LMFDB - Embedded newform 7448.2.a.bu.1.4

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:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 6 x^{13} - 11 x^{12} + 114 x^{11} - 10 x^{10} - 806 x^{9} + 523 x^{8} + 2586 x^{7} - 2226 x^{6} + \cdots + 8 $ x^14 - 6x^13 - 11x^12 + 114x^11 - 10x^10 - 806x^9 + 523x^8 + 2586x^7 - 2226x^6 - 3618x^5 + 3397x^4 + 1570x^3 - 1529x^2 + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.12220$ 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)

$f(q)$	$=$	$q-2.12220 q^{3} +2.73357 q^{5} +1.50371 q^{9} +O(q^{10})$	$q-2.12220 q^{3} +2.73357 q^{5} +1.50371 q^{9} -4.28403 q^{11} +3.77461 q^{13} -5.80118 q^{15} +2.28348 q^{17} -1.00000 q^{19} -3.47494 q^{23} +2.47242 q^{25} +3.17541 q^{27} +2.04853 q^{29} -10.3331 q^{31} +9.09155 q^{33} +4.32089 q^{37} -8.01046 q^{39} +0.564114 q^{41} +5.37741 q^{43} +4.11051 q^{45} +2.75061 q^{47} -4.84599 q^{51} -0.936408 q^{53} -11.7107 q^{55} +2.12220 q^{57} -8.23867 q^{59} -9.53835 q^{61} +10.3182 q^{65} +5.83672 q^{67} +7.37450 q^{69} -12.5768 q^{71} -3.91624 q^{73} -5.24696 q^{75} +13.6695 q^{79} -11.2500 q^{81} +17.1090 q^{83} +6.24206 q^{85} -4.34739 q^{87} -3.64586 q^{89} +21.9288 q^{93} -2.73357 q^{95} -1.40870 q^{97} -6.44195 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 6 q^{3} - 2 q^{5} + 16 q^{9}+O(q^{10}) $ 14 * q - 6 * q^3 - 2 * q^5 + 16 * q^9	$ 14 q - 6 q^{3} - 2 q^{5} + 16 q^{9} - 6 q^{11} - 16 q^{13} + 4 q^{15} + 4 q^{17} - 14 q^{19} - 4 q^{23} + 16 q^{25} - 36 q^{27} - 6 q^{29} - 16 q^{31} + 10 q^{33} + 6 q^{37} + 16 q^{39} + 14 q^{41} - 2 q^{43} - 30 q^{47} + 20 q^{51} - 6 q^{53} - 44 q^{55} + 6 q^{57} - 22 q^{59} - 10 q^{61} - 16 q^{65} + 4 q^{67} - 48 q^{69} + 6 q^{71} - 4 q^{73} - 64 q^{75} + 26 q^{79} + 30 q^{81} - 32 q^{83} - 8 q^{85} - 32 q^{87} + 54 q^{89} - 32 q^{93} + 2 q^{95} - 18 q^{97} - 38 q^{99}+O(q^{100}) $ 14 * q - 6 * q^3 - 2 * q^5 + 16 * q^9 - 6 * q^11 - 16 * q^13 + 4 * q^15 + 4 * q^17 - 14 * q^19 - 4 * q^23 + 16 * q^25 - 36 * q^27 - 6 * q^29 - 16 * q^31 + 10 * q^33 + 6 * q^37 + 16 * q^39 + 14 * q^41 - 2 * q^43 - 30 * q^47 + 20 * q^51 - 6 * q^53 - 44 * q^55 + 6 * q^57 - 22 * q^59 - 10 * q^61 - 16 * q^65 + 4 * q^67 - 48 * q^69 + 6 * q^71 - 4 * q^73 - 64 * q^75 + 26 * q^79 + 30 * q^81 - 32 * q^83 - 8 * q^85 - 32 * q^87 + 54 * q^89 - 32 * q^93 + 2 * q^95 - 18 * q^97 - 38 * 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
$3$	−2.12220	−1.22525	−0.612625	−	0.790374i	$-0.709886\pi$
$3$	−2.12220	−1.22525	−0.612625	+	0.790374i	$0.709886\pi$
$4$	0	0
$5$	2.73357	1.22249	0.611245	−	0.791441i	$-0.290669\pi$
$5$	2.73357	1.22249	0.611245	+	0.791441i	$0.290669\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.50371	0.501238
$10$	0	0
$11$	−4.28403	−1.29168	−0.645842	−	0.763471i	$-0.723493\pi$
$11$	−4.28403	−1.29168	−0.645842	+	0.763471i	$0.723493\pi$
$12$	0	0
$13$	3.77461	1.04689	0.523444	−	0.852060i	$-0.324647\pi$
$13$	3.77461	1.04689	0.523444	+	0.852060i	$0.324647\pi$
$14$	0	0
$15$	−5.80118	−1.49786
$16$	0	0
$17$	2.28348	0.553825	0.276913	−	0.960895i	$-0.410689\pi$
$17$	2.28348	0.553825	0.276913	+	0.960895i	$0.410689\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.47494	−0.724575	−0.362287	−	0.932066i	$-0.618004\pi$
$23$	−3.47494	−0.724575	−0.362287	+	0.932066i	$0.618004\pi$
$24$	0	0
$25$	2.47242	0.494484
$26$	0	0
$27$	3.17541	0.611109
$28$	0	0
$29$	2.04853	0.380403	0.190202	−	0.981745i	$-0.439086\pi$
$29$	2.04853	0.380403	0.190202	+	0.981745i	$0.439086\pi$
$30$	0	0
$31$	−10.3331	−1.85588	−0.927939	−	0.372731i	$-0.878421\pi$
$31$	−10.3331	−1.85588	−0.927939	+	0.372731i	$0.878421\pi$
$32$	0	0
$33$	9.09155	1.58264
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.32089	0.710350	0.355175	−	0.934800i	$-0.384421\pi$
$37$	4.32089	0.710350	0.355175	+	0.934800i	$0.384421\pi$
$38$	0	0
$39$	−8.01046	−1.28270
$40$	0	0
$41$	0.564114	0.0880998	0.0440499	−	0.999029i	$-0.485974\pi$
$41$	0.564114	0.0880998	0.0440499	+	0.999029i	$0.485974\pi$
$42$	0	0
$43$	5.37741	0.820047	0.410023	−	0.912075i	$-0.365521\pi$
$43$	5.37741	0.820047	0.410023	+	0.912075i	$0.365521\pi$
$44$	0	0
$45$	4.11051	0.612758
$46$	0	0
$47$	2.75061	0.401218	0.200609	−	0.979671i	$-0.435708\pi$
$47$	2.75061	0.401218	0.200609	+	0.979671i	$0.435708\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−4.84599	−0.678575
$52$	0	0
$53$	−0.936408	−0.128625	−0.0643127	−	0.997930i	$-0.520486\pi$
$53$	−0.936408	−0.128625	−0.0643127	+	0.997930i	$0.520486\pi$
$54$	0	0
$55$	−11.7107	−1.57907
$56$	0	0
$57$	2.12220	0.281092
$58$	0	0
$59$	−8.23867	−1.07258	−0.536292	−	0.844033i	$-0.680175\pi$
$59$	−8.23867	−1.07258	−0.536292	+	0.844033i	$0.680175\pi$
$60$	0	0
$61$	−9.53835	−1.22126	−0.610630	−	0.791916i	$-0.709084\pi$
$61$	−9.53835	−1.22126	−0.610630	+	0.791916i	$0.709084\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	10.3182	1.27981
$66$	0	0
$67$	5.83672	0.713069	0.356534	−	0.934282i	$-0.383958\pi$
$67$	5.83672	0.713069	0.356534	+	0.934282i	$0.383958\pi$
$68$	0	0
$69$	7.37450	0.887785
$70$	0	0
$71$	−12.5768	−1.49260	−0.746298	−	0.665612i	$-0.768170\pi$
$71$	−12.5768	−1.49260	−0.746298	+	0.665612i	$0.768170\pi$
$72$	0	0
$73$	−3.91624	−0.458362	−0.229181	−	0.973384i	$-0.573605\pi$
$73$	−3.91624	−0.458362	−0.229181	+	0.973384i	$0.573605\pi$
$74$	0	0
$75$	−5.24696	−0.605867
$76$	0	0
$77$	0	0
$78$	0	0
$79$	13.6695	1.53794	0.768970	−	0.639285i	$-0.220770\pi$
$79$	13.6695	1.53794	0.768970	+	0.639285i	$0.220770\pi$
$80$	0	0
$81$	−11.2500	−1.25000
$82$	0	0
$83$	17.1090	1.87795	0.938976	−	0.343982i	$-0.111776\pi$
$83$	17.1090	1.87795	0.938976	+	0.343982i	$0.111776\pi$
$84$	0	0
$85$	6.24206	0.677046
$86$	0	0
$87$	−4.34739	−0.466089
$88$	0	0
$89$	−3.64586	−0.386460	−0.193230	−	0.981153i	$-0.561896\pi$
$89$	−3.64586	−0.386460	−0.193230	+	0.981153i	$0.561896\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	21.9288	2.27392
$94$	0	0
$95$	−2.73357	−0.280459
$96$	0	0
$97$	−1.40870	−0.143032	−0.0715161	−	0.997439i	$-0.522784\pi$
$97$	−1.40870	−0.143032	−0.0715161	+	0.997439i	$0.522784\pi$
$98$	0	0
$99$	−6.44195	−0.647440
$100$	0	0
$101$	−13.7846	−1.37162	−0.685808	−	0.727783i	$-0.740551\pi$
$101$	−13.7846	−1.37162	−0.685808	+	0.727783i	$0.740551\pi$
$102$	0	0
$103$	−16.6569	−1.64125	−0.820625	−	0.571466i	$-0.806375\pi$
$103$	−16.6569	−1.64125	−0.820625	+	0.571466i	$0.806375\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.32544	0.804851	0.402426	−	0.915453i	$-0.368167\pi$
$107$	8.32544	0.804851	0.402426	+	0.915453i	$0.368167\pi$
$108$	0	0
$109$	−8.12891	−0.778609	−0.389304	−	0.921109i	$-0.627285\pi$
$109$	−8.12891	−0.778609	−0.389304	+	0.921109i	$0.627285\pi$
$110$	0	0
$111$	−9.16977	−0.870356
$112$	0	0
$113$	−7.65896	−0.720494	−0.360247	−	0.932857i	$-0.617308\pi$
$113$	−7.65896	−0.720494	−0.360247	+	0.932857i	$0.617308\pi$
$114$	0	0
$115$	−9.49900	−0.885786
$116$	0	0
$117$	5.67593	0.524740
$118$	0	0
$119$	0	0
$120$	0	0
$121$	7.35291	0.668447
$122$	0	0
$123$	−1.19716	−0.107944
$124$	0	0
$125$	−6.90932	−0.617989
$126$	0	0
$127$	−0.339277	−0.0301060	−0.0150530	−	0.999887i	$-0.504792\pi$
$127$	−0.339277	−0.0301060	−0.0150530	+	0.999887i	$0.504792\pi$
$128$	0	0
$129$	−11.4119	−1.00476
$130$	0	0
$131$	−0.285077	−0.0249073	−0.0124536	−	0.999922i	$-0.503964\pi$
$131$	−0.285077	−0.0249073	−0.0124536	+	0.999922i	$0.503964\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	8.68022	0.747075
$136$	0	0
$137$	16.7055	1.42724	0.713622	−	0.700531i	$-0.247053\pi$
$137$	16.7055	1.42724	0.713622	+	0.700531i	$0.247053\pi$
$138$	0	0
$139$	−3.15961	−0.267994	−0.133997	−	0.990982i	$-0.542781\pi$
$139$	−3.15961	−0.267994	−0.133997	+	0.990982i	$0.542781\pi$
$140$	0	0
$141$	−5.83734	−0.491593
$142$	0	0
$143$	−16.1705	−1.35225
$144$	0	0
$145$	5.59982	0.465039
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0.972698	0.0796865	0.0398433	−	0.999206i	$-0.487314\pi$
$149$	0.972698	0.0796865	0.0398433	+	0.999206i	$0.487314\pi$
$150$	0	0
$151$	−1.72381	−0.140281	−0.0701407	−	0.997537i	$-0.522345\pi$
$151$	−1.72381	−0.140281	−0.0701407	+	0.997537i	$0.522345\pi$
$152$	0	0
$153$	3.43370	0.277598
$154$	0	0
$155$	−28.2463	−2.26879
$156$	0	0
$157$	−1.94704	−0.155390	−0.0776952	−	0.996977i	$-0.524756\pi$
$157$	−1.94704	−0.155390	−0.0776952	+	0.996977i	$0.524756\pi$
$158$	0	0
$159$	1.98724	0.157598
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−11.3120	−0.886026	−0.443013	−	0.896515i	$-0.646090\pi$
$163$	−11.3120	−0.886026	−0.443013	+	0.896515i	$0.646090\pi$
$164$	0	0
$165$	24.8524	1.93476
$166$	0	0
$167$	16.7700	1.29770	0.648852	−	0.760915i	$-0.275249\pi$
$167$	16.7700	1.29770	0.648852	+	0.760915i	$0.275249\pi$
$168$	0	0
$169$	1.24769	0.0959763
$170$	0	0
$171$	−1.50371	−0.114992
$172$	0	0
$173$	−19.9188	−1.51440	−0.757201	−	0.653182i	$-0.773434\pi$
$173$	−19.9188	−1.51440	−0.757201	+	0.653182i	$0.773434\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	17.4841	1.31418
$178$	0	0
$179$	10.3198	0.771336	0.385668	−	0.922638i	$-0.373971\pi$
$179$	10.3198	0.771336	0.385668	+	0.922638i	$0.373971\pi$
$180$	0	0
$181$	−20.1813	−1.50006	−0.750031	−	0.661403i	$-0.769961\pi$
$181$	−20.1813	−1.50006	−0.750031	+	0.661403i	$0.769961\pi$
$182$	0	0
$183$	20.2422	1.49635
$184$	0	0
$185$	11.8115	0.868396
$186$	0	0
$187$	−9.78250	−0.715367
$188$	0	0
$189$	0	0
$190$	0	0
$191$	12.7743	0.924317	0.462159	−	0.886797i	$-0.347075\pi$
$191$	12.7743	0.924317	0.462159	+	0.886797i	$0.347075\pi$
$192$	0	0
$193$	13.9646	1.00519	0.502597	−	0.864521i	$-0.332378\pi$
$193$	13.9646	1.00519	0.502597	+	0.864521i	$0.332378\pi$
$194$	0	0
$195$	−21.8972	−1.56809
$196$	0	0
$197$	10.4486	0.744431	0.372216	−	0.928146i	$-0.378598\pi$
$197$	10.4486	0.744431	0.372216	+	0.928146i	$0.378598\pi$
$198$	0	0
$199$	17.9181	1.27018	0.635090	−	0.772438i	$-0.280963\pi$
$199$	17.9181	1.27018	0.635090	+	0.772438i	$0.280963\pi$
$200$	0	0
$201$	−12.3867	−0.873688
$202$	0	0
$203$	0	0
$204$	0	0
$205$	1.54205	0.107701
$206$	0	0
$207$	−5.22531	−0.363184
$208$	0	0
$209$	4.28403	0.296333
$210$	0	0
$211$	−22.4691	−1.54684	−0.773419	−	0.633895i	$-0.781455\pi$
$211$	−22.4691	−1.54684	−0.773419	+	0.633895i	$0.781455\pi$
$212$	0	0
$213$	26.6905	1.82880
$214$	0	0
$215$	14.6995	1.00250
$216$	0	0
$217$	0	0
$218$	0	0
$219$	8.31103	0.561608
$220$	0	0
$221$	8.61925	0.579794
$222$	0	0
$223$	3.72549	0.249477	0.124739	−	0.992190i	$-0.460191\pi$
$223$	3.72549	0.249477	0.124739	+	0.992190i	$0.460191\pi$
$224$	0	0
$225$	3.71781	0.247854
$226$	0	0
$227$	−10.3031	−0.683838	−0.341919	−	0.939729i	$-0.611077\pi$
$227$	−10.3031	−0.683838	−0.341919	+	0.939729i	$0.611077\pi$
$228$	0	0
$229$	−8.15141	−0.538660	−0.269330	−	0.963048i	$-0.586802\pi$
$229$	−8.15141	−0.538660	−0.269330	+	0.963048i	$0.586802\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.4861	−0.686964	−0.343482	−	0.939159i	$-0.611606\pi$
$233$	−10.4861	−0.686964	−0.343482	+	0.939159i	$0.611606\pi$
$234$	0	0
$235$	7.51901	0.490486
$236$	0	0
$237$	−29.0094	−1.88436
$238$	0	0
$239$	−13.4245	−0.868357	−0.434178	−	0.900827i	$-0.642961\pi$
$239$	−13.4245	−0.868357	−0.434178	+	0.900827i	$0.642961\pi$
$240$	0	0
$241$	15.3749	0.990386	0.495193	−	0.868783i	$-0.335097\pi$
$241$	15.3749	0.990386	0.495193	+	0.868783i	$0.335097\pi$
$242$	0	0
$243$	14.3484	0.920452
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3.77461	−0.240173
$248$	0	0
$249$	−36.3086	−2.30096
$250$	0	0
$251$	1.80367	0.113847	0.0569233	−	0.998379i	$-0.481871\pi$
$251$	1.80367	0.113847	0.0569233	+	0.998379i	$0.481871\pi$
$252$	0	0
$253$	14.8867	0.935921
$254$	0	0
$255$	−13.2469	−0.829551
$256$	0	0
$257$	16.6269	1.03716	0.518578	−	0.855030i	$-0.326462\pi$
$257$	16.6269	1.03716	0.518578	+	0.855030i	$0.326462\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	3.08041	0.190672
$262$	0	0
$263$	−12.3832	−0.763582	−0.381791	−	0.924249i	$-0.624693\pi$
$263$	−12.3832	−0.763582	−0.381791	+	0.924249i	$0.624693\pi$
$264$	0	0
$265$	−2.55974	−0.157243
$266$	0	0
$267$	7.73723	0.473511
$268$	0	0
$269$	24.9473	1.52106	0.760532	−	0.649300i	$-0.224938\pi$
$269$	24.9473	1.52106	0.760532	+	0.649300i	$0.224938\pi$
$270$	0	0
$271$	2.82227	0.171441	0.0857203	−	0.996319i	$-0.472681\pi$
$271$	2.82227	0.171441	0.0857203	+	0.996319i	$0.472681\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−10.5919	−0.638717
$276$	0	0
$277$	−15.1057	−0.907611	−0.453805	−	0.891101i	$-0.649934\pi$
$277$	−15.1057	−0.907611	−0.453805	+	0.891101i	$0.649934\pi$
$278$	0	0
$279$	−15.5380	−0.930236
$280$	0	0
$281$	−7.27247	−0.433839	−0.216920	−	0.976189i	$-0.569601\pi$
$281$	−7.27247	−0.433839	−0.216920	+	0.976189i	$0.569601\pi$
$282$	0	0
$283$	1.79564	0.106739	0.0533697	−	0.998575i	$-0.483004\pi$
$283$	1.79564	0.106739	0.0533697	+	0.998575i	$0.483004\pi$
$284$	0	0
$285$	5.80118	0.343632
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−11.7857	−0.693277
$290$	0	0
$291$	2.98954	0.175250
$292$	0	0
$293$	−31.6231	−1.84744	−0.923721	−	0.383067i	$-0.874868\pi$
$293$	−31.6231	−1.84744	−0.923721	+	0.383067i	$0.874868\pi$
$294$	0	0
$295$	−22.5210	−1.31122
$296$	0	0
$297$	−13.6036	−0.789359
$298$	0	0
$299$	−13.1165	−0.758549
$300$	0	0
$301$	0	0
$302$	0	0
$303$	29.2535	1.68057
$304$	0	0
$305$	−26.0738	−1.49298
$306$	0	0
$307$	8.99520	0.513383	0.256691	−	0.966493i	$-0.417368\pi$
$307$	8.99520	0.513383	0.256691	+	0.966493i	$0.417368\pi$
$308$	0	0
$309$	35.3491	2.01094
$310$	0	0
$311$	−23.4072	−1.32730	−0.663650	−	0.748044i	$-0.730993\pi$
$311$	−23.4072	−1.32730	−0.663650	+	0.748044i	$0.730993\pi$
$312$	0	0
$313$	5.84688	0.330485	0.165243	−	0.986253i	$-0.447159\pi$
$313$	5.84688	0.330485	0.165243	+	0.986253i	$0.447159\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.2645	1.02584	0.512919	−	0.858437i	$-0.328564\pi$
$317$	18.2645	1.02584	0.512919	+	0.858437i	$0.328564\pi$
$318$	0	0
$319$	−8.77598	−0.491361
$320$	0	0
$321$	−17.6682	−0.986144
$322$	0	0
$323$	−2.28348	−0.127056
$324$	0	0
$325$	9.33243	0.517670
$326$	0	0
$327$	17.2511	0.953990
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−31.0437	−1.70632	−0.853159	−	0.521651i	$-0.825316\pi$
$331$	−31.0437	−1.70632	−0.853159	+	0.521651i	$0.825316\pi$
$332$	0	0
$333$	6.49738	0.356054
$334$	0	0
$335$	15.9551	0.871720
$336$	0	0
$337$	2.61528	0.142463	0.0712316	−	0.997460i	$-0.477307\pi$
$337$	2.61528	0.142463	0.0712316	+	0.997460i	$0.477307\pi$
$338$	0	0
$339$	16.2538	0.882785
$340$	0	0
$341$	44.2673	2.39721
$342$	0	0
$343$	0	0
$344$	0	0
$345$	20.1587	1.08531
$346$	0	0
$347$	28.3889	1.52400	0.761998	−	0.647579i	$-0.224219\pi$
$347$	28.3889	1.52400	0.761998	+	0.647579i	$0.224219\pi$
$348$	0	0
$349$	−6.57681	−0.352048	−0.176024	−	0.984386i	$-0.556324\pi$
$349$	−6.57681	−0.352048	−0.176024	+	0.984386i	$0.556324\pi$
$350$	0	0
$351$	11.9860	0.639763
$352$	0	0
$353$	−22.7408	−1.21037	−0.605186	−	0.796084i	$-0.706901\pi$
$353$	−22.7408	−1.21037	−0.605186	+	0.796084i	$0.706901\pi$
$354$	0	0
$355$	−34.3797	−1.82468
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.72669	0.249465	0.124733	−	0.992190i	$-0.460193\pi$
$359$	4.72669	0.249465	0.124733	+	0.992190i	$0.460193\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−15.6043	−0.819014
$364$	0	0
$365$	−10.7053	−0.560343
$366$	0	0
$367$	−13.6184	−0.710876	−0.355438	−	0.934700i	$-0.615668\pi$
$367$	−13.6184	−0.710876	−0.355438	+	0.934700i	$0.615668\pi$
$368$	0	0
$369$	0.848266	0.0441589
$370$	0	0
$371$	0	0
$372$	0	0
$373$	7.68363	0.397843	0.198922	−	0.980015i	$-0.436256\pi$
$373$	7.68363	0.397843	0.198922	+	0.980015i	$0.436256\pi$
$374$	0	0
$375$	14.6629	0.757191
$376$	0	0
$377$	7.73242	0.398240
$378$	0	0
$379$	3.53839	0.181755	0.0908775	−	0.995862i	$-0.471033\pi$
$379$	3.53839	0.181755	0.0908775	+	0.995862i	$0.471033\pi$
$380$	0	0
$381$	0.720012	0.0368873
$382$	0	0
$383$	−1.06759	−0.0545512	−0.0272756	−	0.999628i	$-0.508683\pi$
$383$	−1.06759	−0.0545512	−0.0272756	+	0.999628i	$0.508683\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.08608	0.411038
$388$	0	0
$389$	−17.5849	−0.891592	−0.445796	−	0.895135i	$-0.647079\pi$
$389$	−17.5849	−0.891592	−0.445796	+	0.895135i	$0.647079\pi$
$390$	0	0
$391$	−7.93495	−0.401288
$392$	0	0
$393$	0.604988	0.0305176
$394$	0	0
$395$	37.3666	1.88012
$396$	0	0
$397$	−17.2559	−0.866047	−0.433024	−	0.901383i	$-0.642553\pi$
$397$	−17.2559	−0.866047	−0.433024	+	0.901383i	$0.642553\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8.96921	0.447901	0.223951	−	0.974601i	$-0.428105\pi$
$401$	8.96921	0.447901	0.223951	+	0.974601i	$0.428105\pi$
$402$	0	0
$403$	−39.0034	−1.94290
$404$	0	0
$405$	−30.7527	−1.52811
$406$	0	0
$407$	−18.5108	−0.917547
$408$	0	0
$409$	−23.6321	−1.16853	−0.584267	−	0.811562i	$-0.698618\pi$
$409$	−23.6321	−1.16853	−0.584267	+	0.811562i	$0.698618\pi$
$410$	0	0
$411$	−35.4523	−1.74873
$412$	0	0
$413$	0	0
$414$	0	0
$415$	46.7686	2.29578
$416$	0	0
$417$	6.70530	0.328360
$418$	0	0
$419$	−35.9764	−1.75756	−0.878781	−	0.477224i	$-0.841643\pi$
$419$	−35.9764	−1.75756	−0.878781	+	0.477224i	$0.841643\pi$
$420$	0	0
$421$	8.08048	0.393818	0.196909	−	0.980422i	$-0.436910\pi$
$421$	8.08048	0.393818	0.196909	+	0.980422i	$0.436910\pi$
$422$	0	0
$423$	4.13614	0.201106
$424$	0	0
$425$	5.64572	0.273858
$426$	0	0
$427$	0	0
$428$	0	0
$429$	34.3171	1.65684
$430$	0	0
$431$	5.74480	0.276717	0.138359	−	0.990382i	$-0.455817\pi$
$431$	5.74480	0.276717	0.138359	+	0.990382i	$0.455817\pi$
$432$	0	0
$433$	−1.98622	−0.0954519	−0.0477259	−	0.998860i	$-0.515197\pi$
$433$	−1.98622	−0.0954519	−0.0477259	+	0.998860i	$0.515197\pi$
$434$	0	0
$435$	−11.8839	−0.569790
$436$	0	0
$437$	3.47494	0.166229
$438$	0	0
$439$	−28.2563	−1.34860	−0.674300	−	0.738458i	$-0.735554\pi$
$439$	−28.2563	−1.34860	−0.674300	+	0.738458i	$0.735554\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−14.8836	−0.707140	−0.353570	−	0.935408i	$-0.615032\pi$
$443$	−14.8836	−0.707140	−0.353570	+	0.935408i	$0.615032\pi$
$444$	0	0
$445$	−9.96622	−0.472444
$446$	0	0
$447$	−2.06425	−0.0976359
$448$	0	0
$449$	−28.6401	−1.35161	−0.675804	−	0.737081i	$-0.736204\pi$
$449$	−28.6401	−1.35161	−0.675804	+	0.737081i	$0.736204\pi$
$450$	0	0
$451$	−2.41668	−0.113797
$452$	0	0
$453$	3.65826	0.171880
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−31.9232	−1.49330	−0.746652	−	0.665214i	$-0.768340\pi$
$457$	−31.9232	−1.49330	−0.746652	+	0.665214i	$0.768340\pi$
$458$	0	0
$459$	7.25099	0.338447
$460$	0	0
$461$	−29.2353	−1.36163	−0.680813	−	0.732458i	$-0.738373\pi$
$461$	−29.2353	−1.36163	−0.680813	+	0.732458i	$0.738373\pi$
$462$	0	0
$463$	−6.55699	−0.304729	−0.152365	−	0.988324i	$-0.548689\pi$
$463$	−6.55699	−0.304729	−0.152365	+	0.988324i	$0.548689\pi$
$464$	0	0
$465$	59.9441	2.77984
$466$	0	0
$467$	−19.1700	−0.887080	−0.443540	−	0.896255i	$-0.646278\pi$
$467$	−19.1700	−0.887080	−0.443540	+	0.896255i	$0.646278\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	4.13199	0.190392
$472$	0	0
$473$	−23.0370	−1.05924
$474$	0	0
$475$	−2.47242	−0.113442
$476$	0	0
$477$	−1.40809	−0.0644719
$478$	0	0
$479$	10.8689	0.496612	0.248306	−	0.968682i	$-0.420126\pi$
$479$	10.8689	0.496612	0.248306	+	0.968682i	$0.420126\pi$
$480$	0	0
$481$	16.3097	0.743657
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.85079	−0.174856
$486$	0	0
$487$	35.5151	1.60934	0.804672	−	0.593719i	$-0.202341\pi$
$487$	35.5151	1.60934	0.804672	+	0.593719i	$0.202341\pi$
$488$	0	0
$489$	24.0063	1.08560
$490$	0	0
$491$	−19.0084	−0.857837	−0.428918	−	0.903343i	$-0.641105\pi$
$491$	−19.0084	−0.857837	−0.428918	+	0.903343i	$0.641105\pi$
$492$	0	0
$493$	4.67779	0.210677
$494$	0	0
$495$	−17.6095	−0.791490
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−32.9692	−1.47590	−0.737952	−	0.674853i	$-0.764207\pi$
$499$	−32.9692	−1.47590	−0.737952	+	0.674853i	$0.764207\pi$
$500$	0	0
$501$	−35.5893	−1.59001
$502$	0	0
$503$	−14.8249	−0.661010	−0.330505	−	0.943804i	$-0.607219\pi$
$503$	−14.8249	−0.661010	−0.330505	+	0.943804i	$0.607219\pi$
$504$	0	0
$505$	−37.6811	−1.67679
$506$	0	0
$507$	−2.64785	−0.117595
$508$	0	0
$509$	15.1361	0.670897	0.335448	−	0.942059i	$-0.391112\pi$
$509$	15.1361	0.670897	0.335448	+	0.942059i	$0.391112\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−3.17541	−0.140198
$514$	0	0
$515$	−45.5328	−2.00641
$516$	0	0
$517$	−11.7837	−0.518247
$518$	0	0
$519$	42.2717	1.85552
$520$	0	0
$521$	37.9123	1.66097	0.830484	−	0.557042i	$-0.188064\pi$
$521$	37.9123	1.66097	0.830484	+	0.557042i	$0.188064\pi$
$522$	0	0
$523$	−31.3828	−1.37228	−0.686138	−	0.727472i	$-0.740695\pi$
$523$	−31.3828	−1.37228	−0.686138	+	0.727472i	$0.740695\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−23.5954	−1.02783
$528$	0	0
$529$	−10.9248	−0.474991
$530$	0	0
$531$	−12.3886	−0.537619
$532$	0	0
$533$	2.12931	0.0922307
$534$	0	0
$535$	22.7582	0.983923
$536$	0	0
$537$	−21.9006	−0.945080
$538$	0	0
$539$	0	0
$540$	0	0
$541$	45.8949	1.97317	0.986587	−	0.163234i	$-0.0521927\pi$
$541$	45.8949	1.97317	0.986587	+	0.163234i	$0.0521927\pi$
$542$	0	0
$543$	42.8286	1.83795
$544$	0	0
$545$	−22.2210	−0.951842
$546$	0	0
$547$	23.2818	0.995458	0.497729	−	0.867333i	$-0.334167\pi$
$547$	23.2818	0.995458	0.497729	+	0.867333i	$0.334167\pi$
$548$	0	0
$549$	−14.3429	−0.612142
$550$	0	0
$551$	−2.04853	−0.0872705
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−25.0662	−1.06400
$556$	0	0
$557$	6.21130	0.263181	0.131591	−	0.991304i	$-0.457992\pi$
$557$	6.21130	0.263181	0.131591	+	0.991304i	$0.457992\pi$
$558$	0	0
$559$	20.2976	0.858498
$560$	0	0
$561$	20.7604	0.876504
$562$	0	0
$563$	1.33099	0.0560945	0.0280472	−	0.999607i	$-0.491071\pi$
$563$	1.33099	0.0560945	0.0280472	+	0.999607i	$0.491071\pi$
$564$	0	0
$565$	−20.9363	−0.880798
$566$	0	0
$567$	0	0
$568$	0	0
$569$	14.3059	0.599734	0.299867	−	0.953981i	$-0.403058\pi$
$569$	14.3059	0.599734	0.299867	+	0.953981i	$0.403058\pi$
$570$	0	0
$571$	2.15480	0.0901754	0.0450877	−	0.998983i	$-0.485643\pi$
$571$	2.15480	0.0901754	0.0450877	+	0.998983i	$0.485643\pi$
$572$	0	0
$573$	−27.1096	−1.13252
$574$	0	0
$575$	−8.59151	−0.358291
$576$	0	0
$577$	−39.5684	−1.64726	−0.823628	−	0.567130i	$-0.808054\pi$
$577$	−39.5684	−1.64726	−0.823628	+	0.567130i	$0.808054\pi$
$578$	0	0
$579$	−29.6356	−1.23161
$580$	0	0
$581$	0	0
$582$	0	0
$583$	4.01160	0.166143
$584$	0	0
$585$	15.5156	0.641490
$586$	0	0
$587$	−38.3655	−1.58351	−0.791756	−	0.610837i	$-0.790833\pi$
$587$	−38.3655	−1.58351	−0.791756	+	0.610837i	$0.790833\pi$
$588$	0	0
$589$	10.3331	0.425768
$590$	0	0
$591$	−22.1739	−0.912114
$592$	0	0
$593$	−19.9375	−0.818734	−0.409367	−	0.912370i	$-0.634250\pi$
$593$	−19.9375	−0.818734	−0.409367	+	0.912370i	$0.634250\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−38.0257	−1.55629
$598$	0	0
$599$	−13.8283	−0.565010	−0.282505	−	0.959266i	$-0.591165\pi$
$599$	−13.8283	−0.565010	−0.282505	+	0.959266i	$0.591165\pi$
$600$	0	0
$601$	48.1960	1.96596	0.982979	−	0.183719i	$-0.0588137\pi$
$601$	48.1960	1.96596	0.982979	+	0.183719i	$0.0588137\pi$
$602$	0	0
$603$	8.77675	0.357417
$604$	0	0
$605$	20.0997	0.817170
$606$	0	0
$607$	−9.05885	−0.367687	−0.183844	−	0.982955i	$-0.558854\pi$
$607$	−9.05885	−0.367687	−0.183844	+	0.982955i	$0.558854\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	10.3825	0.420031
$612$	0	0
$613$	−21.7822	−0.879776	−0.439888	−	0.898053i	$-0.644982\pi$
$613$	−21.7822	−0.879776	−0.439888	+	0.898053i	$0.644982\pi$
$614$	0	0
$615$	−3.27252	−0.131961
$616$	0	0
$617$	−18.1039	−0.728835	−0.364418	−	0.931236i	$-0.618732\pi$
$617$	−18.1039	−0.728835	−0.364418	+	0.931236i	$0.618732\pi$
$618$	0	0
$619$	7.03758	0.282864	0.141432	−	0.989948i	$-0.454829\pi$
$619$	7.03758	0.282864	0.141432	+	0.989948i	$0.454829\pi$
$620$	0	0
$621$	−11.0344	−0.442794
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−31.2492	−1.24997
$626$	0	0
$627$	−9.09155	−0.363081
$628$	0	0
$629$	9.86666	0.393410
$630$	0	0
$631$	−47.5855	−1.89435	−0.947175	−	0.320718i	$-0.896076\pi$
$631$	−47.5855	−1.89435	−0.947175	+	0.320718i	$0.896076\pi$
$632$	0	0
$633$	47.6839	1.89526
$634$	0	0
$635$	−0.927438	−0.0368043
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−18.9119	−0.748145
$640$	0	0
$641$	−44.4946	−1.75743	−0.878715	−	0.477347i	$-0.841599\pi$
$641$	−44.4946	−1.75743	−0.878715	+	0.477347i	$0.841599\pi$
$642$	0	0
$643$	13.0448	0.514435	0.257217	−	0.966354i	$-0.417194\pi$
$643$	13.0448	0.514435	0.257217	+	0.966354i	$0.417194\pi$
$644$	0	0
$645$	−31.1953	−1.22831
$646$	0	0
$647$	−28.0855	−1.10416	−0.552078	−	0.833793i	$-0.686165\pi$
$647$	−28.0855	−1.10416	−0.552078	+	0.833793i	$0.686165\pi$
$648$	0	0
$649$	35.2947	1.38544
$650$	0	0
$651$	0	0
$652$	0	0
$653$	44.6666	1.74794	0.873970	−	0.485979i	$-0.161537\pi$
$653$	44.6666	1.74794	0.873970	+	0.485979i	$0.161537\pi$
$654$	0	0
$655$	−0.779278	−0.0304489
$656$	0	0
$657$	−5.88891	−0.229748
$658$	0	0
$659$	28.1557	1.09679	0.548395	−	0.836220i	$-0.315239\pi$
$659$	28.1557	1.09679	0.548395	+	0.836220i	$0.315239\pi$
$660$	0	0
$661$	−44.6283	−1.73584	−0.867920	−	0.496704i	$-0.834544\pi$
$661$	−44.6283	−1.73584	−0.867920	+	0.496704i	$0.834544\pi$
$662$	0	0
$663$	−18.2917	−0.710392
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−7.11853	−0.275631
$668$	0	0
$669$	−7.90622	−0.305672
$670$	0	0
$671$	40.8626	1.57748
$672$	0	0
$673$	41.6006	1.60359	0.801793	−	0.597601i	$-0.203879\pi$
$673$	41.6006	1.60359	0.801793	+	0.597601i	$0.203879\pi$
$674$	0	0
$675$	7.85096	0.302183
$676$	0	0
$677$	35.3284	1.35778	0.678891	−	0.734239i	$-0.262461\pi$
$677$	35.3284	1.35778	0.678891	+	0.734239i	$0.262461\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	21.8651	0.837873
$682$	0	0
$683$	14.1262	0.540523	0.270261	−	0.962787i	$-0.412890\pi$
$683$	14.1262	0.540523	0.270261	+	0.962787i	$0.412890\pi$
$684$	0	0
$685$	45.6656	1.74479
$686$	0	0
$687$	17.2989	0.659993
$688$	0	0
$689$	−3.53457	−0.134657
$690$	0	0
$691$	−41.3645	−1.57358	−0.786790	−	0.617221i	$-0.788259\pi$
$691$	−41.3645	−1.57358	−0.786790	+	0.617221i	$0.788259\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−8.63702	−0.327621
$696$	0	0
$697$	1.28814	0.0487919
$698$	0	0
$699$	22.2535	0.841703
$700$	0	0
$701$	−24.7876	−0.936214	−0.468107	−	0.883672i	$-0.655064\pi$
$701$	−24.7876	−0.936214	−0.468107	+	0.883672i	$0.655064\pi$
$702$	0	0
$703$	−4.32089	−0.162965
$704$	0	0
$705$	−15.9568	−0.600968
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−8.24224	−0.309544	−0.154772	−	0.987950i	$-0.549464\pi$
$709$	−8.24224	−0.309544	−0.154772	+	0.987950i	$0.549464\pi$
$710$	0	0
$711$	20.5550	0.770873
$712$	0	0
$713$	35.9069	1.34472
$714$	0	0
$715$	−44.2034	−1.65311
$716$	0	0
$717$	28.4893	1.06395
$718$	0	0
$719$	−30.9085	−1.15269	−0.576346	−	0.817206i	$-0.695522\pi$
$719$	−30.9085	−1.15269	−0.576346	+	0.817206i	$0.695522\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−32.6286	−1.21347
$724$	0	0
$725$	5.06484	0.188103
$726$	0	0
$727$	−34.2915	−1.27180	−0.635900	−	0.771771i	$-0.719371\pi$
$727$	−34.2915	−1.27180	−0.635900	+	0.771771i	$0.719371\pi$
$728$	0	0
$729$	3.29979	0.122214
$730$	0	0
$731$	12.2792	0.454163
$732$	0	0
$733$	6.85832	0.253318	0.126659	−	0.991946i	$-0.459575\pi$
$733$	6.85832	0.253318	0.126659	+	0.991946i	$0.459575\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−25.0047	−0.921059
$738$	0	0
$739$	3.88085	0.142759	0.0713796	−	0.997449i	$-0.477260\pi$
$739$	3.88085	0.142759	0.0713796	+	0.997449i	$0.477260\pi$
$740$	0	0
$741$	8.01046	0.294272
$742$	0	0
$743$	14.4652	0.530676	0.265338	−	0.964155i	$-0.414516\pi$
$743$	14.4652	0.530676	0.265338	+	0.964155i	$0.414516\pi$
$744$	0	0
$745$	2.65894	0.0974160
$746$	0	0
$747$	25.7270	0.941300
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−1.51864	−0.0554161	−0.0277080	−	0.999616i	$-0.508821\pi$
$751$	−1.51864	−0.0554161	−0.0277080	+	0.999616i	$0.508821\pi$
$752$	0	0
$753$	−3.82774	−0.139490
$754$	0	0
$755$	−4.71215	−0.171493
$756$	0	0
$757$	−37.2876	−1.35524	−0.677620	−	0.735412i	$-0.736988\pi$
$757$	−37.2876	−1.35524	−0.677620	+	0.735412i	$0.736988\pi$
$758$	0	0
$759$	−31.5926	−1.14674
$760$	0	0
$761$	29.5449	1.07100	0.535500	−	0.844535i	$-0.320123\pi$
$761$	29.5449	1.07100	0.535500	+	0.844535i	$0.320123\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	9.38627	0.339361
$766$	0	0
$767$	−31.0978	−1.12288
$768$	0	0
$769$	49.9842	1.80247	0.901237	−	0.433326i	$-0.142660\pi$
$769$	49.9842	1.80247	0.901237	+	0.433326i	$0.142660\pi$
$770$	0	0
$771$	−35.2855	−1.27078
$772$	0	0
$773$	−53.8487	−1.93680	−0.968401	−	0.249397i	$-0.919768\pi$
$773$	−53.8487	−1.93680	−0.968401	+	0.249397i	$0.919768\pi$
$774$	0	0
$775$	−25.5478	−0.917702
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.564114	−0.0202115
$780$	0	0
$781$	53.8795	1.92796
$782$	0	0
$783$	6.50494	0.232468
$784$	0	0
$785$	−5.32236	−0.189963
$786$	0	0
$787$	13.8411	0.493381	0.246690	−	0.969094i	$-0.420657\pi$
$787$	13.8411	0.493381	0.246690	+	0.969094i	$0.420657\pi$
$788$	0	0
$789$	26.2796	0.935579
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−36.0036	−1.27852
$794$	0	0
$795$	5.43226	0.192663
$796$	0	0
$797$	−30.5810	−1.08324	−0.541618	−	0.840625i	$-0.682188\pi$
$797$	−30.5810	−1.08324	−0.541618	+	0.840625i	$0.682188\pi$
$798$	0	0
$799$	6.28097	0.222205
$800$	0	0
$801$	−5.48233	−0.193708
$802$	0	0
$803$	16.7773	0.592058
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−52.9431	−1.86368
$808$	0	0
$809$	−11.0989	−0.390216	−0.195108	−	0.980782i	$-0.562506\pi$
$809$	−11.0989	−0.390216	−0.195108	+	0.980782i	$0.562506\pi$
$810$	0	0
$811$	−48.8311	−1.71469	−0.857345	−	0.514742i	$-0.827888\pi$
$811$	−48.8311	−1.71469	−0.857345	+	0.514742i	$0.827888\pi$
$812$	0	0
$813$	−5.98940	−0.210058
$814$	0	0
$815$	−30.9222	−1.08316
$816$	0	0
$817$	−5.37741	−0.188132
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−43.4940	−1.51795	−0.758976	−	0.651118i	$-0.774300\pi$
$821$	−43.4940	−1.51795	−0.758976	+	0.651118i	$0.774300\pi$
$822$	0	0
$823$	28.8140	1.00439	0.502196	−	0.864754i	$-0.332526\pi$
$823$	28.8140	1.00439	0.502196	+	0.864754i	$0.332526\pi$
$824$	0	0
$825$	22.4781	0.782588
$826$	0	0
$827$	23.7113	0.824524	0.412262	−	0.911065i	$-0.364739\pi$
$827$	23.7113	0.824524	0.412262	+	0.911065i	$0.364739\pi$
$828$	0	0
$829$	24.8962	0.864680	0.432340	−	0.901711i	$-0.357688\pi$
$829$	24.8962	0.864680	0.432340	+	0.901711i	$0.357688\pi$
$830$	0	0
$831$	32.0571	1.11205
$832$	0	0
$833$	0	0
$834$	0	0
$835$	45.8421	1.58643
$836$	0	0
$837$	−32.8119	−1.13414
$838$	0	0
$839$	38.8536	1.34138	0.670688	−	0.741740i	$-0.265999\pi$
$839$	38.8536	1.34138	0.670688	+	0.741740i	$0.265999\pi$
$840$	0	0
$841$	−24.8035	−0.855293
$842$	0	0
$843$	15.4336	0.531562
$844$	0	0
$845$	3.41066	0.117330
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−3.81069	−0.130783
$850$	0	0
$851$	−15.0148	−0.514701
$852$	0	0
$853$	−5.24625	−0.179628	−0.0898140	−	0.995959i	$-0.528627\pi$
$853$	−5.24625	−0.179628	−0.0898140	+	0.995959i	$0.528627\pi$
$854$	0	0
$855$	−4.11051	−0.140576
$856$	0	0
$857$	5.07375	0.173316	0.0866580	−	0.996238i	$-0.472381\pi$
$857$	5.07375	0.173316	0.0866580	+	0.996238i	$0.472381\pi$
$858$	0	0
$859$	−42.0845	−1.43590	−0.717952	−	0.696092i	$-0.754920\pi$
$859$	−42.0845	−1.43590	−0.717952	+	0.696092i	$0.754920\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	9.65349	0.328609	0.164304	−	0.986410i	$-0.447462\pi$
$863$	9.65349	0.328609	0.164304	+	0.986410i	$0.447462\pi$
$864$	0	0
$865$	−54.4496	−1.85134
$866$	0	0
$867$	25.0116	0.849438
$868$	0	0
$869$	−58.5606	−1.98653
$870$	0	0
$871$	22.0314	0.746504
$872$	0	0
$873$	−2.11829	−0.0716931
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−24.6914	−0.833770	−0.416885	−	0.908959i	$-0.636878\pi$
$877$	−24.6914	−0.833770	−0.416885	+	0.908959i	$0.636878\pi$
$878$	0	0
$879$	67.1104	2.26358
$880$	0	0
$881$	−26.0753	−0.878499	−0.439249	−	0.898365i	$-0.644756\pi$
$881$	−26.0753	−0.878499	−0.439249	+	0.898365i	$0.644756\pi$
$882$	0	0
$883$	0.743628	0.0250251	0.0125125	−	0.999922i	$-0.496017\pi$
$883$	0.743628	0.0250251	0.0125125	+	0.999922i	$0.496017\pi$
$884$	0	0
$885$	47.7940	1.60658
$886$	0	0
$887$	40.5465	1.36142	0.680710	−	0.732553i	$-0.261671\pi$
$887$	40.5465	1.36142	0.680710	+	0.732553i	$0.261671\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	48.1953	1.61460
$892$	0	0
$893$	−2.75061	−0.0920458
$894$	0	0
$895$	28.2099	0.942951
$896$	0	0
$897$	27.8359	0.929413
$898$	0	0
$899$	−21.1677	−0.705982
$900$	0	0
$901$	−2.13827	−0.0712360
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−55.1670	−1.83381
$906$	0	0
$907$	−39.2142	−1.30209	−0.651043	−	0.759040i	$-0.725668\pi$
$907$	−39.2142	−1.30209	−0.651043	+	0.759040i	$0.725668\pi$
$908$	0	0
$909$	−20.7280	−0.687506
$910$	0	0
$911$	28.5473	0.945814	0.472907	−	0.881112i	$-0.343205\pi$
$911$	28.5473	0.945814	0.472907	+	0.881112i	$0.343205\pi$
$912$	0	0
$913$	−73.2953	−2.42572
$914$	0	0
$915$	55.3336	1.82927
$916$	0	0
$917$	0	0
$918$	0	0
$919$	26.9598	0.889323	0.444661	−	0.895699i	$-0.353324\pi$
$919$	26.9598	0.889323	0.444661	+	0.895699i	$0.353324\pi$
$920$	0	0
$921$	−19.0896	−0.629022
$922$	0	0
$923$	−47.4726	−1.56258
$924$	0	0
$925$	10.6830	0.351256
$926$	0	0
$927$	−25.0472	−0.822657
$928$	0	0
$929$	−11.1488	−0.365780	−0.182890	−	0.983133i	$-0.558545\pi$
$929$	−11.1488	−0.365780	−0.182890	+	0.983133i	$0.558545\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	49.6746	1.62627
$934$	0	0
$935$	−26.7412	−0.874530
$936$	0	0
$937$	−21.1905	−0.692263	−0.346131	−	0.938186i	$-0.612505\pi$
$937$	−21.1905	−0.692263	−0.346131	+	0.938186i	$0.612505\pi$
$938$	0	0
$939$	−12.4082	−0.404927
$940$	0	0
$941$	1.93670	0.0631347	0.0315673	−	0.999502i	$-0.489950\pi$
$941$	1.93670	0.0631347	0.0315673	+	0.999502i	$0.489950\pi$
$942$	0	0
$943$	−1.96026	−0.0638349
$944$	0	0
$945$	0	0
$946$	0	0
$947$	22.5458	0.732640	0.366320	−	0.930489i	$-0.380618\pi$
$947$	22.5458	0.732640	0.366320	+	0.930489i	$0.380618\pi$
$948$	0	0
$949$	−14.7823	−0.479854
$950$	0	0
$951$	−38.7609	−1.25691
$952$	0	0
$953$	−11.0353	−0.357468	−0.178734	−	0.983897i	$-0.557200\pi$
$953$	−11.0353	−0.357468	−0.178734	+	0.983897i	$0.557200\pi$
$954$	0	0
$955$	34.9195	1.12997
$956$	0	0
$957$	18.6243	0.602040
$958$	0	0
$959$	0	0
$960$	0	0
$961$	75.7729	2.44429
$962$	0	0
$963$	12.5191	0.403422
$964$	0	0
$965$	38.1732	1.22884
$966$	0	0
$967$	49.9085	1.60495	0.802475	−	0.596686i	$-0.203516\pi$
$967$	49.9085	1.60495	0.802475	+	0.596686i	$0.203516\pi$
$968$	0	0
$969$	4.84599	0.155676
$970$	0	0
$971$	10.7719	0.345688	0.172844	−	0.984949i	$-0.444704\pi$
$971$	10.7719	0.345688	0.172844	+	0.984949i	$0.444704\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−19.8052	−0.634275
$976$	0	0
$977$	1.18513	0.0379156	0.0189578	−	0.999820i	$-0.493965\pi$
$977$	1.18513	0.0379156	0.0189578	+	0.999820i	$0.493965\pi$
$978$	0	0
$979$	15.6190	0.499184
$980$	0	0
$981$	−12.2236	−0.390268
$982$	0	0
$983$	36.9263	1.17777	0.588883	−	0.808218i	$-0.299568\pi$
$983$	36.9263	1.17777	0.588883	+	0.808218i	$0.299568\pi$
$984$	0	0
$985$	28.5620	0.910060
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−18.6862	−0.594185
$990$	0	0
$991$	47.3901	1.50540	0.752698	−	0.658366i	$-0.228752\pi$
$991$	47.3901	1.50540	0.752698	+	0.658366i	$0.228752\pi$
$992$	0	0
$993$	65.8809	2.09067
$994$	0	0
$995$	48.9804	1.55278
$996$	0	0
$997$	27.2842	0.864099	0.432050	−	0.901850i	$-0.357791\pi$
$997$	27.2842	0.864099	0.432050	+	0.901850i	$0.357791\pi$
$998$	0	0
$999$	13.7206	0.434101

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.bu.1.4	✓	14
7.6	odd	2		7448.2.a.bx.1.11	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7448.2.a.bu.1.4	✓	14	1.1	even	1	trivial
7448.2.a.bx.1.11	yes	14	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)

\(f(q)\)	\(=\)	\(q-2.12220 q^{3} +2.73357 q^{5} +1.50371 q^{9} +O(q^{10})\)	\(q-2.12220 q^{3} +2.73357 q^{5} +1.50371 q^{9} -4.28403 q^{11} +3.77461 q^{13} -5.80118 q^{15} +2.28348 q^{17} -1.00000 q^{19} -3.47494 q^{23} +2.47242 q^{25} +3.17541 q^{27} +2.04853 q^{29} -10.3331 q^{31} +9.09155 q^{33} +4.32089 q^{37} -8.01046 q^{39} +0.564114 q^{41} +5.37741 q^{43} +4.11051 q^{45} +2.75061 q^{47} -4.84599 q^{51} -0.936408 q^{53} -11.7107 q^{55} +2.12220 q^{57} -8.23867 q^{59} -9.53835 q^{61} +10.3182 q^{65} +5.83672 q^{67} +7.37450 q^{69} -12.5768 q^{71} -3.91624 q^{73} -5.24696 q^{75} +13.6695 q^{79} -11.2500 q^{81} +17.1090 q^{83} +6.24206 q^{85} -4.34739 q^{87} -3.64586 q^{89} +21.9288 q^{93} -2.73357 q^{95} -1.40870 q^{97} -6.44195 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 6 q^{3} - 2 q^{5} + 16 q^{9}+O(q^{10}) \) 14 * q - 6 * q^3 - 2 * q^5 + 16 * q^9	\( 14 q - 6 q^{3} - 2 q^{5} + 16 q^{9} - 6 q^{11} - 16 q^{13} + 4 q^{15} + 4 q^{17} - 14 q^{19} - 4 q^{23} + 16 q^{25} - 36 q^{27} - 6 q^{29} - 16 q^{31} + 10 q^{33} + 6 q^{37} + 16 q^{39} + 14 q^{41} - 2 q^{43} - 30 q^{47} + 20 q^{51} - 6 q^{53} - 44 q^{55} + 6 q^{57} - 22 q^{59} - 10 q^{61} - 16 q^{65} + 4 q^{67} - 48 q^{69} + 6 q^{71} - 4 q^{73} - 64 q^{75} + 26 q^{79} + 30 q^{81} - 32 q^{83} - 8 q^{85} - 32 q^{87} + 54 q^{89} - 32 q^{93} + 2 q^{95} - 18 q^{97} - 38 q^{99}+O(q^{100}) \) 14 * q - 6 * q^3 - 2 * q^5 + 16 * q^9 - 6 * q^11 - 16 * q^13 + 4 * q^15 + 4 * q^17 - 14 * q^19 - 4 * q^23 + 16 * q^25 - 36 * q^27 - 6 * q^29 - 16 * q^31 + 10 * q^33 + 6 * q^37 + 16 * q^39 + 14 * q^41 - 2 * q^43 - 30 * q^47 + 20 * q^51 - 6 * q^53 - 44 * q^55 + 6 * q^57 - 22 * q^59 - 10 * q^61 - 16 * q^65 + 4 * q^67 - 48 * q^69 + 6 * q^71 - 4 * q^73 - 64 * q^75 + 26 * q^79 + 30 * q^81 - 32 * q^83 - 8 * q^85 - 32 * q^87 + 54 * q^89 - 32 * q^93 + 2 * q^95 - 18 * q^97 - 38 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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