LMFDB - Embedded newform 7448.2.a.bw.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:	$14$
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} - 2 x^{13} - 27 x^{12} + 46 x^{11} + 286 x^{10} - 386 x^{9} - 1525 x^{8} + 1414 x^{7} + \cdots + 392 $ x^14 - 2x^13 - 27x^12 + 46x^11 + 286x^10 - 386x^9 - 1525x^8 + 1414x^7 + 4390x^6 - 1910x^5 - 6603x^4 - 610x^3 + 4031x^2 + 2352*x + 392
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.516408$ 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)

$f(q)$	$=$	$q-0.516408 q^{3} -0.565648 q^{5} -2.73332 q^{9} +O(q^{10})$	$q-0.516408 q^{3} -0.565648 q^{5} -2.73332 q^{9} -0.707288 q^{11} -5.36821 q^{13} +0.292105 q^{15} -2.10234 q^{17} -1.00000 q^{19} -4.72505 q^{23} -4.68004 q^{25} +2.96073 q^{27} -2.25956 q^{29} -4.11046 q^{31} +0.365249 q^{33} -7.37919 q^{37} +2.77219 q^{39} +5.36541 q^{41} -2.60561 q^{43} +1.54610 q^{45} +9.55271 q^{47} +1.08566 q^{51} -4.05382 q^{53} +0.400076 q^{55} +0.516408 q^{57} -3.50916 q^{59} +4.72438 q^{61} +3.03652 q^{65} +6.24278 q^{67} +2.44005 q^{69} +10.0237 q^{71} +15.1146 q^{73} +2.41681 q^{75} -4.15927 q^{79} +6.67103 q^{81} +3.44346 q^{83} +1.18918 q^{85} +1.16685 q^{87} -10.7383 q^{89} +2.12267 q^{93} +0.565648 q^{95} -2.03910 q^{97} +1.93325 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 2 q^{3} + 2 q^{5} + 16 q^{9}+O(q^{10}) $ 14 * q + 2 * q^3 + 2 * q^5 + 16 * q^9	$ 14 q + 2 q^{3} + 2 q^{5} + 16 q^{9} - 6 q^{11} + 16 q^{13} - 4 q^{15} + 20 q^{17} - 14 q^{19} + 4 q^{23} + 16 q^{25} + 20 q^{27} + 6 q^{29} + 34 q^{33} - 6 q^{37} + 8 q^{39} + 46 q^{41} - 18 q^{43} - 10 q^{47} - 4 q^{51} - 2 q^{53} + 28 q^{55} - 2 q^{57} - 22 q^{59} + 26 q^{61} + 8 q^{65} - 12 q^{67} + 48 q^{69} + 18 q^{71} + 28 q^{73} - 24 q^{75} - 10 q^{79} - 2 q^{81} + 8 q^{83} - 16 q^{85} + 16 q^{87} + 78 q^{89} - 2 q^{95} + 54 q^{97} + 26 q^{99}+O(q^{100}) $ 14 * q + 2 * q^3 + 2 * q^5 + 16 * q^9 - 6 * q^11 + 16 * q^13 - 4 * q^15 + 20 * q^17 - 14 * q^19 + 4 * q^23 + 16 * q^25 + 20 * q^27 + 6 * q^29 + 34 * q^33 - 6 * q^37 + 8 * q^39 + 46 * q^41 - 18 * q^43 - 10 * q^47 - 4 * q^51 - 2 * q^53 + 28 * q^55 - 2 * q^57 - 22 * q^59 + 26 * q^61 + 8 * q^65 - 12 * q^67 + 48 * q^69 + 18 * q^71 + 28 * q^73 - 24 * q^75 - 10 * q^79 - 2 * q^81 + 8 * q^83 - 16 * q^85 + 16 * q^87 + 78 * q^89 - 2 * q^95 + 54 * q^97 + 26 * 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
$3$	−0.516408	−0.298148	−0.149074	−	0.988826i	$-0.547629\pi$
$3$	−0.516408	−0.298148	−0.149074	+	0.988826i	$0.547629\pi$
$4$	0	0
$5$	−0.565648	−0.252965	−0.126483	−	0.991969i	$-0.540369\pi$
$5$	−0.565648	−0.252965	−0.126483	+	0.991969i	$0.540369\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.73332	−0.911108
$10$	0	0
$11$	−0.707288	−0.213255	−0.106628	−	0.994299i	$-0.534005\pi$
$11$	−0.707288	−0.213255	−0.106628	+	0.994299i	$0.534005\pi$
$12$	0	0
$13$	−5.36821	−1.48887	−0.744437	−	0.667692i	$-0.767282\pi$
$13$	−5.36821	−1.48887	−0.744437	+	0.667692i	$0.767282\pi$
$14$	0	0
$15$	0.292105	0.0754212
$16$	0	0
$17$	−2.10234	−0.509892	−0.254946	−	0.966955i	$-0.582058\pi$
$17$	−2.10234	−0.509892	−0.254946	+	0.966955i	$0.582058\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.72505	−0.985240	−0.492620	−	0.870244i	$-0.663961\pi$
$23$	−4.72505	−0.985240	−0.492620	+	0.870244i	$0.663961\pi$
$24$	0	0
$25$	−4.68004	−0.936008
$26$	0	0
$27$	2.96073	0.569793
$28$	0	0
$29$	−2.25956	−0.419590	−0.209795	−	0.977745i	$-0.567280\pi$
$29$	−2.25956	−0.419590	−0.209795	+	0.977745i	$0.567280\pi$
$30$	0	0
$31$	−4.11046	−0.738260	−0.369130	−	0.929378i	$-0.620344\pi$
$31$	−4.11046	−0.738260	−0.369130	+	0.929378i	$0.620344\pi$
$32$	0	0
$33$	0.365249	0.0635817
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−7.37919	−1.21313	−0.606566	−	0.795033i	$-0.707453\pi$
$37$	−7.37919	−1.21313	−0.606566	+	0.795033i	$0.707453\pi$
$38$	0	0
$39$	2.77219	0.443905
$40$	0	0
$41$	5.36541	0.837936	0.418968	−	0.908001i	$-0.362392\pi$
$41$	5.36541	0.837936	0.418968	+	0.908001i	$0.362392\pi$
$42$	0	0
$43$	−2.60561	−0.397352	−0.198676	−	0.980065i	$-0.563664\pi$
$43$	−2.60561	−0.397352	−0.198676	+	0.980065i	$0.563664\pi$
$44$	0	0
$45$	1.54610	0.230479
$46$	0	0
$47$	9.55271	1.39341	0.696703	−	0.717360i	$-0.254650\pi$
$47$	9.55271	1.39341	0.696703	+	0.717360i	$0.254650\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	1.08566	0.152023
$52$	0	0
$53$	−4.05382	−0.556835	−0.278418	−	0.960460i	$-0.589810\pi$
$53$	−4.05382	−0.556835	−0.278418	+	0.960460i	$0.589810\pi$
$54$	0	0
$55$	0.400076	0.0539462
$56$	0	0
$57$	0.516408	0.0683999
$58$	0	0
$59$	−3.50916	−0.456854	−0.228427	−	0.973561i	$-0.573358\pi$
$59$	−3.50916	−0.456854	−0.228427	+	0.973561i	$0.573358\pi$
$60$	0	0
$61$	4.72438	0.604894	0.302447	−	0.953166i	$-0.402196\pi$
$61$	4.72438	0.604894	0.302447	+	0.953166i	$0.402196\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.03652	0.376634
$66$	0	0
$67$	6.24278	0.762677	0.381339	−	0.924435i	$-0.375463\pi$
$67$	6.24278	0.762677	0.381339	+	0.924435i	$0.375463\pi$
$68$	0	0
$69$	2.44005	0.293748
$70$	0	0
$71$	10.0237	1.18959	0.594797	−	0.803876i	$-0.297232\pi$
$71$	10.0237	1.18959	0.594797	+	0.803876i	$0.297232\pi$
$72$	0	0
$73$	15.1146	1.76903	0.884514	−	0.466514i	$-0.154490\pi$
$73$	15.1146	1.76903	0.884514	+	0.466514i	$0.154490\pi$
$74$	0	0
$75$	2.41681	0.279069
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−4.15927	−0.467954	−0.233977	−	0.972242i	$-0.575174\pi$
$79$	−4.15927	−0.467954	−0.233977	+	0.972242i	$0.575174\pi$
$80$	0	0
$81$	6.67103	0.741225
$82$	0	0
$83$	3.44346	0.377969	0.188984	−	0.981980i	$-0.439480\pi$
$83$	3.44346	0.377969	0.188984	+	0.981980i	$0.439480\pi$
$84$	0	0
$85$	1.18918	0.128985
$86$	0	0
$87$	1.16685	0.125100
$88$	0	0
$89$	−10.7383	−1.13825	−0.569126	−	0.822250i	$-0.692718\pi$
$89$	−10.7383	−1.13825	−0.569126	+	0.822250i	$0.692718\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	2.12267	0.220111
$94$	0	0
$95$	0.565648	0.0580343
$96$	0	0
$97$	−2.03910	−0.207039	−0.103519	−	0.994627i	$-0.533010\pi$
$97$	−2.03910	−0.207039	−0.103519	+	0.994627i	$0.533010\pi$
$98$	0	0
$99$	1.93325	0.194299
$100$	0	0
$101$	−3.69110	−0.367278	−0.183639	−	0.982994i	$-0.558788\pi$
$101$	−3.69110	−0.367278	−0.183639	+	0.982994i	$0.558788\pi$
$102$	0	0
$103$	−12.8001	−1.26123	−0.630617	−	0.776094i	$-0.717198\pi$
$103$	−12.8001	−1.26123	−0.630617	+	0.776094i	$0.717198\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.76648	−0.364120	−0.182060	−	0.983287i	$-0.558276\pi$
$107$	−3.76648	−0.364120	−0.182060	+	0.983287i	$0.558276\pi$
$108$	0	0
$109$	−10.7408	−1.02878	−0.514392	−	0.857555i	$-0.671982\pi$
$109$	−10.7408	−1.02878	−0.514392	+	0.857555i	$0.671982\pi$
$110$	0	0
$111$	3.81067	0.361693
$112$	0	0
$113$	7.84583	0.738074	0.369037	−	0.929415i	$-0.379688\pi$
$113$	7.84583	0.738074	0.369037	+	0.929415i	$0.379688\pi$
$114$	0	0
$115$	2.67271	0.249232
$116$	0	0
$117$	14.6731	1.35653
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.4997	−0.954522
$122$	0	0
$123$	−2.77074	−0.249829
$124$	0	0
$125$	5.47550	0.489743
$126$	0	0
$127$	5.85502	0.519549	0.259775	−	0.965669i	$-0.416352\pi$
$127$	5.85502	0.519549	0.259775	+	0.965669i	$0.416352\pi$
$128$	0	0
$129$	1.34556	0.118470
$130$	0	0
$131$	−12.9141	−1.12831	−0.564156	−	0.825668i	$-0.690798\pi$
$131$	−12.9141	−1.12831	−0.564156	+	0.825668i	$0.690798\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.67473	−0.144138
$136$	0	0
$137$	11.1783	0.955029	0.477515	−	0.878624i	$-0.341538\pi$
$137$	11.1783	0.955029	0.477515	+	0.878624i	$0.341538\pi$
$138$	0	0
$139$	22.8377	1.93707	0.968534	−	0.248883i	$-0.0800633\pi$
$139$	22.8377	1.93707	0.968534	+	0.248883i	$0.0800633\pi$
$140$	0	0
$141$	−4.93309	−0.415441
$142$	0	0
$143$	3.79687	0.317510
$144$	0	0
$145$	1.27812	0.106142
$146$	0	0
$147$	0	0
$148$	0	0
$149$	16.1398	1.32223	0.661114	−	0.750286i	$-0.270084\pi$
$149$	16.1398	1.32223	0.661114	+	0.750286i	$0.270084\pi$
$150$	0	0
$151$	−0.893803	−0.0727366	−0.0363683	−	0.999338i	$-0.511579\pi$
$151$	−0.893803	−0.0727366	−0.0363683	+	0.999338i	$0.511579\pi$
$152$	0	0
$153$	5.74637	0.464567
$154$	0	0
$155$	2.32507	0.186754
$156$	0	0
$157$	6.99355	0.558146	0.279073	−	0.960270i	$-0.409973\pi$
$157$	6.99355	0.558146	0.279073	+	0.960270i	$0.409973\pi$
$158$	0	0
$159$	2.09343	0.166019
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−19.4290	−1.52180	−0.760898	−	0.648871i	$-0.775241\pi$
$163$	−19.4290	−1.52180	−0.760898	+	0.648871i	$0.775241\pi$
$164$	0	0
$165$	−0.206602	−0.0160840
$166$	0	0
$167$	1.23293	0.0954070	0.0477035	−	0.998862i	$-0.484810\pi$
$167$	1.23293	0.0954070	0.0477035	+	0.998862i	$0.484810\pi$
$168$	0	0
$169$	15.8177	1.21675
$170$	0	0
$171$	2.73332	0.209022
$172$	0	0
$173$	6.93237	0.527058	0.263529	−	0.964651i	$-0.415113\pi$
$173$	6.93237	0.527058	0.263529	+	0.964651i	$0.415113\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.81216	0.136210
$178$	0	0
$179$	21.1803	1.58309	0.791545	−	0.611110i	$-0.209277\pi$
$179$	21.1803	1.58309	0.791545	+	0.611110i	$0.209277\pi$
$180$	0	0
$181$	1.31687	0.0978824	0.0489412	−	0.998802i	$-0.484415\pi$
$181$	1.31687	0.0978824	0.0489412	+	0.998802i	$0.484415\pi$
$182$	0	0
$183$	−2.43970	−0.180348
$184$	0	0
$185$	4.17402	0.306880
$186$	0	0
$187$	1.48696	0.108737
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−24.3386	−1.76108	−0.880541	−	0.473969i	$-0.842821\pi$
$191$	−24.3386	−1.76108	−0.880541	+	0.473969i	$0.842821\pi$
$192$	0	0
$193$	−8.00734	−0.576381	−0.288191	−	0.957573i	$-0.593054\pi$
$193$	−8.00734	−0.576381	−0.288191	+	0.957573i	$0.593054\pi$
$194$	0	0
$195$	−1.56808	−0.112293
$196$	0	0
$197$	−18.1729	−1.29476	−0.647382	−	0.762166i	$-0.724136\pi$
$197$	−18.1729	−1.29476	−0.647382	+	0.762166i	$0.724136\pi$
$198$	0	0
$199$	4.76164	0.337543	0.168772	−	0.985655i	$-0.446020\pi$
$199$	4.76164	0.337543	0.168772	+	0.985655i	$0.446020\pi$
$200$	0	0
$201$	−3.22382	−0.227391
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−3.03493	−0.211969
$206$	0	0
$207$	12.9151	0.897660
$208$	0	0
$209$	0.707288	0.0489241
$210$	0	0
$211$	−5.50148	−0.378738	−0.189369	−	0.981906i	$-0.560644\pi$
$211$	−5.50148	−0.378738	−0.189369	+	0.981906i	$0.560644\pi$
$212$	0	0
$213$	−5.17632	−0.354675
$214$	0	0
$215$	1.47386	0.100516
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−7.80528	−0.527432
$220$	0	0
$221$	11.2858	0.759166
$222$	0	0
$223$	7.59742	0.508761	0.254381	−	0.967104i	$-0.418128\pi$
$223$	7.59742	0.508761	0.254381	+	0.967104i	$0.418128\pi$
$224$	0	0
$225$	12.7921	0.852805
$226$	0	0
$227$	12.2183	0.810955	0.405477	−	0.914105i	$-0.367105\pi$
$227$	12.2183	0.810955	0.405477	+	0.914105i	$0.367105\pi$
$228$	0	0
$229$	28.5876	1.88912	0.944560	−	0.328338i	$-0.106489\pi$
$229$	28.5876	1.88912	0.944560	+	0.328338i	$0.106489\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−14.9566	−0.979839	−0.489920	−	0.871768i	$-0.662974\pi$
$233$	−14.9566	−0.979839	−0.489920	+	0.871768i	$0.662974\pi$
$234$	0	0
$235$	−5.40347	−0.352484
$236$	0	0
$237$	2.14788	0.139520
$238$	0	0
$239$	−17.6767	−1.14341	−0.571706	−	0.820458i	$-0.693718\pi$
$239$	−17.6767	−1.14341	−0.571706	+	0.820458i	$0.693718\pi$
$240$	0	0
$241$	0.161231	0.0103858	0.00519291	−	0.999987i	$-0.498347\pi$
$241$	0.161231	0.0103858	0.00519291	+	0.999987i	$0.498347\pi$
$242$	0	0
$243$	−12.3272	−0.790788
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.36821	0.341571
$248$	0	0
$249$	−1.77823	−0.112691
$250$	0	0
$251$	1.25200	0.0790256	0.0395128	−	0.999219i	$-0.487419\pi$
$251$	1.25200	0.0790256	0.0395128	+	0.999219i	$0.487419\pi$
$252$	0	0
$253$	3.34197	0.210108
$254$	0	0
$255$	−0.614104	−0.0384567
$256$	0	0
$257$	26.9507	1.68114	0.840571	−	0.541702i	$-0.182220\pi$
$257$	26.9507	1.68114	0.840571	+	0.541702i	$0.182220\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	6.17611	0.382292
$262$	0	0
$263$	−11.1730	−0.688958	−0.344479	−	0.938794i	$-0.611944\pi$
$263$	−11.1730	−0.688958	−0.344479	+	0.938794i	$0.611944\pi$
$264$	0	0
$265$	2.29304	0.140860
$266$	0	0
$267$	5.54532	0.339368
$268$	0	0
$269$	5.31454	0.324034	0.162017	−	0.986788i	$-0.448200\pi$
$269$	5.31454	0.324034	0.162017	+	0.986788i	$0.448200\pi$
$270$	0	0
$271$	−0.972772	−0.0590917	−0.0295458	−	0.999563i	$-0.509406\pi$
$271$	−0.972772	−0.0590917	−0.0295458	+	0.999563i	$0.509406\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.31014	0.199609
$276$	0	0
$277$	4.67247	0.280742	0.140371	−	0.990099i	$-0.455171\pi$
$277$	4.67247	0.280742	0.140371	+	0.990099i	$0.455171\pi$
$278$	0	0
$279$	11.2352	0.672635
$280$	0	0
$281$	16.0316	0.956363	0.478182	−	0.878261i	$-0.341296\pi$
$281$	16.0316	0.956363	0.478182	+	0.878261i	$0.341296\pi$
$282$	0	0
$283$	−5.20958	−0.309678	−0.154839	−	0.987940i	$-0.549486\pi$
$283$	−5.20958	−0.309678	−0.154839	+	0.987940i	$0.549486\pi$
$284$	0	0
$285$	−0.292105	−0.0173028
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−12.5802	−0.740010
$290$	0	0
$291$	1.05300	0.0617282
$292$	0	0
$293$	16.4689	0.962124	0.481062	−	0.876687i	$-0.340251\pi$
$293$	16.4689	0.962124	0.481062	+	0.876687i	$0.340251\pi$
$294$	0	0
$295$	1.98495	0.115568
$296$	0	0
$297$	−2.09409	−0.121511
$298$	0	0
$299$	25.3651	1.46690
$300$	0	0
$301$	0	0
$302$	0	0
$303$	1.90611	0.109503
$304$	0	0
$305$	−2.67233	−0.153017
$306$	0	0
$307$	−20.3284	−1.16020	−0.580102	−	0.814544i	$-0.696987\pi$
$307$	−20.3284	−1.16020	−0.580102	+	0.814544i	$0.696987\pi$
$308$	0	0
$309$	6.61008	0.376034
$310$	0	0
$311$	−24.9559	−1.41512	−0.707559	−	0.706654i	$-0.750204\pi$
$311$	−24.9559	−1.41512	−0.707559	+	0.706654i	$0.750204\pi$
$312$	0	0
$313$	26.0403	1.47189	0.735944	−	0.677043i	$-0.236739\pi$
$313$	26.0403	1.47189	0.735944	+	0.677043i	$0.236739\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−20.4195	−1.14687	−0.573436	−	0.819250i	$-0.694390\pi$
$317$	−20.4195	−1.14687	−0.573436	+	0.819250i	$0.694390\pi$
$318$	0	0
$319$	1.59816	0.0894798
$320$	0	0
$321$	1.94504	0.108562
$322$	0	0
$323$	2.10234	0.116977
$324$	0	0
$325$	25.1235	1.39360
$326$	0	0
$327$	5.54664	0.306730
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−27.0935	−1.48919	−0.744597	−	0.667514i	$-0.767358\pi$
$331$	−27.0935	−1.48919	−0.744597	+	0.667514i	$0.767358\pi$
$332$	0	0
$333$	20.1697	1.10529
$334$	0	0
$335$	−3.53122	−0.192931
$336$	0	0
$337$	27.0954	1.47598	0.737990	−	0.674812i	$-0.235775\pi$
$337$	27.0954	1.47598	0.737990	+	0.674812i	$0.235775\pi$
$338$	0	0
$339$	−4.05165	−0.220055
$340$	0	0
$341$	2.90728	0.157438
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−1.38021	−0.0743080
$346$	0	0
$347$	7.44209	0.399513	0.199756	−	0.979846i	$-0.435985\pi$
$347$	7.44209	0.399513	0.199756	+	0.979846i	$0.435985\pi$
$348$	0	0
$349$	5.19856	0.278272	0.139136	−	0.990273i	$-0.455567\pi$
$349$	5.19856	0.278272	0.139136	+	0.990273i	$0.455567\pi$
$350$	0	0
$351$	−15.8938	−0.848350
$352$	0	0
$353$	−16.6159	−0.884375	−0.442188	−	0.896923i	$-0.645797\pi$
$353$	−16.6159	−0.884375	−0.442188	+	0.896923i	$0.645797\pi$
$354$	0	0
$355$	−5.66989	−0.300926
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.41511	−0.127465	−0.0637324	−	0.997967i	$-0.520300\pi$
$359$	−2.41511	−0.127465	−0.0637324	+	0.997967i	$0.520300\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	5.42215	0.284589
$364$	0	0
$365$	−8.54953	−0.447503
$366$	0	0
$367$	18.7922	0.980942	0.490471	−	0.871457i	$-0.336825\pi$
$367$	18.7922	0.980942	0.490471	+	0.871457i	$0.336825\pi$
$368$	0	0
$369$	−14.6654	−0.763450
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−12.5201	−0.648265	−0.324132	−	0.946012i	$-0.605072\pi$
$373$	−12.5201	−0.648265	−0.324132	+	0.946012i	$0.605072\pi$
$374$	0	0
$375$	−2.82759	−0.146016
$376$	0	0
$377$	12.1298	0.624717
$378$	0	0
$379$	−27.7692	−1.42641	−0.713203	−	0.700958i	$-0.752756\pi$
$379$	−27.7692	−1.42641	−0.713203	+	0.700958i	$0.752756\pi$
$380$	0	0
$381$	−3.02358	−0.154903
$382$	0	0
$383$	−9.77383	−0.499419	−0.249710	−	0.968321i	$-0.580335\pi$
$383$	−9.77383	−0.499419	−0.249710	+	0.968321i	$0.580335\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7.12198	0.362031
$388$	0	0
$389$	−28.0325	−1.42130	−0.710652	−	0.703544i	$-0.751600\pi$
$389$	−28.0325	−1.42130	−0.710652	+	0.703544i	$0.751600\pi$
$390$	0	0
$391$	9.93366	0.502367
$392$	0	0
$393$	6.66895	0.336404
$394$	0	0
$395$	2.35268	0.118376
$396$	0	0
$397$	1.72569	0.0866100	0.0433050	−	0.999062i	$-0.486211\pi$
$397$	1.72569	0.0866100	0.0433050	+	0.999062i	$0.486211\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−7.10556	−0.354835	−0.177417	−	0.984136i	$-0.556774\pi$
$401$	−7.10556	−0.354835	−0.177417	+	0.984136i	$0.556774\pi$
$402$	0	0
$403$	22.0658	1.09918
$404$	0	0
$405$	−3.77345	−0.187504
$406$	0	0
$407$	5.21921	0.258707
$408$	0	0
$409$	−2.48661	−0.122955	−0.0614775	−	0.998108i	$-0.519581\pi$
$409$	−2.48661	−0.122955	−0.0614775	+	0.998108i	$0.519581\pi$
$410$	0	0
$411$	−5.77258	−0.284740
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−1.94779	−0.0956130
$416$	0	0
$417$	−11.7936	−0.577533
$418$	0	0
$419$	−30.5968	−1.49475	−0.747375	−	0.664403i	$-0.768686\pi$
$419$	−30.5968	−1.49475	−0.747375	+	0.664403i	$0.768686\pi$
$420$	0	0
$421$	22.0471	1.07451	0.537254	−	0.843420i	$-0.319462\pi$
$421$	22.0471	1.07451	0.537254	+	0.843420i	$0.319462\pi$
$422$	0	0
$423$	−26.1106	−1.26954
$424$	0	0
$425$	9.83904	0.477264
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1.96073	−0.0946651
$430$	0	0
$431$	40.6574	1.95840	0.979198	−	0.202905i	$-0.0650382\pi$
$431$	40.6574	1.95840	0.979198	+	0.202905i	$0.0650382\pi$
$432$	0	0
$433$	25.6214	1.23129	0.615644	−	0.788025i	$-0.288896\pi$
$433$	25.6214	1.23129	0.615644	+	0.788025i	$0.288896\pi$
$434$	0	0
$435$	−0.660029	−0.0316460
$436$	0	0
$437$	4.72505	0.226030
$438$	0	0
$439$	−16.0646	−0.766724	−0.383362	−	0.923598i	$-0.625234\pi$
$439$	−16.0646	−0.766724	−0.383362	+	0.923598i	$0.625234\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−8.25061	−0.391998	−0.195999	−	0.980604i	$-0.562795\pi$
$443$	−8.25061	−0.391998	−0.195999	+	0.980604i	$0.562795\pi$
$444$	0	0
$445$	6.07407	0.287939
$446$	0	0
$447$	−8.33474	−0.394220
$448$	0	0
$449$	15.6492	0.738531	0.369266	−	0.929324i	$-0.379609\pi$
$449$	15.6492	0.738531	0.369266	+	0.929324i	$0.379609\pi$
$450$	0	0
$451$	−3.79489	−0.178694
$452$	0	0
$453$	0.461567	0.0216863
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7.73275	0.361723	0.180861	−	0.983509i	$-0.442111\pi$
$457$	7.73275	0.361723	0.180861	+	0.983509i	$0.442111\pi$
$458$	0	0
$459$	−6.22446	−0.290533
$460$	0	0
$461$	5.78141	0.269267	0.134634	−	0.990895i	$-0.457014\pi$
$461$	5.78141	0.269267	0.134634	+	0.990895i	$0.457014\pi$
$462$	0	0
$463$	35.0309	1.62803	0.814013	−	0.580847i	$-0.197279\pi$
$463$	35.0309	1.62803	0.814013	+	0.580847i	$0.197279\pi$
$464$	0	0
$465$	−1.20069	−0.0556804
$466$	0	0
$467$	−7.39612	−0.342252	−0.171126	−	0.985249i	$-0.554740\pi$
$467$	−7.39612	−0.342252	−0.171126	+	0.985249i	$0.554740\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−3.61152	−0.166410
$472$	0	0
$473$	1.84292	0.0847374
$474$	0	0
$475$	4.68004	0.214735
$476$	0	0
$477$	11.0804	0.507337
$478$	0	0
$479$	−26.2545	−1.19960	−0.599800	−	0.800150i	$-0.704753\pi$
$479$	−26.2545	−1.19960	−0.599800	+	0.800150i	$0.704753\pi$
$480$	0	0
$481$	39.6131	1.80620
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.15341	0.0523737
$486$	0	0
$487$	40.2747	1.82502	0.912510	−	0.409054i	$-0.134141\pi$
$487$	40.2747	1.82502	0.912510	+	0.409054i	$0.134141\pi$
$488$	0	0
$489$	10.0333	0.453721
$490$	0	0
$491$	39.0791	1.76361	0.881806	−	0.471611i	$-0.156327\pi$
$491$	39.0791	1.76361	0.881806	+	0.471611i	$0.156327\pi$
$492$	0	0
$493$	4.75036	0.213946
$494$	0	0
$495$	−1.09354	−0.0491508
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0.328699	0.0147146	0.00735728	−	0.999973i	$-0.497658\pi$
$499$	0.328699	0.0147146	0.00735728	+	0.999973i	$0.497658\pi$
$500$	0	0
$501$	−0.636695	−0.0284454
$502$	0	0
$503$	15.7949	0.704261	0.352130	−	0.935951i	$-0.385457\pi$
$503$	15.7949	0.704261	0.352130	+	0.935951i	$0.385457\pi$
$504$	0	0
$505$	2.08786	0.0929087
$506$	0	0
$507$	−8.16839	−0.362771
$508$	0	0
$509$	−5.65926	−0.250842	−0.125421	−	0.992104i	$-0.540028\pi$
$509$	−5.65926	−0.250842	−0.125421	+	0.992104i	$0.540028\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−2.96073	−0.130719
$514$	0	0
$515$	7.24036	0.319048
$516$	0	0
$517$	−6.75651	−0.297151
$518$	0	0
$519$	−3.57993	−0.157141
$520$	0	0
$521$	0.694756	0.0304378	0.0152189	−	0.999884i	$-0.495155\pi$
$521$	0.694756	0.0304378	0.0152189	+	0.999884i	$0.495155\pi$
$522$	0	0
$523$	−16.3695	−0.715787	−0.357894	−	0.933762i	$-0.616505\pi$
$523$	−16.3695	−0.715787	−0.357894	+	0.933762i	$0.616505\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.64158	0.376433
$528$	0	0
$529$	−0.673929	−0.0293013
$530$	0	0
$531$	9.59167	0.416243
$532$	0	0
$533$	−28.8026	−1.24758
$534$	0	0
$535$	2.13050	0.0921097
$536$	0	0
$537$	−10.9377	−0.471995
$538$	0	0
$539$	0	0
$540$	0	0
$541$	30.7905	1.32379	0.661893	−	0.749599i	$-0.269753\pi$
$541$	30.7905	1.32379	0.661893	+	0.749599i	$0.269753\pi$
$542$	0	0
$543$	−0.680043	−0.0291835
$544$	0	0
$545$	6.07552	0.260247
$546$	0	0
$547$	28.3689	1.21297	0.606484	−	0.795096i	$-0.292579\pi$
$547$	28.3689	1.21297	0.606484	+	0.795096i	$0.292579\pi$
$548$	0	0
$549$	−12.9132	−0.551124
$550$	0	0
$551$	2.25956	0.0962605
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−2.15550	−0.0914958
$556$	0	0
$557$	−30.8298	−1.30630	−0.653150	−	0.757229i	$-0.726553\pi$
$557$	−30.8298	−1.30630	−0.653150	+	0.757229i	$0.726553\pi$
$558$	0	0
$559$	13.9875	0.591607
$560$	0	0
$561$	−0.767877	−0.0324198
$562$	0	0
$563$	−9.40818	−0.396508	−0.198254	−	0.980151i	$-0.563527\pi$
$563$	−9.40818	−0.396508	−0.198254	+	0.980151i	$0.563527\pi$
$564$	0	0
$565$	−4.43798	−0.186707
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−7.77776	−0.326061	−0.163030	−	0.986621i	$-0.552127\pi$
$569$	−7.77776	−0.326061	−0.163030	+	0.986621i	$0.552127\pi$
$570$	0	0
$571$	−34.9128	−1.46105	−0.730527	−	0.682883i	$-0.760726\pi$
$571$	−34.9128	−1.46105	−0.730527	+	0.682883i	$0.760726\pi$
$572$	0	0
$573$	12.5687	0.525063
$574$	0	0
$575$	22.1134	0.922193
$576$	0	0
$577$	45.6640	1.90102	0.950509	−	0.310698i	$-0.100563\pi$
$577$	45.6640	1.90102	0.950509	+	0.310698i	$0.100563\pi$
$578$	0	0
$579$	4.13505	0.171847
$580$	0	0
$581$	0	0
$582$	0	0
$583$	2.86722	0.118748
$584$	0	0
$585$	−8.29979	−0.343154
$586$	0	0
$587$	30.1300	1.24360	0.621799	−	0.783177i	$-0.286402\pi$
$587$	30.1300	1.24360	0.621799	+	0.783177i	$0.286402\pi$
$588$	0	0
$589$	4.11046	0.169369
$590$	0	0
$591$	9.38461	0.386031
$592$	0	0
$593$	12.1684	0.499698	0.249849	−	0.968285i	$-0.419619\pi$
$593$	12.1684	0.499698	0.249849	+	0.968285i	$0.419619\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−2.45895	−0.100638
$598$	0	0
$599$	1.68661	0.0689132	0.0344566	−	0.999406i	$-0.489030\pi$
$599$	1.68661	0.0689132	0.0344566	+	0.999406i	$0.489030\pi$
$600$	0	0
$601$	−2.69040	−0.109744	−0.0548719	−	0.998493i	$-0.517475\pi$
$601$	−2.69040	−0.109744	−0.0548719	+	0.998493i	$0.517475\pi$
$602$	0	0
$603$	−17.0635	−0.694881
$604$	0	0
$605$	5.93916	0.241461
$606$	0	0
$607$	23.2470	0.943569	0.471784	−	0.881714i	$-0.343610\pi$
$607$	23.2470	0.943569	0.471784	+	0.881714i	$0.343610\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−51.2810	−2.07461
$612$	0	0
$613$	−7.23926	−0.292391	−0.146195	−	0.989256i	$-0.546703\pi$
$613$	−7.23926	−0.292391	−0.146195	+	0.989256i	$0.546703\pi$
$614$	0	0
$615$	1.56726	0.0631981
$616$	0	0
$617$	40.0857	1.61379	0.806895	−	0.590695i	$-0.201146\pi$
$617$	40.0857	1.61379	0.806895	+	0.590695i	$0.201146\pi$
$618$	0	0
$619$	−11.1977	−0.450075	−0.225037	−	0.974350i	$-0.572250\pi$
$619$	−11.1977	−0.450075	−0.225037	+	0.974350i	$0.572250\pi$
$620$	0	0
$621$	−13.9896	−0.561383
$622$	0	0
$623$	0	0
$624$	0	0
$625$	20.3030	0.812120
$626$	0	0
$627$	−0.365249	−0.0145866
$628$	0	0
$629$	15.5136	0.618567
$630$	0	0
$631$	−32.4645	−1.29239	−0.646196	−	0.763172i	$-0.723641\pi$
$631$	−32.4645	−1.29239	−0.646196	+	0.763172i	$0.723641\pi$
$632$	0	0
$633$	2.84101	0.112920
$634$	0	0
$635$	−3.31188	−0.131428
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−27.3980	−1.08385
$640$	0	0
$641$	−10.2514	−0.404904	−0.202452	−	0.979292i	$-0.564891\pi$
$641$	−10.2514	−0.404904	−0.202452	+	0.979292i	$0.564891\pi$
$642$	0	0
$643$	−3.56787	−0.140703	−0.0703515	−	0.997522i	$-0.522412\pi$
$643$	−3.56787	−0.140703	−0.0703515	+	0.997522i	$0.522412\pi$
$644$	0	0
$645$	−0.761112	−0.0299688
$646$	0	0
$647$	4.62213	0.181715	0.0908574	−	0.995864i	$-0.471039\pi$
$647$	4.62213	0.181715	0.0908574	+	0.995864i	$0.471039\pi$
$648$	0	0
$649$	2.48199	0.0974265
$650$	0	0
$651$	0	0
$652$	0	0
$653$	20.5237	0.803153	0.401576	−	0.915826i	$-0.368462\pi$
$653$	20.5237	0.803153	0.401576	+	0.915826i	$0.368462\pi$
$654$	0	0
$655$	7.30484	0.285424
$656$	0	0
$657$	−41.3130	−1.61177
$658$	0	0
$659$	−27.4232	−1.06826	−0.534128	−	0.845404i	$-0.679360\pi$
$659$	−27.4232	−1.06826	−0.534128	+	0.845404i	$0.679360\pi$
$660$	0	0
$661$	10.6857	0.415626	0.207813	−	0.978169i	$-0.433365\pi$
$661$	10.6857	0.415626	0.207813	+	0.978169i	$0.433365\pi$
$662$	0	0
$663$	−5.82808	−0.226344
$664$	0	0
$665$	0	0
$666$	0	0
$667$	10.6765	0.413397
$668$	0	0
$669$	−3.92337	−0.151686
$670$	0	0
$671$	−3.34149	−0.128997
$672$	0	0
$673$	−19.8177	−0.763917	−0.381959	−	0.924179i	$-0.624750\pi$
$673$	−19.8177	−0.763917	−0.381959	+	0.924179i	$0.624750\pi$
$674$	0	0
$675$	−13.8563	−0.533331
$676$	0	0
$677$	−14.2199	−0.546514	−0.273257	−	0.961941i	$-0.588101\pi$
$677$	−14.2199	−0.546514	−0.273257	+	0.961941i	$0.588101\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−6.30961	−0.241785
$682$	0	0
$683$	30.9556	1.18448	0.592242	−	0.805760i	$-0.298243\pi$
$683$	30.9556	1.18448	0.592242	+	0.805760i	$0.298243\pi$
$684$	0	0
$685$	−6.32300	−0.241589
$686$	0	0
$687$	−14.7628	−0.563238
$688$	0	0
$689$	21.7618	0.829058
$690$	0	0
$691$	8.30422	0.315907	0.157954	−	0.987447i	$-0.449510\pi$
$691$	8.30422	0.315907	0.157954	+	0.987447i	$0.449510\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−12.9181	−0.490011
$696$	0	0
$697$	−11.2799	−0.427257
$698$	0	0
$699$	7.72370	0.292137
$700$	0	0
$701$	5.85572	0.221168	0.110584	−	0.993867i	$-0.464728\pi$
$701$	5.85572	0.221168	0.110584	+	0.993867i	$0.464728\pi$
$702$	0	0
$703$	7.37919	0.278312
$704$	0	0
$705$	2.79039	0.105092
$706$	0	0
$707$	0	0
$708$	0	0
$709$	7.06933	0.265494	0.132747	−	0.991150i	$-0.457620\pi$
$709$	7.06933	0.265494	0.132747	+	0.991150i	$0.457620\pi$
$710$	0	0
$711$	11.3686	0.426357
$712$	0	0
$713$	19.4221	0.727364
$714$	0	0
$715$	−2.14769	−0.0803192
$716$	0	0
$717$	9.12840	0.340906
$718$	0	0
$719$	21.4387	0.799528	0.399764	−	0.916618i	$-0.369092\pi$
$719$	21.4387	0.799528	0.399764	+	0.916618i	$0.369092\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−0.0832611	−0.00309651
$724$	0	0
$725$	10.5748	0.392740
$726$	0	0
$727$	9.12930	0.338587	0.169294	−	0.985566i	$-0.445851\pi$
$727$	9.12930	0.338587	0.169294	+	0.985566i	$0.445851\pi$
$728$	0	0
$729$	−13.6472	−0.505453
$730$	0	0
$731$	5.47788	0.202607
$732$	0	0
$733$	−3.88565	−0.143520	−0.0717599	−	0.997422i	$-0.522862\pi$
$733$	−3.88565	−0.143520	−0.0717599	+	0.997422i	$0.522862\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.41544	−0.162645
$738$	0	0
$739$	3.14431	0.115665	0.0578327	−	0.998326i	$-0.481581\pi$
$739$	3.14431	0.115665	0.0578327	+	0.998326i	$0.481581\pi$
$740$	0	0
$741$	−2.77219	−0.101839
$742$	0	0
$743$	7.33586	0.269127	0.134563	−	0.990905i	$-0.457037\pi$
$743$	7.33586	0.269127	0.134563	+	0.990905i	$0.457037\pi$
$744$	0	0
$745$	−9.12947	−0.334478
$746$	0	0
$747$	−9.41208	−0.344370
$748$	0	0
$749$	0	0
$750$	0	0
$751$	52.5132	1.91623	0.958117	−	0.286376i	$-0.0924508\pi$
$751$	52.5132	1.91623	0.958117	+	0.286376i	$0.0924508\pi$
$752$	0	0
$753$	−0.646543	−0.0235613
$754$	0	0
$755$	0.505578	0.0183999
$756$	0	0
$757$	−28.2975	−1.02849	−0.514245	−	0.857644i	$-0.671928\pi$
$757$	−28.2975	−1.02849	−0.514245	+	0.857644i	$0.671928\pi$
$758$	0	0
$759$	−1.72582	−0.0626432
$760$	0	0
$761$	30.4412	1.10349	0.551747	−	0.834012i	$-0.313961\pi$
$761$	30.4412	1.10349	0.551747	+	0.834012i	$0.313961\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−3.25043	−0.117519
$766$	0	0
$767$	18.8379	0.680198
$768$	0	0
$769$	50.6119	1.82511	0.912556	−	0.408953i	$-0.134106\pi$
$769$	50.6119	1.82511	0.912556	+	0.408953i	$0.134106\pi$
$770$	0	0
$771$	−13.9176	−0.501229
$772$	0	0
$773$	18.6587	0.671105	0.335553	−	0.942021i	$-0.391077\pi$
$773$	18.6587	0.671105	0.335553	+	0.942021i	$0.391077\pi$
$774$	0	0
$775$	19.2371	0.691018
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5.36541	−0.192236
$780$	0	0
$781$	−7.08964	−0.253687
$782$	0	0
$783$	−6.68995	−0.239079
$784$	0	0
$785$	−3.95589	−0.141192
$786$	0	0
$787$	−17.1748	−0.612214	−0.306107	−	0.951997i	$-0.599027\pi$
$787$	−17.1748	−0.612214	−0.306107	+	0.951997i	$0.599027\pi$
$788$	0	0
$789$	5.76984	0.205412
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−25.3615	−0.900612
$794$	0	0
$795$	−1.18414	−0.0419972
$796$	0	0
$797$	−45.3566	−1.60661	−0.803306	−	0.595567i	$-0.796928\pi$
$797$	−45.3566	−1.60661	−0.803306	+	0.595567i	$0.796928\pi$
$798$	0	0
$799$	−20.0830	−0.710487
$800$	0	0
$801$	29.3511	1.03707
$802$	0	0
$803$	−10.6904	−0.377254
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−2.74447	−0.0966100
$808$	0	0
$809$	8.45776	0.297359	0.148680	−	0.988885i	$-0.452498\pi$
$809$	8.45776	0.297359	0.148680	+	0.988885i	$0.452498\pi$
$810$	0	0
$811$	8.62878	0.302997	0.151499	−	0.988457i	$-0.451590\pi$
$811$	8.62878	0.302997	0.151499	+	0.988457i	$0.451590\pi$
$812$	0	0
$813$	0.502347	0.0176181
$814$	0	0
$815$	10.9900	0.384962
$816$	0	0
$817$	2.60561	0.0911588
$818$	0	0
$819$	0	0
$820$	0	0
$821$	16.3362	0.570139	0.285069	−	0.958507i	$-0.407983\pi$
$821$	16.3362	0.570139	0.285069	+	0.958507i	$0.407983\pi$
$822$	0	0
$823$	−6.17492	−0.215244	−0.107622	−	0.994192i	$-0.534324\pi$
$823$	−6.17492	−0.215244	−0.107622	+	0.994192i	$0.534324\pi$
$824$	0	0
$825$	−1.70938	−0.0595130
$826$	0	0
$827$	−23.3867	−0.813236	−0.406618	−	0.913598i	$-0.633292\pi$
$827$	−23.3867	−0.813236	−0.406618	+	0.913598i	$0.633292\pi$
$828$	0	0
$829$	−20.1873	−0.701133	−0.350567	−	0.936538i	$-0.614011\pi$
$829$	−20.1873	−0.701133	−0.350567	+	0.936538i	$0.614011\pi$
$830$	0	0
$831$	−2.41290	−0.0837025
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−0.697405	−0.0241347
$836$	0	0
$837$	−12.1700	−0.420656
$838$	0	0
$839$	−5.22946	−0.180541	−0.0902705	−	0.995917i	$-0.528773\pi$
$839$	−5.22946	−0.180541	−0.0902705	+	0.995917i	$0.528773\pi$
$840$	0	0
$841$	−23.8944	−0.823944
$842$	0	0
$843$	−8.27882	−0.285138
$844$	0	0
$845$	−8.94726	−0.307795
$846$	0	0
$847$	0	0
$848$	0	0
$849$	2.69027	0.0923298
$850$	0	0
$851$	34.8670	1.19523
$852$	0	0
$853$	−20.1398	−0.689574	−0.344787	−	0.938681i	$-0.612049\pi$
$853$	−20.1398	−0.689574	−0.344787	+	0.938681i	$0.612049\pi$
$854$	0	0
$855$	−1.54610	−0.0528755
$856$	0	0
$857$	−14.1389	−0.482974	−0.241487	−	0.970404i	$-0.577635\pi$
$857$	−14.1389	−0.482974	−0.241487	+	0.970404i	$0.577635\pi$
$858$	0	0
$859$	13.9130	0.474704	0.237352	−	0.971424i	$-0.423721\pi$
$859$	13.9130	0.474704	0.237352	+	0.971424i	$0.423721\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−2.94906	−0.100387	−0.0501937	−	0.998740i	$-0.515984\pi$
$863$	−2.94906	−0.100387	−0.0501937	+	0.998740i	$0.515984\pi$
$864$	0	0
$865$	−3.92128	−0.133328
$866$	0	0
$867$	6.49649	0.220632
$868$	0	0
$869$	2.94180	0.0997937
$870$	0	0
$871$	−33.5126	−1.13553
$872$	0	0
$873$	5.57351	0.188635
$874$	0	0
$875$	0	0
$876$	0	0
$877$	41.0760	1.38704	0.693519	−	0.720439i	$-0.256059\pi$
$877$	41.0760	1.38704	0.693519	+	0.720439i	$0.256059\pi$
$878$	0	0
$879$	−8.50467	−0.286855
$880$	0	0
$881$	−2.76917	−0.0932957	−0.0466479	−	0.998911i	$-0.514854\pi$
$881$	−2.76917	−0.0932957	−0.0466479	+	0.998911i	$0.514854\pi$
$882$	0	0
$883$	28.8899	0.972222	0.486111	−	0.873897i	$-0.338415\pi$
$883$	28.8899	0.972222	0.486111	+	0.873897i	$0.338415\pi$
$884$	0	0
$885$	−1.02504	−0.0344564
$886$	0	0
$887$	−50.5750	−1.69814	−0.849071	−	0.528280i	$-0.822837\pi$
$887$	−50.5750	−1.69814	−0.849071	+	0.528280i	$0.822837\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−4.71834	−0.158070
$892$	0	0
$893$	−9.55271	−0.319669
$894$	0	0
$895$	−11.9806	−0.400467
$896$	0	0
$897$	−13.0987	−0.437353
$898$	0	0
$899$	9.28783	0.309767
$900$	0	0
$901$	8.52252	0.283926
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−0.744887	−0.0247609
$906$	0	0
$907$	27.0189	0.897147	0.448573	−	0.893746i	$-0.351932\pi$
$907$	27.0189	0.897147	0.448573	+	0.893746i	$0.351932\pi$
$908$	0	0
$909$	10.0890	0.334630
$910$	0	0
$911$	4.34024	0.143798	0.0718992	−	0.997412i	$-0.477094\pi$
$911$	4.34024	0.143798	0.0718992	+	0.997412i	$0.477094\pi$
$912$	0	0
$913$	−2.43552	−0.0806038
$914$	0	0
$915$	1.38001	0.0456218
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−12.1341	−0.400266	−0.200133	−	0.979769i	$-0.564137\pi$
$919$	−12.1341	−0.400266	−0.200133	+	0.979769i	$0.564137\pi$
$920$	0	0
$921$	10.4978	0.345913
$922$	0	0
$923$	−53.8094	−1.77116
$924$	0	0
$925$	34.5349	1.13550
$926$	0	0
$927$	34.9869	1.14912
$928$	0	0
$929$	45.0237	1.47718	0.738589	−	0.674156i	$-0.235492\pi$
$929$	45.0237	1.47718	0.738589	+	0.674156i	$0.235492\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	12.8874	0.421915
$934$	0	0
$935$	−0.841096	−0.0275068
$936$	0	0
$937$	−9.03515	−0.295165	−0.147583	−	0.989050i	$-0.547149\pi$
$937$	−9.03515	−0.295165	−0.147583	+	0.989050i	$0.547149\pi$
$938$	0	0
$939$	−13.4474	−0.438840
$940$	0	0
$941$	−9.60702	−0.313180	−0.156590	−	0.987664i	$-0.550050\pi$
$941$	−9.60702	−0.313180	−0.156590	+	0.987664i	$0.550050\pi$
$942$	0	0
$943$	−25.3518	−0.825568
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.17982	0.0708348	0.0354174	−	0.999373i	$-0.488724\pi$
$947$	2.17982	0.0708348	0.0354174	+	0.999373i	$0.488724\pi$
$948$	0	0
$949$	−81.1383	−2.63386
$950$	0	0
$951$	10.5448	0.341938
$952$	0	0
$953$	−44.1217	−1.42924	−0.714621	−	0.699511i	$-0.753401\pi$
$953$	−44.1217	−1.42924	−0.714621	+	0.699511i	$0.753401\pi$
$954$	0	0
$955$	13.7671	0.445493
$956$	0	0
$957$	−0.825302	−0.0266782
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−14.1041	−0.454972
$962$	0	0
$963$	10.2950	0.331752
$964$	0	0
$965$	4.52934	0.145804
$966$	0	0
$967$	−15.6332	−0.502729	−0.251364	−	0.967893i	$-0.580879\pi$
$967$	−15.6332	−0.502729	−0.251364	+	0.967893i	$0.580879\pi$
$968$	0	0
$969$	−1.08566	−0.0348766
$970$	0	0
$971$	−37.2083	−1.19407	−0.597035	−	0.802215i	$-0.703655\pi$
$971$	−37.2083	−1.19407	−0.597035	+	0.802215i	$0.703655\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−12.9739	−0.415499
$976$	0	0
$977$	32.6718	1.04526	0.522632	−	0.852558i	$-0.324950\pi$
$977$	32.6718	1.04526	0.522632	+	0.852558i	$0.324950\pi$
$978$	0	0
$979$	7.59504	0.242738
$980$	0	0
$981$	29.3581	0.937332
$982$	0	0
$983$	−57.2792	−1.82692	−0.913461	−	0.406925i	$-0.866601\pi$
$983$	−57.2792	−1.82692	−0.913461	+	0.406925i	$0.866601\pi$
$984$	0	0
$985$	10.2794	0.327530
$986$	0	0
$987$	0	0
$988$	0	0
$989$	12.3116	0.391487
$990$	0	0
$991$	−51.4206	−1.63343	−0.816715	−	0.577041i	$-0.804207\pi$
$991$	−51.4206	−1.63343	−0.816715	+	0.577041i	$0.804207\pi$
$992$	0	0
$993$	13.9913	0.444000
$994$	0	0
$995$	−2.69341	−0.0853868
$996$	0	0
$997$	−48.0721	−1.52246	−0.761229	−	0.648483i	$-0.775404\pi$
$997$	−48.0721	−1.52246	−0.761229	+	0.648483i	$0.775404\pi$
$998$	0	0
$999$	−21.8478	−0.691234

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.bw.1.7	yes	14
7.6	odd	2		7448.2.a.bv.1.8	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7448.2.a.bv.1.8	✓	14	7.6	odd	2
7448.2.a.bw.1.7	yes	14	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)

\(f(q)\)	\(=\)	\(q-0.516408 q^{3} -0.565648 q^{5} -2.73332 q^{9} +O(q^{10})\)	\(q-0.516408 q^{3} -0.565648 q^{5} -2.73332 q^{9} -0.707288 q^{11} -5.36821 q^{13} +0.292105 q^{15} -2.10234 q^{17} -1.00000 q^{19} -4.72505 q^{23} -4.68004 q^{25} +2.96073 q^{27} -2.25956 q^{29} -4.11046 q^{31} +0.365249 q^{33} -7.37919 q^{37} +2.77219 q^{39} +5.36541 q^{41} -2.60561 q^{43} +1.54610 q^{45} +9.55271 q^{47} +1.08566 q^{51} -4.05382 q^{53} +0.400076 q^{55} +0.516408 q^{57} -3.50916 q^{59} +4.72438 q^{61} +3.03652 q^{65} +6.24278 q^{67} +2.44005 q^{69} +10.0237 q^{71} +15.1146 q^{73} +2.41681 q^{75} -4.15927 q^{79} +6.67103 q^{81} +3.44346 q^{83} +1.18918 q^{85} +1.16685 q^{87} -10.7383 q^{89} +2.12267 q^{93} +0.565648 q^{95} -2.03910 q^{97} +1.93325 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 2 q^{3} + 2 q^{5} + 16 q^{9}+O(q^{10}) \) 14 * q + 2 * q^3 + 2 * q^5 + 16 * q^9	\( 14 q + 2 q^{3} + 2 q^{5} + 16 q^{9} - 6 q^{11} + 16 q^{13} - 4 q^{15} + 20 q^{17} - 14 q^{19} + 4 q^{23} + 16 q^{25} + 20 q^{27} + 6 q^{29} + 34 q^{33} - 6 q^{37} + 8 q^{39} + 46 q^{41} - 18 q^{43} - 10 q^{47} - 4 q^{51} - 2 q^{53} + 28 q^{55} - 2 q^{57} - 22 q^{59} + 26 q^{61} + 8 q^{65} - 12 q^{67} + 48 q^{69} + 18 q^{71} + 28 q^{73} - 24 q^{75} - 10 q^{79} - 2 q^{81} + 8 q^{83} - 16 q^{85} + 16 q^{87} + 78 q^{89} - 2 q^{95} + 54 q^{97} + 26 q^{99}+O(q^{100}) \) 14 * q + 2 * q^3 + 2 * q^5 + 16 * q^9 - 6 * q^11 + 16 * q^13 - 4 * q^15 + 20 * q^17 - 14 * q^19 + 4 * q^23 + 16 * q^25 + 20 * q^27 + 6 * q^29 + 34 * q^33 - 6 * q^37 + 8 * q^39 + 46 * q^41 - 18 * q^43 - 10 * q^47 - 4 * q^51 - 2 * q^53 + 28 * q^55 - 2 * q^57 - 22 * q^59 + 26 * q^61 + 8 * q^65 - 12 * q^67 + 48 * q^69 + 18 * q^71 + 28 * q^73 - 24 * q^75 - 10 * q^79 - 2 * q^81 + 8 * q^83 - 16 * q^85 + 16 * q^87 + 78 * q^89 - 2 * q^95 + 54 * q^97 + 26 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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