LMFDB - Embedded newform 7448.2.a.bv.1.1

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} - 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			1.1
Root			$3.27248$ 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)

$f(q)$	$=$	$q-3.27248 q^{3} +1.49908 q^{5} +7.70916 q^{9} +O(q^{10})$	$q-3.27248 q^{3} +1.49908 q^{5} +7.70916 q^{9} +5.00142 q^{11} -2.30458 q^{13} -4.90571 q^{15} -5.61729 q^{17} +1.00000 q^{19} +6.90622 q^{23} -2.75277 q^{25} -15.4106 q^{27} +5.82177 q^{29} +7.98931 q^{31} -16.3671 q^{33} -3.77646 q^{37} +7.54169 q^{39} -8.85855 q^{41} -12.4280 q^{43} +11.5566 q^{45} -8.73380 q^{47} +18.3825 q^{51} -11.2438 q^{53} +7.49752 q^{55} -3.27248 q^{57} +7.65749 q^{59} +4.11332 q^{61} -3.45474 q^{65} -5.67543 q^{67} -22.6005 q^{69} -9.70266 q^{71} +2.17782 q^{73} +9.00839 q^{75} -4.54497 q^{79} +27.3036 q^{81} +8.37741 q^{83} -8.42076 q^{85} -19.0516 q^{87} -17.3056 q^{89} -26.1449 q^{93} +1.49908 q^{95} +1.01906 q^{97} +38.5567 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.27248	−1.88937	−0.944685	−	0.327979i	$-0.893632\pi$
$3$	−3.27248	−1.88937	−0.944685	+	0.327979i	$0.893632\pi$
$4$	0	0
$5$	1.49908	0.670408	0.335204	−	0.942146i	$-0.391195\pi$
$5$	1.49908	0.670408	0.335204	+	0.942146i	$0.391195\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	7.70916	2.56972
$10$	0	0
$11$	5.00142	1.50799	0.753993	−	0.656883i	$-0.228125\pi$
$11$	5.00142	1.50799	0.753993	+	0.656883i	$0.228125\pi$
$12$	0	0
$13$	−2.30458	−0.639174	−0.319587	−	0.947557i	$-0.603544\pi$
$13$	−2.30458	−0.639174	−0.319587	+	0.947557i	$0.603544\pi$
$14$	0	0
$15$	−4.90571	−1.26665
$16$	0	0
$17$	−5.61729	−1.36239	−0.681197	−	0.732100i	$-0.738540\pi$
$17$	−5.61729	−1.36239	−0.681197	+	0.732100i	$0.738540\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.90622	1.44005	0.720023	−	0.693950i	$-0.244131\pi$
$23$	6.90622	1.44005	0.720023	+	0.693950i	$0.244131\pi$
$24$	0	0
$25$	−2.75277	−0.550553
$26$	0	0
$27$	−15.4106	−2.96578
$28$	0	0
$29$	5.82177	1.08107	0.540537	−	0.841320i	$-0.318221\pi$
$29$	5.82177	1.08107	0.540537	+	0.841320i	$0.318221\pi$
$30$	0	0
$31$	7.98931	1.43492	0.717461	−	0.696599i	$-0.245304\pi$
$31$	7.98931	1.43492	0.717461	+	0.696599i	$0.245304\pi$
$32$	0	0
$33$	−16.3671	−2.84914
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.77646	−0.620846	−0.310423	−	0.950599i	$-0.600471\pi$
$37$	−3.77646	−0.620846	−0.310423	+	0.950599i	$0.600471\pi$
$38$	0	0
$39$	7.54169	1.20764
$40$	0	0
$41$	−8.85855	−1.38347	−0.691737	−	0.722150i	$-0.743154\pi$
$41$	−8.85855	−1.38347	−0.691737	+	0.722150i	$0.743154\pi$
$42$	0	0
$43$	−12.4280	−1.89525	−0.947626	−	0.319383i	$-0.896524\pi$
$43$	−12.4280	−1.89525	−0.947626	+	0.319383i	$0.896524\pi$
$44$	0	0
$45$	11.5566	1.72276
$46$	0	0
$47$	−8.73380	−1.27396	−0.636978	−	0.770882i	$-0.719816\pi$
$47$	−8.73380	−1.27396	−0.636978	+	0.770882i	$0.719816\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	18.3825	2.57407
$52$	0	0
$53$	−11.2438	−1.54446	−0.772229	−	0.635344i	$-0.780858\pi$
$53$	−11.2438	−1.54446	−0.772229	+	0.635344i	$0.780858\pi$
$54$	0	0
$55$	7.49752	1.01097
$56$	0	0
$57$	−3.27248	−0.433451
$58$	0	0
$59$	7.65749	0.996920	0.498460	−	0.866913i	$-0.333899\pi$
$59$	7.65749	0.996920	0.498460	+	0.866913i	$0.333899\pi$
$60$	0	0
$61$	4.11332	0.526657	0.263328	−	0.964706i	$-0.415180\pi$
$61$	4.11332	0.526657	0.263328	+	0.964706i	$0.415180\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.45474	−0.428507
$66$	0	0
$67$	−5.67543	−0.693364	−0.346682	−	0.937983i	$-0.612692\pi$
$67$	−5.67543	−0.693364	−0.346682	+	0.937983i	$0.612692\pi$
$68$	0	0
$69$	−22.6005	−2.72078
$70$	0	0
$71$	−9.70266	−1.15149	−0.575747	−	0.817628i	$-0.695289\pi$
$71$	−9.70266	−1.15149	−0.575747	+	0.817628i	$0.695289\pi$
$72$	0	0
$73$	2.17782	0.254894	0.127447	−	0.991845i	$-0.459322\pi$
$73$	2.17782	0.254894	0.127447	+	0.991845i	$0.459322\pi$
$74$	0	0
$75$	9.00839	1.04020
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−4.54497	−0.511349	−0.255674	−	0.966763i	$-0.582298\pi$
$79$	−4.54497	−0.511349	−0.255674	+	0.966763i	$0.582298\pi$
$80$	0	0
$81$	27.3036	3.03374
$82$	0	0
$83$	8.37741	0.919541	0.459770	−	0.888038i	$-0.347932\pi$
$83$	8.37741	0.919541	0.459770	+	0.888038i	$0.347932\pi$
$84$	0	0
$85$	−8.42076	−0.913359
$86$	0	0
$87$	−19.0516	−2.04255
$88$	0	0
$89$	−17.3056	−1.83439	−0.917195	−	0.398439i	$-0.869552\pi$
$89$	−17.3056	−1.83439	−0.917195	+	0.398439i	$0.869552\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−26.1449	−2.71110
$94$	0	0
$95$	1.49908	0.153802
$96$	0	0
$97$	1.01906	0.103470	0.0517349	−	0.998661i	$-0.483525\pi$
$97$	1.01906	0.103470	0.0517349	+	0.998661i	$0.483525\pi$
$98$	0	0
$99$	38.5567	3.87510
$100$	0	0
$101$	2.47433	0.246205	0.123102	−	0.992394i	$-0.460716\pi$
$101$	2.47433	0.246205	0.123102	+	0.992394i	$0.460716\pi$
$102$	0	0
$103$	−4.39384	−0.432938	−0.216469	−	0.976290i	$-0.569454\pi$
$103$	−4.39384	−0.432938	−0.216469	+	0.976290i	$0.569454\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.407522	0.0393967	0.0196983	−	0.999806i	$-0.493729\pi$
$107$	0.407522	0.0393967	0.0196983	+	0.999806i	$0.493729\pi$
$108$	0	0
$109$	−12.6564	−1.21226	−0.606130	−	0.795365i	$-0.707279\pi$
$109$	−12.6564	−1.21226	−0.606130	+	0.795365i	$0.707279\pi$
$110$	0	0
$111$	12.3584	1.17301
$112$	0	0
$113$	6.61845	0.622611	0.311306	−	0.950310i	$-0.399234\pi$
$113$	6.61845	0.622611	0.311306	+	0.950310i	$0.399234\pi$
$114$	0	0
$115$	10.3530	0.965419
$116$	0	0
$117$	−17.7663	−1.64250
$118$	0	0
$119$	0	0
$120$	0	0
$121$	14.0142	1.27402
$122$	0	0
$123$	28.9895	2.61389
$124$	0	0
$125$	−11.6220	−1.03950
$126$	0	0
$127$	12.9212	1.14658	0.573288	−	0.819354i	$-0.305668\pi$
$127$	12.9212	1.14658	0.573288	+	0.819354i	$0.305668\pi$
$128$	0	0
$129$	40.6704	3.58083
$130$	0	0
$131$	2.06957	0.180819	0.0904096	−	0.995905i	$-0.471182\pi$
$131$	2.06957	0.180819	0.0904096	+	0.995905i	$0.471182\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−23.1017	−1.98828
$136$	0	0
$137$	9.73661	0.831855	0.415927	−	0.909398i	$-0.363457\pi$
$137$	9.73661	0.831855	0.415927	+	0.909398i	$0.363457\pi$
$138$	0	0
$139$	−7.40811	−0.628347	−0.314174	−	0.949366i	$-0.601727\pi$
$139$	−7.40811	−0.628347	−0.314174	+	0.949366i	$0.601727\pi$
$140$	0	0
$141$	28.5812	2.40697
$142$	0	0
$143$	−11.5262	−0.963865
$144$	0	0
$145$	8.72728	0.724761
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0.0284910	0.00233407	0.00116703	−	0.999999i	$-0.499629\pi$
$149$	0.0284910	0.00233407	0.00116703	+	0.999999i	$0.499629\pi$
$150$	0	0
$151$	3.25628	0.264992	0.132496	−	0.991184i	$-0.457701\pi$
$151$	3.25628	0.264992	0.132496	+	0.991184i	$0.457701\pi$
$152$	0	0
$153$	−43.3046	−3.50097
$154$	0	0
$155$	11.9766	0.961983
$156$	0	0
$157$	16.4129	1.30989	0.654945	−	0.755677i	$-0.272692\pi$
$157$	16.4129	1.30989	0.654945	+	0.755677i	$0.272692\pi$
$158$	0	0
$159$	36.7953	2.91805
$160$	0	0
$161$	0	0
$162$	0	0
$163$	18.0444	1.41335	0.706673	−	0.707541i	$-0.250195\pi$
$163$	18.0444	1.41335	0.706673	+	0.707541i	$0.250195\pi$
$164$	0	0
$165$	−24.5355	−1.91009
$166$	0	0
$167$	2.65830	0.205706	0.102853	−	0.994697i	$-0.467203\pi$
$167$	2.65830	0.205706	0.102853	+	0.994697i	$0.467203\pi$
$168$	0	0
$169$	−7.68893	−0.591456
$170$	0	0
$171$	7.70916	0.589534
$172$	0	0
$173$	−14.0983	−1.07187	−0.535937	−	0.844258i	$-0.680041\pi$
$173$	−14.0983	−1.07187	−0.535937	+	0.844258i	$0.680041\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−25.0590	−1.88355
$178$	0	0
$179$	18.4662	1.38023	0.690114	−	0.723700i	$-0.257560\pi$
$179$	18.4662	1.38023	0.690114	+	0.723700i	$0.257560\pi$
$180$	0	0
$181$	−0.404387	−0.0300579	−0.0150289	−	0.999887i	$-0.504784\pi$
$181$	−0.404387	−0.0300579	−0.0150289	+	0.999887i	$0.504784\pi$
$182$	0	0
$183$	−13.4608	−0.995049
$184$	0	0
$185$	−5.66120	−0.416220
$186$	0	0
$187$	−28.0944	−2.05447
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3.42940	−0.248142	−0.124071	−	0.992273i	$-0.539595\pi$
$191$	−3.42940	−0.248142	−0.124071	+	0.992273i	$0.539595\pi$
$192$	0	0
$193$	−3.84774	−0.276966	−0.138483	−	0.990365i	$-0.544223\pi$
$193$	−3.84774	−0.276966	−0.138483	+	0.990365i	$0.544223\pi$
$194$	0	0
$195$	11.3056	0.809609
$196$	0	0
$197$	−18.5954	−1.32487	−0.662433	−	0.749121i	$-0.730476\pi$
$197$	−18.5954	−1.32487	−0.662433	+	0.749121i	$0.730476\pi$
$198$	0	0
$199$	18.2613	1.29451	0.647255	−	0.762273i	$-0.275917\pi$
$199$	18.2613	1.29451	0.647255	+	0.762273i	$0.275917\pi$
$200$	0	0
$201$	18.5727	1.31002
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−13.2797	−0.927491
$206$	0	0
$207$	53.2411	3.70052
$208$	0	0
$209$	5.00142	0.345956
$210$	0	0
$211$	13.5837	0.935139	0.467569	−	0.883956i	$-0.345130\pi$
$211$	13.5837	0.935139	0.467569	+	0.883956i	$0.345130\pi$
$212$	0	0
$213$	31.7518	2.17560
$214$	0	0
$215$	−18.6305	−1.27059
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−7.12687	−0.481589
$220$	0	0
$221$	12.9455	0.870807
$222$	0	0
$223$	−21.6794	−1.45176	−0.725878	−	0.687823i	$-0.758567\pi$
$223$	−21.6794	−1.45176	−0.725878	+	0.687823i	$0.758567\pi$
$224$	0	0
$225$	−21.2215	−1.41477
$226$	0	0
$227$	8.69858	0.577345	0.288673	−	0.957428i	$-0.406786\pi$
$227$	8.69858	0.577345	0.288673	+	0.957428i	$0.406786\pi$
$228$	0	0
$229$	23.3888	1.54558	0.772789	−	0.634663i	$-0.218861\pi$
$229$	23.3888	1.54558	0.772789	+	0.634663i	$0.218861\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	12.4739	0.817191	0.408595	−	0.912716i	$-0.366019\pi$
$233$	12.4739	0.817191	0.408595	+	0.912716i	$0.366019\pi$
$234$	0	0
$235$	−13.0926	−0.854070
$236$	0	0
$237$	14.8733	0.966127
$238$	0	0
$239$	−12.1858	−0.788232	−0.394116	−	0.919061i	$-0.628949\pi$
$239$	−12.1858	−0.788232	−0.394116	+	0.919061i	$0.628949\pi$
$240$	0	0
$241$	−11.8483	−0.763218	−0.381609	−	0.924324i	$-0.624630\pi$
$241$	−11.8483	−0.763218	−0.381609	+	0.924324i	$0.624630\pi$
$242$	0	0
$243$	−43.1188	−2.76607
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.30458	−0.146637
$248$	0	0
$249$	−27.4150	−1.73735
$250$	0	0
$251$	−2.60750	−0.164584	−0.0822920	−	0.996608i	$-0.526224\pi$
$251$	−2.60750	−0.164584	−0.0822920	+	0.996608i	$0.526224\pi$
$252$	0	0
$253$	34.5409	2.17157
$254$	0	0
$255$	27.5568	1.72567
$256$	0	0
$257$	−15.3433	−0.957090	−0.478545	−	0.878063i	$-0.658836\pi$
$257$	−15.3433	−0.957090	−0.478545	+	0.878063i	$0.658836\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	44.8809	2.77806
$262$	0	0
$263$	−30.5680	−1.88491	−0.942453	−	0.334338i	$-0.891487\pi$
$263$	−30.5680	−1.88491	−0.942453	+	0.334338i	$0.891487\pi$
$264$	0	0
$265$	−16.8554	−1.03542
$266$	0	0
$267$	56.6323	3.46584
$268$	0	0
$269$	−20.3320	−1.23967	−0.619833	−	0.784734i	$-0.712800\pi$
$269$	−20.3320	−1.23967	−0.619833	+	0.784734i	$0.712800\pi$
$270$	0	0
$271$	−8.66654	−0.526455	−0.263227	−	0.964734i	$-0.584787\pi$
$271$	−8.66654	−0.526455	−0.263227	+	0.964734i	$0.584787\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−13.7677	−0.830226
$276$	0	0
$277$	−28.0664	−1.68634	−0.843172	−	0.537644i	$-0.819315\pi$
$277$	−28.0664	−1.68634	−0.843172	+	0.537644i	$0.819315\pi$
$278$	0	0
$279$	61.5908	3.68734
$280$	0	0
$281$	−17.6233	−1.05132	−0.525659	−	0.850696i	$-0.676181\pi$
$281$	−17.6233	−1.05132	−0.525659	+	0.850696i	$0.676181\pi$
$282$	0	0
$283$	14.2362	0.846252	0.423126	−	0.906071i	$-0.360933\pi$
$283$	14.2362	0.846252	0.423126	+	0.906071i	$0.360933\pi$
$284$	0	0
$285$	−4.90571	−0.290589
$286$	0	0
$287$	0	0
$288$	0	0
$289$	14.5540	0.856116
$290$	0	0
$291$	−3.33486	−0.195493
$292$	0	0
$293$	4.56457	0.266665	0.133333	−	0.991071i	$-0.457432\pi$
$293$	4.56457	0.266665	0.133333	+	0.991071i	$0.457432\pi$
$294$	0	0
$295$	11.4792	0.668343
$296$	0	0
$297$	−77.0751	−4.47235
$298$	0	0
$299$	−15.9159	−0.920441
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−8.09720	−0.465172
$304$	0	0
$305$	6.16619	0.353075
$306$	0	0
$307$	−5.05783	−0.288666	−0.144333	−	0.989529i	$-0.546104\pi$
$307$	−5.05783	−0.288666	−0.144333	+	0.989529i	$0.546104\pi$
$308$	0	0
$309$	14.3788	0.817979
$310$	0	0
$311$	−12.4139	−0.703926	−0.351963	−	0.936014i	$-0.614486\pi$
$311$	−12.4139	−0.703926	−0.351963	+	0.936014i	$0.614486\pi$
$312$	0	0
$313$	−9.17271	−0.518472	−0.259236	−	0.965814i	$-0.583471\pi$
$313$	−9.17271	−0.518472	−0.259236	+	0.965814i	$0.583471\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10.1957	−0.572648	−0.286324	−	0.958133i	$-0.592433\pi$
$317$	−10.1957	−0.572648	−0.286324	+	0.958133i	$0.592433\pi$
$318$	0	0
$319$	29.1171	1.63025
$320$	0	0
$321$	−1.33361	−0.0744348
$322$	0	0
$323$	−5.61729	−0.312555
$324$	0	0
$325$	6.34396	0.351900
$326$	0	0
$327$	41.4178	2.29041
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−4.93507	−0.271256	−0.135628	−	0.990760i	$-0.543305\pi$
$331$	−4.93507	−0.271256	−0.135628	+	0.990760i	$0.543305\pi$
$332$	0	0
$333$	−29.1133	−1.59540
$334$	0	0
$335$	−8.50790	−0.464836
$336$	0	0
$337$	−8.13852	−0.443333	−0.221667	−	0.975123i	$-0.571150\pi$
$337$	−8.13852	−0.443333	−0.221667	+	0.975123i	$0.571150\pi$
$338$	0	0
$339$	−21.6588	−1.17634
$340$	0	0
$341$	39.9579	2.16384
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−33.8799	−1.82403
$346$	0	0
$347$	25.2354	1.35471	0.677354	−	0.735657i	$-0.263126\pi$
$347$	25.2354	1.35471	0.677354	+	0.735657i	$0.263126\pi$
$348$	0	0
$349$	−0.211189	−0.0113047	−0.00565235	−	0.999984i	$-0.501799\pi$
$349$	−0.211189	−0.0113047	−0.00565235	+	0.999984i	$0.501799\pi$
$350$	0	0
$351$	35.5150	1.89565
$352$	0	0
$353$	−7.32960	−0.390115	−0.195058	−	0.980792i	$-0.562489\pi$
$353$	−7.32960	−0.390115	−0.195058	+	0.980792i	$0.562489\pi$
$354$	0	0
$355$	−14.5450	−0.771971
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7.07932	−0.373632	−0.186816	−	0.982395i	$-0.559817\pi$
$359$	−7.07932	−0.373632	−0.186816	+	0.982395i	$0.559817\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−45.8613	−2.40709
$364$	0	0
$365$	3.26471	0.170883
$366$	0	0
$367$	−30.0367	−1.56791	−0.783953	−	0.620821i	$-0.786800\pi$
$367$	−30.0367	−1.56791	−0.783953	+	0.620821i	$0.786800\pi$
$368$	0	0
$369$	−68.2919	−3.55514
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−9.04220	−0.468187	−0.234094	−	0.972214i	$-0.575212\pi$
$373$	−9.04220	−0.468187	−0.234094	+	0.972214i	$0.575212\pi$
$374$	0	0
$375$	38.0328	1.96401
$376$	0	0
$377$	−13.4167	−0.690995
$378$	0	0
$379$	−0.456858	−0.0234672	−0.0117336	−	0.999931i	$-0.503735\pi$
$379$	−0.456858	−0.0234672	−0.0117336	+	0.999931i	$0.503735\pi$
$380$	0	0
$381$	−42.2846	−2.16630
$382$	0	0
$383$	34.1435	1.74465	0.872325	−	0.488926i	$-0.162611\pi$
$383$	34.1435	1.74465	0.872325	+	0.488926i	$0.162611\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−95.8093	−4.87026
$388$	0	0
$389$	25.9734	1.31690	0.658451	−	0.752624i	$-0.271212\pi$
$389$	25.9734	1.31690	0.658451	+	0.752624i	$0.271212\pi$
$390$	0	0
$391$	−38.7943	−1.96191
$392$	0	0
$393$	−6.77264	−0.341634
$394$	0	0
$395$	−6.81326	−0.342812
$396$	0	0
$397$	6.02367	0.302319	0.151160	−	0.988509i	$-0.451699\pi$
$397$	6.02367	0.302319	0.151160	+	0.988509i	$0.451699\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−26.4710	−1.32190	−0.660948	−	0.750431i	$-0.729846\pi$
$401$	−26.4710	−1.32190	−0.660948	+	0.750431i	$0.729846\pi$
$402$	0	0
$403$	−18.4120	−0.917165
$404$	0	0
$405$	40.9302	2.03384
$406$	0	0
$407$	−18.8877	−0.936227
$408$	0	0
$409$	−0.397954	−0.0196775	−0.00983877	−	0.999952i	$-0.503132\pi$
$409$	−0.397954	−0.0196775	−0.00983877	+	0.999952i	$0.503132\pi$
$410$	0	0
$411$	−31.8629	−1.57168
$412$	0	0
$413$	0	0
$414$	0	0
$415$	12.5584	0.616467
$416$	0	0
$417$	24.2429	1.18718
$418$	0	0
$419$	−20.9401	−1.02299	−0.511495	−	0.859286i	$-0.670908\pi$
$419$	−20.9401	−1.02299	−0.511495	+	0.859286i	$0.670908\pi$
$420$	0	0
$421$	20.6101	1.00447	0.502236	−	0.864730i	$-0.332511\pi$
$421$	20.6101	1.00447	0.502236	+	0.864730i	$0.332511\pi$
$422$	0	0
$423$	−67.3302	−3.27371
$424$	0	0
$425$	15.4631	0.750070
$426$	0	0
$427$	0	0
$428$	0	0
$429$	37.7192	1.82110
$430$	0	0
$431$	−17.5576	−0.845722	−0.422861	−	0.906195i	$-0.638974\pi$
$431$	−17.5576	−0.845722	−0.422861	+	0.906195i	$0.638974\pi$
$432$	0	0
$433$	23.6561	1.13684	0.568421	−	0.822738i	$-0.307555\pi$
$433$	23.6561	1.13684	0.568421	+	0.822738i	$0.307555\pi$
$434$	0	0
$435$	−28.5599	−1.36934
$436$	0	0
$437$	6.90622	0.330369
$438$	0	0
$439$	−29.8671	−1.42548	−0.712739	−	0.701429i	$-0.752546\pi$
$439$	−29.8671	−1.42548	−0.712739	+	0.701429i	$0.752546\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	5.00422	0.237758	0.118879	−	0.992909i	$-0.462070\pi$
$443$	5.00422	0.237758	0.118879	+	0.992909i	$0.462070\pi$
$444$	0	0
$445$	−25.9424	−1.22979
$446$	0	0
$447$	−0.0932362	−0.00440992
$448$	0	0
$449$	−39.7647	−1.87661	−0.938306	−	0.345805i	$-0.887606\pi$
$449$	−39.7647	−1.87661	−0.938306	+	0.345805i	$0.887606\pi$
$450$	0	0
$451$	−44.3053	−2.08626
$452$	0	0
$453$	−10.6561	−0.500669
$454$	0	0
$455$	0	0
$456$	0	0
$457$	26.7569	1.25164	0.625818	−	0.779969i	$-0.284765\pi$
$457$	26.7569	1.25164	0.625818	+	0.779969i	$0.284765\pi$
$458$	0	0
$459$	86.5661	4.04056
$460$	0	0
$461$	−11.0547	−0.514870	−0.257435	−	0.966296i	$-0.582877\pi$
$461$	−11.0547	−0.514870	−0.257435	+	0.966296i	$0.582877\pi$
$462$	0	0
$463$	11.3689	0.528357	0.264178	−	0.964474i	$-0.414899\pi$
$463$	11.3689	0.528357	0.264178	+	0.964474i	$0.414899\pi$
$464$	0	0
$465$	−39.1932	−1.81754
$466$	0	0
$467$	19.0219	0.880231	0.440115	−	0.897941i	$-0.354938\pi$
$467$	19.0219	0.880231	0.440115	+	0.897941i	$0.354938\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−53.7108	−2.47487
$472$	0	0
$473$	−62.1576	−2.85801
$474$	0	0
$475$	−2.75277	−0.126306
$476$	0	0
$477$	−86.6804	−3.96882
$478$	0	0
$479$	−27.9442	−1.27680	−0.638401	−	0.769704i	$-0.720404\pi$
$479$	−27.9442	−1.27680	−0.638401	+	0.769704i	$0.720404\pi$
$480$	0	0
$481$	8.70314	0.396829
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.52765	0.0693670
$486$	0	0
$487$	−22.2941	−1.01024	−0.505121	−	0.863049i	$-0.668552\pi$
$487$	−22.2941	−1.01024	−0.505121	+	0.863049i	$0.668552\pi$
$488$	0	0
$489$	−59.0500	−2.67033
$490$	0	0
$491$	−41.8778	−1.88992	−0.944960	−	0.327185i	$-0.893900\pi$
$491$	−41.8778	−1.88992	−0.944960	+	0.327185i	$0.893900\pi$
$492$	0	0
$493$	−32.7026	−1.47285
$494$	0	0
$495$	57.7995	2.59790
$496$	0	0
$497$	0	0
$498$	0	0
$499$	4.91769	0.220146	0.110073	−	0.993923i	$-0.464892\pi$
$499$	4.91769	0.220146	0.110073	+	0.993923i	$0.464892\pi$
$500$	0	0
$501$	−8.69925	−0.388654
$502$	0	0
$503$	−30.0878	−1.34155	−0.670773	−	0.741662i	$-0.734038\pi$
$503$	−30.0878	−1.34155	−0.670773	+	0.741662i	$0.734038\pi$
$504$	0	0
$505$	3.70921	0.165058
$506$	0	0
$507$	25.1619	1.11748
$508$	0	0
$509$	36.7466	1.62876	0.814382	−	0.580329i	$-0.197076\pi$
$509$	36.7466	1.62876	0.814382	+	0.580329i	$0.197076\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−15.4106	−0.680396
$514$	0	0
$515$	−6.58670	−0.290245
$516$	0	0
$517$	−43.6814	−1.92111
$518$	0	0
$519$	46.1365	2.02517
$520$	0	0
$521$	9.27348	0.406278	0.203139	−	0.979150i	$-0.434886\pi$
$521$	9.27348	0.406278	0.203139	+	0.979150i	$0.434886\pi$
$522$	0	0
$523$	16.3756	0.716054	0.358027	−	0.933711i	$-0.383450\pi$
$523$	16.3756	0.716054	0.358027	+	0.933711i	$0.383450\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−44.8783	−1.95493
$528$	0	0
$529$	24.6959	1.07373
$530$	0	0
$531$	59.0328	2.56180
$532$	0	0
$533$	20.4152	0.884280
$534$	0	0
$535$	0.610907	0.0264118
$536$	0	0
$537$	−60.4304	−2.60776
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−0.0307910	−0.00132381	−0.000661903	−	1.00000i	$-0.500211\pi$
$541$	−0.0307910	−0.00132381	−0.000661903		1.00000i	$0.500211\pi$
$542$	0	0
$543$	1.32335	0.0567904
$544$	0	0
$545$	−18.9729	−0.812709
$546$	0	0
$547$	−41.2359	−1.76312	−0.881560	−	0.472071i	$-0.843506\pi$
$547$	−41.2359	−1.76312	−0.881560	+	0.472071i	$0.843506\pi$
$548$	0	0
$549$	31.7102	1.35336
$550$	0	0
$551$	5.82177	0.248016
$552$	0	0
$553$	0	0
$554$	0	0
$555$	18.5262	0.786394
$556$	0	0
$557$	12.2236	0.517933	0.258966	−	0.965886i	$-0.416618\pi$
$557$	12.2236	0.517933	0.258966	+	0.965886i	$0.416618\pi$
$558$	0	0
$559$	28.6413	1.21140
$560$	0	0
$561$	91.9387	3.88165
$562$	0	0
$563$	−2.25634	−0.0950936	−0.0475468	−	0.998869i	$-0.515140\pi$
$563$	−2.25634	−0.0950936	−0.0475468	+	0.998869i	$0.515140\pi$
$564$	0	0
$565$	9.92156	0.417403
$566$	0	0
$567$	0	0
$568$	0	0
$569$	13.9784	0.586004	0.293002	−	0.956112i	$-0.405346\pi$
$569$	13.9784	0.586004	0.293002	+	0.956112i	$0.405346\pi$
$570$	0	0
$571$	−8.08951	−0.338535	−0.169268	−	0.985570i	$-0.554140\pi$
$571$	−8.08951	−0.338535	−0.169268	+	0.985570i	$0.554140\pi$
$572$	0	0
$573$	11.2226	0.468833
$574$	0	0
$575$	−19.0112	−0.792823
$576$	0	0
$577$	15.8817	0.661162	0.330581	−	0.943778i	$-0.392755\pi$
$577$	15.8817	0.661162	0.330581	+	0.943778i	$0.392755\pi$
$578$	0	0
$579$	12.5917	0.523291
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−56.2351	−2.32902
$584$	0	0
$585$	−26.6331	−1.10114
$586$	0	0
$587$	30.4238	1.25573	0.627863	−	0.778324i	$-0.283930\pi$
$587$	30.4238	1.25573	0.627863	+	0.778324i	$0.283930\pi$
$588$	0	0
$589$	7.98931	0.329194
$590$	0	0
$591$	60.8531	2.50316
$592$	0	0
$593$	−7.35785	−0.302151	−0.151075	−	0.988522i	$-0.548274\pi$
$593$	−7.35785	−0.302151	−0.151075	+	0.988522i	$0.548274\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−59.7599	−2.44581
$598$	0	0
$599$	−13.0663	−0.533875	−0.266938	−	0.963714i	$-0.586012\pi$
$599$	−13.0663	−0.533875	−0.266938	+	0.963714i	$0.586012\pi$
$600$	0	0
$601$	−4.29987	−0.175395	−0.0876976	−	0.996147i	$-0.527951\pi$
$601$	−4.29987	−0.175395	−0.0876976	+	0.996147i	$0.527951\pi$
$602$	0	0
$603$	−43.7528	−1.78175
$604$	0	0
$605$	21.0084	0.854113
$606$	0	0
$607$	31.4202	1.27531	0.637654	−	0.770323i	$-0.279905\pi$
$607$	31.4202	1.27531	0.637654	+	0.770323i	$0.279905\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	20.1277	0.814280
$612$	0	0
$613$	−26.7948	−1.08223	−0.541116	−	0.840948i	$-0.681998\pi$
$613$	−26.7948	−1.08223	−0.541116	+	0.840948i	$0.681998\pi$
$614$	0	0
$615$	43.4575	1.75237
$616$	0	0
$617$	12.6535	0.509410	0.254705	−	0.967019i	$-0.418022\pi$
$617$	12.6535	0.509410	0.254705	+	0.967019i	$0.418022\pi$
$618$	0	0
$619$	−18.3980	−0.739479	−0.369739	−	0.929135i	$-0.620553\pi$
$619$	−18.3980	−0.739479	−0.369739	+	0.929135i	$0.620553\pi$
$620$	0	0
$621$	−106.429	−4.27086
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−3.65844	−0.146338
$626$	0	0
$627$	−16.3671	−0.653638
$628$	0	0
$629$	21.2135	0.845837
$630$	0	0
$631$	−10.6249	−0.422971	−0.211486	−	0.977381i	$-0.567830\pi$
$631$	−10.6249	−0.422971	−0.211486	+	0.977381i	$0.567830\pi$
$632$	0	0
$633$	−44.4524	−1.76682
$634$	0	0
$635$	19.3699	0.768673
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−74.7993	−2.95902
$640$	0	0
$641$	−29.3679	−1.15996	−0.579982	−	0.814629i	$-0.696940\pi$
$641$	−29.3679	−1.15996	−0.579982	+	0.814629i	$0.696940\pi$
$642$	0	0
$643$	−18.3861	−0.725079	−0.362539	−	0.931968i	$-0.618090\pi$
$643$	−18.3861	−0.725079	−0.362539	+	0.931968i	$0.618090\pi$
$644$	0	0
$645$	60.9681	2.40062
$646$	0	0
$647$	34.2813	1.34774	0.673869	−	0.738851i	$-0.264631\pi$
$647$	34.2813	1.34774	0.673869	+	0.738851i	$0.264631\pi$
$648$	0	0
$649$	38.2983	1.50334
$650$	0	0
$651$	0	0
$652$	0	0
$653$	45.4854	1.77998	0.889991	−	0.455979i	$-0.150711\pi$
$653$	45.4854	1.77998	0.889991	+	0.455979i	$0.150711\pi$
$654$	0	0
$655$	3.10245	0.121223
$656$	0	0
$657$	16.7891	0.655006
$658$	0	0
$659$	−47.1177	−1.83544	−0.917722	−	0.397223i	$-0.869974\pi$
$659$	−47.1177	−1.83544	−0.917722	+	0.397223i	$0.869974\pi$
$660$	0	0
$661$	−43.3330	−1.68546	−0.842729	−	0.538338i	$-0.819052\pi$
$661$	−43.3330	−1.68546	−0.842729	+	0.538338i	$0.819052\pi$
$662$	0	0
$663$	−42.3639	−1.64528
$664$	0	0
$665$	0	0
$666$	0	0
$667$	40.2064	1.55680
$668$	0	0
$669$	70.9454	2.74291
$670$	0	0
$671$	20.5725	0.794191
$672$	0	0
$673$	8.80884	0.339556	0.169778	−	0.985482i	$-0.445695\pi$
$673$	8.80884	0.339556	0.169778	+	0.985482i	$0.445695\pi$
$674$	0	0
$675$	42.4219	1.63282
$676$	0	0
$677$	−16.4151	−0.630884	−0.315442	−	0.948945i	$-0.602153\pi$
$677$	−16.4151	−0.630884	−0.315442	+	0.948945i	$0.602153\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−28.4660	−1.09082
$682$	0	0
$683$	−0.563564	−0.0215642	−0.0107821	−	0.999942i	$-0.503432\pi$
$683$	−0.563564	−0.0215642	−0.0107821	+	0.999942i	$0.503432\pi$
$684$	0	0
$685$	14.5959	0.557682
$686$	0	0
$687$	−76.5396	−2.92017
$688$	0	0
$689$	25.9123	0.987178
$690$	0	0
$691$	−31.9073	−1.21381	−0.606906	−	0.794774i	$-0.707589\pi$
$691$	−31.9073	−1.21381	−0.606906	+	0.794774i	$0.707589\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−11.1053	−0.421249
$696$	0	0
$697$	49.7611	1.88483
$698$	0	0
$699$	−40.8206	−1.54398
$700$	0	0
$701$	7.15536	0.270254	0.135127	−	0.990828i	$-0.456856\pi$
$701$	7.15536	0.270254	0.135127	+	0.990828i	$0.456856\pi$
$702$	0	0
$703$	−3.77646	−0.142432
$704$	0	0
$705$	42.8455	1.61365
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−18.2363	−0.684879	−0.342440	−	0.939540i	$-0.611253\pi$
$709$	−18.2363	−0.684879	−0.342440	+	0.939540i	$0.611253\pi$
$710$	0	0
$711$	−35.0379	−1.31402
$712$	0	0
$713$	55.1759	2.06635
$714$	0	0
$715$	−17.2786	−0.646183
$716$	0	0
$717$	39.8777	1.48926
$718$	0	0
$719$	24.2489	0.904331	0.452166	−	0.891934i	$-0.350652\pi$
$719$	24.2489	0.904331	0.452166	+	0.891934i	$0.350652\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	38.7735	1.44200
$724$	0	0
$725$	−16.0260	−0.595190
$726$	0	0
$727$	24.1905	0.897177	0.448588	−	0.893738i	$-0.351927\pi$
$727$	24.1905	0.897177	0.448588	+	0.893738i	$0.351927\pi$
$728$	0	0
$729$	59.1946	2.19239
$730$	0	0
$731$	69.8117	2.58208
$732$	0	0
$733$	−17.3676	−0.641486	−0.320743	−	0.947166i	$-0.603933\pi$
$733$	−17.3676	−0.641486	−0.320743	+	0.947166i	$0.603933\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−28.3852	−1.04558
$738$	0	0
$739$	−17.6913	−0.650784	−0.325392	−	0.945579i	$-0.605496\pi$
$739$	−17.6913	−0.650784	−0.325392	+	0.945579i	$0.605496\pi$
$740$	0	0
$741$	7.54169	0.277051
$742$	0	0
$743$	37.7422	1.38463	0.692314	−	0.721596i	$-0.256591\pi$
$743$	37.7422	1.38463	0.692314	+	0.721596i	$0.256591\pi$
$744$	0	0
$745$	0.0427101	0.00156478
$746$	0	0
$747$	64.5828	2.36296
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−32.3926	−1.18202	−0.591012	−	0.806663i	$-0.701271\pi$
$751$	−32.3926	−1.18202	−0.591012	+	0.806663i	$0.701271\pi$
$752$	0	0
$753$	8.53300	0.310960
$754$	0	0
$755$	4.88142	0.177653
$756$	0	0
$757$	−14.4341	−0.524615	−0.262307	−	0.964984i	$-0.584483\pi$
$757$	−14.4341	−0.524615	−0.262307	+	0.964984i	$0.584483\pi$
$758$	0	0
$759$	−113.035	−4.10290
$760$	0	0
$761$	42.6312	1.54538	0.772690	−	0.634783i	$-0.218911\pi$
$761$	42.6312	1.54538	0.772690	+	0.634783i	$0.218911\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−64.9169	−2.34708
$766$	0	0
$767$	−17.6473	−0.637206
$768$	0	0
$769$	−18.8097	−0.678296	−0.339148	−	0.940733i	$-0.610139\pi$
$769$	−18.8097	−0.678296	−0.339148	+	0.940733i	$0.610139\pi$
$770$	0	0
$771$	50.2108	1.80830
$772$	0	0
$773$	29.1011	1.04669	0.523347	−	0.852120i	$-0.324683\pi$
$773$	29.1011	1.04669	0.523347	+	0.852120i	$0.324683\pi$
$774$	0	0
$775$	−21.9927	−0.790001
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−8.85855	−0.317390
$780$	0	0
$781$	−48.5271	−1.73644
$782$	0	0
$783$	−89.7172	−3.20623
$784$	0	0
$785$	24.6042	0.878160
$786$	0	0
$787$	36.3075	1.29422	0.647112	−	0.762395i	$-0.275977\pi$
$787$	36.3075	1.29422	0.647112	+	0.762395i	$0.275977\pi$
$788$	0	0
$789$	100.033	3.56129
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−9.47946	−0.336625
$794$	0	0
$795$	55.1589	1.95629
$796$	0	0
$797$	−26.1055	−0.924704	−0.462352	−	0.886697i	$-0.652994\pi$
$797$	−26.1055	−0.924704	−0.462352	+	0.886697i	$0.652994\pi$
$798$	0	0
$799$	49.0603	1.73563
$800$	0	0
$801$	−133.412	−4.71386
$802$	0	0
$803$	10.8922	0.384377
$804$	0	0
$805$	0	0
$806$	0	0
$807$	66.5362	2.34219
$808$	0	0
$809$	53.6208	1.88521	0.942604	−	0.333913i	$-0.108369\pi$
$809$	53.6208	1.88521	0.942604	+	0.333913i	$0.108369\pi$
$810$	0	0
$811$	44.8989	1.57661	0.788307	−	0.615282i	$-0.210958\pi$
$811$	44.8989	1.57661	0.788307	+	0.615282i	$0.210958\pi$
$812$	0	0
$813$	28.3611	0.994667
$814$	0	0
$815$	27.0499	0.947518
$816$	0	0
$817$	−12.4280	−0.434800
$818$	0	0
$819$	0	0
$820$	0	0
$821$	29.6509	1.03483	0.517413	−	0.855736i	$-0.326895\pi$
$821$	29.6509	1.03483	0.517413	+	0.855736i	$0.326895\pi$
$822$	0	0
$823$	−16.4458	−0.573263	−0.286631	−	0.958041i	$-0.592536\pi$
$823$	−16.4458	−0.573263	−0.286631	+	0.958041i	$0.592536\pi$
$824$	0	0
$825$	45.0547	1.56860
$826$	0	0
$827$	−0.222627	−0.00774151	−0.00387076	−	0.999993i	$-0.501232\pi$
$827$	−0.222627	−0.00774151	−0.00387076	+	0.999993i	$0.501232\pi$
$828$	0	0
$829$	16.6051	0.576719	0.288359	−	0.957522i	$-0.406890\pi$
$829$	16.6051	0.576719	0.288359	+	0.957522i	$0.406890\pi$
$830$	0	0
$831$	91.8467	3.18613
$832$	0	0
$833$	0	0
$834$	0	0
$835$	3.98500	0.137907
$836$	0	0
$837$	−123.120	−4.25566
$838$	0	0
$839$	−6.54327	−0.225899	−0.112949	−	0.993601i	$-0.536030\pi$
$839$	−6.54327	−0.225899	−0.112949	+	0.993601i	$0.536030\pi$
$840$	0	0
$841$	4.89297	0.168723
$842$	0	0
$843$	57.6720	1.98633
$844$	0	0
$845$	−11.5263	−0.396517
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−46.5876	−1.59888
$850$	0	0
$851$	−26.0811	−0.894047
$852$	0	0
$853$	26.2381	0.898375	0.449187	−	0.893438i	$-0.351714\pi$
$853$	26.2381	0.898375	0.449187	+	0.893438i	$0.351714\pi$
$854$	0	0
$855$	11.5566	0.395228
$856$	0	0
$857$	15.3381	0.523940	0.261970	−	0.965076i	$-0.415628\pi$
$857$	15.3381	0.523940	0.261970	+	0.965076i	$0.415628\pi$
$858$	0	0
$859$	48.1523	1.64293	0.821467	−	0.570256i	$-0.193156\pi$
$859$	48.1523	1.64293	0.821467	+	0.570256i	$0.193156\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	26.2615	0.893951	0.446975	−	0.894546i	$-0.352501\pi$
$863$	26.2615	0.893951	0.446975	+	0.894546i	$0.352501\pi$
$864$	0	0
$865$	−21.1344	−0.718592
$866$	0	0
$867$	−47.6277	−1.61752
$868$	0	0
$869$	−22.7313	−0.771107
$870$	0	0
$871$	13.0795	0.443180
$872$	0	0
$873$	7.85609	0.265888
$874$	0	0
$875$	0	0
$876$	0	0
$877$	56.3404	1.90248	0.951240	−	0.308452i	$-0.0998110\pi$
$877$	56.3404	1.90248	0.951240	+	0.308452i	$0.0998110\pi$
$878$	0	0
$879$	−14.9375	−0.503829
$880$	0	0
$881$	−50.0489	−1.68619	−0.843095	−	0.537765i	$-0.819269\pi$
$881$	−50.0489	−1.68619	−0.843095	+	0.537765i	$0.819269\pi$
$882$	0	0
$883$	−26.1915	−0.881416	−0.440708	−	0.897651i	$-0.645273\pi$
$883$	−26.1915	−0.881416	−0.440708	+	0.897651i	$0.645273\pi$
$884$	0	0
$885$	−37.5654	−1.26275
$886$	0	0
$887$	−45.6886	−1.53407	−0.767036	−	0.641604i	$-0.778269\pi$
$887$	−45.6886	−1.53407	−0.767036	+	0.641604i	$0.778269\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	136.557	4.57483
$892$	0	0
$893$	−8.73380	−0.292266
$894$	0	0
$895$	27.6823	0.925316
$896$	0	0
$897$	52.0846	1.73905
$898$	0	0
$899$	46.5119	1.55126
$900$	0	0
$901$	63.1599	2.10416
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−0.606208	−0.0201510
$906$	0	0
$907$	30.7685	1.02165	0.510825	−	0.859684i	$-0.329340\pi$
$907$	30.7685	1.02165	0.510825	+	0.859684i	$0.329340\pi$
$908$	0	0
$909$	19.0750	0.632677
$910$	0	0
$911$	−15.6411	−0.518211	−0.259106	−	0.965849i	$-0.583428\pi$
$911$	−15.6411	−0.518211	−0.259106	+	0.965849i	$0.583428\pi$
$912$	0	0
$913$	41.8990	1.38665
$914$	0	0
$915$	−20.1788	−0.667089
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−29.0587	−0.958559	−0.479280	−	0.877662i	$-0.659102\pi$
$919$	−29.0587	−0.958559	−0.479280	+	0.877662i	$0.659102\pi$
$920$	0	0
$921$	16.5517	0.545396
$922$	0	0
$923$	22.3605	0.736006
$924$	0	0
$925$	10.3957	0.341809
$926$	0	0
$927$	−33.8728	−1.11253
$928$	0	0
$929$	−58.5551	−1.92113	−0.960565	−	0.278054i	$-0.910311\pi$
$929$	−58.5551	−1.92113	−0.960565	+	0.278054i	$0.910311\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	40.6242	1.32998
$934$	0	0
$935$	−42.1158	−1.37733
$936$	0	0
$937$	−22.7598	−0.743531	−0.371765	−	0.928327i	$-0.621247\pi$
$937$	−22.7598	−0.743531	−0.371765	+	0.928327i	$0.621247\pi$
$938$	0	0
$939$	30.0176	0.979586
$940$	0	0
$941$	9.31079	0.303523	0.151762	−	0.988417i	$-0.451505\pi$
$941$	9.31079	0.303523	0.151762	+	0.988417i	$0.451505\pi$
$942$	0	0
$943$	−61.1791	−1.99227
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−13.0341	−0.423551	−0.211776	−	0.977318i	$-0.567925\pi$
$947$	−13.0341	−0.423551	−0.211776	+	0.977318i	$0.567925\pi$
$948$	0	0
$949$	−5.01894	−0.162922
$950$	0	0
$951$	33.3653	1.08194
$952$	0	0
$953$	−0.708752	−0.0229587	−0.0114794	−	0.999934i	$-0.503654\pi$
$953$	−0.708752	−0.0229587	−0.0114794	+	0.999934i	$0.503654\pi$
$954$	0	0
$955$	−5.14093	−0.166357
$956$	0	0
$957$	−95.2853	−3.08014
$958$	0	0
$959$	0	0
$960$	0	0
$961$	32.8290	1.05900
$962$	0	0
$963$	3.14165	0.101238
$964$	0	0
$965$	−5.76805	−0.185680
$966$	0	0
$967$	−11.7748	−0.378652	−0.189326	−	0.981914i	$-0.560630\pi$
$967$	−11.7748	−0.378652	−0.189326	+	0.981914i	$0.560630\pi$
$968$	0	0
$969$	18.3825	0.590531
$970$	0	0
$971$	−12.4343	−0.399037	−0.199518	−	0.979894i	$-0.563938\pi$
$971$	−12.4343	−0.399037	−0.199518	+	0.979894i	$0.563938\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−20.7605	−0.664869
$976$	0	0
$977$	−48.9978	−1.56758	−0.783789	−	0.621027i	$-0.786716\pi$
$977$	−48.9978	−1.56758	−0.783789	+	0.621027i	$0.786716\pi$
$978$	0	0
$979$	−86.5526	−2.76623
$980$	0	0
$981$	−97.5700	−3.11517
$982$	0	0
$983$	−7.80916	−0.249073	−0.124537	−	0.992215i	$-0.539744\pi$
$983$	−7.80916	−0.249073	−0.124537	+	0.992215i	$0.539744\pi$
$984$	0	0
$985$	−27.8759	−0.888200
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−85.8305	−2.72925
$990$	0	0
$991$	−47.1560	−1.49796	−0.748979	−	0.662593i	$-0.769456\pi$
$991$	−47.1560	−1.49796	−0.748979	+	0.662593i	$0.769456\pi$
$992$	0	0
$993$	16.1499	0.512503
$994$	0	0
$995$	27.3751	0.867850
$996$	0	0
$997$	34.4501	1.09104	0.545522	−	0.838096i	$-0.316331\pi$
$997$	34.4501	1.09104	0.545522	+	0.838096i	$0.316331\pi$
$998$	0	0
$999$	58.1977	1.84129

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.bv.1.1	✓	14
7.6	odd	2		7448.2.a.bw.1.14	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7448.2.a.bv.1.1	✓	14	1.1	even	1	trivial
7448.2.a.bw.1.14	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)

\(f(q)\)	\(=\)	\(q-3.27248 q^{3} +1.49908 q^{5} +7.70916 q^{9} +O(q^{10})\)	\(q-3.27248 q^{3} +1.49908 q^{5} +7.70916 q^{9} +5.00142 q^{11} -2.30458 q^{13} -4.90571 q^{15} -5.61729 q^{17} +1.00000 q^{19} +6.90622 q^{23} -2.75277 q^{25} -15.4106 q^{27} +5.82177 q^{29} +7.98931 q^{31} -16.3671 q^{33} -3.77646 q^{37} +7.54169 q^{39} -8.85855 q^{41} -12.4280 q^{43} +11.5566 q^{45} -8.73380 q^{47} +18.3825 q^{51} -11.2438 q^{53} +7.49752 q^{55} -3.27248 q^{57} +7.65749 q^{59} +4.11332 q^{61} -3.45474 q^{65} -5.67543 q^{67} -22.6005 q^{69} -9.70266 q^{71} +2.17782 q^{73} +9.00839 q^{75} -4.54497 q^{79} +27.3036 q^{81} +8.37741 q^{83} -8.42076 q^{85} -19.0516 q^{87} -17.3056 q^{89} -26.1449 q^{93} +1.49908 q^{95} +1.01906 q^{97} +38.5567 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 2 q^{3} - 2 q^{5} + 16 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more