LMFDB - Embedded newform 7448.2.a.bq.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7448,2,Mod(1,7448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7448, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7448.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7448 = 2^{3} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7448.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:	$59.4725794254$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 15x^{6} - x^{5} + 66x^{4} + 4x^{3} - 76x^{2} + 8x + 16 $ x^8 - 15x^6 - x^5 + 66x^4 + 4x^3 - 76x^2 + 8*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1064)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$2.39016$ of defining polynomial
Character	$\chi$	$=$	7448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.39016 q^{3} +3.25179 q^{5} +2.71288 q^{9} +O(q^{10})$	$q+2.39016 q^{3} +3.25179 q^{5} +2.71288 q^{9} -0.402893 q^{11} -1.63029 q^{13} +7.77230 q^{15} +4.43555 q^{17} +1.00000 q^{19} +0.837228 q^{23} +5.57412 q^{25} -0.686255 q^{27} -5.33215 q^{29} +3.16041 q^{31} -0.962981 q^{33} +11.4262 q^{37} -3.89665 q^{39} -2.00000 q^{41} +1.93761 q^{43} +8.82172 q^{45} +12.6484 q^{47} +10.6017 q^{51} -11.5064 q^{53} -1.31012 q^{55} +2.39016 q^{57} +5.16962 q^{59} +0.126407 q^{61} -5.30134 q^{65} -2.47276 q^{67} +2.00111 q^{69} -2.17338 q^{71} +12.5196 q^{73} +13.3231 q^{75} -4.40025 q^{79} -9.77891 q^{81} -1.87386 q^{83} +14.4235 q^{85} -12.7447 q^{87} +14.5852 q^{89} +7.55389 q^{93} +3.25179 q^{95} -7.50694 q^{97} -1.09300 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{5} + 6 q^{9}+O(q^{10}) $ 8 * q - q^5 + 6 * q^9	$ 8 q - q^{5} + 6 q^{9} + 9 q^{11} + 8 q^{15} + 4 q^{17} + 8 q^{19} + 25 q^{23} + 15 q^{25} + 3 q^{27} + 6 q^{29} + 8 q^{33} + 13 q^{37} - 11 q^{39} - 16 q^{41} + 17 q^{43} + 17 q^{45} - 24 q^{47} + 5 q^{51} + 2 q^{53} + 5 q^{55} + 2 q^{59} - 13 q^{61} - 26 q^{65} + 2 q^{67} - 11 q^{69} + 10 q^{71} + 5 q^{73} - 20 q^{75} + 16 q^{79} - 12 q^{81} - 43 q^{83} + 24 q^{85} + 20 q^{87} + 8 q^{89} + 2 q^{93} - q^{95} - 12 q^{97} + 37 q^{99}+O(q^{100}) $ 8 * q - q^5 + 6 * q^9 + 9 * q^11 + 8 * q^15 + 4 * q^17 + 8 * q^19 + 25 * q^23 + 15 * q^25 + 3 * q^27 + 6 * q^29 + 8 * q^33 + 13 * q^37 - 11 * q^39 - 16 * q^41 + 17 * q^43 + 17 * q^45 - 24 * q^47 + 5 * q^51 + 2 * q^53 + 5 * q^55 + 2 * q^59 - 13 * q^61 - 26 * q^65 + 2 * q^67 - 11 * q^69 + 10 * q^71 + 5 * q^73 - 20 * q^75 + 16 * q^79 - 12 * q^81 - 43 * q^83 + 24 * q^85 + 20 * q^87 + 8 * q^89 + 2 * q^93 - q^95 - 12 * q^97 + 37 * 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 $

$2$	0	0
$2$	0	0
$3$	2.39016	1.37996	0.689981	−	0.723828i	$-0.257619\pi$
$3$	2.39016	1.37996	0.689981	+	0.723828i	$0.257619\pi$
$4$	0	0
$5$	3.25179	1.45424	0.727122	−	0.686509i	$-0.240858\pi$
$5$	3.25179	1.45424	0.727122	+	0.686509i	$0.240858\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.71288	0.904295
$10$	0	0
$11$	−0.402893	−0.121477	−0.0607385	−	0.998154i	$-0.519346\pi$
$11$	−0.402893	−0.121477	−0.0607385	+	0.998154i	$0.519346\pi$
$12$	0	0
$13$	−1.63029	−0.452160	−0.226080	−	0.974109i	$-0.572591\pi$
$13$	−1.63029	−0.452160	−0.226080	+	0.974109i	$0.572591\pi$
$14$	0	0
$15$	7.77230	2.00680
$16$	0	0
$17$	4.43555	1.07578	0.537890	−	0.843015i	$-0.319222\pi$
$17$	4.43555	1.07578	0.537890	+	0.843015i	$0.319222\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.837228	0.174574	0.0872870	−	0.996183i	$-0.472180\pi$
$23$	0.837228	0.174574	0.0872870	+	0.996183i	$0.472180\pi$
$24$	0	0
$25$	5.57412	1.11482
$26$	0	0
$27$	−0.686255	−0.132070
$28$	0	0
$29$	−5.33215	−0.990156	−0.495078	−	0.868849i	$-0.664860\pi$
$29$	−5.33215	−0.990156	−0.495078	+	0.868849i	$0.664860\pi$
$30$	0	0
$31$	3.16041	0.567626	0.283813	−	0.958880i	$-0.408401\pi$
$31$	3.16041	0.567626	0.283813	+	0.958880i	$0.408401\pi$
$32$	0	0
$33$	−0.962981	−0.167634
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11.4262	1.87846	0.939230	−	0.343289i	$-0.111541\pi$
$37$	11.4262	1.87846	0.939230	+	0.343289i	$0.111541\pi$
$38$	0	0
$39$	−3.89665	−0.623963
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	1.93761	0.295483	0.147742	−	0.989026i	$-0.452800\pi$
$43$	1.93761	0.295483	0.147742	+	0.989026i	$0.452800\pi$
$44$	0	0
$45$	8.82172	1.31506
$46$	0	0
$47$	12.6484	1.84496	0.922479	−	0.386048i	$-0.126160\pi$
$47$	12.6484	1.84496	0.922479	+	0.386048i	$0.126160\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	10.6017	1.48453
$52$	0	0
$53$	−11.5064	−1.58053	−0.790263	−	0.612768i	$-0.790056\pi$
$53$	−11.5064	−1.58053	−0.790263	+	0.612768i	$0.790056\pi$
$54$	0	0
$55$	−1.31012	−0.176657
$56$	0	0
$57$	2.39016	0.316585
$58$	0	0
$59$	5.16962	0.673027	0.336514	−	0.941679i	$-0.390752\pi$
$59$	5.16962	0.673027	0.336514	+	0.941679i	$0.390752\pi$
$60$	0	0
$61$	0.126407	0.0161848	0.00809239	−	0.999967i	$-0.497424\pi$
$61$	0.126407	0.0161848	0.00809239	+	0.999967i	$0.497424\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.30134	−0.657550
$66$	0	0
$67$	−2.47276	−0.302096	−0.151048	−	0.988526i	$-0.548265\pi$
$67$	−2.47276	−0.302096	−0.151048	+	0.988526i	$0.548265\pi$
$68$	0	0
$69$	2.00111	0.240906
$70$	0	0
$71$	−2.17338	−0.257932	−0.128966	−	0.991649i	$-0.541166\pi$
$71$	−2.17338	−0.257932	−0.128966	+	0.991649i	$0.541166\pi$
$72$	0	0
$73$	12.5196	1.46531	0.732657	−	0.680598i	$-0.238280\pi$
$73$	12.5196	1.46531	0.732657	+	0.680598i	$0.238280\pi$
$74$	0	0
$75$	13.3231	1.53841
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−4.40025	−0.495067	−0.247533	−	0.968879i	$-0.579620\pi$
$79$	−4.40025	−0.495067	−0.247533	+	0.968879i	$0.579620\pi$
$80$	0	0
$81$	−9.77891	−1.08655
$82$	0	0
$83$	−1.87386	−0.205683	−0.102841	−	0.994698i	$-0.532793\pi$
$83$	−1.87386	−0.205683	−0.102841	+	0.994698i	$0.532793\pi$
$84$	0	0
$85$	14.4235	1.56445
$86$	0	0
$87$	−12.7447	−1.36638
$88$	0	0
$89$	14.5852	1.54603	0.773016	−	0.634386i	$-0.218747\pi$
$89$	14.5852	1.54603	0.773016	+	0.634386i	$0.218747\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	7.55389	0.783302
$94$	0	0
$95$	3.25179	0.333626
$96$	0	0
$97$	−7.50694	−0.762215	−0.381107	−	0.924531i	$-0.624457\pi$
$97$	−7.50694	−0.762215	−0.381107	+	0.924531i	$0.624457\pi$
$98$	0	0
$99$	−1.09300	−0.109851
$100$	0	0
$101$	−3.20460	−0.318869	−0.159435	−	0.987208i	$-0.550967\pi$
$101$	−3.20460	−0.318869	−0.159435	+	0.987208i	$0.550967\pi$
$102$	0	0
$103$	5.83695	0.575132	0.287566	−	0.957761i	$-0.407154\pi$
$103$	5.83695	0.575132	0.287566	+	0.957761i	$0.407154\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.3526	1.48419	0.742096	−	0.670294i	$-0.233832\pi$
$107$	15.3526	1.48419	0.742096	+	0.670294i	$0.233832\pi$
$108$	0	0
$109$	11.9238	1.14209	0.571046	−	0.820918i	$-0.306538\pi$
$109$	11.9238	1.14209	0.571046	+	0.820918i	$0.306538\pi$
$110$	0	0
$111$	27.3106	2.59220
$112$	0	0
$113$	−19.0112	−1.78842	−0.894210	−	0.447648i	$-0.852262\pi$
$113$	−19.0112	−1.78842	−0.894210	+	0.447648i	$0.852262\pi$
$114$	0	0
$115$	2.72249	0.253873
$116$	0	0
$117$	−4.42277	−0.408886
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.8377	−0.985243
$122$	0	0
$123$	−4.78033	−0.431028
$124$	0	0
$125$	1.86692	0.166982
$126$	0	0
$127$	7.16086	0.635424	0.317712	−	0.948187i	$-0.397086\pi$
$127$	7.16086	0.635424	0.317712	+	0.948187i	$0.397086\pi$
$128$	0	0
$129$	4.63122	0.407756
$130$	0	0
$131$	−1.96032	−0.171274	−0.0856370	−	0.996326i	$-0.527293\pi$
$131$	−1.96032	−0.171274	−0.0856370	+	0.996326i	$0.527293\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−2.23155	−0.192062
$136$	0	0
$137$	−16.1077	−1.37618	−0.688088	−	0.725627i	$-0.741550\pi$
$137$	−16.1077	−1.37618	−0.688088	+	0.725627i	$0.741550\pi$
$138$	0	0
$139$	−2.58462	−0.219224	−0.109612	−	0.993974i	$-0.534961\pi$
$139$	−2.58462	−0.219224	−0.109612	+	0.993974i	$0.534961\pi$
$140$	0	0
$141$	30.2317	2.54597
$142$	0	0
$143$	0.656831	0.0549270
$144$	0	0
$145$	−17.3390	−1.43993
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−9.68077	−0.793079	−0.396540	−	0.918018i	$-0.629789\pi$
$149$	−9.68077	−0.793079	−0.396540	+	0.918018i	$0.629789\pi$
$150$	0	0
$151$	3.90440	0.317735	0.158868	−	0.987300i	$-0.449216\pi$
$151$	3.90440	0.317735	0.158868	+	0.987300i	$0.449216\pi$
$152$	0	0
$153$	12.0331	0.972822
$154$	0	0
$155$	10.2770	0.825466
$156$	0	0
$157$	15.0349	1.19991	0.599956	−	0.800033i	$-0.295185\pi$
$157$	15.0349	1.19991	0.599956	+	0.800033i	$0.295185\pi$
$158$	0	0
$159$	−27.5022	−2.18106
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−17.1071	−1.33993	−0.669964	−	0.742394i	$-0.733691\pi$
$163$	−17.1071	−1.33993	−0.669964	+	0.742394i	$0.733691\pi$
$164$	0	0
$165$	−3.13141	−0.243780
$166$	0	0
$167$	9.06642	0.701580	0.350790	−	0.936454i	$-0.385913\pi$
$167$	9.06642	0.701580	0.350790	+	0.936454i	$0.385913\pi$
$168$	0	0
$169$	−10.3422	−0.795552
$170$	0	0
$171$	2.71288	0.207459
$172$	0	0
$173$	11.2856	0.858025	0.429013	−	0.903299i	$-0.358862\pi$
$173$	11.2856	0.858025	0.429013	+	0.903299i	$0.358862\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	12.3562	0.928752
$178$	0	0
$179$	23.4591	1.75342	0.876709	−	0.481022i	$-0.159734\pi$
$179$	23.4591	1.75342	0.876709	+	0.481022i	$0.159734\pi$
$180$	0	0
$181$	−5.24897	−0.390153	−0.195076	−	0.980788i	$-0.562495\pi$
$181$	−5.24897	−0.390153	−0.195076	+	0.980788i	$0.562495\pi$
$182$	0	0
$183$	0.302134	0.0223344
$184$	0	0
$185$	37.1557	2.73174
$186$	0	0
$187$	−1.78705	−0.130682
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−19.4213	−1.40528	−0.702640	−	0.711546i	$-0.747995\pi$
$191$	−19.4213	−1.40528	−0.702640	+	0.711546i	$0.747995\pi$
$192$	0	0
$193$	−5.89056	−0.424012	−0.212006	−	0.977268i	$-0.568000\pi$
$193$	−5.89056	−0.424012	−0.212006	+	0.977268i	$0.568000\pi$
$194$	0	0
$195$	−12.6711	−0.907394
$196$	0	0
$197$	0.994596	0.0708620	0.0354310	−	0.999372i	$-0.488720\pi$
$197$	0.994596	0.0708620	0.0354310	+	0.999372i	$0.488720\pi$
$198$	0	0
$199$	0.596351	0.0422742	0.0211371	−	0.999777i	$-0.493271\pi$
$199$	0.596351	0.0422742	0.0211371	+	0.999777i	$0.493271\pi$
$200$	0	0
$201$	−5.91031	−0.416881
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−6.50357	−0.454229
$206$	0	0
$207$	2.27130	0.157866
$208$	0	0
$209$	−0.402893	−0.0278687
$210$	0	0
$211$	12.2877	0.845921	0.422960	−	0.906148i	$-0.360991\pi$
$211$	12.2877	0.845921	0.422960	+	0.906148i	$0.360991\pi$
$212$	0	0
$213$	−5.19472	−0.355937
$214$	0	0
$215$	6.30071	0.429705
$216$	0	0
$217$	0	0
$218$	0	0
$219$	29.9240	2.02208
$220$	0	0
$221$	−7.23121	−0.486424
$222$	0	0
$223$	−9.41919	−0.630755	−0.315378	−	0.948966i	$-0.602131\pi$
$223$	−9.41919	−0.630755	−0.315378	+	0.948966i	$0.602131\pi$
$224$	0	0
$225$	15.1219	1.00813
$226$	0	0
$227$	7.22445	0.479504	0.239752	−	0.970834i	$-0.422934\pi$
$227$	7.22445	0.479504	0.239752	+	0.970834i	$0.422934\pi$
$228$	0	0
$229$	5.55832	0.367304	0.183652	−	0.982991i	$-0.441208\pi$
$229$	5.55832	0.367304	0.183652	+	0.982991i	$0.441208\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	11.5357	0.755728	0.377864	−	0.925861i	$-0.376659\pi$
$233$	11.5357	0.755728	0.377864	+	0.925861i	$0.376659\pi$
$234$	0	0
$235$	41.1299	2.68302
$236$	0	0
$237$	−10.5173	−0.683173
$238$	0	0
$239$	28.9484	1.87252	0.936259	−	0.351309i	$-0.114263\pi$
$239$	28.9484	1.87252	0.936259	+	0.351309i	$0.114263\pi$
$240$	0	0
$241$	22.2792	1.43513	0.717566	−	0.696491i	$-0.245256\pi$
$241$	22.2792	1.43513	0.717566	+	0.696491i	$0.245256\pi$
$242$	0	0
$243$	−21.3144	−1.36732
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.63029	−0.103733
$248$	0	0
$249$	−4.47883	−0.283834
$250$	0	0
$251$	−6.67790	−0.421505	−0.210753	−	0.977539i	$-0.567591\pi$
$251$	−6.67790	−0.421505	−0.210753	+	0.977539i	$0.567591\pi$
$252$	0	0
$253$	−0.337314	−0.0212067
$254$	0	0
$255$	34.4745	2.15887
$256$	0	0
$257$	−27.8378	−1.73647	−0.868236	−	0.496151i	$-0.834746\pi$
$257$	−27.8378	−1.73647	−0.868236	+	0.496151i	$0.834746\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−14.4655	−0.895393
$262$	0	0
$263$	−2.85226	−0.175878	−0.0879388	−	0.996126i	$-0.528028\pi$
$263$	−2.85226	−0.175878	−0.0879388	+	0.996126i	$0.528028\pi$
$264$	0	0
$265$	−37.4164	−2.29847
$266$	0	0
$267$	34.8611	2.13347
$268$	0	0
$269$	8.38500	0.511242	0.255621	−	0.966777i	$-0.417720\pi$
$269$	8.38500	0.511242	0.255621	+	0.966777i	$0.417720\pi$
$270$	0	0
$271$	−14.2567	−0.866035	−0.433017	−	0.901386i	$-0.642551\pi$
$271$	−14.2567	−0.866035	−0.433017	+	0.901386i	$0.642551\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.24578	−0.135425
$276$	0	0
$277$	−25.3389	−1.52247	−0.761233	−	0.648479i	$-0.775406\pi$
$277$	−25.3389	−1.52247	−0.761233	+	0.648479i	$0.775406\pi$
$278$	0	0
$279$	8.57381	0.513301
$280$	0	0
$281$	19.0107	1.13408	0.567041	−	0.823689i	$-0.308088\pi$
$281$	19.0107	1.13408	0.567041	+	0.823689i	$0.308088\pi$
$282$	0	0
$283$	−23.5330	−1.39889	−0.699447	−	0.714684i	$-0.746570\pi$
$283$	−23.5330	−1.39889	−0.699447	+	0.714684i	$0.746570\pi$
$284$	0	0
$285$	7.77230	0.460392
$286$	0	0
$287$	0	0
$288$	0	0
$289$	2.67412	0.157301
$290$	0	0
$291$	−17.9428	−1.05183
$292$	0	0
$293$	−11.1260	−0.649989	−0.324995	−	0.945716i	$-0.605362\pi$
$293$	−11.1260	−0.649989	−0.324995	+	0.945716i	$0.605362\pi$
$294$	0	0
$295$	16.8105	0.978746
$296$	0	0
$297$	0.276487	0.0160434
$298$	0	0
$299$	−1.36492	−0.0789354
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−7.65951	−0.440028
$304$	0	0
$305$	0.411049	0.0235366
$306$	0	0
$307$	26.8920	1.53481	0.767404	−	0.641164i	$-0.221548\pi$
$307$	26.8920	1.53481	0.767404	+	0.641164i	$0.221548\pi$
$308$	0	0
$309$	13.9513	0.793660
$310$	0	0
$311$	−25.0433	−1.42007	−0.710036	−	0.704165i	$-0.751322\pi$
$311$	−25.0433	−1.42007	−0.710036	+	0.704165i	$0.751322\pi$
$312$	0	0
$313$	−12.8379	−0.725642	−0.362821	−	0.931859i	$-0.618186\pi$
$313$	−12.8379	−0.725642	−0.362821	+	0.931859i	$0.618186\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−12.4850	−0.701226	−0.350613	−	0.936520i	$-0.614027\pi$
$317$	−12.4850	−0.701226	−0.350613	+	0.936520i	$0.614027\pi$
$318$	0	0
$319$	2.14829	0.120281
$320$	0	0
$321$	36.6952	2.04813
$322$	0	0
$323$	4.43555	0.246801
$324$	0	0
$325$	−9.08741	−0.504079
$326$	0	0
$327$	28.4998	1.57604
$328$	0	0
$329$	0	0
$330$	0	0
$331$	6.22963	0.342412	0.171206	−	0.985235i	$-0.445234\pi$
$331$	6.22963	0.342412	0.171206	+	0.985235i	$0.445234\pi$
$332$	0	0
$333$	30.9980	1.69868
$334$	0	0
$335$	−8.04090	−0.439321
$336$	0	0
$337$	−26.0259	−1.41772	−0.708860	−	0.705349i	$-0.750790\pi$
$337$	−26.0259	−1.41772	−0.708860	+	0.705349i	$0.750790\pi$
$338$	0	0
$339$	−45.4398	−2.46795
$340$	0	0
$341$	−1.27331	−0.0689534
$342$	0	0
$343$	0	0
$344$	0	0
$345$	6.50719	0.350335
$346$	0	0
$347$	9.20710	0.494263	0.247132	−	0.968982i	$-0.420512\pi$
$347$	9.20710	0.494263	0.247132	+	0.968982i	$0.420512\pi$
$348$	0	0
$349$	−26.8773	−1.43871	−0.719354	−	0.694643i	$-0.755562\pi$
$349$	−26.8773	−1.43871	−0.719354	+	0.694643i	$0.755562\pi$
$350$	0	0
$351$	1.11879	0.0597166
$352$	0	0
$353$	−20.7203	−1.10283	−0.551416	−	0.834231i	$-0.685912\pi$
$353$	−20.7203	−1.10283	−0.551416	+	0.834231i	$0.685912\pi$
$354$	0	0
$355$	−7.06735	−0.375096
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−20.6693	−1.09088	−0.545442	−	0.838149i	$-0.683638\pi$
$359$	−20.6693	−1.09088	−0.545442	+	0.838149i	$0.683638\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−25.9038	−1.35960
$364$	0	0
$365$	40.7112	2.13092
$366$	0	0
$367$	−33.8926	−1.76918	−0.884590	−	0.466369i	$-0.845562\pi$
$367$	−33.8926	−1.76918	−0.884590	+	0.466369i	$0.845562\pi$
$368$	0	0
$369$	−5.42577	−0.282454
$370$	0	0
$371$	0	0
$372$	0	0
$373$	27.0170	1.39889	0.699444	−	0.714687i	$-0.253431\pi$
$373$	27.0170	1.39889	0.699444	+	0.714687i	$0.253431\pi$
$374$	0	0
$375$	4.46224	0.230429
$376$	0	0
$377$	8.69293	0.447709
$378$	0	0
$379$	−16.8406	−0.865041	−0.432521	−	0.901624i	$-0.642376\pi$
$379$	−16.8406	−0.865041	−0.432521	+	0.901624i	$0.642376\pi$
$380$	0	0
$381$	17.1156	0.876860
$382$	0	0
$383$	−33.9718	−1.73588	−0.867940	−	0.496670i	$-0.834556\pi$
$383$	−33.9718	−1.73588	−0.867940	+	0.496670i	$0.834556\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.25652	0.267204
$388$	0	0
$389$	−11.6846	−0.592433	−0.296217	−	0.955121i	$-0.595725\pi$
$389$	−11.6846	−0.592433	−0.296217	+	0.955121i	$0.595725\pi$
$390$	0	0
$391$	3.71357	0.187803
$392$	0	0
$393$	−4.68549	−0.236351
$394$	0	0
$395$	−14.3087	−0.719948
$396$	0	0
$397$	−10.0177	−0.502776	−0.251388	−	0.967886i	$-0.580887\pi$
$397$	−10.0177	−0.502776	−0.251388	+	0.967886i	$0.580887\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	10.4698	0.522835	0.261417	−	0.965226i	$-0.415810\pi$
$401$	10.4698	0.522835	0.261417	+	0.965226i	$0.415810\pi$
$402$	0	0
$403$	−5.15236	−0.256657
$404$	0	0
$405$	−31.7989	−1.58010
$406$	0	0
$407$	−4.60355	−0.228190
$408$	0	0
$409$	−24.0093	−1.18719	−0.593593	−	0.804766i	$-0.702291\pi$
$409$	−24.0093	−1.18719	−0.593593	+	0.804766i	$0.702291\pi$
$410$	0	0
$411$	−38.5001	−1.89907
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−6.09339	−0.299113
$416$	0	0
$417$	−6.17766	−0.302521
$418$	0	0
$419$	−15.1723	−0.741216	−0.370608	−	0.928789i	$-0.620851\pi$
$419$	−15.1723	−0.741216	−0.370608	+	0.928789i	$0.620851\pi$
$420$	0	0
$421$	1.77965	0.0867348	0.0433674	−	0.999059i	$-0.486191\pi$
$421$	1.77965	0.0867348	0.0433674	+	0.999059i	$0.486191\pi$
$422$	0	0
$423$	34.3136	1.66838
$424$	0	0
$425$	24.7243	1.19930
$426$	0	0
$427$	0	0
$428$	0	0
$429$	1.56993	0.0757971
$430$	0	0
$431$	32.7349	1.57678	0.788392	−	0.615173i	$-0.210914\pi$
$431$	32.7349	1.57678	0.788392	+	0.615173i	$0.210914\pi$
$432$	0	0
$433$	−4.83058	−0.232143	−0.116071	−	0.993241i	$-0.537030\pi$
$433$	−4.83058	−0.232143	−0.116071	+	0.993241i	$0.537030\pi$
$434$	0	0
$435$	−41.4431	−1.98705
$436$	0	0
$437$	0.837228	0.0400500
$438$	0	0
$439$	28.3595	1.35352	0.676762	−	0.736202i	$-0.263383\pi$
$439$	28.3595	1.35352	0.676762	+	0.736202i	$0.263383\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	19.2801	0.916026	0.458013	−	0.888945i	$-0.348561\pi$
$443$	19.2801	0.916026	0.458013	+	0.888945i	$0.348561\pi$
$444$	0	0
$445$	47.4281	2.24831
$446$	0	0
$447$	−23.1386	−1.09442
$448$	0	0
$449$	−16.0035	−0.755254	−0.377627	−	0.925958i	$-0.623260\pi$
$449$	−16.0035	−0.755254	−0.377627	+	0.925958i	$0.623260\pi$
$450$	0	0
$451$	0.805787	0.0379430
$452$	0	0
$453$	9.33215	0.438463
$454$	0	0
$455$	0	0
$456$	0	0
$457$	24.0347	1.12430	0.562148	−	0.827037i	$-0.309975\pi$
$457$	24.0347	1.12430	0.562148	+	0.827037i	$0.309975\pi$
$458$	0	0
$459$	−3.04392	−0.142078
$460$	0	0
$461$	27.2907	1.27106	0.635528	−	0.772078i	$-0.280782\pi$
$461$	27.2907	1.27106	0.635528	+	0.772078i	$0.280782\pi$
$462$	0	0
$463$	−25.5689	−1.18829	−0.594143	−	0.804359i	$-0.702509\pi$
$463$	−25.5689	−1.18829	−0.594143	+	0.804359i	$0.702509\pi$
$464$	0	0
$465$	24.5636	1.13911
$466$	0	0
$467$	−9.70285	−0.448994	−0.224497	−	0.974475i	$-0.572074\pi$
$467$	−9.70285	−0.448994	−0.224497	+	0.974475i	$0.572074\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	35.9358	1.65583
$472$	0	0
$473$	−0.780652	−0.0358944
$474$	0	0
$475$	5.57412	0.255758
$476$	0	0
$477$	−31.2155	−1.42926
$478$	0	0
$479$	27.5030	1.25665	0.628323	−	0.777953i	$-0.283742\pi$
$479$	27.5030	1.25665	0.628323	+	0.777953i	$0.283742\pi$
$480$	0	0
$481$	−18.6280	−0.849364
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−24.4110	−1.10845
$486$	0	0
$487$	6.77476	0.306993	0.153497	−	0.988149i	$-0.450947\pi$
$487$	6.77476	0.306993	0.153497	+	0.988149i	$0.450947\pi$
$488$	0	0
$489$	−40.8887	−1.84905
$490$	0	0
$491$	−41.5603	−1.87559	−0.937796	−	0.347188i	$-0.887137\pi$
$491$	−41.5603	−1.87559	−0.937796	+	0.347188i	$0.887137\pi$
$492$	0	0
$493$	−23.6510	−1.06519
$494$	0	0
$495$	−3.55421	−0.159750
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−15.7228	−0.703849	−0.351924	−	0.936028i	$-0.614473\pi$
$499$	−15.7228	−0.703849	−0.351924	+	0.936028i	$0.614473\pi$
$500$	0	0
$501$	21.6702	0.968154
$502$	0	0
$503$	14.0804	0.627813	0.313906	−	0.949454i	$-0.398362\pi$
$503$	14.0804	0.627813	0.313906	+	0.949454i	$0.398362\pi$
$504$	0	0
$505$	−10.4207	−0.463714
$506$	0	0
$507$	−24.7195	−1.09783
$508$	0	0
$509$	33.4327	1.48188	0.740939	−	0.671572i	$-0.234381\pi$
$509$	33.4327	1.48188	0.740939	+	0.671572i	$0.234381\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−0.686255	−0.0302989
$514$	0	0
$515$	18.9805	0.836382
$516$	0	0
$517$	−5.09595	−0.224120
$518$	0	0
$519$	26.9743	1.18404
$520$	0	0
$521$	−24.8389	−1.08821	−0.544106	−	0.839017i	$-0.683131\pi$
$521$	−24.8389	−1.08821	−0.544106	+	0.839017i	$0.683131\pi$
$522$	0	0
$523$	−18.6377	−0.814971	−0.407486	−	0.913212i	$-0.633594\pi$
$523$	−18.6377	−0.814971	−0.407486	+	0.913212i	$0.633594\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	14.0181	0.610640
$528$	0	0
$529$	−22.2990	−0.969524
$530$	0	0
$531$	14.0246	0.608615
$532$	0	0
$533$	3.26057	0.141231
$534$	0	0
$535$	49.9234	2.15838
$536$	0	0
$537$	56.0711	2.41965
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−25.3612	−1.09036	−0.545181	−	0.838318i	$-0.683539\pi$
$541$	−25.3612	−1.09036	−0.545181	+	0.838318i	$0.683539\pi$
$542$	0	0
$543$	−12.5459	−0.538396
$544$	0	0
$545$	38.7736	1.66088
$546$	0	0
$547$	30.1315	1.28833	0.644165	−	0.764886i	$-0.277205\pi$
$547$	30.1315	1.28833	0.644165	+	0.764886i	$0.277205\pi$
$548$	0	0
$549$	0.342928	0.0146358
$550$	0	0
$551$	−5.33215	−0.227157
$552$	0	0
$553$	0	0
$554$	0	0
$555$	88.8081	3.76969
$556$	0	0
$557$	34.5323	1.46318	0.731589	−	0.681745i	$-0.238779\pi$
$557$	34.5323	1.46318	0.731589	+	0.681745i	$0.238779\pi$
$558$	0	0
$559$	−3.15886	−0.133606
$560$	0	0
$561$	−4.27135	−0.180337
$562$	0	0
$563$	−7.92050	−0.333809	−0.166905	−	0.985973i	$-0.553377\pi$
$563$	−7.92050	−0.333809	−0.166905	+	0.985973i	$0.553377\pi$
$564$	0	0
$565$	−61.8203	−2.60080
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7.33344	0.307434	0.153717	−	0.988115i	$-0.450876\pi$
$569$	7.33344	0.307434	0.153717	+	0.988115i	$0.450876\pi$
$570$	0	0
$571$	−23.7844	−0.995346	−0.497673	−	0.867365i	$-0.665812\pi$
$571$	−23.7844	−0.995346	−0.497673	+	0.867365i	$0.665812\pi$
$572$	0	0
$573$	−46.4202	−1.93923
$574$	0	0
$575$	4.66681	0.194619
$576$	0	0
$577$	1.79722	0.0748194	0.0374097	−	0.999300i	$-0.488089\pi$
$577$	1.79722	0.0748194	0.0374097	+	0.999300i	$0.488089\pi$
$578$	0	0
$579$	−14.0794	−0.585120
$580$	0	0
$581$	0	0
$582$	0	0
$583$	4.63585	0.191997
$584$	0	0
$585$	−14.3819	−0.594619
$586$	0	0
$587$	−40.2209	−1.66009	−0.830046	−	0.557695i	$-0.811686\pi$
$587$	−40.2209	−1.66009	−0.830046	+	0.557695i	$0.811686\pi$
$588$	0	0
$589$	3.16041	0.130222
$590$	0	0
$591$	2.37725	0.0977869
$592$	0	0
$593$	9.89240	0.406232	0.203116	−	0.979155i	$-0.434893\pi$
$593$	9.89240	0.406232	0.203116	+	0.979155i	$0.434893\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.42538	0.0583368
$598$	0	0
$599$	25.6303	1.04723	0.523613	−	0.851956i	$-0.324584\pi$
$599$	25.6303	1.04723	0.523613	+	0.851956i	$0.324584\pi$
$600$	0	0
$601$	31.9887	1.30485	0.652423	−	0.757855i	$-0.273753\pi$
$601$	31.9887	1.30485	0.652423	+	0.757855i	$0.273753\pi$
$602$	0	0
$603$	−6.70832	−0.273184
$604$	0	0
$605$	−35.2418	−1.43278
$606$	0	0
$607$	−41.4771	−1.68350	−0.841752	−	0.539864i	$-0.818476\pi$
$607$	−41.4771	−1.68350	−0.841752	+	0.539864i	$0.818476\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−20.6205	−0.834215
$612$	0	0
$613$	3.33602	0.134741	0.0673703	−	0.997728i	$-0.478539\pi$
$613$	3.33602	0.134741	0.0673703	+	0.997728i	$0.478539\pi$
$614$	0	0
$615$	−15.5446	−0.626819
$616$	0	0
$617$	1.45553	0.0585976	0.0292988	−	0.999571i	$-0.490673\pi$
$617$	1.45553	0.0585976	0.0292988	+	0.999571i	$0.490673\pi$
$618$	0	0
$619$	40.0031	1.60786	0.803931	−	0.594722i	$-0.202738\pi$
$619$	40.0031	1.60786	0.803931	+	0.594722i	$0.202738\pi$
$620$	0	0
$621$	−0.574552	−0.0230560
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−21.7998	−0.871991
$626$	0	0
$627$	−0.962981	−0.0384578
$628$	0	0
$629$	50.6816	2.02081
$630$	0	0
$631$	29.6609	1.18078	0.590390	−	0.807118i	$-0.298974\pi$
$631$	29.6609	1.18078	0.590390	+	0.807118i	$0.298974\pi$
$632$	0	0
$633$	29.3696	1.16734
$634$	0	0
$635$	23.2856	0.924061
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−5.89611	−0.233247
$640$	0	0
$641$	−31.9823	−1.26322	−0.631612	−	0.775285i	$-0.717606\pi$
$641$	−31.9823	−1.26322	−0.631612	+	0.775285i	$0.717606\pi$
$642$	0	0
$643$	−19.8251	−0.781825	−0.390912	−	0.920428i	$-0.627840\pi$
$643$	−19.8251	−0.781825	−0.390912	+	0.920428i	$0.627840\pi$
$644$	0	0
$645$	15.0597	0.592976
$646$	0	0
$647$	−18.8753	−0.742064	−0.371032	−	0.928620i	$-0.620996\pi$
$647$	−18.8753	−0.742064	−0.371032	+	0.928620i	$0.620996\pi$
$648$	0	0
$649$	−2.08281	−0.0817573
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−2.09633	−0.0820356	−0.0410178	−	0.999158i	$-0.513060\pi$
$653$	−2.09633	−0.0820356	−0.0410178	+	0.999158i	$0.513060\pi$
$654$	0	0
$655$	−6.37454	−0.249074
$656$	0	0
$657$	33.9643	1.32507
$658$	0	0
$659$	−37.1279	−1.44630	−0.723148	−	0.690693i	$-0.757306\pi$
$659$	−37.1279	−1.44630	−0.723148	+	0.690693i	$0.757306\pi$
$660$	0	0
$661$	−26.8649	−1.04492	−0.522461	−	0.852663i	$-0.674986\pi$
$661$	−26.8649	−1.04492	−0.522461	+	0.852663i	$0.674986\pi$
$662$	0	0
$663$	−17.2838	−0.671247
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−4.46423	−0.172856
$668$	0	0
$669$	−22.5134	−0.870418
$670$	0	0
$671$	−0.0509286	−0.00196608
$672$	0	0
$673$	4.79647	0.184890	0.0924452	−	0.995718i	$-0.470532\pi$
$673$	4.79647	0.184890	0.0924452	+	0.995718i	$0.470532\pi$
$674$	0	0
$675$	−3.82527	−0.147235
$676$	0	0
$677$	−45.3193	−1.74176	−0.870882	−	0.491493i	$-0.836451\pi$
$677$	−45.3193	−1.74176	−0.870882	+	0.491493i	$0.836451\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	17.2676	0.661697
$682$	0	0
$683$	13.6586	0.522631	0.261316	−	0.965253i	$-0.415844\pi$
$683$	13.6586	0.522631	0.261316	+	0.965253i	$0.415844\pi$
$684$	0	0
$685$	−52.3789	−2.00129
$686$	0	0
$687$	13.2853	0.506866
$688$	0	0
$689$	18.7587	0.714650
$690$	0	0
$691$	−9.36288	−0.356181	−0.178090	−	0.984014i	$-0.556992\pi$
$691$	−9.36288	−0.356181	−0.178090	+	0.984014i	$0.556992\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−8.40463	−0.318806
$696$	0	0
$697$	−8.87110	−0.336017
$698$	0	0
$699$	27.5722	1.04288
$700$	0	0
$701$	−16.3095	−0.616003	−0.308001	−	0.951386i	$-0.599660\pi$
$701$	−16.3095	−0.616003	−0.308001	+	0.951386i	$0.599660\pi$
$702$	0	0
$703$	11.4262	0.430948
$704$	0	0
$705$	98.3071	3.70246
$706$	0	0
$707$	0	0
$708$	0	0
$709$	35.3417	1.32728	0.663642	−	0.748050i	$-0.269010\pi$
$709$	35.3417	1.32728	0.663642	+	0.748050i	$0.269010\pi$
$710$	0	0
$711$	−11.9374	−0.447686
$712$	0	0
$713$	2.64598	0.0990927
$714$	0	0
$715$	2.13587	0.0798772
$716$	0	0
$717$	69.1915	2.58400
$718$	0	0
$719$	32.5495	1.21389	0.606946	−	0.794743i	$-0.292394\pi$
$719$	32.5495	1.21389	0.606946	+	0.794743i	$0.292394\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	53.2510	1.98043
$724$	0	0
$725$	−29.7221	−1.10385
$726$	0	0
$727$	−12.5070	−0.463859	−0.231929	−	0.972733i	$-0.574504\pi$
$727$	−12.5070	−0.463859	−0.231929	+	0.972733i	$0.574504\pi$
$728$	0	0
$729$	−21.6083	−0.800306
$730$	0	0
$731$	8.59439	0.317875
$732$	0	0
$733$	16.8325	0.621721	0.310861	−	0.950455i	$-0.399383\pi$
$733$	16.8325	0.621721	0.310861	+	0.950455i	$0.399383\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.996260	0.0366977
$738$	0	0
$739$	47.2770	1.73911	0.869556	−	0.493835i	$-0.164405\pi$
$739$	47.2770	1.73911	0.869556	+	0.493835i	$0.164405\pi$
$740$	0	0
$741$	−3.89665	−0.143147
$742$	0	0
$743$	29.4769	1.08140	0.540701	−	0.841215i	$-0.318159\pi$
$743$	29.4769	1.08140	0.540701	+	0.841215i	$0.318159\pi$
$744$	0	0
$745$	−31.4798	−1.15333
$746$	0	0
$747$	−5.08356	−0.185998
$748$	0	0
$749$	0	0
$750$	0	0
$751$	32.0159	1.16828	0.584138	−	0.811654i	$-0.301433\pi$
$751$	32.0159	1.16828	0.584138	+	0.811654i	$0.301433\pi$
$752$	0	0
$753$	−15.9613	−0.581661
$754$	0	0
$755$	12.6963	0.462065
$756$	0	0
$757$	−44.8062	−1.62851	−0.814254	−	0.580509i	$-0.802854\pi$
$757$	−44.8062	−1.62851	−0.814254	+	0.580509i	$0.802854\pi$
$758$	0	0
$759$	−0.806235	−0.0292645
$760$	0	0
$761$	−17.4695	−0.633268	−0.316634	−	0.948548i	$-0.602553\pi$
$761$	−17.4695	−0.633268	−0.316634	+	0.948548i	$0.602553\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	39.1292	1.41472
$766$	0	0
$767$	−8.42796	−0.304316
$768$	0	0
$769$	18.1746	0.655392	0.327696	−	0.944783i	$-0.393728\pi$
$769$	18.1746	0.655392	0.327696	+	0.944783i	$0.393728\pi$
$770$	0	0
$771$	−66.5368	−2.39627
$772$	0	0
$773$	26.7881	0.963500	0.481750	−	0.876309i	$-0.340001\pi$
$773$	26.7881	0.963500	0.481750	+	0.876309i	$0.340001\pi$
$774$	0	0
$775$	17.6165	0.632803
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.00000	−0.0716574
$780$	0	0
$781$	0.875638	0.0313328
$782$	0	0
$783$	3.65921	0.130770
$784$	0	0
$785$	48.8902	1.74496
$786$	0	0
$787$	32.0878	1.14381	0.571904	−	0.820320i	$-0.306205\pi$
$787$	32.0878	1.14381	0.571904	+	0.820320i	$0.306205\pi$
$788$	0	0
$789$	−6.81736	−0.242704
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−0.206080	−0.00731811
$794$	0	0
$795$	−89.4312	−3.17180
$796$	0	0
$797$	−15.2145	−0.538925	−0.269462	−	0.963011i	$-0.586846\pi$
$797$	−15.2145	−0.538925	−0.269462	+	0.963011i	$0.586846\pi$
$798$	0	0
$799$	56.1026	1.98477
$800$	0	0
$801$	39.5681	1.39807
$802$	0	0
$803$	−5.04408	−0.178002
$804$	0	0
$805$	0	0
$806$	0	0
$807$	20.0415	0.705495
$808$	0	0
$809$	−25.7421	−0.905043	−0.452521	−	0.891754i	$-0.649475\pi$
$809$	−25.7421	−0.905043	−0.452521	+	0.891754i	$0.649475\pi$
$810$	0	0
$811$	31.6453	1.11122	0.555608	−	0.831444i	$-0.312486\pi$
$811$	31.6453	1.11122	0.555608	+	0.831444i	$0.312486\pi$
$812$	0	0
$813$	−34.0759	−1.19509
$814$	0	0
$815$	−55.6285	−1.94858
$816$	0	0
$817$	1.93761	0.0677885
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−47.0294	−1.64134	−0.820669	−	0.571404i	$-0.806399\pi$
$821$	−47.0294	−1.64134	−0.820669	+	0.571404i	$0.806399\pi$
$822$	0	0
$823$	6.31892	0.220264	0.110132	−	0.993917i	$-0.464873\pi$
$823$	6.31892	0.220264	0.110132	+	0.993917i	$0.464873\pi$
$824$	0	0
$825$	−5.36777	−0.186882
$826$	0	0
$827$	−46.8959	−1.63073	−0.815364	−	0.578948i	$-0.803463\pi$
$827$	−46.8959	−1.63073	−0.815364	+	0.578948i	$0.803463\pi$
$828$	0	0
$829$	−14.3751	−0.499269	−0.249635	−	0.968340i	$-0.580311\pi$
$829$	−14.3751	−0.499269	−0.249635	+	0.968340i	$0.580311\pi$
$830$	0	0
$831$	−60.5641	−2.10094
$832$	0	0
$833$	0	0
$834$	0	0
$835$	29.4821	1.02027
$836$	0	0
$837$	−2.16884	−0.0749662
$838$	0	0
$839$	33.5325	1.15767	0.578835	−	0.815445i	$-0.303507\pi$
$839$	33.5325	1.15767	0.578835	+	0.815445i	$0.303507\pi$
$840$	0	0
$841$	−0.568151	−0.0195914
$842$	0	0
$843$	45.4387	1.56499
$844$	0	0
$845$	−33.6305	−1.15693
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−56.2478	−1.93042
$850$	0	0
$851$	9.56635	0.327930
$852$	0	0
$853$	3.37139	0.115434	0.0577171	−	0.998333i	$-0.481618\pi$
$853$	3.37139	0.115434	0.0577171	+	0.998333i	$0.481618\pi$
$854$	0	0
$855$	8.82172	0.301696
$856$	0	0
$857$	−38.0628	−1.30020	−0.650101	−	0.759848i	$-0.725273\pi$
$857$	−38.0628	−1.30020	−0.650101	+	0.759848i	$0.725273\pi$
$858$	0	0
$859$	−28.6778	−0.978475	−0.489237	−	0.872151i	$-0.662725\pi$
$859$	−28.6778	−0.978475	−0.489237	+	0.872151i	$0.662725\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	45.8689	1.56139	0.780697	−	0.624910i	$-0.214864\pi$
$863$	45.8689	1.56139	0.780697	+	0.624910i	$0.214864\pi$
$864$	0	0
$865$	36.6982	1.24778
$866$	0	0
$867$	6.39159	0.217070
$868$	0	0
$869$	1.77283	0.0601392
$870$	0	0
$871$	4.03131	0.136596
$872$	0	0
$873$	−20.3655	−0.689267
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−12.0141	−0.405687	−0.202844	−	0.979211i	$-0.565018\pi$
$877$	−12.0141	−0.405687	−0.202844	+	0.979211i	$0.565018\pi$
$878$	0	0
$879$	−26.5930	−0.896960
$880$	0	0
$881$	3.35650	0.113083	0.0565417	−	0.998400i	$-0.481993\pi$
$881$	3.35650	0.113083	0.0565417	+	0.998400i	$0.481993\pi$
$882$	0	0
$883$	6.45095	0.217092	0.108546	−	0.994091i	$-0.465381\pi$
$883$	6.45095	0.217092	0.108546	+	0.994091i	$0.465381\pi$
$884$	0	0
$885$	40.1799	1.35063
$886$	0	0
$887$	34.3674	1.15394	0.576971	−	0.816764i	$-0.304234\pi$
$887$	34.3674	1.15394	0.576971	+	0.816764i	$0.304234\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	3.93986	0.131990
$892$	0	0
$893$	12.6484	0.423262
$894$	0	0
$895$	76.2841	2.54990
$896$	0	0
$897$	−3.26238	−0.108928
$898$	0	0
$899$	−16.8518	−0.562038
$900$	0	0
$901$	−51.0372	−1.70030
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−17.0685	−0.567377
$906$	0	0
$907$	0.322852	0.0107201	0.00536005	−	0.999986i	$-0.498294\pi$
$907$	0.322852	0.0107201	0.00536005	+	0.999986i	$0.498294\pi$
$908$	0	0
$909$	−8.69370	−0.288352
$910$	0	0
$911$	19.3296	0.640419	0.320209	−	0.947347i	$-0.396247\pi$
$911$	19.3296	0.640419	0.320209	+	0.947347i	$0.396247\pi$
$912$	0	0
$913$	0.754966	0.0249857
$914$	0	0
$915$	0.982475	0.0324796
$916$	0	0
$917$	0	0
$918$	0	0
$919$	11.6440	0.384101	0.192050	−	0.981385i	$-0.438486\pi$
$919$	11.6440	0.384101	0.192050	+	0.981385i	$0.438486\pi$
$920$	0	0
$921$	64.2763	2.11798
$922$	0	0
$923$	3.54322	0.116627
$924$	0	0
$925$	63.6912	2.09415
$926$	0	0
$927$	15.8350	0.520089
$928$	0	0
$929$	−10.8240	−0.355123	−0.177562	−	0.984110i	$-0.556821\pi$
$929$	−10.8240	−0.355123	−0.177562	+	0.984110i	$0.556821\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−59.8575	−1.95965
$934$	0	0
$935$	−5.81112	−0.190044
$936$	0	0
$937$	−18.0113	−0.588402	−0.294201	−	0.955744i	$-0.595054\pi$
$937$	−18.0113	−0.588402	−0.294201	+	0.955744i	$0.595054\pi$
$938$	0	0
$939$	−30.6847	−1.00136
$940$	0	0
$941$	−7.35256	−0.239687	−0.119843	−	0.992793i	$-0.538239\pi$
$941$	−7.35256	−0.239687	−0.119843	+	0.992793i	$0.538239\pi$
$942$	0	0
$943$	−1.67446	−0.0545278
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5.30073	0.172251	0.0861253	−	0.996284i	$-0.472551\pi$
$947$	5.30073	0.172251	0.0861253	+	0.996284i	$0.472551\pi$
$948$	0	0
$949$	−20.4106	−0.662556
$950$	0	0
$951$	−29.8411	−0.967666
$952$	0	0
$953$	−46.1846	−1.49607	−0.748033	−	0.663661i	$-0.769002\pi$
$953$	−46.1846	−1.49607	−0.748033	+	0.663661i	$0.769002\pi$
$954$	0	0
$955$	−63.1541	−2.04362
$956$	0	0
$957$	5.13476	0.165983
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−21.0118	−0.677801
$962$	0	0
$963$	41.6498	1.34215
$964$	0	0
$965$	−19.1548	−0.616616
$966$	0	0
$967$	9.76088	0.313889	0.156944	−	0.987607i	$-0.449836\pi$
$967$	9.76088	0.313889	0.156944	+	0.987607i	$0.449836\pi$
$968$	0	0
$969$	10.6017	0.340576
$970$	0	0
$971$	21.2625	0.682347	0.341174	−	0.940000i	$-0.389176\pi$
$971$	21.2625	0.682347	0.341174	+	0.940000i	$0.389176\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−21.7204	−0.695609
$976$	0	0
$977$	−1.88855	−0.0604200	−0.0302100	−	0.999544i	$-0.509618\pi$
$977$	−1.88855	−0.0604200	−0.0302100	+	0.999544i	$0.509618\pi$
$978$	0	0
$979$	−5.87630	−0.187807
$980$	0	0
$981$	32.3479	1.03279
$982$	0	0
$983$	25.8489	0.824453	0.412226	−	0.911081i	$-0.364751\pi$
$983$	25.8489	0.824453	0.412226	+	0.911081i	$0.364751\pi$
$984$	0	0
$985$	3.23421	0.103051
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.62222	0.0515837
$990$	0	0
$991$	−16.9852	−0.539554	−0.269777	−	0.962923i	$-0.586950\pi$
$991$	−16.9852	−0.539554	−0.269777	+	0.962923i	$0.586950\pi$
$992$	0	0
$993$	14.8898	0.472515
$994$	0	0
$995$	1.93921	0.0614770
$996$	0	0
$997$	−11.8086	−0.373984	−0.186992	−	0.982361i	$-0.559874\pi$
$997$	−11.8086	−0.373984	−0.186992	+	0.982361i	$0.559874\pi$
$998$	0	0
$999$	−7.84130	−0.248088

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.bq.1.7		8
7.2	even	3		1064.2.q.n.305.2	✓	16
7.4	even	3		1064.2.q.n.457.2	yes	16
7.6	odd	2		7448.2.a.br.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1064.2.q.n.305.2	✓	16	7.2	even	3
1064.2.q.n.457.2	yes	16	7.4	even	3
7448.2.a.bq.1.7		8	1.1	even	1	trivial
7448.2.a.br.1.2		8	7.6	odd	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:	\( N \)	\(=\)	\( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7448.a (trivial)

\(f(q)\)	\(=\)	\(q+2.39016 q^{3} +3.25179 q^{5} +2.71288 q^{9} +O(q^{10})\)	\(q+2.39016 q^{3} +3.25179 q^{5} +2.71288 q^{9} -0.402893 q^{11} -1.63029 q^{13} +7.77230 q^{15} +4.43555 q^{17} +1.00000 q^{19} +0.837228 q^{23} +5.57412 q^{25} -0.686255 q^{27} -5.33215 q^{29} +3.16041 q^{31} -0.962981 q^{33} +11.4262 q^{37} -3.89665 q^{39} -2.00000 q^{41} +1.93761 q^{43} +8.82172 q^{45} +12.6484 q^{47} +10.6017 q^{51} -11.5064 q^{53} -1.31012 q^{55} +2.39016 q^{57} +5.16962 q^{59} +0.126407 q^{61} -5.30134 q^{65} -2.47276 q^{67} +2.00111 q^{69} -2.17338 q^{71} +12.5196 q^{73} +13.3231 q^{75} -4.40025 q^{79} -9.77891 q^{81} -1.87386 q^{83} +14.4235 q^{85} -12.7447 q^{87} +14.5852 q^{89} +7.55389 q^{93} +3.25179 q^{95} -7.50694 q^{97} -1.09300 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{5} + 6 q^{9}+O(q^{10}) \) 8 * q - q^5 + 6 * q^9	\( 8 q - q^{5} + 6 q^{9} + 9 q^{11} + 8 q^{15} + 4 q^{17} + 8 q^{19} + 25 q^{23} + 15 q^{25} + 3 q^{27} + 6 q^{29} + 8 q^{33} + 13 q^{37} - 11 q^{39} - 16 q^{41} + 17 q^{43} + 17 q^{45} - 24 q^{47} + 5 q^{51} + 2 q^{53} + 5 q^{55} + 2 q^{59} - 13 q^{61} - 26 q^{65} + 2 q^{67} - 11 q^{69} + 10 q^{71} + 5 q^{73} - 20 q^{75} + 16 q^{79} - 12 q^{81} - 43 q^{83} + 24 q^{85} + 20 q^{87} + 8 q^{89} + 2 q^{93} - q^{95} - 12 q^{97} + 37 q^{99}+O(q^{100}) \) 8 * q - q^5 + 6 * q^9 + 9 * q^11 + 8 * q^15 + 4 * q^17 + 8 * q^19 + 25 * q^23 + 15 * q^25 + 3 * q^27 + 6 * q^29 + 8 * q^33 + 13 * q^37 - 11 * q^39 - 16 * q^41 + 17 * q^43 + 17 * q^45 - 24 * q^47 + 5 * q^51 + 2 * q^53 + 5 * q^55 + 2 * q^59 - 13 * q^61 - 26 * q^65 + 2 * q^67 - 11 * q^69 + 10 * q^71 + 5 * q^73 - 20 * q^75 + 16 * q^79 - 12 * q^81 - 43 * q^83 + 24 * q^85 + 20 * q^87 + 8 * q^89 + 2 * q^93 - q^95 - 12 * q^97 + 37 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more