LMFDB - Embedded newform 2736.2.f.g.1025.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(1025,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, 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("2736.1025");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.f (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:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 18x^{6} + 37x^{4} + 22x^{2} + 4 $ x^8 + 18x^6 + 37x^4 + 22*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1368)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1025.3
Root			$-0.626815i$ of defining polynomial
Character	$\chi$	$=$	2736.1025
Dual form			2736.2.f.g.1025.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.32945i q^{5} -4.34658 q^{7} +O(q^{10})$	$q-2.32945i q^{5} -4.34658 q^{7} -0.264508i q^{11} -6.38147i q^{13} +2.56392i q^{17} +(-2.62593 + 3.47915i) q^{19} -2.66784i q^{23} -0.426319 q^{25} -8.92026 q^{29} -1.78265i q^{31} +10.1251i q^{35} +3.08409i q^{37} +7.44131 q^{41} +5.05225 q^{43} +7.45728i q^{47} +11.8928 q^{49} -4.10450 q^{53} -0.616157 q^{55} -8.36157 q^{59} -8.45108 q^{61} -14.8653 q^{65} -2.35948i q^{67} +5.02476 q^{71} +14.9973 q^{73} +1.14971i q^{77} +6.23369i q^{79} +16.0076i q^{83} +5.97251 q^{85} -8.25707 q^{89} +27.7376i q^{91} +(8.10450 + 6.11696i) q^{95} +15.1224i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 4 q^{19} - 8 q^{25} - 32 q^{29} + 24 q^{41} + 28 q^{43} + 4 q^{49} - 8 q^{53} - 12 q^{55} + 8 q^{59} - 8 q^{61} + 24 q^{65} - 24 q^{71} + 4 q^{73} - 4 q^{85} - 16 q^{89} + 40 q^{95}+O(q^{100}) $ 8 * q - 4 * q^19 - 8 * q^25 - 32 * q^29 + 24 * q^41 + 28 * q^43 + 4 * q^49 - 8 * q^53 - 12 * q^55 + 8 * q^59 - 8 * q^61 + 24 * q^65 - 24 * q^71 + 4 * q^73 - 4 * q^85 - 16 * q^89 + 40 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$		−	2.32945i		−	1.04176i	−0.853630	−	0.520880i	$-0.825604\pi$
$5$		−	2.32945i		−	1.04176i	0.853630	−	0.520880i	$-0.174396\pi$
$6$		0			0
$7$	−4.34658			−1.64285			−0.821427	−	0.570314i	$-0.806822\pi$
$7$	−4.34658			−1.64285			−0.821427	+	0.570314i	$0.806822\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	0.264508i		−	0.0797522i	−0.999205	−	0.0398761i	$-0.987304\pi$
$11$		−	0.264508i		−	0.0797522i	0.999205	−	0.0398761i	$-0.0126963\pi$
$12$		0			0
$13$		−	6.38147i		−	1.76990i	−0.465685	−	0.884950i	$-0.654192\pi$
$13$		−	6.38147i		−	1.76990i	0.465685	−	0.884950i	$-0.345808\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			2.56392i			0.621842i	0.950436	+	0.310921i	$0.100637\pi$
$17$			2.56392i			0.621842i	−0.950436	+	0.310921i	$0.899363\pi$
$18$		0			0
$19$	−2.62593	+	3.47915i	−0.602429	+	0.798172i
$19$	−2.62593	+	3.47915i	−0.602429	+	0.798172i
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	2.66784i		−	0.556284i	−0.960540	−	0.278142i	$-0.910281\pi$
$23$		−	2.66784i		−	0.556284i	0.960540	−	0.278142i	$-0.0897186\pi$
$24$		0			0
$25$	−0.426319			−0.0852637
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−8.92026			−1.65645			−0.828226	−	0.560395i	$-0.810649\pi$
$29$	−8.92026			−1.65645			−0.828226	+	0.560395i	$0.810649\pi$
$30$		0			0
$31$		−	1.78265i		−	0.320173i	−0.987103	−	0.160086i	$-0.948823\pi$
$31$		−	1.78265i		−	0.320173i	0.987103	−	0.160086i	$-0.0511773\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			10.1251i			1.71146i
$36$		0			0
$37$			3.08409i			0.507022i	0.967332	+	0.253511i	$0.0815854\pi$
$37$			3.08409i			0.507022i	−0.967332	+	0.253511i	$0.918415\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	7.44131			1.16214			0.581068	−	0.813855i	$-0.302635\pi$
$41$	7.44131			1.16214			0.581068	+	0.813855i	$0.302635\pi$
$42$		0			0
$43$	5.05225			0.770461			0.385230	−	0.922820i	$-0.374122\pi$
$43$	5.05225			0.770461			0.385230	+	0.922820i	$0.374122\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			7.45728i			1.08776i	0.839164	+	0.543878i	$0.183045\pi$
$47$			7.45728i			1.08776i	−0.839164	+	0.543878i	$0.816955\pi$
$48$		0			0
$49$	11.8928			1.69897
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.10450			−0.563796			−0.281898	−	0.959444i	$-0.590964\pi$
$53$	−4.10450			−0.563796			−0.281898	+	0.959444i	$0.590964\pi$
$54$		0			0
$55$	−0.616157			−0.0830826
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−8.36157			−1.08858			−0.544292	−	0.838896i	$-0.683202\pi$
$59$	−8.36157			−1.08858			−0.544292	+	0.838896i	$0.683202\pi$
$60$		0			0
$61$	−8.45108			−1.08205			−0.541025	−	0.841007i	$-0.681963\pi$
$61$	−8.45108			−1.08205			−0.541025	+	0.841007i	$0.681963\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−14.8653			−1.84381
$66$		0			0
$67$		−	2.35948i		−	0.288256i	−0.989559	−	0.144128i	$-0.953962\pi$
$67$		−	2.35948i		−	0.288256i	0.989559	−	0.144128i	$-0.0460378\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	5.02476			0.596329			0.298165	−	0.954514i	$-0.403626\pi$
$71$	5.02476			0.596329			0.298165	+	0.954514i	$0.403626\pi$
$72$		0			0
$73$	14.9973			1.75530			0.877649	−	0.479304i	$-0.159111\pi$
$73$	14.9973			1.75530			0.877649	+	0.479304i	$0.159111\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			1.14971i			0.131021i
$78$		0			0
$79$			6.23369i			0.701345i	0.936498	+	0.350672i	$0.114047\pi$
$79$			6.23369i			0.701345i	−0.936498	+	0.350672i	$0.885953\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			16.0076i			1.75706i	0.477683	+	0.878532i	$0.341477\pi$
$83$			16.0076i			1.75706i	−0.477683	+	0.878532i	$0.658523\pi$
$84$		0			0
$85$	5.97251			0.647810
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−8.25707			−0.875248			−0.437624	−	0.899158i	$-0.644180\pi$
$89$	−8.25707			−0.875248			−0.437624	+	0.899158i	$0.644180\pi$
$90$		0			0
$91$			27.7376i			2.90769i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	8.10450	+	6.11696i	0.831504	+	0.627587i
$96$		0			0
$97$			15.1224i			1.53545i	0.640780	+	0.767724i	$0.278611\pi$
$97$			15.1224i			1.53545i	−0.640780	+	0.767724i	$0.721389\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			0.837379i			0.0833223i	0.999132	+	0.0416611i	$0.0132650\pi$
$101$			0.837379i			0.0833223i	−0.999132	+	0.0416611i	$0.986735\pi$
$102$		0			0
$103$			15.8470i			1.56145i	0.624872	+	0.780727i	$0.285151\pi$
$103$			15.8470i			1.56145i	−0.624872	+	0.780727i	$0.714849\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−18.8653			−1.82378			−0.911888	−	0.410439i	$-0.865375\pi$
$107$	−18.8653			−1.82378			−0.911888	+	0.410439i	$0.865375\pi$
$108$		0			0
$109$		−	0.724614i		−	0.0694054i	−0.999398	−	0.0347027i	$-0.988952\pi$
$109$		−	0.724614i		−	0.0694054i	0.999398	−	0.0347027i	$-0.0110484\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−9.73603			−0.915889			−0.457944	−	0.888981i	$-0.651414\pi$
$113$	−9.73603			−0.915889			−0.457944	+	0.888981i	$0.651414\pi$
$114$		0			0
$115$	−6.21460			−0.579514
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	11.1443i		−	1.02159i
$120$		0			0
$121$	10.9300			0.993640
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	10.6541i		−	0.952936i
$126$		0			0
$127$			0.724614i			0.0642991i	0.999483	+	0.0321495i	$0.0102353\pi$
$127$			0.724614i			0.0642991i	−0.999483	+	0.0321495i	$0.989765\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	9.38771i		−	0.820208i	−0.912039	−	0.410104i	$-0.865492\pi$
$131$		−	9.38771i		−	0.820208i	0.912039	−	0.410104i	$-0.134508\pi$
$132$		0			0
$133$	11.4138	−	15.1224i	0.989703	−	1.31128i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	0.814718i		−	0.0696061i	−0.999394	−	0.0348030i	$-0.988920\pi$
$137$		−	0.814718i		−	0.0696061i	0.999394	−	0.0348030i	$-0.0110804\pi$
$138$		0			0
$139$	1.82554			0.154840			0.0774201	−	0.996999i	$-0.475332\pi$
$139$	1.82554			0.154840			0.0774201	+	0.996999i	$0.475332\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.68795			−0.141153
$144$		0			0
$145$			20.7793i			1.72562i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			11.8339i			0.969470i	0.874661	+	0.484735i	$0.161084\pi$
$149$			11.8339i			0.969470i	−0.874661	+	0.484735i	$0.838916\pi$
$150$		0			0
$151$		−	19.5080i		−	1.58754i	−0.608221	−	0.793768i	$-0.708117\pi$
$151$		−	19.5080i		−	1.58754i	0.608221	−	0.793768i	$-0.291883\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.15258			−0.333543
$156$		0			0
$157$	2.97524			0.237450			0.118725	−	0.992927i	$-0.462119\pi$
$157$	2.97524			0.237450			0.118725	+	0.992927i	$0.462119\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			11.5960i			0.913893i
$162$		0			0
$163$	14.4292			1.13018			0.565091	−	0.825029i	$-0.308841\pi$
$163$	14.4292			1.13018			0.565091	+	0.825029i	$0.308841\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−18.2021			−1.40852			−0.704260	−	0.709942i	$-0.748721\pi$
$167$	−18.2021			−1.40852			−0.704260	+	0.709942i	$0.748721\pi$
$168$		0			0
$169$	−27.7231			−2.13255
$170$		0			0
$171$		0			0
$172$		0			0
$173$	17.2818			1.31391			0.656957	−	0.753928i	$-0.271843\pi$
$173$	17.2818			1.31391			0.656957	+	0.753928i	$0.271843\pi$
$174$		0			0
$175$	1.85303			0.140076
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−8.66319			−0.647517			−0.323759	−	0.946140i	$-0.604947\pi$
$179$	−8.66319			−0.647517			−0.323759	+	0.946140i	$0.604947\pi$
$180$		0			0
$181$			10.0424i			0.746446i	0.927742	+	0.373223i	$0.121747\pi$
$181$			10.0424i			0.746446i	−0.927742	+	0.373223i	$0.878253\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	7.18423			0.528195
$186$		0			0
$187$	0.678177			0.0495932
$188$		0			0
$189$		0			0
$190$		0			0
$191$			11.0581i			0.800139i	0.916485	+	0.400070i	$0.131014\pi$
$191$			11.0581i			0.800139i	−0.916485	+	0.400070i	$0.868986\pi$
$192$		0			0
$193$		−	1.05803i		−	0.0761588i	−0.999275	−	0.0380794i	$-0.987876\pi$
$193$		−	1.05803i		−	0.0761588i	0.999275	−	0.0380794i	$-0.0121240\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			23.7506i			1.69216i	0.533056	+	0.846080i	$0.321044\pi$
$197$			23.7506i			1.69216i	−0.533056	+	0.846080i	$0.678956\pi$
$198$		0			0
$199$	1.05225			0.0745919			0.0372959	−	0.999304i	$-0.488126\pi$
$199$	1.05225			0.0745919			0.0372959	+	0.999304i	$0.488126\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	38.7727			2.72131
$204$		0			0
$205$		−	17.3341i		−	1.21067i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.920263	+	0.694579i	0.0636559	+	0.0480450i
$210$		0			0
$211$		−	13.9687i		−	0.961648i	−0.876817	−	0.480824i	$-0.840338\pi$
$211$		−	13.9687i		−	0.961648i	0.876817	−	0.480824i	$-0.159662\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	11.7689i		−	0.802635i
$216$		0			0
$217$			7.74842i			0.525997i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	16.3616			1.10060
$222$		0			0
$223$		−	20.9926i		−	1.40577i	−0.711306	−	0.702883i	$-0.751896\pi$
$223$		−	20.9926i		−	1.40577i	0.711306	−	0.702883i	$-0.248104\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.79101			0.251618			0.125809	−	0.992054i	$-0.459847\pi$
$227$	3.79101			0.251618			0.125809	+	0.992054i	$0.459847\pi$
$228$		0			0
$229$	−26.7758			−1.76939			−0.884697	−	0.466167i	$-0.845634\pi$
$229$	−26.7758			−1.76939			−0.884697	+	0.466167i	$0.845634\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			5.18450i			0.339648i	0.985474	+	0.169824i	$0.0543199\pi$
$233$			5.18450i			0.339648i	−0.985474	+	0.169824i	$0.945680\pi$
$234$		0			0
$235$	17.3713			1.13318
$236$		0			0
$237$		0			0
$238$		0			0
$239$			5.30565i			0.343194i	0.985167	+	0.171597i	$0.0548927\pi$
$239$			5.30565i			0.343194i	−0.985167	+	0.171597i	$0.945107\pi$
$240$		0			0
$241$		−	0.147779i		−	0.00951929i	−0.999989	−	0.00475965i	$-0.998485\pi$
$241$		−	0.147779i		−	0.00951929i	0.999989	−	0.00475965i	$-0.00151505\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	27.7036i		−	1.76992i
$246$		0			0
$247$	22.2021	+	16.7573i	1.41269	+	1.06624i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	20.8783i		−	1.31783i	−0.752218	−	0.658914i	$-0.771016\pi$
$251$		−	20.8783i		−	1.31783i	0.752218	−	0.658914i	$-0.228984\pi$
$252$		0			0
$253$	−0.705666			−0.0443648
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−29.6434			−1.84910			−0.924552	−	0.381055i	$-0.875561\pi$
$257$	−29.6434			−1.84910			−0.924552	+	0.381055i	$0.875561\pi$
$258$		0			0
$259$		−	13.4053i		−	0.832963i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			17.1696i			1.05872i	0.848397	+	0.529361i	$0.177568\pi$
$263$			17.1696i			1.05872i	−0.848397	+	0.529361i	$0.822432\pi$
$264$		0			0
$265$			9.56120i			0.587340i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	12.2397			0.746271			0.373135	−	0.927777i	$-0.378283\pi$
$269$	12.2397			0.746271			0.373135	+	0.927777i	$0.378283\pi$
$270$		0			0
$271$	3.79101			0.230287			0.115144	−	0.993349i	$-0.463267\pi$
$271$	3.79101			0.230287			0.115144	+	0.993349i	$0.463267\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			0.112765i			0.00679997i
$276$		0			0
$277$	−1.56847			−0.0942400			−0.0471200	−	0.998889i	$-0.515004\pi$
$277$	−1.56847			−0.0942400			−0.0471200	+	0.998889i	$0.515004\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−18.6082			−1.11007			−0.555036	−	0.831826i	$-0.687296\pi$
$281$	−18.6082			−1.11007			−0.555036	+	0.831826i	$0.687296\pi$
$282$		0			0
$283$	−29.7754			−1.76996			−0.884981	−	0.465626i	$-0.845829\pi$
$283$	−29.7754			−1.76996			−0.884981	+	0.465626i	$0.845829\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−32.3442			−1.90922
$288$		0			0
$289$	10.4263			0.613313
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−10.9871			−0.641874			−0.320937	−	0.947101i	$-0.603998\pi$
$293$	−10.9871			−0.641874			−0.320937	+	0.947101i	$0.603998\pi$
$294$		0			0
$295$			19.4778i			1.13404i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−17.0248			−0.984567
$300$		0			0
$301$	−21.9600			−1.26575
$302$		0			0
$303$		0			0
$304$		0			0
$305$			19.6863i			1.12724i
$306$		0			0
$307$		−	23.0186i		−	1.31374i	−0.754003	−	0.656871i	$-0.771880\pi$
$307$		−	23.0186i		−	1.31374i	0.754003	−	0.656871i	$-0.228120\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			0.862537i			0.0489100i	0.999701	+	0.0244550i	$0.00778505\pi$
$311$			0.862537i			0.0489100i	−0.999701	+	0.0244550i	$0.992215\pi$
$312$		0			0
$313$	−1.84053			−0.104033			−0.0520164	−	0.998646i	$-0.516565\pi$
$313$	−1.84053			−0.104033			−0.0520164	+	0.998646i	$0.516565\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	8.10450			0.455194			0.227597	−	0.973755i	$-0.426913\pi$
$317$	8.10450			0.455194			0.227597	+	0.973755i	$0.426913\pi$
$318$		0			0
$319$			2.35948i			0.132106i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−8.92026	−	6.73267i	−0.496337	−	0.374616i
$324$		0			0
$325$			2.72054i			0.150908i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	32.4137i		−	1.78702i
$330$		0			0
$331$		−	20.6315i		−	1.13401i	−0.823715	−	0.567005i	$-0.808102\pi$
$331$		−	20.6315i		−	1.13401i	0.823715	−	0.567005i	$-0.191898\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−5.49628			−0.300294
$336$		0			0
$337$			15.7648i			0.858761i	0.903124	+	0.429380i	$0.141268\pi$
$337$			15.7648i			0.858761i	−0.903124	+	0.429380i	$0.858732\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−0.471524			−0.0255345
$342$		0			0
$343$	−21.2668			−1.14830
$344$		0			0
$345$		0			0
$346$		0			0
$347$			12.5373i			0.673038i	0.941677	+	0.336519i	$0.109250\pi$
$347$			12.5373i			0.673038i	−0.941677	+	0.336519i	$0.890750\pi$
$348$		0			0
$349$	−11.8507			−0.634353			−0.317176	−	0.948367i	$-0.602735\pi$
$349$	−11.8507			−0.634353			−0.317176	+	0.948367i	$0.602735\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			34.7830i			1.85131i	0.378364	+	0.925657i	$0.376487\pi$
$353$			34.7830i			1.85131i	−0.378364	+	0.925657i	$0.623513\pi$
$354$		0			0
$355$		−	11.7049i		−	0.621232i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	23.3698i		−	1.23341i	−0.787194	−	0.616706i	$-0.788467\pi$
$359$		−	23.3698i		−	1.23341i	0.787194	−	0.616706i	$-0.211533\pi$
$360$		0			0
$361$	−5.20899	−	18.2720i	−0.274157	−	0.961685i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	34.9353i		−	1.82860i
$366$		0			0
$367$	12.5141			0.653233			0.326617	−	0.945157i	$-0.394091\pi$
$367$	12.5141			0.653233			0.326617	+	0.945157i	$0.394091\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	17.8405			0.926234
$372$		0			0
$373$			20.9926i			1.08695i	0.839424	+	0.543477i	$0.182892\pi$
$373$			20.9926i			1.08695i	−0.839424	+	0.543477i	$0.817108\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			56.9244i			2.93175i
$378$		0			0
$379$			29.7335i			1.52731i	0.645626	+	0.763654i	$0.276596\pi$
$379$			29.7335i			1.52731i	−0.645626	+	0.763654i	$0.723404\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−22.4111			−1.14515			−0.572577	−	0.819851i	$-0.694056\pi$
$383$	−22.4111			−1.14515			−0.572577	+	0.819851i	$0.694056\pi$
$384$		0			0
$385$	2.67818			0.136493
$386$		0			0
$387$		0			0
$388$		0			0
$389$			17.9064i			0.907892i	0.891029	+	0.453946i	$0.149984\pi$
$389$			17.9064i			0.907892i	−0.891029	+	0.453946i	$0.850016\pi$
$390$		0			0
$391$	6.84014			0.345921
$392$		0			0
$393$		0			0
$394$		0			0
$395$	14.5210			0.730633
$396$		0			0
$397$	−6.57929			−0.330205			−0.165102	−	0.986276i	$-0.552795\pi$
$397$	−6.57929			−0.330205			−0.165102	+	0.986276i	$0.552795\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	11.4413			0.571352			0.285676	−	0.958326i	$-0.407782\pi$
$401$	11.4413			0.571352			0.285676	+	0.958326i	$0.407782\pi$
$402$		0			0
$403$	−11.3759			−0.566674
$404$		0			0
$405$		0			0
$406$		0			0
$407$	0.815768			0.0404361
$408$		0			0
$409$			13.1265i			0.649063i	0.945875	+	0.324532i	$0.105207\pi$
$409$			13.1265i			0.649063i	−0.945875	+	0.324532i	$0.894793\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	36.3442			1.78838
$414$		0			0
$415$	37.2889			1.83044
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	11.6098i		−	0.567177i	−0.958946	−	0.283588i	$-0.908475\pi$
$419$		−	11.6098i		−	0.567177i	0.958946	−	0.283588i	$-0.0915249\pi$
$420$		0			0
$421$		−	31.7074i		−	1.54533i	−0.634817	−	0.772663i	$-0.718924\pi$
$421$		−	31.7074i		−	1.54533i	0.634817	−	0.772663i	$-0.281076\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	1.09305i		−	0.0530205i
$426$		0			0
$427$	36.7333			1.77765
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−25.5953			−1.23288			−0.616442	−	0.787401i	$-0.711426\pi$
$431$	−25.5953			−1.23288			−0.616442	+	0.787401i	$0.711426\pi$
$432$		0			0
$433$		−	18.0286i		−	0.866399i	−0.901298	−	0.433199i	$-0.857385\pi$
$433$		−	18.0286i		−	0.866399i	0.901298	−	0.433199i	$-0.142615\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	9.28183	+	7.00557i	0.444010	+	0.335122i
$438$		0			0
$439$		−	13.0442i		−	0.622566i	−0.950317	−	0.311283i	$-0.899241\pi$
$439$		−	13.0442i		−	0.622566i	0.950317	−	0.311283i	$-0.100759\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	28.8655i		−	1.37144i	−0.727864	−	0.685721i	$-0.759487\pi$
$443$		−	28.8655i		−	1.37144i	0.727864	−	0.685721i	$-0.240513\pi$
$444$		0			0
$445$			19.2344i			0.911798i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−0.466064			−0.0219949			−0.0109975	−	0.999940i	$-0.503501\pi$
$449$	−0.466064			−0.0219949			−0.0109975	+	0.999940i	$0.503501\pi$
$450$		0			0
$451$		−	1.96828i		−	0.0926829i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	64.6132			3.02911
$456$		0			0
$457$	−2.78828			−0.130430			−0.0652151	−	0.997871i	$-0.520773\pi$
$457$	−2.78828			−0.130430			−0.0652151	+	0.997871i	$0.520773\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	32.3276i		−	1.50564i	−0.658224	−	0.752822i	$-0.728692\pi$
$461$		−	32.3276i		−	1.50564i	0.658224	−	0.752822i	$-0.271308\pi$
$462$		0			0
$463$	7.36573			0.342315			0.171157	−	0.985244i	$-0.445249\pi$
$463$	7.36573			0.342315			0.171157	+	0.985244i	$0.445249\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	37.9622i		−	1.75668i	−0.478035	−	0.878341i	$-0.658651\pi$
$467$		−	37.9622i		−	1.75668i	0.478035	−	0.878341i	$-0.341349\pi$
$468$		0			0
$469$			10.2557i			0.473563i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	1.33636i		−	0.0614459i
$474$		0			0
$475$	1.11948	−	1.48323i	0.0513654	−	0.0680551i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			1.21862i			0.0556800i	0.999612	+	0.0278400i	$0.00886289\pi$
$479$			1.21862i			0.0556800i	−0.999612	+	0.0278400i	$0.991137\pi$
$480$		0			0
$481$	19.6811			0.897379
$482$		0			0
$483$		0			0
$484$		0			0
$485$	35.2269			1.59957
$486$		0			0
$487$		−	30.4581i		−	1.38019i	−0.723719	−	0.690094i	$-0.757569\pi$
$487$		−	30.4581i		−	1.38019i	0.723719	−	0.690094i	$-0.242431\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			27.8583i			1.25723i	0.777718	+	0.628613i	$0.216377\pi$
$491$			27.8583i			1.25723i	−0.777718	+	0.628613i	$0.783623\pi$
$492$		0			0
$493$		−	22.8708i		−	1.03005i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−21.8405			−0.979682
$498$		0			0
$499$	−27.9050			−1.24920			−0.624600	−	0.780945i	$-0.714738\pi$
$499$	−27.9050			−1.24920			−0.624600	+	0.780945i	$0.714738\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			29.6931i			1.32395i	0.749527	+	0.661974i	$0.230281\pi$
$503$			29.6931i			1.32395i	−0.749527	+	0.661974i	$0.769719\pi$
$504$		0			0
$505$	1.95063			0.0868018
$506$		0			0
$507$		0			0
$508$		0			0
$509$	16.1471			0.715708			0.357854	−	0.933777i	$-0.383508\pi$
$509$	16.1471			0.715708			0.357854	+	0.933777i	$0.383508\pi$
$510$		0			0
$511$	−65.1869			−2.88370
$512$		0			0
$513$		0			0
$514$		0			0
$515$	36.9148			1.62666
$516$		0			0
$517$	1.97251			0.0867509
$518$		0			0
$519$		0			0
$520$		0			0
$521$	38.5156			1.68740			0.843699	−	0.536816i	$-0.180373\pi$
$521$	38.5156			1.68740			0.843699	+	0.536816i	$0.180373\pi$
$522$		0			0
$523$		−	2.75068i		−	0.120279i	−0.998190	−	0.0601394i	$-0.980845\pi$
$523$		−	2.75068i		−	0.120279i	0.998190	−	0.0601394i	$-0.0191545\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	4.57056			0.199097
$528$		0			0
$529$	15.8826			0.690548
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	47.4865i		−	2.05687i
$534$		0			0
$535$			43.9457i			1.89994i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	3.14573i		−	0.135496i
$540$		0			0
$541$	14.7883			0.635798			0.317899	−	0.948125i	$-0.397023\pi$
$541$	14.7883			0.635798			0.317899	+	0.948125i	$0.397023\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1.68795			−0.0723038
$546$		0			0
$547$			6.20355i			0.265245i	0.991167	+	0.132622i	$0.0423397\pi$
$547$			6.20355i			0.265245i	−0.991167	+	0.132622i	$0.957660\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	23.4240	−	31.0349i	0.997895	−	1.32213i
$552$		0			0
$553$		−	27.0952i		−	1.15221i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	8.92419i		−	0.378130i	−0.981965	−	0.189065i	$-0.939454\pi$
$557$		−	8.92419i		−	0.378130i	0.981965	−	0.189065i	$-0.0605457\pi$
$558$		0			0
$559$		−	32.2408i		−	1.36364i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−16.6805			−0.703000			−0.351500	−	0.936188i	$-0.614328\pi$
$563$	−16.6805			−0.703000			−0.351500	+	0.936188i	$0.614328\pi$
$564$		0			0
$565$			22.6796i			0.954136i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−25.1293			−1.05347			−0.526737	−	0.850029i	$-0.676585\pi$
$569$	−25.1293			−1.05347			−0.526737	+	0.850029i	$0.676585\pi$
$570$		0			0
$571$	−8.51415			−0.356306			−0.178153	−	0.984003i	$-0.557012\pi$
$571$	−8.51415			−0.356306			−0.178153	+	0.984003i	$0.557012\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			1.13735i			0.0474308i
$576$		0			0
$577$	−20.6037			−0.857741			−0.428871	−	0.903366i	$-0.641088\pi$
$577$	−20.6037			−0.857741			−0.428871	+	0.903366i	$0.641088\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	69.5784i		−	2.88660i
$582$		0			0
$583$			1.08567i			0.0449639i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			28.9223i			1.19375i	0.802334	+	0.596875i	$0.203591\pi$
$587$			28.9223i			1.19375i	−0.802334	+	0.596875i	$0.796409\pi$
$588$		0			0
$589$	6.20210	+	4.68110i	0.255553	+	0.192881i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	26.0022i		−	1.06778i	−0.845553	−	0.533891i	$-0.820729\pi$
$593$		−	26.0022i		−	1.06778i	0.845553	−	0.533891i	$-0.179271\pi$
$594$		0			0
$595$	−25.9600			−1.06426
$596$		0			0
$597$		0			0
$598$		0			0
$599$	29.3690			1.19998			0.599992	−	0.800006i	$-0.295170\pi$
$599$	29.3690			1.19998			0.599992	+	0.800006i	$0.295170\pi$
$600$		0			0
$601$		−	17.8731i		−	0.729059i	−0.931192	−	0.364529i	$-0.881230\pi$
$601$		−	17.8731i		−	0.729059i	0.931192	−	0.364529i	$-0.118770\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	25.4609i		−	1.03513i
$606$		0			0
$607$			41.9876i			1.70422i	0.523359	+	0.852112i	$0.324679\pi$
$607$			41.9876i			1.70422i	−0.523359	+	0.852112i	$0.675321\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	47.5884			1.92522
$612$		0			0
$613$	−11.6480			−0.470457			−0.235228	−	0.971940i	$-0.575584\pi$
$613$	−11.6480			−0.470457			−0.235228	+	0.971940i	$0.575584\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	11.5817i		−	0.466263i	−0.972445	−	0.233131i	$-0.925103\pi$
$617$		−	11.5817i		−	0.466263i	0.972445	−	0.233131i	$-0.0748972\pi$
$618$		0			0
$619$	12.6486			0.508391			0.254195	−	0.967153i	$-0.418189\pi$
$619$	12.6486			0.508391			0.254195	+	0.967153i	$0.418189\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	35.8900			1.43790
$624$		0			0
$625$	−26.9498			−1.07799
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−7.90737			−0.315287
$630$		0			0
$631$	−15.3644			−0.611649			−0.305824	−	0.952088i	$-0.598932\pi$
$631$	−15.3644			−0.611649			−0.305824	+	0.952088i	$0.598932\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.68795			0.0669842
$636$		0			0
$637$		−	75.8934i		−	3.00700i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−15.6330			−0.617465			−0.308733	−	0.951149i	$-0.599905\pi$
$641$	−15.6330			−0.617465			−0.308733	+	0.951149i	$0.599905\pi$
$642$		0			0
$643$	−6.62881			−0.261415			−0.130707	−	0.991421i	$-0.541725\pi$
$643$	−6.62881			−0.261415			−0.130707	+	0.991421i	$0.541725\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	34.0200i		−	1.33746i	−0.743504	−	0.668732i	$-0.766837\pi$
$647$		−	34.0200i		−	1.33746i	0.743504	−	0.668732i	$-0.233163\pi$
$648$		0			0
$649$			2.21170i			0.0868169i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			37.5888i			1.47096i	0.677545	+	0.735482i	$0.263044\pi$
$653$			37.5888i			1.47096i	−0.677545	+	0.735482i	$0.736956\pi$
$654$		0			0
$655$	−21.8682			−0.854460
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−8.37889			−0.326395			−0.163198	−	0.986593i	$-0.552181\pi$
$659$	−8.37889			−0.326395			−0.163198	+	0.986593i	$0.552181\pi$
$660$		0			0
$661$		−	31.1281i		−	1.21074i	−0.795943	−	0.605371i	$-0.793025\pi$
$661$		−	31.1281i		−	1.21074i	0.795943	−	0.605371i	$-0.206975\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−35.2269	−	26.5879i	−1.36604	−	1.03103i
$666$		0			0
$667$			23.7979i			0.921457i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			2.23538i			0.0862958i
$672$		0			0
$673$			30.3650i			1.17048i	0.810859	+	0.585242i	$0.199000\pi$
$673$			30.3650i			1.17048i	−0.810859	+	0.585242i	$0.801000\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	23.8955			0.918379			0.459189	−	0.888338i	$-0.348140\pi$
$677$	23.8955			0.918379			0.459189	+	0.888338i	$0.348140\pi$
$678$		0			0
$679$		−	65.7308i		−	2.52252i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−17.7306			−0.678441			−0.339221	−	0.940707i	$-0.610163\pi$
$683$	−17.7306			−0.678441			−0.339221	+	0.940707i	$0.610163\pi$
$684$		0			0
$685$	−1.89784			−0.0725128
$686$		0			0
$687$		0			0
$688$		0			0
$689$			26.1927i			0.997863i
$690$		0			0
$691$	−29.7957			−1.13348			−0.566741	−	0.823896i	$-0.691796\pi$
$691$	−29.7957			−1.13348			−0.566741	+	0.823896i	$0.691796\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	4.25250i		−	0.161306i
$696$		0			0
$697$			19.0789i			0.722665i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	7.13657i		−	0.269544i	−0.990877	−	0.134772i	$-0.956970\pi$
$701$		−	7.13657i		−	0.269544i	0.990877	−	0.134772i	$-0.0430303\pi$
$702$		0			0
$703$	−10.7300	−	8.09861i	−0.404691	−	0.305445i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	3.63974i		−	0.136886i
$708$		0			0
$709$	11.5562			0.434003			0.217002	−	0.976171i	$-0.430372\pi$
$709$	11.5562			0.434003			0.217002	+	0.976171i	$0.430372\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−4.75582			−0.178107
$714$		0			0
$715$			3.93199i			0.147048i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	15.2870i		−	0.570108i	−0.958512	−	0.285054i	$-0.907989\pi$
$719$		−	15.2870i		−	0.570108i	0.958512	−	0.285054i	$-0.0920115\pi$
$720$		0			0
$721$		−	68.8804i		−	2.56524i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3.80287			0.141235
$726$		0			0
$727$	13.5664			0.503150			0.251575	−	0.967838i	$-0.419052\pi$
$727$	13.5664			0.503150			0.251575	+	0.967838i	$0.419052\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			12.9536i			0.479105i
$732$		0			0
$733$	−5.16691			−0.190844			−0.0954220	−	0.995437i	$-0.530420\pi$
$733$	−5.16691			−0.190844			−0.0954220	+	0.995437i	$0.530420\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−0.624101			−0.0229891
$738$		0			0
$739$	4.57290			0.168217			0.0841084	−	0.996457i	$-0.473196\pi$
$739$	4.57290			0.168217			0.0841084	+	0.996457i	$0.473196\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	0.589451			0.0216249			0.0108124	−	0.999942i	$-0.496558\pi$
$743$	0.589451			0.0216249			0.0108124	+	0.999942i	$0.496558\pi$
$744$		0			0
$745$	27.5664			1.00995
$746$		0			0
$747$		0			0
$748$		0			0
$749$	81.9995			2.99620
$750$		0			0
$751$			11.7704i			0.429508i	0.976668	+	0.214754i	$0.0688950\pi$
$751$			11.7704i			0.429508i	−0.976668	+	0.214754i	$0.931105\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−45.4427			−1.65383
$756$		0			0
$757$	18.0526			0.656134			0.328067	−	0.944654i	$-0.393603\pi$
$757$	18.0526			0.656134			0.328067	+	0.944654i	$0.393603\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			7.87400i			0.285432i	0.989764	+	0.142716i	$0.0455836\pi$
$761$			7.87400i			0.285432i	−0.989764	+	0.142716i	$0.954416\pi$
$762$		0			0
$763$			3.14959i			0.114023i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			53.3591i			1.92668i
$768$		0			0
$769$	11.8805			0.428422			0.214211	−	0.976787i	$-0.431282\pi$
$769$	11.8805			0.428422			0.214211	+	0.976787i	$0.431282\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	15.9381			0.573255			0.286627	−	0.958042i	$-0.407466\pi$
$773$	15.9381			0.573255			0.286627	+	0.958042i	$0.407466\pi$
$774$		0			0
$775$			0.759975i			0.0272991i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−19.5403	+	25.8894i	−0.700105	+	0.927585i
$780$		0			0
$781$		−	1.32909i		−	0.0475585i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	6.93066i		−	0.247366i
$786$		0			0
$787$		−	28.4321i		−	1.01349i	−0.862095	−	0.506747i	$-0.830848\pi$
$787$		−	28.4321i		−	1.01349i	0.862095	−	0.506747i	$-0.169152\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	42.3185			1.50467
$792$		0			0
$793$			53.9303i			1.91512i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−46.7246			−1.65507			−0.827535	−	0.561415i	$-0.810257\pi$
$797$	−46.7246			−1.65507			−0.827535	+	0.561415i	$0.810257\pi$
$798$		0			0
$799$	−19.1199			−0.676413
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	3.96690i		−	0.139989i
$804$		0			0
$805$	27.0123			0.952057
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	16.4026i		−	0.576684i	−0.957528	−	0.288342i	$-0.906896\pi$
$809$		−	16.4026i		−	0.576684i	0.957528	−	0.288342i	$-0.0931039\pi$
$810$		0			0
$811$		−	17.3863i		−	0.610514i	−0.952270	−	0.305257i	$-0.901258\pi$
$811$		−	17.3863i		−	0.610514i	0.952270	−	0.305257i	$-0.0987424\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	33.6120i		−	1.17738i
$816$		0			0
$817$	−13.2668	+	17.5775i	−0.464148	+	0.614960i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			33.4165i			1.16625i	0.812384	+	0.583123i	$0.198169\pi$
$821$			33.4165i			1.16625i	−0.812384	+	0.583123i	$0.801831\pi$
$822$		0			0
$823$	27.7134			0.966027			0.483014	−	0.875613i	$-0.339542\pi$
$823$	27.7134			0.966027			0.483014	+	0.875613i	$0.339542\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	8.87218			0.308516			0.154258	−	0.988031i	$-0.450701\pi$
$827$	8.87218			0.308516			0.154258	+	0.988031i	$0.450701\pi$
$828$		0			0
$829$		−	31.3606i		−	1.08920i	−0.838696	−	0.544600i	$-0.816681\pi$
$829$		−	31.3606i		−	1.08920i	0.838696	−	0.544600i	$-0.183319\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			30.4921i			1.05649i
$834$		0			0
$835$			42.4008i			1.46734i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	19.8510			0.685331			0.342666	−	0.939457i	$-0.388670\pi$
$839$	19.8510			0.685331			0.342666	+	0.939457i	$0.388670\pi$
$840$		0			0
$841$	50.5711			1.74383
$842$		0			0
$843$		0			0
$844$		0			0
$845$			64.5796i			2.22160i
$846$		0			0
$847$	−47.5083			−1.63240
$848$		0			0
$849$		0			0
$850$		0			0
$851$	8.22788			0.282048
$852$		0			0
$853$	−53.0570			−1.81664			−0.908318	−	0.418281i	$-0.862633\pi$
$853$	−53.0570			−1.81664			−0.908318	+	0.418281i	$0.862633\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−39.0297			−1.33323			−0.666615	−	0.745402i	$-0.732257\pi$
$857$	−39.0297			−1.33323			−0.666615	+	0.745402i	$0.732257\pi$
$858$		0			0
$859$	47.5861			1.62362			0.811808	−	0.583924i	$-0.198484\pi$
$859$	47.5861			1.62362			0.811808	+	0.583924i	$0.198484\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−39.0179			−1.32818			−0.664092	−	0.747651i	$-0.731182\pi$
$863$	−39.0179			−1.32818			−0.664092	+	0.747651i	$0.731182\pi$
$864$		0			0
$865$		−	40.2571i		−	1.36878i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1.64886			0.0559338
$870$		0			0
$871$	−15.0570			−0.510185
$872$		0			0
$873$		0			0
$874$		0			0
$875$			46.3091i			1.56553i
$876$		0			0
$877$			40.1150i			1.35459i	0.735714	+	0.677293i	$0.236847\pi$
$877$			40.1150i			1.35459i	−0.735714	+	0.677293i	$0.763153\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	21.0749i		−	0.710033i	−0.934860	−	0.355016i	$-0.884475\pi$
$881$		−	21.0749i		−	0.710033i	0.934860	−	0.355016i	$-0.115525\pi$
$882$		0			0
$883$	−0.887168			−0.0298556			−0.0149278	−	0.999889i	$-0.504752\pi$
$883$	−0.887168			−0.0298556			−0.0149278	+	0.999889i	$0.504752\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−55.4185			−1.86077			−0.930386	−	0.366582i	$-0.880528\pi$
$887$	−55.4185			−1.86077			−0.930386	+	0.366582i	$0.880528\pi$
$888$		0			0
$889$		−	3.14959i		−	0.105634i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−25.9450	−	19.5823i	−0.868217	−	0.655297i
$894$		0			0
$895$			20.1804i			0.674558i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			15.9017i			0.530350i
$900$		0			0
$901$		−	10.5236i		−	0.350592i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	23.3932			0.777617
$906$		0			0
$907$			20.7439i			0.688790i	0.938825	+	0.344395i	$0.111916\pi$
$907$			20.7439i			0.688790i	−0.938825	+	0.344395i	$0.888084\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	5.17380			0.171416			0.0857079	−	0.996320i	$-0.472685\pi$
$911$	5.17380			0.171416			0.0857079	+	0.996320i	$0.472685\pi$
$912$		0			0
$913$	4.23414			0.140130
$914$		0			0
$915$		0			0
$916$		0			0
$917$			40.8044i			1.34748i
$918$		0			0
$919$	−40.7050			−1.34274			−0.671368	−	0.741125i	$-0.734293\pi$
$919$	−40.7050			−1.34274			−0.671368	+	0.741125i	$0.734293\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	32.0653i		−	1.05544i
$924$		0			0
$925$		−	1.31481i		−	0.0432306i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	53.5496i		−	1.75691i	−0.477829	−	0.878453i	$-0.658576\pi$
$929$		−	53.5496i		−	1.75691i	0.477829	−	0.878453i	$-0.341424\pi$
$930$		0			0
$931$	−31.2296	+	41.3768i	−1.02351	+	1.35607i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	1.57978i		−	0.0516642i
$936$		0			0
$937$	−53.5060			−1.74796			−0.873982	−	0.485959i	$-0.838470\pi$
$937$	−53.5060			−1.74796			−0.873982	+	0.485959i	$0.838470\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−48.6751			−1.58676			−0.793381	−	0.608726i	$-0.791681\pi$
$941$	−48.6751			−1.58676			−0.793381	+	0.608726i	$0.791681\pi$
$942$		0			0
$943$		−	19.8522i		−	0.646478i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			26.4888i			0.860771i	0.902645	+	0.430386i	$0.141622\pi$
$947$			26.4888i			0.860771i	−0.902645	+	0.430386i	$0.858378\pi$
$948$		0			0
$949$		−	95.7046i		−	3.10670i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−40.0545			−1.29749			−0.648746	−	0.761005i	$-0.724706\pi$
$953$	−40.0545			−1.29749			−0.648746	+	0.761005i	$0.724706\pi$
$954$		0			0
$955$	25.7593			0.833553
$956$		0			0
$957$		0			0
$958$		0			0
$959$			3.54124i			0.114353i
$960$		0			0
$961$	27.8222			0.897489
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2.46463			−0.0793392
$966$		0			0
$967$	19.6629			0.632318			0.316159	−	0.948706i	$-0.397607\pi$
$967$	19.6629			0.632318			0.316159	+	0.948706i	$0.397607\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	32.1590			1.03203			0.516015	−	0.856579i	$-0.327415\pi$
$971$	32.1590			1.03203			0.516015	+	0.856579i	$0.327415\pi$
$972$		0			0
$973$	−7.93486			−0.254380
$974$		0			0
$975$		0			0
$976$		0			0
$977$	2.05498			0.0657446			0.0328723	−	0.999460i	$-0.489535\pi$
$977$	2.05498			0.0657446			0.0328723	+	0.999460i	$0.489535\pi$
$978$		0			0
$979$			2.18406i			0.0698029i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−24.3789			−0.777566			−0.388783	−	0.921329i	$-0.627104\pi$
$983$	−24.3789			−0.777566			−0.388783	+	0.921329i	$0.627104\pi$
$984$		0			0
$985$	55.3257			1.76282
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	13.4786i		−	0.428595i
$990$		0			0
$991$			15.4835i			0.491849i	0.969289	+	0.245924i	$0.0790915\pi$
$991$			15.4835i			0.491849i	−0.969289	+	0.245924i	$0.920909\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	2.45115i		−	0.0777068i
$996$		0			0
$997$	22.6405			0.717033			0.358516	−	0.933523i	$-0.383283\pi$
$997$	22.6405			0.717033			0.358516	+	0.933523i	$0.383283\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.f.g.1025.3		8
3.2	odd	2		2736.2.f.h.1025.6		8
4.3	odd	2		1368.2.f.c.1025.3	✓	8
12.11	even	2		1368.2.f.d.1025.6	yes	8
19.18	odd	2		2736.2.f.h.1025.3		8
57.56	even	2	inner	2736.2.f.g.1025.6		8
76.75	even	2		1368.2.f.d.1025.3	yes	8
228.227	odd	2		1368.2.f.c.1025.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1368.2.f.c.1025.3	✓	8	4.3	odd	2
1368.2.f.c.1025.6	yes	8	228.227	odd	2
1368.2.f.d.1025.3	yes	8	76.75	even	2
1368.2.f.d.1025.6	yes	8	12.11	even	2
2736.2.f.g.1025.3		8	1.1	even	1	trivial
2736.2.f.g.1025.6		8	57.56	even	2	inner
2736.2.f.h.1025.3		8	19.18	odd	2
2736.2.f.h.1025.6		8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.32945i q^{5} -4.34658 q^{7} +O(q^{10})\)	\(q-2.32945i q^{5} -4.34658 q^{7} -0.264508i q^{11} -6.38147i q^{13} +2.56392i q^{17} +(-2.62593 + 3.47915i) q^{19} -2.66784i q^{23} -0.426319 q^{25} -8.92026 q^{29} -1.78265i q^{31} +10.1251i q^{35} +3.08409i q^{37} +7.44131 q^{41} +5.05225 q^{43} +7.45728i q^{47} +11.8928 q^{49} -4.10450 q^{53} -0.616157 q^{55} -8.36157 q^{59} -8.45108 q^{61} -14.8653 q^{65} -2.35948i q^{67} +5.02476 q^{71} +14.9973 q^{73} +1.14971i q^{77} +6.23369i q^{79} +16.0076i q^{83} +5.97251 q^{85} -8.25707 q^{89} +27.7376i q^{91} +(8.10450 + 6.11696i) q^{95} +15.1224i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 4 q^{19} - 8 q^{25} - 32 q^{29} + 24 q^{41} + 28 q^{43} + 4 q^{49} - 8 q^{53} - 12 q^{55} + 8 q^{59} - 8 q^{61} + 24 q^{65} - 24 q^{71} + 4 q^{73} - 4 q^{85} - 16 q^{89} + 40 q^{95}+O(q^{100}) \) 8 * q - 4 * q^19 - 8 * q^25 - 32 * q^29 + 24 * q^41 + 28 * q^43 + 4 * q^49 - 8 * q^53 - 12 * q^55 + 8 * q^59 - 8 * q^61 + 24 * q^65 - 24 * q^71 + 4 * q^73 - 4 * q^85 - 16 * q^89 + 40 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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