LMFDB - Embedded newform 2352.2.k.g.881.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,2,Mod(881,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2352.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2352.k (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:	$18.7808145554$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.3288334336.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 8x^{6} - 8x^{5} + 14x^{4} + 8x^{3} - 16x^{2} + 7 $ x^8 + 8x^6 - 8x^5 + 14x^4 + 8x^3 - 16*x^2 + 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 588)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.7
Root			$0.707107 - 1.43164i$ of defining polynomial
Character	$\chi$	$=$	2352.881
Dual form			2352.2.k.g.881.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.64533 - 0.541196i) q^{3} -0.963811 q^{5} +(2.41421 - 1.78089i) q^{9} +O(q^{10})$	$q+(1.64533 - 0.541196i) q^{3} -0.963811 q^{5} +(2.41421 - 1.78089i) q^{9} -4.29945i q^{11} -3.37849i q^{13} +(-1.58579 + 0.521611i) q^{15} -6.98054 q^{17} -3.06147i q^{19} +6.81801i q^{23} -4.07107 q^{25} +(3.00836 - 4.23671i) q^{27} -7.86123i q^{29} +8.47343i q^{31} +(-2.32685 - 7.07401i) q^{33} -6.24264 q^{37} +(-1.82843 - 5.55873i) q^{39} -3.68988 q^{41} +4.82843 q^{43} +(-2.32685 + 1.71644i) q^{45} -3.29066 q^{47} +(-11.4853 + 3.77784i) q^{51} -1.04322i q^{53} +4.14386i q^{55} +(-1.65685 - 5.03712i) q^{57} -2.72607 q^{59} +3.37849i q^{61} +3.25623i q^{65} +2.00000 q^{67} +(3.68988 + 11.2179i) q^{69} -10.3798i q^{71} -1.39942i q^{73} +(-6.69825 + 2.20325i) q^{75} +11.6569 q^{79} +(2.65685 - 8.59890i) q^{81} -1.36303 q^{83} +6.72792 q^{85} +(-4.25447 - 12.9343i) q^{87} -11.6342 q^{89} +(4.58579 + 13.9416i) q^{93} +2.95068i q^{95} -16.1815i q^{97} +(-7.65685 - 10.3798i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^9	$ 8 q + 8 q^{9} - 24 q^{15} + 24 q^{25} - 16 q^{37} + 8 q^{39} + 16 q^{43} - 24 q^{51} + 32 q^{57} + 16 q^{67} + 48 q^{79} - 24 q^{81} - 48 q^{85} + 48 q^{93} - 16 q^{99}+O(q^{100}) $ 8 * q + 8 * q^9 - 24 * q^15 + 24 * q^25 - 16 * q^37 + 8 * q^39 + 16 * q^43 - 24 * q^51 + 32 * q^57 + 16 * q^67 + 48 * q^79 - 24 * q^81 - 48 * q^85 + 48 * q^93 - 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$3$	1.64533	−	0.541196i	0.949931	−	0.312460i
$3$	1.64533	−	0.541196i	0.949931	−	0.312460i
$4$		0			0
$5$	−0.963811			−0.431030			−0.215515	−	0.976501i	$-0.569143\pi$
$5$	−0.963811			−0.431030			−0.215515	+	0.976501i	$0.569143\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	2.41421	−	1.78089i	0.804738	−	0.593630i
$10$		0			0
$11$		−	4.29945i		−	1.29633i	−0.761499	−	0.648167i	$-0.775536\pi$
$11$		−	4.29945i		−	1.29633i	0.761499	−	0.648167i	$-0.224464\pi$
$12$		0			0
$13$		−	3.37849i		−	0.937025i	−0.883457	−	0.468513i	$-0.844790\pi$
$13$		−	3.37849i		−	0.937025i	0.883457	−	0.468513i	$-0.155210\pi$
$14$		0			0
$15$	−1.58579	+	0.521611i	−0.409448	+	0.134679i
$16$		0			0
$17$	−6.98054			−1.69303			−0.846515	−	0.532365i	$-0.821303\pi$
$17$	−6.98054			−1.69303			−0.846515	+	0.532365i	$0.821303\pi$
$18$		0			0
$19$		−	3.06147i		−	0.702349i	−0.936310	−	0.351174i	$-0.885782\pi$
$19$		−	3.06147i		−	0.702349i	0.936310	−	0.351174i	$-0.114218\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			6.81801i			1.42165i	0.703367	+	0.710827i	$0.251679\pi$
$23$			6.81801i			1.42165i	−0.703367	+	0.710827i	$0.748321\pi$
$24$		0			0
$25$	−4.07107			−0.814214
$26$		0			0
$27$	3.00836	−	4.23671i	0.578960	−	0.815356i
$28$		0			0
$29$		−	7.86123i		−	1.45979i	−0.683557	−	0.729897i	$-0.739568\pi$
$29$		−	7.86123i		−	1.45979i	0.683557	−	0.729897i	$-0.260432\pi$
$30$		0			0
$31$			8.47343i			1.52187i	0.648827	+	0.760936i	$0.275260\pi$
$31$			8.47343i			1.52187i	−0.648827	+	0.760936i	$0.724740\pi$
$32$		0			0
$33$	−2.32685	−	7.07401i	−0.405052	−	1.23143i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−6.24264			−1.02628			−0.513142	−	0.858304i	$-0.671519\pi$
$37$	−6.24264			−1.02628			−0.513142	+	0.858304i	$0.671519\pi$
$38$		0			0
$39$	−1.82843	−	5.55873i	−0.292783	−	0.890109i
$40$		0			0
$41$	−3.68988			−0.576263			−0.288131	−	0.957591i	$-0.593034\pi$
$41$	−3.68988			−0.576263			−0.288131	+	0.957591i	$0.593034\pi$
$42$		0			0
$43$	4.82843			0.736328			0.368164	−	0.929761i	$-0.379986\pi$
$43$	4.82843			0.736328			0.368164	+	0.929761i	$0.379986\pi$
$44$		0			0
$45$	−2.32685	+	1.71644i	−0.346866	+	0.255872i
$46$		0			0
$47$	−3.29066			−0.479992			−0.239996	−	0.970774i	$-0.577146\pi$
$47$	−3.29066			−0.479992			−0.239996	+	0.970774i	$0.577146\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−11.4853	+	3.77784i	−1.60826	+	0.529003i
$52$		0			0
$53$		−	1.04322i		−	0.143298i	−0.997430	−	0.0716488i	$-0.977174\pi$
$53$		−	1.04322i		−	0.143298i	0.997430	−	0.0716488i	$-0.0228261\pi$
$54$		0			0
$55$			4.14386i			0.558758i
$56$		0			0
$57$	−1.65685	−	5.03712i	−0.219456	−	0.667183i
$58$		0			0
$59$	−2.72607			−0.354904			−0.177452	−	0.984129i	$-0.556785\pi$
$59$	−2.72607			−0.354904			−0.177452	+	0.984129i	$0.556785\pi$
$60$		0			0
$61$			3.37849i			0.432572i	0.976330	+	0.216286i	$0.0693943\pi$
$61$			3.37849i			0.432572i	−0.976330	+	0.216286i	$0.930606\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			3.25623i			0.403886i
$66$		0			0
$67$	2.00000			0.244339			0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000			0.244339			0.122169	+	0.992509i	$0.461015\pi$
$68$		0			0
$69$	3.68988	+	11.2179i	0.444210	+	1.35047i
$70$		0			0
$71$		−	10.3798i		−	1.23185i	−0.787803	−	0.615927i	$-0.788781\pi$
$71$		−	10.3798i		−	1.23185i	0.787803	−	0.615927i	$-0.211219\pi$
$72$		0			0
$73$		−	1.39942i		−	0.163789i	−0.996641	−	0.0818947i	$-0.973903\pi$
$73$		−	1.39942i		−	0.163789i	0.996641	−	0.0818947i	$-0.0260971\pi$
$74$		0			0
$75$	−6.69825	+	2.20325i	−0.773447	+	0.254409i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	11.6569			1.31150			0.655749	−	0.754979i	$-0.272353\pi$
$79$	11.6569			1.31150			0.655749	+	0.754979i	$0.272353\pi$
$80$		0			0
$81$	2.65685	−	8.59890i	0.295206	−	0.955434i
$82$		0			0
$83$	−1.36303			−0.149613			−0.0748063	−	0.997198i	$-0.523834\pi$
$83$	−1.36303			−0.149613			−0.0748063	+	0.997198i	$0.523834\pi$
$84$		0			0
$85$	6.72792			0.729746
$86$		0			0
$87$	−4.25447	−	12.9343i	−0.456127	−	1.38670i
$88$		0			0
$89$	−11.6342			−1.23323			−0.616613	−	0.787266i	$-0.711496\pi$
$89$	−11.6342			−1.23323			−0.616613	+	0.787266i	$0.711496\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	4.58579	+	13.9416i	0.475524	+	1.44567i
$94$		0			0
$95$			2.95068i			0.302733i
$96$		0			0
$97$		−	16.1815i		−	1.64298i	−0.570222	−	0.821491i	$-0.693143\pi$
$97$		−	16.1815i		−	1.64298i	0.570222	−	0.821491i	$-0.306857\pi$
$98$		0			0
$99$	−7.65685	−	10.3798i	−0.769543	−	1.04321i
$100$		0			0
$101$	8.90816			0.886395			0.443198	−	0.896424i	$-0.353844\pi$
$101$	8.90816			0.886395			0.443198	+	0.896424i	$0.353844\pi$
$102$		0			0
$103$		−	13.0656i		−	1.28739i	−0.765280	−	0.643697i	$-0.777400\pi$
$103$		−	13.0656i		−	1.28739i	0.765280	−	0.643697i	$-0.222600\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	9.33657i		−	0.902600i	−0.892372	−	0.451300i	$-0.850960\pi$
$107$		−	9.33657i		−	0.902600i	0.892372	−	0.451300i	$-0.149040\pi$
$108$		0			0
$109$	6.58579			0.630804			0.315402	−	0.948958i	$-0.397861\pi$
$109$	6.58579			0.630804			0.315402	+	0.948958i	$0.397861\pi$
$110$		0			0
$111$	−10.2712	+	3.37849i	−0.974899	+	0.320672i
$112$		0			0
$113$			6.08034i			0.571990i	0.958231	+	0.285995i	$0.0923242\pi$
$113$			6.08034i			0.571990i	−0.958231	+	0.285995i	$0.907676\pi$
$114$		0			0
$115$		−	6.57128i		−	0.612775i
$116$		0			0
$117$	−6.01673	−	8.15640i	−0.556247	−	0.754060i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−7.48528			−0.680480
$122$		0			0
$123$	−6.07107	+	1.99695i	−0.547410	+	0.180059i
$124$		0			0
$125$	8.74280			0.781980
$126$		0			0
$127$	11.1716			0.991317			0.495658	−	0.868518i	$-0.334927\pi$
$127$	11.1716			0.991317			0.495658	+	0.868518i	$0.334927\pi$
$128$		0			0
$129$	7.94435	−	2.61313i	0.699461	−	0.230073i
$130$		0			0
$131$	−19.1794			−1.67571			−0.837854	−	0.545894i	$-0.816190\pi$
$131$	−19.1794			−1.67571			−0.837854	+	0.545894i	$0.816190\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−2.89949	+	4.08339i	−0.249549	+	0.351443i
$136$		0			0
$137$		−	20.0219i		−	1.71059i	−0.518143	−	0.855294i	$-0.673377\pi$
$137$		−	20.0219i		−	1.71059i	0.518143	−	0.855294i	$-0.326623\pi$
$138$		0			0
$139$		−	7.83938i		−	0.664927i	−0.943116	−	0.332464i	$-0.892120\pi$
$139$		−	7.83938i		−	0.664927i	0.943116	−	0.332464i	$-0.107880\pi$
$140$		0			0
$141$	−5.41421	+	1.78089i	−0.455959	+	0.149978i
$142$		0			0
$143$	−14.5257			−1.21470
$144$		0			0
$145$			7.57675i			0.629214i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			16.7657i			1.37350i	0.726894	+	0.686749i	$0.240963\pi$
$149$			16.7657i			1.37350i	−0.726894	+	0.686749i	$0.759037\pi$
$150$		0			0
$151$	13.1716			1.07189			0.535944	−	0.844254i	$-0.319956\pi$
$151$	13.1716			1.07189			0.535944	+	0.844254i	$0.319956\pi$
$152$		0			0
$153$	−16.8525	+	12.4316i	−1.36244	+	1.00503i
$154$		0			0
$155$		−	8.16679i		−	0.655972i
$156$		0			0
$157$		−	0.131316i		−	0.0104802i	−0.999986	−	0.00524009i	$-0.998332\pi$
$157$		−	0.131316i		−	0.0104802i	0.999986	−	0.00524009i	$-0.00166798\pi$
$158$		0			0
$159$	−0.564588	−	1.71644i	−0.0447747	−	0.136123i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	2.82843			0.221540			0.110770	−	0.993846i	$-0.464668\pi$
$163$	2.82843			0.221540			0.110770	+	0.993846i	$0.464668\pi$
$164$		0			0
$165$	2.24264	+	6.81801i	0.174589	+	0.530781i
$166$		0			0
$167$	−7.14590			−0.552967			−0.276483	−	0.961019i	$-0.589169\pi$
$167$	−7.14590			−0.552967			−0.276483	+	0.961019i	$0.589169\pi$
$168$		0			0
$169$	1.58579			0.121984
$170$		0			0
$171$	−5.45214	−	7.39104i	−0.416936	−	0.565207i
$172$		0			0
$173$	−5.61750			−0.427091			−0.213545	−	0.976933i	$-0.568501\pi$
$173$	−5.61750			−0.427091			−0.213545	+	0.976933i	$0.568501\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−4.48528	+	1.47534i	−0.337134	+	0.110893i
$178$		0			0
$179$			0.305553i			0.0228381i	0.999935	+	0.0114190i	$0.00363487\pi$
$179$			0.305553i			0.0228381i	−0.999935	+	0.0114190i	$0.996365\pi$
$180$		0			0
$181$			6.62567i			0.492482i	0.969209	+	0.246241i	$0.0791955\pi$
$181$			6.62567i			0.492482i	−0.969209	+	0.246241i	$0.920805\pi$
$182$		0			0
$183$	1.82843	+	5.55873i	0.135161	+	0.410913i
$184$		0			0
$185$	6.01673			0.442359
$186$		0			0
$187$			30.0125i			2.19473i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			8.29335i			0.600086i	0.953926	+	0.300043i	$0.0970010\pi$
$191$			8.29335i			0.600086i	−0.953926	+	0.300043i	$0.902999\pi$
$192$		0			0
$193$	19.6569			1.41493			0.707466	−	0.706748i	$-0.249838\pi$
$193$	19.6569			1.41493			0.707466	+	0.706748i	$0.249838\pi$
$194$		0			0
$195$	1.76226	+	5.35757i	0.126198	+	0.383663i
$196$		0			0
$197$		−	22.2349i		−	1.58417i	−0.610409	−	0.792086i	$-0.708995\pi$
$197$		−	22.2349i		−	1.58417i	0.610409	−	0.792086i	$-0.291005\pi$
$198$		0			0
$199$			17.2095i			1.21995i	0.792421	+	0.609974i	$0.208820\pi$
$199$			17.2095i			1.21995i	−0.792421	+	0.609974i	$0.791180\pi$
$200$		0			0
$201$	3.29066	−	1.08239i	0.232105	−	0.0763461i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	3.55635			0.248386
$206$		0			0
$207$	12.1421	+	16.4601i	0.843937	+	1.14406i
$208$		0			0
$209$	−13.1626			−0.910478
$210$		0			0
$211$	13.3137			0.916553			0.458277	−	0.888810i	$-0.348467\pi$
$211$	13.3137			0.916553			0.458277	+	0.888810i	$0.348467\pi$
$212$		0			0
$213$	−5.61750	−	17.0782i	−0.384905	−	1.17018i
$214$		0			0
$215$	−4.65369			−0.317379
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−0.757359	−	2.30250i	−0.0511776	−	0.155589i
$220$		0			0
$221$			23.5837i			1.58641i
$222$		0			0
$223$		−	0.896683i		−	0.0600463i	−0.999549	−	0.0300232i	$-0.990442\pi$
$223$		−	0.896683i		−	0.0600463i	0.999549	−	0.0300232i	$-0.00955811\pi$
$224$		0			0
$225$	−9.82843	+	7.25013i	−0.655228	+	0.483342i
$226$		0			0
$227$	16.6871			1.10756			0.553782	−	0.832661i	$-0.313184\pi$
$227$	16.6871			1.10756			0.553782	+	0.832661i	$0.313184\pi$
$228$		0			0
$229$			11.8519i			0.783197i	0.920136	+	0.391599i	$0.128078\pi$
$229$			11.8519i			0.783197i	−0.920136	+	0.391599i	$0.871922\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	11.4230i		−	0.748347i	−0.927359	−	0.374173i	$-0.877927\pi$
$233$		−	11.4230i		−	0.748347i	0.927359	−	0.374173i	$-0.122073\pi$
$234$		0			0
$235$	3.17157			0.206891
$236$		0			0
$237$	19.1794	−	6.30864i	1.24583	−	0.409790i
$238$		0			0
$239$			3.86733i			0.250157i	0.992147	+	0.125079i	$0.0399183\pi$
$239$			3.86733i			0.250157i	−0.992147	+	0.125079i	$0.960082\pi$
$240$		0			0
$241$			20.5880i			1.32619i	0.748536	+	0.663094i	$0.230757\pi$
$241$			20.5880i			1.32619i	−0.748536	+	0.663094i	$0.769243\pi$
$242$		0			0
$243$	−0.282294	−	15.5859i	−0.0181092	−	0.999836i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−10.3431			−0.658119
$248$		0			0
$249$	−2.24264	+	0.737669i	−0.142122	+	0.0467479i
$250$		0			0
$251$	9.30739			0.587477			0.293738	−	0.955886i	$-0.405101\pi$
$251$	9.30739			0.587477			0.293738	+	0.955886i	$0.405101\pi$
$252$		0			0
$253$	29.3137			1.84294
$254$		0			0
$255$	11.0696	−	3.64113i	0.693208	−	0.228016i
$256$		0			0
$257$	−12.1988			−0.760941			−0.380471	−	0.924793i	$-0.624238\pi$
$257$	−12.1988			−0.760941			−0.380471	+	0.924793i	$0.624238\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−14.0000	−	18.9787i	−0.866578	−	1.17475i
$262$		0			0
$263$		−	15.4169i		−	0.950648i	−0.879811	−	0.475324i	$-0.842331\pi$
$263$		−	15.4169i		−	0.950648i	0.879811	−	0.475324i	$-0.157669\pi$
$264$		0			0
$265$			1.00547i			0.0617655i
$266$		0			0
$267$	−19.1421	+	6.29640i	−1.17148	+	0.385333i
$268$		0			0
$269$	−16.2879			−0.993092			−0.496546	−	0.868010i	$-0.665399\pi$
$269$	−16.2879			−0.993092			−0.496546	+	0.868010i	$0.665399\pi$
$270$		0			0
$271$		−	2.87576i		−	0.174690i	−0.996178	−	0.0873449i	$-0.972162\pi$
$271$		−	2.87576i		−	0.174690i	0.996178	−	0.0873449i	$-0.0278382\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			17.5034i			1.05549i
$276$		0			0
$277$	−29.4558			−1.76983			−0.884915	−	0.465752i	$-0.845784\pi$
$277$	−29.4558			−1.76983			−0.884915	+	0.465752i	$0.845784\pi$
$278$		0			0
$279$	15.0903	+	20.4567i	0.903430	+	1.22471i
$280$		0			0
$281$			19.4108i			1.15795i	0.815345	+	0.578976i	$0.196548\pi$
$281$			19.4108i			1.15795i	−0.815345	+	0.578976i	$0.803452\pi$
$282$		0			0
$283$			3.32410i			0.197597i	0.995107	+	0.0987986i	$0.0315000\pi$
$283$			3.32410i			0.197597i	−0.995107	+	0.0987986i	$0.968500\pi$
$284$		0			0
$285$	1.59689	+	4.85483i	0.0945919	+	0.287576i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	31.7279			1.86635
$290$		0			0
$291$	−8.75736	−	26.6239i	−0.513365	−	1.56072i
$292$		0			0
$293$	10.8358			0.633033			0.316517	−	0.948587i	$-0.397487\pi$
$293$	10.8358			0.633033			0.316517	+	0.948587i	$0.397487\pi$
$294$		0			0
$295$	2.62742			0.152974
$296$		0			0
$297$	−18.2155	−	12.9343i	−1.05697	−	0.750525i
$298$		0			0
$299$	23.0346			1.33213
$300$		0			0
$301$		0			0
$302$		0			0
$303$	14.6569	−	4.82106i	0.842014	−	0.276963i
$304$		0			0
$305$		−	3.25623i		−	0.186451i
$306$		0			0
$307$		−	11.9832i		−	0.683919i	−0.939715	−	0.341960i	$-0.888909\pi$
$307$		−	11.9832i		−	0.683919i	0.939715	−	0.341960i	$-0.111091\pi$
$308$		0			0
$309$	−7.07107	−	21.4973i	−0.402259	−	1.22294i
$310$		0			0
$311$	24.9622			1.41548			0.707739	−	0.706474i	$-0.249715\pi$
$311$	24.9622			1.41548			0.707739	+	0.706474i	$0.249715\pi$
$312$		0			0
$313$		−	18.8715i		−	1.06668i	−0.845900	−	0.533341i	$-0.820936\pi$
$313$		−	18.8715i		−	1.06668i	0.845900	−	0.533341i	$-0.179064\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	2.08644i		−	0.117186i	−0.998282	−	0.0585932i	$-0.981339\pi$
$317$		−	2.08644i		−	0.117186i	0.998282	−	0.0585932i	$-0.0186615\pi$
$318$		0			0
$319$	−33.7990			−1.89238
$320$		0			0
$321$	−5.05292	−	15.3617i	−0.282026	−	0.857408i
$322$		0			0
$323$			21.3707i			1.18910i
$324$		0			0
$325$			13.7541i			0.762939i
$326$		0			0
$327$	10.8358	−	3.56420i	0.599220	−	0.197101i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−15.7990			−0.868391			−0.434196	−	0.900819i	$-0.642967\pi$
$331$	−15.7990			−0.868391			−0.434196	+	0.900819i	$0.642967\pi$
$332$		0			0
$333$	−15.0711	+	11.1175i	−0.825889	+	0.609233i
$334$		0			0
$335$	−1.92762			−0.105317
$336$		0			0
$337$	14.5858			0.794538			0.397269	−	0.917702i	$-0.369958\pi$
$337$	14.5858			0.794538			0.397269	+	0.917702i	$0.369958\pi$
$338$		0			0
$339$	3.29066	+	10.0042i	0.178724	+	0.543352i
$340$		0			0
$341$	36.4311			1.97285
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−3.55635	−	10.8119i	−0.191467	−	0.582094i
$346$		0			0
$347$			10.9909i			0.590022i	0.955494	+	0.295011i	$0.0953234\pi$
$347$			10.9909i			0.590022i	−0.955494	+	0.295011i	$0.904677\pi$
$348$		0			0
$349$		−	7.89377i		−	0.422544i	−0.977427	−	0.211272i	$-0.932239\pi$
$349$		−	7.89377i		−	0.422544i	0.977427	−	0.211272i	$-0.0677606\pi$
$350$		0			0
$351$	−14.3137	−	10.1637i	−0.764009	−	0.542500i
$352$		0			0
$353$	−17.9817			−0.957069			−0.478534	−	0.878069i	$-0.658832\pi$
$353$	−17.9817			−0.957069			−0.478534	+	0.878069i	$0.658832\pi$
$354$		0			0
$355$			10.0042i			0.530966i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			5.34267i			0.281976i	0.990011	+	0.140988i	$0.0450278\pi$
$359$			5.34267i			0.281976i	−0.990011	+	0.140988i	$0.954972\pi$
$360$		0			0
$361$	9.62742			0.506706
$362$		0			0
$363$	−12.3157	+	4.05101i	−0.646409	+	0.212623i
$364$		0			0
$365$			1.34877i			0.0705981i
$366$		0			0
$367$		−	4.96362i		−	0.259099i	−0.991573	−	0.129549i	$-0.958647\pi$
$367$		−	4.96362i		−	0.259099i	0.991573	−	0.129549i	$-0.0413531\pi$
$368$		0			0
$369$	−8.90816	+	6.57128i	−0.463740	+	0.342087i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	15.7990			0.818041			0.409020	−	0.912525i	$-0.365871\pi$
$373$	15.7990			0.818041			0.409020	+	0.912525i	$0.365871\pi$
$374$		0			0
$375$	14.3848	−	4.73157i	0.742827	−	0.244337i
$376$		0			0
$377$	−26.5591			−1.36786
$378$		0			0
$379$	1.65685			0.0851069			0.0425534	−	0.999094i	$-0.486451\pi$
$379$	1.65685			0.0851069			0.0425534	+	0.999094i	$0.486451\pi$
$380$		0			0
$381$	18.3809	−	6.04601i	0.941683	−	0.309747i
$382$		0			0
$383$	20.5424			1.04967			0.524834	−	0.851205i	$-0.324128\pi$
$383$	20.5424			1.04967			0.524834	+	0.851205i	$0.324128\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	11.6569	−	8.59890i	0.592551	−	0.437107i
$388$		0			0
$389$			16.4601i			0.834562i	0.908778	+	0.417281i	$0.137017\pi$
$389$			16.4601i			0.834562i	−0.908778	+	0.417281i	$0.862983\pi$
$390$		0			0
$391$		−	47.5934i		−	2.40690i
$392$		0			0
$393$	−31.5563	+	10.3798i	−1.59181	+	0.523591i
$394$		0			0
$395$	−11.2350			−0.565295
$396$		0			0
$397$		−	12.3003i		−	0.617332i	−0.951170	−	0.308666i	$-0.900117\pi$
$397$		−	12.3003i		−	0.617332i	0.951170	−	0.308666i	$-0.0998826\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	30.0962i		−	1.50293i	−0.659773	−	0.751465i	$-0.729347\pi$
$401$		−	30.0962i		−	1.50293i	0.659773	−	0.751465i	$-0.270653\pi$
$402$		0			0
$403$	28.6274			1.42603
$404$		0			0
$405$	−2.56071	+	8.28772i	−0.127243	+	0.411820i
$406$		0			0
$407$			26.8399i			1.33041i
$408$		0			0
$409$		−	9.68714i		−	0.478998i	−0.970897	−	0.239499i	$-0.923017\pi$
$409$		−	9.68714i		−	0.478998i	0.970897	−	0.239499i	$-0.0769832\pi$
$410$		0			0
$411$	−10.8358	−	32.9426i	−0.534490	−	1.62494i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	1.31371			0.0644874
$416$		0			0
$417$	−4.24264	−	12.8984i	−0.207763	−	0.631635i
$418$		0			0
$419$	12.3642			0.604030			0.302015	−	0.953303i	$-0.402341\pi$
$419$	12.3642			0.604030			0.302015	+	0.953303i	$0.402341\pi$
$420$		0			0
$421$	−27.7990			−1.35484			−0.677420	−	0.735597i	$-0.736902\pi$
$421$	−27.7990			−1.35484			−0.677420	+	0.735597i	$0.736902\pi$
$422$		0			0
$423$	−7.94435	+	5.86030i	−0.386268	+	0.284938i
$424$		0			0
$425$	28.4182			1.37849
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−23.8995	+	7.86123i	−1.15388	+	0.379544i
$430$		0			0
$431$		−	2.82411i		−	0.136033i	−0.997684	−	0.0680164i	$-0.978333\pi$
$431$		−	2.82411i		−	0.136033i	0.997684	−	0.0680164i	$-0.0216670\pi$
$432$		0			0
$433$			5.91470i			0.284242i	0.989849	+	0.142121i	$0.0453922\pi$
$433$			5.91470i			0.284242i	−0.989849	+	0.142121i	$0.954608\pi$
$434$		0			0
$435$	4.10051	+	12.4662i	0.196604	+	0.597710i
$436$		0			0
$437$	20.8731			0.998497
$438$		0			0
$439$			34.4190i			1.64273i	0.570404	+	0.821364i	$0.306787\pi$
$439$			34.4190i			1.64273i	−0.570404	+	0.821364i	$0.693213\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	29.6640i		−	1.40938i	−0.709515	−	0.704691i	$-0.751086\pi$
$443$		−	29.6640i		−	1.40938i	0.709515	−	0.704691i	$-0.248914\pi$
$444$		0			0
$445$	11.2132			0.531557
$446$		0			0
$447$	9.07353	+	27.5851i	0.429163	+	1.30473i
$448$		0			0
$449$			21.8028i			1.02894i	0.857509	+	0.514469i	$0.172011\pi$
$449$			21.8028i			1.02894i	−0.857509	+	0.514469i	$0.827989\pi$
$450$		0			0
$451$			15.8645i			0.747028i
$452$		0			0
$453$	21.6716	−	7.12840i	1.01822	−	0.334922i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−4.00000			−0.187112			−0.0935561	−	0.995614i	$-0.529823\pi$
$457$	−4.00000			−0.187112			−0.0935561	+	0.995614i	$0.529823\pi$
$458$		0			0
$459$	−21.0000	+	29.5745i	−0.980196	+	1.38042i
$460$		0			0
$461$	40.4517			1.88402			0.942012	−	0.335580i	$-0.108932\pi$
$461$	40.4517			1.88402			0.942012	+	0.335580i	$0.108932\pi$
$462$		0			0
$463$	−16.2843			−0.756794			−0.378397	−	0.925643i	$-0.623525\pi$
$463$	−16.2843			−0.756794			−0.378397	+	0.925643i	$0.623525\pi$
$464$		0			0
$465$	−4.41983	−	13.4370i	−0.204965	−	0.623128i
$466$		0			0
$467$	36.4311			1.68583			0.842915	−	0.538047i	$-0.180838\pi$
$467$	36.4311			1.68583			0.842915	+	0.538047i	$0.180838\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−0.0710678	−	0.216058i	−0.00327463	−	0.00995544i
$472$		0			0
$473$		−	20.7596i		−	0.954527i
$474$		0			0
$475$			12.4634i			0.571862i
$476$		0			0
$477$	−1.85786	−	2.51856i	−0.0850658	−	0.115317i
$478$		0			0
$479$	−27.6883			−1.26511			−0.632555	−	0.774515i	$-0.717994\pi$
$479$	−27.6883			−1.26511			−0.632555	+	0.774515i	$0.717994\pi$
$480$		0			0
$481$			21.0907i			0.961654i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			15.5959i			0.708173i
$486$		0			0
$487$	−38.1421			−1.72839			−0.864193	−	0.503161i	$-0.832170\pi$
$487$	−38.1421			−1.72839			−0.864193	+	0.503161i	$0.832170\pi$
$488$		0			0
$489$	4.65369	−	1.53073i	0.210447	−	0.0692222i
$490$		0			0
$491$		−	14.9848i		−	0.676254i	−0.941100	−	0.338127i	$-0.890207\pi$
$491$		−	14.9848i		−	0.676254i	0.941100	−	0.338127i	$-0.109793\pi$
$492$		0			0
$493$			54.8756i			2.47147i
$494$		0			0
$495$	7.37976	+	10.0042i	0.331696	+	0.449654i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	2.14214			0.0958952			0.0479476	−	0.998850i	$-0.484732\pi$
$499$	2.14214			0.0958952			0.0479476	+	0.998850i	$0.484732\pi$
$500$		0			0
$501$	−11.7574	+	3.86733i	−0.525280	+	0.172780i
$502$		0			0
$503$	−0.798447			−0.0356010			−0.0178005	−	0.999842i	$-0.505666\pi$
$503$	−0.798447			−0.0356010			−0.0178005	+	0.999842i	$0.505666\pi$
$504$		0			0
$505$	−8.58579			−0.382062
$506$		0			0
$507$	2.60914	−	0.858221i	0.115876	−	0.0381150i
$508$		0			0
$509$	7.77899			0.344798			0.172399	−	0.985027i	$-0.444848\pi$
$509$	7.77899			0.344798			0.172399	+	0.985027i	$0.444848\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−12.9706	−	9.21001i	−0.572664	−	0.406632i
$514$		0			0
$515$			12.5928i			0.554905i
$516$		0			0
$517$			14.1480i			0.622229i
$518$		0			0
$519$	−9.24264	+	3.04017i	−0.405707	+	0.133449i
$520$		0			0
$521$	36.2657			1.58883			0.794415	−	0.607375i	$-0.207777\pi$
$521$	36.2657			1.58883			0.794415	+	0.607375i	$0.207777\pi$
$522$		0			0
$523$		−	14.0711i		−	0.615286i	−0.951502	−	0.307643i	$-0.900460\pi$
$523$		−	14.0711i		−	0.615286i	0.951502	−	0.307643i	$-0.0995403\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	59.1491i		−	2.57658i
$528$		0			0
$529$	−23.4853			−1.02110
$530$		0			0
$531$	−6.58132	+	4.85483i	−0.285605	+	0.210682i
$532$		0			0
$533$			12.4662i			0.539973i
$534$		0			0
$535$			8.99869i			0.389047i
$536$		0			0
$537$	0.165364	+	0.502734i	0.00713598	+	0.0216946i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	12.6274			0.542895			0.271448	−	0.962453i	$-0.412498\pi$
$541$	12.6274			0.542895			0.271448	+	0.962453i	$0.412498\pi$
$542$		0			0
$543$	3.58579	+	10.9014i	0.153881	+	0.467824i
$544$		0			0
$545$	−6.34746			−0.271895
$546$		0			0
$547$	6.97056			0.298040			0.149020	−	0.988834i	$-0.452388\pi$
$547$	6.97056			0.298040			0.149020	+	0.988834i	$0.452388\pi$
$548$		0			0
$549$	6.01673	+	8.15640i	0.256788	+	0.348107i
$550$		0			0
$551$	−24.0669			−1.02528
$552$		0			0
$553$		0			0
$554$		0			0
$555$	9.89949	−	3.25623i	0.420210	−	0.138219i
$556$		0			0
$557$			2.08644i			0.0884055i	0.999023	+	0.0442027i	$0.0140748\pi$
$557$			2.08644i			0.0884055i	−0.999023	+	0.0442027i	$0.985925\pi$
$558$		0			0
$559$		−	16.3128i		−	0.689958i
$560$		0			0
$561$	16.2426	+	49.3804i	0.685765	+	2.08484i
$562$		0			0
$563$	1.59689			0.0673011			0.0336505	−	0.999434i	$-0.489287\pi$
$563$	1.59689			0.0673011			0.0336505	+	0.999434i	$0.489287\pi$
$564$		0			0
$565$		−	5.86030i		−	0.246545i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	6.38589i		−	0.267711i	−0.991001	−	0.133855i	$-0.957264\pi$
$569$		−	6.38589i		−	0.267711i	0.991001	−	0.133855i	$-0.0427357\pi$
$570$		0			0
$571$	−0.627417			−0.0262566			−0.0131283	−	0.999914i	$-0.504179\pi$
$571$	−0.627417			−0.0262566			−0.0131283	+	0.999914i	$0.504179\pi$
$572$		0			0
$573$	4.48833	+	13.6453i	0.187503	+	0.570040i
$574$		0			0
$575$		−	27.7566i		−	1.15753i
$576$		0			0
$577$			12.0376i			0.501133i	0.968099	+	0.250567i	$0.0806169\pi$
$577$			12.0376i			0.501133i	−0.968099	+	0.250567i	$0.919383\pi$
$578$		0			0
$579$	32.3420	−	10.6382i	1.34409	−	0.442109i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−4.48528			−0.185761
$584$		0			0
$585$	5.79899	+	7.86123i	0.239759	+	0.325022i
$586$		0			0
$587$	−17.4856			−0.721708			−0.360854	−	0.932622i	$-0.617515\pi$
$587$	−17.4856			−0.721708			−0.360854	+	0.932622i	$0.617515\pi$
$588$		0			0
$589$	25.9411			1.06889
$590$		0			0
$591$	−12.0335	−	36.5838i	−0.494990	−	1.50485i
$592$		0			0
$593$	−0.729951			−0.0299755			−0.0149878	−	0.999888i	$-0.504771\pi$
$593$	−0.729951			−0.0299755			−0.0149878	+	0.999888i	$0.504771\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	9.31371	+	28.3153i	0.381185	+	1.15887i
$598$		0			0
$599$			31.7505i			1.29729i	0.761091	+	0.648645i	$0.224664\pi$
$599$			31.7505i			1.29729i	−0.761091	+	0.648645i	$0.775336\pi$
$600$		0			0
$601$		−	30.9636i		−	1.26303i	−0.775364	−	0.631515i	$-0.782433\pi$
$601$		−	30.9636i		−	1.26303i	0.775364	−	0.631515i	$-0.217567\pi$
$602$		0			0
$603$	4.82843	−	3.56178i	0.196629	−	0.145047i
$604$		0			0
$605$	7.21440			0.293307
$606$		0			0
$607$		−	14.7821i		−	0.599986i	−0.953941	−	0.299993i	$-0.903016\pi$
$607$		−	14.7821i		−	0.599986i	0.953941	−	0.299993i	$-0.0969843\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			11.1175i			0.449764i
$612$		0			0
$613$	24.7279			0.998751			0.499376	−	0.866386i	$-0.333563\pi$
$613$	24.7279			0.998751			0.499376	+	0.866386i	$0.333563\pi$
$614$		0			0
$615$	5.85136	−	1.92468i	0.235950	−	0.0776107i
$616$		0			0
$617$		−	6.38589i		−	0.257086i	−0.991704	−	0.128543i	$-0.958970\pi$
$617$		−	6.38589i		−	0.257086i	0.991704	−	0.128543i	$-0.0410301\pi$
$618$		0			0
$619$		−	42.2584i		−	1.69851i	−0.527986	−	0.849253i	$-0.677052\pi$
$619$		−	42.2584i		−	1.69851i	0.527986	−	0.849253i	$-0.322948\pi$
$620$		0			0
$621$	28.8860	+	20.5111i	1.15915	+	0.823080i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	11.9289			0.477157
$626$		0			0
$627$	−21.6569	+	7.12356i	−0.864891	+	0.284488i
$628$		0			0
$629$	43.5770			1.73753
$630$		0			0
$631$	7.51472			0.299156			0.149578	−	0.988750i	$-0.452208\pi$
$631$	7.51472			0.299156			0.149578	+	0.988750i	$0.452208\pi$
$632$		0			0
$633$	21.9054	−	7.20533i	0.870663	−	0.286386i
$634$		0			0
$635$	−10.7673			−0.427287
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−18.4853	−	25.0590i	−0.731266	−	0.991320i
$640$		0			0
$641$		−	0.126564i		−	0.00499898i	−0.999997	−	0.00249949i	$-0.999204\pi$
$641$		−	0.126564i		−	0.00499898i	0.999997	−	0.00249949i	$-0.000795613\pi$
$642$		0			0
$643$			30.4608i			1.20126i	0.799528	+	0.600629i	$0.205083\pi$
$643$			30.4608i			1.20126i	−0.799528	+	0.600629i	$0.794917\pi$
$644$		0			0
$645$	−7.65685	+	2.51856i	−0.301488	+	0.0991682i
$646$		0			0
$647$	36.1972			1.42306			0.711530	−	0.702656i	$-0.248003\pi$
$647$	36.1972			1.42306			0.711530	+	0.702656i	$0.248003\pi$
$648$		0			0
$649$			11.7206i			0.460074i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			9.33657i			0.365368i	0.983172	+	0.182684i	$0.0584786\pi$
$653$			9.33657i			0.365368i	−0.983172	+	0.182684i	$0.941521\pi$
$654$		0			0
$655$	18.4853			0.722280
$656$		0			0
$657$	−2.49221	−	3.37849i	−0.0972304	−	0.131808i
$658$		0			0
$659$			21.4973i			0.837414i	0.908121	+	0.418707i	$0.137517\pi$
$659$			21.4973i			0.837414i	−0.908121	+	0.418707i	$0.862483\pi$
$660$		0			0
$661$		−	47.2764i		−	1.83884i	−0.393280	−	0.919419i	$-0.628660\pi$
$661$		−	47.2764i		−	1.83884i	0.393280	−	0.919419i	$-0.371340\pi$
$662$		0			0
$663$	12.7634	+	38.8029i	0.495690	+	1.50698i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	53.5980			2.07532
$668$		0			0
$669$	−0.485281	−	1.47534i	−0.0187621	−	0.0570399i
$670$		0			0
$671$	14.5257			0.560757
$672$		0			0
$673$	6.92893			0.267091			0.133545	−	0.991043i	$-0.457364\pi$
$673$	6.92893			0.267091			0.133545	+	0.991043i	$0.457364\pi$
$674$		0			0
$675$	−12.2473	+	17.2480i	−0.471397	+	0.663874i
$676$		0			0
$677$	−28.0875			−1.07949			−0.539746	−	0.841828i	$-0.681480\pi$
$677$	−28.0875			−1.07949			−0.539746	+	0.841828i	$0.681480\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	27.4558	−	9.03102i	1.05211	−	0.346069i
$682$		0			0
$683$			9.76869i			0.373788i	0.982380	+	0.186894i	$0.0598422\pi$
$683$			9.76869i			0.373788i	−0.982380	+	0.186894i	$0.940158\pi$
$684$		0			0
$685$			19.2974i			0.737314i
$686$		0			0
$687$	6.41421	+	19.5003i	0.244718	+	0.743983i
$688$		0			0
$689$	−3.52452			−0.134273
$690$		0			0
$691$			50.3922i			1.91701i	0.285074	+	0.958505i	$0.407982\pi$
$691$			50.3922i			1.91701i	−0.285074	+	0.958505i	$0.592018\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			7.55568i			0.286603i
$696$		0			0
$697$	25.7574			0.975630
$698$		0			0
$699$	−6.18209	−	18.7946i	−0.233828	−	0.710878i
$700$		0			0
$701$			25.9233i			0.979108i	0.871973	+	0.489554i	$0.162840\pi$
$701$			25.9233i			0.979108i	−0.871973	+	0.489554i	$0.837160\pi$
$702$		0			0
$703$			19.1116i			0.720809i
$704$		0			0
$705$	5.21828	−	1.71644i	0.196532	−	0.0646450i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−47.0711			−1.76779			−0.883896	−	0.467684i	$-0.845088\pi$
$709$	−47.0711			−1.76779			−0.883896	+	0.467684i	$0.845088\pi$
$710$		0			0
$711$	28.1421	−	20.7596i	1.05541	−	0.778545i
$712$		0			0
$713$	−57.7719			−2.16358
$714$		0			0
$715$	14.0000			0.523570
$716$		0			0
$717$	2.09299	+	6.36304i	0.0781640	+	0.237632i
$718$		0			0
$719$	−6.91204			−0.257776			−0.128888	−	0.991659i	$-0.541141\pi$
$719$	−6.91204			−0.257776			−0.128888	+	0.991659i	$0.541141\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	11.1421	+	33.8740i	0.414380	+	1.25979i
$724$		0			0
$725$			32.0036i			1.18858i
$726$		0			0
$727$		−	1.97908i		−	0.0733998i	−0.999326	−	0.0366999i	$-0.988315\pi$
$727$		−	1.97908i		−	0.0733998i	0.999326	−	0.0366999i	$-0.0116846\pi$
$728$		0			0
$729$	−8.89949	−	25.4912i	−0.329611	−	0.944117i
$730$		0			0
$731$	−33.7050			−1.24663
$732$		0			0
$733$			46.0083i			1.69935i	0.527303	+	0.849677i	$0.323203\pi$
$733$			46.0083i			1.69935i	−0.527303	+	0.849677i	$0.676797\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	8.59890i		−	0.316745i
$738$		0			0
$739$	38.4264			1.41354			0.706769	−	0.707444i	$-0.250152\pi$
$739$	38.4264			1.41354			0.706769	+	0.707444i	$0.250152\pi$
$740$		0			0
$741$	−17.0179	+	5.59767i	−0.625167	+	0.205636i
$742$		0			0
$743$			28.6208i			1.05000i	0.851103	+	0.524998i	$0.175934\pi$
$743$			28.6208i			1.05000i	−0.851103	+	0.524998i	$0.824066\pi$
$744$		0			0
$745$		−	16.1590i		−	0.592018i
$746$		0			0
$747$	−3.29066	+	2.42742i	−0.120399	+	0.0888145i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−16.3431			−0.596370			−0.298185	−	0.954508i	$-0.596381\pi$
$751$	−16.3431			−0.596370			−0.298185	+	0.954508i	$0.596381\pi$
$752$		0			0
$753$	15.3137	−	5.03712i	0.558063	−	0.183563i
$754$		0			0
$755$	−12.6949			−0.462015
$756$		0			0
$757$	−6.58579			−0.239364			−0.119682	−	0.992812i	$-0.538188\pi$
$757$	−6.58579			−0.239364			−0.119682	+	0.992812i	$0.538188\pi$
$758$		0			0
$759$	48.2307	−	15.8645i	1.75066	−	0.575844i
$760$		0			0
$761$	4.25447			0.154224			0.0771122	−	0.997022i	$-0.475430\pi$
$761$	4.25447			0.154224			0.0771122	+	0.997022i	$0.475430\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	16.2426	−	11.9817i	0.587254	−	0.433199i
$766$		0			0
$767$			9.21001i			0.332554i
$768$		0			0
$769$			2.48181i			0.0894963i	0.998998	+	0.0447482i	$0.0142485\pi$
$769$			2.48181i			0.0894963i	−0.998998	+	0.0447482i	$0.985751\pi$
$770$		0			0
$771$	−20.0711	+	6.60195i	−0.722842	+	0.237764i
$772$		0			0
$773$	−7.54513			−0.271379			−0.135690	−	0.990751i	$-0.543325\pi$
$773$	−7.54513			−0.271379			−0.135690	+	0.990751i	$0.543325\pi$
$774$		0			0
$775$		−	34.4959i		−	1.23913i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			11.2965i			0.404737i
$780$		0			0
$781$	−44.6274			−1.59689
$782$		0			0
$783$	−33.3058	−	23.6494i	−1.19025	−	0.845162i
$784$		0			0
$785$			0.126564i			0.00451726i
$786$		0			0
$787$			35.5014i			1.26549i	0.774361	+	0.632744i	$0.218071\pi$
$787$			35.5014i			1.26549i	−0.774361	+	0.632744i	$0.781929\pi$
$788$		0			0
$789$	−8.34357	−	25.3659i	−0.297039	−	0.903050i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	11.4142			0.405331
$794$		0			0
$795$	0.544156	+	1.65433i	0.0192992	+	0.0586729i
$796$		0			0
$797$	−18.5463			−0.656943			−0.328471	−	0.944514i	$-0.606533\pi$
$797$	−18.5463			−0.656943			−0.328471	+	0.944514i	$0.606533\pi$
$798$		0			0
$799$	22.9706			0.812640
$800$		0			0
$801$	−28.0875	+	20.7193i	−0.992424	+	0.732080i
$802$		0			0
$803$	−6.01673			−0.212326
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−26.7990	+	8.81496i	−0.943369	+	0.310301i
$808$		0			0
$809$			15.1114i			0.531287i	0.964071	+	0.265644i	$0.0855844\pi$
$809$			15.1114i			0.531287i	−0.964071	+	0.265644i	$0.914416\pi$
$810$		0			0
$811$			33.9706i			1.19287i	0.802661	+	0.596435i	$0.203417\pi$
$811$			33.9706i			1.19287i	−0.802661	+	0.596435i	$0.796583\pi$
$812$		0			0
$813$	−1.55635	−	4.73157i	−0.0545835	−	0.165943i
$814$		0			0
$815$	−2.72607			−0.0954901
$816$		0			0
$817$		−	14.7821i		−	0.517159i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	48.0316i		−	1.67632i	−0.545428	−	0.838158i	$-0.683633\pi$
$821$		−	48.0316i		−	1.67632i	0.545428	−	0.838158i	$-0.316367\pi$
$822$		0			0
$823$	−24.3431			−0.848549			−0.424274	−	0.905534i	$-0.639471\pi$
$823$	−24.3431			−0.848549			−0.424274	+	0.905534i	$0.639471\pi$
$824$		0			0
$825$	9.47275	+	28.7988i	0.329799	+	1.00264i
$826$		0			0
$827$			14.9848i			0.521072i	0.965464	+	0.260536i	$0.0838993\pi$
$827$			14.9848i			0.521072i	−0.965464	+	0.260536i	$0.916101\pi$
$828$		0			0
$829$		−	21.1132i		−	0.733293i	−0.930360	−	0.366647i	$-0.880506\pi$
$829$		−	21.1132i		−	0.733293i	0.930360	−	0.366647i	$-0.119494\pi$
$830$		0			0
$831$	−48.4645	+	15.9414i	−1.68122	+	0.553001i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	6.88730			0.238345
$836$		0			0
$837$	35.8995	+	25.4912i	1.24087	+	0.881103i
$838$		0			0
$839$	35.8665			1.23825			0.619125	−	0.785293i	$-0.287488\pi$
$839$	35.8665			1.23825			0.619125	+	0.785293i	$0.287488\pi$
$840$		0			0
$841$	−32.7990			−1.13100
$842$		0			0
$843$	10.5051	+	31.9372i	0.361813	+	1.09997i
$844$		0			0
$845$	−1.52840			−0.0525785
$846$		0			0
$847$		0			0
$848$		0			0
$849$	1.79899	+	5.46924i	0.0617412	+	0.187704i
$850$		0			0
$851$		−	42.5624i		−	1.45902i
$852$		0			0
$853$		−	41.6787i		−	1.42705i	−0.700629	−	0.713526i	$-0.747097\pi$
$853$		−	41.6787i		−	1.42705i	0.700629	−	0.713526i	$-0.252903\pi$
$854$		0			0
$855$	5.25483	+	7.12356i	0.179712	+	0.243621i
$856$		0			0
$857$	−37.9595			−1.29667			−0.648336	−	0.761355i	$-0.724535\pi$
$857$	−37.9595			−1.29667			−0.648336	+	0.761355i	$0.724535\pi$
$858$		0			0
$859$		−	4.59220i		−	0.156684i	−0.996927	−	0.0783419i	$-0.975037\pi$
$859$		−	4.59220i		−	0.156684i	0.996927	−	0.0783419i	$-0.0249626\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			11.4230i			0.388844i	0.980918	+	0.194422i	$0.0622831\pi$
$863$			11.4230i			0.388844i	−0.980918	+	0.194422i	$0.937717\pi$
$864$		0			0
$865$	5.41421			0.184089
$866$		0			0
$867$	52.2029	−	17.1710i	1.77290	−	0.583159i
$868$		0			0
$869$		−	50.1181i		−	1.70014i
$870$		0			0
$871$		−	6.75699i		−	0.228952i
$872$		0			0
$873$	−28.8175	−	39.0656i	−0.975324	−	1.32217i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−15.3553			−0.518513			−0.259256	−	0.965809i	$-0.583477\pi$
$877$	−15.3553			−0.518513			−0.259256	+	0.965809i	$0.583477\pi$
$878$		0			0
$879$	17.8284	−	5.86428i	0.601338	−	0.197797i
$880$		0			0
$881$	11.9650			0.403110			0.201555	−	0.979477i	$-0.435401\pi$
$881$	11.9650			0.403110			0.201555	+	0.979477i	$0.435401\pi$
$882$		0			0
$883$	−35.3137			−1.18840			−0.594200	−	0.804317i	$-0.702531\pi$
$883$	−35.3137			−1.18840			−0.594200	+	0.804317i	$0.702531\pi$
$884$		0			0
$885$	4.32296	−	1.42195i	0.145315	−	0.0477983i
$886$		0			0
$887$	51.2875			1.72207			0.861033	−	0.508550i	$-0.169818\pi$
$887$	51.2875			1.72207			0.861033	+	0.508550i	$0.169818\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−36.9706	−	11.4230i	−1.23856	−	0.382685i
$892$		0			0
$893$			10.0742i			0.337122i
$894$		0			0
$895$		−	0.294495i		−	0.00984388i
$896$		0			0
$897$	37.8995	−	12.4662i	1.26543	−	0.416236i
$898$		0			0
$899$	66.6116			2.22162
$900$		0			0
$901$			7.28225i			0.242607i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	6.38589i		−	0.212274i
$906$		0			0
$907$	33.9411			1.12700			0.563498	−	0.826117i	$-0.309455\pi$
$907$	33.9411			1.12700			0.563498	+	0.826117i	$0.309455\pi$
$908$		0			0
$909$	21.5062	−	15.8645i	0.713316	−	0.526191i
$910$		0			0
$911$		−	30.7073i		−	1.01738i	−0.860951	−	0.508689i	$-0.830130\pi$
$911$		−	30.7073i		−	1.01738i	0.860951	−	0.508689i	$-0.169870\pi$
$912$		0			0
$913$			5.86030i			0.193948i
$914$		0			0
$915$	−1.76226	−	5.35757i	−0.0582585	−	0.177116i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−1.02944			−0.0339580			−0.0169790	−	0.999856i	$-0.505405\pi$
$919$	−1.02944			−0.0339580			−0.0169790	+	0.999856i	$0.505405\pi$
$920$		0			0
$921$	−6.48528	−	19.7164i	−0.213697	−	0.649676i
$922$		0			0
$923$	−35.0681			−1.15428
$924$		0			0
$925$	25.4142			0.835614
$926$		0			0
$927$	−23.2685	−	31.5432i	−0.764237	−	1.03602i
$928$		0			0
$929$	−1.52840			−0.0501451			−0.0250726	−	0.999686i	$-0.507982\pi$
$929$	−1.52840			−0.0501451			−0.0250726	+	0.999686i	$0.507982\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	41.0711	−	13.5095i	1.34461	−	0.442280i
$934$		0			0
$935$		−	28.9264i		−	0.945994i
$936$		0			0
$937$			8.04762i			0.262904i	0.991323	+	0.131452i	$0.0419640\pi$
$937$			8.04762i			0.262904i	−0.991323	+	0.131452i	$0.958036\pi$
$938$		0			0
$939$	−10.2132	−	31.0499i	−0.333295	−	1.01327i
$940$		0			0
$941$	−45.3393			−1.47802			−0.739009	−	0.673696i	$-0.764706\pi$
$941$	−45.3393			−1.47802			−0.739009	+	0.673696i	$0.764706\pi$
$942$		0			0
$943$		−	25.1577i		−	0.819246i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			23.4047i			0.760551i	0.924873	+	0.380275i	$0.124171\pi$
$947$			23.4047i			0.760551i	−0.924873	+	0.380275i	$0.875829\pi$
$948$		0			0
$949$	−4.72792			−0.153475
$950$		0			0
$951$	−1.12918	−	3.43289i	−0.0366160	−	0.111319i
$952$		0			0
$953$		−	5.03712i		−	0.163168i	−0.996666	−	0.0815842i	$-0.974002\pi$
$953$		−	5.03712i		−	0.163168i	0.996666	−	0.0815842i	$-0.0259979\pi$
$954$		0			0
$955$		−	7.99322i		−	0.258655i
$956$		0			0
$957$	−55.6105	+	18.2919i	−1.79763	+	0.591293i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−40.7990			−1.31610
$962$		0			0
$963$	−16.6274	−	22.5405i	−0.535811	−	0.726357i
$964$		0			0
$965$	−18.9455			−0.609877
$966$		0			0
$967$	−51.9411			−1.67031			−0.835157	−	0.550012i	$-0.814623\pi$
$967$	−51.9411			−1.67031			−0.835157	+	0.550012i	$0.814623\pi$
$968$		0			0
$969$	11.5657	+	35.1618i	0.371545	+	1.12956i
$970$		0			0
$971$	−28.1560			−0.903570			−0.451785	−	0.892127i	$-0.649213\pi$
$971$	−28.1560			−0.903570			−0.451785	+	0.892127i	$0.649213\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	7.44365	+	22.6300i	0.238388	+	0.724739i
$976$		0			0
$977$		−	17.0712i		−	0.546157i	−0.961992	−	0.273079i	$-0.911958\pi$
$977$		−	17.0712i		−	0.546157i	0.961992	−	0.273079i	$-0.0880419\pi$
$978$		0			0
$979$			50.0208i			1.59867i
$980$		0			0
$981$	15.8995	−	11.7286i	0.507632	−	0.374464i
$982$		0			0
$983$	24.8654			0.793082			0.396541	−	0.918017i	$-0.370211\pi$
$983$	24.8654			0.793082			0.396541	+	0.918017i	$0.370211\pi$
$984$		0			0
$985$			21.4303i			0.682825i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			32.9203i			1.04680i
$990$		0			0
$991$	−22.2843			−0.707883			−0.353942	−	0.935268i	$-0.615159\pi$
$991$	−22.2843			−0.707883			−0.353942	+	0.935268i	$0.615159\pi$
$992$		0			0
$993$	−25.9945	+	8.55035i	−0.824912	+	0.271337i
$994$		0			0
$995$		−	16.5867i		−	0.525834i
$996$		0			0
$997$		−	36.5612i		−	1.15791i	−0.815361	−	0.578953i	$-0.803461\pi$
$997$		−	36.5612i		−	1.15791i	0.815361	−	0.578953i	$-0.196539\pi$
$998$		0			0
$999$	−18.7801	+	26.4483i	−0.594177	+	0.836787i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.2.k.g.881.7		8
3.2	odd	2	inner	2352.2.k.g.881.1		8
4.3	odd	2		588.2.f.d.293.2	yes	8
7.6	odd	2	inner	2352.2.k.g.881.2		8
12.11	even	2		588.2.f.d.293.8	yes	8
21.20	even	2	inner	2352.2.k.g.881.8		8
28.3	even	6		588.2.k.f.509.2		16
28.11	odd	6		588.2.k.f.509.7		16
28.19	even	6		588.2.k.f.521.4		16
28.23	odd	6		588.2.k.f.521.5		16
28.27	even	2		588.2.f.d.293.7	yes	8
84.11	even	6		588.2.k.f.509.4		16
84.23	even	6		588.2.k.f.521.2		16
84.47	odd	6		588.2.k.f.521.7		16
84.59	odd	6		588.2.k.f.509.5		16
84.83	odd	2		588.2.f.d.293.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.2.f.d.293.1	✓	8	84.83	odd	2
588.2.f.d.293.2	yes	8	4.3	odd	2
588.2.f.d.293.7	yes	8	28.27	even	2
588.2.f.d.293.8	yes	8	12.11	even	2
588.2.k.f.509.2		16	28.3	even	6
588.2.k.f.509.4		16	84.11	even	6
588.2.k.f.509.5		16	84.59	odd	6
588.2.k.f.509.7		16	28.11	odd	6
588.2.k.f.521.2		16	84.23	even	6
588.2.k.f.521.4		16	28.19	even	6
588.2.k.f.521.5		16	28.23	odd	6
588.2.k.f.521.7		16	84.47	odd	6
2352.2.k.g.881.1		8	3.2	odd	2	inner
2352.2.k.g.881.2		8	7.6	odd	2	inner
2352.2.k.g.881.7		8	1.1	even	1	trivial
2352.2.k.g.881.8		8	21.20	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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2352.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.64533 - 0.541196i) q^{3} -0.963811 q^{5} +(2.41421 - 1.78089i) q^{9} +O(q^{10})\)	\(q+(1.64533 - 0.541196i) q^{3} -0.963811 q^{5} +(2.41421 - 1.78089i) q^{9} -4.29945i q^{11} -3.37849i q^{13} +(-1.58579 + 0.521611i) q^{15} -6.98054 q^{17} -3.06147i q^{19} +6.81801i q^{23} -4.07107 q^{25} +(3.00836 - 4.23671i) q^{27} -7.86123i q^{29} +8.47343i q^{31} +(-2.32685 - 7.07401i) q^{33} -6.24264 q^{37} +(-1.82843 - 5.55873i) q^{39} -3.68988 q^{41} +4.82843 q^{43} +(-2.32685 + 1.71644i) q^{45} -3.29066 q^{47} +(-11.4853 + 3.77784i) q^{51} -1.04322i q^{53} +4.14386i q^{55} +(-1.65685 - 5.03712i) q^{57} -2.72607 q^{59} +3.37849i q^{61} +3.25623i q^{65} +2.00000 q^{67} +(3.68988 + 11.2179i) q^{69} -10.3798i q^{71} -1.39942i q^{73} +(-6.69825 + 2.20325i) q^{75} +11.6569 q^{79} +(2.65685 - 8.59890i) q^{81} -1.36303 q^{83} +6.72792 q^{85} +(-4.25447 - 12.9343i) q^{87} -11.6342 q^{89} +(4.58579 + 13.9416i) q^{93} +2.95068i q^{95} -16.1815i q^{97} +(-7.65685 - 10.3798i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^9	\( 8 q + 8 q^{9} - 24 q^{15} + 24 q^{25} - 16 q^{37} + 8 q^{39} + 16 q^{43} - 24 q^{51} + 32 q^{57} + 16 q^{67} + 48 q^{79} - 24 q^{81} - 48 q^{85} + 48 q^{93} - 16 q^{99}+O(q^{100}) \) 8 * q + 8 * q^9 - 24 * q^15 + 24 * q^25 - 16 * q^37 + 8 * q^39 + 16 * q^43 - 24 * q^51 + 32 * q^57 + 16 * q^67 + 48 * q^79 - 24 * q^81 - 48 * q^85 + 48 * q^93 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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