0
$956$	1.51659e7	0.536689
$957$	−7.17252e6	−0.253158
$958$	1.46003e7	0.513981
$959$	2.49498e7	0.876033
$960$	0	0
$961$	−8.44445e6	−0.294960
$962$	1.93402e7	0.673789
$963$	−1.12682e7	−0.391551
$964$	2.42001e7	0.838736
$965$	0	0
$966$	−5.51626e6	−0.190196
$967$	−4.00243e7	−1.37644	−0.688221	−	0.725501i	$-0.741608\pi$
$967$	−4.00243e7	−1.37644	−0.688221	+	0.725501i	$0.741608\pi$
$968$	−2.18464e6	−0.0749360
$969$	−8.31753e6	−0.284567
$970$	0	0
$971$	−3.50158e7	−1.19183	−0.595917	−	0.803046i	$-0.703211\pi$
$971$	−3.50158e7	−1.19183	−0.595917	+	0.803046i	$0.703211\pi$
$972$	1.48765e6	0.0505052
$973$	2.17297e7	0.735821
$974$	−1.10969e6	−0.0374805
$975$	0	0
$976$	1.35036e7	0.453760
$977$	−3.66007e7	−1.22674	−0.613370	−	0.789795i	$-0.710187\pi$
$977$	−3.66007e7	−1.22674	−0.613370	+	0.789795i	$0.710187\pi$
$978$	−5.21698e6	−0.174410
$979$	8.02313e6	0.267539
$980$	0	0
$981$	7.88616e6	0.261633
$982$	−1.45260e7	−0.480692
$983$	4.10117e7	1.35371	0.676853	−	0.736119i	$-0.263343\pi$
$983$	4.10117e7	1.35371	0.676853	+	0.736119i	$0.263343\pi$
$984$	−1.48817e6	−0.0489965
$985$	0	0
$986$	−7.95568e6	−0.260606
$987$	1.43268e7	0.468119
$988$	4.10116e7	1.33664
$989$	−2.71648e7	−0.883111
$990$	0	0
$991$	3.49847e7	1.13160	0.565802	−	0.824541i	$-0.308567\pi$
$991$	3.49847e7	1.13160	0.565802	+	0.824541i	$0.308567\pi$
$992$	−2.63387e7	−0.849797
$993$	4.77451e6	0.153658
$994$	2.89018e7	0.927809
$995$	0	0
$996$	−2.71744e7	−0.867983
$997$	−5.02983e7	−1.60256	−0.801282	−	0.598287i	$-0.795848\pi$
$997$	−5.02983e7	−1.60256	−0.801282	+	0.598287i	$0.795848\pi$
$998$	−1.38620e7	−0.440554
$999$	−6.62658e6	−0.210076

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.o.1.4	yes	7
5.4	even	2		825.6.a.m.1.4	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.6.a.m.1.4	✓	7	5.4	even	2
825.6.a.o.1.4	yes	7	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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

\(f(q)\)	\(=\)	\(q+2.60892 q^{2} -9.00000 q^{3} -25.1935 q^{4} -23.4803 q^{6} -174.969 q^{7} -149.214 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+2.60892 q^{2} -9.00000 q^{3} -25.1935 q^{4} -23.4803 q^{6} -174.969 q^{7} -149.214 q^{8} +81.0000 q^{9} +121.000 q^{11} +226.742 q^{12} +815.527 q^{13} -456.481 q^{14} +416.906 q^{16} -462.991 q^{17} +211.323 q^{18} -1996.09 q^{19} +1574.72 q^{21} +315.680 q^{22} -1342.70 q^{23} +1342.92 q^{24} +2127.65 q^{26} -729.000 q^{27} +4408.08 q^{28} +6586.33 q^{29} -4492.74 q^{31} +5862.51 q^{32} -1089.00 q^{33} -1207.91 q^{34} -2040.67 q^{36} +9089.95 q^{37} -5207.64 q^{38} -7339.74 q^{39} -1108.16 q^{41} +4108.32 q^{42} +20231.4 q^{43} -3048.42 q^{44} -3503.01 q^{46} +9098.00 q^{47} -3752.15 q^{48} +13807.1 q^{49} +4166.92 q^{51} -20546.0 q^{52} +38340.2 q^{53} -1901.91 q^{54} +26107.7 q^{56} +17964.8 q^{57} +17183.2 q^{58} +29309.0 q^{59} +32390.1 q^{61} -11721.2 q^{62} -14172.5 q^{63} +1953.85 q^{64} -2841.12 q^{66} -64412.9 q^{67} +11664.4 q^{68} +12084.3 q^{69} -63314.4 q^{71} -12086.3 q^{72} +4121.83 q^{73} +23715.0 q^{74} +50288.4 q^{76} -21171.2 q^{77} -19148.8 q^{78} -53233.3 q^{79} +6561.00 q^{81} -2891.10 q^{82} -119847. q^{83} -39672.7 q^{84} +52782.1 q^{86} -59277.0 q^{87} -18054.8 q^{88} +66306.8 q^{89} -142692. q^{91} +33827.4 q^{92} +40434.6 q^{93} +23736.0 q^{94} -52762.6 q^{96} -180567. q^{97} +36021.7 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 9 q^{2} - 63 q^{3} + 95 q^{4} - 81 q^{6} + 65 q^{7} + 207 q^{8} + 567 q^{9}+O(q^{10}) \) 7 * q + 9 * q^2 - 63 * q^3 + 95 * q^4 - 81 * q^6 + 65 * q^7 + 207 * q^8 + 567 * q^9	\( 7 q + 9 q^{2} - 63 q^{3} + 95 q^{4} - 81 q^{6} + 65 q^{7} + 207 q^{8} + 567 q^{9} + 847 q^{11} - 855 q^{12} - 113 q^{13} - 1212 q^{14} + 499 q^{16} + 1030 q^{17} + 729 q^{18} - 3803 q^{19} - 585 q^{21} + 1089 q^{22} + 514 q^{23} - 1863 q^{24} - 12111 q^{26} - 5103 q^{27} - 342 q^{28} - 2698 q^{29} - 17233 q^{31} + 9943 q^{32} - 7623 q^{33} + 4090 q^{34} + 7695 q^{36} + 23182 q^{37} - 11943 q^{38} + 1017 q^{39} - 16158 q^{41} + 10908 q^{42} - 4249 q^{43} + 11495 q^{44} - 28769 q^{46} + 7580 q^{47} - 4491 q^{48} + 37140 q^{49} - 9270 q^{51} - 23887 q^{52} + 20574 q^{53} - 6561 q^{54} - 73276 q^{56} + 34227 q^{57} + 8733 q^{58} - 364 q^{59} - 28127 q^{61} - 71917 q^{62} + 5265 q^{63} + 43379 q^{64} - 9801 q^{66} - 21493 q^{67} + 160660 q^{68} - 4626 q^{69} - 177084 q^{71} + 16767 q^{72} + 78670 q^{73} + 196750 q^{74} - 32701 q^{76} + 7865 q^{77} + 108999 q^{78} - 187432 q^{79} + 45927 q^{81} + 179552 q^{82} - 44592 q^{83} + 3078 q^{84} - 110433 q^{86} + 24282 q^{87} + 25047 q^{88} - 151168 q^{89} - 230153 q^{91} - 44767 q^{92} + 155097 q^{93} + 54040 q^{94} - 89487 q^{96} - 55589 q^{97} + 478341 q^{98} + 68607 q^{99}+O(q^{100}) \) 7 * q + 9 * q^2 - 63 * q^3 + 95 * q^4 - 81 * q^6 + 65 * q^7 + 207 * q^8 + 567 * q^9 + 847 * q^11 - 855 * q^12 - 113 * q^13 - 1212 * q^14 + 499 * q^16 + 1030 * q^17 + 729 * q^18 - 3803 * q^19 - 585 * q^21 + 1089 * q^22 + 514 * q^23 - 1863 * q^24 - 12111 * q^26 - 5103 * q^27 - 342 * q^28 - 2698 * q^29 - 17233 * q^31 + 9943 * q^32 - 7623 * q^33 + 4090 * q^34 + 7695 * q^36 + 23182 * q^37 - 11943 * q^38 + 1017 * q^39 - 16158 * q^41 + 10908 * q^42 - 4249 * q^43 + 11495 * q^44 - 28769 * q^46 + 7580 * q^47 - 4491 * q^48 + 37140 * q^49 - 9270 * q^51 - 23887 * q^52 + 20574 * q^53 - 6561 * q^54 - 73276 * q^56 + 34227 * q^57 + 8733 * q^58 - 364 * q^59 - 28127 * q^61 - 71917 * q^62 + 5265 * q^63 + 43379 * q^64 - 9801 * q^66 - 21493 * q^67 + 160660 * q^68 - 4626 * q^69 - 177084 * q^71 + 16767 * q^72 + 78670 * q^73 + 196750 * q^74 - 32701 * q^76 + 7865 * q^77 + 108999 * q^78 - 187432 * q^79 + 45927 * q^81 + 179552 * q^82 - 44592 * q^83 + 3078 * q^84 - 110433 * q^86 + 24282 * q^87 + 25047 * q^88 - 151168 * q^89 - 230153 * q^91 - 44767 * q^92 + 155097 * q^93 + 54040 * q^94 - 89487 * q^96 - 55589 * q^97 + 478341 * q^98 + 68607 * q^99

Introduction

L-functions

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