LMFDB - Embedded newform 1344.4.c.f.673.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(673,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1344.673");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} - x^{10} - 861 x^{8} - 2158 x^{7} + 8654 x^{6} + 118244 x^{5} + 707300 x^{4} + \cdots + 43264 $ x^12 - 2x^11 - x^10 - 861x^8 - 2158x^7 + 8654x^6 + 118244x^5 + 707300x^4 + 1646096x^3 + 1391904x^2 + 174720*x + 43264
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			673.2
Root			$6.03643 + 1.61746i$ of defining polynomial
Character	$\chi$	$=$	1344.673
Dual form			1344.4.c.f.673.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -6.24978i q^{5} -7.00000 q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} -6.24978i q^{5} -7.00000 q^{7} -9.00000 q^{9} -63.4042i q^{11} -82.2923i q^{13} -18.7493 q^{15} +75.4992 q^{17} -125.613i q^{19} +21.0000i q^{21} +155.435 q^{23} +85.9403 q^{25} +27.0000i q^{27} +56.2501i q^{29} +159.142 q^{31} -190.212 q^{33} +43.7484i q^{35} -197.279i q^{37} -246.877 q^{39} +137.974 q^{41} +295.005i q^{43} +56.2480i q^{45} -186.683 q^{47} +49.0000 q^{49} -226.498i q^{51} +409.210i q^{53} -396.262 q^{55} -376.839 q^{57} -311.117i q^{59} +168.801i q^{61} +63.0000 q^{63} -514.308 q^{65} -563.149i q^{67} -466.305i q^{69} -282.950 q^{71} +250.409 q^{73} -257.821i q^{75} +443.829i q^{77} +948.033 q^{79} +81.0000 q^{81} -1289.88i q^{83} -471.853i q^{85} +168.750 q^{87} -655.368 q^{89} +576.046i q^{91} -477.426i q^{93} -785.053 q^{95} +706.813 q^{97} +570.637i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 84 q^{7} - 108 q^{9}+O(q^{10}) $ 12 * q - 84 * q^7 - 108 * q^9	$ 12 q - 84 q^{7} - 108 q^{9} + 24 q^{15} + 376 q^{17} - 336 q^{23} - 180 q^{25} + 192 q^{31} - 168 q^{33} - 504 q^{39} + 488 q^{41} - 448 q^{47} + 588 q^{49} - 3600 q^{55} - 432 q^{57} + 756 q^{63} + 1408 q^{65} - 5104 q^{71} - 1752 q^{73} - 1632 q^{79} + 972 q^{81} - 336 q^{87} - 3688 q^{89} - 2496 q^{95} - 1944 q^{97}+O(q^{100}) $ 12 * q - 84 * q^7 - 108 * q^9 + 24 * q^15 + 376 * q^17 - 336 * q^23 - 180 * q^25 + 192 * q^31 - 168 * q^33 - 504 * q^39 + 488 * q^41 - 448 * q^47 + 588 * q^49 - 3600 * q^55 - 432 * q^57 + 756 * q^63 + 1408 * q^65 - 5104 * q^71 - 1752 * q^73 - 1632 * q^79 + 972 * q^81 - 336 * q^87 - 3688 * q^89 - 2496 * q^95 - 1944 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	0	0
$5$	− 6.24978i	− 0.558997i	−0.960146	−	0.279499i	$-0.909832\pi$
$5$	− 6.24978i	− 0.558997i	0.960146	−	0.279499i	$-0.0901682\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	− 63.4042i	− 1.73792i	−0.494886	−	0.868958i	$-0.664790\pi$
$11$	− 63.4042i	− 1.73792i	0.494886	−	0.868958i	$-0.335210\pi$
$12$	0	0
$13$	− 82.2923i	− 1.75567i	−0.478959	−	0.877837i	$-0.658986\pi$
$13$	− 82.2923i	− 1.75567i	0.478959	−	0.877837i	$-0.341014\pi$
$14$	0	0
$15$	−18.7493	−0.322737
$16$	0	0
$17$	75.4992	1.07713	0.538566	−	0.842583i	$-0.318966\pi$
$17$	75.4992	1.07713	0.538566	+	0.842583i	$0.318966\pi$
$18$	0	0
$19$	− 125.613i	− 1.51672i	−0.651839	−	0.758358i	$-0.726002\pi$
$19$	− 125.613i	− 1.51672i	0.651839	−	0.758358i	$-0.273998\pi$
$20$	0	0
$21$	21.0000i	0.218218i
$22$	0	0
$23$	155.435	1.40915	0.704575	−	0.709629i	$-0.251138\pi$
$23$	155.435	1.40915	0.704575	+	0.709629i	$0.251138\pi$
$24$	0	0
$25$	85.9403	0.687522
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	56.2501i	0.360185i	0.983650	+	0.180093i	$0.0576398\pi$
$29$	56.2501i	0.360185i	−0.983650	+	0.180093i	$0.942360\pi$
$30$	0	0
$31$	159.142	0.922024	0.461012	−	0.887394i	$-0.347487\pi$
$31$	159.142	0.922024	0.461012	+	0.887394i	$0.347487\pi$
$32$	0	0
$33$	−190.212	−1.00339
$34$	0	0
$35$	43.7484i	0.211281i
$36$	0	0
$37$	− 197.279i	− 0.876554i	−0.898840	−	0.438277i	$-0.855589\pi$
$37$	− 197.279i	− 0.876554i	0.898840	−	0.438277i	$-0.144411\pi$
$38$	0	0
$39$	−246.877	−1.01364
$40$	0	0
$41$	137.974	0.525560	0.262780	−	0.964856i	$-0.415361\pi$
$41$	137.974	0.525560	0.262780	+	0.964856i	$0.415361\pi$
$42$	0	0
$43$	295.005i	1.04623i	0.852262	+	0.523115i	$0.175230\pi$
$43$	295.005i	1.04623i	−0.852262	+	0.523115i	$0.824770\pi$
$44$	0	0
$45$	56.2480i	0.186332i
$46$	0	0
$47$	−186.683	−0.579374	−0.289687	−	0.957121i	$-0.593551\pi$
$47$	−186.683	−0.579374	−0.289687	+	0.957121i	$0.593551\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	− 226.498i	− 0.621882i
$52$	0	0
$53$	409.210i	1.06055i	0.847824	+	0.530277i	$0.177912\pi$
$53$	409.210i	1.06055i	−0.847824	+	0.530277i	$0.822088\pi$
$54$	0	0
$55$	−396.262	−0.971490
$56$	0	0
$57$	−376.839	−0.875676
$58$	0	0
$59$	− 311.117i	− 0.686507i	−0.939243	−	0.343254i	$-0.888471\pi$
$59$	− 311.117i	− 0.686507i	0.939243	−	0.343254i	$-0.111529\pi$
$60$	0	0
$61$	168.801i	0.354306i	0.984183	+	0.177153i	$0.0566888\pi$
$61$	168.801i	0.354306i	−0.984183	+	0.177153i	$0.943311\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	−514.308	−0.981417
$66$	0	0
$67$	− 563.149i	− 1.02686i	−0.858132	−	0.513430i	$-0.828375\pi$
$67$	− 563.149i	− 1.02686i	0.858132	−	0.513430i	$-0.171625\pi$
$68$	0	0
$69$	− 466.305i	− 0.813573i
$70$	0	0
$71$	−282.950	−0.472958	−0.236479	−	0.971637i	$-0.575993\pi$
$71$	−282.950	−0.472958	−0.236479	+	0.971637i	$0.575993\pi$
$72$	0	0
$73$	250.409	0.401481	0.200740	−	0.979644i	$-0.435665\pi$
$73$	250.409	0.401481	0.200740	+	0.979644i	$0.435665\pi$
$74$	0	0
$75$	− 257.821i	− 0.396941i
$76$	0	0
$77$	443.829i	0.656870i
$78$	0	0
$79$	948.033	1.35015	0.675076	−	0.737748i	$-0.264111\pi$
$79$	948.033	1.35015	0.675076	+	0.737748i	$0.264111\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 1289.88i	− 1.70582i	−0.522057	−	0.852911i	$-0.674835\pi$
$83$	− 1289.88i	− 1.70582i	0.522057	−	0.852911i	$-0.325165\pi$
$84$	0	0
$85$	− 471.853i	− 0.602114i
$86$	0	0
$87$	168.750	0.207953
$88$	0	0
$89$	−655.368	−0.780549	−0.390275	−	0.920699i	$-0.627620\pi$
$89$	−655.368	−0.780549	−0.390275	+	0.920699i	$0.627620\pi$
$90$	0	0
$91$	576.046i	0.663583i
$92$	0	0
$93$	− 477.426i	− 0.532331i
$94$	0	0
$95$	−785.053	−0.847840
$96$	0	0
$97$	706.813	0.739856	0.369928	−	0.929061i	$-0.379382\pi$
$97$	706.813	0.739856	0.369928	+	0.929061i	$0.379382\pi$
$98$	0	0
$99$	570.637i	0.579305i
$100$	0	0
$101$	− 916.177i	− 0.902604i	−0.892371	−	0.451302i	$-0.850960\pi$
$101$	− 916.177i	− 0.902604i	0.892371	−	0.451302i	$-0.149040\pi$
$102$	0	0
$103$	−1373.56	−1.31399	−0.656993	−	0.753897i	$-0.728172\pi$
$103$	−1373.56	−1.31399	−0.656993	+	0.753897i	$0.728172\pi$
$104$	0	0
$105$	131.245	0.121983
$106$	0	0
$107$	− 1543.06i	− 1.39414i	−0.717001	−	0.697072i	$-0.754486\pi$
$107$	− 1543.06i	− 1.39414i	0.717001	−	0.697072i	$-0.245514\pi$
$108$	0	0
$109$	1798.08i	1.58005i	0.613077	+	0.790023i	$0.289931\pi$
$109$	1798.08i	1.58005i	−0.613077	+	0.790023i	$0.710069\pi$
$110$	0	0
$111$	−591.838	−0.506079
$112$	0	0
$113$	−284.375	−0.236741	−0.118370	−	0.992970i	$-0.537767\pi$
$113$	−284.375	−0.236741	−0.118370	+	0.992970i	$0.537767\pi$
$114$	0	0
$115$	− 971.435i	− 0.787711i
$116$	0	0
$117$	740.630i	0.585225i
$118$	0	0
$119$	−528.494	−0.407117
$120$	0	0
$121$	−2689.09	−2.02035
$122$	0	0
$123$	− 413.922i	− 0.303432i
$124$	0	0
$125$	− 1318.33i	− 0.943320i
$126$	0	0
$127$	842.514	0.588669	0.294335	−	0.955702i	$-0.404902\pi$
$127$	842.514	0.588669	0.294335	+	0.955702i	$0.404902\pi$
$128$	0	0
$129$	885.016	0.604041
$130$	0	0
$131$	2590.38i	1.72765i	0.503790	+	0.863826i	$0.331939\pi$
$131$	2590.38i	1.72765i	−0.503790	+	0.863826i	$0.668061\pi$
$132$	0	0
$133$	879.291i	0.573265i
$134$	0	0
$135$	168.744	0.107579
$136$	0	0
$137$	1917.81	1.19599	0.597993	−	0.801502i	$-0.295965\pi$
$137$	1917.81	1.19599	0.597993	+	0.801502i	$0.295965\pi$
$138$	0	0
$139$	− 2527.67i	− 1.54240i	−0.636590	−	0.771202i	$-0.719656\pi$
$139$	− 2527.67i	− 1.54240i	0.636590	−	0.771202i	$-0.280344\pi$
$140$	0	0
$141$	560.050i	0.334502i
$142$	0	0
$143$	−5217.67	−3.05121
$144$	0	0
$145$	351.550	0.201343
$146$	0	0
$147$	− 147.000i	− 0.0824786i
$148$	0	0
$149$	1340.62i	0.737099i	0.929608	+	0.368549i	$0.120145\pi$
$149$	1340.62i	0.737099i	−0.929608	+	0.368549i	$0.879855\pi$
$150$	0	0
$151$	680.821	0.366917	0.183458	−	0.983027i	$-0.441271\pi$
$151$	680.821	0.366917	0.183458	+	0.983027i	$0.441271\pi$
$152$	0	0
$153$	−679.493	−0.359044
$154$	0	0
$155$	− 994.602i	− 0.515409i
$156$	0	0
$157$	3025.39i	1.53791i	0.639301	+	0.768957i	$0.279224\pi$
$157$	3025.39i	1.53791i	−0.639301	+	0.768957i	$0.720776\pi$
$158$	0	0
$159$	1227.63	0.612311
$160$	0	0
$161$	−1088.05	−0.532609
$162$	0	0
$163$	506.970i	0.243613i	0.992554	+	0.121807i	$0.0388687\pi$
$163$	506.970i	0.243613i	−0.992554	+	0.121807i	$0.961131\pi$
$164$	0	0
$165$	1188.79i	0.560890i
$166$	0	0
$167$	243.431	0.112798	0.0563991	−	0.998408i	$-0.482038\pi$
$167$	243.431	0.112798	0.0563991	+	0.998408i	$0.482038\pi$
$168$	0	0
$169$	−4575.02	−2.08239
$170$	0	0
$171$	1130.52i	0.505572i
$172$	0	0
$173$	73.9661i	0.0325060i	0.999868	+	0.0162530i	$0.00517372\pi$
$173$	73.9661i	0.0325060i	−0.999868	+	0.0162530i	$0.994826\pi$
$174$	0	0
$175$	−601.582	−0.259859
$176$	0	0
$177$	−933.350	−0.396355
$178$	0	0
$179$	926.879i	0.387029i	0.981097	+	0.193514i	$0.0619887\pi$
$179$	926.879i	0.387029i	−0.981097	+	0.193514i	$0.938011\pi$
$180$	0	0
$181$	2615.68i	1.07416i	0.843533	+	0.537078i	$0.180472\pi$
$181$	2615.68i	1.07416i	−0.843533	+	0.537078i	$0.819528\pi$
$182$	0	0
$183$	506.402	0.204559
$184$	0	0
$185$	−1232.95	−0.489991
$186$	0	0
$187$	− 4786.96i	− 1.87196i
$188$	0	0
$189$	− 189.000i	− 0.0727393i
$190$	0	0
$191$	4268.06	1.61689	0.808444	−	0.588573i	$-0.200310\pi$
$191$	4268.06	1.61689	0.808444	+	0.588573i	$0.200310\pi$
$192$	0	0
$193$	−3518.04	−1.31209	−0.656047	−	0.754720i	$-0.727773\pi$
$193$	−3518.04	−1.31209	−0.656047	+	0.754720i	$0.727773\pi$
$194$	0	0
$195$	1542.93i	0.566621i
$196$	0	0
$197$	3490.01i	1.26220i	0.775702	+	0.631099i	$0.217396\pi$
$197$	3490.01i	1.26220i	−0.775702	+	0.631099i	$0.782604\pi$
$198$	0	0
$199$	−1731.09	−0.616652	−0.308326	−	0.951281i	$-0.599769\pi$
$199$	−1731.09	−0.616652	−0.308326	+	0.951281i	$0.599769\pi$
$200$	0	0
$201$	−1689.45	−0.592858
$202$	0	0
$203$	− 393.750i	− 0.136137i
$204$	0	0
$205$	− 862.308i	− 0.293786i
$206$	0	0
$207$	−1398.92	−0.469717
$208$	0	0
$209$	−7964.38	−2.63592
$210$	0	0
$211$	− 2021.06i	− 0.659411i	−0.944084	−	0.329706i	$-0.893050\pi$
$211$	− 2021.06i	− 0.659411i	0.944084	−	0.329706i	$-0.106950\pi$
$212$	0	0
$213$	848.851i	0.273063i
$214$	0	0
$215$	1843.72	0.584839
$216$	0	0
$217$	−1113.99	−0.348492
$218$	0	0
$219$	− 751.226i	− 0.231795i
$220$	0	0
$221$	− 6213.00i	− 1.89109i
$222$	0	0
$223$	4624.54	1.38871	0.694354	−	0.719634i	$-0.255690\pi$
$223$	4624.54	1.38871	0.694354	+	0.719634i	$0.255690\pi$
$224$	0	0
$225$	−773.462	−0.229174
$226$	0	0
$227$	− 4071.85i	− 1.19056i	−0.803517	−	0.595282i	$-0.797040\pi$
$227$	− 4071.85i	− 1.19056i	0.803517	−	0.595282i	$-0.202960\pi$
$228$	0	0
$229$	2670.03i	0.770484i	0.922816	+	0.385242i	$0.125882\pi$
$229$	2670.03i	0.770484i	−0.922816	+	0.385242i	$0.874118\pi$
$230$	0	0
$231$	1331.49	0.379244
$232$	0	0
$233$	6613.19	1.85942	0.929709	−	0.368295i	$-0.120058\pi$
$233$	6613.19	1.85942	0.929709	+	0.368295i	$0.120058\pi$
$234$	0	0
$235$	1166.73i	0.323868i
$236$	0	0
$237$	− 2844.10i	− 0.779511i
$238$	0	0
$239$	−301.154	−0.0815064	−0.0407532	−	0.999169i	$-0.512976\pi$
$239$	−301.154	−0.0815064	−0.0407532	+	0.999169i	$0.512976\pi$
$240$	0	0
$241$	−1311.73	−0.350605	−0.175303	−	0.984515i	$-0.556090\pi$
$241$	−1311.73	−0.350605	−0.175303	+	0.984515i	$0.556090\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	− 306.239i	− 0.0798567i
$246$	0	0
$247$	−10337.0	−2.66286
$248$	0	0
$249$	−3869.65	−0.984856
$250$	0	0
$251$	− 4893.64i	− 1.23061i	−0.788288	−	0.615307i	$-0.789032\pi$
$251$	− 4893.64i	− 1.23061i	0.788288	−	0.615307i	$-0.210968\pi$
$252$	0	0
$253$	− 9855.23i	− 2.44898i
$254$	0	0
$255$	−1415.56	−0.347630
$256$	0	0
$257$	2818.68	0.684141	0.342070	−	0.939674i	$-0.388872\pi$
$257$	2818.68	0.684141	0.342070	+	0.939674i	$0.388872\pi$
$258$	0	0
$259$	1380.95i	0.331306i
$260$	0	0
$261$	− 506.251i	− 0.120062i
$262$	0	0
$263$	−1460.94	−0.342529	−0.171265	−	0.985225i	$-0.554785\pi$
$263$	−1460.94	−0.342529	−0.171265	+	0.985225i	$0.554785\pi$
$264$	0	0
$265$	2557.47	0.592847
$266$	0	0
$267$	1966.10i	0.450650i
$268$	0	0
$269$	2399.05i	0.543765i	0.962330	+	0.271882i	$0.0876462\pi$
$269$	2399.05i	0.543765i	−0.962330	+	0.271882i	$0.912354\pi$
$270$	0	0
$271$	−221.818	−0.0497214	−0.0248607	−	0.999691i	$-0.507914\pi$
$271$	−221.818	−0.0497214	−0.0248607	+	0.999691i	$0.507914\pi$
$272$	0	0
$273$	1728.14	0.383120
$274$	0	0
$275$	− 5448.97i	− 1.19486i
$276$	0	0
$277$	8218.84i	1.78275i	0.453264	+	0.891376i	$0.350259\pi$
$277$	8218.84i	1.78275i	−0.453264	+	0.891376i	$0.649741\pi$
$278$	0	0
$279$	−1432.28	−0.307341
$280$	0	0
$281$	−4275.73	−0.907718	−0.453859	−	0.891073i	$-0.649953\pi$
$281$	−4275.73	−0.907718	−0.453859	+	0.891073i	$0.649953\pi$
$282$	0	0
$283$	1576.10i	0.331059i	0.986205	+	0.165530i	$0.0529333\pi$
$283$	1576.10i	0.331059i	−0.986205	+	0.165530i	$0.947067\pi$
$284$	0	0
$285$	2355.16i	0.489500i
$286$	0	0
$287$	−965.819	−0.198643
$288$	0	0
$289$	787.124	0.160213
$290$	0	0
$291$	− 2120.44i	− 0.427156i
$292$	0	0
$293$	5078.63i	1.01262i	0.862353	+	0.506308i	$0.168990\pi$
$293$	5078.63i	1.01262i	−0.862353	+	0.506308i	$0.831010\pi$
$294$	0	0
$295$	−1944.41	−0.383756
$296$	0	0
$297$	1711.91	0.334462
$298$	0	0
$299$	− 12791.1i	− 2.47401i
$300$	0	0
$301$	− 2065.04i	− 0.395438i
$302$	0	0
$303$	−2748.53	−0.521119
$304$	0	0
$305$	1054.97	0.198056
$306$	0	0
$307$	10138.3i	1.88476i	0.334540	+	0.942382i	$0.391419\pi$
$307$	10138.3i	1.88476i	−0.334540	+	0.942382i	$0.608581\pi$
$308$	0	0
$309$	4120.67i	0.758630i
$310$	0	0
$311$	−2488.16	−0.453667	−0.226833	−	0.973934i	$-0.572837\pi$
$311$	−2488.16	−0.453667	−0.226833	+	0.973934i	$0.572837\pi$
$312$	0	0
$313$	−2857.67	−0.516055	−0.258027	−	0.966138i	$-0.583072\pi$
$313$	−2857.67	−0.516055	−0.258027	+	0.966138i	$0.583072\pi$
$314$	0	0
$315$	− 393.736i	− 0.0704270i
$316$	0	0
$317$	7112.51i	1.26018i	0.776520	+	0.630092i	$0.216983\pi$
$317$	7112.51i	1.26018i	−0.776520	+	0.630092i	$0.783017\pi$
$318$	0	0
$319$	3566.49	0.625972
$320$	0	0
$321$	−4629.18	−0.804909
$322$	0	0
$323$	− 9483.67i	− 1.63370i
$324$	0	0
$325$	− 7072.22i	− 1.20707i
$326$	0	0
$327$	5394.25	0.912240
$328$	0	0
$329$	1306.78	0.218983
$330$	0	0
$331$	6184.60i	1.02700i	0.858090	+	0.513499i	$0.171651\pi$
$331$	6184.60i	1.02700i	−0.858090	+	0.513499i	$0.828349\pi$
$332$	0	0
$333$	1775.51i	0.292185i
$334$	0	0
$335$	−3519.56	−0.574012
$336$	0	0
$337$	−451.244	−0.0729402	−0.0364701	−	0.999335i	$-0.511611\pi$
$337$	−451.244	−0.0729402	−0.0364701	+	0.999335i	$0.511611\pi$
$338$	0	0
$339$	853.124i	0.136682i
$340$	0	0
$341$	− 10090.3i	− 1.60240i
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−2914.30	−0.454785
$346$	0	0
$347$	12493.5i	1.93282i	0.257006	+	0.966410i	$0.417264\pi$
$347$	12493.5i	1.93282i	−0.257006	+	0.966410i	$0.582736\pi$
$348$	0	0
$349$	11188.5i	1.71606i	0.513596	+	0.858032i	$0.328313\pi$
$349$	11188.5i	1.71606i	−0.513596	+	0.858032i	$0.671687\pi$
$350$	0	0
$351$	2221.89	0.337880
$352$	0	0
$353$	1060.98	0.159972	0.0799859	−	0.996796i	$-0.474512\pi$
$353$	1060.98	0.159972	0.0799859	+	0.996796i	$0.474512\pi$
$354$	0	0
$355$	1768.38i	0.264382i
$356$	0	0
$357$	1585.48i	0.235049i
$358$	0	0
$359$	−10700.8	−1.57317	−0.786585	−	0.617482i	$-0.788153\pi$
$359$	−10700.8	−1.57317	−0.786585	+	0.617482i	$0.788153\pi$
$360$	0	0
$361$	−8919.62	−1.30043
$362$	0	0
$363$	8067.26i	1.16645i
$364$	0	0
$365$	− 1565.00i	− 0.224427i
$366$	0	0
$367$	10121.9	1.43967	0.719833	−	0.694147i	$-0.244218\pi$
$367$	10121.9	1.43967	0.719833	+	0.694147i	$0.244218\pi$
$368$	0	0
$369$	−1241.77	−0.175187
$370$	0	0
$371$	− 2864.47i	− 0.400852i
$372$	0	0
$373$	− 2099.58i	− 0.291454i	−0.989325	−	0.145727i	$-0.953448\pi$
$373$	− 2099.58i	− 0.291454i	0.989325	−	0.145727i	$-0.0465521\pi$
$374$	0	0
$375$	−3954.99	−0.544626
$376$	0	0
$377$	4628.94	0.632368
$378$	0	0
$379$	10101.3i	1.36905i	0.728989	+	0.684526i	$0.239991\pi$
$379$	10101.3i	1.36905i	−0.728989	+	0.684526i	$0.760009\pi$
$380$	0	0
$381$	− 2527.54i	− 0.339868i
$382$	0	0
$383$	8403.09	1.12109	0.560546	−	0.828124i	$-0.310591\pi$
$383$	8403.09	1.12109	0.560546	+	0.828124i	$0.310591\pi$
$384$	0	0
$385$	2773.83	0.367189
$386$	0	0
$387$	− 2655.05i	− 0.348743i
$388$	0	0
$389$	411.307i	0.0536095i	0.999641	+	0.0268048i	$0.00853324\pi$
$389$	411.307i	0.0536095i	−0.999641	+	0.0268048i	$0.991467\pi$
$390$	0	0
$391$	11735.2	1.51784
$392$	0	0
$393$	7771.14	0.997461
$394$	0	0
$395$	− 5925.00i	− 0.754731i
$396$	0	0
$397$	− 10734.2i	− 1.35701i	−0.734595	−	0.678506i	$-0.762628\pi$
$397$	− 10734.2i	− 1.35701i	0.734595	−	0.678506i	$-0.237372\pi$
$398$	0	0
$399$	2637.87	0.330974
$400$	0	0
$401$	12415.7	1.54616	0.773080	−	0.634309i	$-0.218715\pi$
$401$	12415.7	1.54616	0.773080	+	0.634309i	$0.218715\pi$
$402$	0	0
$403$	− 13096.1i	− 1.61877i
$404$	0	0
$405$	− 506.232i	− 0.0621108i
$406$	0	0
$407$	−12508.3	−1.52338
$408$	0	0
$409$	2349.10	0.283999	0.141999	−	0.989867i	$-0.454647\pi$
$409$	2349.10	0.283999	0.141999	+	0.989867i	$0.454647\pi$
$410$	0	0
$411$	− 5753.44i	− 0.690502i
$412$	0	0
$413$	2177.82i	0.259475i
$414$	0	0
$415$	−8061.49	−0.953549
$416$	0	0
$417$	−7583.01	−0.890508
$418$	0	0
$419$	1665.84i	0.194228i	0.995273	+	0.0971138i	$0.0309611\pi$
$419$	1665.84i	0.194228i	−0.995273	+	0.0971138i	$0.969039\pi$
$420$	0	0
$421$	− 1154.48i	− 0.133648i	−0.997765	−	0.0668239i	$-0.978713\pi$
$421$	− 1154.48i	− 0.133648i	0.997765	−	0.0668239i	$-0.0212866\pi$
$422$	0	0
$423$	1680.15	0.193125
$424$	0	0
$425$	6488.42	0.740552
$426$	0	0
$427$	− 1181.60i	− 0.133915i
$428$	0	0
$429$	15653.0i	1.76162i
$430$	0	0
$431$	237.625	0.0265569	0.0132784	−	0.999912i	$-0.495773\pi$
$431$	237.625	0.0265569	0.0132784	+	0.999912i	$0.495773\pi$
$432$	0	0
$433$	4763.79	0.528714	0.264357	−	0.964425i	$-0.414840\pi$
$433$	4763.79	0.528714	0.264357	+	0.964425i	$0.414840\pi$
$434$	0	0
$435$	− 1054.65i	− 0.116245i
$436$	0	0
$437$	− 19524.7i	− 2.13728i
$438$	0	0
$439$	1715.57	0.186515	0.0932573	−	0.995642i	$-0.470272\pi$
$439$	1715.57	0.186515	0.0932573	+	0.995642i	$0.470272\pi$
$440$	0	0
$441$	−441.000	−0.0476190
$442$	0	0
$443$	− 6287.39i	− 0.674318i	−0.941448	−	0.337159i	$-0.890534\pi$
$443$	− 6287.39i	− 0.674318i	0.941448	−	0.337159i	$-0.109466\pi$
$444$	0	0
$445$	4095.90i	0.436325i
$446$	0	0
$447$	4021.86	0.425564
$448$	0	0
$449$	9104.58	0.956952	0.478476	−	0.878101i	$-0.341189\pi$
$449$	9104.58	0.956952	0.478476	+	0.878101i	$0.341189\pi$
$450$	0	0
$451$	− 8748.13i	− 0.913378i
$452$	0	0
$453$	− 2042.46i	− 0.211839i
$454$	0	0
$455$	3600.16	0.370941
$456$	0	0
$457$	9817.29	1.00489	0.502444	−	0.864610i	$-0.332434\pi$
$457$	9817.29	1.00489	0.502444	+	0.864610i	$0.332434\pi$
$458$	0	0
$459$	2038.48i	0.207294i
$460$	0	0
$461$	− 3975.83i	− 0.401676i	−0.979624	−	0.200838i	$-0.935633\pi$
$461$	− 3975.83i	− 0.401676i	0.979624	−	0.200838i	$-0.0643665\pi$
$462$	0	0
$463$	−7559.96	−0.758837	−0.379418	−	0.925225i	$-0.623876\pi$
$463$	−7559.96	−0.758837	−0.379418	+	0.925225i	$0.623876\pi$
$464$	0	0
$465$	−2983.81	−0.297571
$466$	0	0
$467$	524.945i	0.0520161i	0.999662	+	0.0260081i	$0.00827956\pi$
$467$	524.945i	0.0520161i	−0.999662	+	0.0260081i	$0.991720\pi$
$468$	0	0
$469$	3942.04i	0.388116i
$470$	0	0
$471$	9076.17	0.887915
$472$	0	0
$473$	18704.6	1.81826
$474$	0	0
$475$	− 10795.2i	− 1.04278i
$476$	0	0
$477$	− 3682.89i	− 0.353518i
$478$	0	0
$479$	78.5348	0.00749133	0.00374567	−	0.999993i	$-0.498808\pi$
$479$	78.5348	0.00749133	0.00374567	+	0.999993i	$0.498808\pi$
$480$	0	0
$481$	−16234.6	−1.53894
$482$	0	0
$483$	3264.14i	0.307502i
$484$	0	0
$485$	− 4417.42i	− 0.413577i
$486$	0	0
$487$	−14048.9	−1.30722	−0.653610	−	0.756832i	$-0.726746\pi$
$487$	−14048.9	−1.30722	−0.653610	+	0.756832i	$0.726746\pi$
$488$	0	0
$489$	1520.91	0.140650
$490$	0	0
$491$	14954.2i	1.37449i	0.726428	+	0.687243i	$0.241179\pi$
$491$	14954.2i	1.37449i	−0.726428	+	0.687243i	$0.758821\pi$
$492$	0	0
$493$	4246.83i	0.387967i
$494$	0	0
$495$	3566.36	0.323830
$496$	0	0
$497$	1980.65	0.178761
$498$	0	0
$499$	11687.5i	1.04851i	0.851562	+	0.524254i	$0.175656\pi$
$499$	11687.5i	1.04851i	−0.851562	+	0.524254i	$0.824344\pi$
$500$	0	0
$501$	− 730.294i	− 0.0651240i
$502$	0	0
$503$	−20229.8	−1.79324	−0.896620	−	0.442800i	$-0.853985\pi$
$503$	−20229.8	−1.79324	−0.896620	+	0.442800i	$0.853985\pi$
$504$	0	0
$505$	−5725.90	−0.504553
$506$	0	0
$507$	13725.0i	1.20227i
$508$	0	0
$509$	− 12175.4i	− 1.06025i	−0.847919	−	0.530125i	$-0.822145\pi$
$509$	− 12175.4i	− 1.06025i	0.847919	−	0.530125i	$-0.177855\pi$
$510$	0	0
$511$	−1752.86	−0.151746
$512$	0	0
$513$	3391.55	0.291892
$514$	0	0
$515$	8584.43i	0.734515i
$516$	0	0
$517$	11836.5i	1.00690i
$518$	0	0
$519$	221.898	0.0187674
$520$	0	0
$521$	−2133.49	−0.179405	−0.0897025	−	0.995969i	$-0.528592\pi$
$521$	−2133.49	−0.179405	−0.0897025	+	0.995969i	$0.528592\pi$
$522$	0	0
$523$	6979.20i	0.583516i	0.956492	+	0.291758i	$0.0942402\pi$
$523$	6979.20i	0.583516i	−0.956492	+	0.291758i	$0.905760\pi$
$524$	0	0
$525$	1804.75i	0.150030i
$526$	0	0
$527$	12015.1	0.993141
$528$	0	0
$529$	11993.1	0.985705
$530$	0	0
$531$	2800.05i	0.228836i
$532$	0	0
$533$	− 11354.2i	− 0.922711i
$534$	0	0
$535$	−9643.79	−0.779322
$536$	0	0
$537$	2780.64	0.223451
$538$	0	0
$539$	− 3106.80i	− 0.248274i
$540$	0	0
$541$	5909.97i	0.469666i	0.972036	+	0.234833i	$0.0754543\pi$
$541$	5909.97i	0.469666i	−0.972036	+	0.234833i	$0.924546\pi$
$542$	0	0
$543$	7847.05	0.620164
$544$	0	0
$545$	11237.6	0.883241
$546$	0	0
$547$	16572.1i	1.29538i	0.761906	+	0.647688i	$0.224264\pi$
$547$	16572.1i	1.29538i	−0.761906	+	0.647688i	$0.775736\pi$
$548$	0	0
$549$	− 1519.20i	− 0.118102i
$550$	0	0
$551$	7065.74	0.546299
$552$	0	0
$553$	−6636.23	−0.510310
$554$	0	0
$555$	3698.85i	0.282897i
$556$	0	0
$557$	− 9525.34i	− 0.724599i	−0.932062	−	0.362300i	$-0.881992\pi$
$557$	− 9525.34i	− 0.724599i	0.932062	−	0.362300i	$-0.118008\pi$
$558$	0	0
$559$	24276.6	1.83684
$560$	0	0
$561$	−14360.9	−1.08078
$562$	0	0
$563$	− 291.473i	− 0.0218190i	−0.999940	−	0.0109095i	$-0.996527\pi$
$563$	− 291.473i	− 0.0218190i	0.999940	−	0.0109095i	$-0.00347268\pi$
$564$	0	0
$565$	1777.28i	0.132337i
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	−852.491	−0.0628089	−0.0314045	−	0.999507i	$-0.509998\pi$
$569$	−852.491	−0.0628089	−0.0314045	+	0.999507i	$0.509998\pi$
$570$	0	0
$571$	13366.5i	0.979634i	0.871825	+	0.489817i	$0.162936\pi$
$571$	13366.5i	0.979634i	−0.871825	+	0.489817i	$0.837064\pi$
$572$	0	0
$573$	− 12804.2i	− 0.933511i
$574$	0	0
$575$	13358.1	0.968822
$576$	0	0
$577$	704.046	0.0507969	0.0253985	−	0.999677i	$-0.491915\pi$
$577$	704.046	0.0507969	0.0253985	+	0.999677i	$0.491915\pi$
$578$	0	0
$579$	10554.1i	0.757538i
$580$	0	0
$581$	9029.19i	0.644740i
$582$	0	0
$583$	25945.6	1.84315
$584$	0	0
$585$	4628.78	0.327139
$586$	0	0
$587$	− 18056.0i	− 1.26959i	−0.772679	−	0.634797i	$-0.781084\pi$
$587$	− 18056.0i	− 1.26959i	0.772679	−	0.634797i	$-0.218916\pi$
$588$	0	0
$589$	− 19990.3i	− 1.39845i
$590$	0	0
$591$	10470.0	0.728730
$592$	0	0
$593$	19109.4	1.32332	0.661660	−	0.749804i	$-0.269852\pi$
$593$	19109.4	1.32332	0.661660	+	0.749804i	$0.269852\pi$
$594$	0	0
$595$	3302.97i	0.227578i
$596$	0	0
$597$	5193.27i	0.356024i
$598$	0	0
$599$	−7474.66	−0.509861	−0.254930	−	0.966959i	$-0.582053\pi$
$599$	−7474.66	−0.509861	−0.254930	+	0.966959i	$0.582053\pi$
$600$	0	0
$601$	2875.13	0.195140	0.0975699	−	0.995229i	$-0.468893\pi$
$601$	2875.13	0.195140	0.0975699	+	0.995229i	$0.468893\pi$
$602$	0	0
$603$	5068.34i	0.342287i
$604$	0	0
$605$	16806.2i	1.12937i
$606$	0	0
$607$	24346.2	1.62797	0.813987	−	0.580883i	$-0.197293\pi$
$607$	24346.2	1.62797	0.813987	+	0.580883i	$0.197293\pi$
$608$	0	0
$609$	−1181.25	−0.0785989
$610$	0	0
$611$	15362.6i	1.01719i
$612$	0	0
$613$	− 19444.9i	− 1.28120i	−0.767877	−	0.640598i	$-0.778687\pi$
$613$	− 19444.9i	− 1.28120i	0.767877	−	0.640598i	$-0.221313\pi$
$614$	0	0
$615$	−2586.92	−0.169618
$616$	0	0
$617$	−9935.81	−0.648299	−0.324150	−	0.946006i	$-0.605078\pi$
$617$	−9935.81	−0.648299	−0.324150	+	0.946006i	$0.605078\pi$
$618$	0	0
$619$	19396.4i	1.25946i	0.776813	+	0.629731i	$0.216835\pi$
$619$	19396.4i	1.25946i	−0.776813	+	0.629731i	$0.783165\pi$
$620$	0	0
$621$	4196.75i	0.271191i
$622$	0	0
$623$	4587.58	0.295020
$624$	0	0
$625$	2503.26	0.160209
$626$	0	0
$627$	23893.2i	1.52185i
$628$	0	0
$629$	− 14894.4i	− 0.944164i
$630$	0	0
$631$	1391.17	0.0877681	0.0438840	−	0.999037i	$-0.486027\pi$
$631$	1391.17	0.0877681	0.0438840	+	0.999037i	$0.486027\pi$
$632$	0	0
$633$	−6063.19	−0.380711
$634$	0	0
$635$	− 5265.53i	− 0.329065i
$636$	0	0
$637$	− 4032.32i	− 0.250811i
$638$	0	0
$639$	2546.55	0.157653
$640$	0	0
$641$	17848.0	1.09977	0.549886	−	0.835240i	$-0.314671\pi$
$641$	17848.0	1.09977	0.549886	+	0.835240i	$0.314671\pi$
$642$	0	0
$643$	− 4152.29i	− 0.254666i	−0.991860	−	0.127333i	$-0.959358\pi$
$643$	− 4152.29i	− 0.254666i	0.991860	−	0.127333i	$-0.0406417\pi$
$644$	0	0
$645$	− 5531.15i	− 0.337657i
$646$	0	0
$647$	−8018.93	−0.487259	−0.243630	−	0.969868i	$-0.578338\pi$
$647$	−8018.93	−0.487259	−0.243630	+	0.969868i	$0.578338\pi$
$648$	0	0
$649$	−19726.1	−1.19309
$650$	0	0
$651$	3341.98i	0.201202i
$652$	0	0
$653$	− 639.444i	− 0.0383206i	−0.999816	−	0.0191603i	$-0.993901\pi$
$653$	− 639.444i	− 0.0383206i	0.999816	−	0.0191603i	$-0.00609929\pi$
$654$	0	0
$655$	16189.3	0.965753
$656$	0	0
$657$	−2253.68	−0.133827
$658$	0	0
$659$	− 24156.4i	− 1.42792i	−0.700187	−	0.713960i	$-0.746900\pi$
$659$	− 24156.4i	− 1.42792i	0.700187	−	0.713960i	$-0.253100\pi$
$660$	0	0
$661$	8195.25i	0.482236i	0.970496	+	0.241118i	$0.0775141\pi$
$661$	8195.25i	0.482236i	−0.970496	+	0.241118i	$0.922486\pi$
$662$	0	0
$663$	−18639.0	−1.09182
$664$	0	0
$665$	5495.37	0.320453
$666$	0	0
$667$	8743.23i	0.507555i
$668$	0	0
$669$	− 13873.6i	− 0.801771i
$670$	0	0
$671$	10702.7	0.615755
$672$	0	0
$673$	−26794.9	−1.53472	−0.767360	−	0.641217i	$-0.778430\pi$
$673$	−26794.9	−1.53472	−0.767360	+	0.641217i	$0.778430\pi$
$674$	0	0
$675$	2320.39i	0.132314i
$676$	0	0
$677$	24832.7i	1.40975i	0.709333	+	0.704874i	$0.248996\pi$
$677$	24832.7i	1.40975i	−0.709333	+	0.704874i	$0.751004\pi$
$678$	0	0
$679$	−4947.69	−0.279639
$680$	0	0
$681$	−12215.5	−0.687373
$682$	0	0
$683$	− 22982.8i	− 1.28758i	−0.765204	−	0.643788i	$-0.777362\pi$
$683$	− 22982.8i	− 1.28758i	0.765204	−	0.643788i	$-0.222638\pi$
$684$	0	0
$685$	− 11985.9i	− 0.668552i
$686$	0	0
$687$	8010.10	0.444839
$688$	0	0
$689$	33674.8	1.86199
$690$	0	0
$691$	2558.77i	0.140868i	0.997516	+	0.0704342i	$0.0224385\pi$
$691$	2558.77i	0.140868i	−0.997516	+	0.0704342i	$0.977562\pi$
$692$	0	0
$693$	− 3994.46i	− 0.218957i
$694$	0	0
$695$	−15797.4	−0.862200
$696$	0	0
$697$	10416.9	0.566097
$698$	0	0
$699$	− 19839.6i	− 1.07354i
$700$	0	0
$701$	− 33784.3i	− 1.82028i	−0.414302	−	0.910140i	$-0.635974\pi$
$701$	− 33784.3i	− 1.82028i	0.414302	−	0.910140i	$-0.364026\pi$
$702$	0	0
$703$	−24780.8	−1.32948
$704$	0	0
$705$	3500.19	0.186985
$706$	0	0
$707$	6413.24i	0.341152i
$708$	0	0
$709$	3291.19i	0.174334i	0.996194	+	0.0871672i	$0.0277814\pi$
$709$	3291.19i	0.174334i	−0.996194	+	0.0871672i	$0.972219\pi$
$710$	0	0
$711$	−8532.30	−0.450051
$712$	0	0
$713$	24736.2	1.29927
$714$	0	0
$715$	32609.3i	1.70562i
$716$	0	0
$717$	903.462i	0.0470577i
$718$	0	0
$719$	19459.7	1.00935	0.504676	−	0.863309i	$-0.331612\pi$
$719$	19459.7	1.00935	0.504676	+	0.863309i	$0.331612\pi$
$720$	0	0
$721$	9614.90	0.496640
$722$	0	0
$723$	3935.19i	0.202422i
$724$	0	0
$725$	4834.15i	0.247635i
$726$	0	0
$727$	15508.8	0.791184	0.395592	−	0.918426i	$-0.370539\pi$
$727$	15508.8	0.791184	0.395592	+	0.918426i	$0.370539\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	22272.6i	1.12693i
$732$	0	0
$733$	26507.3i	1.33570i	0.744295	+	0.667851i	$0.232786\pi$
$733$	26507.3i	1.33570i	−0.744295	+	0.667851i	$0.767214\pi$
$734$	0	0
$735$	−918.717	−0.0461053
$736$	0	0
$737$	−35706.0	−1.78460
$738$	0	0
$739$	13802.6i	0.687057i	0.939142	+	0.343529i	$0.111622\pi$
$739$	13802.6i	0.687057i	−0.939142	+	0.343529i	$0.888378\pi$
$740$	0	0
$741$	31010.9i	1.53740i
$742$	0	0
$743$	−22406.2	−1.10633	−0.553165	−	0.833072i	$-0.686580\pi$
$743$	−22406.2	−1.10633	−0.553165	+	0.833072i	$0.686580\pi$
$744$	0	0
$745$	8378.57	0.412036
$746$	0	0
$747$	11609.0i	0.568607i
$748$	0	0
$749$	10801.4i	0.526937i
$750$	0	0
$751$	27470.5	1.33477	0.667384	−	0.744713i	$-0.267414\pi$
$751$	27470.5	1.33477	0.667384	+	0.744713i	$0.267414\pi$
$752$	0	0
$753$	−14680.9	−0.710495
$754$	0	0
$755$	− 4254.98i	− 0.205105i
$756$	0	0
$757$	10421.8i	0.500379i	0.968197	+	0.250189i	$0.0804929\pi$
$757$	10421.8i	0.500379i	−0.968197	+	0.250189i	$0.919507\pi$
$758$	0	0
$759$	−29565.7	−1.41392
$760$	0	0
$761$	32815.4	1.56315	0.781576	−	0.623810i	$-0.214416\pi$
$761$	32815.4	1.56315	0.781576	+	0.623810i	$0.214416\pi$
$762$	0	0
$763$	− 12586.6i	− 0.597201i
$764$	0	0
$765$	4246.68i	0.200705i
$766$	0	0
$767$	−25602.5	−1.20528
$768$	0	0
$769$	32970.9	1.54611	0.773057	−	0.634336i	$-0.218727\pi$
$769$	32970.9	1.54611	0.773057	+	0.634336i	$0.218727\pi$
$770$	0	0
$771$	− 8456.03i	− 0.394989i
$772$	0	0
$773$	− 19379.4i	− 0.901720i	−0.892595	−	0.450860i	$-0.851117\pi$
$773$	− 19379.4i	− 0.901720i	0.892595	−	0.450860i	$-0.148883\pi$
$774$	0	0
$775$	13676.7	0.633912
$776$	0	0
$777$	4142.86	0.191280
$778$	0	0
$779$	− 17331.3i	− 0.797124i
$780$	0	0
$781$	17940.2i	0.821961i
$782$	0	0
$783$	−1518.75	−0.0693177
$784$	0	0
$785$	18908.0	0.859689
$786$	0	0
$787$	39440.8i	1.78642i	0.449640	+	0.893210i	$0.351552\pi$
$787$	39440.8i	1.78642i	−0.449640	+	0.893210i	$0.648448\pi$
$788$	0	0
$789$	4382.81i	0.197759i
$790$	0	0
$791$	1990.62	0.0894797
$792$	0	0
$793$	13891.0	0.622047
$794$	0	0
$795$	− 7672.42i	− 0.342280i
$796$	0	0
$797$	− 1593.15i	− 0.0708058i	−0.999373	−	0.0354029i	$-0.988729\pi$
$797$	− 1593.15i	− 0.0708058i	0.999373	−	0.0354029i	$-0.0112714\pi$
$798$	0	0
$799$	−14094.4	−0.624062
$800$	0	0
$801$	5898.31	0.260183
$802$	0	0
$803$	− 15876.9i	− 0.697740i
$804$	0	0
$805$	6800.04i	0.297727i
$806$	0	0
$807$	7197.15	0.313943
$808$	0	0
$809$	−24472.3	−1.06354	−0.531768	−	0.846890i	$-0.678472\pi$
$809$	−24472.3	−1.06354	−0.531768	+	0.846890i	$0.678472\pi$
$810$	0	0
$811$	− 12522.8i	− 0.542215i	−0.962549	−	0.271107i	$-0.912610\pi$
$811$	− 12522.8i	− 0.542215i	0.962549	−	0.271107i	$-0.0873898\pi$
$812$	0	0
$813$	665.455i	0.0287067i
$814$	0	0
$815$	3168.45	0.136179
$816$	0	0
$817$	37056.5	1.58683
$818$	0	0
$819$	− 5184.41i	− 0.221194i
$820$	0	0
$821$	− 42179.5i	− 1.79303i	−0.443016	−	0.896514i	$-0.646091\pi$
$821$	− 42179.5i	− 1.79303i	0.443016	−	0.896514i	$-0.353909\pi$
$822$	0	0
$823$	−19287.4	−0.816909	−0.408454	−	0.912779i	$-0.633932\pi$
$823$	−19287.4	−0.816909	−0.408454	+	0.912779i	$0.633932\pi$
$824$	0	0
$825$	−16346.9	−0.689850
$826$	0	0
$827$	25707.3i	1.08093i	0.841367	+	0.540465i	$0.181752\pi$
$827$	25707.3i	1.08093i	−0.841367	+	0.540465i	$0.818248\pi$
$828$	0	0
$829$	23979.6i	1.00464i	0.864682	+	0.502320i	$0.167520\pi$
$829$	23979.6i	1.00464i	−0.864682	+	0.502320i	$0.832480\pi$
$830$	0	0
$831$	24656.5	1.02927
$832$	0	0
$833$	3699.46	0.153876
$834$	0	0
$835$	− 1521.39i	− 0.0630539i
$836$	0	0
$837$	4296.83i	0.177444i
$838$	0	0
$839$	17884.0	0.735907	0.367953	−	0.929844i	$-0.380059\pi$
$839$	17884.0	0.735907	0.367953	+	0.929844i	$0.380059\pi$
$840$	0	0
$841$	21224.9	0.870267
$842$	0	0
$843$	12827.2i	0.524071i
$844$	0	0
$845$	28592.8i	1.16405i
$846$	0	0
$847$	18823.6	0.763621
$848$	0	0
$849$	4728.31	0.191137
$850$	0	0
$851$	− 30664.1i	− 1.23520i
$852$	0	0
$853$	− 38267.4i	− 1.53605i	−0.640421	−	0.768024i	$-0.721240\pi$
$853$	− 38267.4i	− 1.53605i	0.640421	−	0.768024i	$-0.278760\pi$
$854$	0	0
$855$	7065.48	0.282613
$856$	0	0
$857$	−15924.9	−0.634753	−0.317376	−	0.948300i	$-0.602802\pi$
$857$	−15924.9	−0.634753	−0.317376	+	0.948300i	$0.602802\pi$
$858$	0	0
$859$	− 5525.23i	− 0.219462i	−0.993961	−	0.109731i	$-0.965001\pi$
$859$	− 5525.23i	− 0.219462i	0.993961	−	0.109731i	$-0.0349990\pi$
$860$	0	0
$861$	2897.46i	0.114686i
$862$	0	0
$863$	6293.58	0.248245	0.124123	−	0.992267i	$-0.460388\pi$
$863$	6293.58	0.248245	0.124123	+	0.992267i	$0.460388\pi$
$864$	0	0
$865$	462.272	0.0181708
$866$	0	0
$867$	− 2361.37i	− 0.0924988i
$868$	0	0
$869$	− 60109.2i	− 2.34645i
$870$	0	0
$871$	−46342.8	−1.80283
$872$	0	0
$873$	−6361.32	−0.246619
$874$	0	0
$875$	9228.31i	0.356541i
$876$	0	0
$877$	− 20722.2i	− 0.797879i	−0.916977	−	0.398940i	$-0.869378\pi$
$877$	− 20722.2i	− 0.797879i	0.916977	−	0.398940i	$-0.130622\pi$
$878$	0	0
$879$	15235.9	0.584634
$880$	0	0
$881$	776.203	0.0296832	0.0148416	−	0.999890i	$-0.495276\pi$
$881$	776.203	0.0296832	0.0148416	+	0.999890i	$0.495276\pi$
$882$	0	0
$883$	− 28411.0i	− 1.08279i	−0.840767	−	0.541397i	$-0.817896\pi$
$883$	− 28411.0i	− 1.08279i	0.840767	−	0.541397i	$-0.182104\pi$
$884$	0	0
$885$	5833.23i	0.221561i
$886$	0	0
$887$	−8007.06	−0.303101	−0.151551	−	0.988449i	$-0.548427\pi$
$887$	−8007.06	−0.303101	−0.151551	+	0.988449i	$0.548427\pi$
$888$	0	0
$889$	−5897.60	−0.222496
$890$	0	0
$891$	− 5135.74i	− 0.193102i
$892$	0	0
$893$	23449.9i	0.878745i
$894$	0	0
$895$	5792.79	0.216348
$896$	0	0
$897$	−38373.3	−1.42837
$898$	0	0
$899$	8951.74i	0.332099i
$900$	0	0
$901$	30895.0i	1.14236i
$902$	0	0
$903$	−6195.11	−0.228306
$904$	0	0
$905$	16347.4	0.600450
$906$	0	0
$907$	5330.38i	0.195140i	0.995229	+	0.0975701i	$0.0311070\pi$
$907$	5330.38i	0.195140i	−0.995229	+	0.0975701i	$0.968893\pi$
$908$	0	0
$909$	8245.59i	0.300868i
$910$	0	0
$911$	−43468.8	−1.58089	−0.790443	−	0.612536i	$-0.790150\pi$
$911$	−43468.8	−1.58089	−0.790443	+	0.612536i	$0.790150\pi$
$912$	0	0
$913$	−81784.0	−2.96457
$914$	0	0
$915$	− 3164.90i	− 0.114348i
$916$	0	0
$917$	− 18132.7i	− 0.652991i
$918$	0	0
$919$	35514.2	1.27476	0.637381	−	0.770549i	$-0.280018\pi$
$919$	35514.2	1.27476	0.637381	+	0.770549i	$0.280018\pi$
$920$	0	0
$921$	30414.8	1.08817
$922$	0	0
$923$	23284.6i	0.830361i
$924$	0	0
$925$	− 16954.2i	− 0.602651i
$926$	0	0
$927$	12362.0	0.437995
$928$	0	0
$929$	−38350.8	−1.35441	−0.677206	−	0.735793i	$-0.736810\pi$
$929$	−38350.8	−1.35441	−0.677206	+	0.735793i	$0.736810\pi$
$930$	0	0
$931$	− 6155.04i	− 0.216674i
$932$	0	0
$933$	7464.47i	0.261925i
$934$	0	0
$935$	−29917.4	−1.04642
$936$	0	0
$937$	−39488.4	−1.37676	−0.688382	−	0.725348i	$-0.741679\pi$
$937$	−39488.4	−1.37676	−0.688382	+	0.725348i	$0.741679\pi$
$938$	0	0
$939$	8573.01i	0.297944i
$940$	0	0
$941$	− 46022.8i	− 1.59437i	−0.603737	−	0.797184i	$-0.706322\pi$
$941$	− 46022.8i	− 1.59437i	0.603737	−	0.797184i	$-0.293678\pi$
$942$	0	0
$943$	21446.0	0.740592
$944$	0	0
$945$	−1181.21	−0.0406611
$946$	0	0
$947$	− 34780.0i	− 1.19345i	−0.802446	−	0.596725i	$-0.796468\pi$
$947$	− 34780.0i	− 1.19345i	0.802446	−	0.596725i	$-0.203532\pi$
$948$	0	0
$949$	− 20606.7i	− 0.704870i
$950$	0	0
$951$	21337.5	0.727568
$952$	0	0
$953$	−1554.63	−0.0528431	−0.0264215	−	0.999651i	$-0.508411\pi$
$953$	−1554.63	−0.0528431	−0.0264215	+	0.999651i	$0.508411\pi$
$954$	0	0
$955$	− 26674.4i	− 0.903836i
$956$	0	0
$957$	− 10699.5i	− 0.361405i
$958$	0	0
$959$	−13424.7	−0.452040
$960$	0	0
$961$	−4464.85	−0.149872
$962$	0	0
$963$	13887.5i	0.464714i
$964$	0	0
$965$	21987.0i	0.733457i
$966$	0	0
$967$	−32939.5	−1.09541	−0.547706	−	0.836671i	$-0.684499\pi$
$967$	−32939.5	−1.09541	−0.547706	+	0.836671i	$0.684499\pi$
$968$	0	0
$969$	−28451.0	−0.943218
$970$	0	0
$971$	− 10059.9i	− 0.332480i	−0.986085	−	0.166240i	$-0.946837\pi$
$971$	− 10059.9i	− 0.332480i	0.986085	−	0.166240i	$-0.0531627\pi$
$972$	0	0
$973$	17693.7i	0.582974i
$974$	0	0
$975$	−21216.7	−0.696899
$976$	0	0
$977$	−25682.5	−0.840999	−0.420500	−	0.907293i	$-0.638145\pi$
$977$	−25682.5	−0.840999	−0.420500	+	0.907293i	$0.638145\pi$
$978$	0	0
$979$	41553.1i	1.35653i
$980$	0	0
$981$	− 16182.7i	− 0.526682i
$982$	0	0
$983$	35827.0	1.16246	0.581232	−	0.813738i	$-0.302571\pi$
$983$	35827.0	1.16246	0.581232	+	0.813738i	$0.302571\pi$
$984$	0	0
$985$	21811.8	0.705565
$986$	0	0
$987$	− 3920.35i	− 0.126430i
$988$	0	0
$989$	45854.2i	1.47429i
$990$	0	0
$991$	−28383.9	−0.909832	−0.454916	−	0.890534i	$-0.650331\pi$
$991$	−28383.9	−0.909832	−0.454916	+	0.890534i	$0.650331\pi$
$992$	0	0
$993$	18553.8	0.592938
$994$	0	0
$995$	10818.9i	0.344707i
$996$	0	0
$997$	− 22391.1i	− 0.711268i	−0.934625	−	0.355634i	$-0.884265\pi$
$997$	− 22391.1i	− 0.711268i	0.934625	−	0.355634i	$-0.115735\pi$
$998$	0	0
$999$	5326.54	0.168693

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.c.f.673.2	✓	12
4.3	odd	2		1344.4.c.g.673.8	yes	12
8.3	odd	2		1344.4.c.g.673.5	yes	12
8.5	even	2	inner	1344.4.c.f.673.11	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1344.4.c.f.673.2	✓	12	1.1	even	1	trivial
1344.4.c.f.673.11	yes	12	8.5	even	2	inner
1344.4.c.g.673.5	yes	12	8.3	odd	2
1344.4.c.g.673.8	yes	12	4.3	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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -6.24978i q^{5} -7.00000 q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} -6.24978i q^{5} -7.00000 q^{7} -9.00000 q^{9} -63.4042i q^{11} -82.2923i q^{13} -18.7493 q^{15} +75.4992 q^{17} -125.613i q^{19} +21.0000i q^{21} +155.435 q^{23} +85.9403 q^{25} +27.0000i q^{27} +56.2501i q^{29} +159.142 q^{31} -190.212 q^{33} +43.7484i q^{35} -197.279i q^{37} -246.877 q^{39} +137.974 q^{41} +295.005i q^{43} +56.2480i q^{45} -186.683 q^{47} +49.0000 q^{49} -226.498i q^{51} +409.210i q^{53} -396.262 q^{55} -376.839 q^{57} -311.117i q^{59} +168.801i q^{61} +63.0000 q^{63} -514.308 q^{65} -563.149i q^{67} -466.305i q^{69} -282.950 q^{71} +250.409 q^{73} -257.821i q^{75} +443.829i q^{77} +948.033 q^{79} +81.0000 q^{81} -1289.88i q^{83} -471.853i q^{85} +168.750 q^{87} -655.368 q^{89} +576.046i q^{91} -477.426i q^{93} -785.053 q^{95} +706.813 q^{97} +570.637i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 84 q^{7} - 108 q^{9}+O(q^{10}) \) 12 * q - 84 * q^7 - 108 * q^9	\( 12 q - 84 q^{7} - 108 q^{9} + 24 q^{15} + 376 q^{17} - 336 q^{23} - 180 q^{25} + 192 q^{31} - 168 q^{33} - 504 q^{39} + 488 q^{41} - 448 q^{47} + 588 q^{49} - 3600 q^{55} - 432 q^{57} + 756 q^{63} + 1408 q^{65} - 5104 q^{71} - 1752 q^{73} - 1632 q^{79} + 972 q^{81} - 336 q^{87} - 3688 q^{89} - 2496 q^{95} - 1944 q^{97}+O(q^{100}) \) 12 * q - 84 * q^7 - 108 * q^9 + 24 * q^15 + 376 * q^17 - 336 * q^23 - 180 * q^25 + 192 * q^31 - 168 * q^33 - 504 * q^39 + 488 * q^41 - 448 * q^47 + 588 * q^49 - 3600 * q^55 - 432 * q^57 + 756 * q^63 + 1408 * q^65 - 5104 * q^71 - 1752 * q^73 - 1632 * q^79 + 972 * q^81 - 336 * q^87 - 3688 * q^89 - 2496 * q^95 - 1944 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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