LMFDB - Embedded newform 825.6.a.m.1.1

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			1.1
Root			$9.48549$ 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)

$f(q)$	$=$	$q-10.4855 q^{2} +9.00000 q^{3} +77.9456 q^{4} -94.3694 q^{6} +152.972 q^{7} -481.762 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-10.4855 q^{2} +9.00000 q^{3} +77.9456 q^{4} -94.3694 q^{6} +152.972 q^{7} -481.762 q^{8} +81.0000 q^{9} +121.000 q^{11} +701.510 q^{12} +260.154 q^{13} -1603.99 q^{14} +2557.25 q^{16} -586.683 q^{17} -849.325 q^{18} -614.788 q^{19} +1376.75 q^{21} -1268.74 q^{22} +1376.50 q^{23} -4335.86 q^{24} -2727.84 q^{26} +729.000 q^{27} +11923.5 q^{28} -4061.47 q^{29} -2689.85 q^{31} -11397.7 q^{32} +1089.00 q^{33} +6151.66 q^{34} +6313.59 q^{36} -5302.71 q^{37} +6446.35 q^{38} +2341.39 q^{39} -3301.79 q^{41} -14435.9 q^{42} +19595.9 q^{43} +9431.41 q^{44} -14433.3 q^{46} -15415.5 q^{47} +23015.3 q^{48} +6593.51 q^{49} -5280.15 q^{51} +20277.8 q^{52} -17365.5 q^{53} -7643.92 q^{54} -73696.2 q^{56} -5533.09 q^{57} +42586.5 q^{58} -51539.1 q^{59} -30001.8 q^{61} +28204.4 q^{62} +12390.8 q^{63} +37678.2 q^{64} -11418.7 q^{66} +33552.3 q^{67} -45729.3 q^{68} +12388.5 q^{69} -74159.6 q^{71} -39022.7 q^{72} -39434.2 q^{73} +55601.6 q^{74} -47920.0 q^{76} +18509.6 q^{77} -24550.6 q^{78} -26312.5 q^{79} +6561.00 q^{81} +34620.9 q^{82} -55494.3 q^{83} +107312. q^{84} -205472. q^{86} -36553.2 q^{87} -58293.2 q^{88} -11955.8 q^{89} +39796.3 q^{91} +107292. q^{92} -24208.6 q^{93} +161639. q^{94} -102579. q^{96} -123351. q^{97} -69136.2 q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−10.4855	−1.85359	−0.926795	−	0.375567i	$-0.877448\pi$
$2$	−10.4855	−1.85359	−0.926795	+	0.375567i	$0.877448\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	77.9456	2.43580
$5$	0	0
$5$	0	0
$6$	−94.3694	−1.07017
$7$	152.972	1.17996	0.589980	−	0.807418i	$-0.299135\pi$
$7$	152.972	1.17996	0.589980	+	0.807418i	$0.299135\pi$
$8$	−481.762	−2.66138
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	701.510	1.40631
$13$	260.154	0.426945	0.213473	−	0.976949i	$-0.431523\pi$
$13$	260.154	0.426945	0.213473	+	0.976949i	$0.431523\pi$
$14$	−1603.99	−2.18716
$15$	0	0
$16$	2557.25	2.49732
$17$	−586.683	−0.492358	−0.246179	−	0.969224i	$-0.579175\pi$
$17$	−586.683	−0.492358	−0.246179	+	0.969224i	$0.579175\pi$
$18$	−849.325	−0.617864
$19$	−614.788	−0.390698	−0.195349	−	0.980734i	$-0.562584\pi$
$19$	−614.788	−0.390698	−0.195349	+	0.980734i	$0.562584\pi$
$20$	0	0
$21$	1376.75	0.681251
$22$	−1268.74	−0.558879
$23$	1376.50	0.542573	0.271287	−	0.962499i	$-0.412551\pi$
$23$	1376.50	0.542573	0.271287	+	0.962499i	$0.412551\pi$
$24$	−4335.86	−1.53655
$25$	0	0
$26$	−2727.84	−0.791381
$27$	729.000	0.192450
$28$	11923.5	2.87415
$29$	−4061.47	−0.896784	−0.448392	−	0.893837i	$-0.648003\pi$
$29$	−4061.47	−0.896784	−0.448392	+	0.893837i	$0.648003\pi$
$30$	0	0
$31$	−2689.85	−0.502717	−0.251359	−	0.967894i	$-0.580877\pi$
$31$	−2689.85	−0.502717	−0.251359	+	0.967894i	$0.580877\pi$
$32$	−11397.7	−1.96762
$33$	1089.00	0.174078
$34$	6151.66	0.912631
$35$	0	0
$36$	6313.59	0.811933
$37$	−5302.71	−0.636786	−0.318393	−	0.947959i	$-0.603143\pi$
$37$	−5302.71	−0.636786	−0.318393	+	0.947959i	$0.603143\pi$
$38$	6446.35	0.724194
$39$	2341.39	0.246497
$40$	0	0
$41$	−3301.79	−0.306754	−0.153377	−	0.988168i	$-0.549015\pi$
$41$	−3301.79	−0.306754	−0.153377	+	0.988168i	$0.549015\pi$
$42$	−14435.9	−1.26276
$43$	19595.9	1.61619	0.808096	−	0.589050i	$-0.200498\pi$
$43$	19595.9	1.61619	0.808096	+	0.589050i	$0.200498\pi$
$44$	9431.41	0.734421
$45$	0	0
$46$	−14433.3	−1.00571
$47$	−15415.5	−1.01792	−0.508958	−	0.860791i	$-0.669969\pi$
$47$	−15415.5	−1.01792	−0.508958	+	0.860791i	$0.669969\pi$
$48$	23015.3	1.44183
$49$	6593.51	0.392307
$50$	0	0
$51$	−5280.15	−0.284263
$52$	20277.8	1.03995
$53$	−17365.5	−0.849177	−0.424589	−	0.905386i	$-0.639581\pi$
$53$	−17365.5	−0.849177	−0.424589	+	0.905386i	$0.639581\pi$
$54$	−7643.92	−0.356724
$55$	0	0
$56$	−73696.2	−3.14033
$57$	−5533.09	−0.225570
$58$	42586.5	1.66227
$59$	−51539.1	−1.92755	−0.963777	−	0.266710i	$-0.914063\pi$
$59$	−51539.1	−1.92755	−0.963777	+	0.266710i	$0.914063\pi$
$60$	0	0
$61$	−30001.8	−1.03234	−0.516169	−	0.856487i	$-0.672642\pi$
$61$	−30001.8	−1.03234	−0.516169	+	0.856487i	$0.672642\pi$
$62$	28204.4	0.931832
$63$	12390.8	0.393320
$64$	37678.2	1.14985
$65$	0	0
$66$	−11418.7	−0.322669
$67$	33552.3	0.913136	0.456568	−	0.889689i	$-0.349079\pi$
$67$	33552.3	0.913136	0.456568	+	0.889689i	$0.349079\pi$
$68$	−45729.3	−1.19929
$69$	12388.5	0.313255
$70$	0	0
$71$	−74159.6	−1.74591	−0.872955	−	0.487801i	$-0.837799\pi$
$71$	−74159.6	−1.74591	−0.872955	+	0.487801i	$0.837799\pi$
$72$	−39022.7	−0.887128
$73$	−39434.2	−0.866095	−0.433047	−	0.901371i	$-0.642562\pi$
$73$	−39434.2	−0.866095	−0.433047	+	0.901371i	$0.642562\pi$
$74$	55601.6	1.18034
$75$	0	0
$76$	−47920.0	−0.951662
$77$	18509.6	0.355772
$78$	−24550.6	−0.456904
$79$	−26312.5	−0.474344	−0.237172	−	0.971468i	$-0.576221\pi$
$79$	−26312.5	−0.474344	−0.237172	+	0.971468i	$0.576221\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	34620.9	0.568596
$83$	−55494.3	−0.884205	−0.442103	−	0.896964i	$-0.645767\pi$
$83$	−55494.3	−0.884205	−0.442103	+	0.896964i	$0.645767\pi$
$84$	107312.	1.65939
$85$	0	0
$86$	−205472.	−2.99576
$87$	−36553.2	−0.517759
$88$	−58293.2	−0.802437
$89$	−11955.8	−0.159994	−0.0799970	−	0.996795i	$-0.525491\pi$
$89$	−11955.8	−0.159994	−0.0799970	+	0.996795i	$0.525491\pi$
$90$	0	0
$91$	39796.3	0.503778
$92$	107292.	1.32160
$93$	−24208.6	−0.290244
$94$	161639.	1.88680
$95$	0	0
$96$	−102579.	−1.13601
$97$	−123351.	−1.33110	−0.665552	−	0.746351i	$-0.731804\pi$
$97$	−123351.	−1.33110	−0.665552	+	0.746351i	$0.731804\pi$
$98$	−69136.2	−0.727177
$99$	9801.00	0.100504
$100$	0	0
$101$	−114557.	−1.11743	−0.558713	−	0.829361i	$-0.688705\pi$
$101$	−114557.	−1.11743	−0.558713	+	0.829361i	$0.688705\pi$
$102$	55364.9	0.526908
$103$	74036.9	0.687630	0.343815	−	0.939037i	$-0.388281\pi$
$103$	74036.9	0.687630	0.343815	+	0.939037i	$0.388281\pi$
$104$	−125332.	−1.13626
$105$	0	0
$106$	182086.	1.57403
$107$	124007.	1.04710	0.523549	−	0.851995i	$-0.324608\pi$
$107$	124007.	1.04710	0.523549	+	0.851995i	$0.324608\pi$
$108$	56822.3	0.468770
$109$	−70576.5	−0.568976	−0.284488	−	0.958680i	$-0.591824\pi$
$109$	−70576.5	−0.568976	−0.284488	+	0.958680i	$0.591824\pi$
$110$	0	0
$111$	−47724.4	−0.367649
$112$	391189.	2.94674
$113$	48851.6	0.359901	0.179950	−	0.983676i	$-0.442406\pi$
$113$	48851.6	0.359901	0.179950	+	0.983676i	$0.442406\pi$
$114$	58017.2	0.418114
$115$	0	0
$116$	−316573.	−2.18439
$117$	21072.5	0.142315
$118$	540412.	3.57290
$119$	−89746.2	−0.580963
$120$	0	0
$121$	14641.0	0.0909091
$122$	314583.	1.91353
$123$	−29716.1	−0.177104
$124$	−209662.	−1.22452
$125$	0	0
$126$	−129923.	−0.729055
$127$	−13694.5	−0.0753421	−0.0376711	−	0.999290i	$-0.511994\pi$
$127$	−13694.5	−0.0753421	−0.0376711	+	0.999290i	$0.511994\pi$
$128$	−30348.5	−0.163724
$129$	176363.	0.933109
$130$	0	0
$131$	−30218.0	−0.153847	−0.0769233	−	0.997037i	$-0.524510\pi$
$131$	−30218.0	−0.153847	−0.0769233	+	0.997037i	$0.524510\pi$
$132$	84882.7	0.424018
$133$	−94045.5	−0.461008
$134$	−351812.	−1.69258
$135$	0	0
$136$	282642.	1.31035
$137$	250788.	1.14158	0.570788	−	0.821097i	$-0.306638\pi$
$137$	250788.	1.14158	0.570788	+	0.821097i	$0.306638\pi$
$138$	−129900.	−0.580646
$139$	334866.	1.47006	0.735028	−	0.678036i	$-0.237169\pi$
$139$	334866.	1.47006	0.735028	+	0.678036i	$0.237169\pi$
$140$	0	0
$141$	−138739.	−0.587694
$142$	777600.	3.23620
$143$	31478.6	0.128729
$144$	207137.	0.832439
$145$	0	0
$146$	413487.	1.60539
$147$	59341.6	0.226499
$148$	−413323.	−1.55108
$149$	341827.	1.26136	0.630682	−	0.776041i	$-0.282775\pi$
$149$	341827.	1.26136	0.630682	+	0.776041i	$0.282775\pi$
$150$	0	0
$151$	275735.	0.984122	0.492061	−	0.870561i	$-0.336244\pi$
$151$	275735.	0.984122	0.492061	+	0.870561i	$0.336244\pi$
$152$	296181.	1.03980
$153$	−47521.3	−0.164119
$154$	−194083.	−0.659455
$155$	0	0
$156$	182501.	0.600417
$157$	−214582.	−0.694776	−0.347388	−	0.937722i	$-0.612931\pi$
$157$	−214582.	−0.694776	−0.347388	+	0.937722i	$0.612931\pi$
$158$	275899.	0.879240
$159$	−156290.	−0.490273
$160$	0	0
$161$	210567.	0.640215
$162$	−68795.3	−0.205955
$163$	−46462.0	−0.136971	−0.0684855	−	0.997652i	$-0.521817\pi$
$163$	−46462.0	−0.136971	−0.0684855	+	0.997652i	$0.521817\pi$
$164$	−257360.	−0.747191
$165$	0	0
$166$	581885.	1.63895
$167$	186190.	0.516612	0.258306	−	0.966063i	$-0.416836\pi$
$167$	186190.	0.516612	0.258306	+	0.966063i	$0.416836\pi$
$168$	−663266.	−1.81307
$169$	−303613.	−0.817718
$170$	0	0
$171$	−49797.8	−0.130233
$172$	1.52741e6	3.93672
$173$	168791.	0.428780	0.214390	−	0.976748i	$-0.431224\pi$
$173$	168791.	0.428780	0.214390	+	0.976748i	$0.431224\pi$
$174$	383278.	0.959713
$175$	0	0
$176$	309428.	0.752969
$177$	−463852.	−1.11287
$178$	125362.	0.296563
$179$	740758.	1.72800	0.864000	−	0.503492i	$-0.167952\pi$
$179$	740758.	1.72800	0.864000	+	0.503492i	$0.167952\pi$
$180$	0	0
$181$	−480671.	−1.09056	−0.545282	−	0.838253i	$-0.683577\pi$
$181$	−480671.	−1.09056	−0.545282	+	0.838253i	$0.683577\pi$
$182$	−417284.	−0.933799
$183$	−270016.	−0.596021
$184$	−663148.	−1.44399
$185$	0	0
$186$	253839.	0.537993
$187$	−70988.7	−0.148452
$188$	−1.20157e6	−2.47944
$189$	111517.	0.227084
$190$	0	0
$191$	−940136.	−1.86469	−0.932346	−	0.361566i	$-0.882242\pi$
$191$	−940136.	−1.86469	−0.932346	+	0.361566i	$0.882242\pi$
$192$	339103.	0.663864
$193$	833050.	1.60982	0.804911	−	0.593396i	$-0.202213\pi$
$193$	833050.	1.60982	0.804911	+	0.593396i	$0.202213\pi$
$194$	1.29339e6	2.46732
$195$	0	0
$196$	513935.	0.955582
$197$	−1.08690e6	−1.99537	−0.997687	−	0.0679736i	$-0.978347\pi$
$197$	−1.08690e6	−1.99537	−0.997687	+	0.0679736i	$0.978347\pi$
$198$	−102768.	−0.186293
$199$	683817.	1.22407	0.612037	−	0.790829i	$-0.290351\pi$
$199$	683817.	1.22407	0.612037	+	0.790829i	$0.290351\pi$
$200$	0	0
$201$	301971.	0.527199
$202$	1.20119e6	2.07125
$203$	−621292.	−1.05817
$204$	−411564.	−0.692408
$205$	0	0
$206$	−776313.	−1.27459
$207$	111497.	0.180858
$208$	665279.	1.06622
$209$	−74389.3	−0.117800
$210$	0	0
$211$	−348007.	−0.538124	−0.269062	−	0.963123i	$-0.586714\pi$
$211$	−348007.	−0.538124	−0.269062	+	0.963123i	$0.586714\pi$
$212$	−1.35357e6	−2.06842
$213$	−667437.	−1.00800
$214$	−1.30028e6	−1.94089
$215$	0	0
$216$	−351204.	−0.512183
$217$	−411472.	−0.593186
$218$	740030.	1.05465
$219$	−354907.	−0.500040
$220$	0	0
$221$	−152628.	−0.210210
$222$	500414.	0.681470
$223$	−451991.	−0.608649	−0.304325	−	0.952568i	$-0.598431\pi$
$223$	−451991.	−0.608649	−0.304325	+	0.952568i	$0.598431\pi$
$224$	−1.74353e6	−2.32172
$225$	0	0
$226$	−512233.	−0.667109
$227$	−1.11441e6	−1.43543	−0.717715	−	0.696337i	$-0.754812\pi$
$227$	−1.11441e6	−1.43543	−0.717715	+	0.696337i	$0.754812\pi$
$228$	−431280.	−0.549442
$229$	186206.	0.234642	0.117321	−	0.993094i	$-0.462569\pi$
$229$	186206.	0.234642	0.117321	+	0.993094i	$0.462569\pi$
$230$	0	0
$231$	166587.	0.205405
$232$	1.95666e6	2.38669
$233$	−1.06753e6	−1.28822	−0.644111	−	0.764932i	$-0.722773\pi$
$233$	−1.06753e6	−1.28822	−0.644111	+	0.764932i	$0.722773\pi$
$234$	−220955.	−0.263794
$235$	0	0
$236$	−4.01724e6	−4.69513
$237$	−236812.	−0.273863
$238$	941033.	1.07687
$239$	−492371.	−0.557567	−0.278784	−	0.960354i	$-0.589931\pi$
$239$	−492371.	−0.557567	−0.278784	+	0.960354i	$0.589931\pi$
$240$	0	0
$241$	1.18780e6	1.31734	0.658672	−	0.752430i	$-0.271118\pi$
$241$	1.18780e6	1.31734	0.658672	+	0.752430i	$0.271118\pi$
$242$	−153518.	−0.168508
$243$	59049.0	0.0641500
$244$	−2.33850e6	−2.51457
$245$	0	0
$246$	311588.	0.328279
$247$	−159939.	−0.166807
$248$	1.29587e6	1.33792
$249$	−499449.	−0.510496
$250$	0	0
$251$	−581875.	−0.582969	−0.291484	−	0.956576i	$-0.594149\pi$
$251$	−581875.	−0.582969	−0.291484	+	0.956576i	$0.594149\pi$
$252$	965804.	0.958049
$253$	166557.	0.163592
$254$	143594.	0.139653
$255$	0	0
$256$	−887483.	−0.846369
$257$	1.98329e6	1.87306	0.936532	−	0.350581i	$-0.114016\pi$
$257$	1.98329e6	1.87306	0.936532	+	0.350581i	$0.114016\pi$
$258$	−1.84925e6	−1.72960
$259$	−811168.	−0.751383
$260$	0	0
$261$	−328979.	−0.298928
$262$	316851.	0.285169
$263$	1.87292e6	1.66967	0.834833	−	0.550504i	$-0.185564\pi$
$263$	1.87292e6	1.66967	0.834833	+	0.550504i	$0.185564\pi$
$264$	−524639.	−0.463287
$265$	0	0
$266$	986113.	0.854521
$267$	−107602.	−0.0923726
$268$	2.61525e6	2.22421
$269$	1.37049e6	1.15477	0.577384	−	0.816473i	$-0.304074\pi$
$269$	1.37049e6	1.15477	0.577384	+	0.816473i	$0.304074\pi$
$270$	0	0
$271$	−273433.	−0.226166	−0.113083	−	0.993586i	$-0.536073\pi$
$271$	−273433.	−0.226166	−0.113083	+	0.993586i	$0.536073\pi$
$272$	−1.50030e6	−1.22957
$273$	358167.	0.290857
$274$	−2.62963e6	−2.11601
$275$	0	0
$276$	965632.	0.763025
$277$	−866335.	−0.678401	−0.339200	−	0.940714i	$-0.610156\pi$
$277$	−866335.	−0.678401	−0.339200	+	0.940714i	$0.610156\pi$
$278$	−3.51123e6	−2.72488
$279$	−217878.	−0.167572
$280$	0	0
$281$	−906552.	−0.684900	−0.342450	−	0.939536i	$-0.611257\pi$
$281$	−906552.	−0.684900	−0.342450	+	0.939536i	$0.611257\pi$
$282$	1.45475e6	1.08934
$283$	924307.	0.686041	0.343021	−	0.939328i	$-0.388550\pi$
$283$	924307.	0.686041	0.343021	+	0.939328i	$0.388550\pi$
$284$	−5.78042e6	−4.25268
$285$	0	0
$286$	−330069.	−0.238610
$287$	−505083.	−0.361958
$288$	−923212.	−0.655874
$289$	−1.07566e6	−0.757583
$290$	0	0
$291$	−1.11016e6	−0.768514
$292$	−3.07372e6	−2.10963
$293$	1.27764e6	0.869443	0.434721	−	0.900565i	$-0.356847\pi$
$293$	1.27764e6	0.869443	0.434721	+	0.900565i	$0.356847\pi$
$294$	−622226.	−0.419836
$295$	0	0
$296$	2.55464e6	1.69473
$297$	88209.0	0.0580259
$298$	−3.58422e6	−2.33805
$299$	358103.	0.231649
$300$	0	0
$301$	2.99762e6	1.90704
$302$	−2.89121e6	−1.82416
$303$	−1.03101e6	−0.645146
$304$	−1.57217e6	−0.975697
$305$	0	0
$306$	498285.	0.304210
$307$	−38104.9	−0.0230746	−0.0115373	−	0.999933i	$-0.503673\pi$
$307$	−38104.9	−0.0230746	−0.0115373	+	0.999933i	$0.503673\pi$
$308$	1.44274e6	0.866588
$309$	666332.	0.397004
$310$	0	0
$311$	−1.15601e6	−0.677738	−0.338869	−	0.940834i	$-0.610044\pi$
$311$	−1.15601e6	−0.677738	−0.338869	+	0.940834i	$0.610044\pi$
$312$	−1.12799e6	−0.656023
$313$	−3.13458e6	−1.80850	−0.904250	−	0.427004i	$-0.859569\pi$
$313$	−3.13458e6	−1.80850	−0.904250	+	0.427004i	$0.859569\pi$
$314$	2.25000e6	1.28783
$315$	0	0
$316$	−2.05094e6	−1.15541
$317$	−2.14261e6	−1.19755	−0.598777	−	0.800915i	$-0.704347\pi$
$317$	−2.14261e6	−1.19755	−0.598777	+	0.800915i	$0.704347\pi$
$318$	1.63878e6	0.908765
$319$	−491438.	−0.270391
$320$	0	0
$321$	1.11606e6	0.604542
$322$	−2.20790e6	−1.18670
$323$	360686.	0.192363
$324$	511401.	0.270644
$325$	0	0
$326$	487177.	0.253888
$327$	−635189.	−0.328499
$328$	1.59068e6	0.816390
$329$	−2.35814e6	−1.20110
$330$	0	0
$331$	1.82848e6	0.917317	0.458659	−	0.888613i	$-0.348330\pi$
$331$	1.82848e6	0.917317	0.458659	+	0.888613i	$0.348330\pi$
$332$	−4.32553e6	−2.15375
$333$	−429520.	−0.212262
$334$	−1.95229e6	−0.957587
$335$	0	0
$336$	3.52070e6	1.70130
$337$	690005.	0.330961	0.165481	−	0.986213i	$-0.447082\pi$
$337$	690005.	0.330961	0.165481	+	0.986213i	$0.447082\pi$
$338$	3.18353e6	1.51571
$339$	439665.	0.207789
$340$	0	0
$341$	−325472.	−0.151575
$342$	522155.	0.241398
$343$	−1.56238e6	−0.717053
$344$	−9.44054e6	−4.30131
$345$	0	0
$346$	−1.76986e6	−0.794782
$347$	470628.	0.209824	0.104912	−	0.994482i	$-0.466544\pi$
$347$	470628.	0.209824	0.104912	+	0.994482i	$0.466544\pi$
$348$	−2.84916e6	−1.26116
$349$	−1.58105e6	−0.694836	−0.347418	−	0.937710i	$-0.612941\pi$
$349$	−1.58105e6	−0.694836	−0.347418	+	0.937710i	$0.612941\pi$
$350$	0	0
$351$	189652.	0.0821656
$352$	−1.37912e6	−0.593260
$353$	−2.15723e6	−0.921426	−0.460713	−	0.887549i	$-0.652406\pi$
$353$	−2.15723e6	−0.921426	−0.460713	+	0.887549i	$0.652406\pi$
$354$	4.86371e6	2.06281
$355$	0	0
$356$	−931902.	−0.389713
$357$	−807716.	−0.335419
$358$	−7.76721e6	−3.20300
$359$	344763.	0.141184	0.0705918	−	0.997505i	$-0.477511\pi$
$359$	344763.	0.141184	0.0705918	+	0.997505i	$0.477511\pi$
$360$	0	0
$361$	−2.09814e6	−0.847355
$362$	5.04007e6	2.02146
$363$	131769.	0.0524864
$364$	3.10195e6	1.22710
$365$	0	0
$366$	2.83125e6	1.10478
$367$	−2.47624e6	−0.959684	−0.479842	−	0.877355i	$-0.659306\pi$
$367$	−2.47624e6	−0.959684	−0.479842	+	0.877355i	$0.659306\pi$
$368$	3.52007e6	1.35498
$369$	−267445.	−0.102251
$370$	0	0
$371$	−2.65644e6	−1.00200
$372$	−1.88696e6	−0.706976
$373$	−2.12942e6	−0.792480	−0.396240	−	0.918147i	$-0.629685\pi$
$373$	−2.12942e6	−0.792480	−0.396240	+	0.918147i	$0.629685\pi$
$374$	744351.	0.275169
$375$	0	0
$376$	7.42658e6	2.70906
$377$	−1.05661e6	−0.382878
$378$	−1.16931e6	−0.420920
$379$	4.31155e6	1.54183	0.770914	−	0.636939i	$-0.219800\pi$
$379$	4.31155e6	1.54183	0.770914	+	0.636939i	$0.219800\pi$
$380$	0	0
$381$	−123251.	−0.0434988
$382$	9.85779e6	3.45638
$383$	−154808.	−0.0539257	−0.0269629	−	0.999636i	$-0.508584\pi$
$383$	−154808.	−0.0539257	−0.0269629	+	0.999636i	$0.508584\pi$
$384$	−273136.	−0.0945260
$385$	0	0
$386$	−8.73494e6	−2.98395
$387$	1.58726e6	0.538731
$388$	−9.61464e6	−3.24230
$389$	−2.35631e6	−0.789510	−0.394755	−	0.918786i	$-0.629171\pi$
$389$	−2.35631e6	−0.789510	−0.394755	+	0.918786i	$0.629171\pi$
$390$	0	0
$391$	−807572.	−0.267140
$392$	−3.17650e6	−1.04408
$393$	−271962.	−0.0888234
$394$	1.13967e7	3.69861
$395$	0	0
$396$	763944.	0.244807
$397$	−1.24799e6	−0.397405	−0.198703	−	0.980060i	$-0.563673\pi$
$397$	−1.24799e6	−0.397405	−0.198703	+	0.980060i	$0.563673\pi$
$398$	−7.17016e6	−2.26893
$399$	−846409.	−0.266163
$400$	0	0
$401$	572862.	0.177906	0.0889528	−	0.996036i	$-0.471648\pi$
$401$	572862.	0.177906	0.0889528	+	0.996036i	$0.471648\pi$
$402$	−3.16631e6	−0.977211
$403$	−699775.	−0.214633
$404$	−8.92922e6	−2.72183
$405$	0	0
$406$	6.51455e6	1.96141
$407$	−641628.	−0.191998
$408$	2.54377e6	0.756533
$409$	6.47752e6	1.91470	0.957349	−	0.288933i	$-0.0933006\pi$
$409$	6.47752e6	1.91470	0.957349	+	0.288933i	$0.0933006\pi$
$410$	0	0
$411$	2.25709e6	0.659089
$412$	5.77085e6	1.67493
$413$	−7.88405e6	−2.27444
$414$	−1.16910e6	−0.335236
$415$	0	0
$416$	−2.96515e6	−0.840066
$417$	3.01379e6	0.848738
$418$	780009.	0.218353
$419$	1.53285e6	0.426545	0.213272	−	0.976993i	$-0.431588\pi$
$419$	1.53285e6	0.426545	0.213272	+	0.976993i	$0.431588\pi$
$420$	0	0
$421$	−4.02142e6	−1.10579	−0.552897	−	0.833249i	$-0.686478\pi$
$421$	−4.02142e6	−1.10579	−0.552897	+	0.833249i	$0.686478\pi$
$422$	3.64903e6	0.997462
$423$	−1.24865e6	−0.339305
$424$	8.36605e6	2.25999
$425$	0	0
$426$	6.99840e6	1.86842
$427$	−4.58944e6	−1.21812
$428$	9.66581e6	2.55052
$429$	283308.	0.0743216
$430$	0	0
$431$	3.62573e6	0.940161	0.470080	−	0.882624i	$-0.344225\pi$
$431$	3.62573e6	0.940161	0.470080	+	0.882624i	$0.344225\pi$
$432$	1.86424e6	0.480609
$433$	2.57549e6	0.660147	0.330073	−	0.943955i	$-0.392927\pi$
$433$	2.57549e6	0.660147	0.330073	+	0.943955i	$0.392927\pi$
$434$	4.31449e6	1.09952
$435$	0	0
$436$	−5.50113e6	−1.38591
$437$	−846258.	−0.211982
$438$	3.72138e6	0.926870
$439$	5.26276e6	1.30332	0.651662	−	0.758510i	$-0.274072\pi$
$439$	5.26276e6	1.30332	0.651662	+	0.758510i	$0.274072\pi$
$440$	0	0
$441$	534074.	0.130769
$442$	1.60038e6	0.389643
$443$	2.56225e6	0.620316	0.310158	−	0.950685i	$-0.399618\pi$
$443$	2.56225e6	0.620316	0.310158	+	0.950685i	$0.399618\pi$
$444$	−3.71991e6	−0.895519
$445$	0	0
$446$	4.73934e6	1.12819
$447$	3.07644e6	0.728249
$448$	5.76371e6	1.35677
$449$	−1.74666e6	−0.408877	−0.204439	−	0.978879i	$-0.565537\pi$
$449$	−1.74666e6	−0.408877	−0.204439	+	0.978879i	$0.565537\pi$
$450$	0	0
$451$	−399517.	−0.0924898
$452$	3.80777e6	0.876646
$453$	2.48161e6	0.568183
$454$	1.16852e7	2.66070
$455$	0	0
$456$	2.66563e6	0.600327
$457$	1.84479e6	0.413197	0.206598	−	0.978426i	$-0.433761\pi$
$457$	1.84479e6	0.413197	0.206598	+	0.978426i	$0.433761\pi$
$458$	−1.95246e6	−0.434930
$459$	−427692.	−0.0947544
$460$	0	0
$461$	7.37402e6	1.61604	0.808020	−	0.589155i	$-0.200539\pi$
$461$	7.37402e6	1.61604	0.808020	+	0.589155i	$0.200539\pi$
$462$	−1.74674e6	−0.380736
$463$	−6.04649e6	−1.31084	−0.655422	−	0.755263i	$-0.727509\pi$
$463$	−6.04649e6	−1.31084	−0.655422	+	0.755263i	$0.727509\pi$
$464$	−1.03862e7	−2.23955
$465$	0	0
$466$	1.11936e7	2.38784
$467$	2.42753e6	0.515077	0.257539	−	0.966268i	$-0.417089\pi$
$467$	2.42753e6	0.515077	0.257539	+	0.966268i	$0.417089\pi$
$468$	1.64251e6	0.346651
$469$	5.13257e6	1.07746
$470$	0	0
$471$	−1.93124e6	−0.401129
$472$	2.48296e7	5.12996
$473$	2.37110e6	0.487300
$474$	2.48309e6	0.507630
$475$	0	0
$476$	−6.99532e6	−1.41511
$477$	−1.40661e6	−0.283059
$478$	5.16275e6	1.03350
$479$	3.37363e6	0.671829	0.335914	−	0.941893i	$-0.390955\pi$
$479$	3.37363e6	0.671829	0.335914	+	0.941893i	$0.390955\pi$
$480$	0	0
$481$	−1.37952e6	−0.271873
$482$	−1.24546e7	−2.44182
$483$	1.89510e6	0.369628
$484$	1.14120e6	0.221436
$485$	0	0
$486$	−619158.	−0.118908
$487$	3.47446e6	0.663842	0.331921	−	0.943307i	$-0.392303\pi$
$487$	3.47446e6	0.663842	0.331921	+	0.943307i	$0.392303\pi$
$488$	1.44537e7	2.74745
$489$	−418158.	−0.0790803
$490$	0	0
$491$	−1.80904e6	−0.338645	−0.169322	−	0.985561i	$-0.554158\pi$
$491$	−1.80904e6	−0.338645	−0.169322	+	0.985561i	$0.554158\pi$
$492$	−2.31624e6	−0.431391
$493$	2.38279e6	0.441539
$494$	1.67704e6	0.309191
$495$	0	0
$496$	−6.87862e6	−1.25544
$497$	−1.13444e7	−2.06010
$498$	5.23696e6	0.946251
$499$	−5.23886e6	−0.941859	−0.470929	−	0.882171i	$-0.656081\pi$
$499$	−5.23886e6	−0.941859	−0.470929	+	0.882171i	$0.656081\pi$
$500$	0	0
$501$	1.67571e6	0.298266
$502$	6.10125e6	1.08059
$503$	−2.72935e6	−0.480993	−0.240497	−	0.970650i	$-0.577310\pi$
$503$	−2.72935e6	−0.480993	−0.240497	+	0.970650i	$0.577310\pi$
$504$	−5.96939e6	−1.04678
$505$	0	0
$506$	−1.74643e6	−0.303232
$507$	−2.73252e6	−0.472110
$508$	−1.06743e6	−0.183518
$509$	−4.71280e6	−0.806277	−0.403138	−	0.915139i	$-0.632081\pi$
$509$	−4.71280e6	−0.806277	−0.403138	+	0.915139i	$0.632081\pi$
$510$	0	0
$511$	−6.03233e6	−1.02196
$512$	1.02768e7	1.73255
$513$	−448180.	−0.0751899
$514$	−2.07957e7	−3.47190
$515$	0	0
$516$	1.37467e7	2.27287
$517$	−1.86527e6	−0.306913
$518$	8.50549e6	1.39276
$519$	1.51912e6	0.247556
$520$	0	0
$521$	−1.10696e7	−1.78665	−0.893324	−	0.449414i	$-0.851633\pi$
$521$	−1.10696e7	−1.78665	−0.893324	+	0.449414i	$0.851633\pi$
$522$	3.44951e6	0.554090
$523$	−4.15857e6	−0.664798	−0.332399	−	0.943139i	$-0.607858\pi$
$523$	−4.15857e6	−0.664798	−0.332399	+	0.943139i	$0.607858\pi$
$524$	−2.35536e6	−0.374739
$525$	0	0
$526$	−1.96385e7	−3.09488
$527$	1.57809e6	0.247517
$528$	2.78485e6	0.434727
$529$	−4.54158e6	−0.705615
$530$	0	0
$531$	−4.17466e6	−0.642518
$532$	−7.33043e6	−1.12292
$533$	−858974.	−0.130967
$534$	1.12826e6	0.171221
$535$	0	0
$536$	−1.61642e7	−2.43020
$537$	6.66682e6	0.997661
$538$	−1.43702e7	−2.14047
$539$	797815.	0.118285
$540$	0	0
$541$	−1.22934e6	−0.180583	−0.0902917	−	0.995915i	$-0.528780\pi$
$541$	−1.22934e6	−0.180583	−0.0902917	+	0.995915i	$0.528780\pi$
$542$	2.86708e6	0.419219
$543$	−4.32604e6	−0.629638
$544$	6.68682e6	0.968774
$545$	0	0
$546$	−3.75556e6	−0.539129
$547$	−6.35582e6	−0.908245	−0.454123	−	0.890939i	$-0.650047\pi$
$547$	−6.35582e6	−0.908245	−0.454123	+	0.890939i	$0.650047\pi$
$548$	1.95478e7	2.78065
$549$	−2.43014e6	−0.344113
$550$	0	0
$551$	2.49694e6	0.350372
$552$	−5.96833e6	−0.833691
$553$	−4.02508e6	−0.559708
$554$	9.08395e6	1.25748
$555$	0	0
$556$	2.61013e7	3.58076
$557$	−8.78735e6	−1.20011	−0.600053	−	0.799960i	$-0.704854\pi$
$557$	−8.78735e6	−1.20011	−0.600053	+	0.799960i	$0.704854\pi$
$558$	2.28456e6	0.310611
$559$	5.09794e6	0.690025
$560$	0	0
$561$	−638898.	−0.0857086
$562$	9.50564e6	1.26952
$563$	7.23126e6	0.961487	0.480743	−	0.876861i	$-0.340367\pi$
$563$	7.23126e6	0.961487	0.480743	+	0.876861i	$0.340367\pi$
$564$	−1.08141e7	−1.43150
$565$	0	0
$566$	−9.69182e6	−1.27164
$567$	1.00365e6	0.131107
$568$	3.57273e7	4.64654
$569$	1.15946e7	1.50133	0.750663	−	0.660685i	$-0.229734\pi$
$569$	1.15946e7	1.50133	0.750663	+	0.660685i	$0.229734\pi$
$570$	0	0
$571$	−1.29729e7	−1.66512	−0.832561	−	0.553933i	$-0.813126\pi$
$571$	−1.29729e7	−1.66512	−0.832561	+	0.553933i	$0.813126\pi$
$572$	2.45362e6	0.313557
$573$	−8.46123e6	−1.07658
$574$	5.29604e6	0.670921
$575$	0	0
$576$	3.05193e6	0.383282
$577$	−1.17422e7	−1.46829	−0.734143	−	0.678994i	$-0.762416\pi$
$577$	−1.17422e7	−1.46829	−0.734143	+	0.678994i	$0.762416\pi$
$578$	1.12788e7	1.40425
$579$	7.49745e6	0.929431
$580$	0	0
$581$	−8.48909e6	−1.04333
$582$	1.16405e7	1.42451
$583$	−2.10123e6	−0.256037
$584$	1.89979e7	2.30501
$585$	0	0
$586$	−1.33967e7	−1.61159
$587$	1.27874e7	1.53175	0.765874	−	0.642991i	$-0.222307\pi$
$587$	1.27874e7	1.53175	0.765874	+	0.642991i	$0.222307\pi$
$588$	4.62541e6	0.551705
$589$	1.65369e6	0.196411
$590$	0	0
$591$	−9.78211e6	−1.15203
$592$	−1.35604e7	−1.59026
$593$	−5.13034e6	−0.599113	−0.299557	−	0.954079i	$-0.596839\pi$
$593$	−5.13034e6	−0.599113	−0.299557	+	0.954079i	$0.596839\pi$
$594$	−924915.	−0.107556
$595$	0	0
$596$	2.66439e7	3.07243
$597$	6.15436e6	0.706719
$598$	−3.75489e6	−0.429382
$599$	−5.19929e6	−0.592075	−0.296038	−	0.955176i	$-0.595665\pi$
$599$	−5.19929e6	−0.592075	−0.296038	+	0.955176i	$0.595665\pi$
$600$	0	0
$601$	7.63338e6	0.862046	0.431023	−	0.902341i	$-0.358153\pi$
$601$	7.63338e6	0.862046	0.431023	+	0.902341i	$0.358153\pi$
$602$	−3.14315e7	−3.53488
$603$	2.71774e6	0.304379
$604$	2.14923e7	2.39712
$605$	0	0
$606$	1.08107e7	1.19584
$607$	1.02992e7	1.13457	0.567286	−	0.823521i	$-0.307994\pi$
$607$	1.02992e7	1.13457	0.567286	+	0.823521i	$0.307994\pi$
$608$	7.00715e6	0.768745
$609$	−5.59163e6	−0.610935
$610$	0	0
$611$	−4.01039e6	−0.434594
$612$	−3.70408e6	−0.399762
$613$	−1.96574e6	−0.211288	−0.105644	−	0.994404i	$-0.533690\pi$
$613$	−1.96574e6	−0.211288	−0.105644	+	0.994404i	$0.533690\pi$
$614$	399549.	0.0427709
$615$	0	0
$616$	−8.91724e6	−0.946845
$617$	−8.99693e6	−0.951441	−0.475720	−	0.879597i	$-0.657813\pi$
$617$	−8.99693e6	−0.951441	−0.475720	+	0.879597i	$0.657813\pi$
$618$	−6.98682e6	−0.735882
$619$	1.38807e7	1.45608	0.728039	−	0.685536i	$-0.240432\pi$
$619$	1.38807e7	1.45608	0.728039	+	0.685536i	$0.240432\pi$
$620$	0	0
$621$	1.00347e6	0.104418
$622$	1.21214e7	1.25625
$623$	−1.82891e6	−0.188787
$624$	5.98751e6	0.615581
$625$	0	0
$626$	3.28676e7	3.35222
$627$	−669504.	−0.0680118
$628$	−1.67257e7	−1.69233
$629$	3.11101e6	0.313527
$630$	0	0
$631$	−9.66294e6	−0.966131	−0.483065	−	0.875584i	$-0.660477\pi$
$631$	−9.66294e6	−0.966131	−0.483065	+	0.875584i	$0.660477\pi$
$632$	1.26763e7	1.26241
$633$	−3.13207e6	−0.310686
$634$	2.24663e7	2.21978
$635$	0	0
$636$	−1.21821e7	−1.19421
$637$	1.71533e6	0.167494
$638$	5.15297e6	0.501194
$639$	−6.00693e6	−0.581970
$640$	0	0
$641$	−1.41222e7	−1.35756	−0.678778	−	0.734343i	$-0.737490\pi$
$641$	−1.41222e7	−1.35756	−0.678778	+	0.734343i	$0.737490\pi$
$642$	−1.17025e7	−1.12057
$643$	2.48278e6	0.236816	0.118408	−	0.992965i	$-0.462221\pi$
$643$	2.48278e6	0.236816	0.118408	+	0.992965i	$0.462221\pi$
$644$	1.64128e7	1.55943
$645$	0	0
$646$	−3.78197e6	−0.356563
$647$	−1.18689e7	−1.11468	−0.557342	−	0.830283i	$-0.688179\pi$
$647$	−1.18689e7	−1.11468	−0.557342	+	0.830283i	$0.688179\pi$
$648$	−3.16084e6	−0.295709
$649$	−6.23623e6	−0.581179
$650$	0	0
$651$	−3.70325e6	−0.342476
$652$	−3.62151e6	−0.333634
$653$	−1.22343e7	−1.12279	−0.561394	−	0.827549i	$-0.689735\pi$
$653$	−1.22343e7	−1.12279	−0.561394	+	0.827549i	$0.689735\pi$
$654$	6.66027e6	0.608902
$655$	0	0
$656$	−8.44352e6	−0.766062
$657$	−3.19417e6	−0.288698
$658$	2.47262e7	2.22635
$659$	1.68488e7	1.51131	0.755657	−	0.654968i	$-0.227318\pi$
$659$	1.68488e7	1.51131	0.755657	+	0.654968i	$0.227318\pi$
$660$	0	0
$661$	−1.67317e7	−1.48948	−0.744741	−	0.667354i	$-0.767427\pi$
$661$	−1.67317e7	−1.48948	−0.744741	+	0.667354i	$0.767427\pi$
$662$	−1.91725e7	−1.70033
$663$	−1.37365e6	−0.121365
$664$	2.67350e7	2.35321
$665$	0	0
$666$	4.50373e6	0.393447
$667$	−5.59063e6	−0.486571
$668$	1.45127e7	1.25836
$669$	−4.06792e6	−0.351404
$670$	0	0
$671$	−3.63021e6	−0.311262
$672$	−1.56918e7	−1.34044
$673$	−9.76022e6	−0.830658	−0.415329	−	0.909671i	$-0.636334\pi$
$673$	−9.76022e6	−0.830658	−0.415329	+	0.909671i	$0.636334\pi$
$674$	−7.23504e6	−0.613467
$675$	0	0
$676$	−2.36653e7	−1.99180
$677$	6.32241e6	0.530165	0.265082	−	0.964226i	$-0.414601\pi$
$677$	6.32241e6	0.530165	0.265082	+	0.964226i	$0.414601\pi$
$678$	−4.61010e6	−0.385156
$679$	−1.88692e7	−1.57065
$680$	0	0
$681$	−1.00297e7	−0.828746
$682$	3.41273e6	0.280958
$683$	−9.86643e6	−0.809298	−0.404649	−	0.914472i	$-0.632606\pi$
$683$	−9.86643e6	−0.809298	−0.404649	+	0.914472i	$0.632606\pi$
$684$	−3.88152e6	−0.317221
$685$	0	0
$686$	1.63823e7	1.32912
$687$	1.67586e6	0.135471
$688$	5.01116e7	4.03615
$689$	−4.51771e6	−0.362552
$690$	0	0
$691$	2.15044e6	0.171330	0.0856648	−	0.996324i	$-0.472699\pi$
$691$	2.15044e6	0.171330	0.0856648	+	0.996324i	$0.472699\pi$
$692$	1.31565e7	1.04442
$693$	1.49928e6	0.118591
$694$	−4.93477e6	−0.388927
$695$	0	0
$696$	1.76099e7	1.37795
$697$	1.93711e6	0.151033
$698$	1.65781e7	1.28794
$699$	−9.60778e6	−0.743755
$700$	0	0
$701$	−4.64729e6	−0.357194	−0.178597	−	0.983922i	$-0.557156\pi$
$701$	−4.64729e6	−0.357194	−0.178597	+	0.983922i	$0.557156\pi$
$702$	−1.98860e6	−0.152301
$703$	3.26004e6	0.248791
$704$	4.55906e6	0.346692
$705$	0	0
$706$	2.26197e7	1.70795
$707$	−1.75241e7	−1.31852
$708$	−3.61552e7	−2.71074
$709$	−1.39002e7	−1.03849	−0.519247	−	0.854624i	$-0.673788\pi$
$709$	−1.39002e7	−1.03849	−0.519247	+	0.854624i	$0.673788\pi$
$710$	0	0
$711$	−2.13131e6	−0.158115
$712$	5.75985e6	0.425805
$713$	−3.70259e6	−0.272761
$714$	8.46930e6	0.621730
$715$	0	0
$716$	5.77388e7	4.20906
$717$	−4.43133e6	−0.321912
$718$	−3.61501e6	−0.261696
$719$	−3.42677e6	−0.247208	−0.123604	−	0.992332i	$-0.539445\pi$
$719$	−3.42677e6	−0.247208	−0.123604	+	0.992332i	$0.539445\pi$
$720$	0	0
$721$	1.13256e7	0.811377
$722$	2.20000e7	1.57065
$723$	1.06902e7	0.760569
$724$	−3.74661e7	−2.65639
$725$	0	0
$726$	−1.38166e6	−0.0972883
$727$	2.43422e7	1.70814	0.854069	−	0.520159i	$-0.174127\pi$
$727$	2.43422e7	1.70814	0.854069	+	0.520159i	$0.174127\pi$
$728$	−1.91724e7	−1.34075
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.14966e7	−0.795746
$732$	−2.10465e7	−1.45179
$733$	2.13288e7	1.46624	0.733121	−	0.680098i	$-0.238063\pi$
$733$	2.13288e7	1.46624	0.733121	+	0.680098i	$0.238063\pi$
$734$	2.59646e7	1.77886
$735$	0	0
$736$	−1.56890e7	−1.06758
$737$	4.05983e6	0.275321
$738$	2.80429e6	0.189532
$739$	−2.53646e7	−1.70851	−0.854255	−	0.519854i	$-0.825987\pi$
$739$	−2.53646e7	−1.70851	−0.854255	+	0.519854i	$0.825987\pi$
$740$	0	0
$741$	−1.43945e6	−0.0963058
$742$	2.78541e7	1.85729
$743$	9.15836e6	0.608619	0.304310	−	0.952573i	$-0.401574\pi$
$743$	9.15836e6	0.608619	0.304310	+	0.952573i	$0.401574\pi$
$744$	1.16628e7	0.772450
$745$	0	0
$746$	2.23280e7	1.46893
$747$	−4.49504e6	−0.294735
$748$	−5.53325e6	−0.361598
$749$	1.89697e7	1.23553
$750$	0	0
$751$	2.19605e7	1.42083	0.710415	−	0.703783i	$-0.248507\pi$
$751$	2.19605e7	1.42083	0.710415	+	0.703783i	$0.248507\pi$
$752$	−3.94212e7	−2.54206
$753$	−5.23687e6	−0.336577
$754$	1.10790e7	0.709698
$755$	0	0
$756$	8.69224e6	0.553130
$757$	−2.54426e7	−1.61370	−0.806849	−	0.590758i	$-0.798829\pi$
$757$	−2.54426e7	−1.61370	−0.806849	+	0.590758i	$0.798829\pi$
$758$	−4.52088e7	−2.85792
$759$	1.49901e6	0.0944498
$760$	0	0
$761$	1.57773e7	0.987574	0.493787	−	0.869583i	$-0.335612\pi$
$761$	1.57773e7	0.987574	0.493787	+	0.869583i	$0.335612\pi$
$762$	1.29235e6	0.0806290
$763$	−1.07962e7	−0.671369
$764$	−7.32794e7	−4.54202
$765$	0	0
$766$	1.62324e6	0.0999562
$767$	−1.34081e7	−0.822959
$768$	−7.98734e6	−0.488652
$769$	2.75351e7	1.67908	0.839540	−	0.543298i	$-0.182825\pi$
$769$	2.75351e7	1.67908	0.839540	+	0.543298i	$0.182825\pi$
$770$	0	0
$771$	1.78496e7	1.08141
$772$	6.49325e7	3.92120
$773$	1.77917e7	1.07095	0.535474	−	0.844552i	$-0.320133\pi$
$773$	1.77917e7	1.07095	0.535474	+	0.844552i	$0.320133\pi$
$774$	−1.66433e7	−0.998587
$775$	0	0
$776$	5.94257e7	3.54258
$777$	−7.30051e6	−0.433811
$778$	2.47070e7	1.46343
$779$	2.02990e6	0.119848
$780$	0	0
$781$	−8.97332e6	−0.526412
$782$	8.46779e6	0.495169
$783$	−2.96081e6	−0.172586
$784$	1.68613e7	0.979716
$785$	0	0
$786$	2.85166e6	0.164642
$787$	−1.08596e7	−0.624994	−0.312497	−	0.949919i	$-0.601165\pi$
$787$	−1.08596e7	−0.624994	−0.312497	+	0.949919i	$0.601165\pi$
$788$	−8.47191e7	−4.86033
$789$	1.68563e7	0.963982
$790$	0	0
$791$	7.47294e6	0.424669
$792$	−4.72175e6	−0.267479
$793$	−7.80508e6	−0.440752
$794$	1.30857e7	0.736626
$795$	0	0
$796$	5.33005e7	2.98160
$797$	1.72960e7	0.964493	0.482246	−	0.876036i	$-0.339821\pi$
$797$	1.72960e7	0.964493	0.482246	+	0.876036i	$0.339821\pi$
$798$	8.87502e6	0.493358
$799$	9.04399e6	0.501179
$800$	0	0
$801$	−968420.	−0.0533313
$802$	−6.00674e6	−0.329764
$803$	−4.77153e6	−0.261137
$804$	2.35373e7	1.28415
$805$	0	0
$806$	7.33748e6	0.397841
$807$	1.23344e7	0.666705
$808$	5.51893e7	2.97390
$809$	6.41368e6	0.344537	0.172268	−	0.985050i	$-0.444890\pi$
$809$	6.41368e6	0.344537	0.172268	+	0.985050i	$0.444890\pi$
$810$	0	0
$811$	−3.11308e7	−1.66203	−0.831014	−	0.556251i	$-0.812239\pi$
$811$	−3.11308e7	−1.66203	−0.831014	+	0.556251i	$0.812239\pi$
$812$	−4.84269e7	−2.57749
$813$	−2.46089e6	−0.130577
$814$	6.72779e6	0.355886
$815$	0	0
$816$	−1.35027e7	−0.709895
$817$	−1.20473e7	−0.631443
$818$	−6.79200e7	−3.54907
$819$	3.22350e6	0.167926
$820$	0	0
$821$	−2.37141e7	−1.22786	−0.613930	−	0.789360i	$-0.710412\pi$
$821$	−2.37141e7	−1.22786	−0.613930	+	0.789360i	$0.710412\pi$
$822$	−2.36667e7	−1.22168
$823$	1.87730e7	0.966128	0.483064	−	0.875585i	$-0.339524\pi$
$823$	1.87730e7	0.966128	0.483064	+	0.875585i	$0.339524\pi$
$824$	−3.56681e7	−1.83005
$825$	0	0
$826$	8.26681e7	4.21588
$827$	8.78741e6	0.446783	0.223392	−	0.974729i	$-0.428287\pi$
$827$	8.78741e6	0.446783	0.223392	+	0.974729i	$0.428287\pi$
$828$	8.69069e6	0.440533
$829$	−4.19733e6	−0.212123	−0.106061	−	0.994360i	$-0.533824\pi$
$829$	−4.19733e6	−0.212123	−0.106061	+	0.994360i	$0.533824\pi$
$830$	0	0
$831$	−7.79701e6	−0.391675
$832$	9.80212e6	0.490921
$833$	−3.86830e6	−0.193156
$834$	−3.16011e7	−1.57321
$835$	0	0
$836$	−5.79832e6	−0.286937
$837$	−1.96090e6	−0.0967479
$838$	−1.60727e7	−0.790640
$839$	−3.07480e6	−0.150804	−0.0754018	−	0.997153i	$-0.524024\pi$
$839$	−3.07480e6	−0.150804	−0.0754018	+	0.997153i	$0.524024\pi$
$840$	0	0
$841$	−4.01563e6	−0.195778
$842$	4.21666e7	2.04969
$843$	−8.15897e6	−0.395427
$844$	−2.71256e7	−1.31076
$845$	0	0
$846$	1.30927e7	0.628933
$847$	2.23967e6	0.107269
$848$	−4.44080e7	−2.12066
$849$	8.31877e6	0.396086
$850$	0	0
$851$	−7.29921e6	−0.345503
$852$	−5.20237e7	−2.45529
$853$	−729879.	−0.0343462	−0.0171731	−	0.999853i	$-0.505467\pi$
$853$	−729879.	−0.0343462	−0.0171731	+	0.999853i	$0.505467\pi$
$854$	4.81225e7	2.25789
$855$	0	0
$856$	−5.97419e7	−2.78673
$857$	−1.21921e7	−0.567057	−0.283528	−	0.958964i	$-0.591505\pi$
$857$	−1.21921e7	−0.567057	−0.283528	+	0.958964i	$0.591505\pi$
$858$	−2.97062e6	−0.137762
$859$	−3.85802e7	−1.78395	−0.891973	−	0.452089i	$-0.850679\pi$
$859$	−3.85802e7	−1.78395	−0.891973	+	0.452089i	$0.850679\pi$
$860$	0	0
$861$	−4.54574e6	−0.208976
$862$	−3.80175e7	−1.74267
$863$	2.98448e7	1.36408	0.682042	−	0.731313i	$-0.261092\pi$
$863$	2.98448e7	1.36408	0.682042	+	0.731313i	$0.261092\pi$
$864$	−8.30891e6	−0.378669
$865$	0	0
$866$	−2.70053e7	−1.22364
$867$	−9.68094e6	−0.437391
$868$	−3.20724e7	−1.44488
$869$	−3.18381e6	−0.143020
$870$	0	0
$871$	8.72876e6	0.389859
$872$	3.40011e7	1.51426
$873$	−9.99141e6	−0.443702
$874$	8.87344e6	0.392928
$875$	0	0
$876$	−2.76635e7	−1.21800
$877$	−1.22501e7	−0.537826	−0.268913	−	0.963164i	$-0.586664\pi$
$877$	−1.22501e7	−0.537826	−0.268913	+	0.963164i	$0.586664\pi$
$878$	−5.51826e7	−2.41583
$879$	1.14988e7	0.501973
$880$	0	0
$881$	1.57963e7	0.685671	0.342835	−	0.939395i	$-0.388613\pi$
$881$	1.57963e7	0.685671	0.342835	+	0.939395i	$0.388613\pi$
$882$	−5.60003e6	−0.242392
$883$	3.86988e7	1.67030	0.835151	−	0.550020i	$-0.185380\pi$
$883$	3.86988e7	1.67030	0.835151	+	0.550020i	$0.185380\pi$
$884$	−1.18967e7	−0.512029
$885$	0	0
$886$	−2.68665e7	−1.14981
$887$	1.51666e7	0.647259	0.323629	−	0.946184i	$-0.395097\pi$
$887$	1.51666e7	0.647259	0.323629	+	0.946184i	$0.395097\pi$
$888$	2.29918e7	0.978454
$889$	−2.09488e6	−0.0889008
$890$	0	0
$891$	793881.	0.0335013
$892$	−3.52307e7	−1.48255
$893$	9.47723e6	0.397698
$894$	−3.22580e7	−1.34988
$895$	0	0
$896$	−4.64247e6	−0.193188
$897$	3.22293e6	0.133743
$898$	1.83146e7	0.757891
$899$	1.09247e7	0.450829
$900$	0	0
$901$	1.01881e7	0.418099
$902$	4.18913e6	0.171438
$903$	2.69786e7	1.10103
$904$	−2.35348e7	−0.957834
$905$	0	0
$906$	−2.60209e7	−1.05318
$907$	2.24786e7	0.907302	0.453651	−	0.891179i	$-0.350121\pi$
$907$	2.24786e7	0.907302	0.453651	+	0.891179i	$0.350121\pi$
$908$	−8.68637e7	−3.49642
$909$	−9.27913e6	−0.372475
$910$	0	0
$911$	−2.24682e7	−0.896959	−0.448479	−	0.893793i	$-0.648034\pi$
$911$	−2.24682e7	−0.896959	−0.448479	+	0.893793i	$0.648034\pi$
$912$	−1.41495e7	−0.563319
$913$	−6.71481e6	−0.266598
$914$	−1.93436e7	−0.765898
$915$	0	0
$916$	1.45140e7	0.571541
$917$	−4.62252e6	−0.181533
$918$	4.48456e6	0.175636
$919$	−3.33845e7	−1.30394	−0.651968	−	0.758246i	$-0.726057\pi$
$919$	−3.33845e7	−1.30394	−0.651968	+	0.758246i	$0.726057\pi$
$920$	0	0
$921$	−342944.	−0.0133222
$922$	−7.73203e7	−2.99548
$923$	−1.92929e7	−0.745408
$924$	1.29847e7	0.500325
$925$	0	0
$926$	6.34004e7	2.42977
$927$	5.99699e6	0.229210
$928$	4.62913e7	1.76453
$929$	−4.81423e6	−0.183015	−0.0915077	−	0.995804i	$-0.529169\pi$
$929$	−4.81423e6	−0.183015	−0.0915077	+	0.995804i	$0.529169\pi$
$930$	0	0
$931$	−4.05361e6	−0.153274
$932$	−8.32093e7	−3.13785
$933$	−1.04041e7	−0.391292
$934$	−2.54538e7	−0.954742
$935$	0	0
$936$	−1.01519e7	−0.378755
$937$	−4.35019e7	−1.61868	−0.809338	−	0.587344i	$-0.800174\pi$
$937$	−4.35019e7	−1.61868	−0.809338	+	0.587344i	$0.800174\pi$
$938$	−5.38175e7	−1.99718
$939$	−2.82112e7	−1.04414
$940$	0	0
$941$	−1.33711e7	−0.492259	−0.246129	−	0.969237i	$-0.579159\pi$
$941$	−1.33711e7	−0.492259	−0.246129	+	0.969237i	$0.579159\pi$
$942$	2.02500e7	0.743529
$943$	−4.54493e6	−0.166436
$944$	−1.31798e8	−4.81371
$945$	0	0
$946$	−2.48621e7	−0.903256
$947$	9.61836e6	0.348519	0.174259	−	0.984700i	$-0.444247\pi$
$947$	9.61836e6	0.348519	0.174259	+	0.984700i	$0.444247\pi$
$948$	−1.84585e7	−0.667075
$949$	−1.02590e7	−0.369775
$950$	0	0
$951$	−1.92835e7	−0.691409
$952$	4.32363e7	1.54617
$953$	3.55856e7	1.26923	0.634617	−	0.772827i	$-0.281158\pi$
$953$	3.55856e7	1.26923	0.634617	+	0.772827i	$0.281158\pi$
$954$	1.47490e7	0.524676
$955$	0	0
$956$	−3.83781e7	−1.35812
$957$	−4.42294e6	−0.156110
$958$	−3.53742e7	−1.24530
$959$	3.83636e7	1.34701
$960$	0	0
$961$	−2.13939e7	−0.747276
$962$	1.44650e7	0.503941
$963$	1.00446e7	0.349033
$964$	9.25835e7	3.20879
$965$	0	0
$966$	−1.98711e7	−0.685139
$967$	2.13660e6	0.0734781	0.0367390	−	0.999325i	$-0.488303\pi$
$967$	2.13660e6	0.0734781	0.0367390	+	0.999325i	$0.488303\pi$
$968$	−7.05348e6	−0.241944
$969$	3.24617e6	0.111061
$970$	0	0
$971$	3.22980e7	1.09933	0.549664	−	0.835386i	$-0.314756\pi$
$971$	3.22980e7	1.09933	0.549664	+	0.835386i	$0.314756\pi$
$972$	4.60261e6	0.156257
$973$	5.12252e7	1.73461
$974$	−3.64314e7	−1.23049
$975$	0	0
$976$	−7.67221e7	−2.57808
$977$	−4.28757e7	−1.43706	−0.718530	−	0.695496i	$-0.755185\pi$
$977$	−4.28757e7	−1.43706	−0.718530	+	0.695496i	$0.755185\pi$
$978$	4.38459e6	0.146583
$979$	−1.44665e6	−0.0482400
$980$	0	0
$981$	−5.71670e6	−0.189659
$982$	1.89687e7	0.627709
$983$	4.12347e7	1.36107	0.680533	−	0.732718i	$-0.261748\pi$
$983$	4.12347e7	1.36107	0.680533	+	0.732718i	$0.261748\pi$
$984$	1.43161e7	0.471343
$985$	0	0
$986$	−2.49848e7	−0.818433
$987$	−2.12232e7	−0.693456
$988$	−1.24666e7	−0.406307
$989$	2.69738e7	0.876903
$990$	0	0
$991$	1.37710e7	0.445433	0.222717	−	0.974883i	$-0.428508\pi$
$991$	1.37710e7	0.445433	0.222717	+	0.974883i	$0.428508\pi$
$992$	3.06580e7	0.989156
$993$	1.64563e7	0.529613
$994$	1.18951e8	3.81859
$995$	0	0
$996$	−3.89298e7	−1.24347
$997$	3.84270e7	1.22433	0.612166	−	0.790730i	$-0.290299\pi$
$997$	3.84270e7	1.22433	0.612166	+	0.790730i	$0.290299\pi$
$998$	5.49321e7	1.74582
$999$	−3.86568e6	−0.122550

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.m.1.1	✓	7
5.4	even	2		825.6.a.o.1.7	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.6.a.m.1.1	✓	7	1.1	even	1	trivial
825.6.a.o.1.7	yes	7	5.4	even	2

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)

\(f(q)\)	\(=\)	\(q-10.4855 q^{2} +9.00000 q^{3} +77.9456 q^{4} -94.3694 q^{6} +152.972 q^{7} -481.762 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-10.4855 q^{2} +9.00000 q^{3} +77.9456 q^{4} -94.3694 q^{6} +152.972 q^{7} -481.762 q^{8} +81.0000 q^{9} +121.000 q^{11} +701.510 q^{12} +260.154 q^{13} -1603.99 q^{14} +2557.25 q^{16} -586.683 q^{17} -849.325 q^{18} -614.788 q^{19} +1376.75 q^{21} -1268.74 q^{22} +1376.50 q^{23} -4335.86 q^{24} -2727.84 q^{26} +729.000 q^{27} +11923.5 q^{28} -4061.47 q^{29} -2689.85 q^{31} -11397.7 q^{32} +1089.00 q^{33} +6151.66 q^{34} +6313.59 q^{36} -5302.71 q^{37} +6446.35 q^{38} +2341.39 q^{39} -3301.79 q^{41} -14435.9 q^{42} +19595.9 q^{43} +9431.41 q^{44} -14433.3 q^{46} -15415.5 q^{47} +23015.3 q^{48} +6593.51 q^{49} -5280.15 q^{51} +20277.8 q^{52} -17365.5 q^{53} -7643.92 q^{54} -73696.2 q^{56} -5533.09 q^{57} +42586.5 q^{58} -51539.1 q^{59} -30001.8 q^{61} +28204.4 q^{62} +12390.8 q^{63} +37678.2 q^{64} -11418.7 q^{66} +33552.3 q^{67} -45729.3 q^{68} +12388.5 q^{69} -74159.6 q^{71} -39022.7 q^{72} -39434.2 q^{73} +55601.6 q^{74} -47920.0 q^{76} +18509.6 q^{77} -24550.6 q^{78} -26312.5 q^{79} +6561.00 q^{81} +34620.9 q^{82} -55494.3 q^{83} +107312. q^{84} -205472. q^{86} -36553.2 q^{87} -58293.2 q^{88} -11955.8 q^{89} +39796.3 q^{91} +107292. q^{92} -24208.6 q^{93} +161639. q^{94} -102579. q^{96} -123351. q^{97} -69136.2 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 9 q^{2} + 63 q^{3} + 95 q^{4} - 81 q^{6} - 65 q^{7} - 207 q^{8} + 567 q^{9}+O(q^{10}) \) 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

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more