LMFDB - Embedded newform 7248.2.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7248,2,Mod(1,7248)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7248, 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("7248.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7248 = 2^{4} \cdot 3 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7248.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:	$57.8755713850$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 453)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	7248.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.73205 q^{11} +3.46410 q^{13} +2.00000 q^{15} -1.00000 q^{21} -2.53590 q^{23} -1.00000 q^{25} +1.00000 q^{27} -3.46410 q^{29} -9.46410 q^{31} -3.73205 q^{33} -2.00000 q^{35} -5.92820 q^{37} +3.46410 q^{39} -11.1962 q^{41} +3.46410 q^{43} +2.00000 q^{45} -0.267949 q^{47} -6.00000 q^{49} +4.26795 q^{53} -7.46410 q^{55} +6.66025 q^{59} -7.46410 q^{61} -1.00000 q^{63} +6.92820 q^{65} -5.00000 q^{67} -2.53590 q^{69} -12.9282 q^{71} -14.9282 q^{73} -1.00000 q^{75} +3.73205 q^{77} +11.9282 q^{79} +1.00000 q^{81} +5.46410 q^{83} -3.46410 q^{87} +6.92820 q^{89} -3.46410 q^{91} -9.46410 q^{93} +5.92820 q^{97} -3.73205 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 4 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 4 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 4 q^{5} - 2 q^{7} + 2 q^{9} - 4 q^{11} + 4 q^{15} - 2 q^{21} - 12 q^{23} - 2 q^{25} + 2 q^{27} - 12 q^{31} - 4 q^{33} - 4 q^{35} + 2 q^{37} - 12 q^{41} + 4 q^{45} - 4 q^{47} - 12 q^{49} + 12 q^{53} - 8 q^{55} - 4 q^{59} - 8 q^{61} - 2 q^{63} - 10 q^{67} - 12 q^{69} - 12 q^{71} - 16 q^{73} - 2 q^{75} + 4 q^{77} + 10 q^{79} + 2 q^{81} + 4 q^{83} - 12 q^{93} - 2 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 4 * q^5 - 2 * q^7 + 2 * q^9 - 4 * q^11 + 4 * q^15 - 2 * q^21 - 12 * q^23 - 2 * q^25 + 2 * q^27 - 12 * q^31 - 4 * q^33 - 4 * q^35 + 2 * q^37 - 12 * q^41 + 4 * q^45 - 4 * q^47 - 12 * q^49 + 12 * q^53 - 8 * q^55 - 4 * q^59 - 8 * q^61 - 2 * q^63 - 10 * q^67 - 12 * q^69 - 12 * q^71 - 16 * q^73 - 2 * q^75 + 4 * q^77 + 10 * q^79 + 2 * q^81 + 4 * q^83 - 12 * q^93 - 2 * q^97 - 4 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.73205	−1.12526	−0.562628	−	0.826710i	$-0.690210\pi$
$11$	−3.73205	−1.12526	−0.562628	+	0.826710i	$0.690210\pi$
$12$	0	0
$13$	3.46410	0.960769	0.480384	−	0.877058i	$-0.340497\pi$
$13$	3.46410	0.960769	0.480384	+	0.877058i	$0.340497\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−2.53590	−0.528771	−0.264386	−	0.964417i	$-0.585169\pi$
$23$	−2.53590	−0.528771	−0.264386	+	0.964417i	$0.585169\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.46410	−0.643268	−0.321634	−	0.946864i	$-0.604232\pi$
$29$	−3.46410	−0.643268	−0.321634	+	0.946864i	$0.604232\pi$
$30$	0	0
$31$	−9.46410	−1.69980	−0.849901	−	0.526942i	$-0.823339\pi$
$31$	−9.46410	−1.69980	−0.849901	+	0.526942i	$0.823339\pi$
$32$	0	0
$33$	−3.73205	−0.649667
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	−5.92820	−0.974591	−0.487295	−	0.873237i	$-0.662016\pi$
$37$	−5.92820	−0.974591	−0.487295	+	0.873237i	$0.662016\pi$
$38$	0	0
$39$	3.46410	0.554700
$40$	0	0
$41$	−11.1962	−1.74855	−0.874273	−	0.485435i	$-0.838661\pi$
$41$	−11.1962	−1.74855	−0.874273	+	0.485435i	$0.838661\pi$
$42$	0	0
$43$	3.46410	0.528271	0.264135	−	0.964486i	$-0.414913\pi$
$43$	3.46410	0.528271	0.264135	+	0.964486i	$0.414913\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	0	0
$47$	−0.267949	−0.0390844	−0.0195422	−	0.999809i	$-0.506221\pi$
$47$	−0.267949	−0.0390844	−0.0195422	+	0.999809i	$0.506221\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.26795	0.586248	0.293124	−	0.956074i	$-0.405305\pi$
$53$	4.26795	0.586248	0.293124	+	0.956074i	$0.405305\pi$
$54$	0	0
$55$	−7.46410	−1.00646
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.66025	0.867091	0.433546	−	0.901132i	$-0.357262\pi$
$59$	6.66025	0.867091	0.433546	+	0.901132i	$0.357262\pi$
$60$	0	0
$61$	−7.46410	−0.955680	−0.477840	−	0.878447i	$-0.658580\pi$
$61$	−7.46410	−0.955680	−0.477840	+	0.878447i	$0.658580\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	6.92820	0.859338
$66$	0	0
$67$	−5.00000	−0.610847	−0.305424	−	0.952217i	$-0.598798\pi$
$67$	−5.00000	−0.610847	−0.305424	+	0.952217i	$0.598798\pi$
$68$	0	0
$69$	−2.53590	−0.305286
$70$	0	0
$71$	−12.9282	−1.53430	−0.767148	−	0.641470i	$-0.778325\pi$
$71$	−12.9282	−1.53430	−0.767148	+	0.641470i	$0.778325\pi$
$72$	0	0
$73$	−14.9282	−1.74721	−0.873607	−	0.486632i	$-0.838225\pi$
$73$	−14.9282	−1.74721	−0.873607	+	0.486632i	$0.838225\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	3.73205	0.425307
$78$	0	0
$79$	11.9282	1.34203	0.671014	−	0.741445i	$-0.265859\pi$
$79$	11.9282	1.34203	0.671014	+	0.741445i	$0.265859\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.46410	0.599763	0.299882	−	0.953976i	$-0.403053\pi$
$83$	5.46410	0.599763	0.299882	+	0.953976i	$0.403053\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−3.46410	−0.371391
$88$	0	0
$89$	6.92820	0.734388	0.367194	−	0.930144i	$-0.380318\pi$
$89$	6.92820	0.734388	0.367194	+	0.930144i	$0.380318\pi$
$90$	0	0
$91$	−3.46410	−0.363137
$92$	0	0
$93$	−9.46410	−0.981382
$94$	0	0
$95$	0	0
$96$	0	0
$97$	5.92820	0.601918	0.300959	−	0.953637i	$-0.402693\pi$
$97$	5.92820	0.601918	0.300959	+	0.953637i	$0.402693\pi$
$98$	0	0
$99$	−3.73205	−0.375085
$100$	0	0
$101$	2.66025	0.264705	0.132353	−	0.991203i	$-0.457747\pi$
$101$	2.66025	0.264705	0.132353	+	0.991203i	$0.457747\pi$
$102$	0	0
$103$	5.46410	0.538394	0.269197	−	0.963085i	$-0.413242\pi$
$103$	5.46410	0.538394	0.269197	+	0.963085i	$0.413242\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	0	0
$107$	−2.00000	−0.193347	−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000	−0.193347	−0.0966736	+	0.995316i	$0.530820\pi$
$108$	0	0
$109$	−8.39230	−0.803837	−0.401919	−	0.915675i	$-0.631656\pi$
$109$	−8.39230	−0.803837	−0.401919	+	0.915675i	$0.631656\pi$
$110$	0	0
$111$	−5.92820	−0.562680
$112$	0	0
$113$	12.0000	1.12887	0.564433	−	0.825479i	$-0.309095\pi$
$113$	12.0000	1.12887	0.564433	+	0.825479i	$0.309095\pi$
$114$	0	0
$115$	−5.07180	−0.472947
$116$	0	0
$117$	3.46410	0.320256
$118$	0	0
$119$	0	0
$120$	0	0
$121$	2.92820	0.266200
$122$	0	0
$123$	−11.1962	−1.00952
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	8.39230	0.744697	0.372348	−	0.928093i	$-0.378553\pi$
$127$	8.39230	0.744697	0.372348	+	0.928093i	$0.378553\pi$
$128$	0	0
$129$	3.46410	0.304997
$130$	0	0
$131$	−4.53590	−0.396303	−0.198152	−	0.980171i	$-0.563494\pi$
$131$	−4.53590	−0.396303	−0.198152	+	0.980171i	$0.563494\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	2.00000	0.172133
$136$	0	0
$137$	15.3205	1.30892	0.654460	−	0.756097i	$-0.272896\pi$
$137$	15.3205	1.30892	0.654460	+	0.756097i	$0.272896\pi$
$138$	0	0
$139$	10.5359	0.893643	0.446822	−	0.894623i	$-0.352556\pi$
$139$	10.5359	0.893643	0.446822	+	0.894623i	$0.352556\pi$
$140$	0	0
$141$	−0.267949	−0.0225654
$142$	0	0
$143$	−12.9282	−1.08111
$144$	0	0
$145$	−6.92820	−0.575356
$146$	0	0
$147$	−6.00000	−0.494872
$148$	0	0
$149$	4.00000	0.327693	0.163846	−	0.986486i	$-0.447610\pi$
$149$	4.00000	0.327693	0.163846	+	0.986486i	$0.447610\pi$
$150$	0	0
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−18.9282	−1.52035
$156$	0	0
$157$	16.5359	1.31971	0.659854	−	0.751394i	$-0.270618\pi$
$157$	16.5359	1.31971	0.659854	+	0.751394i	$0.270618\pi$
$158$	0	0
$159$	4.26795	0.338470
$160$	0	0
$161$	2.53590	0.199857
$162$	0	0
$163$	−14.9282	−1.16927	−0.584634	−	0.811297i	$-0.698762\pi$
$163$	−14.9282	−1.16927	−0.584634	+	0.811297i	$0.698762\pi$
$164$	0	0
$165$	−7.46410	−0.581080
$166$	0	0
$167$	−14.1244	−1.09298	−0.546488	−	0.837467i	$-0.684035\pi$
$167$	−14.1244	−1.09298	−0.546488	+	0.837467i	$0.684035\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	4.00000	0.304114	0.152057	−	0.988372i	$-0.451410\pi$
$173$	4.00000	0.304114	0.152057	+	0.988372i	$0.451410\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	6.66025	0.500615
$178$	0	0
$179$	−0.535898	−0.0400549	−0.0200275	−	0.999799i	$-0.506375\pi$
$179$	−0.535898	−0.0400549	−0.0200275	+	0.999799i	$0.506375\pi$
$180$	0	0
$181$	8.39230	0.623795	0.311898	−	0.950116i	$-0.399035\pi$
$181$	8.39230	0.623795	0.311898	+	0.950116i	$0.399035\pi$
$182$	0	0
$183$	−7.46410	−0.551762
$184$	0	0
$185$	−11.8564	−0.871700
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−17.0526	−1.23388	−0.616940	−	0.787010i	$-0.711628\pi$
$191$	−17.0526	−1.23388	−0.616940	+	0.787010i	$0.711628\pi$
$192$	0	0
$193$	−16.8564	−1.21335	−0.606675	−	0.794950i	$-0.707497\pi$
$193$	−16.8564	−1.21335	−0.606675	+	0.794950i	$0.707497\pi$
$194$	0	0
$195$	6.92820	0.496139
$196$	0	0
$197$	5.85641	0.417252	0.208626	−	0.977996i	$-0.433101\pi$
$197$	5.85641	0.417252	0.208626	+	0.977996i	$0.433101\pi$
$198$	0	0
$199$	10.9282	0.774680	0.387340	−	0.921937i	$-0.373394\pi$
$199$	10.9282	0.774680	0.387340	+	0.921937i	$0.373394\pi$
$200$	0	0
$201$	−5.00000	−0.352673
$202$	0	0
$203$	3.46410	0.243132
$204$	0	0
$205$	−22.3923	−1.56395
$206$	0	0
$207$	−2.53590	−0.176257
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−9.92820	−0.683486	−0.341743	−	0.939794i	$-0.611017\pi$
$211$	−9.92820	−0.683486	−0.341743	+	0.939794i	$0.611017\pi$
$212$	0	0
$213$	−12.9282	−0.885826
$214$	0	0
$215$	6.92820	0.472500
$216$	0	0
$217$	9.46410	0.642465
$218$	0	0
$219$	−14.9282	−1.00875
$220$	0	0
$221$	0	0
$222$	0	0
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	6.39230	0.424272	0.212136	−	0.977240i	$-0.431958\pi$
$227$	6.39230	0.424272	0.212136	+	0.977240i	$0.431958\pi$
$228$	0	0
$229$	−13.9282	−0.920402	−0.460201	−	0.887815i	$-0.652223\pi$
$229$	−13.9282	−0.920402	−0.460201	+	0.887815i	$0.652223\pi$
$230$	0	0
$231$	3.73205	0.245551
$232$	0	0
$233$	−10.6603	−0.698376	−0.349188	−	0.937053i	$-0.613543\pi$
$233$	−10.6603	−0.698376	−0.349188	+	0.937053i	$0.613543\pi$
$234$	0	0
$235$	−0.535898	−0.0349582
$236$	0	0
$237$	11.9282	0.774820
$238$	0	0
$239$	−27.4641	−1.77651	−0.888253	−	0.459355i	$-0.848080\pi$
$239$	−27.4641	−1.77651	−0.888253	+	0.459355i	$0.848080\pi$
$240$	0	0
$241$	−25.0000	−1.61039	−0.805196	−	0.593009i	$-0.797940\pi$
$241$	−25.0000	−1.61039	−0.805196	+	0.593009i	$0.797940\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−12.0000	−0.766652
$246$	0	0
$247$	0	0
$248$	0	0
$249$	5.46410	0.346273
$250$	0	0
$251$	12.5359	0.791259	0.395629	−	0.918410i	$-0.370526\pi$
$251$	12.5359	0.791259	0.395629	+	0.918410i	$0.370526\pi$
$252$	0	0
$253$	9.46410	0.595003
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−9.85641	−0.614826	−0.307413	−	0.951576i	$-0.599463\pi$
$257$	−9.85641	−0.614826	−0.307413	+	0.951576i	$0.599463\pi$
$258$	0	0
$259$	5.92820	0.368361
$260$	0	0
$261$	−3.46410	−0.214423
$262$	0	0
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	0	0
$265$	8.53590	0.524356
$266$	0	0
$267$	6.92820	0.423999
$268$	0	0
$269$	1.07180	0.0653486	0.0326743	−	0.999466i	$-0.489598\pi$
$269$	1.07180	0.0653486	0.0326743	+	0.999466i	$0.489598\pi$
$270$	0	0
$271$	−28.7128	−1.74418	−0.872090	−	0.489346i	$-0.837235\pi$
$271$	−28.7128	−1.74418	−0.872090	+	0.489346i	$0.837235\pi$
$272$	0	0
$273$	−3.46410	−0.209657
$274$	0	0
$275$	3.73205	0.225051
$276$	0	0
$277$	18.7846	1.12866	0.564329	−	0.825550i	$-0.309135\pi$
$277$	18.7846	1.12866	0.564329	+	0.825550i	$0.309135\pi$
$278$	0	0
$279$	−9.46410	−0.566601
$280$	0	0
$281$	9.58846	0.571999	0.286000	−	0.958230i	$-0.407674\pi$
$281$	9.58846	0.571999	0.286000	+	0.958230i	$0.407674\pi$
$282$	0	0
$283$	5.00000	0.297219	0.148610	−	0.988896i	$-0.452520\pi$
$283$	5.00000	0.297219	0.148610	+	0.988896i	$0.452520\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	11.1962	0.660888
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	5.92820	0.347517
$292$	0	0
$293$	−6.66025	−0.389096	−0.194548	−	0.980893i	$-0.562324\pi$
$293$	−6.66025	−0.389096	−0.194548	+	0.980893i	$0.562324\pi$
$294$	0	0
$295$	13.3205	0.775550
$296$	0	0
$297$	−3.73205	−0.216556
$298$	0	0
$299$	−8.78461	−0.508027
$300$	0	0
$301$	−3.46410	−0.199667
$302$	0	0
$303$	2.66025	0.152828
$304$	0	0
$305$	−14.9282	−0.854786
$306$	0	0
$307$	1.07180	0.0611707	0.0305853	−	0.999532i	$-0.490263\pi$
$307$	1.07180	0.0611707	0.0305853	+	0.999532i	$0.490263\pi$
$308$	0	0
$309$	5.46410	0.310842
$310$	0	0
$311$	21.3205	1.20898	0.604488	−	0.796615i	$-0.293378\pi$
$311$	21.3205	1.20898	0.604488	+	0.796615i	$0.293378\pi$
$312$	0	0
$313$	28.8564	1.63106	0.815530	−	0.578714i	$-0.196445\pi$
$313$	28.8564	1.63106	0.815530	+	0.578714i	$0.196445\pi$
$314$	0	0
$315$	−2.00000	−0.112687
$316$	0	0
$317$	9.58846	0.538541	0.269271	−	0.963065i	$-0.413217\pi$
$317$	9.58846	0.538541	0.269271	+	0.963065i	$0.413217\pi$
$318$	0	0
$319$	12.9282	0.723840
$320$	0	0
$321$	−2.00000	−0.111629
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−3.46410	−0.192154
$326$	0	0
$327$	−8.39230	−0.464096
$328$	0	0
$329$	0.267949	0.0147725
$330$	0	0
$331$	−12.9282	−0.710598	−0.355299	−	0.934753i	$-0.615621\pi$
$331$	−12.9282	−0.710598	−0.355299	+	0.934753i	$0.615621\pi$
$332$	0	0
$333$	−5.92820	−0.324864
$334$	0	0
$335$	−10.0000	−0.546358
$336$	0	0
$337$	19.3205	1.05246	0.526228	−	0.850344i	$-0.323606\pi$
$337$	19.3205	1.05246	0.526228	+	0.850344i	$0.323606\pi$
$338$	0	0
$339$	12.0000	0.651751
$340$	0	0
$341$	35.3205	1.91271
$342$	0	0
$343$	13.0000	0.701934
$344$	0	0
$345$	−5.07180	−0.273056
$346$	0	0
$347$	23.7321	1.27400	0.637002	−	0.770862i	$-0.280174\pi$
$347$	23.7321	1.27400	0.637002	+	0.770862i	$0.280174\pi$
$348$	0	0
$349$	6.85641	0.367015	0.183508	−	0.983018i	$-0.441255\pi$
$349$	6.85641	0.367015	0.183508	+	0.983018i	$0.441255\pi$
$350$	0	0
$351$	3.46410	0.184900
$352$	0	0
$353$	−21.0526	−1.12051	−0.560257	−	0.828319i	$-0.689298\pi$
$353$	−21.0526	−1.12051	−0.560257	+	0.828319i	$0.689298\pi$
$354$	0	0
$355$	−25.8564	−1.37232
$356$	0	0
$357$	0	0
$358$	0	0
$359$	27.3205	1.44192	0.720961	−	0.692976i	$-0.243701\pi$
$359$	27.3205	1.44192	0.720961	+	0.692976i	$0.243701\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	2.92820	0.153691
$364$	0	0
$365$	−29.8564	−1.56276
$366$	0	0
$367$	−3.92820	−0.205051	−0.102525	−	0.994730i	$-0.532692\pi$
$367$	−3.92820	−0.205051	−0.102525	+	0.994730i	$0.532692\pi$
$368$	0	0
$369$	−11.1962	−0.582848
$370$	0	0
$371$	−4.26795	−0.221581
$372$	0	0
$373$	33.8564	1.75302	0.876509	−	0.481385i	$-0.159866\pi$
$373$	33.8564	1.75302	0.876509	+	0.481385i	$0.159866\pi$
$374$	0	0
$375$	−12.0000	−0.619677
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	−1.92820	−0.0990451	−0.0495226	−	0.998773i	$-0.515770\pi$
$379$	−1.92820	−0.0990451	−0.0495226	+	0.998773i	$0.515770\pi$
$380$	0	0
$381$	8.39230	0.429951
$382$	0	0
$383$	28.2679	1.44442	0.722212	−	0.691671i	$-0.243125\pi$
$383$	28.2679	1.44442	0.722212	+	0.691671i	$0.243125\pi$
$384$	0	0
$385$	7.46410	0.380406
$386$	0	0
$387$	3.46410	0.176090
$388$	0	0
$389$	−25.0526	−1.27022	−0.635108	−	0.772424i	$-0.719044\pi$
$389$	−25.0526	−1.27022	−0.635108	+	0.772424i	$0.719044\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−4.53590	−0.228806
$394$	0	0
$395$	23.8564	1.20035
$396$	0	0
$397$	−15.0000	−0.752828	−0.376414	−	0.926451i	$-0.622843\pi$
$397$	−15.0000	−0.752828	−0.376414	+	0.926451i	$0.622843\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−8.39230	−0.419092	−0.209546	−	0.977799i	$-0.567199\pi$
$401$	−8.39230	−0.419092	−0.209546	+	0.977799i	$0.567199\pi$
$402$	0	0
$403$	−32.7846	−1.63312
$404$	0	0
$405$	2.00000	0.0993808
$406$	0	0
$407$	22.1244	1.09666
$408$	0	0
$409$	−18.9282	−0.935939	−0.467970	−	0.883745i	$-0.655014\pi$
$409$	−18.9282	−0.935939	−0.467970	+	0.883745i	$0.655014\pi$
$410$	0	0
$411$	15.3205	0.755705
$412$	0	0
$413$	−6.66025	−0.327730
$414$	0	0
$415$	10.9282	0.536444
$416$	0	0
$417$	10.5359	0.515945
$418$	0	0
$419$	1.60770	0.0785410	0.0392705	−	0.999229i	$-0.487497\pi$
$419$	1.60770	0.0785410	0.0392705	+	0.999229i	$0.487497\pi$
$420$	0	0
$421$	−37.1769	−1.81189	−0.905946	−	0.423393i	$-0.860839\pi$
$421$	−37.1769	−1.81189	−0.905946	+	0.423393i	$0.860839\pi$
$422$	0	0
$423$	−0.267949	−0.0130281
$424$	0	0
$425$	0	0
$426$	0	0
$427$	7.46410	0.361213
$428$	0	0
$429$	−12.9282	−0.624180
$430$	0	0
$431$	−21.7128	−1.04587	−0.522935	−	0.852373i	$-0.675163\pi$
$431$	−21.7128	−1.04587	−0.522935	+	0.852373i	$0.675163\pi$
$432$	0	0
$433$	25.8564	1.24258	0.621290	−	0.783581i	$-0.286609\pi$
$433$	25.8564	1.24258	0.621290	+	0.783581i	$0.286609\pi$
$434$	0	0
$435$	−6.92820	−0.332182
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−2.53590	−0.121032	−0.0605159	−	0.998167i	$-0.519275\pi$
$439$	−2.53590	−0.121032	−0.0605159	+	0.998167i	$0.519275\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	−29.3205	−1.39306	−0.696530	−	0.717528i	$-0.745274\pi$
$443$	−29.3205	−1.39306	−0.696530	+	0.717528i	$0.745274\pi$
$444$	0	0
$445$	13.8564	0.656857
$446$	0	0
$447$	4.00000	0.189194
$448$	0	0
$449$	26.6603	1.25818	0.629088	−	0.777334i	$-0.283429\pi$
$449$	26.6603	1.25818	0.629088	+	0.777334i	$0.283429\pi$
$450$	0	0
$451$	41.7846	1.96756
$452$	0	0
$453$	−1.00000	−0.0469841
$454$	0	0
$455$	−6.92820	−0.324799
$456$	0	0
$457$	−19.0718	−0.892141	−0.446071	−	0.894998i	$-0.647177\pi$
$457$	−19.0718	−0.892141	−0.446071	+	0.894998i	$0.647177\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	29.4641	1.37228	0.686140	−	0.727470i	$-0.259304\pi$
$461$	29.4641	1.37228	0.686140	+	0.727470i	$0.259304\pi$
$462$	0	0
$463$	14.2487	0.662194	0.331097	−	0.943597i	$-0.392581\pi$
$463$	14.2487	0.662194	0.331097	+	0.943597i	$0.392581\pi$
$464$	0	0
$465$	−18.9282	−0.877774
$466$	0	0
$467$	−19.0718	−0.882538	−0.441269	−	0.897375i	$-0.645471\pi$
$467$	−19.0718	−0.882538	−0.441269	+	0.897375i	$0.645471\pi$
$468$	0	0
$469$	5.00000	0.230879
$470$	0	0
$471$	16.5359	0.761934
$472$	0	0
$473$	−12.9282	−0.594439
$474$	0	0
$475$	0	0
$476$	0	0
$477$	4.26795	0.195416
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−20.5359	−0.936356
$482$	0	0
$483$	2.53590	0.115387
$484$	0	0
$485$	11.8564	0.538372
$486$	0	0
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	0	0
$489$	−14.9282	−0.675077
$490$	0	0
$491$	−10.3923	−0.468998	−0.234499	−	0.972116i	$-0.575345\pi$
$491$	−10.3923	−0.468998	−0.234499	+	0.972116i	$0.575345\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−7.46410	−0.335486
$496$	0	0
$497$	12.9282	0.579909
$498$	0	0
$499$	−14.8564	−0.665064	−0.332532	−	0.943092i	$-0.607903\pi$
$499$	−14.8564	−0.665064	−0.332532	+	0.943092i	$0.607903\pi$
$500$	0	0
$501$	−14.1244	−0.631030
$502$	0	0
$503$	35.4449	1.58041	0.790204	−	0.612844i	$-0.209974\pi$
$503$	35.4449	1.58041	0.790204	+	0.612844i	$0.209974\pi$
$504$	0	0
$505$	5.32051	0.236760
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	0	0
$511$	14.9282	0.660385
$512$	0	0
$513$	0	0
$514$	0	0
$515$	10.9282	0.481554
$516$	0	0
$517$	1.00000	0.0439799
$518$	0	0
$519$	4.00000	0.175581
$520$	0	0
$521$	−41.1769	−1.80399	−0.901997	−	0.431743i	$-0.857899\pi$
$521$	−41.1769	−1.80399	−0.901997	+	0.431743i	$0.857899\pi$
$522$	0	0
$523$	41.6410	1.82083	0.910417	−	0.413691i	$-0.135761\pi$
$523$	41.6410	1.82083	0.910417	+	0.413691i	$0.135761\pi$
$524$	0	0
$525$	1.00000	0.0436436
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−16.5692	−0.720401
$530$	0	0
$531$	6.66025	0.289030
$532$	0	0
$533$	−38.7846	−1.67995
$534$	0	0
$535$	−4.00000	−0.172935
$536$	0	0
$537$	−0.535898	−0.0231257
$538$	0	0
$539$	22.3923	0.964505
$540$	0	0
$541$	17.0000	0.730887	0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000	0.730887	0.365444	+	0.930834i	$0.380917\pi$
$542$	0	0
$543$	8.39230	0.360148
$544$	0	0
$545$	−16.7846	−0.718974
$546$	0	0
$547$	−26.5359	−1.13459	−0.567297	−	0.823514i	$-0.692011\pi$
$547$	−26.5359	−1.13459	−0.567297	+	0.823514i	$0.692011\pi$
$548$	0	0
$549$	−7.46410	−0.318560
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−11.9282	−0.507239
$554$	0	0
$555$	−11.8564	−0.503276
$556$	0	0
$557$	38.1244	1.61538	0.807690	−	0.589607i	$-0.200717\pi$
$557$	38.1244	1.61538	0.807690	+	0.589607i	$0.200717\pi$
$558$	0	0
$559$	12.0000	0.507546
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−27.7321	−1.16877	−0.584383	−	0.811478i	$-0.698664\pi$
$563$	−27.7321	−1.16877	−0.584383	+	0.811478i	$0.698664\pi$
$564$	0	0
$565$	24.0000	1.00969
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	44.3923	1.86102	0.930511	−	0.366264i	$-0.119363\pi$
$569$	44.3923	1.86102	0.930511	+	0.366264i	$0.119363\pi$
$570$	0	0
$571$	16.9282	0.708423	0.354212	−	0.935165i	$-0.384749\pi$
$571$	16.9282	0.708423	0.354212	+	0.935165i	$0.384749\pi$
$572$	0	0
$573$	−17.0526	−0.712381
$574$	0	0
$575$	2.53590	0.105754
$576$	0	0
$577$	−19.9282	−0.829622	−0.414811	−	0.909908i	$-0.636152\pi$
$577$	−19.9282	−0.829622	−0.414811	+	0.909908i	$0.636152\pi$
$578$	0	0
$579$	−16.8564	−0.700528
$580$	0	0
$581$	−5.46410	−0.226689
$582$	0	0
$583$	−15.9282	−0.659679
$584$	0	0
$585$	6.92820	0.286446
$586$	0	0
$587$	−38.7846	−1.60081	−0.800406	−	0.599458i	$-0.795383\pi$
$587$	−38.7846	−1.60081	−0.800406	+	0.599458i	$0.795383\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	5.85641	0.240900
$592$	0	0
$593$	6.12436	0.251497	0.125749	−	0.992062i	$-0.459867\pi$
$593$	6.12436	0.251497	0.125749	+	0.992062i	$0.459867\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	10.9282	0.447262
$598$	0	0
$599$	−15.0718	−0.615817	−0.307908	−	0.951416i	$-0.599629\pi$
$599$	−15.0718	−0.615817	−0.307908	+	0.951416i	$0.599629\pi$
$600$	0	0
$601$	3.78461	0.154377	0.0771887	−	0.997016i	$-0.475406\pi$
$601$	3.78461	0.154377	0.0771887	+	0.997016i	$0.475406\pi$
$602$	0	0
$603$	−5.00000	−0.203616
$604$	0	0
$605$	5.85641	0.238097
$606$	0	0
$607$	−27.7128	−1.12483	−0.562414	−	0.826856i	$-0.690127\pi$
$607$	−27.7128	−1.12483	−0.562414	+	0.826856i	$0.690127\pi$
$608$	0	0
$609$	3.46410	0.140372
$610$	0	0
$611$	−0.928203	−0.0375511
$612$	0	0
$613$	13.0000	0.525065	0.262533	−	0.964923i	$-0.415442\pi$
$613$	13.0000	0.525065	0.262533	+	0.964923i	$0.415442\pi$
$614$	0	0
$615$	−22.3923	−0.902945
$616$	0	0
$617$	6.92820	0.278919	0.139459	−	0.990228i	$-0.455464\pi$
$617$	6.92820	0.278919	0.139459	+	0.990228i	$0.455464\pi$
$618$	0	0
$619$	0.856406	0.0344219	0.0172109	−	0.999852i	$-0.494521\pi$
$619$	0.856406	0.0344219	0.0172109	+	0.999852i	$0.494521\pi$
$620$	0	0
$621$	−2.53590	−0.101762
$622$	0	0
$623$	−6.92820	−0.277573
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	29.8564	1.18856	0.594282	−	0.804256i	$-0.297436\pi$
$631$	29.8564	1.18856	0.594282	+	0.804256i	$0.297436\pi$
$632$	0	0
$633$	−9.92820	−0.394611
$634$	0	0
$635$	16.7846	0.666077
$636$	0	0
$637$	−20.7846	−0.823516
$638$	0	0
$639$	−12.9282	−0.511432
$640$	0	0
$641$	21.8564	0.863276	0.431638	−	0.902047i	$-0.357936\pi$
$641$	21.8564	0.863276	0.431638	+	0.902047i	$0.357936\pi$
$642$	0	0
$643$	−39.8564	−1.57178	−0.785892	−	0.618364i	$-0.787796\pi$
$643$	−39.8564	−1.57178	−0.785892	+	0.618364i	$0.787796\pi$
$644$	0	0
$645$	6.92820	0.272798
$646$	0	0
$647$	40.2487	1.58234	0.791170	−	0.611596i	$-0.209472\pi$
$647$	40.2487	1.58234	0.791170	+	0.611596i	$0.209472\pi$
$648$	0	0
$649$	−24.8564	−0.975699
$650$	0	0
$651$	9.46410	0.370927
$652$	0	0
$653$	38.1051	1.49117	0.745584	−	0.666411i	$-0.232171\pi$
$653$	38.1051	1.49117	0.745584	+	0.666411i	$0.232171\pi$
$654$	0	0
$655$	−9.07180	−0.354464
$656$	0	0
$657$	−14.9282	−0.582405
$658$	0	0
$659$	−41.8372	−1.62974	−0.814872	−	0.579640i	$-0.803193\pi$
$659$	−41.8372	−1.62974	−0.814872	+	0.579640i	$0.803193\pi$
$660$	0	0
$661$	−17.4641	−0.679275	−0.339637	−	0.940556i	$-0.610304\pi$
$661$	−17.4641	−0.679275	−0.339637	+	0.940556i	$0.610304\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8.78461	0.340141
$668$	0	0
$669$	4.00000	0.154649
$670$	0	0
$671$	27.8564	1.07538
$672$	0	0
$673$	8.71281	0.335854	0.167927	−	0.985799i	$-0.446293\pi$
$673$	8.71281	0.335854	0.167927	+	0.985799i	$0.446293\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	19.1962	0.737768	0.368884	−	0.929475i	$-0.379740\pi$
$677$	19.1962	0.737768	0.368884	+	0.929475i	$0.379740\pi$
$678$	0	0
$679$	−5.92820	−0.227504
$680$	0	0
$681$	6.39230	0.244954
$682$	0	0
$683$	−21.4641	−0.821301	−0.410651	−	0.911793i	$-0.634698\pi$
$683$	−21.4641	−0.821301	−0.410651	+	0.911793i	$0.634698\pi$
$684$	0	0
$685$	30.6410	1.17073
$686$	0	0
$687$	−13.9282	−0.531394
$688$	0	0
$689$	14.7846	0.563249
$690$	0	0
$691$	−39.9282	−1.51894	−0.759470	−	0.650542i	$-0.774542\pi$
$691$	−39.9282	−1.51894	−0.759470	+	0.650542i	$0.774542\pi$
$692$	0	0
$693$	3.73205	0.141769
$694$	0	0
$695$	21.0718	0.799299
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−10.6603	−0.403208
$700$	0	0
$701$	−40.7846	−1.54041	−0.770207	−	0.637794i	$-0.779847\pi$
$701$	−40.7846	−1.54041	−0.770207	+	0.637794i	$0.779847\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−0.535898	−0.0201831
$706$	0	0
$707$	−2.66025	−0.100049
$708$	0	0
$709$	5.21539	0.195868	0.0979340	−	0.995193i	$-0.468777\pi$
$709$	5.21539	0.195868	0.0979340	+	0.995193i	$0.468777\pi$
$710$	0	0
$711$	11.9282	0.447343
$712$	0	0
$713$	24.0000	0.898807
$714$	0	0
$715$	−25.8564	−0.966975
$716$	0	0
$717$	−27.4641	−1.02567
$718$	0	0
$719$	46.2487	1.72479	0.862393	−	0.506239i	$-0.168965\pi$
$719$	46.2487	1.72479	0.862393	+	0.506239i	$0.168965\pi$
$720$	0	0
$721$	−5.46410	−0.203494
$722$	0	0
$723$	−25.0000	−0.929760
$724$	0	0
$725$	3.46410	0.128654
$726$	0	0
$727$	−33.7128	−1.25034	−0.625170	−	0.780489i	$-0.714970\pi$
$727$	−33.7128	−1.25034	−0.625170	+	0.780489i	$0.714970\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−18.9282	−0.699129	−0.349565	−	0.936912i	$-0.613670\pi$
$733$	−18.9282	−0.699129	−0.349565	+	0.936912i	$0.613670\pi$
$734$	0	0
$735$	−12.0000	−0.442627
$736$	0	0
$737$	18.6603	0.687359
$738$	0	0
$739$	−40.0000	−1.47142	−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000	−1.47142	−0.735712	+	0.677295i	$0.763152\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	10.4115	0.381962	0.190981	−	0.981594i	$-0.438833\pi$
$743$	10.4115	0.381962	0.190981	+	0.981594i	$0.438833\pi$
$744$	0	0
$745$	8.00000	0.293097
$746$	0	0
$747$	5.46410	0.199921
$748$	0	0
$749$	2.00000	0.0730784
$750$	0	0
$751$	−24.7846	−0.904403	−0.452202	−	0.891916i	$-0.649361\pi$
$751$	−24.7846	−0.904403	−0.452202	+	0.891916i	$0.649361\pi$
$752$	0	0
$753$	12.5359	0.456834
$754$	0	0
$755$	−2.00000	−0.0727875
$756$	0	0
$757$	25.6410	0.931939	0.465969	−	0.884801i	$-0.345706\pi$
$757$	25.6410	0.931939	0.465969	+	0.884801i	$0.345706\pi$
$758$	0	0
$759$	9.46410	0.343525
$760$	0	0
$761$	−47.4449	−1.71987	−0.859937	−	0.510399i	$-0.829498\pi$
$761$	−47.4449	−1.71987	−0.859937	+	0.510399i	$0.829498\pi$
$762$	0	0
$763$	8.39230	0.303822
$764$	0	0
$765$	0	0
$766$	0	0
$767$	23.0718	0.833074
$768$	0	0
$769$	−26.9282	−0.971056	−0.485528	−	0.874221i	$-0.661373\pi$
$769$	−26.9282	−0.971056	−0.485528	+	0.874221i	$0.661373\pi$
$770$	0	0
$771$	−9.85641	−0.354970
$772$	0	0
$773$	22.6410	0.814341	0.407170	−	0.913352i	$-0.366516\pi$
$773$	22.6410	0.814341	0.407170	+	0.913352i	$0.366516\pi$
$774$	0	0
$775$	9.46410	0.339961
$776$	0	0
$777$	5.92820	0.212673
$778$	0	0
$779$	0	0
$780$	0	0
$781$	48.2487	1.72647
$782$	0	0
$783$	−3.46410	−0.123797
$784$	0	0
$785$	33.0718	1.18038
$786$	0	0
$787$	−2.14359	−0.0764109	−0.0382054	−	0.999270i	$-0.512164\pi$
$787$	−2.14359	−0.0764109	−0.0382054	+	0.999270i	$0.512164\pi$
$788$	0	0
$789$	−8.00000	−0.284808
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	−25.8564	−0.918188
$794$	0	0
$795$	8.53590	0.302737
$796$	0	0
$797$	−3.32051	−0.117618	−0.0588092	−	0.998269i	$-0.518730\pi$
$797$	−3.32051	−0.117618	−0.0588092	+	0.998269i	$0.518730\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	6.92820	0.244796
$802$	0	0
$803$	55.7128	1.96606
$804$	0	0
$805$	5.07180	0.178757
$806$	0	0
$807$	1.07180	0.0377290
$808$	0	0
$809$	−26.3731	−0.927228	−0.463614	−	0.886037i	$-0.653447\pi$
$809$	−26.3731	−0.927228	−0.463614	+	0.886037i	$0.653447\pi$
$810$	0	0
$811$	−16.7128	−0.586866	−0.293433	−	0.955980i	$-0.594798\pi$
$811$	−16.7128	−0.586866	−0.293433	+	0.955980i	$0.594798\pi$
$812$	0	0
$813$	−28.7128	−1.00700
$814$	0	0
$815$	−29.8564	−1.04582
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−3.46410	−0.121046
$820$	0	0
$821$	25.0526	0.874340	0.437170	−	0.899379i	$-0.355981\pi$
$821$	25.0526	0.874340	0.437170	+	0.899379i	$0.355981\pi$
$822$	0	0
$823$	−41.1769	−1.43534	−0.717669	−	0.696385i	$-0.754791\pi$
$823$	−41.1769	−1.43534	−0.717669	+	0.696385i	$0.754791\pi$
$824$	0	0
$825$	3.73205	0.129933
$826$	0	0
$827$	40.8038	1.41889	0.709444	−	0.704761i	$-0.248946\pi$
$827$	40.8038	1.41889	0.709444	+	0.704761i	$0.248946\pi$
$828$	0	0
$829$	−14.7128	−0.510997	−0.255499	−	0.966809i	$-0.582240\pi$
$829$	−14.7128	−0.510997	−0.255499	+	0.966809i	$0.582240\pi$
$830$	0	0
$831$	18.7846	0.651631
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−28.2487	−0.977587
$836$	0	0
$837$	−9.46410	−0.327127
$838$	0	0
$839$	12.5359	0.432787	0.216394	−	0.976306i	$-0.430571\pi$
$839$	12.5359	0.432787	0.216394	+	0.976306i	$0.430571\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	0	0
$843$	9.58846	0.330244
$844$	0	0
$845$	−2.00000	−0.0688021
$846$	0	0
$847$	−2.92820	−0.100614
$848$	0	0
$849$	5.00000	0.171600
$850$	0	0
$851$	15.0333	0.515336
$852$	0	0
$853$	3.85641	0.132041	0.0660204	−	0.997818i	$-0.478970\pi$
$853$	3.85641	0.132041	0.0660204	+	0.997818i	$0.478970\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11.1962	0.382453	0.191227	−	0.981546i	$-0.438753\pi$
$857$	11.1962	0.382453	0.191227	+	0.981546i	$0.438753\pi$
$858$	0	0
$859$	4.71281	0.160799	0.0803996	−	0.996763i	$-0.474380\pi$
$859$	4.71281	0.160799	0.0803996	+	0.996763i	$0.474380\pi$
$860$	0	0
$861$	11.1962	0.381564
$862$	0	0
$863$	20.1051	0.684386	0.342193	−	0.939630i	$-0.388830\pi$
$863$	20.1051	0.684386	0.342193	+	0.939630i	$0.388830\pi$
$864$	0	0
$865$	8.00000	0.272008
$866$	0	0
$867$	−17.0000	−0.577350
$868$	0	0
$869$	−44.5167	−1.51012
$870$	0	0
$871$	−17.3205	−0.586883
$872$	0	0
$873$	5.92820	0.200639
$874$	0	0
$875$	12.0000	0.405674
$876$	0	0
$877$	−37.1769	−1.25538	−0.627688	−	0.778465i	$-0.715998\pi$
$877$	−37.1769	−1.25538	−0.627688	+	0.778465i	$0.715998\pi$
$878$	0	0
$879$	−6.66025	−0.224645
$880$	0	0
$881$	25.8756	0.871773	0.435886	−	0.900002i	$-0.356435\pi$
$881$	25.8756	0.871773	0.435886	+	0.900002i	$0.356435\pi$
$882$	0	0
$883$	−50.1051	−1.68617	−0.843086	−	0.537779i	$-0.819263\pi$
$883$	−50.1051	−1.68617	−0.843086	+	0.537779i	$0.819263\pi$
$884$	0	0
$885$	13.3205	0.447764
$886$	0	0
$887$	−51.7128	−1.73635	−0.868173	−	0.496261i	$-0.834706\pi$
$887$	−51.7128	−1.73635	−0.868173	+	0.496261i	$0.834706\pi$
$888$	0	0
$889$	−8.39230	−0.281469
$890$	0	0
$891$	−3.73205	−0.125028
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−1.07180	−0.0358262
$896$	0	0
$897$	−8.78461	−0.293310
$898$	0	0
$899$	32.7846	1.09343
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−3.46410	−0.115278
$904$	0	0
$905$	16.7846	0.557939
$906$	0	0
$907$	−9.07180	−0.301224	−0.150612	−	0.988593i	$-0.548124\pi$
$907$	−9.07180	−0.301224	−0.150612	+	0.988593i	$0.548124\pi$
$908$	0	0
$909$	2.66025	0.0882351
$910$	0	0
$911$	23.1769	0.767885	0.383943	−	0.923357i	$-0.374566\pi$
$911$	23.1769	0.767885	0.383943	+	0.923357i	$0.374566\pi$
$912$	0	0
$913$	−20.3923	−0.674887
$914$	0	0
$915$	−14.9282	−0.493511
$916$	0	0
$917$	4.53590	0.149789
$918$	0	0
$919$	22.1436	0.730450	0.365225	−	0.930919i	$-0.380992\pi$
$919$	22.1436	0.730450	0.365225	+	0.930919i	$0.380992\pi$
$920$	0	0
$921$	1.07180	0.0353169
$922$	0	0
$923$	−44.7846	−1.47410
$924$	0	0
$925$	5.92820	0.194918
$926$	0	0
$927$	5.46410	0.179465
$928$	0	0
$929$	41.8372	1.37263	0.686316	−	0.727303i	$-0.259227\pi$
$929$	41.8372	1.37263	0.686316	+	0.727303i	$0.259227\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	21.3205	0.698002
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−51.8564	−1.69407	−0.847037	−	0.531533i	$-0.821616\pi$
$937$	−51.8564	−1.69407	−0.847037	+	0.531533i	$0.821616\pi$
$938$	0	0
$939$	28.8564	0.941693
$940$	0	0
$941$	−43.9808	−1.43373	−0.716866	−	0.697211i	$-0.754424\pi$
$941$	−43.9808	−1.43373	−0.716866	+	0.697211i	$0.754424\pi$
$942$	0	0
$943$	28.3923	0.924581
$944$	0	0
$945$	−2.00000	−0.0650600
$946$	0	0
$947$	−13.4641	−0.437525	−0.218762	−	0.975778i	$-0.570202\pi$
$947$	−13.4641	−0.437525	−0.218762	+	0.975778i	$0.570202\pi$
$948$	0	0
$949$	−51.7128	−1.67867
$950$	0	0
$951$	9.58846	0.310927
$952$	0	0
$953$	−12.9282	−0.418786	−0.209393	−	0.977832i	$-0.567149\pi$
$953$	−12.9282	−0.418786	−0.209393	+	0.977832i	$0.567149\pi$
$954$	0	0
$955$	−34.1051	−1.10362
$956$	0	0
$957$	12.9282	0.417909
$958$	0	0
$959$	−15.3205	−0.494725
$960$	0	0
$961$	58.5692	1.88933
$962$	0	0
$963$	−2.00000	−0.0644491
$964$	0	0
$965$	−33.7128	−1.08525
$966$	0	0
$967$	35.7128	1.14845	0.574223	−	0.818699i	$-0.305304\pi$
$967$	35.7128	1.14845	0.574223	+	0.818699i	$0.305304\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	22.6795	0.727820	0.363910	−	0.931434i	$-0.381442\pi$
$971$	22.6795	0.727820	0.363910	+	0.931434i	$0.381442\pi$
$972$	0	0
$973$	−10.5359	−0.337765
$974$	0	0
$975$	−3.46410	−0.110940
$976$	0	0
$977$	37.5885	1.20256	0.601281	−	0.799038i	$-0.294657\pi$
$977$	37.5885	1.20256	0.601281	+	0.799038i	$0.294657\pi$
$978$	0	0
$979$	−25.8564	−0.826374
$980$	0	0
$981$	−8.39230	−0.267946
$982$	0	0
$983$	−7.17691	−0.228908	−0.114454	−	0.993429i	$-0.536512\pi$
$983$	−7.17691	−0.228908	−0.114454	+	0.993429i	$0.536512\pi$
$984$	0	0
$985$	11.7128	0.373201
$986$	0	0
$987$	0.267949	0.00852892
$988$	0	0
$989$	−8.78461	−0.279334
$990$	0	0
$991$	16.1436	0.512818	0.256409	−	0.966568i	$-0.417461\pi$
$991$	16.1436	0.512818	0.256409	+	0.966568i	$0.417461\pi$
$992$	0	0
$993$	−12.9282	−0.410264
$994$	0	0
$995$	21.8564	0.692895
$996$	0	0
$997$	−31.7846	−1.00663	−0.503314	−	0.864103i	$-0.667886\pi$
$997$	−31.7846	−1.00663	−0.503314	+	0.864103i	$0.667886\pi$
$998$	0	0
$999$	−5.92820	−0.187560

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7248.2.a.t.1.1		2
4.3	odd	2		453.2.a.c.1.2	✓	2
12.11	even	2		1359.2.a.b.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
453.2.a.c.1.2	✓	2	4.3	odd	2
1359.2.a.b.1.1		2	12.11	even	2
7248.2.a.t.1.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7248 = 2^{4} \cdot 3 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7248.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.73205 q^{11} +3.46410 q^{13} +2.00000 q^{15} -1.00000 q^{21} -2.53590 q^{23} -1.00000 q^{25} +1.00000 q^{27} -3.46410 q^{29} -9.46410 q^{31} -3.73205 q^{33} -2.00000 q^{35} -5.92820 q^{37} +3.46410 q^{39} -11.1962 q^{41} +3.46410 q^{43} +2.00000 q^{45} -0.267949 q^{47} -6.00000 q^{49} +4.26795 q^{53} -7.46410 q^{55} +6.66025 q^{59} -7.46410 q^{61} -1.00000 q^{63} +6.92820 q^{65} -5.00000 q^{67} -2.53590 q^{69} -12.9282 q^{71} -14.9282 q^{73} -1.00000 q^{75} +3.73205 q^{77} +11.9282 q^{79} +1.00000 q^{81} +5.46410 q^{83} -3.46410 q^{87} +6.92820 q^{89} -3.46410 q^{91} -9.46410 q^{93} +5.92820 q^{97} -3.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 4 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 4 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 4 q^{5} - 2 q^{7} + 2 q^{9} - 4 q^{11} + 4 q^{15} - 2 q^{21} - 12 q^{23} - 2 q^{25} + 2 q^{27} - 12 q^{31} - 4 q^{33} - 4 q^{35} + 2 q^{37} - 12 q^{41} + 4 q^{45} - 4 q^{47} - 12 q^{49} + 12 q^{53} - 8 q^{55} - 4 q^{59} - 8 q^{61} - 2 q^{63} - 10 q^{67} - 12 q^{69} - 12 q^{71} - 16 q^{73} - 2 q^{75} + 4 q^{77} + 10 q^{79} + 2 q^{81} + 4 q^{83} - 12 q^{93} - 2 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 4 * q^5 - 2 * q^7 + 2 * q^9 - 4 * q^11 + 4 * q^15 - 2 * q^21 - 12 * q^23 - 2 * q^25 + 2 * q^27 - 12 * q^31 - 4 * q^33 - 4 * q^35 + 2 * q^37 - 12 * q^41 + 4 * q^45 - 4 * q^47 - 12 * q^49 + 12 * q^53 - 8 * q^55 - 4 * q^59 - 8 * q^61 - 2 * q^63 - 10 * q^67 - 12 * q^69 - 12 * q^71 - 16 * q^73 - 2 * q^75 + 4 * q^77 + 10 * q^79 + 2 * q^81 + 4 * q^83 - 12 * q^93 - 2 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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