LMFDB - Embedded newform 7448.2.a.br.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:	$0$
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			1.4
Root			$0.684189$ 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)

$f(q)$	$=$	$q-0.684189 q^{3} +3.65096 q^{5} -2.53189 q^{9} +O(q^{10})$	$q-0.684189 q^{3} +3.65096 q^{5} -2.53189 q^{9} +2.37730 q^{11} -5.46860 q^{13} -2.49795 q^{15} +4.86063 q^{17} -1.00000 q^{19} -1.77549 q^{23} +8.32954 q^{25} +3.78485 q^{27} -6.04283 q^{29} -1.69856 q^{31} -1.62652 q^{33} +5.62635 q^{37} +3.74155 q^{39} +2.00000 q^{41} +3.10972 q^{43} -9.24382 q^{45} +5.23017 q^{47} -3.32559 q^{51} -6.46513 q^{53} +8.67942 q^{55} +0.684189 q^{57} +7.52266 q^{59} +7.75592 q^{61} -19.9657 q^{65} -2.62090 q^{67} +1.21477 q^{69} +11.1529 q^{71} -14.5305 q^{73} -5.69898 q^{75} +0.961803 q^{79} +5.00610 q^{81} +15.0476 q^{83} +17.7460 q^{85} +4.13444 q^{87} -12.0321 q^{89} +1.16213 q^{93} -3.65096 q^{95} +6.26438 q^{97} -6.01904 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.684189	−0.395017	−0.197508	−	0.980301i	$-0.563285\pi$
$3$	−0.684189	−0.395017	−0.197508	+	0.980301i	$0.563285\pi$
$4$	0	0
$5$	3.65096	1.63276	0.816380	−	0.577515i	$-0.195977\pi$
$5$	3.65096	1.63276	0.816380	+	0.577515i	$0.195977\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.53189	−0.843962
$10$	0	0
$11$	2.37730	0.716782	0.358391	−	0.933572i	$-0.383325\pi$
$11$	2.37730	0.716782	0.358391	+	0.933572i	$0.383325\pi$
$12$	0	0
$13$	−5.46860	−1.51672	−0.758358	−	0.651838i	$-0.773998\pi$
$13$	−5.46860	−1.51672	−0.758358	+	0.651838i	$0.773998\pi$
$14$	0	0
$15$	−2.49795	−0.644968
$16$	0	0
$17$	4.86063	1.17888	0.589438	−	0.807813i	$-0.299349\pi$
$17$	4.86063	1.17888	0.589438	+	0.807813i	$0.299349\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.77549	−0.370216	−0.185108	−	0.982718i	$-0.559263\pi$
$23$	−1.77549	−0.370216	−0.185108	+	0.982718i	$0.559263\pi$
$24$	0	0
$25$	8.32954	1.66591
$26$	0	0
$27$	3.78485	0.728396
$28$	0	0
$29$	−6.04283	−1.12213	−0.561063	−	0.827773i	$-0.689607\pi$
$29$	−6.04283	−1.12213	−0.561063	+	0.827773i	$0.689607\pi$
$30$	0	0
$31$	−1.69856	−0.305070	−0.152535	−	0.988298i	$-0.548744\pi$
$31$	−1.69856	−0.305070	−0.152535	+	0.988298i	$0.548744\pi$
$32$	0	0
$33$	−1.62652	−0.283141
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.62635	0.924967	0.462483	−	0.886628i	$-0.346958\pi$
$37$	5.62635	0.924967	0.462483	+	0.886628i	$0.346958\pi$
$38$	0	0
$39$	3.74155	0.599128
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	3.10972	0.474227	0.237114	−	0.971482i	$-0.423799\pi$
$43$	3.10972	0.474227	0.237114	+	0.971482i	$0.423799\pi$
$44$	0	0
$45$	−9.24382	−1.37799
$46$	0	0
$47$	5.23017	0.762898	0.381449	−	0.924390i	$-0.375425\pi$
$47$	5.23017	0.762898	0.381449	+	0.924390i	$0.375425\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−3.32559	−0.465676
$52$	0	0
$53$	−6.46513	−0.888054	−0.444027	−	0.896013i	$-0.646451\pi$
$53$	−6.46513	−0.888054	−0.444027	+	0.896013i	$0.646451\pi$
$54$	0	0
$55$	8.67942	1.17033
$56$	0	0
$57$	0.684189	0.0906230
$58$	0	0
$59$	7.52266	0.979367	0.489684	−	0.871900i	$-0.337112\pi$
$59$	7.52266	0.979367	0.489684	+	0.871900i	$0.337112\pi$
$60$	0	0
$61$	7.75592	0.993044	0.496522	−	0.868024i	$-0.334610\pi$
$61$	7.75592	0.993044	0.496522	+	0.868024i	$0.334610\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−19.9657	−2.47644
$66$	0	0
$67$	−2.62090	−0.320194	−0.160097	−	0.987101i	$-0.551181\pi$
$67$	−2.62090	−0.320194	−0.160097	+	0.987101i	$0.551181\pi$
$68$	0	0
$69$	1.21477	0.146241
$70$	0	0
$71$	11.1529	1.32361	0.661805	−	0.749676i	$-0.269791\pi$
$71$	11.1529	1.32361	0.661805	+	0.749676i	$0.269791\pi$
$72$	0	0
$73$	−14.5305	−1.70067	−0.850333	−	0.526244i	$-0.823600\pi$
$73$	−14.5305	−1.70067	−0.850333	+	0.526244i	$0.823600\pi$
$74$	0	0
$75$	−5.69898	−0.658061
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0.961803	0.108211	0.0541057	−	0.998535i	$-0.482769\pi$
$79$	0.961803	0.108211	0.0541057	+	0.998535i	$0.482769\pi$
$80$	0	0
$81$	5.00610	0.556234
$82$	0	0
$83$	15.0476	1.65169	0.825844	−	0.563898i	$-0.190699\pi$
$83$	15.0476	1.65169	0.825844	+	0.563898i	$0.190699\pi$
$84$	0	0
$85$	17.7460	1.92482
$86$	0	0
$87$	4.13444	0.443258
$88$	0	0
$89$	−12.0321	−1.27540	−0.637701	−	0.770284i	$-0.720114\pi$
$89$	−12.0321	−1.27540	−0.637701	+	0.770284i	$0.720114\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.16213	0.120508
$94$	0	0
$95$	−3.65096	−0.374581
$96$	0	0
$97$	6.26438	0.636052	0.318026	−	0.948082i	$-0.396980\pi$
$97$	6.26438	0.636052	0.318026	+	0.948082i	$0.396980\pi$
$98$	0	0
$99$	−6.01904	−0.604937
$100$	0	0
$101$	5.71595	0.568759	0.284379	−	0.958712i	$-0.408213\pi$
$101$	5.71595	0.568759	0.284379	+	0.958712i	$0.408213\pi$
$102$	0	0
$103$	1.69879	0.167387	0.0836935	−	0.996492i	$-0.473328\pi$
$103$	1.69879	0.167387	0.0836935	+	0.996492i	$0.473328\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.25842	0.701698	0.350849	−	0.936432i	$-0.385893\pi$
$107$	7.25842	0.701698	0.350849	+	0.936432i	$0.385893\pi$
$108$	0	0
$109$	18.4449	1.76670	0.883351	−	0.468712i	$-0.155282\pi$
$109$	18.4449	1.76670	0.883351	+	0.468712i	$0.155282\pi$
$110$	0	0
$111$	−3.84949	−0.365377
$112$	0	0
$113$	17.9243	1.68618	0.843090	−	0.537772i	$-0.180734\pi$
$113$	17.9243	1.68618	0.843090	+	0.537772i	$0.180734\pi$
$114$	0	0
$115$	−6.48226	−0.604474
$116$	0	0
$117$	13.8459	1.28005
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−5.34846	−0.486224
$122$	0	0
$123$	−1.36838	−0.123382
$124$	0	0
$125$	12.1560	1.08727
$126$	0	0
$127$	16.8044	1.49115	0.745575	−	0.666421i	$-0.232175\pi$
$127$	16.8044	1.49115	0.745575	+	0.666421i	$0.232175\pi$
$128$	0	0
$129$	−2.12763	−0.187328
$130$	0	0
$131$	10.1064	0.882999	0.441500	−	0.897261i	$-0.354447\pi$
$131$	10.1064	0.882999	0.441500	+	0.897261i	$0.354447\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	13.8184	1.18930
$136$	0	0
$137$	−5.55210	−0.474348	−0.237174	−	0.971467i	$-0.576221\pi$
$137$	−5.55210	−0.474348	−0.237174	+	0.971467i	$0.576221\pi$
$138$	0	0
$139$	−4.35122	−0.369066	−0.184533	−	0.982826i	$-0.559077\pi$
$139$	−4.35122	−0.369066	−0.184533	+	0.982826i	$0.559077\pi$
$140$	0	0
$141$	−3.57842	−0.301357
$142$	0	0
$143$	−13.0005	−1.08716
$144$	0	0
$145$	−22.0622	−1.83216
$146$	0	0
$147$	0	0
$148$	0	0
$149$	12.1443	0.994896	0.497448	−	0.867494i	$-0.334271\pi$
$149$	12.1443	0.994896	0.497448	+	0.867494i	$0.334271\pi$
$150$	0	0
$151$	−15.7921	−1.28514	−0.642571	−	0.766226i	$-0.722132\pi$
$151$	−15.7921	−1.28514	−0.642571	+	0.766226i	$0.722132\pi$
$152$	0	0
$153$	−12.3066	−0.994927
$154$	0	0
$155$	−6.20138	−0.498106
$156$	0	0
$157$	−7.83836	−0.625569	−0.312785	−	0.949824i	$-0.601262\pi$
$157$	−7.83836	−0.625569	−0.312785	+	0.949824i	$0.601262\pi$
$158$	0	0
$159$	4.42337	0.350796
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−4.86466	−0.381030	−0.190515	−	0.981684i	$-0.561016\pi$
$163$	−4.86466	−0.381030	−0.190515	+	0.981684i	$0.561016\pi$
$164$	0	0
$165$	−5.93836	−0.462301
$166$	0	0
$167$	13.6708	1.05788	0.528939	−	0.848660i	$-0.322590\pi$
$167$	13.6708	1.05788	0.528939	+	0.848660i	$0.322590\pi$
$168$	0	0
$169$	16.9056	1.30043
$170$	0	0
$171$	2.53189	0.193618
$172$	0	0
$173$	5.34907	0.406682	0.203341	−	0.979108i	$-0.434820\pi$
$173$	5.34907	0.406682	0.203341	+	0.979108i	$0.434820\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−5.14692	−0.386866
$178$	0	0
$179$	−0.687824	−0.0514104	−0.0257052	−	0.999670i	$-0.508183\pi$
$179$	−0.687824	−0.0514104	−0.0257052	+	0.999670i	$0.508183\pi$
$180$	0	0
$181$	−7.91388	−0.588234	−0.294117	−	0.955769i	$-0.595026\pi$
$181$	−7.91388	−0.588234	−0.294117	+	0.955769i	$0.595026\pi$
$182$	0	0
$183$	−5.30652	−0.392269
$184$	0	0
$185$	20.5416	1.51025
$186$	0	0
$187$	11.5552	0.844998
$188$	0	0
$189$	0	0
$190$	0	0
$191$	24.0896	1.74306	0.871530	−	0.490341i	$-0.163128\pi$
$191$	24.0896	1.74306	0.871530	+	0.490341i	$0.163128\pi$
$192$	0	0
$193$	−4.47773	−0.322314	−0.161157	−	0.986929i	$-0.551523\pi$
$193$	−4.47773	−0.322314	−0.161157	+	0.986929i	$0.551523\pi$
$194$	0	0
$195$	13.6603	0.978233
$196$	0	0
$197$	15.0277	1.07068	0.535339	−	0.844637i	$-0.320184\pi$
$197$	15.0277	1.07068	0.535339	+	0.844637i	$0.320184\pi$
$198$	0	0
$199$	27.3915	1.94173	0.970866	−	0.239623i	$-0.0770240\pi$
$199$	27.3915	1.94173	0.970866	+	0.239623i	$0.0770240\pi$
$200$	0	0
$201$	1.79319	0.126482
$202$	0	0
$203$	0	0
$204$	0	0
$205$	7.30193	0.509989
$206$	0	0
$207$	4.49534	0.312448
$208$	0	0
$209$	−2.37730	−0.164441
$210$	0	0
$211$	−1.37592	−0.0947219	−0.0473610	−	0.998878i	$-0.515081\pi$
$211$	−1.37592	−0.0947219	−0.0473610	+	0.998878i	$0.515081\pi$
$212$	0	0
$213$	−7.63071	−0.522848
$214$	0	0
$215$	11.3535	0.774300
$216$	0	0
$217$	0	0
$218$	0	0
$219$	9.94161	0.671792
$220$	0	0
$221$	−26.5809	−1.78802
$222$	0	0
$223$	−27.7782	−1.86016	−0.930082	−	0.367352i	$-0.880265\pi$
$223$	−27.7782	−1.86016	−0.930082	+	0.367352i	$0.880265\pi$
$224$	0	0
$225$	−21.0894	−1.40596
$226$	0	0
$227$	−20.5905	−1.36664	−0.683320	−	0.730119i	$-0.739465\pi$
$227$	−20.5905	−1.36664	−0.683320	+	0.730119i	$0.739465\pi$
$228$	0	0
$229$	−26.1587	−1.72861	−0.864307	−	0.502964i	$-0.832243\pi$
$229$	−26.1587	−1.72861	−0.864307	+	0.502964i	$0.832243\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−12.5845	−0.824439	−0.412219	−	0.911085i	$-0.635246\pi$
$233$	−12.5845	−0.824439	−0.412219	+	0.911085i	$0.635246\pi$
$234$	0	0
$235$	19.0952	1.24563
$236$	0	0
$237$	−0.658055	−0.0427453
$238$	0	0
$239$	28.5846	1.84898	0.924492	−	0.381202i	$-0.124490\pi$
$239$	28.5846	1.84898	0.924492	+	0.381202i	$0.124490\pi$
$240$	0	0
$241$	−4.64546	−0.299241	−0.149620	−	0.988744i	$-0.547805\pi$
$241$	−4.64546	−0.299241	−0.149620	+	0.988744i	$0.547805\pi$
$242$	0	0
$243$	−14.7797	−0.948117
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.46860	0.347959
$248$	0	0
$249$	−10.2954	−0.652444
$250$	0	0
$251$	12.0953	0.763446	0.381723	−	0.924277i	$-0.375331\pi$
$251$	12.0953	0.763446	0.381723	+	0.924277i	$0.375331\pi$
$252$	0	0
$253$	−4.22087	−0.265364
$254$	0	0
$255$	−12.1416	−0.760337
$256$	0	0
$257$	−1.89239	−0.118044	−0.0590219	−	0.998257i	$-0.518798\pi$
$257$	−1.89239	−0.118044	−0.0590219	+	0.998257i	$0.518798\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	15.2998	0.947031
$262$	0	0
$263$	23.1801	1.42935	0.714674	−	0.699457i	$-0.246575\pi$
$263$	23.1801	1.42935	0.714674	+	0.699457i	$0.246575\pi$
$264$	0	0
$265$	−23.6040	−1.44998
$266$	0	0
$267$	8.23224	0.503805
$268$	0	0
$269$	−24.2982	−1.48149	−0.740744	−	0.671787i	$-0.765527\pi$
$269$	−24.2982	−1.48149	−0.740744	+	0.671787i	$0.765527\pi$
$270$	0	0
$271$	−12.4096	−0.753831	−0.376915	−	0.926248i	$-0.623015\pi$
$271$	−12.4096	−0.753831	−0.376915	+	0.926248i	$0.623015\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	19.8018	1.19409
$276$	0	0
$277$	−2.33798	−0.140475	−0.0702377	−	0.997530i	$-0.522376\pi$
$277$	−2.33798	−0.140475	−0.0702377	+	0.997530i	$0.522376\pi$
$278$	0	0
$279$	4.30056	0.257468
$280$	0	0
$281$	3.65566	0.218078	0.109039	−	0.994037i	$-0.465223\pi$
$281$	3.65566	0.218078	0.109039	+	0.994037i	$0.465223\pi$
$282$	0	0
$283$	−9.58475	−0.569754	−0.284877	−	0.958564i	$-0.591953\pi$
$283$	−9.58475	−0.569754	−0.284877	+	0.958564i	$0.591953\pi$
$284$	0	0
$285$	2.49795	0.147966
$286$	0	0
$287$	0	0
$288$	0	0
$289$	6.62575	0.389750
$290$	0	0
$291$	−4.28602	−0.251251
$292$	0	0
$293$	10.0249	0.585662	0.292831	−	0.956164i	$-0.405403\pi$
$293$	10.0249	0.585662	0.292831	+	0.956164i	$0.405403\pi$
$294$	0	0
$295$	27.4650	1.59907
$296$	0	0
$297$	8.99772	0.522101
$298$	0	0
$299$	9.70946	0.561512
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−3.91079	−0.224669
$304$	0	0
$305$	28.3166	1.62140
$306$	0	0
$307$	−31.0911	−1.77446	−0.887230	−	0.461327i	$-0.847373\pi$
$307$	−31.0911	−1.77446	−0.887230	+	0.461327i	$0.847373\pi$
$308$	0	0
$309$	−1.16230	−0.0661207
$310$	0	0
$311$	2.59990	0.147427	0.0737134	−	0.997279i	$-0.476515\pi$
$311$	2.59990	0.147427	0.0737134	+	0.997279i	$0.476515\pi$
$312$	0	0
$313$	14.5732	0.823726	0.411863	−	0.911246i	$-0.364878\pi$
$313$	14.5732	0.823726	0.411863	+	0.911246i	$0.364878\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	11.3005	0.634700	0.317350	−	0.948309i	$-0.397207\pi$
$317$	11.3005	0.634700	0.317350	+	0.948309i	$0.397207\pi$
$318$	0	0
$319$	−14.3656	−0.804319
$320$	0	0
$321$	−4.96613	−0.277182
$322$	0	0
$323$	−4.86063	−0.270453
$324$	0	0
$325$	−45.5509	−2.52671
$326$	0	0
$327$	−12.6198	−0.697877
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−15.3378	−0.843042	−0.421521	−	0.906819i	$-0.638504\pi$
$331$	−15.3378	−0.843042	−0.421521	+	0.906819i	$0.638504\pi$
$332$	0	0
$333$	−14.2453	−0.780637
$334$	0	0
$335$	−9.56882	−0.522801
$336$	0	0
$337$	7.62537	0.415380	0.207690	−	0.978195i	$-0.433405\pi$
$337$	7.62537	0.415380	0.207690	+	0.978195i	$0.433405\pi$
$338$	0	0
$339$	−12.2636	−0.666069
$340$	0	0
$341$	−4.03798	−0.218669
$342$	0	0
$343$	0	0
$344$	0	0
$345$	4.43509	0.238777
$346$	0	0
$347$	24.8757	1.33540	0.667700	−	0.744431i	$-0.267279\pi$
$347$	24.8757	1.33540	0.667700	+	0.744431i	$0.267279\pi$
$348$	0	0
$349$	−31.0832	−1.66384	−0.831922	−	0.554893i	$-0.812759\pi$
$349$	−31.0832	−1.66384	−0.831922	+	0.554893i	$0.812759\pi$
$350$	0	0
$351$	−20.6979	−1.10477
$352$	0	0
$353$	−14.7050	−0.782668	−0.391334	−	0.920249i	$-0.627986\pi$
$353$	−14.7050	−0.782668	−0.391334	+	0.920249i	$0.627986\pi$
$354$	0	0
$355$	40.7189	2.16114
$356$	0	0
$357$	0	0
$358$	0	0
$359$	20.0904	1.06033	0.530165	−	0.847895i	$-0.322130\pi$
$359$	20.0904	1.06033	0.530165	+	0.847895i	$0.322130\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	3.65936	0.192066
$364$	0	0
$365$	−53.0503	−2.77678
$366$	0	0
$367$	34.5138	1.80160	0.900802	−	0.434230i	$-0.142980\pi$
$367$	34.5138	1.80160	0.900802	+	0.434230i	$0.142980\pi$
$368$	0	0
$369$	−5.06377	−0.263609
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−27.5909	−1.42860	−0.714301	−	0.699838i	$-0.753255\pi$
$373$	−27.5909	−1.42860	−0.714301	+	0.699838i	$0.753255\pi$
$374$	0	0
$375$	−8.31701	−0.429489
$376$	0	0
$377$	33.0458	1.70195
$378$	0	0
$379$	9.40171	0.482933	0.241467	−	0.970409i	$-0.422372\pi$
$379$	9.40171	0.482933	0.241467	+	0.970409i	$0.422372\pi$
$380$	0	0
$381$	−11.4974	−0.589029
$382$	0	0
$383$	−14.3201	−0.731723	−0.365861	−	0.930669i	$-0.619226\pi$
$383$	−14.3201	−0.731723	−0.365861	+	0.930669i	$0.619226\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−7.87345	−0.400230
$388$	0	0
$389$	−1.63413	−0.0828538	−0.0414269	−	0.999142i	$-0.513190\pi$
$389$	−1.63413	−0.0828538	−0.0414269	+	0.999142i	$0.513190\pi$
$390$	0	0
$391$	−8.63002	−0.436439
$392$	0	0
$393$	−6.91468	−0.348799
$394$	0	0
$395$	3.51151	0.176683
$396$	0	0
$397$	−7.30979	−0.366868	−0.183434	−	0.983032i	$-0.558721\pi$
$397$	−7.30979	−0.366868	−0.183434	+	0.983032i	$0.558721\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	26.9972	1.34818	0.674088	−	0.738651i	$-0.264537\pi$
$401$	26.9972	1.34818	0.674088	+	0.738651i	$0.264537\pi$
$402$	0	0
$403$	9.28874	0.462705
$404$	0	0
$405$	18.2771	0.908196
$406$	0	0
$407$	13.3755	0.663000
$408$	0	0
$409$	−25.3639	−1.25416	−0.627082	−	0.778953i	$-0.715751\pi$
$409$	−25.3639	−1.25416	−0.627082	+	0.778953i	$0.715751\pi$
$410$	0	0
$411$	3.79868	0.187375
$412$	0	0
$413$	0	0
$414$	0	0
$415$	54.9382	2.69681
$416$	0	0
$417$	2.97706	0.145787
$418$	0	0
$419$	−8.43567	−0.412109	−0.206055	−	0.978540i	$-0.566062\pi$
$419$	−8.43567	−0.412109	−0.206055	+	0.978540i	$0.566062\pi$
$420$	0	0
$421$	10.5892	0.516087	0.258044	−	0.966133i	$-0.416922\pi$
$421$	10.5892	0.516087	0.258044	+	0.966133i	$0.416922\pi$
$422$	0	0
$423$	−13.2422	−0.643857
$424$	0	0
$425$	40.4868	1.96390
$426$	0	0
$427$	0	0
$428$	0	0
$429$	8.89479	0.429444
$430$	0	0
$431$	−29.0468	−1.39913	−0.699567	−	0.714567i	$-0.746624\pi$
$431$	−29.0468	−1.39913	−0.699567	+	0.714567i	$0.746624\pi$
$432$	0	0
$433$	15.4759	0.743725	0.371862	−	0.928288i	$-0.378719\pi$
$433$	15.4759	0.743725	0.371862	+	0.928288i	$0.378719\pi$
$434$	0	0
$435$	15.0947	0.723734
$436$	0	0
$437$	1.77549	0.0849333
$438$	0	0
$439$	24.6189	1.17500	0.587499	−	0.809225i	$-0.300113\pi$
$439$	24.6189	1.17500	0.587499	+	0.809225i	$0.300113\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8.44795	0.401374	0.200687	−	0.979655i	$-0.435683\pi$
$443$	8.44795	0.401374	0.200687	+	0.979655i	$0.435683\pi$
$444$	0	0
$445$	−43.9288	−2.08242
$446$	0	0
$447$	−8.30896	−0.393000
$448$	0	0
$449$	−12.8063	−0.604368	−0.302184	−	0.953250i	$-0.597716\pi$
$449$	−12.8063	−0.604368	−0.302184	+	0.953250i	$0.597716\pi$
$450$	0	0
$451$	4.75459	0.223885
$452$	0	0
$453$	10.8048	0.507652
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−28.5157	−1.33391	−0.666954	−	0.745098i	$-0.732402\pi$
$457$	−28.5157	−1.33391	−0.666954	+	0.745098i	$0.732402\pi$
$458$	0	0
$459$	18.3968	0.858688
$460$	0	0
$461$	37.8648	1.76354	0.881771	−	0.471679i	$-0.156352\pi$
$461$	37.8648	1.76354	0.881771	+	0.471679i	$0.156352\pi$
$462$	0	0
$463$	24.6922	1.14754	0.573772	−	0.819015i	$-0.305480\pi$
$463$	24.6922	1.14754	0.573772	+	0.819015i	$0.305480\pi$
$464$	0	0
$465$	4.24291	0.196760
$466$	0	0
$467$	2.34498	0.108513	0.0542563	−	0.998527i	$-0.482721\pi$
$467$	2.34498	0.108513	0.0542563	+	0.998527i	$0.482721\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	5.36292	0.247110
$472$	0	0
$473$	7.39272	0.339918
$474$	0	0
$475$	−8.32954	−0.382185
$476$	0	0
$477$	16.3690	0.749484
$478$	0	0
$479$	−29.3129	−1.33934	−0.669670	−	0.742659i	$-0.733564\pi$
$479$	−29.3129	−1.33934	−0.669670	+	0.742659i	$0.733564\pi$
$480$	0	0
$481$	−30.7683	−1.40291
$482$	0	0
$483$	0	0
$484$	0	0
$485$	22.8710	1.03852
$486$	0	0
$487$	14.0645	0.637321	0.318661	−	0.947869i	$-0.396767\pi$
$487$	14.0645	0.637321	0.318661	+	0.947869i	$0.396767\pi$
$488$	0	0
$489$	3.32835	0.150513
$490$	0	0
$491$	33.8667	1.52838	0.764191	−	0.644991i	$-0.223139\pi$
$491$	33.8667	1.52838	0.764191	+	0.644991i	$0.223139\pi$
$492$	0	0
$493$	−29.3720	−1.32285
$494$	0	0
$495$	−21.9753	−0.987717
$496$	0	0
$497$	0	0
$498$	0	0
$499$	26.2914	1.17696	0.588482	−	0.808511i	$-0.299726\pi$
$499$	26.2914	1.17696	0.588482	+	0.808511i	$0.299726\pi$
$500$	0	0
$501$	−9.35340	−0.417879
$502$	0	0
$503$	−36.7282	−1.63763	−0.818815	−	0.574058i	$-0.805368\pi$
$503$	−36.7282	−1.63763	−0.818815	+	0.574058i	$0.805368\pi$
$504$	0	0
$505$	20.8687	0.928646
$506$	0	0
$507$	−11.5666	−0.513691
$508$	0	0
$509$	−6.77695	−0.300383	−0.150192	−	0.988657i	$-0.547989\pi$
$509$	−6.77695	−0.300383	−0.150192	+	0.988657i	$0.547989\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−3.78485	−0.167105
$514$	0	0
$515$	6.20223	0.273303
$516$	0	0
$517$	12.4337	0.546832
$518$	0	0
$519$	−3.65977	−0.160646
$520$	0	0
$521$	31.3361	1.37286	0.686431	−	0.727195i	$-0.259176\pi$
$521$	31.3361	1.37286	0.686431	+	0.727195i	$0.259176\pi$
$522$	0	0
$523$	31.8391	1.39223	0.696114	−	0.717931i	$-0.254911\pi$
$523$	31.8391	1.39223	0.696114	+	0.717931i	$0.254911\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.25607	−0.359640
$528$	0	0
$529$	−19.8476	−0.862940
$530$	0	0
$531$	−19.0465	−0.826549
$532$	0	0
$533$	−10.9372	−0.473743
$534$	0	0
$535$	26.5002	1.14570
$536$	0	0
$537$	0.470602	0.0203080
$538$	0	0
$539$	0	0
$540$	0	0
$541$	8.96765	0.385549	0.192775	−	0.981243i	$-0.438251\pi$
$541$	8.96765	0.385549	0.192775	+	0.981243i	$0.438251\pi$
$542$	0	0
$543$	5.41459	0.232362
$544$	0	0
$545$	67.3417	2.88460
$546$	0	0
$547$	−26.0700	−1.11467	−0.557337	−	0.830287i	$-0.688177\pi$
$547$	−26.0700	−1.11467	−0.557337	+	0.830287i	$0.688177\pi$
$548$	0	0
$549$	−19.6371	−0.838092
$550$	0	0
$551$	6.04283	0.257433
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−14.0543	−0.596574
$556$	0	0
$557$	24.2751	1.02857	0.514284	−	0.857620i	$-0.328058\pi$
$557$	24.2751	1.02857	0.514284	+	0.857620i	$0.328058\pi$
$558$	0	0
$559$	−17.0058	−0.719268
$560$	0	0
$561$	−7.90592	−0.333788
$562$	0	0
$563$	−31.2151	−1.31556	−0.657779	−	0.753211i	$-0.728504\pi$
$563$	−31.2151	−1.31556	−0.657779	+	0.753211i	$0.728504\pi$
$564$	0	0
$565$	65.4411	2.75313
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−16.9648	−0.711203	−0.355602	−	0.934638i	$-0.615724\pi$
$569$	−16.9648	−0.711203	−0.355602	+	0.934638i	$0.615724\pi$
$570$	0	0
$571$	6.28139	0.262868	0.131434	−	0.991325i	$-0.458042\pi$
$571$	6.28139	0.262868	0.131434	+	0.991325i	$0.458042\pi$
$572$	0	0
$573$	−16.4818	−0.688538
$574$	0	0
$575$	−14.7890	−0.616745
$576$	0	0
$577$	16.2518	0.676573	0.338287	−	0.941043i	$-0.390153\pi$
$577$	16.2518	0.676573	0.338287	+	0.941043i	$0.390153\pi$
$578$	0	0
$579$	3.06361	0.127319
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−15.3695	−0.636541
$584$	0	0
$585$	50.5508	2.09002
$586$	0	0
$587$	45.7490	1.88826	0.944131	−	0.329571i	$-0.106904\pi$
$587$	45.7490	1.88826	0.944131	+	0.329571i	$0.106904\pi$
$588$	0	0
$589$	1.69856	0.0699879
$590$	0	0
$591$	−10.2818	−0.422936
$592$	0	0
$593$	−6.94183	−0.285067	−0.142533	−	0.989790i	$-0.545525\pi$
$593$	−6.94183	−0.285067	−0.142533	+	0.989790i	$0.545525\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−18.7410	−0.767016
$598$	0	0
$599$	14.5903	0.596142	0.298071	−	0.954544i	$-0.403657\pi$
$599$	14.5903	0.596142	0.298071	+	0.954544i	$0.403657\pi$
$600$	0	0
$601$	43.5586	1.77679	0.888396	−	0.459077i	$-0.151820\pi$
$601$	43.5586	1.77679	0.888396	+	0.459077i	$0.151820\pi$
$602$	0	0
$603$	6.63583	0.270232
$604$	0	0
$605$	−19.5270	−0.793887
$606$	0	0
$607$	23.4431	0.951526	0.475763	−	0.879573i	$-0.342172\pi$
$607$	23.4431	0.951526	0.475763	+	0.879573i	$0.342172\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−28.6017	−1.15710
$612$	0	0
$613$	−3.58627	−0.144848	−0.0724240	−	0.997374i	$-0.523073\pi$
$613$	−3.58627	−0.144848	−0.0724240	+	0.997374i	$0.523073\pi$
$614$	0	0
$615$	−4.99590	−0.201454
$616$	0	0
$617$	5.93981	0.239128	0.119564	−	0.992827i	$-0.461850\pi$
$617$	5.93981	0.239128	0.119564	+	0.992827i	$0.461850\pi$
$618$	0	0
$619$	−27.5236	−1.10627	−0.553134	−	0.833093i	$-0.686568\pi$
$619$	−27.5236	−1.10627	−0.553134	+	0.833093i	$0.686568\pi$
$620$	0	0
$621$	−6.71998	−0.269663
$622$	0	0
$623$	0	0
$624$	0	0
$625$	2.73349	0.109340
$626$	0	0
$627$	1.62652	0.0649569
$628$	0	0
$629$	27.3476	1.09042
$630$	0	0
$631$	−45.2443	−1.80115	−0.900573	−	0.434705i	$-0.856853\pi$
$631$	−45.2443	−1.80115	−0.900573	+	0.434705i	$0.856853\pi$
$632$	0	0
$633$	0.941386	0.0374167
$634$	0	0
$635$	61.3523	2.43469
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−28.2379	−1.11708
$640$	0	0
$641$	−28.4998	−1.12568	−0.562838	−	0.826567i	$-0.690290\pi$
$641$	−28.4998	−1.12568	−0.562838	+	0.826567i	$0.690290\pi$
$642$	0	0
$643$	22.8611	0.901553	0.450776	−	0.892637i	$-0.351147\pi$
$643$	22.8611	0.901553	0.450776	+	0.892637i	$0.351147\pi$
$644$	0	0
$645$	−7.76791	−0.305861
$646$	0	0
$647$	24.0834	0.946816	0.473408	−	0.880843i	$-0.343024\pi$
$647$	24.0834	0.946816	0.473408	+	0.880843i	$0.343024\pi$
$648$	0	0
$649$	17.8836	0.701993
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.05259	0.158590	0.0792950	−	0.996851i	$-0.474733\pi$
$653$	4.05259	0.158590	0.0792950	+	0.996851i	$0.474733\pi$
$654$	0	0
$655$	36.8980	1.44173
$656$	0	0
$657$	36.7896	1.43530
$658$	0	0
$659$	23.7165	0.923864	0.461932	−	0.886915i	$-0.347156\pi$
$659$	23.7165	0.923864	0.461932	+	0.886915i	$0.347156\pi$
$660$	0	0
$661$	17.2174	0.669681	0.334840	−	0.942275i	$-0.391318\pi$
$661$	17.2174	0.669681	0.334840	+	0.942275i	$0.391318\pi$
$662$	0	0
$663$	18.1863	0.706298
$664$	0	0
$665$	0	0
$666$	0	0
$667$	10.7290	0.415428
$668$	0	0
$669$	19.0055	0.734796
$670$	0	0
$671$	18.4381	0.711796
$672$	0	0
$673$	−20.1890	−0.778229	−0.389115	−	0.921189i	$-0.627219\pi$
$673$	−20.1890	−0.778229	−0.389115	+	0.921189i	$0.627219\pi$
$674$	0	0
$675$	31.5261	1.21344
$676$	0	0
$677$	−23.4527	−0.901362	−0.450681	−	0.892685i	$-0.648819\pi$
$677$	−23.4527	−0.901362	−0.450681	+	0.892685i	$0.648819\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	14.0878	0.539846
$682$	0	0
$683$	25.7238	0.984295	0.492148	−	0.870512i	$-0.336212\pi$
$683$	25.7238	0.984295	0.492148	+	0.870512i	$0.336212\pi$
$684$	0	0
$685$	−20.2705	−0.774496
$686$	0	0
$687$	17.8975	0.682832
$688$	0	0
$689$	35.3552	1.34693
$690$	0	0
$691$	−50.8091	−1.93287	−0.966434	−	0.256915i	$-0.917294\pi$
$691$	−50.8091	−1.93287	−0.966434	+	0.256915i	$0.917294\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−15.8861	−0.602596
$696$	0	0
$697$	9.72127	0.368219
$698$	0	0
$699$	8.61018	0.325667
$700$	0	0
$701$	46.4663	1.75501	0.877504	−	0.479570i	$-0.159207\pi$
$701$	46.4663	1.75501	0.877504	+	0.479570i	$0.159207\pi$
$702$	0	0
$703$	−5.62635	−0.212202
$704$	0	0
$705$	−13.0647	−0.492045
$706$	0	0
$707$	0	0
$708$	0	0
$709$	2.92358	0.109797	0.0548986	−	0.998492i	$-0.482516\pi$
$709$	2.92358	0.109797	0.0548986	+	0.998492i	$0.482516\pi$
$710$	0	0
$711$	−2.43518	−0.0913263
$712$	0	0
$713$	3.01578	0.112942
$714$	0	0
$715$	−47.4643	−1.77506
$716$	0	0
$717$	−19.5573	−0.730379
$718$	0	0
$719$	−29.6006	−1.10392	−0.551959	−	0.833871i	$-0.686119\pi$
$719$	−29.6006	−1.10392	−0.551959	+	0.833871i	$0.686119\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	3.17837	0.118205
$724$	0	0
$725$	−50.3340	−1.86936
$726$	0	0
$727$	−39.1285	−1.45119	−0.725597	−	0.688120i	$-0.758436\pi$
$727$	−39.1285	−1.45119	−0.725597	+	0.688120i	$0.758436\pi$
$728$	0	0
$729$	−4.90622	−0.181712
$730$	0	0
$731$	15.1152	0.559055
$732$	0	0
$733$	−14.6944	−0.542750	−0.271375	−	0.962474i	$-0.587478\pi$
$733$	−14.6944	−0.542750	−0.271375	+	0.962474i	$0.587478\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6.23067	−0.229510
$738$	0	0
$739$	16.1624	0.594542	0.297271	−	0.954793i	$-0.403924\pi$
$739$	16.1624	0.594542	0.297271	+	0.954793i	$0.403924\pi$
$740$	0	0
$741$	−3.74155	−0.137449
$742$	0	0
$743$	21.2412	0.779265	0.389632	−	0.920971i	$-0.372602\pi$
$743$	21.2412	0.779265	0.389632	+	0.920971i	$0.372602\pi$
$744$	0	0
$745$	44.3382	1.62443
$746$	0	0
$747$	−38.0988	−1.39396
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−52.0353	−1.89880	−0.949398	−	0.314075i	$-0.898306\pi$
$751$	−52.0353	−1.89880	−0.949398	+	0.314075i	$0.898306\pi$
$752$	0	0
$753$	−8.27544	−0.301574
$754$	0	0
$755$	−57.6563	−2.09833
$756$	0	0
$757$	30.6379	1.11355	0.556777	−	0.830662i	$-0.312038\pi$
$757$	30.6379	1.11355	0.556777	+	0.830662i	$0.312038\pi$
$758$	0	0
$759$	2.88787	0.104823
$760$	0	0
$761$	19.5926	0.710230	0.355115	−	0.934823i	$-0.384442\pi$
$761$	19.5926	0.710230	0.355115	+	0.934823i	$0.384442\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−44.9308	−1.62448
$766$	0	0
$767$	−41.1384	−1.48542
$768$	0	0
$769$	6.70349	0.241734	0.120867	−	0.992669i	$-0.461433\pi$
$769$	6.70349	0.241734	0.120867	+	0.992669i	$0.461433\pi$
$770$	0	0
$771$	1.29475	0.0466293
$772$	0	0
$773$	13.5509	0.487394	0.243697	−	0.969851i	$-0.421640\pi$
$773$	13.5509	0.487394	0.243697	+	0.969851i	$0.421640\pi$
$774$	0	0
$775$	−14.1482	−0.508219
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.00000	−0.0716574
$780$	0	0
$781$	26.5138	0.948739
$782$	0	0
$783$	−22.8712	−0.817351
$784$	0	0
$785$	−28.6176	−1.02140
$786$	0	0
$787$	36.3319	1.29509	0.647547	−	0.762026i	$-0.275795\pi$
$787$	36.3319	1.29509	0.647547	+	0.762026i	$0.275795\pi$
$788$	0	0
$789$	−15.8596	−0.564616
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−42.4141	−1.50617
$794$	0	0
$795$	16.1496	0.572766
$796$	0	0
$797$	55.0232	1.94902	0.974511	−	0.224338i	$-0.0720219\pi$
$797$	55.0232	1.94902	0.974511	+	0.224338i	$0.0720219\pi$
$798$	0	0
$799$	25.4219	0.899363
$800$	0	0
$801$	30.4639	1.07639
$802$	0	0
$803$	−34.5433	−1.21901
$804$	0	0
$805$	0	0
$806$	0	0
$807$	16.6246	0.585213
$808$	0	0
$809$	−27.1162	−0.953354	−0.476677	−	0.879079i	$-0.658159\pi$
$809$	−27.1162	−0.953354	−0.476677	+	0.879079i	$0.658159\pi$
$810$	0	0
$811$	20.7862	0.729903	0.364952	−	0.931027i	$-0.381086\pi$
$811$	20.7862	0.729903	0.364952	+	0.931027i	$0.381086\pi$
$812$	0	0
$813$	8.49052	0.297776
$814$	0	0
$815$	−17.7607	−0.622130
$816$	0	0
$817$	−3.10972	−0.108795
$818$	0	0
$819$	0	0
$820$	0	0
$821$	42.1674	1.47165	0.735826	−	0.677170i	$-0.236794\pi$
$821$	42.1674	1.47165	0.735826	+	0.677170i	$0.236794\pi$
$822$	0	0
$823$	27.9529	0.974377	0.487189	−	0.873297i	$-0.338022\pi$
$823$	27.9529	0.974377	0.487189	+	0.873297i	$0.338022\pi$
$824$	0	0
$825$	−13.5482	−0.471686
$826$	0	0
$827$	−46.6553	−1.62236	−0.811182	−	0.584794i	$-0.801175\pi$
$827$	−46.6553	−1.62236	−0.811182	+	0.584794i	$0.801175\pi$
$828$	0	0
$829$	29.5816	1.02741	0.513706	−	0.857966i	$-0.328272\pi$
$829$	29.5816	1.02741	0.513706	+	0.857966i	$0.328272\pi$
$830$	0	0
$831$	1.59962	0.0554901
$832$	0	0
$833$	0	0
$834$	0	0
$835$	49.9116	1.72726
$836$	0	0
$837$	−6.42880	−0.222212
$838$	0	0
$839$	27.0614	0.934263	0.467132	−	0.884188i	$-0.345287\pi$
$839$	27.0614	0.934263	0.467132	+	0.884188i	$0.345287\pi$
$840$	0	0
$841$	7.51580	0.259166
$842$	0	0
$843$	−2.50116	−0.0861446
$844$	0	0
$845$	61.7217	2.12329
$846$	0	0
$847$	0	0
$848$	0	0
$849$	6.55778	0.225062
$850$	0	0
$851$	−9.98955	−0.342437
$852$	0	0
$853$	1.19742	0.0409989	0.0204995	−	0.999790i	$-0.493474\pi$
$853$	1.19742	0.0409989	0.0204995	+	0.999790i	$0.493474\pi$
$854$	0	0
$855$	9.24382	0.316132
$856$	0	0
$857$	−34.4417	−1.17651	−0.588253	−	0.808677i	$-0.700184\pi$
$857$	−34.4417	−1.17651	−0.588253	+	0.808677i	$0.700184\pi$
$858$	0	0
$859$	3.74549	0.127794	0.0638972	−	0.997956i	$-0.479647\pi$
$859$	3.74549	0.127794	0.0638972	+	0.997956i	$0.479647\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−29.8671	−1.01669	−0.508343	−	0.861154i	$-0.669742\pi$
$863$	−29.8671	−1.01669	−0.508343	+	0.861154i	$0.669742\pi$
$864$	0	0
$865$	19.5293	0.664015
$866$	0	0
$867$	−4.53327	−0.153958
$868$	0	0
$869$	2.28649	0.0775639
$870$	0	0
$871$	14.3327	0.485644
$872$	0	0
$873$	−15.8607	−0.536803
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−9.48210	−0.320188	−0.160094	−	0.987102i	$-0.551180\pi$
$877$	−9.48210	−0.320188	−0.160094	+	0.987102i	$0.551180\pi$
$878$	0	0
$879$	−6.85894	−0.231346
$880$	0	0
$881$	−33.1936	−1.11832	−0.559160	−	0.829060i	$-0.688876\pi$
$881$	−33.1936	−1.11832	−0.559160	+	0.829060i	$0.688876\pi$
$882$	0	0
$883$	−27.9645	−0.941079	−0.470540	−	0.882379i	$-0.655941\pi$
$883$	−27.9645	−0.941079	−0.470540	+	0.882379i	$0.655941\pi$
$884$	0	0
$885$	−18.7912	−0.631660
$886$	0	0
$887$	−31.3624	−1.05305	−0.526523	−	0.850161i	$-0.676505\pi$
$887$	−31.3624	−1.05305	−0.526523	+	0.850161i	$0.676505\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	11.9010	0.398698
$892$	0	0
$893$	−5.23017	−0.175021
$894$	0	0
$895$	−2.51122	−0.0839409
$896$	0	0
$897$	−6.64310	−0.221807
$898$	0	0
$899$	10.2641	0.342327
$900$	0	0
$901$	−31.4246	−1.04691
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−28.8933	−0.960446
$906$	0	0
$907$	8.73561	0.290061	0.145031	−	0.989427i	$-0.453672\pi$
$907$	8.73561	0.290061	0.145031	+	0.989427i	$0.453672\pi$
$908$	0	0
$909$	−14.4721	−0.480011
$910$	0	0
$911$	6.50534	0.215532	0.107766	−	0.994176i	$-0.465630\pi$
$911$	6.50534	0.215532	0.107766	+	0.994176i	$0.465630\pi$
$912$	0	0
$913$	35.7726	1.18390
$914$	0	0
$915$	−19.3739	−0.640481
$916$	0	0
$917$	0	0
$918$	0	0
$919$	32.4502	1.07043	0.535216	−	0.844715i	$-0.320230\pi$
$919$	32.4502	1.07043	0.535216	+	0.844715i	$0.320230\pi$
$920$	0	0
$921$	21.2722	0.700941
$922$	0	0
$923$	−60.9909	−2.00754
$924$	0	0
$925$	46.8649	1.54091
$926$	0	0
$927$	−4.30115	−0.141268
$928$	0	0
$929$	21.9512	0.720196	0.360098	−	0.932914i	$-0.382743\pi$
$929$	21.9512	0.720196	0.360098	+	0.932914i	$0.382743\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−1.77882	−0.0582360
$934$	0	0
$935$	42.1875	1.37968
$936$	0	0
$937$	20.0361	0.654552	0.327276	−	0.944929i	$-0.393869\pi$
$937$	20.0361	0.654552	0.327276	+	0.944929i	$0.393869\pi$
$938$	0	0
$939$	−9.97083	−0.325386
$940$	0	0
$941$	6.51229	0.212295	0.106147	−	0.994350i	$-0.466148\pi$
$941$	6.51229	0.212295	0.106147	+	0.994350i	$0.466148\pi$
$942$	0	0
$943$	−3.55098	−0.115636
$944$	0	0
$945$	0	0
$946$	0	0
$947$	56.1061	1.82320	0.911602	−	0.411073i	$-0.134846\pi$
$947$	56.1061	1.82320	0.911602	+	0.411073i	$0.134846\pi$
$948$	0	0
$949$	79.4615	2.57943
$950$	0	0
$951$	−7.73168	−0.250717
$952$	0	0
$953$	23.9841	0.776920	0.388460	−	0.921466i	$-0.373007\pi$
$953$	23.9841	0.776920	0.388460	+	0.921466i	$0.373007\pi$
$954$	0	0
$955$	87.9502	2.84600
$956$	0	0
$957$	9.82878	0.317719
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−28.1149	−0.906932
$962$	0	0
$963$	−18.3775	−0.592206
$964$	0	0
$965$	−16.3480	−0.526261
$966$	0	0
$967$	25.6539	0.824974	0.412487	−	0.910963i	$-0.364660\pi$
$967$	25.6539	0.824974	0.412487	+	0.910963i	$0.364660\pi$
$968$	0	0
$969$	3.32559	0.106833
$970$	0	0
$971$	−7.50346	−0.240797	−0.120399	−	0.992726i	$-0.538417\pi$
$971$	−7.50346	−0.240797	−0.120399	+	0.992726i	$0.538417\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	31.1654	0.998092
$976$	0	0
$977$	−58.6541	−1.87651	−0.938255	−	0.345943i	$-0.887559\pi$
$977$	−58.6541	−1.87651	−0.938255	+	0.345943i	$0.887559\pi$
$978$	0	0
$979$	−28.6039	−0.914185
$980$	0	0
$981$	−46.7004	−1.49103
$982$	0	0
$983$	16.3863	0.522640	0.261320	−	0.965252i	$-0.415842\pi$
$983$	16.3863	0.522640	0.261320	+	0.965252i	$0.415842\pi$
$984$	0	0
$985$	54.8655	1.74816
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−5.52128	−0.175566
$990$	0	0
$991$	−45.0420	−1.43080	−0.715402	−	0.698713i	$-0.753757\pi$
$991$	−45.0420	−1.43080	−0.715402	+	0.698713i	$0.753757\pi$
$992$	0	0
$993$	10.4940	0.333015
$994$	0	0
$995$	100.005	3.17038
$996$	0	0
$997$	3.40836	0.107944	0.0539719	−	0.998542i	$-0.482812\pi$
$997$	3.40836	0.107944	0.0539719	+	0.998542i	$0.482812\pi$
$998$	0	0
$999$	21.2949	0.673742

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.br.1.4		8
7.3	odd	6		1064.2.q.n.457.4	yes	16
7.5	odd	6		1064.2.q.n.305.4	✓	16
7.6	odd	2		7448.2.a.bq.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1064.2.q.n.305.4	✓	16	7.5	odd	6
1064.2.q.n.457.4	yes	16	7.3	odd	6
7448.2.a.bq.1.5		8	7.6	odd	2
7448.2.a.br.1.4		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7448.a (trivial)

\(f(q)\)	\(=\)	\(q-0.684189 q^{3} +3.65096 q^{5} -2.53189 q^{9} +O(q^{10})\)	\(q-0.684189 q^{3} +3.65096 q^{5} -2.53189 q^{9} +2.37730 q^{11} -5.46860 q^{13} -2.49795 q^{15} +4.86063 q^{17} -1.00000 q^{19} -1.77549 q^{23} +8.32954 q^{25} +3.78485 q^{27} -6.04283 q^{29} -1.69856 q^{31} -1.62652 q^{33} +5.62635 q^{37} +3.74155 q^{39} +2.00000 q^{41} +3.10972 q^{43} -9.24382 q^{45} +5.23017 q^{47} -3.32559 q^{51} -6.46513 q^{53} +8.67942 q^{55} +0.684189 q^{57} +7.52266 q^{59} +7.75592 q^{61} -19.9657 q^{65} -2.62090 q^{67} +1.21477 q^{69} +11.1529 q^{71} -14.5305 q^{73} -5.69898 q^{75} +0.961803 q^{79} +5.00610 q^{81} +15.0476 q^{83} +17.7460 q^{85} +4.13444 q^{87} -12.0321 q^{89} +1.16213 q^{93} -3.65096 q^{95} +6.26438 q^{97} -6.01904 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{5} + 6 q^{9}+O(q^{10}) \) 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

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