LMFDB - Embedded newform 4304.2.a.h.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4304,2,Mod(1,4304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4304, 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("4304.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4304 = 2^{4} \cdot 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4304.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:	$34.3676130300$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 15x^{5} + 16x^{4} + 49x^{3} - 53x^{2} - 44x + 48 $ x^7 - x^6 - 15x^5 + 16x^4 + 49x^3 - 53x^2 - 44*x + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 538)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$3.22542$ of defining polynomial
Character	$\chi$	$=$	4304.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.383827 q^{3} +4.15424 q^{5} -4.87818 q^{7} -2.85268 q^{9} +1.62910 q^{11} -2.16682 q^{13} +1.59451 q^{15} -6.78448 q^{17} +8.61766 q^{19} -1.87238 q^{21} -2.76765 q^{23} +12.2577 q^{25} -2.24641 q^{27} +2.10510 q^{29} +3.62414 q^{31} +0.625290 q^{33} -20.2651 q^{35} +1.19760 q^{37} -0.831683 q^{39} +5.43715 q^{41} +9.63402 q^{43} -11.8507 q^{45} +7.07613 q^{47} +16.7967 q^{49} -2.60406 q^{51} -3.03890 q^{53} +6.76765 q^{55} +3.30769 q^{57} -4.21827 q^{59} +8.74744 q^{61} +13.9159 q^{63} -9.00148 q^{65} +1.50761 q^{67} -1.06230 q^{69} -11.0270 q^{71} +1.37039 q^{73} +4.70482 q^{75} -7.94703 q^{77} +14.9749 q^{79} +7.69580 q^{81} +8.08276 q^{83} -28.1843 q^{85} +0.807993 q^{87} +3.31299 q^{89} +10.5701 q^{91} +1.39104 q^{93} +35.7998 q^{95} +16.8474 q^{97} -4.64729 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{3} + 7 q^{5} - 6 q^{7} + 12 q^{9} + 3 q^{11} - 9 q^{13} - 8 q^{15} + 8 q^{17} + 11 q^{19} - 6 q^{21} - 12 q^{23} + 22 q^{25} + 14 q^{27} - 5 q^{29} - 14 q^{31} - 4 q^{33} + 4 q^{35} + 13 q^{37}+ \cdots + 37 q^{99}+O(q^{100}) $ 7 * q - q^3 + 7 * q^5 - 6 * q^7 + 12 * q^9 + 3 * q^11 - 9 * q^13 - 8 * q^15 + 8 * q^17 + 11 * q^19 - 6 * q^21 - 12 * q^23 + 22 * q^25 + 14 * q^27 - 5 * q^29 - 14 * q^31 - 4 * q^33 + 4 * q^35 + 13 * q^37 + 18 * q^39 + 12 * q^41 + 11 * q^43 + 3 * q^45 - 2 * q^47 + 15 * q^49 + 26 * q^51 + 19 * q^53 + 40 * q^55 - 12 * q^57 + 9 * q^59 - 3 * q^61 + 26 * q^63 - 10 * q^65 + 33 * q^67 - 64 * q^69 - 28 * q^71 - 14 * q^73 + 45 * q^75 + 10 * q^77 - 2 * q^79 + 15 * q^81 + 7 * q^83 - 16 * q^85 - 16 * q^87 + 18 * q^89 + 26 * q^91 + 6 * q^93 + 34 * q^95 - 4 * q^97 + 37 * q^99

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
$3$	0.383827	0.221602	0.110801	−	0.993843i	$-0.464658\pi$
$3$	0.383827	0.221602	0.110801	+	0.993843i	$0.464658\pi$
$4$	0	0
$5$	4.15424	1.85783	0.928916	−	0.370291i	$-0.120742\pi$
$5$	4.15424	1.85783	0.928916	+	0.370291i	$0.120742\pi$
$6$	0	0
$7$	−4.87818	−1.84378	−0.921890	−	0.387452i	$-0.873355\pi$
$7$	−4.87818	−1.84378	−0.921890	+	0.387452i	$0.873355\pi$
$8$	0	0
$9$	−2.85268	−0.950892
$10$	0	0
$11$	1.62910	0.491191	0.245596	−	0.969372i	$-0.421016\pi$
$11$	1.62910	0.491191	0.245596	+	0.969372i	$0.421016\pi$
$12$	0	0
$13$	−2.16682	−0.600968	−0.300484	−	0.953787i	$-0.597148\pi$
$13$	−2.16682	−0.600968	−0.300484	+	0.953787i	$0.597148\pi$
$14$	0	0
$15$	1.59451	0.411700
$16$	0	0
$17$	−6.78448	−1.64548	−0.822739	−	0.568420i	$-0.807555\pi$
$17$	−6.78448	−1.64548	−0.822739	+	0.568420i	$0.807555\pi$
$18$	0	0
$19$	8.61766	1.97703	0.988513	−	0.151136i	$-0.0482931\pi$
$19$	8.61766	1.97703	0.988513	+	0.151136i	$0.0482931\pi$
$20$	0	0
$21$	−1.87238	−0.408586
$22$	0	0
$23$	−2.76765	−0.577096	−0.288548	−	0.957465i	$-0.593172\pi$
$23$	−2.76765	−0.577096	−0.288548	+	0.957465i	$0.593172\pi$
$24$	0	0
$25$	12.2577	2.45154
$26$	0	0
$27$	−2.24641	−0.432322
$28$	0	0
$29$	2.10510	0.390907	0.195454	−	0.980713i	$-0.437382\pi$
$29$	2.10510	0.390907	0.195454	+	0.980713i	$0.437382\pi$
$30$	0	0
$31$	3.62414	0.650914	0.325457	−	0.945557i	$-0.394482\pi$
$31$	3.62414	0.650914	0.325457	+	0.945557i	$0.394482\pi$
$32$	0	0
$33$	0.625290	0.108849
$34$	0	0
$35$	−20.2651	−3.42543
$36$	0	0
$37$	1.19760	0.196885	0.0984424	−	0.995143i	$-0.468614\pi$
$37$	1.19760	0.196885	0.0984424	+	0.995143i	$0.468614\pi$
$38$	0	0
$39$	−0.831683	−0.133176
$40$	0	0
$41$	5.43715	0.849140	0.424570	−	0.905395i	$-0.360425\pi$
$41$	5.43715	0.849140	0.424570	+	0.905395i	$0.360425\pi$
$42$	0	0
$43$	9.63402	1.46917	0.734587	−	0.678514i	$-0.237376\pi$
$43$	9.63402	1.46917	0.734587	+	0.678514i	$0.237376\pi$
$44$	0	0
$45$	−11.8507	−1.76660
$46$	0	0
$47$	7.07613	1.03216	0.516080	−	0.856541i	$-0.327391\pi$
$47$	7.07613	1.03216	0.516080	+	0.856541i	$0.327391\pi$
$48$	0	0
$49$	16.7967	2.39952
$50$	0	0
$51$	−2.60406	−0.364642
$52$	0	0
$53$	−3.03890	−0.417425	−0.208712	−	0.977977i	$-0.566927\pi$
$53$	−3.03890	−0.417425	−0.208712	+	0.977977i	$0.566927\pi$
$54$	0	0
$55$	6.76765	0.912550
$56$	0	0
$57$	3.30769	0.438114
$58$	0	0
$59$	−4.21827	−0.549172	−0.274586	−	0.961563i	$-0.588541\pi$
$59$	−4.21827	−0.549172	−0.274586	+	0.961563i	$0.588541\pi$
$60$	0	0
$61$	8.74744	1.11999	0.559997	−	0.828494i	$-0.310802\pi$
$61$	8.74744	1.11999	0.559997	+	0.828494i	$0.310802\pi$
$62$	0	0
$63$	13.9159	1.75324
$64$	0	0
$65$	−9.00148	−1.11650
$66$	0	0
$67$	1.50761	0.184184	0.0920921	−	0.995750i	$-0.470645\pi$
$67$	1.50761	0.184184	0.0920921	+	0.995750i	$0.470645\pi$
$68$	0	0
$69$	−1.06230	−0.127886
$70$	0	0
$71$	−11.0270	−1.30866	−0.654331	−	0.756208i	$-0.727050\pi$
$71$	−11.0270	−1.30866	−0.654331	+	0.756208i	$0.727050\pi$
$72$	0	0
$73$	1.37039	0.160391	0.0801957	−	0.996779i	$-0.474445\pi$
$73$	1.37039	0.160391	0.0801957	+	0.996779i	$0.474445\pi$
$74$	0	0
$75$	4.70482	0.543266
$76$	0	0
$77$	−7.94703	−0.905648
$78$	0	0
$79$	14.9749	1.68480	0.842401	−	0.538850i	$-0.181141\pi$
$79$	14.9749	1.68480	0.842401	+	0.538850i	$0.181141\pi$
$80$	0	0
$81$	7.69580	0.855089
$82$	0	0
$83$	8.08276	0.887198	0.443599	−	0.896225i	$-0.353701\pi$
$83$	8.08276	0.887198	0.443599	+	0.896225i	$0.353701\pi$
$84$	0	0
$85$	−28.1843	−3.05702
$86$	0	0
$87$	0.807993	0.0866259
$88$	0	0
$89$	3.31299	0.351176	0.175588	−	0.984464i	$-0.443817\pi$
$89$	3.31299	0.351176	0.175588	+	0.984464i	$0.443817\pi$
$90$	0	0
$91$	10.5701	1.10805
$92$	0	0
$93$	1.39104	0.144244
$94$	0	0
$95$	35.7998	3.67298
$96$	0	0
$97$	16.8474	1.71059	0.855296	−	0.518139i	$-0.173375\pi$
$97$	16.8474	1.71059	0.855296	+	0.518139i	$0.173375\pi$
$98$	0	0
$99$	−4.64729	−0.467070
$100$	0	0
$101$	2.93034	0.291580	0.145790	−	0.989316i	$-0.453428\pi$
$101$	2.93034	0.291580	0.145790	+	0.989316i	$0.453428\pi$
$102$	0	0
$103$	−1.25819	−0.123973	−0.0619867	−	0.998077i	$-0.519744\pi$
$103$	−1.25819	−0.123973	−0.0619867	+	0.998077i	$0.519744\pi$
$104$	0	0
$105$	−7.77829	−0.759084
$106$	0	0
$107$	−9.35946	−0.904813	−0.452407	−	0.891812i	$-0.649434\pi$
$107$	−9.35946	−0.904813	−0.452407	+	0.891812i	$0.649434\pi$
$108$	0	0
$109$	11.4192	1.09376	0.546882	−	0.837210i	$-0.315815\pi$
$109$	11.4192	1.09376	0.546882	+	0.837210i	$0.315815\pi$
$110$	0	0
$111$	0.459672	0.0436301
$112$	0	0
$113$	0.316816	0.0298036	0.0149018	−	0.999889i	$-0.495256\pi$
$113$	0.316816	0.0298036	0.0149018	+	0.999889i	$0.495256\pi$
$114$	0	0
$115$	−11.4975	−1.07215
$116$	0	0
$117$	6.18124	0.571456
$118$	0	0
$119$	33.0959	3.03390
$120$	0	0
$121$	−8.34604	−0.758731
$122$	0	0
$123$	2.08692	0.188171
$124$	0	0
$125$	30.1501	2.69671
$126$	0	0
$127$	2.08772	0.185255	0.0926276	−	0.995701i	$-0.470473\pi$
$127$	2.08772	0.185255	0.0926276	+	0.995701i	$0.470473\pi$
$128$	0	0
$129$	3.69779	0.325572
$130$	0	0
$131$	5.17397	0.452052	0.226026	−	0.974121i	$-0.427427\pi$
$131$	5.17397	0.452052	0.226026	+	0.974121i	$0.427427\pi$
$132$	0	0
$133$	−42.0385	−3.64520
$134$	0	0
$135$	−9.33213	−0.803182
$136$	0	0
$137$	6.35949	0.543328	0.271664	−	0.962392i	$-0.412426\pi$
$137$	6.35949	0.543328	0.271664	+	0.962392i	$0.412426\pi$
$138$	0	0
$139$	10.6786	0.905746	0.452873	−	0.891575i	$-0.350399\pi$
$139$	10.6786	0.905746	0.452873	+	0.891575i	$0.350399\pi$
$140$	0	0
$141$	2.71601	0.228729
$142$	0	0
$143$	−3.52996	−0.295190
$144$	0	0
$145$	8.74508	0.726239
$146$	0	0
$147$	6.44701	0.531740
$148$	0	0
$149$	−3.47426	−0.284622	−0.142311	−	0.989822i	$-0.545453\pi$
$149$	−3.47426	−0.284622	−0.142311	+	0.989822i	$0.545453\pi$
$150$	0	0
$151$	−11.2407	−0.914752	−0.457376	−	0.889273i	$-0.651211\pi$
$151$	−11.2407	−0.914752	−0.457376	+	0.889273i	$0.651211\pi$
$152$	0	0
$153$	19.3539	1.56467
$154$	0	0
$155$	15.0555	1.20929
$156$	0	0
$157$	8.59533	0.685982	0.342991	−	0.939339i	$-0.388560\pi$
$157$	8.59533	0.685982	0.342991	+	0.939339i	$0.388560\pi$
$158$	0	0
$159$	−1.16641	−0.0925024
$160$	0	0
$161$	13.5011	1.06404
$162$	0	0
$163$	−10.1530	−0.795241	−0.397621	−	0.917550i	$-0.630164\pi$
$163$	−10.1530	−0.795241	−0.397621	+	0.917550i	$0.630164\pi$
$164$	0	0
$165$	2.59760	0.202223
$166$	0	0
$167$	−4.37572	−0.338604	−0.169302	−	0.985564i	$-0.554151\pi$
$167$	−4.37572	−0.338604	−0.169302	+	0.985564i	$0.554151\pi$
$168$	0	0
$169$	−8.30489	−0.638838
$170$	0	0
$171$	−24.5834	−1.87994
$172$	0	0
$173$	−22.7651	−1.73080	−0.865398	−	0.501085i	$-0.832934\pi$
$173$	−22.7651	−1.73080	−0.865398	+	0.501085i	$0.832934\pi$
$174$	0	0
$175$	−59.7952	−4.52009
$176$	0	0
$177$	−1.61908	−0.121698
$178$	0	0
$179$	−12.1634	−0.909139	−0.454569	−	0.890711i	$-0.650207\pi$
$179$	−12.1634	−0.909139	−0.454569	+	0.890711i	$0.650207\pi$
$180$	0	0
$181$	−4.61048	−0.342694	−0.171347	−	0.985211i	$-0.554812\pi$
$181$	−4.61048	−0.342694	−0.171347	+	0.985211i	$0.554812\pi$
$182$	0	0
$183$	3.35750	0.248193
$184$	0	0
$185$	4.97513	0.365779
$186$	0	0
$187$	−11.0526	−0.808244
$188$	0	0
$189$	10.9584	0.797107
$190$	0	0
$191$	−3.30553	−0.239180	−0.119590	−	0.992823i	$-0.538158\pi$
$191$	−3.30553	−0.239180	−0.119590	+	0.992823i	$0.538158\pi$
$192$	0	0
$193$	−18.5157	−1.33279	−0.666394	−	0.745600i	$-0.732163\pi$
$193$	−18.5157	−1.33279	−0.666394	+	0.745600i	$0.732163\pi$
$194$	0	0
$195$	−3.45501	−0.247418
$196$	0	0
$197$	8.78109	0.625627	0.312814	−	0.949815i	$-0.398729\pi$
$197$	8.78109	0.625627	0.312814	+	0.949815i	$0.398729\pi$
$198$	0	0
$199$	12.4370	0.881634	0.440817	−	0.897597i	$-0.354689\pi$
$199$	12.4370	0.881634	0.440817	+	0.897597i	$0.354689\pi$
$200$	0	0
$201$	0.578661	0.0408156
$202$	0	0
$203$	−10.2691	−0.720746
$204$	0	0
$205$	22.5872	1.57756
$206$	0	0
$207$	7.89522	0.548756
$208$	0	0
$209$	14.0390	0.971098
$210$	0	0
$211$	10.9290	0.752381	0.376191	−	0.926542i	$-0.377234\pi$
$211$	10.9290	0.752381	0.376191	+	0.926542i	$0.377234\pi$
$212$	0	0
$213$	−4.23245	−0.290003
$214$	0	0
$215$	40.0220	2.72948
$216$	0	0
$217$	−17.6792	−1.20014
$218$	0	0
$219$	0.525990	0.0355431
$220$	0	0
$221$	14.7007	0.988879
$222$	0	0
$223$	−8.79372	−0.588871	−0.294436	−	0.955671i	$-0.595132\pi$
$223$	−8.79372	−0.588871	−0.294436	+	0.955671i	$0.595132\pi$
$224$	0	0
$225$	−34.9672	−2.33115
$226$	0	0
$227$	−6.74871	−0.447928	−0.223964	−	0.974597i	$-0.571900\pi$
$227$	−6.74871	−0.447928	−0.223964	+	0.974597i	$0.571900\pi$
$228$	0	0
$229$	−23.2606	−1.53711	−0.768553	−	0.639787i	$-0.779023\pi$
$229$	−23.2606	−1.53711	−0.768553	+	0.639787i	$0.779023\pi$
$230$	0	0
$231$	−3.05028	−0.200694
$232$	0	0
$233$	11.0721	0.725357	0.362679	−	0.931914i	$-0.381862\pi$
$233$	11.0721	0.725357	0.362679	+	0.931914i	$0.381862\pi$
$234$	0	0
$235$	29.3959	1.91758
$236$	0	0
$237$	5.74775	0.373356
$238$	0	0
$239$	3.99072	0.258138	0.129069	−	0.991636i	$-0.458801\pi$
$239$	3.99072	0.258138	0.129069	+	0.991636i	$0.458801\pi$
$240$	0	0
$241$	−16.5408	−1.06549	−0.532745	−	0.846276i	$-0.678839\pi$
$241$	−16.5408	−1.06549	−0.532745	+	0.846276i	$0.678839\pi$
$242$	0	0
$243$	9.69309	0.621812
$244$	0	0
$245$	69.7773	4.45791
$246$	0	0
$247$	−18.6729	−1.18813
$248$	0	0
$249$	3.10238	0.196605
$250$	0	0
$251$	−24.0533	−1.51823	−0.759117	−	0.650955i	$-0.774369\pi$
$251$	−24.0533	−1.51823	−0.759117	+	0.650955i	$0.774369\pi$
$252$	0	0
$253$	−4.50877	−0.283464
$254$	0	0
$255$	−10.8179	−0.677443
$256$	0	0
$257$	−13.6822	−0.853470	−0.426735	−	0.904377i	$-0.640336\pi$
$257$	−13.6822	−0.853470	−0.426735	+	0.904377i	$0.640336\pi$
$258$	0	0
$259$	−5.84213	−0.363012
$260$	0	0
$261$	−6.00517	−0.371711
$262$	0	0
$263$	−11.3942	−0.702596	−0.351298	−	0.936264i	$-0.614260\pi$
$263$	−11.3942	−0.702596	−0.351298	+	0.936264i	$0.614260\pi$
$264$	0	0
$265$	−12.6243	−0.775505
$266$	0	0
$267$	1.27161	0.0778214
$268$	0	0
$269$	−1.00000	−0.0609711
$269$	−1.00000	−0.0609711
$270$	0	0
$271$	24.6167	1.49536	0.747679	−	0.664061i	$-0.231168\pi$
$271$	24.6167	1.49536	0.747679	+	0.664061i	$0.231168\pi$
$272$	0	0
$273$	4.05710	0.245547
$274$	0	0
$275$	19.9690	1.20417
$276$	0	0
$277$	4.70762	0.282853	0.141427	−	0.989949i	$-0.454831\pi$
$277$	4.70762	0.282853	0.141427	+	0.989949i	$0.454831\pi$
$278$	0	0
$279$	−10.3385	−0.618949
$280$	0	0
$281$	6.51540	0.388676	0.194338	−	0.980935i	$-0.437744\pi$
$281$	6.51540	0.388676	0.194338	+	0.980935i	$0.437744\pi$
$282$	0	0
$283$	8.67417	0.515626	0.257813	−	0.966195i	$-0.416998\pi$
$283$	8.67417	0.515626	0.257813	+	0.966195i	$0.416998\pi$
$284$	0	0
$285$	13.7409	0.813941
$286$	0	0
$287$	−26.5234	−1.56563
$288$	0	0
$289$	29.0291	1.70760
$290$	0	0
$291$	6.46647	0.379071
$292$	0	0
$293$	20.3798	1.19060	0.595300	−	0.803504i	$-0.297033\pi$
$293$	20.3798	1.19060	0.595300	+	0.803504i	$0.297033\pi$
$294$	0	0
$295$	−17.5237	−1.02027
$296$	0	0
$297$	−3.65962	−0.212353
$298$	0	0
$299$	5.99701	0.346816
$300$	0	0
$301$	−46.9965	−2.70883
$302$	0	0
$303$	1.12474	0.0646147
$304$	0	0
$305$	36.3389	2.08076
$306$	0	0
$307$	17.5028	0.998939	0.499470	−	0.866331i	$-0.333528\pi$
$307$	17.5028	0.998939	0.499470	+	0.866331i	$0.333528\pi$
$308$	0	0
$309$	−0.482928	−0.0274728
$310$	0	0
$311$	−22.4715	−1.27424	−0.637120	−	0.770764i	$-0.719875\pi$
$311$	−22.4715	−1.27424	−0.637120	+	0.770764i	$0.719875\pi$
$312$	0	0
$313$	−32.6321	−1.84447	−0.922236	−	0.386626i	$-0.873640\pi$
$313$	−32.6321	−1.84447	−0.922236	+	0.386626i	$0.873640\pi$
$314$	0	0
$315$	57.8099	3.25722
$316$	0	0
$317$	−19.5532	−1.09822	−0.549110	−	0.835750i	$-0.685033\pi$
$317$	−19.5532	−1.09822	−0.549110	+	0.835750i	$0.685033\pi$
$318$	0	0
$319$	3.42941	0.192010
$320$	0	0
$321$	−3.59241	−0.200509
$322$	0	0
$323$	−58.4663	−3.25315
$324$	0	0
$325$	−26.5602	−1.47329
$326$	0	0
$327$	4.38300	0.242381
$328$	0	0
$329$	−34.5186	−1.90307
$330$	0	0
$331$	20.1680	1.10853	0.554266	−	0.832340i	$-0.312999\pi$
$331$	20.1680	1.10853	0.554266	+	0.832340i	$0.312999\pi$
$332$	0	0
$333$	−3.41638	−0.187216
$334$	0	0
$335$	6.26298	0.342183
$336$	0	0
$337$	29.1409	1.58741	0.793704	−	0.608305i	$-0.208150\pi$
$337$	29.1409	1.58741	0.793704	+	0.608305i	$0.208150\pi$
$338$	0	0
$339$	0.121603	0.00660454
$340$	0	0
$341$	5.90407	0.319723
$342$	0	0
$343$	−47.7899	−2.58041
$344$	0	0
$345$	−4.41304	−0.237590
$346$	0	0
$347$	−12.2141	−0.655686	−0.327843	−	0.944732i	$-0.606322\pi$
$347$	−12.2141	−0.655686	−0.327843	+	0.944732i	$0.606322\pi$
$348$	0	0
$349$	15.3346	0.820843	0.410421	−	0.911896i	$-0.365382\pi$
$349$	15.3346	0.820843	0.410421	+	0.911896i	$0.365382\pi$
$350$	0	0
$351$	4.86757	0.259812
$352$	0	0
$353$	30.6859	1.63324	0.816622	−	0.577173i	$-0.195844\pi$
$353$	30.6859	1.63324	0.816622	+	0.577173i	$0.195844\pi$
$354$	0	0
$355$	−45.8087	−2.43127
$356$	0	0
$357$	12.7031	0.672319
$358$	0	0
$359$	−0.344824	−0.0181991	−0.00909956	−	0.999959i	$-0.502897\pi$
$359$	−0.344824	−0.0181991	−0.00909956	+	0.999959i	$0.502897\pi$
$360$	0	0
$361$	55.2640	2.90863
$362$	0	0
$363$	−3.20343	−0.168137
$364$	0	0
$365$	5.69291	0.297980
$366$	0	0
$367$	−22.3072	−1.16443	−0.582214	−	0.813035i	$-0.697814\pi$
$367$	−22.3072	−1.16443	−0.582214	+	0.813035i	$0.697814\pi$
$368$	0	0
$369$	−15.5104	−0.807440
$370$	0	0
$371$	14.8243	0.769640
$372$	0	0
$373$	−5.92843	−0.306962	−0.153481	−	0.988152i	$-0.549048\pi$
$373$	−5.92843	−0.306962	−0.153481	+	0.988152i	$0.549048\pi$
$374$	0	0
$375$	11.5724	0.597597
$376$	0	0
$377$	−4.56137	−0.234922
$378$	0	0
$379$	2.01119	0.103308	0.0516540	−	0.998665i	$-0.483551\pi$
$379$	2.01119	0.103308	0.0516540	+	0.998665i	$0.483551\pi$
$380$	0	0
$381$	0.801322	0.0410530
$382$	0	0
$383$	−4.21696	−0.215477	−0.107738	−	0.994179i	$-0.534361\pi$
$383$	−4.21696	−0.215477	−0.107738	+	0.994179i	$0.534361\pi$
$384$	0	0
$385$	−33.0138	−1.68254
$386$	0	0
$387$	−27.4827	−1.39703
$388$	0	0
$389$	4.54939	0.230663	0.115332	−	0.993327i	$-0.463207\pi$
$389$	4.54939	0.230663	0.115332	+	0.993327i	$0.463207\pi$
$390$	0	0
$391$	18.7771	0.949598
$392$	0	0
$393$	1.98591	0.100176
$394$	0	0
$395$	62.2091	3.13008
$396$	0	0
$397$	−15.2162	−0.763677	−0.381839	−	0.924229i	$-0.624709\pi$
$397$	−15.2162	−0.763677	−0.381839	+	0.924229i	$0.624709\pi$
$398$	0	0
$399$	−16.1355	−0.807785
$400$	0	0
$401$	18.4280	0.920250	0.460125	−	0.887854i	$-0.347805\pi$
$401$	18.4280	0.920250	0.460125	+	0.887854i	$0.347805\pi$
$402$	0	0
$403$	−7.85285	−0.391178
$404$	0	0
$405$	31.9702	1.58861
$406$	0	0
$407$	1.95101	0.0967081
$408$	0	0
$409$	36.3293	1.79637	0.898184	−	0.439619i	$-0.144887\pi$
$409$	36.3293	1.79637	0.898184	+	0.439619i	$0.144887\pi$
$410$	0	0
$411$	2.44094	0.120403
$412$	0	0
$413$	20.5775	1.01255
$414$	0	0
$415$	33.5777	1.64826
$416$	0	0
$417$	4.09872	0.200715
$418$	0	0
$419$	6.63242	0.324015	0.162007	−	0.986790i	$-0.448203\pi$
$419$	6.63242	0.324015	0.162007	+	0.986790i	$0.448203\pi$
$420$	0	0
$421$	−33.1090	−1.61363	−0.806817	−	0.590801i	$-0.798812\pi$
$421$	−33.1090	−1.61363	−0.806817	+	0.590801i	$0.798812\pi$
$422$	0	0
$423$	−20.1859	−0.981472
$424$	0	0
$425$	−83.1620	−4.03395
$426$	0	0
$427$	−42.6716	−2.06502
$428$	0	0
$429$	−1.35489	−0.0654148
$430$	0	0
$431$	16.6498	0.801995	0.400997	−	0.916079i	$-0.368664\pi$
$431$	16.6498	0.801995	0.400997	+	0.916079i	$0.368664\pi$
$432$	0	0
$433$	5.36131	0.257648	0.128824	−	0.991667i	$-0.458880\pi$
$433$	5.36131	0.257648	0.128824	+	0.991667i	$0.458880\pi$
$434$	0	0
$435$	3.35659	0.160936
$436$	0	0
$437$	−23.8507	−1.14093
$438$	0	0
$439$	5.42961	0.259141	0.129571	−	0.991570i	$-0.458640\pi$
$439$	5.42961	0.259141	0.129571	+	0.991570i	$0.458640\pi$
$440$	0	0
$441$	−47.9155	−2.28169
$442$	0	0
$443$	−3.66946	−0.174341	−0.0871706	−	0.996193i	$-0.527783\pi$
$443$	−3.66946	−0.174341	−0.0871706	+	0.996193i	$0.527783\pi$
$444$	0	0
$445$	13.7629	0.652426
$446$	0	0
$447$	−1.33351	−0.0630729
$448$	0	0
$449$	6.43513	0.303693	0.151846	−	0.988404i	$-0.451478\pi$
$449$	6.43513	0.303693	0.151846	+	0.988404i	$0.451478\pi$
$450$	0	0
$451$	8.85764	0.417090
$452$	0	0
$453$	−4.31446	−0.202711
$454$	0	0
$455$	43.9109	2.05857
$456$	0	0
$457$	2.97606	0.139214	0.0696071	−	0.997574i	$-0.477825\pi$
$457$	2.97606	0.139214	0.0696071	+	0.997574i	$0.477825\pi$
$458$	0	0
$459$	15.2407	0.711377
$460$	0	0
$461$	9.73958	0.453617	0.226809	−	0.973939i	$-0.427171\pi$
$461$	9.73958	0.453617	0.226809	+	0.973939i	$0.427171\pi$
$462$	0	0
$463$	19.9810	0.928598	0.464299	−	0.885679i	$-0.346306\pi$
$463$	19.9810	0.928598	0.464299	+	0.885679i	$0.346306\pi$
$464$	0	0
$465$	5.77871	0.267981
$466$	0	0
$467$	14.3949	0.666115	0.333057	−	0.942907i	$-0.391920\pi$
$467$	14.3949	0.666115	0.333057	+	0.942907i	$0.391920\pi$
$468$	0	0
$469$	−7.35441	−0.339595
$470$	0	0
$471$	3.29912	0.152015
$472$	0	0
$473$	15.6947	0.721645
$474$	0	0
$475$	105.633	4.84675
$476$	0	0
$477$	8.66900	0.396926
$478$	0	0
$479$	22.4679	1.02659	0.513293	−	0.858213i	$-0.328425\pi$
$479$	22.4679	1.02659	0.513293	+	0.858213i	$0.328425\pi$
$480$	0	0
$481$	−2.59499	−0.118321
$482$	0	0
$483$	5.18209	0.235793
$484$	0	0
$485$	69.9880	3.17799
$486$	0	0
$487$	15.3289	0.694618	0.347309	−	0.937751i	$-0.387096\pi$
$487$	15.3289	0.694618	0.347309	+	0.937751i	$0.387096\pi$
$488$	0	0
$489$	−3.89698	−0.176227
$490$	0	0
$491$	23.8945	1.07834	0.539172	−	0.842196i	$-0.318737\pi$
$491$	23.8945	1.07834	0.539172	+	0.842196i	$0.318737\pi$
$492$	0	0
$493$	−14.2820	−0.643229
$494$	0	0
$495$	−19.3059	−0.867737
$496$	0	0
$497$	53.7917	2.41289
$498$	0	0
$499$	15.7627	0.705635	0.352818	−	0.935692i	$-0.385224\pi$
$499$	15.7627	0.705635	0.352818	+	0.935692i	$0.385224\pi$
$500$	0	0
$501$	−1.67952	−0.0750353
$502$	0	0
$503$	−0.635388	−0.0283305	−0.0141653	−	0.999900i	$-0.504509\pi$
$503$	−0.635388	−0.0283305	−0.0141653	+	0.999900i	$0.504509\pi$
$504$	0	0
$505$	12.1733	0.541706
$506$	0	0
$507$	−3.18764	−0.141568
$508$	0	0
$509$	1.74035	0.0771399	0.0385699	−	0.999256i	$-0.487720\pi$
$509$	1.74035	0.0771399	0.0385699	+	0.999256i	$0.487720\pi$
$510$	0	0
$511$	−6.68499	−0.295727
$512$	0	0
$513$	−19.3588	−0.854712
$514$	0	0
$515$	−5.22683	−0.230322
$516$	0	0
$517$	11.5277	0.506987
$518$	0	0
$519$	−8.73784	−0.383548
$520$	0	0
$521$	10.5683	0.463004	0.231502	−	0.972834i	$-0.425636\pi$
$521$	10.5683	0.463004	0.231502	+	0.972834i	$0.425636\pi$
$522$	0	0
$523$	−10.5329	−0.460572	−0.230286	−	0.973123i	$-0.573966\pi$
$523$	−10.5329	−0.460572	−0.230286	+	0.973123i	$0.573966\pi$
$524$	0	0
$525$	−22.9510	−1.00166
$526$	0	0
$527$	−24.5879	−1.07106
$528$	0	0
$529$	−15.3401	−0.666961
$530$	0	0
$531$	12.0334	0.522203
$532$	0	0
$533$	−11.7813	−0.510305
$534$	0	0
$535$	−38.8814	−1.68099
$536$	0	0
$537$	−4.66865	−0.201467
$538$	0	0
$539$	27.3634	1.17862
$540$	0	0
$541$	−0.0839881	−0.00361093	−0.00180546	−	0.999998i	$-0.500575\pi$
$541$	−0.0839881	−0.00361093	−0.00180546	+	0.999998i	$0.500575\pi$
$542$	0	0
$543$	−1.76963	−0.0759419
$544$	0	0
$545$	47.4382	2.03203
$546$	0	0
$547$	−1.38614	−0.0592669	−0.0296334	−	0.999561i	$-0.509434\pi$
$547$	−1.38614	−0.0592669	−0.0296334	+	0.999561i	$0.509434\pi$
$548$	0	0
$549$	−24.9536	−1.06499
$550$	0	0
$551$	18.1410	0.772833
$552$	0	0
$553$	−73.0501	−3.10641
$554$	0	0
$555$	1.90959	0.0810575
$556$	0	0
$557$	17.1840	0.728111	0.364056	−	0.931377i	$-0.381392\pi$
$557$	17.1840	0.728111	0.364056	+	0.931377i	$0.381392\pi$
$558$	0	0
$559$	−20.8752	−0.882926
$560$	0	0
$561$	−4.24227	−0.179109
$562$	0	0
$563$	12.1968	0.514033	0.257017	−	0.966407i	$-0.417260\pi$
$563$	12.1968	0.514033	0.257017	+	0.966407i	$0.417260\pi$
$564$	0	0
$565$	1.31613	0.0553700
$566$	0	0
$567$	−37.5415	−1.57660
$568$	0	0
$569$	23.4816	0.984401	0.492201	−	0.870482i	$-0.336193\pi$
$569$	23.4816	0.984401	0.492201	+	0.870482i	$0.336193\pi$
$570$	0	0
$571$	2.61940	0.109619	0.0548093	−	0.998497i	$-0.482545\pi$
$571$	2.61940	0.109619	0.0548093	+	0.998497i	$0.482545\pi$
$572$	0	0
$573$	−1.26875	−0.0530028
$574$	0	0
$575$	−33.9250	−1.41477
$576$	0	0
$577$	12.1035	0.503874	0.251937	−	0.967744i	$-0.418932\pi$
$577$	12.1035	0.503874	0.251937	+	0.967744i	$0.418932\pi$
$578$	0	0
$579$	−7.10681	−0.295349
$580$	0	0
$581$	−39.4292	−1.63580
$582$	0	0
$583$	−4.95066	−0.205035
$584$	0	0
$585$	25.6783	1.06167
$586$	0	0
$587$	3.09578	0.127776	0.0638882	−	0.997957i	$-0.479650\pi$
$587$	3.09578	0.127776	0.0638882	+	0.997957i	$0.479650\pi$
$588$	0	0
$589$	31.2316	1.28687
$590$	0	0
$591$	3.37042	0.138640
$592$	0	0
$593$	−24.1730	−0.992666	−0.496333	−	0.868132i	$-0.665321\pi$
$593$	−24.1730	−0.992666	−0.496333	+	0.868132i	$0.665321\pi$
$594$	0	0
$595$	137.488	5.63647
$596$	0	0
$597$	4.77364	0.195372
$598$	0	0
$599$	−39.0984	−1.59752	−0.798759	−	0.601651i	$-0.794510\pi$
$599$	−39.0984	−1.59752	−0.798759	+	0.601651i	$0.794510\pi$
$600$	0	0
$601$	−29.5069	−1.20361	−0.601807	−	0.798642i	$-0.705552\pi$
$601$	−29.5069	−1.20361	−0.601807	+	0.798642i	$0.705552\pi$
$602$	0	0
$603$	−4.30073	−0.175139
$604$	0	0
$605$	−34.6714	−1.40959
$606$	0	0
$607$	−17.2674	−0.700861	−0.350431	−	0.936589i	$-0.613965\pi$
$607$	−17.2674	−0.700861	−0.350431	+	0.936589i	$0.613965\pi$
$608$	0	0
$609$	−3.94154	−0.159719
$610$	0	0
$611$	−15.3327	−0.620294
$612$	0	0
$613$	−7.94254	−0.320796	−0.160398	−	0.987052i	$-0.551278\pi$
$613$	−7.94254	−0.320796	−0.160398	+	0.987052i	$0.551278\pi$
$614$	0	0
$615$	8.66956	0.349591
$616$	0	0
$617$	18.0886	0.728218	0.364109	−	0.931356i	$-0.381374\pi$
$617$	18.0886	0.728218	0.364109	+	0.931356i	$0.381374\pi$
$618$	0	0
$619$	33.4949	1.34627	0.673136	−	0.739518i	$-0.264947\pi$
$619$	33.4949	1.34627	0.673136	+	0.739518i	$0.264947\pi$
$620$	0	0
$621$	6.21729	0.249491
$622$	0	0
$623$	−16.1614	−0.647491
$624$	0	0
$625$	63.9624	2.55850
$626$	0	0
$627$	5.38854	0.215197
$628$	0	0
$629$	−8.12512	−0.323970
$630$	0	0
$631$	−37.3162	−1.48554	−0.742768	−	0.669549i	$-0.766488\pi$
$631$	−37.3162	−1.48554	−0.742768	+	0.669549i	$0.766488\pi$
$632$	0	0
$633$	4.19483	0.166729
$634$	0	0
$635$	8.67288	0.344173
$636$	0	0
$637$	−36.3954	−1.44204
$638$	0	0
$639$	31.4564	1.24440
$640$	0	0
$641$	−18.0393	−0.712509	−0.356254	−	0.934389i	$-0.615946\pi$
$641$	−18.0393	−0.712509	−0.356254	+	0.934389i	$0.615946\pi$
$642$	0	0
$643$	33.0496	1.30335	0.651675	−	0.758498i	$-0.274067\pi$
$643$	33.0496	1.30335	0.651675	+	0.758498i	$0.274067\pi$
$644$	0	0
$645$	15.3615	0.604859
$646$	0	0
$647$	40.3067	1.58462	0.792310	−	0.610118i	$-0.208878\pi$
$647$	40.3067	1.58462	0.792310	+	0.610118i	$0.208878\pi$
$648$	0	0
$649$	−6.87196	−0.269748
$650$	0	0
$651$	−6.78575	−0.265954
$652$	0	0
$653$	−21.3784	−0.836601	−0.418300	−	0.908309i	$-0.637374\pi$
$653$	−21.3784	−0.836601	−0.418300	+	0.908309i	$0.637374\pi$
$654$	0	0
$655$	21.4939	0.839836
$656$	0	0
$657$	−3.90927	−0.152515
$658$	0	0
$659$	−23.1777	−0.902875	−0.451438	−	0.892303i	$-0.649089\pi$
$659$	−23.1777	−0.902875	−0.451438	+	0.892303i	$0.649089\pi$
$660$	0	0
$661$	−3.96791	−0.154334	−0.0771669	−	0.997018i	$-0.524587\pi$
$661$	−3.96791	−0.154334	−0.0771669	+	0.997018i	$0.524587\pi$
$662$	0	0
$663$	5.64253	0.219138
$664$	0	0
$665$	−174.638	−6.77217
$666$	0	0
$667$	−5.82618	−0.225591
$668$	0	0
$669$	−3.37526	−0.130495
$670$	0	0
$671$	14.2504	0.550131
$672$	0	0
$673$	32.9589	1.27047	0.635235	−	0.772319i	$-0.280903\pi$
$673$	32.9589	1.27047	0.635235	+	0.772319i	$0.280903\pi$
$674$	0	0
$675$	−27.5358	−1.05985
$676$	0	0
$677$	17.1159	0.657816	0.328908	−	0.944362i	$-0.393319\pi$
$677$	17.1159	0.657816	0.328908	+	0.944362i	$0.393319\pi$
$678$	0	0
$679$	−82.1846	−3.15396
$680$	0	0
$681$	−2.59033	−0.0992618
$682$	0	0
$683$	−33.2813	−1.27347	−0.636736	−	0.771082i	$-0.719716\pi$
$683$	−33.2813	−1.27347	−0.636736	+	0.771082i	$0.719716\pi$
$684$	0	0
$685$	26.4188	1.00941
$686$	0	0
$687$	−8.92804	−0.340626
$688$	0	0
$689$	6.58475	0.250859
$690$	0	0
$691$	21.3236	0.811189	0.405594	−	0.914053i	$-0.367065\pi$
$691$	21.3236	0.811189	0.405594	+	0.914053i	$0.367065\pi$
$692$	0	0
$693$	22.6703	0.861174
$694$	0	0
$695$	44.3614	1.68272
$696$	0	0
$697$	−36.8882	−1.39724
$698$	0	0
$699$	4.24976	0.160741
$700$	0	0
$701$	−28.5333	−1.07769	−0.538844	−	0.842406i	$-0.681139\pi$
$701$	−28.5333	−1.07769	−0.538844	+	0.842406i	$0.681139\pi$
$702$	0	0
$703$	10.3205	0.389246
$704$	0	0
$705$	11.2829	0.424940
$706$	0	0
$707$	−14.2947	−0.537609
$708$	0	0
$709$	4.38294	0.164605	0.0823024	−	0.996607i	$-0.473773\pi$
$709$	4.38294	0.164605	0.0823024	+	0.996607i	$0.473773\pi$
$710$	0	0
$711$	−42.7184	−1.60207
$712$	0	0
$713$	−10.0304	−0.375640
$714$	0	0
$715$	−14.6643	−0.548413
$716$	0	0
$717$	1.53174	0.0572040
$718$	0	0
$719$	36.1805	1.34930	0.674652	−	0.738136i	$-0.264294\pi$
$719$	36.1805	1.34930	0.674652	+	0.738136i	$0.264294\pi$
$720$	0	0
$721$	6.13770	0.228580
$722$	0	0
$723$	−6.34881	−0.236115
$724$	0	0
$725$	25.8036	0.958323
$726$	0	0
$727$	23.0348	0.854312	0.427156	−	0.904178i	$-0.359515\pi$
$727$	23.0348	0.854312	0.427156	+	0.904178i	$0.359515\pi$
$728$	0	0
$729$	−19.3669	−0.717294
$730$	0	0
$731$	−65.3618	−2.41749
$732$	0	0
$733$	−41.3032	−1.52557	−0.762785	−	0.646652i	$-0.776169\pi$
$733$	−41.3032	−1.52557	−0.762785	+	0.646652i	$0.776169\pi$
$734$	0	0
$735$	26.7824	0.987884
$736$	0	0
$737$	2.45605	0.0904696
$738$	0	0
$739$	15.2014	0.559191	0.279596	−	0.960118i	$-0.409800\pi$
$739$	15.2014	0.559191	0.279596	+	0.960118i	$0.409800\pi$
$740$	0	0
$741$	−7.16716	−0.263292
$742$	0	0
$743$	−51.4130	−1.88616	−0.943079	−	0.332568i	$-0.892085\pi$
$743$	−51.4130	−1.88616	−0.943079	+	0.332568i	$0.892085\pi$
$744$	0	0
$745$	−14.4329	−0.528780
$746$	0	0
$747$	−23.0575	−0.843630
$748$	0	0
$749$	45.6572	1.66828
$750$	0	0
$751$	−37.2197	−1.35816	−0.679082	−	0.734062i	$-0.737622\pi$
$751$	−37.2197	−1.35816	−0.679082	+	0.734062i	$0.737622\pi$
$752$	0	0
$753$	−9.23231	−0.336444
$754$	0	0
$755$	−46.6964	−1.69945
$756$	0	0
$757$	51.7191	1.87976	0.939882	−	0.341499i	$-0.110935\pi$
$757$	51.7191	1.87976	0.939882	+	0.341499i	$0.110935\pi$
$758$	0	0
$759$	−1.73059	−0.0628163
$760$	0	0
$761$	16.8448	0.610623	0.305312	−	0.952253i	$-0.401239\pi$
$761$	16.8448	0.610623	0.305312	+	0.952253i	$0.401239\pi$
$762$	0	0
$763$	−55.7051	−2.01666
$764$	0	0
$765$	80.4008	2.90690
$766$	0	0
$767$	9.14022	0.330034
$768$	0	0
$769$	10.2220	0.368614	0.184307	−	0.982869i	$-0.440996\pi$
$769$	10.2220	0.368614	0.184307	+	0.982869i	$0.440996\pi$
$770$	0	0
$771$	−5.25158	−0.189131
$772$	0	0
$773$	−31.6879	−1.13973	−0.569867	−	0.821737i	$-0.693005\pi$
$773$	−31.6879	−1.13973	−0.569867	+	0.821737i	$0.693005\pi$
$774$	0	0
$775$	44.4235	1.59574
$776$	0	0
$777$	−2.24236	−0.0804444
$778$	0	0
$779$	46.8555	1.67877
$780$	0	0
$781$	−17.9640	−0.642803
$782$	0	0
$783$	−4.72892	−0.168998
$784$	0	0
$785$	35.7070	1.27444
$786$	0	0
$787$	−21.1252	−0.753033	−0.376516	−	0.926410i	$-0.622878\pi$
$787$	−21.1252	−0.753033	−0.376516	+	0.926410i	$0.622878\pi$
$788$	0	0
$789$	−4.37339	−0.155697
$790$	0	0
$791$	−1.54549	−0.0549512
$792$	0	0
$793$	−18.9541	−0.673080
$794$	0	0
$795$	−4.84554	−0.171854
$796$	0	0
$797$	−4.27630	−0.151474	−0.0757371	−	0.997128i	$-0.524131\pi$
$797$	−4.27630	−0.151474	−0.0757371	+	0.997128i	$0.524131\pi$
$798$	0	0
$799$	−48.0078	−1.69839
$800$	0	0
$801$	−9.45089	−0.333931
$802$	0	0
$803$	2.23249	0.0787829
$804$	0	0
$805$	56.0868	1.97680
$806$	0	0
$807$	−0.383827	−0.0135113
$808$	0	0
$809$	4.78386	0.168192	0.0840958	−	0.996458i	$-0.473200\pi$
$809$	4.78386	0.168192	0.0840958	+	0.996458i	$0.473200\pi$
$810$	0	0
$811$	34.4127	1.20839	0.604197	−	0.796835i	$-0.293494\pi$
$811$	34.4127	1.20839	0.604197	+	0.796835i	$0.293494\pi$
$812$	0	0
$813$	9.44854	0.331375
$814$	0	0
$815$	−42.1778	−1.47742
$816$	0	0
$817$	83.0227	2.90460
$818$	0	0
$819$	−30.1532	−1.05364
$820$	0	0
$821$	−38.7973	−1.35404	−0.677018	−	0.735967i	$-0.736728\pi$
$821$	−38.7973	−1.35404	−0.677018	+	0.735967i	$0.736728\pi$
$822$	0	0
$823$	8.52567	0.297186	0.148593	−	0.988898i	$-0.452526\pi$
$823$	8.52567	0.297186	0.148593	+	0.988898i	$0.452526\pi$
$824$	0	0
$825$	7.66461	0.266848
$826$	0	0
$827$	−3.84932	−0.133854	−0.0669271	−	0.997758i	$-0.521319\pi$
$827$	−3.84932	−0.133854	−0.0669271	+	0.997758i	$0.521319\pi$
$828$	0	0
$829$	1.76719	0.0613771	0.0306886	−	0.999529i	$-0.490230\pi$
$829$	1.76719	0.0613771	0.0306886	+	0.999529i	$0.490230\pi$
$830$	0	0
$831$	1.80691	0.0626810
$832$	0	0
$833$	−113.957	−3.94836
$834$	0	0
$835$	−18.1778	−0.629068
$836$	0	0
$837$	−8.14131	−0.281405
$838$	0	0
$839$	−48.4527	−1.67277	−0.836386	−	0.548141i	$-0.815336\pi$
$839$	−48.4527	−1.67277	−0.836386	+	0.548141i	$0.815336\pi$
$840$	0	0
$841$	−24.5686	−0.847192
$842$	0	0
$843$	2.50078	0.0861315
$844$	0	0
$845$	−34.5005	−1.18685
$846$	0	0
$847$	40.7135	1.39893
$848$	0	0
$849$	3.32938	0.114264
$850$	0	0
$851$	−3.31455	−0.113621
$852$	0	0
$853$	35.2024	1.20531	0.602654	−	0.798002i	$-0.294110\pi$
$853$	35.2024	1.20531	0.602654	+	0.798002i	$0.294110\pi$
$854$	0	0
$855$	−102.125	−3.49261
$856$	0	0
$857$	8.30685	0.283757	0.141878	−	0.989884i	$-0.454686\pi$
$857$	8.30685	0.283757	0.141878	+	0.989884i	$0.454686\pi$
$858$	0	0
$859$	−48.5886	−1.65782	−0.828911	−	0.559381i	$-0.811039\pi$
$859$	−48.5886	−1.65782	−0.828911	+	0.559381i	$0.811039\pi$
$860$	0	0
$861$	−10.1804	−0.346947
$862$	0	0
$863$	−50.0516	−1.70378	−0.851888	−	0.523723i	$-0.824543\pi$
$863$	−50.0516	−1.70378	−0.851888	+	0.523723i	$0.824543\pi$
$864$	0	0
$865$	−94.5715	−3.21553
$866$	0	0
$867$	11.1421	0.378407
$868$	0	0
$869$	24.3955	0.827560
$870$	0	0
$871$	−3.26672	−0.110689
$872$	0	0
$873$	−48.0601	−1.62659
$874$	0	0
$875$	−147.078	−4.97214
$876$	0	0
$877$	−35.1921	−1.18835	−0.594177	−	0.804334i	$-0.702522\pi$
$877$	−35.1921	−1.18835	−0.594177	+	0.804334i	$0.702522\pi$
$878$	0	0
$879$	7.82230	0.263840
$880$	0	0
$881$	−29.4895	−0.993527	−0.496764	−	0.867886i	$-0.665478\pi$
$881$	−29.4895	−0.993527	−0.496764	+	0.867886i	$0.665478\pi$
$882$	0	0
$883$	28.1353	0.946829	0.473415	−	0.880840i	$-0.343021\pi$
$883$	28.1353	0.946829	0.473415	+	0.880840i	$0.343021\pi$
$884$	0	0
$885$	−6.72605	−0.226094
$886$	0	0
$887$	12.3917	0.416072	0.208036	−	0.978121i	$-0.433293\pi$
$887$	12.3917	0.416072	0.208036	+	0.978121i	$0.433293\pi$
$888$	0	0
$889$	−10.1843	−0.341570
$890$	0	0
$891$	12.5372	0.420012
$892$	0	0
$893$	60.9796	2.04061
$894$	0	0
$895$	−50.5298	−1.68903
$896$	0	0
$897$	2.30181	0.0768552
$898$	0	0
$899$	7.62916	0.254447
$900$	0	0
$901$	20.6173	0.686863
$902$	0	0
$903$	−18.0385	−0.600284
$904$	0	0
$905$	−19.1530	−0.636669
$906$	0	0
$907$	14.9432	0.496181	0.248090	−	0.968737i	$-0.420197\pi$
$907$	14.9432	0.496181	0.248090	+	0.968737i	$0.420197\pi$
$908$	0	0
$909$	−8.35931	−0.277261
$910$	0	0
$911$	−29.3458	−0.972269	−0.486134	−	0.873884i	$-0.661593\pi$
$911$	−29.3458	−0.972269	−0.486134	+	0.873884i	$0.661593\pi$
$912$	0	0
$913$	13.1676	0.435784
$914$	0	0
$915$	13.9478	0.461101
$916$	0	0
$917$	−25.2396	−0.833485
$918$	0	0
$919$	−24.7551	−0.816595	−0.408297	−	0.912849i	$-0.633877\pi$
$919$	−24.7551	−0.816595	−0.408297	+	0.912849i	$0.633877\pi$
$920$	0	0
$921$	6.71805	0.221367
$922$	0	0
$923$	23.8935	0.786464
$924$	0	0
$925$	14.6799	0.482671
$926$	0	0
$927$	3.58922	0.117885
$928$	0	0
$929$	−27.1303	−0.890117	−0.445059	−	0.895501i	$-0.646817\pi$
$929$	−27.1303	−0.890117	−0.445059	+	0.895501i	$0.646817\pi$
$930$	0	0
$931$	144.748	4.74392
$932$	0	0
$933$	−8.62515	−0.282375
$934$	0	0
$935$	−45.9150	−1.50158
$936$	0	0
$937$	44.2931	1.44699	0.723496	−	0.690328i	$-0.242534\pi$
$937$	44.2931	1.44699	0.723496	+	0.690328i	$0.242534\pi$
$938$	0	0
$939$	−12.5250	−0.408740
$940$	0	0
$941$	−27.0894	−0.883090	−0.441545	−	0.897239i	$-0.645569\pi$
$941$	−27.0894	−0.883090	−0.441545	+	0.897239i	$0.645569\pi$
$942$	0	0
$943$	−15.0481	−0.490035
$944$	0	0
$945$	45.5238	1.48089
$946$	0	0
$947$	−44.9196	−1.45969	−0.729845	−	0.683613i	$-0.760408\pi$
$947$	−44.9196	−1.45969	−0.729845	+	0.683613i	$0.760408\pi$
$948$	0	0
$949$	−2.96938	−0.0963901
$950$	0	0
$951$	−7.50505	−0.243368
$952$	0	0
$953$	−21.0858	−0.683036	−0.341518	−	0.939875i	$-0.610941\pi$
$953$	−21.0858	−0.683036	−0.341518	+	0.939875i	$0.610941\pi$
$954$	0	0
$955$	−13.7320	−0.444356
$956$	0	0
$957$	1.31630	0.0425499
$958$	0	0
$959$	−31.0227	−1.00178
$960$	0	0
$961$	−17.8656	−0.576311
$962$	0	0
$963$	26.6995	0.860380
$964$	0	0
$965$	−76.9185	−2.47609
$966$	0	0
$967$	−25.2415	−0.811711	−0.405855	−	0.913937i	$-0.633026\pi$
$967$	−25.2415	−0.811711	−0.405855	+	0.913937i	$0.633026\pi$
$968$	0	0
$969$	−22.4409	−0.720906
$970$	0	0
$971$	34.5382	1.10838	0.554192	−	0.832389i	$-0.313027\pi$
$971$	34.5382	1.10838	0.554192	+	0.832389i	$0.313027\pi$
$972$	0	0
$973$	−52.0921	−1.67000
$974$	0	0
$975$	−10.1945	−0.326486
$976$	0	0
$977$	61.0213	1.95224	0.976122	−	0.217225i	$-0.0697004\pi$
$977$	61.0213	1.95224	0.976122	+	0.217225i	$0.0697004\pi$
$978$	0	0
$979$	5.39718	0.172495
$980$	0	0
$981$	−32.5754	−1.04005
$982$	0	0
$983$	47.5260	1.51585	0.757923	−	0.652345i	$-0.226214\pi$
$983$	47.5260	1.51585	0.757923	+	0.652345i	$0.226214\pi$
$984$	0	0
$985$	36.4787	1.16231
$986$	0	0
$987$	−13.2492	−0.421726
$988$	0	0
$989$	−26.6636	−0.847854
$990$	0	0
$991$	35.6855	1.13359	0.566794	−	0.823860i	$-0.308184\pi$
$991$	35.6855	1.13359	0.566794	+	0.823860i	$0.308184\pi$
$992$	0	0
$993$	7.74100	0.245653
$994$	0	0
$995$	51.6662	1.63793
$996$	0	0
$997$	12.2019	0.386438	0.193219	−	0.981156i	$-0.438107\pi$
$997$	12.2019	0.386438	0.193219	+	0.981156i	$0.438107\pi$
$998$	0	0
$999$	−2.69031	−0.0851177

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4304.2.a.h.1.5		7
4.3	odd	2		538.2.a.e.1.3	✓	7
12.11	even	2		4842.2.a.n.1.1		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.2.a.e.1.3	✓	7	4.3	odd	2
4304.2.a.h.1.5		7	1.1	even	1	trivial
4842.2.a.n.1.1		7	12.11	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:	\( N \)	\(=\)	\( 4304 = 2^{4} \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4304.a (trivial)

\(f(q)\)	\(=\)	\(q+0.383827 q^{3} +4.15424 q^{5} -4.87818 q^{7} -2.85268 q^{9} +1.62910 q^{11} -2.16682 q^{13} +1.59451 q^{15} -6.78448 q^{17} +8.61766 q^{19} -1.87238 q^{21} -2.76765 q^{23} +12.2577 q^{25} -2.24641 q^{27} +2.10510 q^{29} +3.62414 q^{31} +0.625290 q^{33} -20.2651 q^{35} +1.19760 q^{37} -0.831683 q^{39} +5.43715 q^{41} +9.63402 q^{43} -11.8507 q^{45} +7.07613 q^{47} +16.7967 q^{49} -2.60406 q^{51} -3.03890 q^{53} +6.76765 q^{55} +3.30769 q^{57} -4.21827 q^{59} +8.74744 q^{61} +13.9159 q^{63} -9.00148 q^{65} +1.50761 q^{67} -1.06230 q^{69} -11.0270 q^{71} +1.37039 q^{73} +4.70482 q^{75} -7.94703 q^{77} +14.9749 q^{79} +7.69580 q^{81} +8.08276 q^{83} -28.1843 q^{85} +0.807993 q^{87} +3.31299 q^{89} +10.5701 q^{91} +1.39104 q^{93} +35.7998 q^{95} +16.8474 q^{97} -4.64729 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{3} + 7 q^{5} - 6 q^{7} + 12 q^{9} + 3 q^{11} - 9 q^{13} - 8 q^{15} + 8 q^{17} + 11 q^{19} - 6 q^{21} - 12 q^{23} + 22 q^{25} + 14 q^{27} - 5 q^{29} - 14 q^{31} - 4 q^{33} + 4 q^{35} + 13 q^{37}+ \cdots + 37 q^{99}+O(q^{100}) \) 7 * q - q^3 + 7 * q^5 - 6 * q^7 + 12 * q^9 + 3 * q^11 - 9 * q^13 - 8 * q^15 + 8 * q^17 + 11 * q^19 - 6 * q^21 - 12 * q^23 + 22 * q^25 + 14 * q^27 - 5 * q^29 - 14 * q^31 - 4 * q^33 + 4 * q^35 + 13 * q^37 + 18 * q^39 + 12 * q^41 + 11 * q^43 + 3 * q^45 - 2 * q^47 + 15 * q^49 + 26 * q^51 + 19 * q^53 + 40 * q^55 - 12 * q^57 + 9 * q^59 - 3 * q^61 + 26 * q^63 - 10 * q^65 + 33 * q^67 - 64 * q^69 - 28 * q^71 - 14 * q^73 + 45 * q^75 + 10 * q^77 - 2 * q^79 + 15 * q^81 + 7 * q^83 - 16 * q^85 - 16 * q^87 + 18 * q^89 + 26 * q^91 + 6 * q^93 + 34 * q^95 - 4 * q^97 + 37 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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