LMFDB - Embedded newform 4368.2.h.s.337.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(337,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4368.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.h (of order $2$, degree $1$, not 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:	$34.8786556029$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 21x^{8} + 124x^{6} + 212x^{4} + 116x^{2} + 16 $ x^10 + 21x^8 + 124x^6 + 212x^4 + 116x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 2184)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.8
Root			$-1.21919i$ of defining polynomial
Character	$\chi$	$=$	4368.337
Dual form			4368.2.h.s.337.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.64044i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.64044i q^{5} -1.00000i q^{7} +1.00000 q^{9} -0.797934i q^{11} +(-3.39317 - 1.21919i) q^{13} -1.64044i q^{15} -6.81442 q^{17} -3.89776i q^{19} +1.00000i q^{21} +3.97192 q^{23} +2.30896 q^{25} -1.00000 q^{27} -4.81442 q^{29} +0.797934i q^{33} +1.64044 q^{35} +0.466457i q^{37} +(3.39317 + 1.21919i) q^{39} +6.05073i q^{41} +11.1786 q^{43} +1.64044i q^{45} +4.01649i q^{47} -1.00000 q^{49} +6.81442 q^{51} +6.36422 q^{53} +1.30896 q^{55} +3.89776i q^{57} -0.585184i q^{59} +3.45939 q^{61} -1.00000i q^{63} +(2.00000 - 5.56628i) q^{65} +0.150425i q^{67} -3.97192 q^{69} +11.0437i q^{71} +10.3361i q^{73} -2.30896 q^{75} -0.797934 q^{77} -1.41120 q^{79} +1.00000 q^{81} -5.76278i q^{83} -11.1786i q^{85} +4.81442 q^{87} -4.51103i q^{89} +(-1.21919 + 3.39317i) q^{91} +6.39403 q^{95} +8.06721i q^{97} -0.797934i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{3} + 10 q^{9}+O(q^{10}) $ 10 * q - 10 * q^3 + 10 * q^9	$ 10 q - 10 q^{3} + 10 q^{9} - 22 q^{17} + 18 q^{23} - 8 q^{25} - 10 q^{27} - 2 q^{29} - 10 q^{35} + 2 q^{43} - 10 q^{49} + 22 q^{51} - 18 q^{55} + 6 q^{61} + 20 q^{65} - 18 q^{69} + 8 q^{75} - 6 q^{77} - 40 q^{79} + 10 q^{81} + 2 q^{87} + 2 q^{91} + 38 q^{95}+O(q^{100}) $ 10 * q - 10 * q^3 + 10 * q^9 - 22 * q^17 + 18 * q^23 - 8 * q^25 - 10 * q^27 - 2 * q^29 - 10 * q^35 + 2 * q^43 - 10 * q^49 + 22 * q^51 - 18 * q^55 + 6 * q^61 + 20 * q^65 - 18 * q^69 + 8 * q^75 - 6 * q^77 - 40 * q^79 + 10 * q^81 + 2 * q^87 + 2 * q^91 + 38 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4368\mathbb{Z}\right)^\times$.

$n$	$1093$	$1249$	$1457$	$2017$	$3823$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

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
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$			1.64044i			0.733626i	0.930295	+	0.366813i	$0.119551\pi$
$5$			1.64044i			0.733626i	−0.930295	+	0.366813i	$0.880449\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	0.797934i		−	0.240586i	−0.992738	−	0.120293i	$-0.961617\pi$
$11$		−	0.797934i		−	0.240586i	0.992738	−	0.120293i	$-0.0383834\pi$
$12$		0			0
$13$	−3.39317	−	1.21919i	−0.941095	−	0.338141i
$13$	−3.39317	−	1.21919i	−0.941095	−	0.338141i
$14$		0			0
$15$		−	1.64044i		−	0.423559i
$16$		0			0
$17$	−6.81442			−1.65274			−0.826370	−	0.563128i	$-0.809598\pi$
$17$	−6.81442			−1.65274			−0.826370	+	0.563128i	$0.809598\pi$
$18$		0			0
$19$		−	3.89776i		−	0.894207i	−0.894482	−	0.447104i	$-0.852456\pi$
$19$		−	3.89776i		−	0.894207i	0.894482	−	0.447104i	$-0.147544\pi$
$20$		0			0
$21$			1.00000i			0.218218i
$22$		0			0
$23$	3.97192			0.828202			0.414101	−	0.910231i	$-0.364096\pi$
$23$	3.97192			0.828202			0.414101	+	0.910231i	$0.364096\pi$
$24$		0			0
$25$	2.30896			0.461792
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	−4.81442			−0.894015			−0.447008	−	0.894530i	$-0.647510\pi$
$29$	−4.81442			−0.894015			−0.447008	+	0.894530i	$0.647510\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$			0.797934i			0.138902i
$34$		0			0
$35$	1.64044			0.277285
$36$		0			0
$37$			0.466457i			0.0766851i	0.999265	+	0.0383426i	$0.0122078\pi$
$37$			0.466457i			0.0766851i	−0.999265	+	0.0383426i	$0.987792\pi$
$38$		0			0
$39$	3.39317	+	1.21919i	0.543342	+	0.195226i
$40$		0			0
$41$			6.05073i			0.944965i	0.881340	+	0.472482i	$0.156642\pi$
$41$			6.05073i			0.944965i	−0.881340	+	0.472482i	$0.843358\pi$
$42$		0			0
$43$	11.1786			1.70473			0.852363	−	0.522951i	$-0.175169\pi$
$43$	11.1786			1.70473			0.852363	+	0.522951i	$0.175169\pi$
$44$		0			0
$45$			1.64044i			0.244542i
$46$		0			0
$47$			4.01649i			0.585865i	0.956133	+	0.292932i	$0.0946311\pi$
$47$			4.01649i			0.585865i	−0.956133	+	0.292932i	$0.905369\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	6.81442			0.954210
$52$		0			0
$53$	6.36422			0.874192			0.437096	−	0.899415i	$-0.356007\pi$
$53$	6.36422			0.874192			0.437096	+	0.899415i	$0.356007\pi$
$54$		0			0
$55$	1.30896			0.176500
$56$		0			0
$57$			3.89776i			0.516271i
$58$		0			0
$59$		−	0.585184i		−	0.0761845i	−0.999274	−	0.0380922i	$-0.987872\pi$
$59$		−	0.585184i		−	0.0761845i	0.999274	−	0.0380922i	$-0.0121281\pi$
$60$		0			0
$61$	3.45939			0.442929			0.221465	−	0.975168i	$-0.428916\pi$
$61$	3.45939			0.442929			0.221465	+	0.975168i	$0.428916\pi$
$62$		0			0
$63$		−	1.00000i		−	0.125988i
$64$		0			0
$65$	2.00000	−	5.56628i	0.248069	−	0.690412i
$66$		0			0
$67$			0.150425i			0.0183773i	0.999958	+	0.00918865i	$0.00292488\pi$
$67$			0.150425i			0.0183773i	−0.999958	+	0.00918865i	$0.997075\pi$
$68$		0			0
$69$	−3.97192			−0.478162
$70$		0			0
$71$			11.0437i			1.31064i	0.755351	+	0.655320i	$0.227466\pi$
$71$			11.0437i			1.31064i	−0.755351	+	0.655320i	$0.772534\pi$
$72$		0			0
$73$			10.3361i			1.20975i	0.796319	+	0.604876i	$0.206777\pi$
$73$			10.3361i			1.20975i	−0.796319	+	0.604876i	$0.793223\pi$
$74$		0			0
$75$	−2.30896			−0.266616
$76$		0			0
$77$	−0.797934			−0.0909330
$78$		0			0
$79$	−1.41120			−0.158773			−0.0793864	−	0.996844i	$-0.525296\pi$
$79$	−1.41120			−0.158773			−0.0793864	+	0.996844i	$0.525296\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	5.76278i		−	0.632547i	−0.948668	−	0.316274i	$-0.897568\pi$
$83$		−	5.76278i		−	0.632547i	0.948668	−	0.316274i	$-0.102432\pi$
$84$		0			0
$85$		−	11.1786i		−	1.21249i
$86$		0			0
$87$	4.81442			0.516160
$88$		0			0
$89$		−	4.51103i		−	0.478168i	−0.970999	−	0.239084i	$-0.923153\pi$
$89$		−	4.51103i		−	0.478168i	0.970999	−	0.239084i	$-0.0768471\pi$
$90$		0			0
$91$	−1.21919	+	3.39317i	−0.127805	+	0.355701i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	6.39403			0.656014
$96$		0			0
$97$			8.06721i			0.819101i	0.912287	+	0.409551i	$0.134314\pi$
$97$			8.06721i			0.819101i	−0.912287	+	0.409551i	$0.865686\pi$
$98$		0			0
$99$		−	0.797934i		−	0.0801954i
$100$		0			0
$101$	−9.11631			−0.907107			−0.453553	−	0.891229i	$-0.649844\pi$
$101$	−9.11631			−0.907107			−0.453553	+	0.891229i	$0.649844\pi$
$102$		0			0
$103$	1.82149			0.179477			0.0897384	−	0.995965i	$-0.471397\pi$
$103$	1.82149			0.179477			0.0897384	+	0.995965i	$0.471397\pi$
$104$		0			0
$105$	−1.64044			−0.160090
$106$		0			0
$107$	6.78633			0.656060			0.328030	−	0.944667i	$-0.393615\pi$
$107$	6.78633			0.656060			0.328030	+	0.944667i	$0.393615\pi$
$108$		0			0
$109$		−	4.41029i		−	0.422429i	−0.977440	−	0.211214i	$-0.932258\pi$
$109$		−	4.41029i		−	0.422429i	0.977440	−	0.211214i	$-0.0677419\pi$
$110$		0			0
$111$		−	0.466457i		−	0.0442742i
$112$		0			0
$113$	−13.7393			−1.29249			−0.646245	−	0.763130i	$-0.723661\pi$
$113$	−13.7393			−1.29249			−0.646245	+	0.763130i	$0.723661\pi$
$114$		0			0
$115$			6.51568i			0.607591i
$116$		0			0
$117$	−3.39317	−	1.21919i	−0.313698	−	0.112714i
$118$		0			0
$119$			6.81442i			0.624677i
$120$		0			0
$121$	10.3633			0.942118
$122$		0			0
$123$		−	6.05073i		−	0.545576i
$124$		0			0
$125$			11.9899i			1.07241i
$126$		0			0
$127$	−0.273806			−0.0242964			−0.0121482	−	0.999926i	$-0.503867\pi$
$127$	−0.273806			−0.0242964			−0.0121482	+	0.999926i	$0.503867\pi$
$128$		0			0
$129$	−11.1786			−0.984224
$130$		0			0
$131$	12.0554			1.05328			0.526642	−	0.850087i	$-0.323451\pi$
$131$	12.0554			1.05328			0.526642	+	0.850087i	$0.323451\pi$
$132$		0			0
$133$	−3.89776			−0.337979
$134$		0			0
$135$		−	1.64044i		−	0.141186i
$136$		0			0
$137$			14.1118i			1.20565i	0.797873	+	0.602825i	$0.205958\pi$
$137$			14.1118i			1.20565i	−0.797873	+	0.602825i	$0.794042\pi$
$138$		0			0
$139$	−9.51266			−0.806853			−0.403426	−	0.915012i	$-0.632181\pi$
$139$	−9.51266			−0.806853			−0.403426	+	0.915012i	$0.632181\pi$
$140$		0			0
$141$		−	4.01649i		−	0.338249i
$142$		0			0
$143$	−0.972830	+	2.70752i	−0.0813522	+	0.226415i
$144$		0			0
$145$		−	7.89776i		−	0.655873i
$146$		0			0
$147$	1.00000			0.0824786
$148$		0			0
$149$			12.3245i			1.00967i	0.863217	+	0.504833i	$0.168446\pi$
$149$			12.3245i			1.00967i	−0.863217	+	0.504833i	$0.831554\pi$
$150$		0			0
$151$			17.0120i			1.38441i	0.721699	+	0.692207i	$0.243361\pi$
$151$			17.0120i			1.38441i	−0.721699	+	0.692207i	$0.756639\pi$
$152$		0			0
$153$	−6.81442			−0.550913
$154$		0			0
$155$		0			0
$156$		0			0
$157$	14.4433			1.15270			0.576349	−	0.817204i	$-0.304477\pi$
$157$	14.4433			1.15270			0.576349	+	0.817204i	$0.304477\pi$
$158$		0			0
$159$	−6.36422			−0.504715
$160$		0			0
$161$		−	3.97192i		−	0.313031i
$162$		0			0
$163$		−	14.7122i		−	1.15235i	−0.817327	−	0.576173i	$-0.804545\pi$
$163$		−	14.7122i		−	1.15235i	0.817327	−	0.576173i	$-0.195455\pi$
$164$		0			0
$165$	−1.30896			−0.101903
$166$		0			0
$167$		−	12.8472i		−	0.994143i	−0.867710	−	0.497072i	$-0.834409\pi$
$167$		−	12.8472i		−	0.994143i	0.867710	−	0.497072i	$-0.165591\pi$
$168$		0			0
$169$	10.0272	+	8.27381i	0.771321	+	0.636447i
$170$		0			0
$171$		−	3.89776i		−	0.298069i
$172$		0			0
$173$	−5.74629			−0.436883			−0.218441	−	0.975850i	$-0.570097\pi$
$173$	−5.74629			−0.436883			−0.218441	+	0.975850i	$0.570097\pi$
$174$		0			0
$175$		−	2.30896i		−	0.174541i
$176$		0			0
$177$			0.585184i			0.0439851i
$178$		0			0
$179$	3.92373			0.293273			0.146637	−	0.989190i	$-0.453155\pi$
$179$	3.92373			0.293273			0.146637	+	0.989190i	$0.453155\pi$
$180$		0			0
$181$	3.38927			0.251923			0.125961	−	0.992035i	$-0.459798\pi$
$181$	3.38927			0.251923			0.125961	+	0.992035i	$0.459798\pi$
$182$		0			0
$183$	−3.45939			−0.255725
$184$		0			0
$185$	−0.765195			−0.0562582
$186$		0			0
$187$			5.43746i			0.397626i
$188$		0			0
$189$			1.00000i			0.0727393i
$190$		0			0
$191$	−2.94686			−0.213227			−0.106614	−	0.994301i	$-0.534001\pi$
$191$	−2.94686			−0.213227			−0.106614	+	0.994301i	$0.534001\pi$
$192$		0			0
$193$		−	8.21379i		−	0.591242i	−0.955305	−	0.295621i	$-0.904474\pi$
$193$		−	8.21379i		−	0.591242i	0.955305	−	0.295621i	$-0.0955265\pi$
$194$		0			0
$195$	−2.00000	+	5.56628i	−0.143223	+	0.398610i
$196$		0			0
$197$			24.5383i			1.74828i	0.485671	+	0.874142i	$0.338575\pi$
$197$			24.5383i			1.74828i	−0.485671	+	0.874142i	$0.661425\pi$
$198$		0			0
$199$	−5.88150			−0.416929			−0.208464	−	0.978030i	$-0.566847\pi$
$199$	−5.88150			−0.416929			−0.208464	+	0.978030i	$0.566847\pi$
$200$		0			0
$201$		−	0.150425i		−	0.0106101i
$202$		0			0
$203$			4.81442i			0.337906i
$204$		0			0
$205$	−9.92584			−0.693251
$206$		0			0
$207$	3.97192			0.276067
$208$		0			0
$209$	−3.11015			−0.215134
$210$		0			0
$211$	−9.97192			−0.686495			−0.343247	−	0.939245i	$-0.611527\pi$
$211$	−9.97192			−0.686495			−0.343247	+	0.939245i	$0.611527\pi$
$212$		0			0
$213$		−	11.0437i		−	0.756699i
$214$		0			0
$215$			18.3379i			1.25063i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		−	10.3361i		−	0.698451i
$220$		0			0
$221$	23.1225	+	8.30805i	1.55539	+	0.558860i
$222$		0			0
$223$		−	8.03297i		−	0.537928i	−0.963150	−	0.268964i	$-0.913319\pi$
$223$		−	8.03297i		−	0.537928i	0.963150	−	0.268964i	$-0.0866812\pi$
$224$		0			0
$225$	2.30896			0.153931
$226$		0			0
$227$			8.52901i			0.566091i	0.959107	+	0.283045i	$0.0913447\pi$
$227$			8.52901i			0.566091i	−0.959107	+	0.283045i	$0.908655\pi$
$228$		0			0
$229$			3.96576i			0.262065i	0.991378	+	0.131032i	$0.0418292\pi$
$229$			3.96576i			0.262065i	−0.991378	+	0.131032i	$0.958171\pi$
$230$		0			0
$231$	0.797934			0.0525002
$232$		0			0
$233$	10.8767			0.712559			0.356280	−	0.934379i	$-0.384045\pi$
$233$	10.8767			0.712559			0.356280	+	0.934379i	$0.384045\pi$
$234$		0			0
$235$	−6.58880			−0.429806
$236$		0			0
$237$	1.41120			0.0916675
$238$		0			0
$239$			19.2342i			1.24416i	0.782954	+	0.622080i	$0.213712\pi$
$239$			19.2342i			1.24416i	−0.782954	+	0.622080i	$0.786288\pi$
$240$		0			0
$241$			20.5819i			1.32579i	0.748711	+	0.662897i	$0.230673\pi$
$241$			20.5819i			1.32579i	−0.748711	+	0.662897i	$0.769327\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$		−	1.64044i		−	0.104804i
$246$		0			0
$247$	−4.75209	+	13.2257i	−0.302369	+	0.841534i
$248$		0			0
$249$			5.76278i			0.365201i
$250$		0			0
$251$	7.23481			0.456657			0.228328	−	0.973584i	$-0.426674\pi$
$251$	7.23481			0.456657			0.228328	+	0.973584i	$0.426674\pi$
$252$		0			0
$253$		−	3.16933i		−	0.199254i
$254$		0			0
$255$			11.1786i			0.700033i
$256$		0			0
$257$	−23.6781			−1.47700			−0.738499	−	0.674255i	$-0.764465\pi$
$257$	−23.6781			−1.47700			−0.738499	+	0.674255i	$0.764465\pi$
$258$		0			0
$259$	0.466457			0.0289842
$260$		0			0
$261$	−4.81442			−0.298005
$262$		0			0
$263$	7.09438			0.437458			0.218729	−	0.975786i	$-0.429809\pi$
$263$	7.09438			0.437458			0.218729	+	0.975786i	$0.429809\pi$
$264$		0			0
$265$			10.4401i			0.641331i
$266$		0			0
$267$			4.51103i			0.276070i
$268$		0			0
$269$	24.1036			1.46962			0.734810	−	0.678273i	$-0.237271\pi$
$269$	24.1036			1.46962			0.734810	+	0.678273i	$0.237271\pi$
$270$		0			0
$271$			30.4824i			1.85167i	0.377924	+	0.925837i	$0.376638\pi$
$271$			30.4824i			1.85167i	−0.377924	+	0.925837i	$0.623362\pi$
$272$		0			0
$273$	1.21919	−	3.39317i	0.0737885	−	0.205364i
$274$		0			0
$275$		−	1.84240i		−	0.111101i
$276$		0			0
$277$	−1.09959			−0.0660682			−0.0330341	−	0.999454i	$-0.510517\pi$
$277$	−1.09959			−0.0660682			−0.0330341	+	0.999454i	$0.510517\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			27.6616i			1.65015i	0.565022	+	0.825076i	$0.308868\pi$
$281$			27.6616i			1.65015i	−0.565022	+	0.825076i	$0.691132\pi$
$282$		0			0
$283$	14.9188			0.886829			0.443414	−	0.896317i	$-0.353767\pi$
$283$	14.9188			0.886829			0.443414	+	0.896317i	$0.353767\pi$
$284$		0			0
$285$	−6.39403			−0.378750
$286$		0			0
$287$	6.05073			0.357163
$288$		0			0
$289$	29.4363			1.73155
$290$		0			0
$291$		−	8.06721i		−	0.472908i
$292$		0			0
$293$			18.3917i			1.07446i	0.843437	+	0.537229i	$0.180529\pi$
$293$			18.3917i			1.07446i	−0.843437	+	0.537229i	$0.819471\pi$
$294$		0			0
$295$	0.959958			0.0558909
$296$		0			0
$297$			0.797934i			0.0463008i
$298$		0			0
$299$	−13.4774	−	4.84250i	−0.779417	−	0.280049i
$300$		0			0
$301$		−	11.1786i		−	0.644326i
$302$		0			0
$303$	9.11631			0.523718
$304$		0			0
$305$			5.67491i			0.324944i
$306$		0			0
$307$		−	20.9427i		−	1.19526i	−0.801771	−	0.597631i	$-0.796109\pi$
$307$		−	20.9427i		−	1.19526i	0.801771	−	0.597631i	$-0.203891\pi$
$308$		0			0
$309$	−1.82149			−0.103621
$310$		0			0
$311$	20.3152			1.15197			0.575986	−	0.817460i	$-0.304618\pi$
$311$	20.3152			1.15197			0.575986	+	0.817460i	$0.304618\pi$
$312$		0			0
$313$	7.26289			0.410523			0.205261	−	0.978707i	$-0.434196\pi$
$313$	7.26289			0.410523			0.205261	+	0.978707i	$0.434196\pi$
$314$		0			0
$315$	1.64044			0.0924282
$316$		0			0
$317$		−	12.0799i		−	0.678472i	−0.940701	−	0.339236i	$-0.889831\pi$
$317$		−	12.0799i		−	0.678472i	0.940701	−	0.339236i	$-0.110169\pi$
$318$		0			0
$319$			3.84159i			0.215088i
$320$		0			0
$321$	−6.78633			−0.378776
$322$		0			0
$323$			26.5610i			1.47789i
$324$		0			0
$325$	−7.83469	−	2.81505i	−0.434591	−	0.156151i
$326$		0			0
$327$			4.41029i			0.243889i
$328$		0			0
$329$	4.01649			0.221436
$330$		0			0
$331$			9.54181i			0.524465i	0.965005	+	0.262233i	$0.0844588\pi$
$331$			9.54181i			0.524465i	−0.965005	+	0.262233i	$0.915541\pi$
$332$		0			0
$333$			0.466457i			0.0255617i
$334$		0			0
$335$	−0.246763			−0.0134821
$336$		0			0
$337$	16.8816			0.919601			0.459801	−	0.888022i	$-0.347921\pi$
$337$	16.8816			0.919601			0.459801	+	0.888022i	$0.347921\pi$
$338$		0			0
$339$	13.7393			0.746219
$340$		0			0
$341$		0			0
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$		−	6.51568i		−	0.350793i
$346$		0			0
$347$	−3.41840			−0.183509			−0.0917547	−	0.995782i	$-0.529248\pi$
$347$	−3.41840			−0.183509			−0.0917547	+	0.995782i	$0.529248\pi$
$348$		0			0
$349$			7.79725i			0.417377i	0.977982	+	0.208689i	$0.0669195\pi$
$349$			7.79725i			0.417377i	−0.977982	+	0.208689i	$0.933080\pi$
$350$		0			0
$351$	3.39317	+	1.21919i	0.181114	+	0.0650753i
$352$		0			0
$353$			11.7357i			0.624630i	0.949979	+	0.312315i	$0.101104\pi$
$353$			11.7357i			0.624630i	−0.949979	+	0.312315i	$0.898896\pi$
$354$		0			0
$355$	−18.1164			−0.961521
$356$		0			0
$357$		−	6.81442i		−	0.360657i
$358$		0			0
$359$			34.9643i			1.84534i	0.385585	+	0.922672i	$0.374000\pi$
$359$			34.9643i			1.84534i	−0.385585	+	0.922672i	$0.626000\pi$
$360$		0			0
$361$	3.80748			0.200393
$362$		0			0
$363$	−10.3633			−0.543932
$364$		0			0
$365$	−16.9558			−0.887506
$366$		0			0
$367$	−27.8627			−1.45442			−0.727211	−	0.686414i	$-0.759184\pi$
$367$	−27.8627			−1.45442			−0.727211	+	0.686414i	$0.759184\pi$
$368$		0			0
$369$			6.05073i			0.314988i
$370$		0			0
$371$		−	6.36422i		−	0.330414i
$372$		0			0
$373$	−26.5561			−1.37502			−0.687511	−	0.726174i	$-0.741297\pi$
$373$	−26.5561			−1.37502			−0.687511	+	0.726174i	$0.741297\pi$
$374$		0			0
$375$		−	11.9899i		−	0.619156i
$376$		0			0
$377$	16.3361	+	5.86967i	0.841354	+	0.302304i
$378$		0			0
$379$		−	12.1834i		−	0.625819i	−0.949783	−	0.312910i	$-0.898696\pi$
$379$		−	12.1834i		−	0.625819i	0.949783	−	0.312910i	$-0.101304\pi$
$380$		0			0
$381$	0.273806			0.0140275
$382$		0			0
$383$		−	17.0516i		−	0.871298i	−0.900117	−	0.435649i	$-0.856519\pi$
$383$		−	17.0516i		−	0.871298i	0.900117	−	0.435649i	$-0.143481\pi$
$384$		0			0
$385$		−	1.30896i		−	0.0667109i
$386$		0			0
$387$	11.1786			0.568242
$388$		0			0
$389$	27.3754			1.38799			0.693994	−	0.719981i	$-0.255850\pi$
$389$	27.3754			1.38799			0.693994	+	0.719981i	$0.255850\pi$
$390$		0			0
$391$	−27.0663			−1.36880
$392$		0			0
$393$	−12.0554			−0.608114
$394$		0			0
$395$		−	2.31499i		−	0.116480i
$396$		0			0
$397$		−	32.7070i		−	1.64152i	−0.571276	−	0.820758i	$-0.693552\pi$
$397$		−	32.7070i		−	1.64152i	0.571276	−	0.820758i	$-0.306448\pi$
$398$		0			0
$399$	3.89776			0.195132
$400$		0			0
$401$		−	2.60655i		−	0.130165i	−0.997880	−	0.0650825i	$-0.979269\pi$
$401$		−	2.60655i		−	0.130165i	0.997880	−	0.0650825i	$-0.0207311\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$			1.64044i			0.0815140i
$406$		0			0
$407$	0.372202			0.0184494
$408$		0			0
$409$		−	13.3702i		−	0.661116i	−0.943786	−	0.330558i	$-0.892763\pi$
$409$		−	13.3702i		−	0.661116i	0.943786	−	0.330558i	$-0.107237\pi$
$410$		0			0
$411$		−	14.1118i		−	0.696083i
$412$		0			0
$413$	−0.585184			−0.0287950
$414$		0			0
$415$	9.45348			0.464053
$416$		0			0
$417$	9.51266			0.465837
$418$		0			0
$419$	−2.63102			−0.128534			−0.0642670	−	0.997933i	$-0.520471\pi$
$419$	−2.63102			−0.128534			−0.0642670	+	0.997933i	$0.520471\pi$
$420$		0			0
$421$		−	1.99306i		−	0.0971356i	−0.998820	−	0.0485678i	$-0.984534\pi$
$421$		−	1.99306i		−	0.0971356i	0.998820	−	0.0485678i	$-0.0154657\pi$
$422$		0			0
$423$			4.01649i			0.195288i
$424$		0			0
$425$	−15.7342			−0.763222
$426$		0			0
$427$		−	3.45939i		−	0.167411i
$428$		0			0
$429$	0.972830	−	2.70752i	0.0469687	−	0.130720i
$430$		0			0
$431$			0.326362i			0.0157203i	0.999969	+	0.00786016i	$0.00250199\pi$
$431$			0.326362i			0.0157203i	−0.999969	+	0.00786016i	$0.997498\pi$
$432$		0			0
$433$	16.9602			0.815056			0.407528	−	0.913193i	$-0.366391\pi$
$433$	16.9602			0.815056			0.407528	+	0.913193i	$0.366391\pi$
$434$		0			0
$435$			7.89776i			0.378669i
$436$		0			0
$437$		−	15.4816i		−	0.740584i
$438$		0			0
$439$	2.41550			0.115286			0.0576428	−	0.998337i	$-0.481642\pi$
$439$	2.41550			0.115286			0.0576428	+	0.998337i	$0.481642\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$		0			0
$443$	−4.07627			−0.193669			−0.0968347	−	0.995300i	$-0.530872\pi$
$443$	−4.07627			−0.193669			−0.0968347	+	0.995300i	$0.530872\pi$
$444$		0			0
$445$	7.40006			0.350797
$446$		0			0
$447$		−	12.3245i		−	0.582930i
$448$		0			0
$449$			3.76382i			0.177626i	0.996048	+	0.0888128i	$0.0283073\pi$
$449$			3.76382i			0.177626i	−0.996048	+	0.0888128i	$0.971693\pi$
$450$		0			0
$451$	4.82808			0.227345
$452$		0			0
$453$		−	17.0120i		−	0.799291i
$454$		0			0
$455$	−5.56628	−	2.00000i	−0.260951	−	0.0937614i
$456$		0			0
$457$		−	40.2359i		−	1.88215i	−0.338191	−	0.941077i	$-0.609815\pi$
$457$		−	40.2359i		−	1.88215i	0.338191	−	0.941077i	$-0.390185\pi$
$458$		0			0
$459$	6.81442			0.318070
$460$		0			0
$461$		−	3.77415i		−	0.175780i	−0.996130	−	0.0878898i	$-0.971988\pi$
$461$		−	3.77415i		−	0.175780i	0.996130	−	0.0878898i	$-0.0280123\pi$
$462$		0			0
$463$			34.4033i			1.59886i	0.600760	+	0.799429i	$0.294865\pi$
$463$			34.4033i			1.59886i	−0.600760	+	0.799429i	$0.705135\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−0.283525			−0.0131199			−0.00655997	−	0.999978i	$-0.502088\pi$
$467$	−0.283525			−0.0131199			−0.00655997	+	0.999978i	$0.502088\pi$
$468$		0			0
$469$	0.150425			0.00694597
$470$		0			0
$471$	−14.4433			−0.665510
$472$		0			0
$473$		−	8.91981i		−	0.410133i
$474$		0			0
$475$		−	8.99978i		−	0.412938i
$476$		0			0
$477$	6.36422			0.291397
$478$		0			0
$479$		−	6.37455i		−	0.291260i	−0.989339	−	0.145630i	$-0.953479\pi$
$479$		−	6.37455i		−	0.291260i	0.989339	−	0.145630i	$-0.0465210\pi$
$480$		0			0
$481$	0.568698	−	1.58277i	0.0259304	−	0.0721680i
$482$		0			0
$483$			3.97192i			0.180728i
$484$		0			0
$485$	−13.2338			−0.600914
$486$		0			0
$487$		−	37.4244i		−	1.69586i	−0.530109	−	0.847930i	$-0.677849\pi$
$487$		−	37.4244i		−	1.69586i	0.530109	−	0.847930i	$-0.322151\pi$
$488$		0			0
$489$			14.7122i			0.665308i
$490$		0			0
$491$	9.15750			0.413272			0.206636	−	0.978418i	$-0.433748\pi$
$491$	9.15750			0.413272			0.206636	+	0.978418i	$0.433748\pi$
$492$		0			0
$493$	32.8075			1.47757
$494$		0			0
$495$	1.30896			0.0588335
$496$		0			0
$497$	11.0437			0.495376
$498$		0			0
$499$			27.6781i			1.23904i	0.784981	+	0.619520i	$0.212673\pi$
$499$			27.6781i			1.23904i	−0.784981	+	0.619520i	$0.787327\pi$
$500$		0			0
$501$			12.8472i			0.573969i
$502$		0			0
$503$	7.49467			0.334171			0.167085	−	0.985942i	$-0.446564\pi$
$503$	7.49467			0.334171			0.167085	+	0.985942i	$0.446564\pi$
$504$		0			0
$505$		−	14.9547i		−	0.665477i
$506$		0			0
$507$	−10.0272	−	8.27381i	−0.445322	−	0.367453i
$508$		0			0
$509$			37.7187i			1.67185i	0.548842	+	0.835926i	$0.315069\pi$
$509$			37.7187i			1.67185i	−0.548842	+	0.835926i	$0.684931\pi$
$510$		0			0
$511$	10.3361			0.457244
$512$		0			0
$513$			3.89776i			0.172090i
$514$		0			0
$515$			2.98804i			0.131669i
$516$		0			0
$517$	3.20489			0.140951
$518$		0			0
$519$	5.74629			0.252234
$520$		0			0
$521$	26.7042			1.16993			0.584965	−	0.811058i	$-0.301108\pi$
$521$	26.7042			1.16993			0.584965	+	0.811058i	$0.301108\pi$
$522$		0			0
$523$	−27.0168			−1.18136			−0.590682	−	0.806904i	$-0.701141\pi$
$523$	−27.0168			−1.18136			−0.590682	+	0.806904i	$0.701141\pi$
$524$		0			0
$525$			2.30896i			0.100771i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−7.22389			−0.314082
$530$		0			0
$531$		−	0.585184i		−	0.0253948i
$532$		0			0
$533$	7.37696	−	20.5311i	0.319532	−	0.889302i
$534$		0			0
$535$			11.1326i			0.481303i
$536$		0			0
$537$	−3.92373			−0.169322
$538$		0			0
$539$			0.797934i			0.0343695i
$540$		0			0
$541$			6.72228i			0.289013i	0.989504	+	0.144507i	$0.0461595\pi$
$541$			6.72228i			0.289013i	−0.989504	+	0.144507i	$0.953840\pi$
$542$		0			0
$543$	−3.38927			−0.145448
$544$		0			0
$545$	7.23481			0.309905
$546$		0			0
$547$	−34.4494			−1.47295			−0.736475	−	0.676464i	$-0.763511\pi$
$547$	−34.4494			−1.47295			−0.736475	+	0.676464i	$0.763511\pi$
$548$		0			0
$549$	3.45939			0.147643
$550$		0			0
$551$			18.7654i			0.799435i
$552$		0			0
$553$			1.41120i			0.0600104i
$554$		0			0
$555$	0.765195			0.0324807
$556$		0			0
$557$			10.6040i			0.449306i	0.974439	+	0.224653i	$0.0721249\pi$
$557$			10.6040i			0.449306i	−0.974439	+	0.224653i	$0.927875\pi$
$558$		0			0
$559$	−37.9310	−	13.6288i	−1.60431	−	0.576438i
$560$		0			0
$561$		−	5.43746i		−	0.229570i
$562$		0			0
$563$	−0.468570			−0.0197479			−0.00987393	−	0.999951i	$-0.503143\pi$
$563$	−0.468570			−0.0197479			−0.00987393	+	0.999951i	$0.503143\pi$
$564$		0			0
$565$		−	22.5386i		−	0.948204i
$566$		0			0
$567$		−	1.00000i		−	0.0419961i
$568$		0			0
$569$	12.6277			0.529381			0.264690	−	0.964333i	$-0.414730\pi$
$569$	12.6277			0.529381			0.264690	+	0.964333i	$0.414730\pi$
$570$		0			0
$571$	1.15055			0.0481491			0.0240745	−	0.999710i	$-0.492336\pi$
$571$	1.15055			0.0481491			0.0240745	+	0.999710i	$0.492336\pi$
$572$		0			0
$573$	2.94686			0.123107
$574$		0			0
$575$	9.17100			0.382457
$576$		0			0
$577$			23.0422i			0.959257i	0.877472	+	0.479629i	$0.159229\pi$
$577$			23.0422i			0.959257i	−0.877472	+	0.479629i	$0.840771\pi$
$578$		0			0
$579$			8.21379i			0.341354i
$580$		0			0
$581$	−5.76278			−0.239080
$582$		0			0
$583$		−	5.07822i		−	0.210319i
$584$		0			0
$585$	2.00000	−	5.56628i	0.0826898	−	0.230137i
$586$		0			0
$587$			16.1994i			0.668622i	0.942463	+	0.334311i	$0.108504\pi$
$587$			16.1994i			0.668622i	−0.942463	+	0.334311i	$0.891496\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		−	24.5383i		−	1.00937i
$592$		0			0
$593$			14.6122i			0.600052i	0.953931	+	0.300026i	$0.0969954\pi$
$593$			14.6122i			0.600052i	−0.953931	+	0.300026i	$0.903005\pi$
$594$		0			0
$595$	−11.1786			−0.458279
$596$		0			0
$597$	5.88150			0.240714
$598$		0			0
$599$	23.9580			0.978898			0.489449	−	0.872032i	$-0.337198\pi$
$599$	23.9580			0.978898			0.489449	+	0.872032i	$0.337198\pi$
$600$		0			0
$601$	24.0835			0.982385			0.491193	−	0.871051i	$-0.336561\pi$
$601$	24.0835			0.982385			0.491193	+	0.871051i	$0.336561\pi$
$602$		0			0
$603$			0.150425i			0.00612577i
$604$		0			0
$605$			17.0004i			0.691163i
$606$		0			0
$607$	15.5627			0.631669			0.315835	−	0.948814i	$-0.397716\pi$
$607$	15.5627			0.631669			0.315835	+	0.948814i	$0.397716\pi$
$608$		0			0
$609$		−	4.81442i		−	0.195090i
$610$		0			0
$611$	4.89684	−	13.6286i	0.198105	−	0.551354i
$612$		0			0
$613$			5.90170i			0.238367i	0.992872	+	0.119184i	$0.0380278\pi$
$613$			5.90170i			0.238367i	−0.992872	+	0.119184i	$0.961972\pi$
$614$		0			0
$615$	9.92584			0.400249
$616$		0			0
$617$		−	49.0124i		−	1.97317i	−0.163259	−	0.986583i	$-0.552201\pi$
$617$		−	49.0124i		−	1.97317i	0.163259	−	0.986583i	$-0.447799\pi$
$618$		0			0
$619$		−	12.7843i		−	0.513843i	−0.966432	−	0.256922i	$-0.917292\pi$
$619$		−	12.7843i		−	0.513843i	0.966432	−	0.256922i	$-0.0827083\pi$
$620$		0			0
$621$	−3.97192			−0.159387
$622$		0			0
$623$	−4.51103			−0.180731
$624$		0			0
$625$	−8.12389			−0.324955
$626$		0			0
$627$	3.11015			0.124208
$628$		0			0
$629$		−	3.17864i		−	0.126741i
$630$		0			0
$631$			17.6749i			0.703627i	0.936070	+	0.351814i	$0.114435\pi$
$631$			17.6749i			0.703627i	−0.936070	+	0.351814i	$0.885565\pi$
$632$		0			0
$633$	9.97192			0.396348
$634$		0			0
$635$		−	0.449162i		−	0.0178244i
$636$		0			0
$637$	3.39317	+	1.21919i	0.134442	+	0.0483059i
$638$		0			0
$639$			11.0437i			0.436880i
$640$		0			0
$641$	12.3932			0.489504			0.244752	−	0.969586i	$-0.421293\pi$
$641$	12.3932			0.489504			0.244752	+	0.969586i	$0.421293\pi$
$642$		0			0
$643$		−	38.8056i		−	1.53035i	−0.643825	−	0.765173i	$-0.722654\pi$
$643$		−	38.8056i		−	1.53035i	0.643825	−	0.765173i	$-0.277346\pi$
$644$		0			0
$645$		−	18.3379i		−	0.722053i
$646$		0			0
$647$	6.67201			0.262304			0.131152	−	0.991362i	$-0.458132\pi$
$647$	6.67201			0.262304			0.131152	+	0.991362i	$0.458132\pi$
$648$		0			0
$649$	−0.466938			−0.0183289
$650$		0			0
$651$		0			0
$652$		0			0
$653$	17.2961			0.676848			0.338424	−	0.940994i	$-0.390106\pi$
$653$	17.2961			0.676848			0.338424	+	0.940994i	$0.390106\pi$
$654$		0			0
$655$			19.7761i			0.772717i
$656$		0			0
$657$			10.3361i			0.403251i
$658$		0			0
$659$	−29.4140			−1.14581			−0.572904	−	0.819622i	$-0.694183\pi$
$659$	−29.4140			−1.14581			−0.572904	+	0.819622i	$0.694183\pi$
$660$		0			0
$661$			6.97668i			0.271361i	0.990753	+	0.135681i	$0.0433221\pi$
$661$			6.97668i			0.271361i	−0.990753	+	0.135681i	$0.956678\pi$
$662$		0			0
$663$	−23.1225	−	8.30805i	−0.898002	−	0.322658i
$664$		0			0
$665$		−	6.39403i		−	0.247950i
$666$		0			0
$667$	−19.1225			−0.740425
$668$		0			0
$669$			8.03297i			0.310573i
$670$		0			0
$671$		−	2.76036i		−	0.106563i
$672$		0			0
$673$	−34.2037			−1.31846			−0.659228	−	0.751943i	$-0.729117\pi$
$673$	−34.2037			−1.31846			−0.659228	+	0.751943i	$0.729117\pi$
$674$		0			0
$675$	−2.30896			−0.0888720
$676$		0			0
$677$	12.3317			0.473946			0.236973	−	0.971516i	$-0.423845\pi$
$677$	12.3317			0.473946			0.236973	+	0.971516i	$0.423845\pi$
$678$		0			0
$679$	8.06721			0.309591
$680$		0			0
$681$		−	8.52901i		−	0.326833i
$682$		0			0
$683$		−	44.2462i		−	1.69303i	−0.532362	−	0.846517i	$-0.678695\pi$
$683$		−	44.2462i		−	1.69303i	0.532362	−	0.846517i	$-0.321305\pi$
$684$		0			0
$685$	−23.1495			−0.884497
$686$		0			0
$687$		−	3.96576i		−	0.151303i
$688$		0			0
$689$	−21.5949	−	7.75916i	−0.822698	−	0.295601i
$690$		0			0
$691$			17.2388i			0.655797i	0.944713	+	0.327898i	$0.106340\pi$
$691$			17.2388i			0.655797i	−0.944713	+	0.327898i	$0.893660\pi$
$692$		0			0
$693$	−0.797934			−0.0303110
$694$		0			0
$695$		−	15.6049i		−	0.591929i
$696$		0			0
$697$		−	41.2322i		−	1.56178i
$698$		0			0
$699$	−10.8767			−0.411396
$700$		0			0
$701$	−16.0167			−0.604943			−0.302472	−	0.953158i	$-0.597812\pi$
$701$	−16.0167			−0.604943			−0.302472	+	0.953158i	$0.597812\pi$
$702$		0			0
$703$	1.81814			0.0685724
$704$		0			0
$705$	6.58880			0.248148
$706$		0			0
$707$			9.11631i			0.342854i
$708$		0			0
$709$		−	6.54367i		−	0.245753i	−0.992422	−	0.122876i	$-0.960788\pi$
$709$		−	6.54367i		−	0.245753i	0.992422	−	0.122876i	$-0.0392119\pi$
$710$		0			0
$711$	−1.41120			−0.0529242
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−4.44153	−	1.59587i	−0.166104	−	0.0596821i
$716$		0			0
$717$		−	19.2342i		−	0.718316i
$718$		0			0
$719$	34.1126			1.27219			0.636093	−	0.771613i	$-0.280550\pi$
$719$	34.1126			1.27219			0.636093	+	0.771613i	$0.280550\pi$
$720$		0			0
$721$		−	1.82149i		−	0.0678358i
$722$		0			0
$723$		−	20.5819i		−	0.765447i
$724$		0			0
$725$	−11.1163			−0.412849
$726$		0			0
$727$	−53.3323			−1.97799			−0.988993	−	0.147962i	$-0.952729\pi$
$727$	−53.3323			−1.97799			−0.988993	+	0.147962i	$0.952729\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−76.1759			−2.81747
$732$		0			0
$733$			47.9967i			1.77280i	0.462922	+	0.886399i	$0.346801\pi$
$733$			47.9967i			1.77280i	−0.462922	+	0.886399i	$0.653199\pi$
$734$		0			0
$735$			1.64044i			0.0605085i
$736$		0			0
$737$	0.120029			0.00442133
$738$		0			0
$739$		−	40.3533i		−	1.48442i	−0.670167	−	0.742211i	$-0.733777\pi$
$739$		−	40.3533i		−	1.48442i	0.670167	−	0.742211i	$-0.266223\pi$
$740$		0			0
$741$	4.75209	−	13.2257i	0.174573	−	0.485860i
$742$		0			0
$743$			28.5602i			1.04777i	0.851789	+	0.523885i	$0.175518\pi$
$743$			28.5602i			1.04777i	−0.851789	+	0.523885i	$0.824482\pi$
$744$		0			0
$745$	−20.2176			−0.740717
$746$		0			0
$747$		−	5.76278i		−	0.210849i
$748$		0			0
$749$		−	6.78633i		−	0.247967i
$750$		0			0
$751$	37.6412			1.37355			0.686773	−	0.726872i	$-0.259027\pi$
$751$	37.6412			1.37355			0.686773	+	0.726872i	$0.259027\pi$
$752$		0			0
$753$	−7.23481			−0.263651
$754$		0			0
$755$	−27.9071			−1.01564
$756$		0			0
$757$	−11.2520			−0.408961			−0.204481	−	0.978871i	$-0.565551\pi$
$757$	−11.2520			−0.408961			−0.204481	+	0.978871i	$0.565551\pi$
$758$		0			0
$759$			3.16933i			0.115039i
$760$		0			0
$761$		−	47.3504i		−	1.71645i	−0.513272	−	0.858226i	$-0.671567\pi$
$761$		−	47.3504i		−	1.71645i	0.513272	−	0.858226i	$-0.328433\pi$
$762$		0			0
$763$	−4.41029			−0.159663
$764$		0			0
$765$		−	11.1786i		−	0.404164i
$766$		0			0
$767$	−0.713448	+	1.98563i	−0.0257611	+	0.0716968i
$768$		0			0
$769$			44.7717i			1.61451i	0.590205	+	0.807254i	$0.299047\pi$
$769$			44.7717i			1.61451i	−0.590205	+	0.807254i	$0.700953\pi$
$770$		0			0
$771$	23.6781			0.852745
$772$		0			0
$773$		−	30.1881i		−	1.08579i	−0.839801	−	0.542894i	$-0.817328\pi$
$773$		−	30.1881i		−	1.08579i	0.839801	−	0.542894i	$-0.182672\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−0.466457			−0.0167341
$778$		0			0
$779$	23.5843			0.844994
$780$		0			0
$781$	8.81211			0.315322
$782$		0			0
$783$	4.81442			0.172053
$784$		0			0
$785$			23.6933i			0.845649i
$786$		0			0
$787$		−	16.3850i		−	0.584061i	−0.956409	−	0.292031i	$-0.905669\pi$
$787$		−	16.3850i		−	0.584061i	0.956409	−	0.292031i	$-0.0943309\pi$
$788$		0			0
$789$	−7.09438			−0.252567
$790$		0			0
$791$			13.7393i			0.488515i
$792$		0			0
$793$	−11.7383	−	4.21764i	−0.416838	−	0.149773i
$794$		0			0
$795$		−	10.4401i		−	0.370272i
$796$		0			0
$797$	7.44570			0.263740			0.131870	−	0.991267i	$-0.457902\pi$
$797$	7.44570			0.263740			0.131870	+	0.991267i	$0.457902\pi$
$798$		0			0
$799$		−	27.3700i		−	0.968282i
$800$		0			0
$801$		−	4.51103i		−	0.159389i
$802$		0			0
$803$	8.24755			0.291050
$804$		0			0
$805$	6.51568			0.229648
$806$		0			0
$807$	−24.1036			−0.848486
$808$		0			0
$809$	−1.49467			−0.0525497			−0.0262749	−	0.999655i	$-0.508365\pi$
$809$	−1.49467			−0.0525497			−0.0262749	+	0.999655i	$0.508365\pi$
$810$		0			0
$811$		−	46.3663i		−	1.62814i	−0.580767	−	0.814070i	$-0.697247\pi$
$811$		−	46.3663i		−	1.62814i	0.580767	−	0.814070i	$-0.302753\pi$
$812$		0			0
$813$		−	30.4824i		−	1.06906i
$814$		0			0
$815$	24.1344			0.845392
$816$		0			0
$817$		−	43.5716i		−	1.52438i
$818$		0			0
$819$	−1.21919	+	3.39317i	−0.0426018	+	0.118567i
$820$		0			0
$821$		−	10.6488i		−	0.371647i	−0.982583	−	0.185823i	$-0.940505\pi$
$821$		−	10.6488i		−	0.371647i	0.982583	−	0.185823i	$-0.0594952\pi$
$822$		0			0
$823$	55.0655			1.91946			0.959731	−	0.280919i	$-0.0906392\pi$
$823$	55.0655			1.91946			0.959731	+	0.280919i	$0.0906392\pi$
$824$		0			0
$825$			1.84240i			0.0641441i
$826$		0			0
$827$		−	6.18978i		−	0.215240i	−0.994192	−	0.107620i	$-0.965677\pi$
$827$		−	6.18978i		−	0.215240i	0.994192	−	0.107620i	$-0.0343230\pi$
$828$		0			0
$829$	−3.87258			−0.134500			−0.0672500	−	0.997736i	$-0.521423\pi$
$829$	−3.87258			−0.134500			−0.0672500	+	0.997736i	$0.521423\pi$
$830$		0			0
$831$	1.09959			0.0381445
$832$		0			0
$833$	6.81442			0.236106
$834$		0			0
$835$	21.0750			0.729330
$836$		0			0
$837$		0			0
$838$		0			0
$839$			48.0098i			1.65748i	0.559631	+	0.828742i	$0.310943\pi$
$839$			48.0098i			1.65748i	−0.559631	+	0.828742i	$0.689057\pi$
$840$		0			0
$841$	−5.82136			−0.200737
$842$		0			0
$843$		−	27.6616i		−	0.952715i
$844$		0			0
$845$	−13.5727	+	16.4490i	−0.466914	+	0.565861i
$846$		0			0
$847$		−	10.3633i		−	0.356087i
$848$		0			0
$849$	−14.9188			−0.512011
$850$		0			0
$851$			1.85273i			0.0635107i
$852$		0			0
$853$			30.9019i			1.05806i	0.848603	+	0.529031i	$0.177444\pi$
$853$			30.9019i			1.05806i	−0.848603	+	0.529031i	$0.822556\pi$
$854$		0			0
$855$	6.39403			0.218671
$856$		0			0
$857$	0.801067			0.0273639			0.0136820	−	0.999906i	$-0.495645\pi$
$857$	0.801067			0.0273639			0.0136820	+	0.999906i	$0.495645\pi$
$858$		0			0
$859$	−32.5599			−1.11093			−0.555465	−	0.831540i	$-0.687460\pi$
$859$	−32.5599			−1.11093			−0.555465	+	0.831540i	$0.687460\pi$
$860$		0			0
$861$	−6.05073			−0.206208
$862$		0			0
$863$			30.1762i			1.02721i	0.858027	+	0.513605i	$0.171690\pi$
$863$			30.1762i			1.02721i	−0.858027	+	0.513605i	$0.828310\pi$
$864$		0			0
$865$		−	9.42644i		−	0.320509i
$866$		0			0
$867$	−29.4363			−0.999710
$868$		0			0
$869$			1.12605i			0.0381985i
$870$		0			0
$871$	0.183396	−	0.510416i	0.00621413	−	0.0172948i
$872$		0			0
$873$			8.06721i			0.273034i
$874$		0			0
$875$	11.9899			0.405333
$876$		0			0
$877$		−	36.5223i		−	1.23327i	−0.787249	−	0.616635i	$-0.788495\pi$
$877$		−	36.5223i		−	1.23327i	0.787249	−	0.616635i	$-0.211505\pi$
$878$		0			0
$879$		−	18.3917i		−	0.620338i
$880$		0			0
$881$	−22.8829			−0.770945			−0.385472	−	0.922719i	$-0.625961\pi$
$881$	−22.8829			−0.770945			−0.385472	+	0.922719i	$0.625961\pi$
$882$		0			0
$883$	−44.3854			−1.49369			−0.746843	−	0.665000i	$-0.768431\pi$
$883$	−44.3854			−1.49369			−0.746843	+	0.665000i	$0.768431\pi$
$884$		0			0
$885$	−0.959958			−0.0322686
$886$		0			0
$887$	−24.3029			−0.816013			−0.408006	−	0.912979i	$-0.633776\pi$
$887$	−24.3029			−0.816013			−0.408006	+	0.912979i	$0.633776\pi$
$888$		0			0
$889$			0.273806i			0.00918316i
$890$		0			0
$891$		−	0.797934i		−	0.0267318i
$892$		0			0
$893$	15.6553			0.523884
$894$		0			0
$895$			6.43664i			0.215153i
$896$		0			0
$897$	13.4774	+	4.84250i	0.449996	+	0.161687i
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−43.3684			−1.44481
$902$		0			0
$903$			11.1786i			0.372002i
$904$		0			0
$905$			5.55989i			0.184817i
$906$		0			0
$907$	53.4633			1.77522			0.887609	−	0.460597i	$-0.152365\pi$
$907$	53.4633			1.77522			0.887609	+	0.460597i	$0.152365\pi$
$908$		0			0
$909$	−9.11631			−0.302369
$910$		0			0
$911$	−36.6776			−1.21518			−0.607592	−	0.794249i	$-0.707864\pi$
$911$	−36.6776			−1.21518			−0.607592	+	0.794249i	$0.707864\pi$
$912$		0			0
$913$	−4.59832			−0.152182
$914$		0			0
$915$		−	5.67491i		−	0.187607i
$916$		0			0
$917$		−	12.0554i		−	0.398104i
$918$		0			0
$919$	21.5761			0.711728			0.355864	−	0.934538i	$-0.384187\pi$
$919$	21.5761			0.711728			0.355864	+	0.934538i	$0.384187\pi$
$920$		0			0
$921$			20.9427i			0.690085i
$922$		0			0
$923$	13.4643	−	37.4730i	0.443182	−	1.23344i
$924$		0			0
$925$			1.07703i			0.0354126i
$926$		0			0
$927$	1.82149			0.0598256
$928$		0			0
$929$			27.5853i			0.905046i	0.891753	+	0.452523i	$0.149476\pi$
$929$			27.5853i			0.905046i	−0.891753	+	0.452523i	$0.850524\pi$
$930$		0			0
$931$			3.89776i			0.127744i
$932$		0			0
$933$	−20.3152			−0.665091
$934$		0			0
$935$	−8.91981			−0.291709
$936$		0			0
$937$	−40.3523			−1.31825			−0.659126	−	0.752032i	$-0.729074\pi$
$937$	−40.3523			−1.31825			−0.659126	+	0.752032i	$0.729074\pi$
$938$		0			0
$939$	−7.26289			−0.237015
$940$		0			0
$941$			21.6963i			0.707279i	0.935382	+	0.353639i	$0.115056\pi$
$941$			21.6963i			0.707279i	−0.935382	+	0.353639i	$0.884944\pi$
$942$		0			0
$943$			24.0330i			0.782621i
$944$		0			0
$945$	−1.64044			−0.0533635
$946$		0			0
$947$		−	7.32672i		−	0.238086i	−0.992889	−	0.119043i	$-0.962017\pi$
$947$		−	7.32672i		−	0.238086i	0.992889	−	0.119043i	$-0.0379827\pi$
$948$		0			0
$949$	12.6017	−	35.0722i	0.409067	−	1.13849i
$950$		0			0
$951$			12.0799i			0.391716i
$952$		0			0
$953$	−47.8879			−1.55124			−0.775621	−	0.631199i	$-0.782563\pi$
$953$	−47.8879			−1.55124			−0.775621	+	0.631199i	$0.782563\pi$
$954$		0			0
$955$		−	4.83414i		−	0.156429i
$956$		0			0
$957$		−	3.84159i		−	0.124181i
$958$		0			0
$959$	14.1118			0.455693
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$	6.78633			0.218687
$964$		0			0
$965$	13.4742			0.433750
$966$		0			0
$967$		−	28.6622i		−	0.921713i	−0.887475	−	0.460857i	$-0.847542\pi$
$967$		−	28.6622i		−	0.921713i	0.887475	−	0.460857i	$-0.152458\pi$
$968$		0			0
$969$		−	26.5610i		−	0.853261i
$970$		0			0
$971$	−34.8770			−1.11926			−0.559628	−	0.828744i	$-0.689056\pi$
$971$	−34.8770			−1.11926			−0.559628	+	0.828744i	$0.689056\pi$
$972$		0			0
$973$			9.51266i			0.304962i
$974$		0			0
$975$	7.83469	+	2.81505i	0.250911	+	0.0901539i
$976$		0			0
$977$			7.39749i			0.236667i	0.992974	+	0.118333i	$0.0377551\pi$
$977$			7.39749i			0.236667i	−0.992974	+	0.118333i	$0.962245\pi$
$978$		0			0
$979$	−3.59950			−0.115041
$980$		0			0
$981$		−	4.41029i		−	0.140810i
$982$		0			0
$983$			46.5681i			1.48529i	0.669684	+	0.742647i	$0.266430\pi$
$983$			46.5681i			1.48529i	−0.669684	+	0.742647i	$0.733570\pi$
$984$		0			0
$985$	−40.2536			−1.28259
$986$		0			0
$987$	−4.01649			−0.127846
$988$		0			0
$989$	44.4006			1.41186
$990$		0			0
$991$	15.8738			0.504247			0.252123	−	0.967695i	$-0.418871\pi$
$991$	15.8738			0.504247			0.252123	+	0.967695i	$0.418871\pi$
$992$		0			0
$993$		−	9.54181i		−	0.302800i
$994$		0			0
$995$		−	9.64825i		−	0.305870i
$996$		0			0
$997$	17.2268			0.545579			0.272789	−	0.962074i	$-0.412054\pi$
$997$	17.2268			0.545579			0.272789	+	0.962074i	$0.412054\pi$
$998$		0			0
$999$		−	0.466457i		−	0.0147581i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.h.s.337.8		10
4.3	odd	2		2184.2.h.g.337.8	yes	10
13.12	even	2	inner	4368.2.h.s.337.3		10
52.51	odd	2		2184.2.h.g.337.3	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2184.2.h.g.337.3	✓	10	52.51	odd	2
2184.2.h.g.337.8	yes	10	4.3	odd	2
4368.2.h.s.337.3		10	13.12	even	2	inner
4368.2.h.s.337.8		10	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:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.64044i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.64044i q^{5} -1.00000i q^{7} +1.00000 q^{9} -0.797934i q^{11} +(-3.39317 - 1.21919i) q^{13} -1.64044i q^{15} -6.81442 q^{17} -3.89776i q^{19} +1.00000i q^{21} +3.97192 q^{23} +2.30896 q^{25} -1.00000 q^{27} -4.81442 q^{29} +0.797934i q^{33} +1.64044 q^{35} +0.466457i q^{37} +(3.39317 + 1.21919i) q^{39} +6.05073i q^{41} +11.1786 q^{43} +1.64044i q^{45} +4.01649i q^{47} -1.00000 q^{49} +6.81442 q^{51} +6.36422 q^{53} +1.30896 q^{55} +3.89776i q^{57} -0.585184i q^{59} +3.45939 q^{61} -1.00000i q^{63} +(2.00000 - 5.56628i) q^{65} +0.150425i q^{67} -3.97192 q^{69} +11.0437i q^{71} +10.3361i q^{73} -2.30896 q^{75} -0.797934 q^{77} -1.41120 q^{79} +1.00000 q^{81} -5.76278i q^{83} -11.1786i q^{85} +4.81442 q^{87} -4.51103i q^{89} +(-1.21919 + 3.39317i) q^{91} +6.39403 q^{95} +8.06721i q^{97} -0.797934i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{3} + 10 q^{9}+O(q^{10}) \) 10 * q - 10 * q^3 + 10 * q^9	\( 10 q - 10 q^{3} + 10 q^{9} - 22 q^{17} + 18 q^{23} - 8 q^{25} - 10 q^{27} - 2 q^{29} - 10 q^{35} + 2 q^{43} - 10 q^{49} + 22 q^{51} - 18 q^{55} + 6 q^{61} + 20 q^{65} - 18 q^{69} + 8 q^{75} - 6 q^{77} - 40 q^{79} + 10 q^{81} + 2 q^{87} + 2 q^{91} + 38 q^{95}+O(q^{100}) \) 10 * q - 10 * q^3 + 10 * q^9 - 22 * q^17 + 18 * q^23 - 8 * q^25 - 10 * q^27 - 2 * q^29 - 10 * q^35 + 2 * q^43 - 10 * q^49 + 22 * q^51 - 18 * q^55 + 6 * q^61 + 20 * q^65 - 18 * q^69 + 8 * q^75 - 6 * q^77 - 40 * q^79 + 10 * q^81 + 2 * q^87 + 2 * q^91 + 38 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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