LMFDB - Embedded newform 1148.2.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1148,2,Mod(1,1148)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1148, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1148.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1148 = 2^{2} \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1148.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:	$9.16682615204$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1935333.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 8x^{3} + 10x^{2} + 13x - 11 $ x^5 - 2x^4 - 8x^3 + 10x^2 + 13x - 11
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			1.3
Root			$0.704110$ of defining polynomial
Character	$\chi$	$=$	1148.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.704110 q^{3} +3.89333 q^{5} +1.00000 q^{7} -2.50423 q^{9} +O(q^{10})$	$q+0.704110 q^{3} +3.89333 q^{5} +1.00000 q^{7} -2.50423 q^{9} +1.15200 q^{11} +4.05634 q^{13} +2.74133 q^{15} -5.09356 q^{17} +4.35223 q^{19} +0.704110 q^{21} +2.70411 q^{23} +10.1580 q^{25} -3.87558 q^{27} -2.74133 q^{29} -1.89333 q^{31} +0.811133 q^{33} +3.89333 q^{35} -2.57832 q^{37} +2.85611 q^{39} +1.00000 q^{41} +7.09356 q^{43} -9.74979 q^{45} -0.312561 q^{47} +1.00000 q^{49} -3.58643 q^{51} -3.89333 q^{53} +4.48511 q^{55} +3.06445 q^{57} +0.811133 q^{59} -15.1640 q^{61} -2.50423 q^{63} +15.7927 q^{65} -5.41423 q^{67} +1.90399 q^{69} -2.88624 q^{71} -6.93866 q^{73} +7.15235 q^{75} +1.15200 q^{77} +13.8249 q^{79} +4.78386 q^{81} +3.11376 q^{83} -19.8309 q^{85} -1.93020 q^{87} +4.61125 q^{89} +4.05634 q^{91} -1.33311 q^{93} +16.9447 q^{95} -0.112329 q^{97} -2.88487 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{3} + 3 q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q + 2 * q^3 + 3 * q^5 + 5 * q^7 + 5 * q^9	$ 5 q + 2 q^{3} + 3 q^{5} + 5 q^{7} + 5 q^{9} + 7 q^{13} + 3 q^{15} - 3 q^{17} + 10 q^{19} + 2 q^{21} + 12 q^{23} + 2 q^{25} + 14 q^{27} - 3 q^{29} + 7 q^{31} - 3 q^{33} + 3 q^{35} + q^{37} + 7 q^{39} + 5 q^{41} + 13 q^{43} - 3 q^{45} + 9 q^{47} + 5 q^{49} + 9 q^{51} - 3 q^{53} + 9 q^{55} - 11 q^{57} - 3 q^{59} + 16 q^{61} + 5 q^{63} + 3 q^{65} + 19 q^{67} + 24 q^{69} - 12 q^{71} + 4 q^{73} + 8 q^{75} + 28 q^{79} + 5 q^{81} + 18 q^{83} - 15 q^{85} - 6 q^{87} + 7 q^{91} + q^{93} + 3 q^{95} + 4 q^{97} - 33 q^{99}+O(q^{100}) $ 5 * q + 2 * q^3 + 3 * q^5 + 5 * q^7 + 5 * q^9 + 7 * q^13 + 3 * q^15 - 3 * q^17 + 10 * q^19 + 2 * q^21 + 12 * q^23 + 2 * q^25 + 14 * q^27 - 3 * q^29 + 7 * q^31 - 3 * q^33 + 3 * q^35 + q^37 + 7 * q^39 + 5 * q^41 + 13 * q^43 - 3 * q^45 + 9 * q^47 + 5 * q^49 + 9 * q^51 - 3 * q^53 + 9 * q^55 - 11 * q^57 - 3 * q^59 + 16 * q^61 + 5 * q^63 + 3 * q^65 + 19 * q^67 + 24 * q^69 - 12 * q^71 + 4 * q^73 + 8 * q^75 + 28 * q^79 + 5 * q^81 + 18 * q^83 - 15 * q^85 - 6 * q^87 + 7 * q^91 + q^93 + 3 * q^95 + 4 * q^97 - 33 * 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
$3$	0.704110	0.406518	0.203259	−	0.979125i	$-0.434847\pi$
$3$	0.704110	0.406518	0.203259	+	0.979125i	$0.434847\pi$
$4$	0	0
$5$	3.89333	1.74115	0.870575	−	0.492036i	$-0.163747\pi$
$5$	3.89333	1.74115	0.870575	+	0.492036i	$0.163747\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.50423	−0.834743
$10$	0	0
$11$	1.15200	0.347341	0.173670	−	0.984804i	$-0.444437\pi$
$11$	1.15200	0.347341	0.173670	+	0.984804i	$0.444437\pi$
$12$	0	0
$13$	4.05634	1.12503	0.562513	−	0.826788i	$-0.309835\pi$
$13$	4.05634	1.12503	0.562513	+	0.826788i	$0.309835\pi$
$14$	0	0
$15$	2.74133	0.707808
$16$	0	0
$17$	−5.09356	−1.23537	−0.617685	−	0.786426i	$-0.711929\pi$
$17$	−5.09356	−1.23537	−0.617685	+	0.786426i	$0.711929\pi$
$18$	0	0
$19$	4.35223	0.998470	0.499235	−	0.866467i	$-0.333614\pi$
$19$	4.35223	0.998470	0.499235	+	0.866467i	$0.333614\pi$
$20$	0	0
$21$	0.704110	0.153649
$22$	0	0
$23$	2.70411	0.563846	0.281923	−	0.959437i	$-0.409028\pi$
$23$	2.70411	0.563846	0.281923	+	0.959437i	$0.409028\pi$
$24$	0	0
$25$	10.1580	2.03160
$26$	0	0
$27$	−3.87558	−0.745856
$28$	0	0
$29$	−2.74133	−0.509052	−0.254526	−	0.967066i	$-0.581919\pi$
$29$	−2.74133	−0.509052	−0.254526	+	0.967066i	$0.581919\pi$
$30$	0	0
$31$	−1.89333	−0.340052	−0.170026	−	0.985440i	$-0.554385\pi$
$31$	−1.89333	−0.340052	−0.170026	+	0.985440i	$0.554385\pi$
$32$	0	0
$33$	0.811133	0.141200
$34$	0	0
$35$	3.89333	0.658093
$36$	0	0
$37$	−2.57832	−0.423873	−0.211936	−	0.977283i	$-0.567977\pi$
$37$	−2.57832	−0.423873	−0.211936	+	0.977283i	$0.567977\pi$
$38$	0	0
$39$	2.85611	0.457343
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	7.09356	1.08176	0.540879	−	0.841100i	$-0.318092\pi$
$43$	7.09356	1.08176	0.540879	+	0.841100i	$0.318092\pi$
$44$	0	0
$45$	−9.74979	−1.45341
$46$	0	0
$47$	−0.312561	−0.0455917	−0.0227959	−	0.999740i	$-0.507257\pi$
$47$	−0.312561	−0.0455917	−0.0227959	+	0.999740i	$0.507257\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.58643	−0.502200
$52$	0	0
$53$	−3.89333	−0.534790	−0.267395	−	0.963587i	$-0.586163\pi$
$53$	−3.89333	−0.534790	−0.267395	+	0.963587i	$0.586163\pi$
$54$	0	0
$55$	4.48511	0.604772
$56$	0	0
$57$	3.06445	0.405896
$58$	0	0
$59$	0.811133	0.105601	0.0528003	−	0.998605i	$-0.483185\pi$
$59$	0.811133	0.105601	0.0528003	+	0.998605i	$0.483185\pi$
$60$	0	0
$61$	−15.1640	−1.94155	−0.970777	−	0.239984i	$-0.922858\pi$
$61$	−15.1640	−1.94155	−0.970777	+	0.239984i	$0.922858\pi$
$62$	0	0
$63$	−2.50423	−0.315503
$64$	0	0
$65$	15.7927	1.95884
$66$	0	0
$67$	−5.41423	−0.661453	−0.330727	−	0.943727i	$-0.607294\pi$
$67$	−5.41423	−0.661453	−0.330727	+	0.943727i	$0.607294\pi$
$68$	0	0
$69$	1.90399	0.229213
$70$	0	0
$71$	−2.88624	−0.342534	−0.171267	−	0.985225i	$-0.554786\pi$
$71$	−2.88624	−0.342534	−0.171267	+	0.985225i	$0.554786\pi$
$72$	0	0
$73$	−6.93866	−0.812108	−0.406054	−	0.913849i	$-0.633096\pi$
$73$	−6.93866	−0.812108	−0.406054	+	0.913849i	$0.633096\pi$
$74$	0	0
$75$	7.15235	0.825882
$76$	0	0
$77$	1.15200	0.131282
$78$	0	0
$79$	13.8249	1.55542	0.777711	−	0.628622i	$-0.216381\pi$
$79$	13.8249	1.55542	0.777711	+	0.628622i	$0.216381\pi$
$80$	0	0
$81$	4.78386	0.531540
$82$	0	0
$83$	3.11376	0.341779	0.170890	−	0.985290i	$-0.445336\pi$
$83$	3.11376	0.341779	0.170890	+	0.985290i	$0.445336\pi$
$84$	0	0
$85$	−19.8309	−2.15096
$86$	0	0
$87$	−1.93020	−0.206939
$88$	0	0
$89$	4.61125	0.488792	0.244396	−	0.969676i	$-0.421410\pi$
$89$	4.61125	0.488792	0.244396	+	0.969676i	$0.421410\pi$
$90$	0	0
$91$	4.05634	0.425220
$92$	0	0
$93$	−1.33311	−0.138237
$94$	0	0
$95$	16.9447	1.73849
$96$	0	0
$97$	−0.112329	−0.0114053	−0.00570263	−	0.999984i	$-0.501815\pi$
$97$	−0.112329	−0.0114053	−0.00570263	+	0.999984i	$0.501815\pi$
$98$	0	0
$99$	−2.88487	−0.289940
$100$	0	0
$101$	−14.7579	−1.46847	−0.734233	−	0.678898i	$-0.762458\pi$
$101$	−14.7579	−1.46847	−0.734233	+	0.678898i	$0.762458\pi$
$102$	0	0
$103$	2.15445	0.212284	0.106142	−	0.994351i	$-0.466150\pi$
$103$	2.15445	0.212284	0.106142	+	0.994351i	$0.466150\pi$
$104$	0	0
$105$	2.74133	0.267526
$106$	0	0
$107$	0.420662	0.0406669	0.0203335	−	0.999793i	$-0.493527\pi$
$107$	0.420662	0.0406669	0.0203335	+	0.999793i	$0.493527\pi$
$108$	0	0
$109$	6.89088	0.660027	0.330013	−	0.943976i	$-0.392947\pi$
$109$	6.89088	0.660027	0.330013	+	0.943976i	$0.392947\pi$
$110$	0	0
$111$	−1.81542	−0.172312
$112$	0	0
$113$	8.83234	0.830876	0.415438	−	0.909621i	$-0.363628\pi$
$113$	8.83234	0.830876	0.415438	+	0.909621i	$0.363628\pi$
$114$	0	0
$115$	10.5280	0.981740
$116$	0	0
$117$	−10.1580	−0.939108
$118$	0	0
$119$	−5.09356	−0.466926
$120$	0	0
$121$	−9.67290	−0.879354
$122$	0	0
$123$	0.704110	0.0634874
$124$	0	0
$125$	20.0818	1.79617
$126$	0	0
$127$	2.50851	0.222595	0.111297	−	0.993787i	$-0.464499\pi$
$127$	2.50851	0.222595	0.111297	+	0.993787i	$0.464499\pi$
$128$	0	0
$129$	4.99464	0.439754
$130$	0	0
$131$	13.0500	1.14018	0.570090	−	0.821582i	$-0.306908\pi$
$131$	13.0500	1.14018	0.570090	+	0.821582i	$0.306908\pi$
$132$	0	0
$133$	4.35223	0.377386
$134$	0	0
$135$	−15.0889	−1.29865
$136$	0	0
$137$	7.48867	0.639800	0.319900	−	0.947451i	$-0.396351\pi$
$137$	7.48867	0.639800	0.319900	+	0.947451i	$0.396351\pi$
$138$	0	0
$139$	11.8727	1.00703	0.503514	−	0.863987i	$-0.332040\pi$
$139$	11.8727	1.00703	0.503514	+	0.863987i	$0.332040\pi$
$140$	0	0
$141$	−0.220077	−0.0185338
$142$	0	0
$143$	4.67290	0.390767
$144$	0	0
$145$	−10.6729	−0.886336
$146$	0	0
$147$	0.704110	0.0580740
$148$	0	0
$149$	5.38619	0.441254	0.220627	−	0.975358i	$-0.429190\pi$
$149$	5.38619	0.441254	0.220627	+	0.975358i	$0.429190\pi$
$150$	0	0
$151$	−0.499951	−0.0406855	−0.0203427	−	0.999793i	$-0.506476\pi$
$151$	−0.499951	−0.0406855	−0.0203427	+	0.999793i	$0.506476\pi$
$152$	0	0
$153$	12.7554	1.03122
$154$	0	0
$155$	−7.37135	−0.592081
$156$	0	0
$157$	19.8954	1.58782	0.793911	−	0.608034i	$-0.208042\pi$
$157$	19.8954	1.58782	0.793911	+	0.608034i	$0.208042\pi$
$158$	0	0
$159$	−2.74133	−0.217402
$160$	0	0
$161$	2.70411	0.213114
$162$	0	0
$163$	−18.6888	−1.46382	−0.731910	−	0.681402i	$-0.761371\pi$
$163$	−18.6888	−1.46382	−0.731910	+	0.681402i	$0.761371\pi$
$164$	0	0
$165$	3.15801	0.245851
$166$	0	0
$167$	−15.3706	−1.18942	−0.594708	−	0.803942i	$-0.702732\pi$
$167$	−15.3706	−1.18942	−0.594708	+	0.803942i	$0.702732\pi$
$168$	0	0
$169$	3.45390	0.265685
$170$	0	0
$171$	−10.8990	−0.833466
$172$	0	0
$173$	−22.3638	−1.70029	−0.850144	−	0.526550i	$-0.823485\pi$
$173$	−22.3638	−1.70029	−0.850144	+	0.526550i	$0.823485\pi$
$174$	0	0
$175$	10.1580	0.767873
$176$	0	0
$177$	0.571127	0.0429285
$178$	0	0
$179$	−25.8833	−1.93461	−0.967305	−	0.253615i	$-0.918380\pi$
$179$	−25.8833	−1.93461	−0.967305	+	0.253615i	$0.918380\pi$
$180$	0	0
$181$	−14.0064	−1.04108	−0.520542	−	0.853836i	$-0.674270\pi$
$181$	−14.0064	−1.04108	−0.520542	+	0.853836i	$0.674270\pi$
$182$	0	0
$183$	−10.6771	−0.789276
$184$	0	0
$185$	−10.0382	−0.738026
$186$	0	0
$187$	−5.86778	−0.429094
$188$	0	0
$189$	−3.87558	−0.281907
$190$	0	0
$191$	13.8200	0.999980	0.499990	−	0.866031i	$-0.333337\pi$
$191$	13.8200	0.999980	0.499990	+	0.866031i	$0.333337\pi$
$192$	0	0
$193$	−24.7480	−1.78140	−0.890702	−	0.454588i	$-0.849786\pi$
$193$	−24.7480	−1.78140	−0.890702	+	0.454588i	$0.849786\pi$
$194$	0	0
$195$	11.1198	0.796303
$196$	0	0
$197$	−23.6980	−1.68841	−0.844207	−	0.536017i	$-0.819928\pi$
$197$	−23.6980	−1.68841	−0.844207	+	0.536017i	$0.819928\pi$
$198$	0	0
$199$	−16.8132	−1.19186	−0.595929	−	0.803037i	$-0.703216\pi$
$199$	−16.8132	−1.19186	−0.595929	+	0.803037i	$0.703216\pi$
$200$	0	0
$201$	−3.81221	−0.268893
$202$	0	0
$203$	−2.74133	−0.192404
$204$	0	0
$205$	3.89333	0.271922
$206$	0	0
$207$	−6.77171	−0.470666
$208$	0	0
$209$	5.01376	0.346809
$210$	0	0
$211$	−12.9057	−0.888466	−0.444233	−	0.895911i	$-0.646524\pi$
$211$	−12.9057	−0.888466	−0.444233	+	0.895911i	$0.646524\pi$
$212$	0	0
$213$	−2.03223	−0.139246
$214$	0	0
$215$	27.6176	1.88350
$216$	0	0
$217$	−1.89333	−0.128528
$218$	0	0
$219$	−4.88557	−0.330137
$220$	0	0
$221$	−20.6612	−1.38982
$222$	0	0
$223$	6.76799	0.453218	0.226609	−	0.973986i	$-0.427236\pi$
$223$	6.76799	0.453218	0.226609	+	0.973986i	$0.427236\pi$
$224$	0	0
$225$	−25.4380	−1.69587
$226$	0	0
$227$	8.08182	0.536409	0.268205	−	0.963362i	$-0.413570\pi$
$227$	8.08182	0.536409	0.268205	+	0.963362i	$0.413570\pi$
$228$	0	0
$229$	12.5981	0.832509	0.416254	−	0.909248i	$-0.363343\pi$
$229$	12.5981	0.832509	0.416254	+	0.909248i	$0.363343\pi$
$230$	0	0
$231$	0.811133	0.0533687
$232$	0	0
$233$	−10.2671	−0.672622	−0.336311	−	0.941751i	$-0.609179\pi$
$233$	−10.2671	−0.672622	−0.336311	+	0.941751i	$0.609179\pi$
$234$	0	0
$235$	−1.21690	−0.0793820
$236$	0	0
$237$	9.73424	0.632307
$238$	0	0
$239$	−10.8589	−0.702405	−0.351202	−	0.936300i	$-0.614227\pi$
$239$	−10.8589	−0.702405	−0.351202	+	0.936300i	$0.614227\pi$
$240$	0	0
$241$	6.86885	0.442462	0.221231	−	0.975221i	$-0.428993\pi$
$241$	6.86885	0.442462	0.221231	+	0.975221i	$0.428993\pi$
$242$	0	0
$243$	14.9951	0.961936
$244$	0	0
$245$	3.89333	0.248736
$246$	0	0
$247$	17.6541	1.12331
$248$	0	0
$249$	2.19243	0.138939
$250$	0	0
$251$	−10.1556	−0.641013	−0.320507	−	0.947246i	$-0.603853\pi$
$251$	−10.1556	−0.641013	−0.320507	+	0.947246i	$0.603853\pi$
$252$	0	0
$253$	3.11513	0.195847
$254$	0	0
$255$	−13.9631	−0.874405
$256$	0	0
$257$	−9.64807	−0.601830	−0.300915	−	0.953651i	$-0.597292\pi$
$257$	−9.64807	−0.601830	−0.300915	+	0.953651i	$0.597292\pi$
$258$	0	0
$259$	−2.57832	−0.160209
$260$	0	0
$261$	6.86492	0.424928
$262$	0	0
$263$	−7.49149	−0.461945	−0.230973	−	0.972960i	$-0.574191\pi$
$263$	−7.49149	−0.461945	−0.230973	+	0.972960i	$0.574191\pi$
$264$	0	0
$265$	−15.1580	−0.931149
$266$	0	0
$267$	3.24683	0.198703
$268$	0	0
$269$	−6.48975	−0.395687	−0.197843	−	0.980234i	$-0.563394\pi$
$269$	−6.48975	−0.395687	−0.197843	+	0.980234i	$0.563394\pi$
$270$	0	0
$271$	21.9040	1.33057	0.665286	−	0.746589i	$-0.268310\pi$
$271$	21.9040	1.33057	0.665286	+	0.746589i	$0.268310\pi$
$272$	0	0
$273$	2.85611	0.172860
$274$	0	0
$275$	11.7020	0.705658
$276$	0	0
$277$	16.3936	0.984998	0.492499	−	0.870313i	$-0.336084\pi$
$277$	16.3936	0.984998	0.492499	+	0.870313i	$0.336084\pi$
$278$	0	0
$279$	4.74133	0.283856
$280$	0	0
$281$	−0.318642	−0.0190086	−0.00950428	−	0.999955i	$-0.503025\pi$
$281$	−0.318642	−0.0190086	−0.00950428	+	0.999955i	$0.503025\pi$
$282$	0	0
$283$	−24.0127	−1.42741	−0.713704	−	0.700447i	$-0.752984\pi$
$283$	−24.0127	−1.42741	−0.713704	+	0.700447i	$0.752984\pi$
$284$	0	0
$285$	11.9309	0.706726
$286$	0	0
$287$	1.00000	0.0590281
$288$	0	0
$289$	8.94436	0.526139
$290$	0	0
$291$	−0.0790917	−0.00463644
$292$	0	0
$293$	−26.8493	−1.56855	−0.784277	−	0.620411i	$-0.786966\pi$
$293$	−26.8493	−1.56855	−0.784277	+	0.620411i	$0.786966\pi$
$294$	0	0
$295$	3.15801	0.183866
$296$	0	0
$297$	−4.46466	−0.259066
$298$	0	0
$299$	10.9688	0.634341
$300$	0	0
$301$	7.09356	0.408866
$302$	0	0
$303$	−10.3912	−0.596957
$304$	0	0
$305$	−59.0385	−3.38054
$306$	0	0
$307$	33.0567	1.88664	0.943322	−	0.331878i	$-0.107682\pi$
$307$	33.0567	1.88664	0.943322	+	0.331878i	$0.107682\pi$
$308$	0	0
$309$	1.51697	0.0862973
$310$	0	0
$311$	−17.5652	−0.996031	−0.498016	−	0.867168i	$-0.665938\pi$
$311$	−17.5652	−0.996031	−0.498016	+	0.867168i	$0.665938\pi$
$312$	0	0
$313$	5.26285	0.297474	0.148737	−	0.988877i	$-0.452479\pi$
$313$	5.26285	0.297474	0.148737	+	0.988877i	$0.452479\pi$
$314$	0	0
$315$	−9.74979	−0.549338
$316$	0	0
$317$	7.63180	0.428645	0.214322	−	0.976763i	$-0.431246\pi$
$317$	7.63180	0.428645	0.214322	+	0.976763i	$0.431246\pi$
$318$	0	0
$319$	−3.15801	−0.176815
$320$	0	0
$321$	0.296192	0.0165318
$322$	0	0
$323$	−22.1684	−1.23348
$324$	0	0
$325$	41.2043	2.28561
$326$	0	0
$327$	4.85193	0.268313
$328$	0	0
$329$	−0.312561	−0.0172320
$330$	0	0
$331$	30.2536	1.66289	0.831444	−	0.555609i	$-0.187515\pi$
$331$	30.2536	1.66289	0.831444	+	0.555609i	$0.187515\pi$
$332$	0	0
$333$	6.45670	0.353825
$334$	0	0
$335$	−21.0794	−1.15169
$336$	0	0
$337$	−28.4748	−1.55112	−0.775559	−	0.631275i	$-0.782532\pi$
$337$	−28.4748	−1.55112	−0.775559	+	0.631275i	$0.782532\pi$
$338$	0	0
$339$	6.21893	0.337766
$340$	0	0
$341$	−2.18111	−0.118114
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	7.41286	0.399095
$346$	0	0
$347$	−12.2416	−0.657162	−0.328581	−	0.944476i	$-0.606570\pi$
$347$	−12.2416	−0.657162	−0.328581	+	0.944476i	$0.606570\pi$
$348$	0	0
$349$	16.0791	0.860693	0.430347	−	0.902664i	$-0.358391\pi$
$349$	16.0791	0.860693	0.430347	+	0.902664i	$0.358391\pi$
$350$	0	0
$351$	−15.7207	−0.839108
$352$	0	0
$353$	13.4776	0.717338	0.358669	−	0.933465i	$-0.383231\pi$
$353$	13.4776	0.717338	0.358669	+	0.933465i	$0.383231\pi$
$354$	0	0
$355$	−11.2371	−0.596403
$356$	0	0
$357$	−3.58643	−0.189814
$358$	0	0
$359$	27.9603	1.47569	0.737845	−	0.674971i	$-0.235844\pi$
$359$	27.9603	1.47569	0.737845	+	0.674971i	$0.235844\pi$
$360$	0	0
$361$	−0.0580852	−0.00305711
$362$	0	0
$363$	−6.81078	−0.357473
$364$	0	0
$365$	−27.0145	−1.41400
$366$	0	0
$367$	−23.5123	−1.22733	−0.613665	−	0.789566i	$-0.710306\pi$
$367$	−23.5123	−1.22733	−0.613665	+	0.789566i	$0.710306\pi$
$368$	0	0
$369$	−2.50423	−0.130365
$370$	0	0
$371$	−3.89333	−0.202132
$372$	0	0
$373$	10.9398	0.566442	0.283221	−	0.959055i	$-0.408597\pi$
$373$	10.9398	0.566442	0.283221	+	0.959055i	$0.408597\pi$
$374$	0	0
$375$	14.1398	0.730176
$376$	0	0
$377$	−11.1198	−0.572697
$378$	0	0
$379$	−13.1317	−0.674530	−0.337265	−	0.941410i	$-0.609502\pi$
$379$	−13.1317	−0.674530	−0.337265	+	0.941410i	$0.609502\pi$
$380$	0	0
$381$	1.76627	0.0904887
$382$	0	0
$383$	34.7861	1.77749	0.888744	−	0.458404i	$-0.151579\pi$
$383$	34.7861	1.77749	0.888744	+	0.458404i	$0.151579\pi$
$384$	0	0
$385$	4.48511	0.228582
$386$	0	0
$387$	−17.7639	−0.902990
$388$	0	0
$389$	−5.51096	−0.279417	−0.139708	−	0.990193i	$-0.544617\pi$
$389$	−5.51096	−0.279417	−0.139708	+	0.990193i	$0.544617\pi$
$390$	0	0
$391$	−13.7735	−0.696558
$392$	0	0
$393$	9.18861	0.463504
$394$	0	0
$395$	53.8249	2.70822
$396$	0	0
$397$	10.6353	0.533772	0.266886	−	0.963728i	$-0.414005\pi$
$397$	10.6353	0.533772	0.266886	+	0.963728i	$0.414005\pi$
$398$	0	0
$399$	3.06445	0.153414
$400$	0	0
$401$	−14.6279	−0.730481	−0.365241	−	0.930913i	$-0.619013\pi$
$401$	−14.6279	−0.730481	−0.365241	+	0.930913i	$0.619013\pi$
$402$	0	0
$403$	−7.67999	−0.382567
$404$	0	0
$405$	18.6251	0.925490
$406$	0	0
$407$	−2.97022	−0.147228
$408$	0	0
$409$	14.5036	0.717157	0.358579	−	0.933500i	$-0.383261\pi$
$409$	14.5036	0.717157	0.358579	+	0.933500i	$0.383261\pi$
$410$	0	0
$411$	5.27284	0.260090
$412$	0	0
$413$	0.811133	0.0399133
$414$	0	0
$415$	12.1229	0.595089
$416$	0	0
$417$	8.35966	0.409375
$418$	0	0
$419$	−25.9893	−1.26966	−0.634831	−	0.772651i	$-0.718930\pi$
$419$	−25.9893	−1.26966	−0.634831	+	0.772651i	$0.718930\pi$
$420$	0	0
$421$	−15.4854	−0.754712	−0.377356	−	0.926068i	$-0.623167\pi$
$421$	−15.4854	−0.754712	−0.377356	+	0.926068i	$0.623167\pi$
$422$	0	0
$423$	0.782725	0.0380574
$424$	0	0
$425$	−51.7404	−2.50978
$426$	0	0
$427$	−15.1640	−0.733838
$428$	0	0
$429$	3.29023	0.158854
$430$	0	0
$431$	−17.1271	−0.824984	−0.412492	−	0.910961i	$-0.635342\pi$
$431$	−17.1271	−0.824984	−0.412492	+	0.910961i	$0.635342\pi$
$432$	0	0
$433$	−16.2455	−0.780711	−0.390356	−	0.920664i	$-0.627648\pi$
$433$	−16.2455	−0.780711	−0.390356	+	0.920664i	$0.627648\pi$
$434$	0	0
$435$	−7.51489	−0.360311
$436$	0	0
$437$	11.7689	0.562983
$438$	0	0
$439$	34.1030	1.62765	0.813823	−	0.581112i	$-0.197382\pi$
$439$	34.1030	1.62765	0.813823	+	0.581112i	$0.197382\pi$
$440$	0	0
$441$	−2.50423	−0.119249
$442$	0	0
$443$	−29.2102	−1.38782	−0.693908	−	0.720063i	$-0.744113\pi$
$443$	−29.2102	−1.38782	−0.693908	+	0.720063i	$0.744113\pi$
$444$	0	0
$445$	17.9531	0.851060
$446$	0	0
$447$	3.79247	0.179378
$448$	0	0
$449$	−4.97792	−0.234922	−0.117461	−	0.993077i	$-0.537476\pi$
$449$	−4.97792	−0.234922	−0.117461	+	0.993077i	$0.537476\pi$
$450$	0	0
$451$	1.15200	0.0542455
$452$	0	0
$453$	−0.352020	−0.0165394
$454$	0	0
$455$	15.7927	0.740372
$456$	0	0
$457$	31.7489	1.48515	0.742575	−	0.669763i	$-0.233604\pi$
$457$	31.7489	1.48515	0.742575	+	0.669763i	$0.233604\pi$
$458$	0	0
$459$	19.7405	0.921408
$460$	0	0
$461$	−12.9700	−0.604071	−0.302036	−	0.953297i	$-0.597666\pi$
$461$	−12.9700	−0.604071	−0.302036	+	0.953297i	$0.597666\pi$
$462$	0	0
$463$	−6.79374	−0.315732	−0.157866	−	0.987461i	$-0.550461\pi$
$463$	−6.79374	−0.315732	−0.157866	+	0.987461i	$0.550461\pi$
$464$	0	0
$465$	−5.19024	−0.240692
$466$	0	0
$467$	41.0967	1.90173	0.950864	−	0.309608i	$-0.100198\pi$
$467$	41.0967	1.90173	0.950864	+	0.309608i	$0.100198\pi$
$468$	0	0
$469$	−5.41423	−0.250006
$470$	0	0
$471$	14.0085	0.645478
$472$	0	0
$473$	8.17177	0.375739
$474$	0	0
$475$	44.2100	2.02849
$476$	0	0
$477$	9.74979	0.446412
$478$	0	0
$479$	−11.1457	−0.509259	−0.254629	−	0.967039i	$-0.581953\pi$
$479$	−11.1457	−0.509259	−0.254629	+	0.967039i	$0.581953\pi$
$480$	0	0
$481$	−10.4585	−0.476868
$482$	0	0
$483$	1.90399	0.0866345
$484$	0	0
$485$	−0.437333	−0.0198583
$486$	0	0
$487$	37.6190	1.70468	0.852340	−	0.522989i	$-0.175183\pi$
$487$	37.6190	1.70468	0.852340	+	0.522989i	$0.175183\pi$
$488$	0	0
$489$	−13.1590	−0.595069
$490$	0	0
$491$	39.4824	1.78182	0.890909	−	0.454183i	$-0.150069\pi$
$491$	39.4824	1.78182	0.890909	+	0.454183i	$0.150069\pi$
$492$	0	0
$493$	13.9631	0.628868
$494$	0	0
$495$	−11.2317	−0.504829
$496$	0	0
$497$	−2.88624	−0.129466
$498$	0	0
$499$	−36.8547	−1.64984	−0.824920	−	0.565249i	$-0.808780\pi$
$499$	−36.8547	−1.64984	−0.824920	+	0.565249i	$0.808780\pi$
$500$	0	0
$501$	−10.8226	−0.483519
$502$	0	0
$503$	10.1084	0.450711	0.225356	−	0.974277i	$-0.427646\pi$
$503$	10.1084	0.450711	0.225356	+	0.974277i	$0.427646\pi$
$504$	0	0
$505$	−57.4573	−2.55682
$506$	0	0
$507$	2.43192	0.108006
$508$	0	0
$509$	−2.84545	−0.126122	−0.0630611	−	0.998010i	$-0.520086\pi$
$509$	−2.84545	−0.126122	−0.0630611	+	0.998010i	$0.520086\pi$
$510$	0	0
$511$	−6.93866	−0.306948
$512$	0	0
$513$	−16.8674	−0.744715
$514$	0	0
$515$	8.38797	0.369618
$516$	0	0
$517$	−0.360070	−0.0158359
$518$	0	0
$519$	−15.7466	−0.691197
$520$	0	0
$521$	−41.4488	−1.81590	−0.907952	−	0.419074i	$-0.862355\pi$
$521$	−41.4488	−1.81590	−0.907952	+	0.419074i	$0.862355\pi$
$522$	0	0
$523$	35.5418	1.55413	0.777066	−	0.629419i	$-0.216707\pi$
$523$	35.5418	1.55413	0.777066	+	0.629419i	$0.216707\pi$
$524$	0	0
$525$	7.15235	0.312154
$526$	0	0
$527$	9.64379	0.420090
$528$	0	0
$529$	−15.6878	−0.682078
$530$	0	0
$531$	−2.03126	−0.0881494
$532$	0	0
$533$	4.05634	0.175700
$534$	0	0
$535$	1.63778	0.0708072
$536$	0	0
$537$	−18.2247	−0.786454
$538$	0	0
$539$	1.15200	0.0496201
$540$	0	0
$541$	−22.0879	−0.949632	−0.474816	−	0.880085i	$-0.657485\pi$
$541$	−22.0879	−0.949632	−0.474816	+	0.880085i	$0.657485\pi$
$542$	0	0
$543$	−9.86201	−0.423220
$544$	0	0
$545$	26.8285	1.14920
$546$	0	0
$547$	3.81170	0.162977	0.0814883	−	0.996674i	$-0.474033\pi$
$547$	3.81170	0.162977	0.0814883	+	0.996674i	$0.474033\pi$
$548$	0	0
$549$	37.9742	1.62070
$550$	0	0
$551$	−11.9309	−0.508273
$552$	0	0
$553$	13.8249	0.587894
$554$	0	0
$555$	−7.06802	−0.300021
$556$	0	0
$557$	−14.1233	−0.598425	−0.299212	−	0.954187i	$-0.596724\pi$
$557$	−14.1233	−0.598425	−0.299212	+	0.954187i	$0.596724\pi$
$558$	0	0
$559$	28.7739	1.21701
$560$	0	0
$561$	−4.13156	−0.174434
$562$	0	0
$563$	−22.0733	−0.930277	−0.465138	−	0.885238i	$-0.653995\pi$
$563$	−22.0733	−0.930277	−0.465138	+	0.885238i	$0.653995\pi$
$564$	0	0
$565$	34.3872	1.44668
$566$	0	0
$567$	4.78386	0.200903
$568$	0	0
$569$	26.0985	1.09411	0.547053	−	0.837098i	$-0.315750\pi$
$569$	26.0985	1.09411	0.547053	+	0.837098i	$0.315750\pi$
$570$	0	0
$571$	37.5464	1.57127	0.785634	−	0.618692i	$-0.212337\pi$
$571$	37.5464	1.57127	0.785634	+	0.618692i	$0.212337\pi$
$572$	0	0
$573$	9.73079	0.406510
$574$	0	0
$575$	27.4684	1.14551
$576$	0	0
$577$	−10.9081	−0.454111	−0.227055	−	0.973882i	$-0.572910\pi$
$577$	−10.9081	−0.454111	−0.227055	+	0.973882i	$0.572910\pi$
$578$	0	0
$579$	−17.4253	−0.724172
$580$	0	0
$581$	3.11376	0.129180
$582$	0	0
$583$	−4.48511	−0.185754
$584$	0	0
$585$	−39.5485	−1.63513
$586$	0	0
$587$	9.73867	0.401958	0.200979	−	0.979596i	$-0.435588\pi$
$587$	9.73867	0.401958	0.200979	+	0.979596i	$0.435588\pi$
$588$	0	0
$589$	−8.24020	−0.339532
$590$	0	0
$591$	−16.6860	−0.686371
$592$	0	0
$593$	45.9148	1.88550	0.942748	−	0.333507i	$-0.108232\pi$
$593$	45.9148	1.88550	0.942748	+	0.333507i	$0.108232\pi$
$594$	0	0
$595$	−19.8309	−0.812988
$596$	0	0
$597$	−11.8383	−0.484511
$598$	0	0
$599$	−30.9488	−1.26453	−0.632267	−	0.774751i	$-0.717875\pi$
$599$	−30.9488	−1.26453	−0.632267	+	0.774751i	$0.717875\pi$
$600$	0	0
$601$	40.1307	1.63696	0.818482	−	0.574532i	$-0.194816\pi$
$601$	40.1307	1.63696	0.818482	+	0.574532i	$0.194816\pi$
$602$	0	0
$603$	13.5585	0.552144
$604$	0	0
$605$	−37.6598	−1.53109
$606$	0	0
$607$	24.8551	1.00884	0.504419	−	0.863459i	$-0.331707\pi$
$607$	24.8551	1.00884	0.504419	+	0.863459i	$0.331707\pi$
$608$	0	0
$609$	−1.93020	−0.0782155
$610$	0	0
$611$	−1.26785	−0.0512919
$612$	0	0
$613$	36.8103	1.48676	0.743378	−	0.668872i	$-0.233222\pi$
$613$	36.8103	1.48676	0.743378	+	0.668872i	$0.233222\pi$
$614$	0	0
$615$	2.74133	0.110541
$616$	0	0
$617$	7.26796	0.292597	0.146298	−	0.989240i	$-0.453264\pi$
$617$	7.26796	0.292597	0.146298	+	0.989240i	$0.453264\pi$
$618$	0	0
$619$	42.5131	1.70875	0.854373	−	0.519661i	$-0.173942\pi$
$619$	42.5131	1.70875	0.854373	+	0.519661i	$0.173942\pi$
$620$	0	0
$621$	−10.4800	−0.420548
$622$	0	0
$623$	4.61125	0.184746
$624$	0	0
$625$	27.3951	1.09580
$626$	0	0
$627$	3.53024	0.140984
$628$	0	0
$629$	13.1328	0.523640
$630$	0	0
$631$	25.8119	1.02755	0.513777	−	0.857924i	$-0.328246\pi$
$631$	25.8119	1.02755	0.513777	+	0.857924i	$0.328246\pi$
$632$	0	0
$633$	−9.08704	−0.361177
$634$	0	0
$635$	9.76647	0.387571
$636$	0	0
$637$	4.05634	0.160718
$638$	0	0
$639$	7.22781	0.285928
$640$	0	0
$641$	6.02737	0.238067	0.119033	−	0.992890i	$-0.462020\pi$
$641$	6.02737	0.238067	0.119033	+	0.992890i	$0.462020\pi$
$642$	0	0
$643$	−0.681057	−0.0268583	−0.0134291	−	0.999910i	$-0.504275\pi$
$643$	−0.681057	−0.0268583	−0.0134291	+	0.999910i	$0.504275\pi$
$644$	0	0
$645$	19.4458	0.765677
$646$	0	0
$647$	−2.42132	−0.0951918	−0.0475959	−	0.998867i	$-0.515156\pi$
$647$	−2.42132	−0.0951918	−0.0475959	+	0.998867i	$0.515156\pi$
$648$	0	0
$649$	0.934425	0.0366794
$650$	0	0
$651$	−1.33311	−0.0522487
$652$	0	0
$653$	11.3602	0.444559	0.222279	−	0.974983i	$-0.428650\pi$
$653$	11.3602	0.444559	0.222279	+	0.974983i	$0.428650\pi$
$654$	0	0
$655$	50.8078	1.98523
$656$	0	0
$657$	17.3760	0.677902
$658$	0	0
$659$	−43.7947	−1.70600	−0.853001	−	0.521910i	$-0.825220\pi$
$659$	−43.7947	−1.70600	−0.853001	+	0.521910i	$0.825220\pi$
$660$	0	0
$661$	9.29554	0.361555	0.180777	−	0.983524i	$-0.442139\pi$
$661$	9.29554	0.361555	0.180777	+	0.983524i	$0.442139\pi$
$662$	0	0
$663$	−14.5478	−0.564988
$664$	0	0
$665$	16.9447	0.657086
$666$	0	0
$667$	−7.41286	−0.287027
$668$	0	0
$669$	4.76541	0.184241
$670$	0	0
$671$	−17.4689	−0.674381
$672$	0	0
$673$	17.3801	0.669954	0.334977	−	0.942226i	$-0.391271\pi$
$673$	17.3801	0.669954	0.334977	+	0.942226i	$0.391271\pi$
$674$	0	0
$675$	−39.3682	−1.51528
$676$	0	0
$677$	40.9393	1.57343	0.786713	−	0.617320i	$-0.211781\pi$
$677$	40.9393	1.57343	0.786713	+	0.617320i	$0.211781\pi$
$678$	0	0
$679$	−0.112329	−0.00431078
$680$	0	0
$681$	5.69049	0.218060
$682$	0	0
$683$	14.5992	0.558621	0.279311	−	0.960201i	$-0.409894\pi$
$683$	14.5992	0.558621	0.279311	+	0.960201i	$0.409894\pi$
$684$	0	0
$685$	29.1559	1.11399
$686$	0	0
$687$	8.87047	0.338430
$688$	0	0
$689$	−15.7927	−0.601653
$690$	0	0
$691$	−44.6610	−1.69899	−0.849493	−	0.527600i	$-0.823092\pi$
$691$	−44.6610	−1.69899	−0.849493	+	0.527600i	$0.823092\pi$
$692$	0	0
$693$	−2.88487	−0.109587
$694$	0	0
$695$	46.2242	1.75338
$696$	0	0
$697$	−5.09356	−0.192932
$698$	0	0
$699$	−7.22918	−0.273433
$700$	0	0
$701$	24.3717	0.920506	0.460253	−	0.887788i	$-0.347759\pi$
$701$	24.3717	0.920506	0.460253	+	0.887788i	$0.347759\pi$
$702$	0	0
$703$	−11.2214	−0.423224
$704$	0	0
$705$	−0.856833	−0.0322702
$706$	0	0
$707$	−14.7579	−0.555028
$708$	0	0
$709$	−24.6351	−0.925191	−0.462595	−	0.886570i	$-0.653082\pi$
$709$	−24.6351	−0.925191	−0.462595	+	0.886570i	$0.653082\pi$
$710$	0	0
$711$	−34.6207	−1.29838
$712$	0	0
$713$	−5.11977	−0.191737
$714$	0	0
$715$	18.1931	0.680385
$716$	0	0
$717$	−7.64586	−0.285540
$718$	0	0
$719$	−45.8557	−1.71013	−0.855065	−	0.518521i	$-0.826483\pi$
$719$	−45.8557	−1.71013	−0.855065	+	0.518521i	$0.826483\pi$
$720$	0	0
$721$	2.15445	0.0802358
$722$	0	0
$723$	4.83643	0.179869
$724$	0	0
$725$	−27.8465	−1.03419
$726$	0	0
$727$	−22.5566	−0.836579	−0.418290	−	0.908314i	$-0.637370\pi$
$727$	−22.5566	−0.836579	−0.418290	+	0.908314i	$0.637370\pi$
$728$	0	0
$729$	−3.79337	−0.140495
$730$	0	0
$731$	−36.1315	−1.33637
$732$	0	0
$733$	23.8353	0.880379	0.440189	−	0.897905i	$-0.354911\pi$
$733$	23.8353	0.880379	0.440189	+	0.897905i	$0.354911\pi$
$734$	0	0
$735$	2.74133	0.101115
$736$	0	0
$737$	−6.23718	−0.229750
$738$	0	0
$739$	21.0105	0.772885	0.386443	−	0.922313i	$-0.373704\pi$
$739$	21.0105	0.772885	0.386443	+	0.922313i	$0.373704\pi$
$740$	0	0
$741$	12.4304	0.456644
$742$	0	0
$743$	10.9707	0.402475	0.201238	−	0.979542i	$-0.435504\pi$
$743$	10.9707	0.402475	0.201238	+	0.979542i	$0.435504\pi$
$744$	0	0
$745$	20.9702	0.768289
$746$	0	0
$747$	−7.79757	−0.285298
$748$	0	0
$749$	0.420662	0.0153707
$750$	0	0
$751$	−4.26203	−0.155524	−0.0777619	−	0.996972i	$-0.524777\pi$
$751$	−4.26203	−0.155524	−0.0777619	+	0.996972i	$0.524777\pi$
$752$	0	0
$753$	−7.15063	−0.260583
$754$	0	0
$755$	−1.94647	−0.0708395
$756$	0	0
$757$	−5.36222	−0.194893	−0.0974466	−	0.995241i	$-0.531068\pi$
$757$	−5.36222	−0.194893	−0.0974466	+	0.995241i	$0.531068\pi$
$758$	0	0
$759$	2.19339	0.0796151
$760$	0	0
$761$	29.5422	1.07090	0.535452	−	0.844566i	$-0.320141\pi$
$761$	29.5422	1.07090	0.535452	+	0.844566i	$0.320141\pi$
$762$	0	0
$763$	6.89088	0.249467
$764$	0	0
$765$	49.6611	1.79550
$766$	0	0
$767$	3.29023	0.118803
$768$	0	0
$769$	46.2585	1.66812	0.834062	−	0.551671i	$-0.186009\pi$
$769$	46.2585	1.66812	0.834062	+	0.551671i	$0.186009\pi$
$770$	0	0
$771$	−6.79330	−0.244655
$772$	0	0
$773$	7.18513	0.258431	0.129216	−	0.991617i	$-0.458754\pi$
$773$	7.18513	0.258431	0.129216	+	0.991617i	$0.458754\pi$
$774$	0	0
$775$	−19.2324	−0.690850
$776$	0	0
$777$	−1.81542	−0.0651278
$778$	0	0
$779$	4.35223	0.155935
$780$	0	0
$781$	−3.32495	−0.118976
$782$	0	0
$783$	10.6242	0.379680
$784$	0	0
$785$	77.4592	2.76464
$786$	0	0
$787$	−44.5448	−1.58785	−0.793926	−	0.608014i	$-0.791966\pi$
$787$	−44.5448	−1.58785	−0.793926	+	0.608014i	$0.791966\pi$
$788$	0	0
$789$	−5.27483	−0.187789
$790$	0	0
$791$	8.83234	0.314042
$792$	0	0
$793$	−61.5104	−2.18430
$794$	0	0
$795$	−10.6729	−0.378529
$796$	0	0
$797$	44.4347	1.57396	0.786980	−	0.616979i	$-0.211644\pi$
$797$	44.4347	1.57396	0.786980	+	0.616979i	$0.211644\pi$
$798$	0	0
$799$	1.59205	0.0563226
$800$	0	0
$801$	−11.5476	−0.408016
$802$	0	0
$803$	−7.99332	−0.282078
$804$	0	0
$805$	10.5280	0.371063
$806$	0	0
$807$	−4.56949	−0.160854
$808$	0	0
$809$	35.0953	1.23389	0.616943	−	0.787008i	$-0.288371\pi$
$809$	35.0953	1.23389	0.616943	+	0.787008i	$0.288371\pi$
$810$	0	0
$811$	−26.2002	−0.920014	−0.460007	−	0.887915i	$-0.652153\pi$
$811$	−26.2002	−0.920014	−0.460007	+	0.887915i	$0.652153\pi$
$812$	0	0
$813$	15.4228	0.540901
$814$	0	0
$815$	−72.7616	−2.54873
$816$	0	0
$817$	30.8728	1.08010
$818$	0	0
$819$	−10.1580	−0.354950
$820$	0	0
$821$	−14.0445	−0.490156	−0.245078	−	0.969503i	$-0.578814\pi$
$821$	−14.0445	−0.490156	−0.245078	+	0.969503i	$0.578814\pi$
$822$	0	0
$823$	19.3473	0.674404	0.337202	−	0.941432i	$-0.390519\pi$
$823$	19.3473	0.674404	0.337202	+	0.941432i	$0.390519\pi$
$824$	0	0
$825$	8.23950	0.286863
$826$	0	0
$827$	12.3738	0.430279	0.215140	−	0.976583i	$-0.430979\pi$
$827$	12.3738	0.430279	0.215140	+	0.976583i	$0.430979\pi$
$828$	0	0
$829$	9.39798	0.326405	0.163203	−	0.986593i	$-0.447818\pi$
$829$	9.39798	0.326405	0.163203	+	0.986593i	$0.447818\pi$
$830$	0	0
$831$	11.5429	0.400419
$832$	0	0
$833$	−5.09356	−0.176481
$834$	0	0
$835$	−59.8429	−2.07095
$836$	0	0
$837$	7.33775	0.253630
$838$	0	0
$839$	−22.2311	−0.767504	−0.383752	−	0.923436i	$-0.625368\pi$
$839$	−22.2311	−0.767504	−0.383752	+	0.923436i	$0.625368\pi$
$840$	0	0
$841$	−21.4851	−0.740866
$842$	0	0
$843$	−0.224359	−0.00772732
$844$	0	0
$845$	13.4472	0.462597
$846$	0	0
$847$	−9.67290	−0.332365
$848$	0	0
$849$	−16.9076	−0.580267
$850$	0	0
$851$	−6.97205	−0.238999
$852$	0	0
$853$	43.2629	1.48129	0.740646	−	0.671895i	$-0.234519\pi$
$853$	43.2629	1.48129	0.740646	+	0.671895i	$0.234519\pi$
$854$	0	0
$855$	−42.4333	−1.45119
$856$	0	0
$857$	4.07295	0.139129	0.0695647	−	0.997577i	$-0.477839\pi$
$857$	4.07295	0.139129	0.0695647	+	0.997577i	$0.477839\pi$
$858$	0	0
$859$	25.5489	0.871718	0.435859	−	0.900015i	$-0.356445\pi$
$859$	25.5489	0.871718	0.435859	+	0.900015i	$0.356445\pi$
$860$	0	0
$861$	0.704110	0.0239960
$862$	0	0
$863$	48.7756	1.66034	0.830170	−	0.557510i	$-0.188243\pi$
$863$	48.7756	1.66034	0.830170	+	0.557510i	$0.188243\pi$
$864$	0	0
$865$	−87.0696	−2.96046
$866$	0	0
$867$	6.29781	0.213885
$868$	0	0
$869$	15.9263	0.540262
$870$	0	0
$871$	−21.9620	−0.744153
$872$	0	0
$873$	0.281297	0.00952046
$874$	0	0
$875$	20.0818	0.678890
$876$	0	0
$877$	−11.0318	−0.372518	−0.186259	−	0.982501i	$-0.559636\pi$
$877$	−11.0318	−0.372518	−0.186259	+	0.982501i	$0.559636\pi$
$878$	0	0
$879$	−18.9049	−0.637645
$880$	0	0
$881$	−24.0665	−0.810820	−0.405410	−	0.914135i	$-0.632871\pi$
$881$	−24.0665	−0.810820	−0.405410	+	0.914135i	$0.632871\pi$
$882$	0	0
$883$	−13.6253	−0.458529	−0.229264	−	0.973364i	$-0.573632\pi$
$883$	−13.6253	−0.458529	−0.229264	+	0.973364i	$0.573632\pi$
$884$	0	0
$885$	2.22358	0.0747450
$886$	0	0
$887$	−41.1806	−1.38271	−0.691354	−	0.722516i	$-0.742986\pi$
$887$	−41.1806	−1.38271	−0.691354	+	0.722516i	$0.742986\pi$
$888$	0	0
$889$	2.50851	0.0841329
$890$	0	0
$891$	5.51100	0.184625
$892$	0	0
$893$	−1.36034	−0.0455220
$894$	0	0
$895$	−100.772	−3.36845
$896$	0	0
$897$	7.72323	0.257871
$898$	0	0
$899$	5.19024	0.173104
$900$	0	0
$901$	19.8309	0.660663
$902$	0	0
$903$	4.99464	0.166211
$904$	0	0
$905$	−54.5314	−1.81268
$906$	0	0
$907$	29.9766	0.995356	0.497678	−	0.867362i	$-0.334186\pi$
$907$	29.9766	0.995356	0.497678	+	0.867362i	$0.334186\pi$
$908$	0	0
$909$	36.9572	1.22579
$910$	0	0
$911$	17.2220	0.570590	0.285295	−	0.958440i	$-0.407908\pi$
$911$	17.2220	0.570590	0.285295	+	0.958440i	$0.407908\pi$
$912$	0	0
$913$	3.58705	0.118714
$914$	0	0
$915$	−41.5696	−1.37425
$916$	0	0
$917$	13.0500	0.430948
$918$	0	0
$919$	−26.1444	−0.862426	−0.431213	−	0.902250i	$-0.641914\pi$
$919$	−26.1444	−0.862426	−0.431213	+	0.902250i	$0.641914\pi$
$920$	0	0
$921$	23.2755	0.766955
$922$	0	0
$923$	−11.7076	−0.385360
$924$	0	0
$925$	−26.1906	−0.861141
$926$	0	0
$927$	−5.39523	−0.177203
$928$	0	0
$929$	−54.0538	−1.77345	−0.886724	−	0.462300i	$-0.847024\pi$
$929$	−54.0538	−1.77345	−0.886724	+	0.462300i	$0.847024\pi$
$930$	0	0
$931$	4.35223	0.142639
$932$	0	0
$933$	−12.3678	−0.404904
$934$	0	0
$935$	−22.8452	−0.747117
$936$	0	0
$937$	−22.7495	−0.743193	−0.371597	−	0.928394i	$-0.621190\pi$
$937$	−22.7495	−0.743193	−0.371597	+	0.928394i	$0.621190\pi$
$938$	0	0
$939$	3.70562	0.120928
$940$	0	0
$941$	27.1185	0.884037	0.442019	−	0.897006i	$-0.354263\pi$
$941$	27.1185	0.884037	0.442019	+	0.897006i	$0.354263\pi$
$942$	0	0
$943$	2.70411	0.0880579
$944$	0	0
$945$	−15.0889	−0.490842
$946$	0	0
$947$	−7.58464	−0.246468	−0.123234	−	0.992378i	$-0.539327\pi$
$947$	−7.58464	−0.246468	−0.123234	+	0.992378i	$0.539327\pi$
$948$	0	0
$949$	−28.1456	−0.913643
$950$	0	0
$951$	5.37363	0.174252
$952$	0	0
$953$	−35.9220	−1.16363	−0.581814	−	0.813322i	$-0.697657\pi$
$953$	−35.9220	−1.16363	−0.581814	+	0.813322i	$0.697657\pi$
$954$	0	0
$955$	53.8058	1.74111
$956$	0	0
$957$	−2.22358	−0.0718783
$958$	0	0
$959$	7.48867	0.241822
$960$	0	0
$961$	−27.4153	−0.884365
$962$	0	0
$963$	−1.05343	−0.0339464
$964$	0	0
$965$	−96.3523	−3.10169
$966$	0	0
$967$	1.31175	0.0421831	0.0210915	−	0.999778i	$-0.493286\pi$
$967$	1.31175	0.0421831	0.0210915	+	0.999778i	$0.493286\pi$
$968$	0	0
$969$	−15.6090	−0.501432
$970$	0	0
$971$	−30.6124	−0.982399	−0.491200	−	0.871047i	$-0.663441\pi$
$971$	−30.6124	−0.982399	−0.491200	+	0.871047i	$0.663441\pi$
$972$	0	0
$973$	11.8727	0.380620
$974$	0	0
$975$	29.0124	0.929140
$976$	0	0
$977$	−10.8256	−0.346341	−0.173170	−	0.984892i	$-0.555401\pi$
$977$	−10.8256	−0.346341	−0.173170	+	0.984892i	$0.555401\pi$
$978$	0	0
$979$	5.31216	0.169777
$980$	0	0
$981$	−17.2563	−0.550953
$982$	0	0
$983$	14.7636	0.470884	0.235442	−	0.971888i	$-0.424346\pi$
$983$	14.7636	0.470884	0.235442	+	0.971888i	$0.424346\pi$
$984$	0	0
$985$	−92.2642	−2.93978
$986$	0	0
$987$	−0.220077	−0.00700514
$988$	0	0
$989$	19.1818	0.609945
$990$	0	0
$991$	−42.5954	−1.35309	−0.676543	−	0.736403i	$-0.736523\pi$
$991$	−42.5954	−1.35309	−0.676543	+	0.736403i	$0.736523\pi$
$992$	0	0
$993$	21.3018	0.675994
$994$	0	0
$995$	−65.4594	−2.07520
$996$	0	0
$997$	29.5392	0.935517	0.467759	−	0.883856i	$-0.345062\pi$
$997$	29.5392	0.935517	0.467759	+	0.883856i	$0.345062\pi$
$998$	0	0
$999$	9.99248	0.316148

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1148.2.a.e.1.3	✓	5
4.3	odd	2		4592.2.a.bd.1.3		5
7.6	odd	2		8036.2.a.j.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.a.e.1.3	✓	5	1.1	even	1	trivial
4592.2.a.bd.1.3		5	4.3	odd	2
8036.2.a.j.1.3		5	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:	\( N \)	\(=\)	\( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1148.a (trivial)

\(f(q)\)	\(=\)	\(q+0.704110 q^{3} +3.89333 q^{5} +1.00000 q^{7} -2.50423 q^{9} +O(q^{10})\)	\(q+0.704110 q^{3} +3.89333 q^{5} +1.00000 q^{7} -2.50423 q^{9} +1.15200 q^{11} +4.05634 q^{13} +2.74133 q^{15} -5.09356 q^{17} +4.35223 q^{19} +0.704110 q^{21} +2.70411 q^{23} +10.1580 q^{25} -3.87558 q^{27} -2.74133 q^{29} -1.89333 q^{31} +0.811133 q^{33} +3.89333 q^{35} -2.57832 q^{37} +2.85611 q^{39} +1.00000 q^{41} +7.09356 q^{43} -9.74979 q^{45} -0.312561 q^{47} +1.00000 q^{49} -3.58643 q^{51} -3.89333 q^{53} +4.48511 q^{55} +3.06445 q^{57} +0.811133 q^{59} -15.1640 q^{61} -2.50423 q^{63} +15.7927 q^{65} -5.41423 q^{67} +1.90399 q^{69} -2.88624 q^{71} -6.93866 q^{73} +7.15235 q^{75} +1.15200 q^{77} +13.8249 q^{79} +4.78386 q^{81} +3.11376 q^{83} -19.8309 q^{85} -1.93020 q^{87} +4.61125 q^{89} +4.05634 q^{91} -1.33311 q^{93} +16.9447 q^{95} -0.112329 q^{97} -2.88487 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{3} + 3 q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q + 2 * q^3 + 3 * q^5 + 5 * q^7 + 5 * q^9	\( 5 q + 2 q^{3} + 3 q^{5} + 5 q^{7} + 5 q^{9} + 7 q^{13} + 3 q^{15} - 3 q^{17} + 10 q^{19} + 2 q^{21} + 12 q^{23} + 2 q^{25} + 14 q^{27} - 3 q^{29} + 7 q^{31} - 3 q^{33} + 3 q^{35} + q^{37} + 7 q^{39} + 5 q^{41} + 13 q^{43} - 3 q^{45} + 9 q^{47} + 5 q^{49} + 9 q^{51} - 3 q^{53} + 9 q^{55} - 11 q^{57} - 3 q^{59} + 16 q^{61} + 5 q^{63} + 3 q^{65} + 19 q^{67} + 24 q^{69} - 12 q^{71} + 4 q^{73} + 8 q^{75} + 28 q^{79} + 5 q^{81} + 18 q^{83} - 15 q^{85} - 6 q^{87} + 7 q^{91} + q^{93} + 3 q^{95} + 4 q^{97} - 33 q^{99}+O(q^{100}) \) 5 * q + 2 * q^3 + 3 * q^5 + 5 * q^7 + 5 * q^9 + 7 * q^13 + 3 * q^15 - 3 * q^17 + 10 * q^19 + 2 * q^21 + 12 * q^23 + 2 * q^25 + 14 * q^27 - 3 * q^29 + 7 * q^31 - 3 * q^33 + 3 * q^35 + q^37 + 7 * q^39 + 5 * q^41 + 13 * q^43 - 3 * q^45 + 9 * q^47 + 5 * q^49 + 9 * q^51 - 3 * q^53 + 9 * q^55 - 11 * q^57 - 3 * q^59 + 16 * q^61 + 5 * q^63 + 3 * q^65 + 19 * q^67 + 24 * q^69 - 12 * q^71 + 4 * q^73 + 8 * q^75 + 28 * q^79 + 5 * q^81 + 18 * q^83 - 15 * q^85 - 6 * q^87 + 7 * q^91 + q^93 + 3 * q^95 + 4 * q^97 - 33 * q^99

Introduction

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