LMFDB - Embedded newform 7448.2.a.bk.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:	$4$
Coefficient field:	4.4.18097.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 7x^{2} + 6x + 4 $ x^4 - x^3 - 7x^2 + 6x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1064)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.50155$ of defining polynomial
Character	$\chi$	$=$	7448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.70205 q^{3} +3.25776 q^{5} +4.30105 q^{9} +O(q^{10})$	$q+2.70205 q^{3} +3.25776 q^{5} +4.30105 q^{9} -1.05726 q^{11} +6.64789 q^{13} +8.80261 q^{15} +7.80261 q^{17} +1.00000 q^{19} +0.240694 q^{23} +5.61297 q^{25} +3.51551 q^{27} -9.41557 q^{29} -5.71601 q^{31} -2.85676 q^{33} +5.96818 q^{37} +17.9629 q^{39} +2.29795 q^{41} -3.61297 q^{43} +14.0118 q^{45} -4.49845 q^{47} +21.0830 q^{51} +6.30105 q^{53} -3.44429 q^{55} +2.70205 q^{57} -5.25776 q^{59} -12.1952 q^{61} +21.6572 q^{65} -14.2098 q^{67} +0.650368 q^{69} -0.612968 q^{71} +2.26613 q^{73} +15.1665 q^{75} +9.41805 q^{79} -3.40409 q^{81} -1.75683 q^{83} +25.4190 q^{85} -25.4413 q^{87} +1.56409 q^{89} -15.4449 q^{93} +3.25776 q^{95} +15.9087 q^{97} -4.54733 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{5} + 6 q^{9}+O(q^{10}) $ 4 * q + 3 * q^5 + 6 * q^9	$ 4 q + 3 q^{5} + 6 q^{9} + 6 q^{11} + 4 q^{13} + 13 q^{15} + 9 q^{17} + 4 q^{19} + 22 q^{23} + 13 q^{25} - 6 q^{27} - 6 q^{29} - 3 q^{31} - q^{33} + 15 q^{37} + 29 q^{39} + 20 q^{41} - 5 q^{43} - 9 q^{45} - 29 q^{47} + 14 q^{53} - 13 q^{55} - 11 q^{59} - 15 q^{61} - 2 q^{65} + 9 q^{67} - 19 q^{69} + 7 q^{71} + 11 q^{73} - 4 q^{75} + 7 q^{79} + 8 q^{81} + 15 q^{83} - 7 q^{85} + 4 q^{87} + 19 q^{89} - 38 q^{93} + 3 q^{95} + 9 q^{97} - 7 q^{99}+O(q^{100}) $ 4 * q + 3 * q^5 + 6 * q^9 + 6 * q^11 + 4 * q^13 + 13 * q^15 + 9 * q^17 + 4 * q^19 + 22 * q^23 + 13 * q^25 - 6 * q^27 - 6 * q^29 - 3 * q^31 - q^33 + 15 * q^37 + 29 * q^39 + 20 * q^41 - 5 * q^43 - 9 * q^45 - 29 * q^47 + 14 * q^53 - 13 * q^55 - 11 * q^59 - 15 * q^61 - 2 * q^65 + 9 * q^67 - 19 * q^69 + 7 * q^71 + 11 * q^73 - 4 * q^75 + 7 * q^79 + 8 * q^81 + 15 * q^83 - 7 * q^85 + 4 * q^87 + 19 * q^89 - 38 * q^93 + 3 * q^95 + 9 * q^97 - 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.70205	1.56003	0.780014	−	0.625763i	$-0.215212\pi$
$3$	2.70205	1.56003	0.780014	+	0.625763i	$0.215212\pi$
$4$	0	0
$5$	3.25776	1.45691	0.728456	−	0.685092i	$-0.240238\pi$
$5$	3.25776	1.45691	0.728456	+	0.685092i	$0.240238\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	4.30105	1.43368
$10$	0	0
$11$	−1.05726	−0.318776	−0.159388	−	0.987216i	$-0.550952\pi$
$11$	−1.05726	−0.318776	−0.159388	+	0.987216i	$0.550952\pi$
$12$	0	0
$13$	6.64789	1.84379	0.921896	−	0.387437i	$-0.126639\pi$
$13$	6.64789	1.84379	0.921896	+	0.387437i	$0.126639\pi$
$14$	0	0
$15$	8.80261	2.27282
$16$	0	0
$17$	7.80261	1.89241	0.946205	−	0.323568i	$-0.104883\pi$
$17$	7.80261	1.89241	0.946205	+	0.323568i	$0.104883\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.240694	0.0501883	0.0250941	−	0.999685i	$-0.492011\pi$
$23$	0.240694	0.0501883	0.0250941	+	0.999685i	$0.492011\pi$
$24$	0	0
$25$	5.61297	1.12259
$26$	0	0
$27$	3.51551	0.676560
$28$	0	0
$29$	−9.41557	−1.74843	−0.874214	−	0.485541i	$-0.838623\pi$
$29$	−9.41557	−1.74843	−0.874214	+	0.485541i	$0.838623\pi$
$30$	0	0
$31$	−5.71601	−1.02662	−0.513312	−	0.858202i	$-0.671582\pi$
$31$	−5.71601	−1.02662	−0.513312	+	0.858202i	$0.671582\pi$
$32$	0	0
$33$	−2.85676	−0.497299
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.96818	0.981163	0.490581	−	0.871395i	$-0.336784\pi$
$37$	5.96818	0.981163	0.490581	+	0.871395i	$0.336784\pi$
$38$	0	0
$39$	17.9629	2.87637
$40$	0	0
$41$	2.29795	0.358880	0.179440	−	0.983769i	$-0.442571\pi$
$41$	2.29795	0.358880	0.179440	+	0.983769i	$0.442571\pi$
$42$	0	0
$43$	−3.61297	−0.550972	−0.275486	−	0.961305i	$-0.588839\pi$
$43$	−3.61297	−0.550972	−0.275486	+	0.961305i	$0.588839\pi$
$44$	0	0
$45$	14.0118	2.08875
$46$	0	0
$47$	−4.49845	−0.656166	−0.328083	−	0.944649i	$-0.606403\pi$
$47$	−4.49845	−0.656166	−0.328083	+	0.944649i	$0.606403\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	21.0830	2.95221
$52$	0	0
$53$	6.30105	0.865516	0.432758	−	0.901510i	$-0.357540\pi$
$53$	6.30105	0.865516	0.432758	+	0.901510i	$0.357540\pi$
$54$	0	0
$55$	−3.44429	−0.464428
$56$	0	0
$57$	2.70205	0.357895
$58$	0	0
$59$	−5.25776	−0.684501	−0.342251	−	0.939609i	$-0.611189\pi$
$59$	−5.25776	−0.684501	−0.342251	+	0.939609i	$0.611189\pi$
$60$	0	0
$61$	−12.1952	−1.56144	−0.780719	−	0.624882i	$-0.785147\pi$
$61$	−12.1952	−1.56144	−0.780719	+	0.624882i	$0.785147\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	21.6572	2.68624
$66$	0	0
$67$	−14.2098	−1.73600	−0.868002	−	0.496561i	$-0.834596\pi$
$67$	−14.2098	−1.73600	−0.868002	+	0.496561i	$0.834596\pi$
$68$	0	0
$69$	0.650368	0.0782951
$70$	0	0
$71$	−0.612968	−0.0727459	−0.0363729	−	0.999338i	$-0.511580\pi$
$71$	−0.612968	−0.0727459	−0.0363729	+	0.999338i	$0.511580\pi$
$72$	0	0
$73$	2.26613	0.265231	0.132615	−	0.991168i	$-0.457662\pi$
$73$	2.26613	0.265231	0.132615	+	0.991168i	$0.457662\pi$
$74$	0	0
$75$	15.1665	1.75128
$76$	0	0
$77$	0	0
$78$	0	0
$79$	9.41805	1.05961	0.529807	−	0.848118i	$-0.322264\pi$
$79$	9.41805	1.05961	0.529807	+	0.848118i	$0.322264\pi$
$80$	0	0
$81$	−3.40409	−0.378233
$82$	0	0
$83$	−1.75683	−0.192837	−0.0964184	−	0.995341i	$-0.530739\pi$
$83$	−1.75683	−0.192837	−0.0964184	+	0.995341i	$0.530739\pi$
$84$	0	0
$85$	25.4190	2.75708
$86$	0	0
$87$	−25.4413	−2.72760
$88$	0	0
$89$	1.56409	0.165793	0.0828965	−	0.996558i	$-0.473583\pi$
$89$	1.56409	0.165793	0.0828965	+	0.996558i	$0.473583\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−15.4449	−1.60156
$94$	0	0
$95$	3.25776	0.334239
$96$	0	0
$97$	15.9087	1.61529	0.807644	−	0.589670i	$-0.200742\pi$
$97$	15.9087	1.61529	0.807644	+	0.589670i	$0.200742\pi$
$98$	0	0
$99$	−4.54733	−0.457024
$100$	0	0
$101$	2.26861	0.225735	0.112868	−	0.993610i	$-0.463996\pi$
$101$	2.26861	0.225735	0.112868	+	0.993610i	$0.463996\pi$
$102$	0	0
$103$	−9.35831	−0.922102	−0.461051	−	0.887374i	$-0.652527\pi$
$103$	−9.35831	−0.922102	−0.461051	+	0.887374i	$0.652527\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−15.3645	−1.48534	−0.742672	−	0.669655i	$-0.766442\pi$
$107$	−15.3645	−1.48534	−0.742672	+	0.669655i	$0.766442\pi$
$108$	0	0
$109$	−8.30974	−0.795928	−0.397964	−	0.917401i	$-0.630283\pi$
$109$	−8.30974	−0.795928	−0.397964	+	0.917401i	$0.630283\pi$
$110$	0	0
$111$	16.1263	1.53064
$112$	0	0
$113$	−0.817364	−0.0768911	−0.0384456	−	0.999261i	$-0.512241\pi$
$113$	−0.817364	−0.0768911	−0.0384456	+	0.999261i	$0.512241\pi$
$114$	0	0
$115$	0.784124	0.0731199
$116$	0	0
$117$	28.5929	2.64342
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−9.88220	−0.898382
$122$	0	0
$123$	6.20918	0.559863
$124$	0	0
$125$	1.99690	0.178608
$126$	0	0
$127$	−11.9545	−1.06079	−0.530396	−	0.847750i	$-0.677957\pi$
$127$	−11.9545	−1.06079	−0.530396	+	0.847750i	$0.677957\pi$
$128$	0	0
$129$	−9.76241	−0.859532
$130$	0	0
$131$	−1.68809	−0.147489	−0.0737444	−	0.997277i	$-0.523495\pi$
$131$	−1.68809	−0.147489	−0.0737444	+	0.997277i	$0.523495\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	11.4527	0.985689
$136$	0	0
$137$	−1.10366	−0.0942920	−0.0471460	−	0.998888i	$-0.515013\pi$
$137$	−1.10366	−0.0942920	−0.0471460	+	0.998888i	$0.515013\pi$
$138$	0	0
$139$	−14.1833	−1.20301	−0.601504	−	0.798870i	$-0.705432\pi$
$139$	−14.1833	−1.20301	−0.601504	+	0.798870i	$0.705432\pi$
$140$	0	0
$141$	−12.1550	−1.02364
$142$	0	0
$143$	−7.02854	−0.587756
$144$	0	0
$145$	−30.6736	−2.54731
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−20.1944	−1.65439	−0.827196	−	0.561914i	$-0.810065\pi$
$149$	−20.1944	−1.65439	−0.827196	+	0.561914i	$0.810065\pi$
$150$	0	0
$151$	11.7160	0.953435	0.476718	−	0.879057i	$-0.341826\pi$
$151$	11.7160	0.953435	0.476718	+	0.879057i	$0.341826\pi$
$152$	0	0
$153$	33.5594	2.71312
$154$	0	0
$155$	−18.6213	−1.49570
$156$	0	0
$157$	2.71991	0.217072	0.108536	−	0.994093i	$-0.465384\pi$
$157$	2.71991	0.217072	0.108536	+	0.994093i	$0.465384\pi$
$158$	0	0
$159$	17.0257	1.35023
$160$	0	0
$161$	0	0
$162$	0	0
$163$	22.5999	1.77016	0.885082	−	0.465436i	$-0.154102\pi$
$163$	22.5999	1.77016	0.885082	+	0.465436i	$0.154102\pi$
$164$	0	0
$165$	−9.30664	−0.724521
$166$	0	0
$167$	3.47283	0.268736	0.134368	−	0.990932i	$-0.457100\pi$
$167$	3.47283	0.268736	0.134368	+	0.990932i	$0.457100\pi$
$168$	0	0
$169$	31.1944	2.39957
$170$	0	0
$171$	4.30105	0.328910
$172$	0	0
$173$	−0.824944	−0.0627193	−0.0313597	−	0.999508i	$-0.509984\pi$
$173$	−0.824944	−0.0627193	−0.0313597	+	0.999508i	$0.509984\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−14.2067	−1.06784
$178$	0	0
$179$	−24.4159	−1.82493	−0.912464	−	0.409157i	$-0.865823\pi$
$179$	−24.4159	−1.82493	−0.912464	+	0.409157i	$0.865823\pi$
$180$	0	0
$181$	−20.5025	−1.52394	−0.761968	−	0.647614i	$-0.775767\pi$
$181$	−20.5025	−1.52394	−0.761968	+	0.647614i	$0.775767\pi$
$182$	0	0
$183$	−32.9520	−2.43589
$184$	0	0
$185$	19.4429	1.42947
$186$	0	0
$187$	−8.24938	−0.603254
$188$	0	0
$189$	0	0
$190$	0	0
$191$	15.6535	1.13265	0.566323	−	0.824184i	$-0.308366\pi$
$191$	15.6535	1.13265	0.566323	+	0.824184i	$0.308366\pi$
$192$	0	0
$193$	15.0285	1.08178	0.540889	−	0.841094i	$-0.318088\pi$
$193$	15.0285	1.08178	0.540889	+	0.841094i	$0.318088\pi$
$194$	0	0
$195$	58.5187	4.19061
$196$	0	0
$197$	4.07742	0.290504	0.145252	−	0.989395i	$-0.453601\pi$
$197$	4.07742	0.290504	0.145252	+	0.989395i	$0.453601\pi$
$198$	0	0
$199$	19.6712	1.39445	0.697225	−	0.716852i	$-0.254418\pi$
$199$	19.6712	1.39445	0.697225	+	0.716852i	$0.254418\pi$
$200$	0	0
$201$	−38.3955	−2.70821
$202$	0	0
$203$	0	0
$204$	0	0
$205$	7.48617	0.522857
$206$	0	0
$207$	1.03524	0.0719542
$208$	0	0
$209$	−1.05726	−0.0731321
$210$	0	0
$211$	8.15782	0.561607	0.280804	−	0.959765i	$-0.409399\pi$
$211$	8.15782	0.561607	0.280804	+	0.959765i	$0.409399\pi$
$212$	0	0
$213$	−1.65627	−0.113486
$214$	0	0
$215$	−11.7702	−0.802719
$216$	0	0
$217$	0	0
$218$	0	0
$219$	6.12320	0.413767
$220$	0	0
$221$	51.8708	3.48921
$222$	0	0
$223$	27.3198	1.82947	0.914736	−	0.404052i	$-0.132399\pi$
$223$	27.3198	1.82947	0.914736	+	0.404052i	$0.132399\pi$
$224$	0	0
$225$	24.1417	1.60945
$226$	0	0
$227$	7.57357	0.502675	0.251338	−	0.967899i	$-0.419130\pi$
$227$	7.57357	0.502675	0.251338	+	0.967899i	$0.419130\pi$
$228$	0	0
$229$	−7.81409	−0.516369	−0.258185	−	0.966096i	$-0.583124\pi$
$229$	−7.81409	−0.516369	−0.258185	+	0.966096i	$0.583124\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−21.2695	−1.39341	−0.696707	−	0.717356i	$-0.745352\pi$
$233$	−21.2695	−1.39341	−0.696707	+	0.717356i	$0.745352\pi$
$234$	0	0
$235$	−14.6548	−0.955977
$236$	0	0
$237$	25.4480	1.65303
$238$	0	0
$239$	−18.7345	−1.21183	−0.605917	−	0.795528i	$-0.707194\pi$
$239$	−18.7345	−1.21183	−0.605917	+	0.795528i	$0.707194\pi$
$240$	0	0
$241$	7.99224	0.514826	0.257413	−	0.966302i	$-0.417130\pi$
$241$	7.99224	0.514826	0.257413	+	0.966302i	$0.417130\pi$
$242$	0	0
$243$	−19.7445	−1.26661
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.64789	0.422995
$248$	0	0
$249$	−4.74703	−0.300831
$250$	0	0
$251$	14.9839	0.945773	0.472887	−	0.881123i	$-0.343212\pi$
$251$	14.9839	0.945773	0.472887	+	0.881123i	$0.343212\pi$
$252$	0	0
$253$	−0.254476	−0.0159988
$254$	0	0
$255$	68.6833	4.30111
$256$	0	0
$257$	5.85118	0.364987	0.182493	−	0.983207i	$-0.441583\pi$
$257$	5.85118	0.364987	0.182493	+	0.983207i	$0.441583\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−40.4969	−2.50669
$262$	0	0
$263$	24.2277	1.49394	0.746971	−	0.664857i	$-0.231508\pi$
$263$	24.2277	1.49394	0.746971	+	0.664857i	$0.231508\pi$
$264$	0	0
$265$	20.5273	1.26098
$266$	0	0
$267$	4.22624	0.258642
$268$	0	0
$269$	6.55261	0.399520	0.199760	−	0.979845i	$-0.435984\pi$
$269$	6.55261	0.399520	0.199760	+	0.979845i	$0.435984\pi$
$270$	0	0
$271$	5.44349	0.330669	0.165334	−	0.986238i	$-0.447130\pi$
$271$	5.44349	0.330669	0.165334	+	0.986238i	$0.447130\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.93436	−0.357855
$276$	0	0
$277$	−5.53537	−0.332588	−0.166294	−	0.986076i	$-0.553180\pi$
$277$	−5.53537	−0.332588	−0.166294	+	0.986076i	$0.553180\pi$
$278$	0	0
$279$	−24.5849	−1.47186
$280$	0	0
$281$	6.84776	0.408503	0.204252	−	0.978918i	$-0.434524\pi$
$281$	6.84776	0.408503	0.204252	+	0.978918i	$0.434524\pi$
$282$	0	0
$283$	−17.6826	−1.05112	−0.525562	−	0.850756i	$-0.676145\pi$
$283$	−17.6826	−1.05112	−0.525562	+	0.850756i	$0.676145\pi$
$284$	0	0
$285$	8.80261	0.521421
$286$	0	0
$287$	0	0
$288$	0	0
$289$	43.8806	2.58121
$290$	0	0
$291$	42.9862	2.51989
$292$	0	0
$293$	15.1802	0.886834	0.443417	−	0.896315i	$-0.353766\pi$
$293$	15.1802	0.886834	0.443417	+	0.896315i	$0.353766\pi$
$294$	0	0
$295$	−17.1285	−0.997259
$296$	0	0
$297$	−3.71681	−0.215671
$298$	0	0
$299$	1.60011	0.0925367
$300$	0	0
$301$	0	0
$302$	0	0
$303$	6.12990	0.352153
$304$	0	0
$305$	−39.7290	−2.27488
$306$	0	0
$307$	27.4267	1.56533	0.782663	−	0.622445i	$-0.213861\pi$
$307$	27.4267	1.56533	0.782663	+	0.622445i	$0.213861\pi$
$308$	0	0
$309$	−25.2866	−1.43850
$310$	0	0
$311$	9.79560	0.555458	0.277729	−	0.960660i	$-0.410418\pi$
$311$	9.79560	0.555458	0.277729	+	0.960660i	$0.410418\pi$
$312$	0	0
$313$	−22.9506	−1.29725	−0.648623	−	0.761110i	$-0.724655\pi$
$313$	−22.9506	−1.29725	−0.648623	+	0.761110i	$0.724655\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.74845	−0.322865	−0.161432	−	0.986884i	$-0.551611\pi$
$317$	−5.74845	−0.322865	−0.161432	+	0.986884i	$0.551611\pi$
$318$	0	0
$319$	9.95470	0.557356
$320$	0	0
$321$	−41.5156	−2.31718
$322$	0	0
$323$	7.80261	0.434149
$324$	0	0
$325$	37.3144	2.06983
$326$	0	0
$327$	−22.4533	−1.24167
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−12.4722	−0.685535	−0.342767	−	0.939420i	$-0.611364\pi$
$331$	−12.4722	−0.685535	−0.342767	+	0.939420i	$0.611364\pi$
$332$	0	0
$333$	25.6695	1.40668
$334$	0	0
$335$	−46.2920	−2.52920
$336$	0	0
$337$	2.00372	0.109150	0.0545749	−	0.998510i	$-0.482620\pi$
$337$	2.00372	0.109150	0.0545749	+	0.998510i	$0.482620\pi$
$338$	0	0
$339$	−2.20856	−0.119952
$340$	0	0
$341$	6.04330	0.327263
$342$	0	0
$343$	0	0
$344$	0	0
$345$	2.11874	0.114069
$346$	0	0
$347$	22.7012	1.21867	0.609333	−	0.792914i	$-0.291437\pi$
$347$	22.7012	1.21867	0.609333	+	0.792914i	$0.291437\pi$
$348$	0	0
$349$	−34.1702	−1.82909	−0.914545	−	0.404484i	$-0.867451\pi$
$349$	−34.1702	−1.82909	−0.914545	+	0.404484i	$0.867451\pi$
$350$	0	0
$351$	23.3707	1.24744
$352$	0	0
$353$	14.1170	0.751372	0.375686	−	0.926747i	$-0.377407\pi$
$353$	14.1170	0.751372	0.375686	+	0.926747i	$0.377407\pi$
$354$	0	0
$355$	−1.99690	−0.105984
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−9.02234	−0.476181	−0.238090	−	0.971243i	$-0.576521\pi$
$359$	−9.02234	−0.476181	−0.238090	+	0.971243i	$0.576521\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−26.7022	−1.40150
$364$	0	0
$365$	7.38251	0.386418
$366$	0	0
$367$	−25.1347	−1.31202	−0.656010	−	0.754752i	$-0.727757\pi$
$367$	−25.1347	−1.31202	−0.656010	+	0.754752i	$0.727757\pi$
$368$	0	0
$369$	9.88362	0.514521
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−29.6966	−1.53763	−0.768816	−	0.639471i	$-0.779154\pi$
$373$	−29.6966	−1.53763	−0.768816	+	0.639471i	$0.779154\pi$
$374$	0	0
$375$	5.39571	0.278633
$376$	0	0
$377$	−62.5937	−3.22374
$378$	0	0
$379$	4.81984	0.247579	0.123789	−	0.992309i	$-0.460495\pi$
$379$	4.81984	0.247579	0.123789	+	0.992309i	$0.460495\pi$
$380$	0	0
$381$	−32.3017	−1.65487
$382$	0	0
$383$	−28.2581	−1.44392	−0.721960	−	0.691935i	$-0.756758\pi$
$383$	−28.2581	−1.44392	−0.721960	+	0.691935i	$0.756758\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−15.5396	−0.789921
$388$	0	0
$389$	6.83363	0.346479	0.173239	−	0.984880i	$-0.444577\pi$
$389$	6.83363	0.346479	0.173239	+	0.984880i	$0.444577\pi$
$390$	0	0
$391$	1.87804	0.0949768
$392$	0	0
$393$	−4.56129	−0.230087
$394$	0	0
$395$	30.6817	1.54376
$396$	0	0
$397$	−29.1308	−1.46203	−0.731017	−	0.682359i	$-0.760954\pi$
$397$	−29.1308	−1.46203	−0.731017	+	0.682359i	$0.760954\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.22841	−0.0613441	−0.0306721	−	0.999530i	$-0.509765\pi$
$401$	−1.22841	−0.0613441	−0.0306721	+	0.999530i	$0.509765\pi$
$402$	0	0
$403$	−37.9994	−1.89288
$404$	0	0
$405$	−11.0897	−0.551052
$406$	0	0
$407$	−6.30991	−0.312771
$408$	0	0
$409$	16.5133	0.816532	0.408266	−	0.912863i	$-0.366134\pi$
$409$	16.5133	0.816532	0.408266	+	0.912863i	$0.366134\pi$
$410$	0	0
$411$	−2.98214	−0.147098
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−5.72331	−0.280946
$416$	0	0
$417$	−38.3238	−1.87673
$418$	0	0
$419$	−0.583006	−0.0284817	−0.0142409	−	0.999899i	$-0.504533\pi$
$419$	−0.583006	−0.0284817	−0.0142409	+	0.999899i	$0.504533\pi$
$420$	0	0
$421$	21.6448	1.05490	0.527451	−	0.849585i	$-0.323148\pi$
$421$	21.6448	1.05490	0.527451	+	0.849585i	$0.323148\pi$
$422$	0	0
$423$	−19.3481	−0.940736
$424$	0	0
$425$	43.7958	2.12441
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−18.9914	−0.916916
$430$	0	0
$431$	26.4930	1.27612	0.638061	−	0.769986i	$-0.279737\pi$
$431$	26.4930	1.27612	0.638061	+	0.769986i	$0.279737\pi$
$432$	0	0
$433$	8.65565	0.415964	0.207982	−	0.978133i	$-0.433310\pi$
$433$	8.65565	0.415964	0.207982	+	0.978133i	$0.433310\pi$
$434$	0	0
$435$	−82.8816	−3.97387
$436$	0	0
$437$	0.240694	0.0115140
$438$	0	0
$439$	24.1102	1.15072	0.575358	−	0.817902i	$-0.304863\pi$
$439$	24.1102	1.15072	0.575358	+	0.817902i	$0.304863\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	25.2497	1.19965	0.599824	−	0.800132i	$-0.295237\pi$
$443$	25.2497	1.19965	0.599824	+	0.800132i	$0.295237\pi$
$444$	0	0
$445$	5.09541	0.241546
$446$	0	0
$447$	−54.5663	−2.58090
$448$	0	0
$449$	−19.0358	−0.898357	−0.449179	−	0.893442i	$-0.648283\pi$
$449$	−19.0358	−0.898357	−0.449179	+	0.893442i	$0.648283\pi$
$450$	0	0
$451$	−2.42953	−0.114402
$452$	0	0
$453$	31.6572	1.48738
$454$	0	0
$455$	0	0
$456$	0	0
$457$	22.6216	1.05820	0.529098	−	0.848561i	$-0.322530\pi$
$457$	22.6216	1.05820	0.529098	+	0.848561i	$0.322530\pi$
$458$	0	0
$459$	27.4301	1.28033
$460$	0	0
$461$	0.811606	0.0378003	0.0189001	−	0.999821i	$-0.493984\pi$
$461$	0.811606	0.0378003	0.0189001	+	0.999821i	$0.493984\pi$
$462$	0	0
$463$	−17.1355	−0.796353	−0.398177	−	0.917309i	$-0.630357\pi$
$463$	−17.1355	−0.796353	−0.398177	+	0.917309i	$0.630357\pi$
$464$	0	0
$465$	−50.3157	−2.33334
$466$	0	0
$467$	−5.60879	−0.259544	−0.129772	−	0.991544i	$-0.541425\pi$
$467$	−5.60879	−0.259544	−0.129772	+	0.991544i	$0.541425\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	7.34931	0.338638
$472$	0	0
$473$	3.81984	0.175637
$474$	0	0
$475$	5.61297	0.257541
$476$	0	0
$477$	27.1012	1.24088
$478$	0	0
$479$	20.6589	0.943928	0.471964	−	0.881618i	$-0.343545\pi$
$479$	20.6589	0.943928	0.471964	+	0.881618i	$0.343545\pi$
$480$	0	0
$481$	39.6758	1.80906
$482$	0	0
$483$	0	0
$484$	0	0
$485$	51.8268	2.35333
$486$	0	0
$487$	36.4899	1.65351	0.826757	−	0.562559i	$-0.190183\pi$
$487$	36.4899	1.65351	0.826757	+	0.562559i	$0.190183\pi$
$488$	0	0
$489$	61.0661	2.76150
$490$	0	0
$491$	38.2352	1.72553	0.862766	−	0.505603i	$-0.168730\pi$
$491$	38.2352	1.72553	0.862766	+	0.505603i	$0.168730\pi$
$492$	0	0
$493$	−73.4660	−3.30874
$494$	0	0
$495$	−14.8141	−0.665844
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−7.26644	−0.325290	−0.162645	−	0.986685i	$-0.552003\pi$
$499$	−7.26644	−0.325290	−0.162645	+	0.986685i	$0.552003\pi$
$500$	0	0
$501$	9.38375	0.419235
$502$	0	0
$503$	1.42425	0.0635044	0.0317522	−	0.999496i	$-0.489891\pi$
$503$	1.42425	0.0635044	0.0317522	+	0.999496i	$0.489891\pi$
$504$	0	0
$505$	7.39059	0.328877
$506$	0	0
$507$	84.2888	3.74340
$508$	0	0
$509$	−15.4907	−0.686613	−0.343306	−	0.939223i	$-0.611547\pi$
$509$	−15.4907	−0.686613	−0.343306	+	0.939223i	$0.611547\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	3.51551	0.155214
$514$	0	0
$515$	−30.4871	−1.34342
$516$	0	0
$517$	4.75603	0.209170
$518$	0	0
$519$	−2.22904	−0.0978438
$520$	0	0
$521$	31.6684	1.38742	0.693708	−	0.720256i	$-0.255976\pi$
$521$	31.6684	1.38742	0.693708	+	0.720256i	$0.255976\pi$
$522$	0	0
$523$	21.2438	0.928926	0.464463	−	0.885592i	$-0.346247\pi$
$523$	21.2438	0.928926	0.464463	+	0.885592i	$0.346247\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−44.5997	−1.94280
$528$	0	0
$529$	−22.9421	−0.997481
$530$	0	0
$531$	−22.6139	−0.981359
$532$	0	0
$533$	15.2765	0.661700
$534$	0	0
$535$	−50.0538	−2.16402
$536$	0	0
$537$	−65.9728	−2.84694
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−5.09466	−0.219036	−0.109518	−	0.993985i	$-0.534931\pi$
$541$	−5.09466	−0.219036	−0.109518	+	0.993985i	$0.534931\pi$
$542$	0	0
$543$	−55.3986	−2.37738
$544$	0	0
$545$	−27.0711	−1.15960
$546$	0	0
$547$	16.3589	0.699458	0.349729	−	0.936851i	$-0.386274\pi$
$547$	16.3589	0.699458	0.349729	+	0.936851i	$0.386274\pi$
$548$	0	0
$549$	−52.4523	−2.23861
$550$	0	0
$551$	−9.41557	−0.401117
$552$	0	0
$553$	0	0
$554$	0	0
$555$	52.5355	2.23001
$556$	0	0
$557$	−6.89726	−0.292246	−0.146123	−	0.989266i	$-0.546680\pi$
$557$	−6.89726	−0.292246	−0.146123	+	0.989266i	$0.546680\pi$
$558$	0	0
$559$	−24.0186	−1.01588
$560$	0	0
$561$	−22.2902	−0.941093
$562$	0	0
$563$	5.72934	0.241463	0.120732	−	0.992685i	$-0.461476\pi$
$563$	5.72934	0.241463	0.120732	+	0.992685i	$0.461476\pi$
$564$	0	0
$565$	−2.66277	−0.112024
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−29.5538	−1.23896	−0.619481	−	0.785012i	$-0.712657\pi$
$569$	−29.5538	−1.23896	−0.619481	+	0.785012i	$0.712657\pi$
$570$	0	0
$571$	14.0121	0.586388	0.293194	−	0.956053i	$-0.405282\pi$
$571$	14.0121	0.586388	0.293194	+	0.956053i	$0.405282\pi$
$572$	0	0
$573$	42.2964	1.76696
$574$	0	0
$575$	1.35101	0.0563410
$576$	0	0
$577$	−12.7928	−0.532571	−0.266286	−	0.963894i	$-0.585796\pi$
$577$	−12.7928	−0.532571	−0.266286	+	0.963894i	$0.585796\pi$
$578$	0	0
$579$	40.6078	1.68760
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−6.66185	−0.275906
$584$	0	0
$585$	93.1488	3.85123
$586$	0	0
$587$	2.29888	0.0948848	0.0474424	−	0.998874i	$-0.484893\pi$
$587$	2.29888	0.0948848	0.0474424	+	0.998874i	$0.484893\pi$
$588$	0	0
$589$	−5.71601	−0.235524
$590$	0	0
$591$	11.0174	0.453194
$592$	0	0
$593$	8.82605	0.362442	0.181221	−	0.983442i	$-0.441995\pi$
$593$	8.82605	0.362442	0.181221	+	0.983442i	$0.441995\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	53.1524	2.17538
$598$	0	0
$599$	28.2765	1.15535	0.577673	−	0.816268i	$-0.303961\pi$
$599$	28.2765	1.15535	0.577673	+	0.816268i	$0.303961\pi$
$600$	0	0
$601$	−20.3422	−0.829775	−0.414887	−	0.909873i	$-0.636179\pi$
$601$	−20.3422	−0.829775	−0.414887	+	0.909873i	$0.636179\pi$
$602$	0	0
$603$	−61.1171	−2.48888
$604$	0	0
$605$	−32.1938	−1.30886
$606$	0	0
$607$	−12.1045	−0.491305	−0.245652	−	0.969358i	$-0.579002\pi$
$607$	−12.1045	−0.491305	−0.245652	+	0.969358i	$0.579002\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−29.9052	−1.20983
$612$	0	0
$613$	−7.94614	−0.320942	−0.160471	−	0.987041i	$-0.551301\pi$
$613$	−7.94614	−0.320942	−0.160471	+	0.987041i	$0.551301\pi$
$614$	0	0
$615$	20.2280	0.815671
$616$	0	0
$617$	−24.1014	−0.970287	−0.485144	−	0.874434i	$-0.661233\pi$
$617$	−24.1014	−0.970287	−0.485144	+	0.874434i	$0.661233\pi$
$618$	0	0
$619$	−15.2972	−0.614845	−0.307422	−	0.951573i	$-0.599466\pi$
$619$	−15.2972	−0.614845	−0.307422	+	0.951573i	$0.599466\pi$
$620$	0	0
$621$	0.846164	0.0339554
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−21.5594	−0.862377
$626$	0	0
$627$	−2.85676	−0.114088
$628$	0	0
$629$	46.5674	1.85676
$630$	0	0
$631$	−21.4778	−0.855017	−0.427509	−	0.904011i	$-0.640609\pi$
$631$	−21.4778	−0.855017	−0.427509	+	0.904011i	$0.640609\pi$
$632$	0	0
$633$	22.0428	0.876123
$634$	0	0
$635$	−38.9449	−1.54548
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−2.63641	−0.104295
$640$	0	0
$641$	35.4022	1.39830	0.699152	−	0.714973i	$-0.253561\pi$
$641$	35.4022	1.39830	0.699152	+	0.714973i	$0.253561\pi$
$642$	0	0
$643$	0.0470224	0.00185438	0.000927192	−	1.00000i	$-0.499705\pi$
$643$	0.0470224	0.00185438	0.000927192		1.00000i	$0.499705\pi$
$644$	0	0
$645$	−31.8035	−1.25226
$646$	0	0
$647$	−7.32264	−0.287883	−0.143941	−	0.989586i	$-0.545978\pi$
$647$	−7.32264	−0.287883	−0.143941	+	0.989586i	$0.545978\pi$
$648$	0	0
$649$	5.55881	0.218202
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−14.4697	−0.566244	−0.283122	−	0.959084i	$-0.591370\pi$
$653$	−14.4697	−0.566244	−0.283122	+	0.959084i	$0.591370\pi$
$654$	0	0
$655$	−5.49937	−0.214878
$656$	0	0
$657$	9.74677	0.380258
$658$	0	0
$659$	28.5428	1.11187	0.555935	−	0.831226i	$-0.312360\pi$
$659$	28.5428	1.11187	0.555935	+	0.831226i	$0.312360\pi$
$660$	0	0
$661$	21.4395	0.833898	0.416949	−	0.908930i	$-0.363099\pi$
$661$	21.4395	0.833898	0.416949	+	0.908930i	$0.363099\pi$
$662$	0	0
$663$	140.157	5.44326
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.26628	−0.0877506
$668$	0	0
$669$	73.8195	2.85403
$670$	0	0
$671$	12.8935	0.497748
$672$	0	0
$673$	−22.5929	−0.870893	−0.435447	−	0.900214i	$-0.643410\pi$
$673$	−22.5929	−0.870893	−0.435447	+	0.900214i	$0.643410\pi$
$674$	0	0
$675$	19.7324	0.759502
$676$	0	0
$677$	−17.0576	−0.655575	−0.327788	−	0.944751i	$-0.606303\pi$
$677$	−17.0576	−0.655575	−0.327788	+	0.944751i	$0.606303\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	20.4641	0.784187
$682$	0	0
$683$	−24.5780	−0.940453	−0.470226	−	0.882546i	$-0.655828\pi$
$683$	−24.5780	−0.940453	−0.470226	+	0.882546i	$0.655828\pi$
$684$	0	0
$685$	−3.59545	−0.137375
$686$	0	0
$687$	−21.1140	−0.805550
$688$	0	0
$689$	41.8887	1.59583
$690$	0	0
$691$	−16.9289	−0.644006	−0.322003	−	0.946739i	$-0.604356\pi$
$691$	−16.9289	−0.644006	−0.322003	+	0.946739i	$0.604356\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−46.2056	−1.75268
$696$	0	0
$697$	17.9300	0.679148
$698$	0	0
$699$	−57.4713	−2.17376
$700$	0	0
$701$	45.2445	1.70886	0.854431	−	0.519564i	$-0.173906\pi$
$701$	45.2445	1.70886	0.854431	+	0.519564i	$0.173906\pi$
$702$	0	0
$703$	5.96818	0.225094
$704$	0	0
$705$	−39.5981	−1.49135
$706$	0	0
$707$	0	0
$708$	0	0
$709$	48.2794	1.81317	0.906586	−	0.422022i	$-0.138679\pi$
$709$	48.2794	1.81317	0.906586	+	0.422022i	$0.138679\pi$
$710$	0	0
$711$	40.5076	1.51915
$712$	0	0
$713$	−1.37581	−0.0515245
$714$	0	0
$715$	−22.8973	−0.856309
$716$	0	0
$717$	−50.6215	−1.89049
$718$	0	0
$719$	−20.9752	−0.782242	−0.391121	−	0.920339i	$-0.627913\pi$
$719$	−20.9752	−0.782242	−0.391121	+	0.920339i	$0.627913\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	21.5954	0.803142
$724$	0	0
$725$	−52.8493	−1.96277
$726$	0	0
$727$	−28.9739	−1.07458	−0.537291	−	0.843397i	$-0.680552\pi$
$727$	−28.9739	−1.07458	−0.537291	+	0.843397i	$0.680552\pi$
$728$	0	0
$729$	−43.1384	−1.59772
$730$	0	0
$731$	−28.1906	−1.04267
$732$	0	0
$733$	−9.85601	−0.364040	−0.182020	−	0.983295i	$-0.558264\pi$
$733$	−9.85601	−0.364040	−0.182020	+	0.983295i	$0.558264\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	15.0234	0.553396
$738$	0	0
$739$	22.3894	0.823607	0.411803	−	0.911273i	$-0.364899\pi$
$739$	22.3894	0.823607	0.411803	+	0.911273i	$0.364899\pi$
$740$	0	0
$741$	17.9629	0.659884
$742$	0	0
$743$	49.3322	1.80982	0.904911	−	0.425600i	$-0.139937\pi$
$743$	49.3322	1.80982	0.904911	+	0.425600i	$0.139937\pi$
$744$	0	0
$745$	−65.7885	−2.41030
$746$	0	0
$747$	−7.55621	−0.276467
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−50.7975	−1.85363	−0.926813	−	0.375524i	$-0.877463\pi$
$751$	−50.7975	−1.85363	−0.926813	+	0.375524i	$0.877463\pi$
$752$	0	0
$753$	40.4871	1.47543
$754$	0	0
$755$	38.1679	1.38907
$756$	0	0
$757$	2.25855	0.0820886	0.0410443	−	0.999157i	$-0.486932\pi$
$757$	2.25855	0.0820886	0.0410443	+	0.999157i	$0.486932\pi$
$758$	0	0
$759$	−0.687607	−0.0249586
$760$	0	0
$761$	−29.2093	−1.05884	−0.529418	−	0.848361i	$-0.677590\pi$
$761$	−29.2093	−1.05884	−0.529418	+	0.848361i	$0.677590\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	109.328	3.95278
$766$	0	0
$767$	−34.9530	−1.26208
$768$	0	0
$769$	25.0031	0.901635	0.450817	−	0.892616i	$-0.351133\pi$
$769$	25.0031	0.901635	0.450817	+	0.892616i	$0.351133\pi$
$770$	0	0
$771$	15.8102	0.569389
$772$	0	0
$773$	18.4770	0.664573	0.332286	−	0.943179i	$-0.392180\pi$
$773$	18.4770	0.664573	0.332286	+	0.943179i	$0.392180\pi$
$774$	0	0
$775$	−32.0838	−1.15248
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.29795	0.0823327
$780$	0	0
$781$	0.648066	0.0231896
$782$	0	0
$783$	−33.1005	−1.18292
$784$	0	0
$785$	8.86079	0.316255
$786$	0	0
$787$	1.79608	0.0640235	0.0320117	−	0.999487i	$-0.489809\pi$
$787$	1.79608	0.0640235	0.0320117	+	0.999487i	$0.489809\pi$
$788$	0	0
$789$	65.4643	2.33059
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−81.0724	−2.87897
$794$	0	0
$795$	55.4657	1.96717
$796$	0	0
$797$	−33.8876	−1.20036	−0.600180	−	0.799865i	$-0.704905\pi$
$797$	−33.8876	−1.20036	−0.600180	+	0.799865i	$0.704905\pi$
$798$	0	0
$799$	−35.0996	−1.24174
$800$	0	0
$801$	6.72723	0.237695
$802$	0	0
$803$	−2.39589	−0.0845492
$804$	0	0
$805$	0	0
$806$	0	0
$807$	17.7054	0.623261
$808$	0	0
$809$	−23.3661	−0.821507	−0.410754	−	0.911746i	$-0.634734\pi$
$809$	−23.3661	−0.821507	−0.410754	+	0.911746i	$0.634734\pi$
$810$	0	0
$811$	33.5597	1.17844	0.589221	−	0.807972i	$-0.299435\pi$
$811$	33.5597	1.17844	0.589221	+	0.807972i	$0.299435\pi$
$812$	0	0
$813$	14.7086	0.515852
$814$	0	0
$815$	73.6250	2.57897
$816$	0	0
$817$	−3.61297	−0.126402
$818$	0	0
$819$	0	0
$820$	0	0
$821$	7.29578	0.254624	0.127312	−	0.991863i	$-0.459365\pi$
$821$	7.29578	0.254624	0.127312	+	0.991863i	$0.459365\pi$
$822$	0	0
$823$	20.4568	0.713080	0.356540	−	0.934280i	$-0.383956\pi$
$823$	20.4568	0.713080	0.356540	+	0.934280i	$0.383956\pi$
$824$	0	0
$825$	−16.0349	−0.558264
$826$	0	0
$827$	−24.3754	−0.847616	−0.423808	−	0.905752i	$-0.639307\pi$
$827$	−24.3754	−0.847616	−0.423808	+	0.905752i	$0.639307\pi$
$828$	0	0
$829$	47.4953	1.64958	0.824790	−	0.565439i	$-0.191293\pi$
$829$	47.4953	1.64958	0.824790	+	0.565439i	$0.191293\pi$
$830$	0	0
$831$	−14.9568	−0.518847
$832$	0	0
$833$	0	0
$834$	0	0
$835$	11.3136	0.391525
$836$	0	0
$837$	−20.0947	−0.694574
$838$	0	0
$839$	−38.9519	−1.34477	−0.672384	−	0.740203i	$-0.734730\pi$
$839$	−38.9519	−1.34477	−0.672384	+	0.740203i	$0.734730\pi$
$840$	0	0
$841$	59.6530	2.05700
$842$	0	0
$843$	18.5030	0.637276
$844$	0	0
$845$	101.624	3.49596
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−47.7793	−1.63978
$850$	0	0
$851$	1.43651	0.0492429
$852$	0	0
$853$	−3.54671	−0.121437	−0.0607185	−	0.998155i	$-0.519339\pi$
$853$	−3.54671	−0.121437	−0.0607185	+	0.998155i	$0.519339\pi$
$854$	0	0
$855$	14.0118	0.479193
$856$	0	0
$857$	19.7964	0.676232	0.338116	−	0.941104i	$-0.390210\pi$
$857$	19.7964	0.676232	0.338116	+	0.941104i	$0.390210\pi$
$858$	0	0
$859$	−29.7983	−1.01670	−0.508352	−	0.861150i	$-0.669745\pi$
$859$	−29.7983	−1.01670	−0.508352	+	0.861150i	$0.669745\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−40.1501	−1.36672	−0.683362	−	0.730080i	$-0.739483\pi$
$863$	−40.1501	−1.36672	−0.683362	+	0.730080i	$0.739483\pi$
$864$	0	0
$865$	−2.68747	−0.0913766
$866$	0	0
$867$	118.568	4.02676
$868$	0	0
$869$	−9.95732	−0.337779
$870$	0	0
$871$	−94.4652	−3.20083
$872$	0	0
$873$	68.4244	2.31581
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−55.5647	−1.87629	−0.938143	−	0.346248i	$-0.887456\pi$
$877$	−55.5647	−1.87629	−0.938143	+	0.346248i	$0.887456\pi$
$878$	0	0
$879$	41.0175	1.38349
$880$	0	0
$881$	−6.53213	−0.220073	−0.110037	−	0.993928i	$-0.535097\pi$
$881$	−6.53213	−0.220073	−0.110037	+	0.993928i	$0.535097\pi$
$882$	0	0
$883$	23.0621	0.776101	0.388050	−	0.921638i	$-0.373149\pi$
$883$	23.0621	0.776101	0.388050	+	0.921638i	$0.373149\pi$
$884$	0	0
$885$	−46.2819	−1.55575
$886$	0	0
$887$	−24.9517	−0.837797	−0.418898	−	0.908033i	$-0.637584\pi$
$887$	−24.9517	−0.837797	−0.418898	+	0.908033i	$0.637584\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	3.59901	0.120571
$892$	0	0
$893$	−4.49845	−0.150535
$894$	0	0
$895$	−79.5409	−2.65876
$896$	0	0
$897$	4.32357	0.144360
$898$	0	0
$899$	53.8195	1.79498
$900$	0	0
$901$	49.1646	1.63791
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−66.7920	−2.22024
$906$	0	0
$907$	13.3114	0.441999	0.220999	−	0.975274i	$-0.429068\pi$
$907$	13.3114	0.441999	0.220999	+	0.975274i	$0.429068\pi$
$908$	0	0
$909$	9.75743	0.323634
$910$	0	0
$911$	−25.0741	−0.830743	−0.415372	−	0.909652i	$-0.636348\pi$
$911$	−25.0741	−0.830743	−0.415372	+	0.909652i	$0.636348\pi$
$912$	0	0
$913$	1.85742	0.0614717
$914$	0	0
$915$	−107.350	−3.54887
$916$	0	0
$917$	0	0
$918$	0	0
$919$	11.1913	0.369167	0.184584	−	0.982817i	$-0.440906\pi$
$919$	11.1913	0.369167	0.184584	+	0.982817i	$0.440906\pi$
$920$	0	0
$921$	74.1083	2.44195
$922$	0	0
$923$	−4.07494	−0.134128
$924$	0	0
$925$	33.4992	1.10145
$926$	0	0
$927$	−40.2506	−1.32200
$928$	0	0
$929$	−3.33412	−0.109389	−0.0546944	−	0.998503i	$-0.517418\pi$
$929$	−3.33412	−0.109389	−0.0546944	+	0.998503i	$0.517418\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	26.4682	0.866529
$934$	0	0
$935$	−26.8744	−0.878888
$936$	0	0
$937$	35.4463	1.15798	0.578990	−	0.815335i	$-0.303447\pi$
$937$	35.4463	1.15798	0.578990	+	0.815335i	$0.303447\pi$
$938$	0	0
$939$	−62.0136	−2.02374
$940$	0	0
$941$	−53.5947	−1.74714	−0.873568	−	0.486702i	$-0.838200\pi$
$941$	−53.5947	−1.74714	−0.873568	+	0.486702i	$0.838200\pi$
$942$	0	0
$943$	0.553105	0.0180116
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−29.8153	−0.968867	−0.484434	−	0.874828i	$-0.660974\pi$
$947$	−29.8153	−0.968867	−0.484434	+	0.874828i	$0.660974\pi$
$948$	0	0
$949$	15.0650	0.489031
$950$	0	0
$951$	−15.5326	−0.503678
$952$	0	0
$953$	36.6260	1.18643	0.593215	−	0.805044i	$-0.297858\pi$
$953$	36.6260	1.18643	0.593215	+	0.805044i	$0.297858\pi$
$954$	0	0
$955$	50.9952	1.65016
$956$	0	0
$957$	26.8981	0.869491
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1.67272	0.0539588
$962$	0	0
$963$	−66.0836	−2.12951
$964$	0	0
$965$	48.9593	1.57606
$966$	0	0
$967$	−1.76564	−0.0567792	−0.0283896	−	0.999597i	$-0.509038\pi$
$967$	−1.76564	−0.0567792	−0.0283896	+	0.999597i	$0.509038\pi$
$968$	0	0
$969$	21.0830	0.677284
$970$	0	0
$971$	32.2970	1.03646	0.518230	−	0.855242i	$-0.326591\pi$
$971$	32.2970	1.03646	0.518230	+	0.855242i	$0.326591\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	100.825	3.22899
$976$	0	0
$977$	19.3125	0.617863	0.308931	−	0.951084i	$-0.400029\pi$
$977$	19.3125	0.617863	0.308931	+	0.951084i	$0.400029\pi$
$978$	0	0
$979$	−1.65365	−0.0528508
$980$	0	0
$981$	−35.7406	−1.14111
$982$	0	0
$983$	−48.9195	−1.56029	−0.780144	−	0.625600i	$-0.784854\pi$
$983$	−48.9195	−1.56029	−0.780144	+	0.625600i	$0.784854\pi$
$984$	0	0
$985$	13.2832	0.423239
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−0.869621	−0.0276523
$990$	0	0
$991$	−16.2528	−0.516287	−0.258144	−	0.966107i	$-0.583111\pi$
$991$	−16.2528	−0.516287	−0.258144	+	0.966107i	$0.583111\pi$
$992$	0	0
$993$	−33.7005	−1.06945
$994$	0	0
$995$	64.0838	2.03159
$996$	0	0
$997$	43.5464	1.37913	0.689563	−	0.724225i	$-0.257802\pi$
$997$	43.5464	1.37913	0.689563	+	0.724225i	$0.257802\pi$
$998$	0	0
$999$	20.9812	0.663816

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.bk.1.4		4
7.6	odd	2		1064.2.a.g.1.1	✓	4
21.20	even	2		9576.2.a.ck.1.3		4
28.27	even	2		2128.2.a.u.1.4		4
56.13	odd	2		8512.2.a.bs.1.4		4
56.27	even	2		8512.2.a.bt.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1064.2.a.g.1.1	✓	4	7.6	odd	2
2128.2.a.u.1.4		4	28.27	even	2
7448.2.a.bk.1.4		4	1.1	even	1	trivial
8512.2.a.bs.1.4		4	56.13	odd	2
8512.2.a.bt.1.1		4	56.27	even	2
9576.2.a.ck.1.3		4	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.70205 q^{3} +3.25776 q^{5} +4.30105 q^{9} +O(q^{10})\)	\(q+2.70205 q^{3} +3.25776 q^{5} +4.30105 q^{9} -1.05726 q^{11} +6.64789 q^{13} +8.80261 q^{15} +7.80261 q^{17} +1.00000 q^{19} +0.240694 q^{23} +5.61297 q^{25} +3.51551 q^{27} -9.41557 q^{29} -5.71601 q^{31} -2.85676 q^{33} +5.96818 q^{37} +17.9629 q^{39} +2.29795 q^{41} -3.61297 q^{43} +14.0118 q^{45} -4.49845 q^{47} +21.0830 q^{51} +6.30105 q^{53} -3.44429 q^{55} +2.70205 q^{57} -5.25776 q^{59} -12.1952 q^{61} +21.6572 q^{65} -14.2098 q^{67} +0.650368 q^{69} -0.612968 q^{71} +2.26613 q^{73} +15.1665 q^{75} +9.41805 q^{79} -3.40409 q^{81} -1.75683 q^{83} +25.4190 q^{85} -25.4413 q^{87} +1.56409 q^{89} -15.4449 q^{93} +3.25776 q^{95} +15.9087 q^{97} -4.54733 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{5} + 6 q^{9}+O(q^{10}) \) 4 * q + 3 * q^5 + 6 * q^9	\( 4 q + 3 q^{5} + 6 q^{9} + 6 q^{11} + 4 q^{13} + 13 q^{15} + 9 q^{17} + 4 q^{19} + 22 q^{23} + 13 q^{25} - 6 q^{27} - 6 q^{29} - 3 q^{31} - q^{33} + 15 q^{37} + 29 q^{39} + 20 q^{41} - 5 q^{43} - 9 q^{45} - 29 q^{47} + 14 q^{53} - 13 q^{55} - 11 q^{59} - 15 q^{61} - 2 q^{65} + 9 q^{67} - 19 q^{69} + 7 q^{71} + 11 q^{73} - 4 q^{75} + 7 q^{79} + 8 q^{81} + 15 q^{83} - 7 q^{85} + 4 q^{87} + 19 q^{89} - 38 q^{93} + 3 q^{95} + 9 q^{97} - 7 q^{99}+O(q^{100}) \) 4 * q + 3 * q^5 + 6 * q^9 + 6 * q^11 + 4 * q^13 + 13 * q^15 + 9 * q^17 + 4 * q^19 + 22 * q^23 + 13 * q^25 - 6 * q^27 - 6 * q^29 - 3 * q^31 - q^33 + 15 * q^37 + 29 * q^39 + 20 * q^41 - 5 * q^43 - 9 * q^45 - 29 * q^47 + 14 * q^53 - 13 * q^55 - 11 * q^59 - 15 * q^61 - 2 * q^65 + 9 * q^67 - 19 * q^69 + 7 * q^71 + 11 * q^73 - 4 * q^75 + 7 * q^79 + 8 * q^81 + 15 * q^83 - 7 * q^85 + 4 * q^87 + 19 * q^89 - 38 * q^93 + 3 * q^95 + 9 * q^97 - 7 * q^99

Introduction

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