LMFDB - Embedded newform 4368.2.h.s.337.5

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			337.5
Root			$3.49183i$ of defining polynomial
Character	$\chi$	$=$	4368.337
Dual form			4368.2.h.s.337.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -0.572766i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.572766i q^{5} -1.00000i q^{7} +1.00000 q^{9} +6.41089i q^{11} +(0.898409 + 3.49183i) q^{13} +0.572766i q^{15} -5.02065 q^{17} +5.73013i q^{19} +1.00000i q^{21} -2.81747 q^{23} +4.67194 q^{25} -1.00000 q^{27} -3.02065 q^{29} -6.41089i q^{33} -0.572766 q^{35} -2.16618i q^{37} +(-0.898409 - 3.49183i) q^{39} -12.3739i q^{41} -2.87566 q^{43} -0.572766i q^{45} +9.43154i q^{47} -1.00000 q^{49} +5.02065 q^{51} -5.89631 q^{53} +3.67194 q^{55} -5.73013i q^{57} -12.9955i q^{59} +3.25353 q^{61} -1.00000i q^{63} +(2.00000 - 0.514578i) q^{65} -2.41841i q^{67} +2.81747 q^{69} -4.95418i q^{71} -8.71378i q^{73} -4.67194 q^{75} +6.41089 q^{77} -13.4021 q^{79} +1.00000 q^{81} +5.80865i q^{83} +2.87566i q^{85} +3.02065 q^{87} -14.0828i q^{89} +(3.49183 - 0.898409i) q^{91} +3.28202 q^{95} -4.94235i q^{97} +6.41089i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$		−	0.572766i		−	0.256149i	−0.991765	−	0.128074i	$-0.959120\pi$
$5$		−	0.572766i		−	0.256149i	0.991765	−	0.128074i	$-0.0408796\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			6.41089i			1.93296i	0.256750	+	0.966478i	$0.417348\pi$
$11$			6.41089i			1.93296i	−0.256750	+	0.966478i	$0.582652\pi$
$12$		0			0
$13$	0.898409	+	3.49183i	0.249174	+	0.968459i
$13$	0.898409	+	3.49183i	0.249174	+	0.968459i
$14$		0			0
$15$			0.572766i			0.147888i
$16$		0			0
$17$	−5.02065			−1.21769			−0.608844	−	0.793290i	$-0.708366\pi$
$17$	−5.02065			−1.21769			−0.608844	+	0.793290i	$0.708366\pi$
$18$		0			0
$19$			5.73013i			1.31458i	0.753637	+	0.657291i	$0.228298\pi$
$19$			5.73013i			1.31458i	−0.753637	+	0.657291i	$0.771702\pi$
$20$		0			0
$21$			1.00000i			0.218218i
$22$		0			0
$23$	−2.81747			−0.587483			−0.293742	−	0.955885i	$-0.594901\pi$
$23$	−2.81747			−0.587483			−0.293742	+	0.955885i	$0.594901\pi$
$24$		0			0
$25$	4.67194			0.934388
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	−3.02065			−0.560921			−0.280461	−	0.959866i	$-0.590487\pi$
$29$	−3.02065			−0.560921			−0.280461	+	0.959866i	$0.590487\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		−	6.41089i		−	1.11599i
$34$		0			0
$35$	−0.572766			−0.0968151
$36$		0			0
$37$		−	2.16618i		−	0.356119i	−0.984020	−	0.178059i	$-0.943018\pi$
$37$		−	2.16618i		−	0.356119i	0.984020	−	0.178059i	$-0.0569819\pi$
$38$		0			0
$39$	−0.898409	−	3.49183i	−0.143861	−	0.559140i
$40$		0			0
$41$		−	12.3739i		−	1.93248i	−0.257649	−	0.966239i	$-0.582948\pi$
$41$		−	12.3739i		−	1.93248i	0.257649	−	0.966239i	$-0.417052\pi$
$42$		0			0
$43$	−2.87566			−0.438534			−0.219267	−	0.975665i	$-0.570367\pi$
$43$	−2.87566			−0.438534			−0.219267	+	0.975665i	$0.570367\pi$
$44$		0			0
$45$		−	0.572766i		−	0.0853829i
$46$		0			0
$47$			9.43154i			1.37573i	0.725838	+	0.687866i	$0.241452\pi$
$47$			9.43154i			1.37573i	−0.725838	+	0.687866i	$0.758548\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	5.02065			0.703032
$52$		0			0
$53$	−5.89631			−0.809921			−0.404960	−	0.914334i	$-0.632715\pi$
$53$	−5.89631			−0.809921			−0.404960	+	0.914334i	$0.632715\pi$
$54$		0			0
$55$	3.67194			0.495124
$56$		0			0
$57$		−	5.73013i		−	0.758974i
$58$		0			0
$59$		−	12.9955i		−	1.69187i	−0.533288	−	0.845934i	$-0.679044\pi$
$59$		−	12.9955i		−	1.69187i	0.533288	−	0.845934i	$-0.320956\pi$
$60$		0			0
$61$	3.25353			0.416572			0.208286	−	0.978068i	$-0.433212\pi$
$61$	3.25353			0.416572			0.208286	+	0.978068i	$0.433212\pi$
$62$		0			0
$63$		−	1.00000i		−	0.125988i
$64$		0			0
$65$	2.00000	−	0.514578i	0.248069	−	0.0638256i
$66$		0			0
$67$		−	2.41841i		−	0.295456i	−0.989028	−	0.147728i	$-0.952804\pi$
$67$		−	2.41841i		−	0.295456i	0.989028	−	0.147728i	$-0.0471960\pi$
$68$		0			0
$69$	2.81747			0.339184
$70$		0			0
$71$		−	4.95418i		−	0.587953i	−0.955813	−	0.293976i	$-0.905021\pi$
$71$		−	4.95418i		−	0.587953i	0.955813	−	0.293976i	$-0.0949787\pi$
$72$		0			0
$73$		−	8.71378i		−	1.01987i	−0.860213	−	0.509936i	$-0.829669\pi$
$73$		−	8.71378i		−	1.01987i	0.860213	−	0.509936i	$-0.170331\pi$
$74$		0			0
$75$	−4.67194			−0.539469
$76$		0			0
$77$	6.41089			0.730589
$78$		0			0
$79$	−13.4021			−1.50785			−0.753925	−	0.656960i	$-0.771842\pi$
$79$	−13.4021			−1.50785			−0.753925	+	0.656960i	$0.771842\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			5.80865i			0.637582i	0.947825	+	0.318791i	$0.103277\pi$
$83$			5.80865i			0.637582i	−0.947825	+	0.318791i	$0.896723\pi$
$84$		0			0
$85$			2.87566i			0.311909i
$86$		0			0
$87$	3.02065			0.323848
$88$		0			0
$89$		−	14.0828i		−	1.49278i	−0.665511	−	0.746388i	$-0.731786\pi$
$89$		−	14.0828i		−	1.49278i	0.665511	−	0.746388i	$-0.268214\pi$
$90$		0			0
$91$	3.49183	−	0.898409i	0.366043	−	0.0941789i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.28202			0.336728
$96$		0			0
$97$		−	4.94235i		−	0.501820i	−0.968010	−	0.250910i	$-0.919270\pi$
$97$		−	4.94235i		−	0.501820i	0.968010	−	0.250910i	$-0.0807298\pi$
$98$		0			0
$99$			6.41089i			0.644319i
$100$		0			0
$101$	−12.1123			−1.20522			−0.602610	−	0.798036i	$-0.705872\pi$
$101$	−12.1123			−1.20522			−0.602610	+	0.798036i	$0.705872\pi$
$102$		0			0
$103$	−2.39906			−0.236386			−0.118193	−	0.992991i	$-0.537710\pi$
$103$	−2.39906			−0.236386			−0.118193	+	0.992991i	$0.537710\pi$
$104$		0			0
$105$	0.572766			0.0558962
$106$		0			0
$107$	−1.79682			−0.173705			−0.0868525	−	0.996221i	$-0.527681\pi$
$107$	−1.79682			−0.173705			−0.0868525	+	0.996221i	$0.527681\pi$
$108$		0			0
$109$			11.8011i			1.13034i	0.824973	+	0.565171i	$0.191190\pi$
$109$			11.8011i			1.13034i	−0.824973	+	0.565171i	$0.808810\pi$
$110$		0			0
$111$			2.16618i			0.205605i
$112$		0			0
$113$	19.0952			1.79632			0.898162	−	0.439664i	$-0.144902\pi$
$113$	19.0952			1.79632			0.898162	+	0.439664i	$0.144902\pi$
$114$		0			0
$115$			1.61375i			0.150483i
$116$		0			0
$117$	0.898409	+	3.49183i	0.0830580	+	0.322820i
$118$		0			0
$119$			5.02065i			0.460242i
$120$		0			0
$121$	−30.0995			−2.73632
$122$		0			0
$123$			12.3739i			1.11572i
$124$		0			0
$125$		−	5.53976i		−	0.495491i
$126$		0			0
$127$	1.72582			0.153142			0.0765708	−	0.997064i	$-0.475603\pi$
$127$	1.72582			0.153142			0.0765708	+	0.997064i	$0.475603\pi$
$128$		0			0
$129$	2.87566			0.253188
$130$		0			0
$131$	−20.8430			−1.82106			−0.910529	−	0.413444i	$-0.864326\pi$
$131$	−20.8430			−1.82106			−0.910529	+	0.413444i	$0.864326\pi$
$132$		0			0
$133$	5.73013			0.496865
$134$		0			0
$135$			0.572766i			0.0492958i
$136$		0			0
$137$			13.3067i			1.13686i	0.822730	+	0.568432i	$0.192450\pi$
$137$			13.3067i			1.13686i	−0.822730	+	0.568432i	$0.807550\pi$
$138$		0			0
$139$	15.3457			1.30161			0.650803	−	0.759246i	$-0.274432\pi$
$139$	15.3457			1.30161			0.650803	+	0.759246i	$0.274432\pi$
$140$		0			0
$141$		−	9.43154i		−	0.794279i
$142$		0			0
$143$	−22.3857	+	5.75960i	−1.87199	+	0.481642i
$144$		0			0
$145$			1.73013i			0.143679i
$146$		0			0
$147$	1.00000			0.0824786
$148$		0			0
$149$		−	8.09971i		−	0.663554i	−0.943358	−	0.331777i	$-0.892352\pi$
$149$		−	8.09971i		−	0.663554i	0.943358	−	0.331777i	$-0.107648\pi$
$150$		0			0
$151$			18.6259i			1.51575i	0.652397	+	0.757877i	$0.273763\pi$
$151$			18.6259i			1.51575i	−0.652397	+	0.757877i	$0.726237\pi$
$152$		0			0
$153$	−5.02065			−0.405896
$154$		0			0
$155$		0			0
$156$		0			0
$157$	9.06196			0.723223			0.361611	−	0.932329i	$-0.382227\pi$
$157$	9.06196			0.723223			0.361611	+	0.932329i	$0.382227\pi$
$158$		0			0
$159$	5.89631			0.467608
$160$		0			0
$161$			2.81747i			0.222048i
$162$		0			0
$163$		−	3.29053i		−	0.257734i	−0.991662	−	0.128867i	$-0.958866\pi$
$163$		−	3.29053i		−	0.257734i	0.991662	−	0.128867i	$-0.0411340\pi$
$164$		0			0
$165$	−3.67194			−0.285860
$166$		0			0
$167$		−	3.36905i		−	0.260705i	−0.991468	−	0.130352i	$-0.958389\pi$
$167$		−	3.36905i		−	0.260705i	0.991468	−	0.130352i	$-0.0416108\pi$
$168$		0			0
$169$	−11.3857	+	6.27418i	−0.875825	+	0.482629i
$170$		0			0
$171$			5.73013i			0.438194i
$172$		0			0
$173$	11.2402			0.854576			0.427288	−	0.904116i	$-0.359469\pi$
$173$	11.2402			0.854576			0.427288	+	0.904116i	$0.359469\pi$
$174$		0			0
$175$		−	4.67194i		−	0.353165i
$176$		0			0
$177$			12.9955i			0.976800i
$178$		0			0
$179$	9.33107			0.697437			0.348718	−	0.937228i	$-0.386617\pi$
$179$	9.33107			0.697437			0.348718	+	0.937228i	$0.386617\pi$
$180$		0			0
$181$	−22.0383			−1.63809			−0.819047	−	0.573726i	$-0.805497\pi$
$181$	−22.0383			−1.63809			−0.819047	+	0.573726i	$0.805497\pi$
$182$		0			0
$183$	−3.25353			−0.240508
$184$		0			0
$185$	−1.24072			−0.0912193
$186$		0			0
$187$		−	32.1868i		−	2.35374i
$188$		0			0
$189$			1.00000i			0.0727393i
$190$		0			0
$191$	−9.32453			−0.674699			−0.337350	−	0.941379i	$-0.609530\pi$
$191$	−9.32453			−0.674699			−0.337350	+	0.941379i	$0.609530\pi$
$192$		0			0
$193$			1.47790i			0.106382i	0.998584	+	0.0531908i	$0.0169392\pi$
$193$			1.47790i			0.106382i	−0.998584	+	0.0531908i	$0.983061\pi$
$194$		0			0
$195$	−2.00000	+	0.514578i	−0.143223	+	0.0368497i
$196$		0			0
$197$		−	5.57761i		−	0.397388i	−0.980062	−	0.198694i	$-0.936330\pi$
$197$		−	5.57761i		−	0.397388i	0.980062	−	0.198694i	$-0.0636701\pi$
$198$		0			0
$199$	−9.35302			−0.663018			−0.331509	−	0.943452i	$-0.607558\pi$
$199$	−9.35302			−0.663018			−0.331509	+	0.943452i	$0.607558\pi$
$200$		0			0
$201$			2.41841i			0.170582i
$202$		0			0
$203$			3.02065i			0.212008i
$204$		0			0
$205$	−7.08734			−0.495002
$206$		0			0
$207$	−2.81747			−0.195828
$208$		0			0
$209$	−36.7352			−2.54103
$210$		0			0
$211$	−3.18253			−0.219094			−0.109547	−	0.993982i	$-0.534940\pi$
$211$	−3.18253			−0.219094			−0.109547	+	0.993982i	$0.534940\pi$
$212$		0			0
$213$			4.95418i			0.339455i
$214$		0			0
$215$			1.64708i			0.112330i
$216$		0			0
$217$		0			0
$218$		0			0
$219$			8.71378i			0.588823i
$220$		0			0
$221$	−4.51060	−	17.5313i	−0.303416	−	1.17928i
$222$		0			0
$223$		−	18.8631i		−	1.26317i	−0.775308	−	0.631583i	$-0.782406\pi$
$223$		−	18.8631i		−	1.26317i	0.775308	−	0.631583i	$-0.217594\pi$
$224$		0			0
$225$	4.67194			0.311463
$226$		0			0
$227$			7.36054i			0.488536i	0.969708	+	0.244268i	$0.0785477\pi$
$227$			7.36054i			0.488536i	−0.969708	+	0.244268i	$0.921452\pi$
$228$		0			0
$229$			27.8054i			1.83743i	0.394916	+	0.918717i	$0.370774\pi$
$229$			27.8054i			1.83743i	−0.394916	+	0.918717i	$0.629226\pi$
$230$		0			0
$231$	−6.41089			−0.421806
$232$		0			0
$233$	−7.96731			−0.521956			−0.260978	−	0.965345i	$-0.584045\pi$
$233$	−7.96731			−0.521956			−0.260978	+	0.965345i	$0.584045\pi$
$234$		0			0
$235$	5.40207			0.352392
$236$		0			0
$237$	13.4021			0.870558
$238$		0			0
$239$		−	9.20394i		−	0.595353i	−0.954667	−	0.297677i	$-0.903788\pi$
$239$		−	9.20394i		−	0.595353i	0.954667	−	0.297677i	$-0.0962117\pi$
$240$		0			0
$241$		−	7.25707i		−	0.467469i	−0.972300	−	0.233735i	$-0.924905\pi$
$241$		−	7.25707i		−	0.467469i	0.972300	−	0.233735i	$-0.0750947\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$			0.572766i			0.0365927i
$246$		0			0
$247$	−20.0086	+	5.14800i	−1.27312	+	0.327559i
$248$		0			0
$249$		−	5.80865i		−	0.368108i
$250$		0			0
$251$	6.75928			0.426642			0.213321	−	0.976982i	$-0.431572\pi$
$251$	6.75928			0.426642			0.213321	+	0.976982i	$0.431572\pi$
$252$		0			0
$253$		−	18.0625i		−	1.13558i
$254$		0			0
$255$		−	2.87566i		−	0.180081i
$256$		0			0
$257$	−17.8212			−1.11166			−0.555829	−	0.831296i	$-0.687599\pi$
$257$	−17.8212			−1.11166			−0.555829	+	0.831296i	$0.687599\pi$
$258$		0			0
$259$	−2.16618			−0.134600
$260$		0			0
$261$	−3.02065			−0.186974
$262$		0			0
$263$	−27.3281			−1.68512			−0.842561	−	0.538602i	$-0.818953\pi$
$263$	−27.3281			−1.68512			−0.842561	+	0.538602i	$0.818953\pi$
$264$		0			0
$265$			3.37721i			0.207460i
$266$		0			0
$267$			14.0828i			0.861855i
$268$		0			0
$269$	−20.9915			−1.27987			−0.639937	−	0.768427i	$-0.721040\pi$
$269$	−20.9915			−1.27987			−0.639937	+	0.768427i	$0.721040\pi$
$270$		0			0
$271$			5.30214i			0.322082i	0.986948	+	0.161041i	$0.0514851\pi$
$271$			5.30214i			0.322082i	−0.986948	+	0.161041i	$0.948515\pi$
$272$		0			0
$273$	−3.49183	+	0.898409i	−0.211335	+	0.0543742i
$274$		0			0
$275$			29.9513i			1.80613i
$276$		0			0
$277$	19.8339			1.19171			0.595853	−	0.803094i	$-0.296814\pi$
$277$	19.8339			1.19171			0.595853	+	0.803094i	$0.296814\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			16.3897i			0.977727i	0.872360	+	0.488864i	$0.162588\pi$
$281$			16.3897i			0.977727i	−0.872360	+	0.488864i	$0.837412\pi$
$282$		0			0
$283$	14.5071			0.862355			0.431178	−	0.902267i	$-0.358098\pi$
$283$	14.5071			0.862355			0.431178	+	0.902267i	$0.358098\pi$
$284$		0			0
$285$	−3.28202			−0.194410
$286$		0			0
$287$	−12.3739			−0.730408
$288$		0			0
$289$	8.20695			0.482762
$290$		0			0
$291$			4.94235i			0.289726i
$292$		0			0
$293$		−	15.0421i		−	0.878767i	−0.898300	−	0.439383i	$-0.855197\pi$
$293$		−	15.0421i		−	0.878767i	0.898300	−	0.439383i	$-0.144803\pi$
$294$		0			0
$295$	−7.44337			−0.433370
$296$		0			0
$297$		−	6.41089i		−	0.371998i
$298$		0			0
$299$	−2.53124	−	9.83812i	−0.146386	−	0.568953i
$300$		0			0
$301$			2.87566i			0.165750i
$302$		0			0
$303$	12.1123			0.695834
$304$		0			0
$305$		−	1.86351i		−	0.106704i
$306$		0			0
$307$		−	23.7589i		−	1.35599i	−0.735067	−	0.677995i	$-0.762849\pi$
$307$		−	23.7589i		−	1.35599i	0.735067	−	0.677995i	$-0.237151\pi$
$308$		0			0
$309$	2.39906			0.136478
$310$		0			0
$311$	−26.2257			−1.48712			−0.743561	−	0.668668i	$-0.766865\pi$
$311$	−26.2257			−1.48712			−0.743561	+	0.668668i	$0.766865\pi$
$312$		0			0
$313$	13.5768			0.767404			0.383702	−	0.923457i	$-0.374649\pi$
$313$	13.5768			0.767404			0.383702	+	0.923457i	$0.374649\pi$
$314$		0			0
$315$	−0.572766			−0.0322717
$316$		0			0
$317$		−	10.3721i		−	0.582552i	−0.956639	−	0.291276i	$-0.905920\pi$
$317$		−	10.3721i		−	0.582552i	0.956639	−	0.291276i	$-0.0940799\pi$
$318$		0			0
$319$		−	19.3651i		−	1.08424i
$320$		0			0
$321$	1.79682			0.100289
$322$		0			0
$323$		−	28.7690i		−	1.60075i
$324$		0			0
$325$	4.19731	+	16.3136i	0.232825	+	0.904916i
$326$		0			0
$327$		−	11.8011i		−	0.652604i
$328$		0			0
$329$	9.43154			0.519978
$330$		0			0
$331$		−	26.7004i		−	1.46759i	−0.679371	−	0.733795i	$-0.737747\pi$
$331$		−	26.7004i		−	1.46759i	0.679371	−	0.733795i	$-0.262253\pi$
$332$		0			0
$333$		−	2.16618i		−	0.118706i
$334$		0			0
$335$	−1.38518			−0.0756806
$336$		0			0
$337$	2.07830			0.113212			0.0566062	−	0.998397i	$-0.481972\pi$
$337$	2.07830			0.113212			0.0566062	+	0.998397i	$0.481972\pi$
$338$		0			0
$339$	−19.0952			−1.03711
$340$		0			0
$341$		0			0
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$		−	1.61375i		−	0.0868815i
$346$		0			0
$347$	5.29237			0.284109			0.142055	−	0.989859i	$-0.454629\pi$
$347$	5.29237			0.284109			0.142055	+	0.989859i	$0.454629\pi$
$348$		0			0
$349$		−	9.09939i		−	0.487079i	−0.969891	−	0.243540i	$-0.921691\pi$
$349$		−	9.09939i		−	0.487079i	0.969891	−	0.243540i	$-0.0783086\pi$
$350$		0			0
$351$	−0.898409	−	3.49183i	−0.0479535	−	0.186380i
$352$		0			0
$353$			3.30235i			0.175767i	0.996131	+	0.0878833i	$0.0280103\pi$
$353$			3.30235i			0.175767i	−0.996131	+	0.0878833i	$0.971990\pi$
$354$		0			0
$355$	−2.83759			−0.150603
$356$		0			0
$357$		−	5.02065i		−	0.265721i
$358$		0			0
$359$			2.63902i			0.139282i	0.997572	+	0.0696411i	$0.0221854\pi$
$359$			2.63902i			0.139282i	−0.997572	+	0.0696411i	$0.977815\pi$
$360$		0			0
$361$	−13.8344			−0.728124
$362$		0			0
$363$	30.0995			1.57981
$364$		0			0
$365$	−4.99096			−0.261239
$366$		0			0
$367$	4.40260			0.229814			0.114907	−	0.993376i	$-0.463343\pi$
$367$	4.40260			0.229814			0.114907	+	0.993376i	$0.463343\pi$
$368$		0			0
$369$		−	12.3739i		−	0.644159i
$370$		0			0
$371$			5.89631i			0.306121i
$372$		0			0
$373$	32.8146			1.69907			0.849537	−	0.527528i	$-0.176881\pi$
$373$	32.8146			1.69907			0.849537	+	0.527528i	$0.176881\pi$
$374$		0			0
$375$			5.53976i			0.286072i
$376$		0			0
$377$	−2.71378	−	10.5476i	−0.139767	−	0.543229i
$378$		0			0
$379$		−	20.4447i		−	1.05017i	−0.851049	−	0.525086i	$-0.824033\pi$
$379$		−	20.4447i		−	1.05017i	0.851049	−	0.525086i	$-0.175967\pi$
$380$		0			0
$381$	−1.72582			−0.0884164
$382$		0			0
$383$		−	26.8293i		−	1.37091i	−0.728113	−	0.685457i	$-0.759603\pi$
$383$		−	26.8293i		−	1.37091i	0.728113	−	0.685457i	$-0.240397\pi$
$384$		0			0
$385$		−	3.67194i		−	0.187139i
$386$		0			0
$387$	−2.87566			−0.146178
$388$		0			0
$389$	−29.7483			−1.50830			−0.754150	−	0.656702i	$-0.771951\pi$
$389$	−29.7483			−1.50830			−0.754150	+	0.656702i	$0.771951\pi$
$390$		0			0
$391$	14.1455			0.715371
$392$		0			0
$393$	20.8430			1.05139
$394$		0			0
$395$			7.67625i			0.386234i
$396$		0			0
$397$		−	7.79638i		−	0.391289i	−0.980675	−	0.195645i	$-0.937320\pi$
$397$		−	7.79638i		−	0.391289i	0.980675	−	0.195645i	$-0.0626799\pi$
$398$		0			0
$399$	−5.73013			−0.286865
$400$		0			0
$401$			38.6390i			1.92954i	0.263092	+	0.964771i	$0.415258\pi$
$401$			38.6390i			1.92954i	−0.263092	+	0.964771i	$0.584742\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		−	0.572766i		−	0.0284610i
$406$		0			0
$407$	13.8872			0.688361
$408$		0			0
$409$			11.2445i			0.556005i	0.960580	+	0.278002i	$0.0896723\pi$
$409$			11.2445i			0.556005i	−0.960580	+	0.278002i	$0.910328\pi$
$410$		0			0
$411$		−	13.3067i		−	0.656369i
$412$		0			0
$413$	−12.9955			−0.639466
$414$		0			0
$415$	3.32700			0.163316
$416$		0			0
$417$	−15.3457			−0.751483
$418$		0			0
$419$	7.42402			0.362687			0.181344	−	0.983420i	$-0.441955\pi$
$419$	7.42402			0.362687			0.181344	+	0.983420i	$0.441955\pi$
$420$		0			0
$421$			13.8550i			0.675252i	0.941280	+	0.337626i	$0.109624\pi$
$421$			13.8550i			0.675252i	−0.941280	+	0.337626i	$0.890376\pi$
$422$		0			0
$423$			9.43154i			0.458577i
$424$		0			0
$425$	−23.4562			−1.13779
$426$		0			0
$427$		−	3.25353i		−	0.157449i
$428$		0			0
$429$	22.3857	−	5.75960i	1.08079	−	0.278076i
$430$		0			0
$431$			36.3067i			1.74883i	0.485178	+	0.874415i	$0.338755\pi$
$431$			36.3067i			1.74883i	−0.485178	+	0.874415i	$0.661245\pi$
$432$		0			0
$433$	−27.9928			−1.34525			−0.672624	−	0.739984i	$-0.734833\pi$
$433$	−27.9928			−1.34525			−0.672624	+	0.739984i	$0.734833\pi$
$434$		0			0
$435$		−	1.73013i		−	0.0829532i
$436$		0			0
$437$		−	16.1445i		−	0.772295i
$438$		0			0
$439$	−0.306982			−0.0146514			−0.00732572	−	0.999973i	$-0.502332\pi$
$439$	−0.306982			−0.0146514			−0.00732572	+	0.999973i	$0.502332\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$		0			0
$443$	1.33107			0.0632409			0.0316204	−	0.999500i	$-0.489933\pi$
$443$	1.33107			0.0632409			0.0316204	+	0.999500i	$0.489933\pi$
$444$		0			0
$445$	−8.06616			−0.382373
$446$		0			0
$447$			8.09971i			0.383103i
$448$		0			0
$449$			2.11983i			0.100041i	0.998748	+	0.0500204i	$0.0159286\pi$
$449$			2.11983i			0.100041i	−0.998748	+	0.0500204i	$0.984071\pi$
$450$		0			0
$451$	79.3277			3.73539
$452$		0			0
$453$		−	18.6259i		−	0.875121i
$454$		0			0
$455$	−0.514578	−	2.00000i	−0.0241238	−	0.0937614i
$456$		0			0
$457$			22.6325i			1.05870i	0.848403	+	0.529352i	$0.177565\pi$
$457$			22.6325i			1.05870i	−0.848403	+	0.529352i	$0.822435\pi$
$458$		0			0
$459$	5.02065			0.234344
$460$		0			0
$461$			24.9374i			1.16145i	0.814099	+	0.580725i	$0.197231\pi$
$461$			24.9374i			1.16145i	−0.814099	+	0.580725i	$0.802769\pi$
$462$		0			0
$463$			2.34387i			0.108929i	0.998516	+	0.0544644i	$0.0173451\pi$
$463$			2.34387i			0.108929i	−0.998516	+	0.0544644i	$0.982655\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−26.4185			−1.22250			−0.611252	−	0.791436i	$-0.709334\pi$
$467$	−26.4185			−1.22250			−0.611252	+	0.791436i	$0.709334\pi$
$468$		0			0
$469$	−2.41841			−0.111672
$470$		0			0
$471$	−9.06196			−0.417553
$472$		0			0
$473$		−	18.4355i		−	0.847667i
$474$		0			0
$475$			26.7708i			1.22833i
$476$		0			0
$477$	−5.89631			−0.269974
$478$		0			0
$479$		−	30.1581i		−	1.37796i	−0.724780	−	0.688980i	$-0.758059\pi$
$479$		−	30.1581i		−	1.37796i	0.724780	−	0.688980i	$-0.241941\pi$
$480$		0			0
$481$	7.56394	−	1.94612i	0.344886	−	0.0887354i
$482$		0			0
$483$		−	2.81747i		−	0.128199i
$484$		0			0
$485$	−2.83081			−0.128540
$486$		0			0
$487$		−	14.5811i		−	0.660730i	−0.943853	−	0.330365i	$-0.892828\pi$
$487$		−	14.5811i		−	0.660730i	0.943853	−	0.330365i	$-0.107172\pi$
$488$		0			0
$489$			3.29053i			0.148803i
$490$		0			0
$491$	4.16188			0.187823			0.0939114	−	0.995581i	$-0.470063\pi$
$491$	4.16188			0.187823			0.0939114	+	0.995581i	$0.470063\pi$
$492$		0			0
$493$	15.1656			0.683026
$494$		0			0
$495$	3.67194			0.165041
$496$		0			0
$497$	−4.95418			−0.222225
$498$		0			0
$499$			21.8212i			0.976853i	0.872605	+	0.488426i	$0.162429\pi$
$499$			21.8212i			0.976853i	−0.872605	+	0.488426i	$0.837571\pi$
$500$		0			0
$501$			3.36905i			0.150518i
$502$		0			0
$503$	−6.62343			−0.295324			−0.147662	−	0.989038i	$-0.547175\pi$
$503$	−6.62343			−0.295324			−0.147662	+	0.989038i	$0.547175\pi$
$504$		0			0
$505$			6.93752i			0.308715i
$506$		0			0
$507$	11.3857	−	6.27418i	0.505658	−	0.278646i
$508$		0			0
$509$		−	4.09241i		−	0.181393i	−0.995879	−	0.0906964i	$-0.971091\pi$
$509$		−	4.09241i		−	0.181393i	0.995879	−	0.0906964i	$-0.0289093\pi$
$510$		0			0
$511$	−8.71378			−0.385475
$512$		0			0
$513$		−	5.73013i		−	0.252991i
$514$		0			0
$515$			1.37410i			0.0605501i
$516$		0			0
$517$	−60.4646			−2.65923
$518$		0			0
$519$	−11.2402			−0.493390
$520$		0			0
$521$	17.6077			0.771408			0.385704	−	0.922623i	$-0.373959\pi$
$521$	17.6077			0.771408			0.385704	+	0.922623i	$0.373959\pi$
$522$		0			0
$523$	−32.6715			−1.42863			−0.714313	−	0.699827i	$-0.753261\pi$
$523$	−32.6715			−1.42863			−0.714313	+	0.699827i	$0.753261\pi$
$524$		0			0
$525$			4.67194i			0.203900i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−15.0619			−0.654863
$530$		0			0
$531$		−	12.9955i		−	0.563956i
$532$		0			0
$533$	43.2075	−	11.1168i	1.87152	−	0.481523i
$534$		0			0
$535$			1.02916i			0.0444943i
$536$		0			0
$537$	−9.33107			−0.402665
$538$		0			0
$539$		−	6.41089i		−	0.276137i
$540$		0			0
$541$			12.8303i			0.551617i	0.961213	+	0.275808i	$0.0889456\pi$
$541$			12.8303i			0.551617i	−0.961213	+	0.275808i	$0.911054\pi$
$542$		0			0
$543$	22.0383			0.945754
$544$		0			0
$545$	6.75928			0.289536
$546$		0			0
$547$	1.56095			0.0667413			0.0333707	−	0.999443i	$-0.489376\pi$
$547$	1.56095			0.0667413			0.0333707	+	0.999443i	$0.489376\pi$
$548$		0			0
$549$	3.25353			0.138857
$550$		0			0
$551$		−	17.3087i		−	0.737376i
$552$		0			0
$553$			13.4021i			0.569914i
$554$		0			0
$555$	1.24072			0.0526655
$556$		0			0
$557$			38.6980i			1.63969i	0.572588	+	0.819843i	$0.305940\pi$
$557$			38.6980i			1.63969i	−0.572588	+	0.819843i	$0.694060\pi$
$558$		0			0
$559$	−2.58352	−	10.0413i	−0.109271	−	0.424702i
$560$		0			0
$561$			32.1868i			1.35893i
$562$		0			0
$563$	10.4099			0.438725			0.219363	−	0.975643i	$-0.429602\pi$
$563$	10.4099			0.438725			0.219363	+	0.975643i	$0.429602\pi$
$564$		0			0
$565$		−	10.9371i		−	0.460126i
$566$		0			0
$567$		−	1.00000i		−	0.0419961i
$568$		0			0
$569$	25.4351			1.06630			0.533148	−	0.846022i	$-0.321009\pi$
$569$	25.4351			1.06630			0.533148	+	0.846022i	$0.321009\pi$
$570$		0			0
$571$	−19.6931			−0.824132			−0.412066	−	0.911154i	$-0.635193\pi$
$571$	−19.6931			−0.824132			−0.412066	+	0.911154i	$0.635193\pi$
$572$		0			0
$573$	9.32453			0.389538
$574$		0			0
$575$	−13.1631			−0.548937
$576$		0			0
$577$			23.1996i			0.965814i	0.875672	+	0.482907i	$0.160419\pi$
$577$			23.1996i			0.965814i	−0.875672	+	0.482907i	$0.839581\pi$
$578$		0			0
$579$		−	1.47790i		−	0.0614195i
$580$		0			0
$581$	5.80865			0.240983
$582$		0			0
$583$		−	37.8006i		−	1.56554i
$584$		0			0
$585$	2.00000	−	0.514578i	0.0826898	−	0.0212752i
$586$		0			0
$587$		−	7.15316i		−	0.295243i	−0.989044	−	0.147621i	$-0.952838\pi$
$587$		−	7.15316i		−	0.295243i	0.989044	−	0.147621i	$-0.0471617\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$			5.57761i			0.229432i
$592$		0			0
$593$			23.8845i			0.980818i	0.871492	+	0.490409i	$0.163153\pi$
$593$			23.8845i			0.980818i	−0.871492	+	0.490409i	$0.836847\pi$
$594$		0			0
$595$	2.87566			0.117891
$596$		0			0
$597$	9.35302			0.382794
$598$		0			0
$599$	−14.5275			−0.593577			−0.296788	−	0.954943i	$-0.595916\pi$
$599$	−14.5275			−0.593577			−0.296788	+	0.954943i	$0.595916\pi$
$600$		0			0
$601$	−2.02550			−0.0826218			−0.0413109	−	0.999146i	$-0.513153\pi$
$601$	−2.02550			−0.0826218			−0.0413109	+	0.999146i	$0.513153\pi$
$602$		0			0
$603$		−	2.41841i		−	0.0984853i
$604$		0			0
$605$			17.2400i			0.700904i
$606$		0			0
$607$	34.9121			1.41704			0.708520	−	0.705691i	$-0.249363\pi$
$607$	34.9121			1.41704			0.708520	+	0.705691i	$0.249363\pi$
$608$		0			0
$609$		−	3.02065i		−	0.122403i
$610$		0			0
$611$	−32.9333	+	8.47339i	−1.33234	+	0.342796i
$612$		0			0
$613$			44.4325i			1.79461i	0.441408	+	0.897307i	$0.354479\pi$
$613$			44.4325i			1.79461i	−0.441408	+	0.897307i	$0.645521\pi$
$614$		0			0
$615$	7.08734			0.285789
$616$		0			0
$617$		−	32.5912i		−	1.31207i	−0.754730	−	0.656035i	$-0.772232\pi$
$617$		−	32.5912i		−	1.31207i	0.754730	−	0.656035i	$-0.227768\pi$
$618$		0			0
$619$			7.60621i			0.305719i	0.988248	+	0.152860i	$0.0488483\pi$
$619$			7.60621i			0.305719i	−0.988248	+	0.152860i	$0.951152\pi$
$620$		0			0
$621$	2.81747			0.113061
$622$		0			0
$623$	−14.0828			−0.564217
$624$		0			0
$625$	20.1867			0.807468
$626$		0			0
$627$	36.7352			1.46706
$628$		0			0
$629$			10.8757i			0.433641i
$630$		0			0
$631$			10.1365i			0.403527i	0.979434	+	0.201764i	$0.0646673\pi$
$631$			10.1365i			0.403527i	−0.979434	+	0.201764i	$0.935333\pi$
$632$		0			0
$633$	3.18253			0.126494
$634$		0			0
$635$		−	0.988490i		−	0.0392270i
$636$		0			0
$637$	−0.898409	−	3.49183i	−0.0355963	−	0.138351i
$638$		0			0
$639$		−	4.95418i		−	0.195984i
$640$		0			0
$641$	−37.1959			−1.46915			−0.734575	−	0.678528i	$-0.762618\pi$
$641$	−37.1959			−1.46915			−0.734575	+	0.678528i	$0.762618\pi$
$642$		0			0
$643$			35.2407i			1.38976i	0.719126	+	0.694880i	$0.244542\pi$
$643$			35.2407i			1.38976i	−0.719126	+	0.694880i	$0.755458\pi$
$644$		0			0
$645$		−	1.64708i		−	0.0648537i
$646$		0			0
$647$	5.12187			0.201362			0.100681	−	0.994919i	$-0.467898\pi$
$647$	5.12187			0.201362			0.100681	+	0.994919i	$0.467898\pi$
$648$		0			0
$649$	83.3126			3.27031
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−10.1572			−0.397480			−0.198740	−	0.980052i	$-0.563685\pi$
$653$	−10.1572			−0.397480			−0.198740	+	0.980052i	$0.563685\pi$
$654$		0			0
$655$			11.9381i			0.466462i
$656$		0			0
$657$		−	8.71378i		−	0.339957i
$658$		0			0
$659$	−33.6383			−1.31036			−0.655181	−	0.755472i	$-0.727408\pi$
$659$	−33.6383			−1.31036			−0.655181	+	0.755472i	$0.727408\pi$
$660$		0			0
$661$			22.5029i			0.875260i	0.899155	+	0.437630i	$0.144182\pi$
$661$			22.5029i			0.875260i	−0.899155	+	0.437630i	$0.855818\pi$
$662$		0			0
$663$	4.51060	+	17.5313i	0.175177	+	0.680857i
$664$		0			0
$665$		−	3.28202i		−	0.127271i
$666$		0			0
$667$	8.51060			0.329532
$668$		0			0
$669$			18.8631i			0.729289i
$670$		0			0
$671$			20.8580i			0.805215i
$672$		0			0
$673$	−6.98234			−0.269150			−0.134575	−	0.990903i	$-0.542967\pi$
$673$	−6.98234			−0.269150			−0.134575	+	0.990903i	$0.542967\pi$
$674$		0			0
$675$	−4.67194			−0.179823
$676$		0			0
$677$	26.2700			1.00964			0.504819	−	0.863225i	$-0.331559\pi$
$677$	26.2700			1.00964			0.504819	+	0.863225i	$0.331559\pi$
$678$		0			0
$679$	−4.94235			−0.189670
$680$		0			0
$681$		−	7.36054i		−	0.282057i
$682$		0			0
$683$		−	17.4220i		−	0.666633i	−0.942815	−	0.333316i	$-0.891832\pi$
$683$		−	17.4220i		−	0.666633i	0.942815	−	0.333316i	$-0.108168\pi$
$684$		0			0
$685$	7.62160			0.291206
$686$		0			0
$687$		−	27.8054i		−	1.06084i
$688$		0			0
$689$	−5.29730	−	20.5889i	−0.201811	−	0.784375i
$690$		0			0
$691$		−	5.61990i		−	0.213791i	−0.994270	−	0.106896i	$-0.965909\pi$
$691$		−	5.61990i		−	0.213791i	0.994270	−	0.106896i	$-0.0340910\pi$
$692$		0			0
$693$	6.41089			0.243530
$694$		0			0
$695$		−	8.78951i		−	0.333405i
$696$		0			0
$697$			62.1250i			2.35315i
$698$		0			0
$699$	7.96731			0.301351
$700$		0			0
$701$	−39.9462			−1.50875			−0.754374	−	0.656445i	$-0.772060\pi$
$701$	−39.9462			−1.50875			−0.754374	+	0.656445i	$0.772060\pi$
$702$		0			0
$703$	12.4125			0.468147
$704$		0			0
$705$	−5.40207			−0.203454
$706$		0			0
$707$			12.1123i			0.455530i
$708$		0			0
$709$			45.6143i			1.71308i	0.516080	+	0.856541i	$0.327391\pi$
$709$			45.6143i			1.71308i	−0.516080	+	0.856541i	$0.672609\pi$
$710$		0			0
$711$	−13.4021			−0.502617
$712$		0			0
$713$		0			0
$714$		0			0
$715$	3.29890	+	12.8218i	0.123372	+	0.479507i
$716$		0			0
$717$			9.20394i			0.343727i
$718$		0			0
$719$	24.7204			0.921917			0.460959	−	0.887422i	$-0.347506\pi$
$719$	24.7204			0.921917			0.460959	+	0.887422i	$0.347506\pi$
$720$		0			0
$721$			2.39906i			0.0893457i
$722$		0			0
$723$			7.25707i			0.269893i
$724$		0			0
$725$	−14.1123			−0.524118
$726$		0			0
$727$	32.1512			1.19242			0.596210	−	0.802828i	$-0.296672\pi$
$727$	32.1512			1.19242			0.596210	+	0.802828i	$0.296672\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	14.4377			0.533997
$732$		0			0
$733$		−	47.3167i		−	1.74768i	−0.486212	−	0.873841i	$-0.661622\pi$
$733$		−	47.3167i		−	1.74768i	0.486212	−	0.873841i	$-0.338378\pi$
$734$		0			0
$735$		−	0.572766i		−	0.0211268i
$736$		0			0
$737$	15.5042			0.571103
$738$		0			0
$739$			35.9140i			1.32112i	0.750775	+	0.660559i	$0.229680\pi$
$739$			35.9140i			1.32112i	−0.750775	+	0.660559i	$0.770320\pi$
$740$		0			0
$741$	20.0086	−	5.14800i	0.735035	−	0.189116i
$742$		0			0
$743$		−	18.1828i		−	0.667061i	−0.942739	−	0.333530i	$-0.891760\pi$
$743$		−	18.1828i		−	0.667061i	0.942739	−	0.333530i	$-0.108240\pi$
$744$		0			0
$745$	−4.63924			−0.169969
$746$		0			0
$747$			5.80865i			0.212527i
$748$		0			0
$749$			1.79682i			0.0656543i
$750$		0			0
$751$	−27.2045			−0.992707			−0.496353	−	0.868121i	$-0.665328\pi$
$751$	−27.2045			−0.992707			−0.496353	+	0.868121i	$0.665328\pi$
$752$		0			0
$753$	−6.75928			−0.246322
$754$		0			0
$755$	10.6683			0.388259
$756$		0			0
$757$	46.4409			1.68792			0.843962	−	0.536403i	$-0.180217\pi$
$757$	46.4409			1.68792			0.843962	+	0.536403i	$0.180217\pi$
$758$		0			0
$759$			18.0625i			0.655627i
$760$		0			0
$761$		−	40.1831i		−	1.45664i	−0.685239	−	0.728318i	$-0.740302\pi$
$761$		−	40.1831i		−	1.45664i	0.685239	−	0.728318i	$-0.259698\pi$
$762$		0			0
$763$	11.8011			0.427229
$764$		0			0
$765$			2.87566i			0.103970i
$766$		0			0
$767$	45.3780	−	11.6753i	1.63850	−	0.421569i
$768$		0			0
$769$		−	41.9847i		−	1.51401i	−0.653410	−	0.757004i	$-0.726662\pi$
$769$		−	41.9847i		−	1.51401i	0.653410	−	0.757004i	$-0.273338\pi$
$770$		0			0
$771$	17.8212			0.641816
$772$		0			0
$773$		−	23.9756i		−	0.862342i	−0.902270	−	0.431171i	$-0.858101\pi$
$773$		−	23.9756i		−	0.862342i	0.902270	−	0.431171i	$-0.141899\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	2.16618			0.0777114
$778$		0			0
$779$	70.9040			2.54040
$780$		0			0
$781$	31.7607			1.13649
$782$		0			0
$783$	3.02065			0.107949
$784$		0			0
$785$		−	5.19038i		−	0.185253i
$786$		0			0
$787$		−	5.67074i		−	0.202140i	−0.994879	−	0.101070i	$-0.967773\pi$
$787$		−	5.67074i		−	0.202140i	0.994879	−	0.101070i	$-0.0322266\pi$
$788$		0			0
$789$	27.3281			0.972905
$790$		0			0
$791$		−	19.0952i		−	0.678947i
$792$		0			0
$793$	2.92300	+	11.3608i	0.103799	+	0.403432i
$794$		0			0
$795$		−	3.37721i		−	0.119777i
$796$		0			0
$797$	−40.9528			−1.45062			−0.725311	−	0.688421i	$-0.758304\pi$
$797$	−40.9528			−1.45062			−0.725311	+	0.688421i	$0.758304\pi$
$798$		0			0
$799$		−	47.3525i		−	1.67521i
$800$		0			0
$801$		−	14.0828i		−	0.497592i
$802$		0			0
$803$	55.8631			1.97137
$804$		0			0
$805$	1.61375			0.0568773
$806$		0			0
$807$	20.9915			0.738936
$808$		0			0
$809$	12.6234			0.443816			0.221908	−	0.975068i	$-0.428771\pi$
$809$	12.6234			0.443816			0.221908	+	0.975068i	$0.428771\pi$
$810$		0			0
$811$			20.1380i			0.707141i	0.935408	+	0.353570i	$0.115033\pi$
$811$			20.1380i			0.707141i	−0.935408	+	0.353570i	$0.884967\pi$
$812$		0			0
$813$		−	5.30214i		−	0.185954i
$814$		0			0
$815$	−1.88470			−0.0660182
$816$		0			0
$817$		−	16.4779i		−	0.576488i
$818$		0			0
$819$	3.49183	−	0.898409i	0.122014	−	0.0313930i
$820$		0			0
$821$			48.7141i			1.70013i	0.526674	+	0.850067i	$0.323439\pi$
$821$			48.7141i			1.70013i	−0.526674	+	0.850067i	$0.676561\pi$
$822$		0			0
$823$	−32.6235			−1.13718			−0.568591	−	0.822620i	$-0.692511\pi$
$823$	−32.6235			−1.13718			−0.568591	+	0.822620i	$0.692511\pi$
$824$		0			0
$825$		−	29.9513i		−	1.04277i
$826$		0			0
$827$		−	2.33646i		−	0.0812466i	−0.999175	−	0.0406233i	$-0.987066\pi$
$827$		−	2.33646i		−	0.0812466i	0.999175	−	0.0406233i	$-0.0129344\pi$
$828$		0			0
$829$	−25.6866			−0.892132			−0.446066	−	0.895000i	$-0.647175\pi$
$829$	−25.6866			−0.892132			−0.446066	+	0.895000i	$0.647175\pi$
$830$		0			0
$831$	−19.8339			−0.688031
$832$		0			0
$833$	5.02065			0.173955
$834$		0			0
$835$	−1.92968			−0.0667792
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	27.0736i		−	0.934683i	−0.884077	−	0.467342i	$-0.845212\pi$
$839$		−	27.0736i		−	0.934683i	0.884077	−	0.467342i	$-0.154788\pi$
$840$		0			0
$841$	−19.8757			−0.685368
$842$		0			0
$843$		−	16.3897i		−	0.564491i
$844$		0			0
$845$	3.59364	+	6.52135i	0.123625	+	0.224341i
$846$		0			0
$847$			30.0995i			1.03423i
$848$		0			0
$849$	−14.5071			−0.497881
$850$		0			0
$851$			6.10316i			0.209214i
$852$		0			0
$853$		−	19.3840i		−	0.663697i	−0.943333	−	0.331848i	$-0.892328\pi$
$853$		−	19.3840i		−	0.663697i	0.943333	−	0.331848i	$-0.107672\pi$
$854$		0			0
$855$	3.28202			0.112243
$856$		0			0
$857$	50.3380			1.71951			0.859756	−	0.510704i	$-0.170615\pi$
$857$	50.3380			1.71951			0.859756	+	0.510704i	$0.170615\pi$
$858$		0			0
$859$	32.6974			1.11562			0.557811	−	0.829968i	$-0.311641\pi$
$859$	32.6974			1.11562			0.557811	+	0.829968i	$0.311641\pi$
$860$		0			0
$861$	12.3739			0.421701
$862$		0			0
$863$			4.07498i			0.138714i	0.997592	+	0.0693569i	$0.0220947\pi$
$863$			4.07498i			0.138714i	−0.997592	+	0.0693569i	$0.977905\pi$
$864$		0			0
$865$		−	6.43800i		−	0.218898i
$866$		0			0
$867$	−8.20695			−0.278723
$868$		0			0
$869$		−	85.9192i		−	2.91461i
$870$		0			0
$871$	8.44467	−	2.17272i	0.286137	−	0.0736199i
$872$		0			0
$873$		−	4.94235i		−	0.167273i
$874$		0			0
$875$	−5.53976			−0.187278
$876$		0			0
$877$		−	38.0202i		−	1.28385i	−0.766767	−	0.641926i	$-0.778136\pi$
$877$		−	38.0202i		−	1.28385i	0.766767	−	0.641926i	$-0.221864\pi$
$878$		0			0
$879$			15.0421i			0.507356i
$880$		0			0
$881$	26.5902			0.895847			0.447924	−	0.894072i	$-0.352164\pi$
$881$	26.5902			0.895847			0.447924	+	0.894072i	$0.352164\pi$
$882$		0			0
$883$	−23.0662			−0.776238			−0.388119	−	0.921609i	$-0.626875\pi$
$883$	−23.0662			−0.776238			−0.388119	+	0.921609i	$0.626875\pi$
$884$		0			0
$885$	7.44337			0.250206
$886$		0			0
$887$	−39.0201			−1.31017			−0.655084	−	0.755556i	$-0.727367\pi$
$887$	−39.0201			−1.31017			−0.655084	+	0.755556i	$0.727367\pi$
$888$		0			0
$889$		−	1.72582i		−	0.0578821i
$890$		0			0
$891$			6.41089i			0.214773i
$892$		0			0
$893$	−54.0439			−1.80851
$894$		0			0
$895$		−	5.34452i		−	0.178648i
$896$		0			0
$897$	2.53124	+	9.83812i	0.0845157	+	0.328485i
$898$		0			0
$899$		0			0
$900$		0			0
$901$	29.6033			0.986230
$902$		0			0
$903$		−	2.87566i		−	0.0956959i
$904$		0			0
$905$			12.6228i			0.419596i
$906$		0			0
$907$	55.4182			1.84013			0.920066	−	0.391764i	$-0.128135\pi$
$907$	55.4182			1.84013			0.920066	+	0.391764i	$0.128135\pi$
$908$		0			0
$909$	−12.1123			−0.401740
$910$		0			0
$911$	−39.6474			−1.31358			−0.656789	−	0.754074i	$-0.728086\pi$
$911$	−39.6474			−1.31358			−0.656789	+	0.754074i	$0.728086\pi$
$912$		0			0
$913$	−37.2386			−1.23242
$914$		0			0
$915$			1.86351i			0.0616058i
$916$		0			0
$917$			20.8430i			0.688295i
$918$		0			0
$919$	−38.5059			−1.27019			−0.635096	−	0.772433i	$-0.719039\pi$
$919$	−38.5059			−1.27019			−0.635096	+	0.772433i	$0.719039\pi$
$920$		0			0
$921$			23.7589i			0.782881i
$922$		0			0
$923$	17.2991	−	4.45088i	0.569408	−	0.146503i
$924$		0			0
$925$		−	10.1203i		−	0.332753i
$926$		0			0
$927$	−2.39906			−0.0787955
$928$		0			0
$929$		−	6.37990i		−	0.209318i	−0.994508	−	0.104659i	$-0.966625\pi$
$929$		−	6.37990i		−	0.209318i	0.994508	−	0.104659i	$-0.0333751\pi$
$930$		0			0
$931$		−	5.73013i		−	0.187797i
$932$		0			0
$933$	26.2257			0.858590
$934$		0			0
$935$	−18.4355			−0.602906
$936$		0			0
$937$	37.7949			1.23471			0.617353	−	0.786686i	$-0.288205\pi$
$937$	37.7949			1.23471			0.617353	+	0.786686i	$0.288205\pi$
$938$		0			0
$939$	−13.5768			−0.443061
$940$		0			0
$941$			23.6136i			0.769783i	0.922962	+	0.384891i	$0.125761\pi$
$941$			23.6136i			0.769783i	−0.922962	+	0.384891i	$0.874239\pi$
$942$		0			0
$943$			34.8631i			1.13530i
$944$		0			0
$945$	0.572766			0.0186321
$946$		0			0
$947$			19.5650i			0.635778i	0.948128	+	0.317889i	$0.102974\pi$
$947$			19.5650i			0.635778i	−0.948128	+	0.317889i	$0.897026\pi$
$948$		0			0
$949$	30.4270	−	7.82854i	0.987703	−	0.254125i
$950$		0			0
$951$			10.3721i			0.336337i
$952$		0			0
$953$	15.8193			0.512438			0.256219	−	0.966619i	$-0.417523\pi$
$953$	15.8193			0.512438			0.256219	+	0.966619i	$0.417523\pi$
$954$		0			0
$955$			5.34077i			0.172823i
$956$		0			0
$957$			19.3651i			0.625984i
$958$		0			0
$959$	13.3067			0.429695
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$	−1.79682			−0.0579017
$964$		0			0
$965$	0.846491			0.0272495
$966$		0			0
$967$			26.9673i			0.867211i	0.901103	+	0.433605i	$0.142759\pi$
$967$			26.9673i			0.867211i	−0.901103	+	0.433605i	$0.857241\pi$
$968$		0			0
$969$			28.7690i			0.924193i
$970$		0			0
$971$	20.5167			0.658414			0.329207	−	0.944258i	$-0.393219\pi$
$971$	20.5167			0.658414			0.329207	+	0.944258i	$0.393219\pi$
$972$		0			0
$973$		−	15.3457i		−	0.491961i
$974$		0			0
$975$	−4.19731	−	16.3136i	−0.134422	−	0.522454i
$976$		0			0
$977$			26.2599i			0.840127i	0.907495	+	0.420064i	$0.137992\pi$
$977$			26.2599i			0.840127i	−0.907495	+	0.420064i	$0.862008\pi$
$978$		0			0
$979$	90.2835			2.88547
$980$		0			0
$981$			11.8011i			0.376781i
$982$		0			0
$983$			25.6007i			0.816536i	0.912862	+	0.408268i	$0.133867\pi$
$983$			25.6007i			0.816536i	−0.912862	+	0.408268i	$0.866133\pi$
$984$		0			0
$985$	−3.19467			−0.101791
$986$		0			0
$987$	−9.43154			−0.300209
$988$		0			0
$989$	8.10209			0.257631
$990$		0			0
$991$	−42.9799			−1.36530			−0.682651	−	0.730745i	$-0.739173\pi$
$991$	−42.9799			−1.36530			−0.682651	+	0.730745i	$0.739173\pi$
$992$		0			0
$993$			26.7004i			0.847313i
$994$		0			0
$995$			5.35709i			0.169831i
$996$		0			0
$997$	−9.02420			−0.285799			−0.142900	−	0.989737i	$-0.545643\pi$
$997$	−9.02420			−0.285799			−0.142900	+	0.989737i	$0.545643\pi$
$998$		0			0
$999$			2.16618i			0.0685350i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.h.s.337.5		10
4.3	odd	2		2184.2.h.g.337.5	✓	10
13.12	even	2	inner	4368.2.h.s.337.6		10
52.51	odd	2		2184.2.h.g.337.6	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2184.2.h.g.337.5	✓	10	4.3	odd	2
2184.2.h.g.337.6	yes	10	52.51	odd	2
4368.2.h.s.337.5		10	1.1	even	1	trivial
4368.2.h.s.337.6		10	13.12	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.572766i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.572766i q^{5} -1.00000i q^{7} +1.00000 q^{9} +6.41089i q^{11} +(0.898409 + 3.49183i) q^{13} +0.572766i q^{15} -5.02065 q^{17} +5.73013i q^{19} +1.00000i q^{21} -2.81747 q^{23} +4.67194 q^{25} -1.00000 q^{27} -3.02065 q^{29} -6.41089i q^{33} -0.572766 q^{35} -2.16618i q^{37} +(-0.898409 - 3.49183i) q^{39} -12.3739i q^{41} -2.87566 q^{43} -0.572766i q^{45} +9.43154i q^{47} -1.00000 q^{49} +5.02065 q^{51} -5.89631 q^{53} +3.67194 q^{55} -5.73013i q^{57} -12.9955i q^{59} +3.25353 q^{61} -1.00000i q^{63} +(2.00000 - 0.514578i) q^{65} -2.41841i q^{67} +2.81747 q^{69} -4.95418i q^{71} -8.71378i q^{73} -4.67194 q^{75} +6.41089 q^{77} -13.4021 q^{79} +1.00000 q^{81} +5.80865i q^{83} +2.87566i q^{85} +3.02065 q^{87} -14.0828i q^{89} +(3.49183 - 0.898409i) q^{91} +3.28202 q^{95} -4.94235i q^{97} +6.41089i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{3} + 10 q^{9}+O(q^{10}) \) 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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