LMFDB - Embedded newform 3168.2.o.e.703.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3168,2,Mod(703,3168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3168, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("3168.703");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3168 = 2^{5} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3168.o (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$25.2966073603$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.653473922154496.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4x^{10} + 13x^{8} - 28x^{6} + 52x^{4} - 64x^{2} + 64 $ x^12 - 4x^10 + 13x^8 - 28x^6 + 52x^4 - 64*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{15} $
Twist minimal:	no (minimal twist has level 352)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			703.5
Root			$0.892524 - 1.09700i$ of defining polynomial
Character	$\chi$	$=$	3168.703
Dual form			3168.2.o.e.703.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.289169 q^{5} -4.38799 q^{7} +O(q^{10})$	$q-0.289169 q^{5} -4.38799 q^{7} +(2.56483 - 2.10278i) q^{11} -1.55956i q^{13} -3.57009i q^{17} +6.39852 q^{19} +3.39194i q^{23} -4.91638 q^{25} -3.64632i q^{29} +9.01916i q^{31} +1.26887 q^{35} +0.289169 q^{37} -6.68921i q^{41} -5.12965 q^{43} +7.04888i q^{47} +12.2544 q^{49} -9.25443 q^{53} +(-0.741667 + 0.608056i) q^{55} +0.440820i q^{59} +10.7865i q^{61} +0.450976i q^{65} +3.97028i q^{67} -5.76473i q^{71} -6.68921i q^{73} +(-11.2544 + 9.22695i) q^{77} -11.3137 q^{79} -13.9056 q^{83} +1.03236i q^{85} -12.5925 q^{89} +6.84333i q^{91} -1.85025 q^{95} -8.75971 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q - 4 q^{25} + 44 q^{49} - 8 q^{53} - 32 q^{77} - 64 q^{97}+O(q^{100}) $ 12 * q - 4 * q^25 + 44 * q^49 - 8 * q^53 - 32 * q^77 - 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3168\mathbb{Z}\right)^\times$.

$n$	$353$	$991$	$1189$	$1729$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$5$	−0.289169			−0.129320			−0.0646601	−	0.997907i	$-0.520596\pi$
$5$	−0.289169			−0.129320			−0.0646601	+	0.997907i	$0.520596\pi$
$6$		0			0
$7$	−4.38799			−1.65850			−0.829252	−	0.558876i	$-0.811233\pi$
$7$	−4.38799			−1.65850			−0.829252	+	0.558876i	$0.811233\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.56483	−	2.10278i	0.773324	−	0.634011i
$11$	2.56483	−	2.10278i	0.773324	−	0.634011i
$12$		0			0
$13$		−	1.55956i		−	0.432544i	−0.976333	−	0.216272i	$-0.930610\pi$
$13$		−	1.55956i		−	0.432544i	0.976333	−	0.216272i	$-0.0693898\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	3.57009i		−	0.865875i	−0.901424	−	0.432938i	$-0.857477\pi$
$17$		−	3.57009i		−	0.865875i	0.901424	−	0.432938i	$-0.142523\pi$
$18$		0			0
$19$	6.39852			1.46792			0.733961	−	0.679192i	$-0.237670\pi$
$19$	6.39852			1.46792			0.733961	+	0.679192i	$0.237670\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			3.39194i			0.707269i	0.935384	+	0.353635i	$0.115054\pi$
$23$			3.39194i			0.707269i	−0.935384	+	0.353635i	$0.884946\pi$
$24$		0			0
$25$	−4.91638			−0.983276
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	3.64632i		−	0.677104i	−0.940948	−	0.338552i	$-0.890063\pi$
$29$		−	3.64632i		−	0.677104i	0.940948	−	0.338552i	$-0.109937\pi$
$30$		0			0
$31$			9.01916i			1.61989i	0.586507	+	0.809944i	$0.300503\pi$
$31$			9.01916i			1.61989i	−0.586507	+	0.809944i	$0.699497\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.26887			0.214478
$36$		0			0
$37$	0.289169			0.0475390			0.0237695	−	0.999717i	$-0.492433\pi$
$37$	0.289169			0.0475390			0.0237695	+	0.999717i	$0.492433\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	6.68921i		−	1.04468i	−0.852737	−	0.522340i	$-0.825059\pi$
$41$		−	6.68921i		−	1.04468i	0.852737	−	0.522340i	$-0.174941\pi$
$42$		0			0
$43$	−5.12965			−0.782265			−0.391132	−	0.920334i	$-0.627917\pi$
$43$	−5.12965			−0.782265			−0.391132	+	0.920334i	$0.627917\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			7.04888i			1.02818i	0.857735	+	0.514092i	$0.171871\pi$
$47$			7.04888i			1.02818i	−0.857735	+	0.514092i	$0.828129\pi$
$48$		0			0
$49$	12.2544			1.75063
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−9.25443			−1.27119			−0.635597	−	0.772021i	$-0.719246\pi$
$53$	−9.25443			−1.27119			−0.635597	+	0.772021i	$0.719246\pi$
$54$		0			0
$55$	−0.741667	+	0.608056i	−0.100006	+	0.0819903i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			0.440820i			0.0573898i	0.999588	+	0.0286949i	$0.00913513\pi$
$59$			0.440820i			0.0573898i	−0.999588	+	0.0286949i	$0.990865\pi$
$60$		0			0
$61$			10.7865i			1.38107i	0.723299	+	0.690535i	$0.242625\pi$
$61$			10.7865i			1.38107i	−0.723299	+	0.690535i	$0.757375\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			0.450976i			0.0559366i
$66$		0			0
$67$			3.97028i			0.485047i	0.970146	+	0.242523i	$0.0779751\pi$
$67$			3.97028i			0.485047i	−0.970146	+	0.242523i	$0.922025\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	5.76473i		−	0.684148i	−0.939673	−	0.342074i	$-0.888871\pi$
$71$		−	5.76473i		−	0.684148i	0.939673	−	0.342074i	$-0.111129\pi$
$72$		0			0
$73$		−	6.68921i		−	0.782913i	−0.920197	−	0.391457i	$-0.871971\pi$
$73$		−	6.68921i		−	0.782913i	0.920197	−	0.391457i	$-0.128029\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−11.2544	+	9.22695i	−1.28256	+	1.05151i
$78$		0			0
$79$	−11.3137			−1.27289			−0.636446	−	0.771321i	$-0.719596\pi$
$79$	−11.3137			−1.27289			−0.636446	+	0.771321i	$0.719596\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−13.9056			−1.52634			−0.763170	−	0.646197i	$-0.776358\pi$
$83$	−13.9056			−1.52634			−0.763170	+	0.646197i	$0.776358\pi$
$84$		0			0
$85$			1.03236i			0.111975i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−12.5925			−1.33480			−0.667400	−	0.744700i	$-0.732593\pi$
$89$	−12.5925			−1.33480			−0.667400	+	0.744700i	$0.732593\pi$
$90$		0			0
$91$			6.84333i			0.717375i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.85025			−0.189832
$96$		0			0
$97$	−8.75971			−0.889414			−0.444707	−	0.895676i	$-0.646692\pi$
$97$	−8.75971			−0.889414			−0.444707	+	0.895676i	$0.646692\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	12.8733i		−	1.28094i	−0.767984	−	0.640469i	$-0.778740\pi$
$101$		−	12.8733i		−	1.28094i	0.767984	−	0.640469i	$-0.221260\pi$
$102$		0			0
$103$		−	16.2056i		−	1.59678i	−0.602140	−	0.798390i	$-0.705685\pi$
$103$		−	16.2056i		−	1.59678i	0.602140	−	0.798390i	$-0.294315\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−13.6912			−1.32357			−0.661787	−	0.749692i	$-0.730202\pi$
$107$	−13.6912			−1.32357			−0.661787	+	0.749692i	$0.730202\pi$
$108$		0			0
$109$		−	0.527200i		−	0.0504966i	−0.999681	−	0.0252483i	$-0.991962\pi$
$109$		−	0.527200i		−	0.0504966i	0.999681	−	0.0252483i	$-0.00803764\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−2.49472			−0.234683			−0.117342	−	0.993092i	$-0.537437\pi$
$113$	−2.49472			−0.234683			−0.117342	+	0.993092i	$0.537437\pi$
$114$		0			0
$115$		−	0.980843i		−	0.0914641i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			15.6655i			1.43606i
$120$		0			0
$121$	2.15667	−	10.7865i	0.196061	−	0.980592i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	2.86751			0.256478
$126$		0			0
$127$	−2.90465			−0.257746			−0.128873	−	0.991661i	$-0.541136\pi$
$127$	−2.90465			−0.257746			−0.128873	+	0.991661i	$0.541136\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6.39852			−0.559041			−0.279521	−	0.960140i	$-0.590176\pi$
$131$	−6.39852			−0.559041			−0.279521	+	0.960140i	$0.590176\pi$
$132$		0			0
$133$	−28.0766			−2.43455
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.33804			−0.114317			−0.0571584	−	0.998365i	$-0.518204\pi$
$137$	−1.33804			−0.114317			−0.0571584	+	0.998365i	$0.518204\pi$
$138$		0			0
$139$	2.37745			0.201653			0.100826	−	0.994904i	$-0.467851\pi$
$139$	2.37745			0.201653			0.100826	+	0.994904i	$0.467851\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.27940	−	4.00000i	−0.274237	−	0.334497i
$144$		0			0
$145$			1.05440i			0.0875632i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	1.55956i		−	0.127764i	−0.997957	−	0.0638820i	$-0.979652\pi$
$149$		−	1.55956i		−	0.127764i	0.997957	−	0.0638820i	$-0.0203481\pi$
$150$		0			0
$151$	−7.72157			−0.628373			−0.314186	−	0.949361i	$-0.601732\pi$
$151$	−7.72157			−0.628373			−0.314186	+	0.949361i	$0.601732\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	2.60806i		−	0.209484i
$156$		0			0
$157$	−4.12193			−0.328966			−0.164483	−	0.986380i	$-0.552596\pi$
$157$	−4.12193			−0.328966			−0.164483	+	0.986380i	$0.552596\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	14.8838i		−	1.17301i
$162$		0			0
$163$			18.7144i			1.46583i	0.680323	+	0.732913i	$0.261840\pi$
$163$			18.7144i			1.46583i	−0.680323	+	0.732913i	$0.738160\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.92572			0.535928			0.267964	−	0.963429i	$-0.413649\pi$
$167$	6.92572			0.535928			0.267964	+	0.963429i	$0.413649\pi$
$168$		0			0
$169$	10.5678			0.812906
$170$		0			0
$171$		0			0
$172$		0			0
$173$			18.0791i			1.37453i	0.726406	+	0.687266i	$0.241189\pi$
$173$			18.0791i			1.37453i	−0.726406	+	0.687266i	$0.758811\pi$
$174$		0			0
$175$	21.5730			1.63077
$176$		0			0
$177$		0			0
$178$		0			0
$179$			11.6952i			0.874144i	0.899427	+	0.437072i	$0.143984\pi$
$179$			11.6952i			0.874144i	−0.899427	+	0.437072i	$0.856016\pi$
$180$		0			0
$181$	−2.96526			−0.220406			−0.110203	−	0.993909i	$-0.535150\pi$
$181$	−2.96526			−0.220406			−0.110203	+	0.993909i	$0.535150\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.0836184			−0.00614775
$186$		0			0
$187$	−7.50711	−	9.15667i	−0.548974	−	0.669602i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			6.98084i			0.505116i	0.967582	+	0.252558i	$0.0812719\pi$
$191$			6.98084i			0.505116i	−0.967582	+	0.252558i	$0.918728\pi$
$192$		0			0
$193$			4.17352i			0.300417i	0.988654	+	0.150208i	$0.0479944\pi$
$193$			4.17352i			0.300417i	−0.988654	+	0.150208i	$0.952006\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	13.9056i		−	0.990735i	−0.868684	−	0.495367i	$-0.835033\pi$
$197$		−	13.9056i		−	0.990735i	0.868684	−	0.495367i	$-0.164967\pi$
$198$		0			0
$199$			15.4600i			1.09593i	0.836502	+	0.547964i	$0.184597\pi$
$199$			15.4600i			1.09593i	−0.836502	+	0.547964i	$0.815403\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			16.0000i			1.12298i
$204$		0			0
$205$			1.93431i			0.135098i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	16.4111	−	13.4547i	1.13518	−	0.930678i
$210$		0			0
$211$	−6.18405			−0.425728			−0.212864	−	0.977082i	$-0.568279\pi$
$211$	−6.18405			−0.425728			−0.212864	+	0.977082i	$0.568279\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.48333			0.101163
$216$		0			0
$217$		−	39.5759i		−	2.68659i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−5.56777			−0.374529
$222$		0			0
$223$		−	6.98084i		−	0.467472i	−0.972300	−	0.233736i	$-0.924905\pi$
$223$		−	6.98084i		−	0.467472i	0.972300	−	0.233736i	$-0.0750951\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	13.6912			0.908714			0.454357	−	0.890820i	$-0.349869\pi$
$227$	13.6912			0.908714			0.454357	+	0.890820i	$0.349869\pi$
$228$		0			0
$229$	−21.2786			−1.40613			−0.703065	−	0.711126i	$-0.748186\pi$
$229$	−21.2786			−1.40613			−0.703065	+	0.711126i	$0.748186\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			27.8113i			1.82198i	0.412433	+	0.910988i	$0.364679\pi$
$233$			27.8113i			1.82198i	−0.412433	+	0.910988i	$0.635321\pi$
$234$		0			0
$235$		−	2.03831i		−	0.132965i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	5.50440			0.356050			0.178025	−	0.984026i	$-0.443029\pi$
$239$	5.50440			0.356050			0.178025	+	0.984026i	$0.443029\pi$
$240$		0			0
$241$		−	10.4118i		−	0.670680i	−0.942097	−	0.335340i	$-0.891149\pi$
$241$		−	10.4118i		−	0.670680i	0.942097	−	0.335340i	$-0.108851\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.54359			−0.226392
$246$		0			0
$247$		−	9.97887i		−	0.634941i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			15.2841i			0.964727i	0.875971	+	0.482363i	$0.160222\pi$
$251$			15.2841i			0.964727i	−0.875971	+	0.482363i	$0.839778\pi$
$252$		0			0
$253$	7.13249	+	8.69975i	0.448416	+	0.546948i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	11.5678			0.721578			0.360789	−	0.932647i	$-0.382507\pi$
$257$	11.5678			0.721578			0.360789	+	0.932647i	$0.382507\pi$
$258$		0			0
$259$	−1.26887			−0.0788436
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−27.4443			−1.69229			−0.846145	−	0.532952i	$-0.821082\pi$
$263$	−27.4443			−1.69229			−0.846145	+	0.532952i	$0.821082\pi$
$264$		0			0
$265$	2.67609			0.164391
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.25443			−0.0764837			−0.0382419	−	0.999269i	$-0.512176\pi$
$269$	−1.25443			−0.0764837			−0.0382419	+	0.999269i	$0.512176\pi$
$270$		0			0
$271$	25.9610			1.57702			0.788509	−	0.615023i	$-0.210853\pi$
$271$	25.9610			1.57702			0.788509	+	0.615023i	$0.210853\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−12.6097	+	10.3380i	−0.760392	+	0.623408i
$276$		0			0
$277$			0.527200i			0.0316764i	0.999875	+	0.0158382i	$0.00504167\pi$
$277$			0.527200i			0.0316764i	−0.999875	+	0.0158382i	$0.994958\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	10.2593i		−	0.612019i	−0.952028	−	0.306009i	$-0.901006\pi$
$281$		−	10.2593i		−	0.612019i	0.952028	−	0.306009i	$-0.0989939\pi$
$282$		0			0
$283$	−2.59192			−0.154074			−0.0770368	−	0.997028i	$-0.524546\pi$
$283$	−2.59192			−0.154074			−0.0770368	+	0.997028i	$0.524546\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			29.3522i			1.73260i
$288$		0			0
$289$	4.25443			0.250260
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	5.73308i		−	0.334930i	−0.985878	−	0.167465i	$-0.946442\pi$
$293$		−	5.73308i		−	0.334930i	0.985878	−	0.167465i	$-0.0535581\pi$
$294$		0			0
$295$		−	0.127471i		−	0.00742166i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	5.28994			0.305925
$300$		0			0
$301$	22.5089			1.29739
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	3.11912i		−	0.178600i
$306$		0			0
$307$	15.1745			0.866054			0.433027	−	0.901381i	$-0.357445\pi$
$307$	15.1745			0.866054			0.433027	+	0.901381i	$0.357445\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	13.1466i		−	0.745477i	−0.927936	−	0.372738i	$-0.878419\pi$
$311$		−	13.1466i		−	0.745477i	0.927936	−	0.372738i	$-0.121581\pi$
$312$		0			0
$313$	−15.6030			−0.881936			−0.440968	−	0.897523i	$-0.645365\pi$
$313$	−15.6030			−0.881936			−0.440968	+	0.897523i	$0.645365\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.44584			−0.0812066			−0.0406033	−	0.999175i	$-0.512928\pi$
$317$	−1.44584			−0.0812066			−0.0406033	+	0.999175i	$0.512928\pi$
$318$		0			0
$319$	−7.66739	−	9.35218i	−0.429291	−	0.523621i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	22.8433i		−	1.27104i
$324$		0			0
$325$			7.66739i			0.425310i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	30.9304i		−	1.70525i
$330$		0			0
$331$			14.0680i			0.773249i	0.922237	+	0.386624i	$0.126359\pi$
$331$			14.0680i			0.773249i	−0.922237	+	0.386624i	$0.873641\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	1.14808i		−	0.0627263i
$336$		0			0
$337$			2.66814i			0.145343i	0.997356	+	0.0726715i	$0.0231525\pi$
$337$			2.66814i			0.145343i	−0.997356	+	0.0726715i	$0.976848\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	18.9653	+	23.1326i	1.02703	+	1.25270i
$342$		0			0
$343$	−23.0564			−1.24493
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−19.9294			−1.06987			−0.534933	−	0.844894i	$-0.679663\pi$
$347$	−19.9294			−1.06987			−0.534933	+	0.844894i	$0.679663\pi$
$348$		0			0
$349$		−	9.60170i		−	0.513967i	−0.966416	−	0.256984i	$-0.917271\pi$
$349$		−	9.60170i		−	0.513967i	0.966416	−	0.256984i	$-0.0827286\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−7.24029			−0.385362			−0.192681	−	0.981261i	$-0.561718\pi$
$353$	−7.24029			−0.385362			−0.192681	+	0.981261i	$0.561718\pi$
$354$		0			0
$355$			1.66698i			0.0884740i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−7.35466			−0.388164			−0.194082	−	0.980985i	$-0.562173\pi$
$359$	−7.35466			−0.388164			−0.194082	+	0.980985i	$0.562173\pi$
$360$		0			0
$361$	21.9411			1.15479
$362$		0			0
$363$		0			0
$364$		0			0
$365$			1.93431i			0.101246i
$366$		0			0
$367$			15.5280i			0.810555i	0.914194	+	0.405278i	$0.132825\pi$
$367$			15.5280i			0.810555i	−0.914194	+	0.405278i	$0.867175\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	40.6083			2.10828
$372$		0			0
$373$			33.3919i			1.72897i	0.502662	+	0.864483i	$0.332354\pi$
$373$			33.3919i			1.72897i	−0.502662	+	0.864483i	$0.667646\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−5.68665			−0.292877
$378$		0			0
$379$		−	14.1275i		−	0.725679i	−0.931852	−	0.362840i	$-0.881807\pi$
$379$		−	14.1275i		−	0.725679i	0.931852	−	0.362840i	$-0.118193\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	38.3713i		−	1.96068i	−0.197307	−	0.980342i	$-0.563219\pi$
$383$		−	38.3713i		−	1.96068i	0.197307	−	0.980342i	$-0.436781\pi$
$384$		0			0
$385$	3.25443	−	2.66814i	0.165861	−	0.135981i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−28.3658			−1.43820			−0.719101	−	0.694905i	$-0.755446\pi$
$389$	−28.3658			−1.43820			−0.719101	+	0.694905i	$0.755446\pi$
$390$		0			0
$391$	12.1096			0.612407
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3.27157			0.164610
$396$		0			0
$397$	−21.2544			−1.06673			−0.533365	−	0.845885i	$-0.679073\pi$
$397$	−21.2544			−1.06673			−0.533365	+	0.845885i	$0.679073\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	18.0766			0.902704			0.451352	−	0.892346i	$-0.350942\pi$
$401$	18.0766			0.902704			0.451352	+	0.892346i	$0.350942\pi$
$402$		0			0
$403$	14.0659			0.700673
$404$		0			0
$405$		0			0
$406$		0			0
$407$	0.741667	−	0.608056i	0.0367631	−	0.0301402i
$408$		0			0
$409$			14.4328i			0.713657i	0.934170	+	0.356829i	$0.116142\pi$
$409$			14.4328i			0.713657i	−0.934170	+	0.356829i	$0.883858\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	1.93431i		−	0.0951812i
$414$		0			0
$415$	4.02107			0.197387
$416$		0			0
$417$		0			0
$418$		0			0
$419$			4.61665i			0.225538i	0.993621	+	0.112769i	$0.0359720\pi$
$419$			4.61665i			0.225538i	−0.993621	+	0.112769i	$0.964028\pi$
$420$		0			0
$421$	−13.2544			−0.645981			−0.322991	−	0.946402i	$-0.604688\pi$
$421$	−13.2544			−0.645981			−0.322991	+	0.946402i	$0.604688\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			17.5519i			0.851394i
$426$		0			0
$427$		−	47.3311i		−	2.29051i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	3.33359			0.160573			0.0802866	−	0.996772i	$-0.474416\pi$
$431$	3.33359			0.160573			0.0802866	+	0.996772i	$0.474416\pi$
$432$		0			0
$433$	−0.759707			−0.0365092			−0.0182546	−	0.999833i	$-0.505811\pi$
$433$	−0.759707			−0.0365092			−0.0182546	+	0.999833i	$0.505811\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			21.7034i			1.03822i
$438$		0			0
$439$	−10.2593			−0.489650			−0.244825	−	0.969567i	$-0.578731\pi$
$439$	−10.2593			−0.489650			−0.244825	+	0.969567i	$0.578731\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	29.7336i		−	1.41268i	−0.707871	−	0.706342i	$-0.750344\pi$
$443$		−	29.7336i		−	1.41268i	0.707871	−	0.706342i	$-0.249656\pi$
$444$		0			0
$445$	3.64135			0.172616
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−17.0036			−0.802448			−0.401224	−	0.915980i	$-0.631415\pi$
$449$	−17.0036			−0.802448			−0.401224	+	0.915980i	$0.631415\pi$
$450$		0			0
$451$	−14.0659	−	17.1567i	−0.662338	−	0.807876i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	1.97887i		−	0.0927711i
$456$		0			0
$457$			15.3348i			0.717331i	0.933466	+	0.358665i	$0.116768\pi$
$457$			15.3348i			0.717331i	−0.933466	+	0.358665i	$0.883232\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			27.2841i			1.27075i	0.772206	+	0.635373i	$0.219153\pi$
$461$			27.2841i			1.27075i	−0.772206	+	0.635373i	$0.780847\pi$
$462$		0			0
$463$			15.8625i			0.737192i	0.929590	+	0.368596i	$0.120161\pi$
$463$			15.8625i			0.737192i	−0.929590	+	0.368596i	$0.879839\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	1.32246i		−	0.0611961i	−0.999532	−	0.0305980i	$-0.990259\pi$
$467$		−	1.32246i		−	0.0611961i	0.999532	−	0.0305980i	$-0.00974118\pi$
$468$		0			0
$469$		−	17.4215i		−	0.804452i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−13.1567	+	10.7865i	−0.604945	+	0.495964i
$474$		0			0
$475$	−31.4576			−1.44337
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−37.2747			−1.70313			−0.851563	−	0.524253i	$-0.824345\pi$
$479$	−37.2747			−1.70313			−0.851563	+	0.524253i	$0.824345\pi$
$480$		0			0
$481$		−	0.450976i		−	0.0205627i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.53303			0.115019
$486$		0			0
$487$			5.43026i			0.246068i	0.992402	+	0.123034i	$0.0392625\pi$
$487$			5.43026i			0.246068i	−0.992402	+	0.123034i	$0.960738\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−0.374751			−0.0169123			−0.00845613	−	0.999964i	$-0.502692\pi$
$491$	−0.374751			−0.0169123			−0.00845613	+	0.999964i	$0.502692\pi$
$492$		0			0
$493$	−13.0177			−0.586288
$494$		0			0
$495$		0			0
$496$		0			0
$497$			25.2956i			1.13466i
$498$		0			0
$499$		−	4.61665i		−	0.206670i	−0.994647	−	0.103335i	$-0.967049\pi$
$499$		−	4.61665i		−	0.206670i	0.994647	−	0.103335i	$-0.0329513\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−1.48333			−0.0661386			−0.0330693	−	0.999453i	$-0.510528\pi$
$503$	−1.48333			−0.0661386			−0.0330693	+	0.999453i	$0.510528\pi$
$504$		0			0
$505$			3.72254i			0.165651i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−17.4458			−0.773273			−0.386637	−	0.922232i	$-0.626363\pi$
$509$	−17.4458			−0.773273			−0.386637	+	0.922232i	$0.626363\pi$
$510$		0			0
$511$			29.3522i			1.29846i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			4.68614i			0.206496i
$516$		0			0
$517$	14.8222	+	18.0791i	0.651880	+	0.795120i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	33.3663			1.46180			0.730902	−	0.682482i	$-0.239099\pi$
$521$	33.3663			1.46180			0.730902	+	0.682482i	$0.239099\pi$
$522$		0			0
$523$	−20.9838			−0.917557			−0.458779	−	0.888551i	$-0.651713\pi$
$523$	−20.9838			−0.917557			−0.458779	+	0.888551i	$0.651713\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	32.1992			1.40262
$528$		0			0
$529$	11.4947			0.499770
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−10.4322			−0.451870
$534$		0			0
$535$	3.95905			0.171165
$536$		0			0
$537$		0			0
$538$		0			0
$539$	31.4305	−	25.7683i	1.35381	−	1.10992i
$540$		0			0
$541$		−	23.2850i		−	1.00110i	−0.865707	−	0.500551i	$-0.833131\pi$
$541$		−	23.2850i		−	1.00110i	0.865707	−	0.500551i	$-0.166869\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			0.152450i			0.00653023i
$546$		0			0
$547$	22.6816			0.969795			0.484898	−	0.874571i	$-0.338857\pi$
$547$	22.6816			0.969795			0.484898	+	0.874571i	$0.338857\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	23.3311i		−	0.993936i
$552$		0			0
$553$	49.6444			2.11109
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	40.6845i		−	1.72386i	−0.507029	−	0.861929i	$-0.669256\pi$
$557$		−	40.6845i		−	1.72386i	0.507029	−	0.861929i	$-0.330744\pi$
$558$		0			0
$559$			8.00000i			0.338364i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	19.0895			0.804525			0.402262	−	0.915524i	$-0.368224\pi$
$563$	19.0895			0.804525			0.402262	+	0.915524i	$0.368224\pi$
$564$		0			0
$565$	0.721394			0.0303493
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	26.6485i		−	1.11716i	−0.829450	−	0.558581i	$-0.811346\pi$
$569$		−	26.6485i		−	1.11716i	0.829450	−	0.558581i	$-0.188654\pi$
$570$		0			0
$571$	18.6605			0.780919			0.390459	−	0.920620i	$-0.372316\pi$
$571$	18.6605			0.780919			0.390459	+	0.920620i	$0.372316\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	16.6761i		−	0.695441i
$576$		0			0
$577$	−18.5230			−0.771122			−0.385561	−	0.922682i	$-0.625992\pi$
$577$	−18.5230			−0.771122			−0.385561	+	0.922682i	$0.625992\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	61.0177			2.53144
$582$		0			0
$583$	−23.7360	+	19.4600i	−0.983045	+	0.805950i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	13.3622i		−	0.551518i	−0.961227	−	0.275759i	$-0.911071\pi$
$587$		−	13.3622i		−	0.551518i	0.961227	−	0.275759i	$-0.0889292\pi$
$588$		0			0
$589$			57.7093i			2.37787i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	9.24899i		−	0.379811i	−0.981802	−	0.189905i	$-0.939182\pi$
$593$		−	9.24899i		−	0.379811i	0.981802	−	0.189905i	$-0.0608181\pi$
$594$		0			0
$595$		−	4.52998i		−	0.185711i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	33.7733i		−	1.37994i	−0.723838	−	0.689970i	$-0.757624\pi$
$599$		−	33.7733i		−	1.37994i	0.723838	−	0.689970i	$-0.242376\pi$
$600$		0			0
$601$		−	33.4901i		−	1.36609i	−0.730375	−	0.683046i	$-0.760655\pi$
$601$		−	33.4901i		−	1.36609i	0.730375	−	0.683046i	$-0.239345\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.623642	+	3.11912i	−0.0253547	+	0.126810i
$606$		0			0
$607$	17.5519			0.712412			0.356206	−	0.934408i	$-0.384070\pi$
$607$	17.5519			0.712412			0.356206	+	0.934408i	$0.384070\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	10.9931			0.444735
$612$		0			0
$613$			35.3262i			1.42681i	0.700751	+	0.713406i	$0.252848\pi$
$613$			35.3262i			1.42681i	−0.700751	+	0.713406i	$0.747152\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	25.2544			1.01670			0.508352	−	0.861149i	$-0.330255\pi$
$617$	25.2544			1.01670			0.508352	+	0.861149i	$0.330255\pi$
$618$		0			0
$619$			19.7547i			0.794008i	0.917817	+	0.397004i	$0.129950\pi$
$619$			19.7547i			0.794008i	−0.917817	+	0.397004i	$0.870050\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	55.2556			2.21377
$624$		0			0
$625$	23.7527			0.950109
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	1.03236i		−	0.0411628i
$630$		0			0
$631$		−	19.0575i		−	0.758666i	−0.925260	−	0.379333i	$-0.876154\pi$
$631$		−	19.0575i		−	0.758666i	0.925260	−	0.379333i	$-0.123846\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	0.839934			0.0333318
$636$		0			0
$637$		−	19.1115i		−	0.757225i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	22.3275			0.881883			0.440941	−	0.897536i	$-0.354645\pi$
$641$	22.3275			0.881883			0.440941	+	0.897536i	$0.354645\pi$
$642$		0			0
$643$			11.9703i			0.472062i	0.971746	+	0.236031i	$0.0758467\pi$
$643$			11.9703i			0.472062i	−0.971746	+	0.236031i	$0.924153\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			10.2353i			0.402390i	0.979551	+	0.201195i	$0.0644825\pi$
$647$			10.2353i			0.402390i	−0.979551	+	0.201195i	$0.935518\pi$
$648$		0			0
$649$	0.926944	+	1.13063i	0.0363857	+	0.0443809i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−15.1325			−0.592180			−0.296090	−	0.955160i	$-0.595683\pi$
$653$	−15.1325			−0.592180			−0.296090	+	0.955160i	$0.595683\pi$
$654$		0			0
$655$	1.85025			0.0722953
$656$		0			0
$657$		0			0
$658$		0			0
$659$	0.374751			0.0145982			0.00729911	−	0.999973i	$-0.497677\pi$
$659$	0.374751			0.0145982			0.00729911	+	0.999973i	$0.497677\pi$
$660$		0			0
$661$	47.6202			1.85221			0.926106	−	0.377263i	$-0.123135\pi$
$661$	47.6202			1.85221			0.926106	+	0.377263i	$0.123135\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	8.11888			0.314837
$666$		0			0
$667$	12.3681			0.478895
$668$		0			0
$669$		0			0
$670$		0			0
$671$	22.6816	+	27.6655i	0.875613	+	1.06802i
$672$		0			0
$673$		−	39.8745i		−	1.53705i	−0.639821	−	0.768524i	$-0.720992\pi$
$673$		−	39.8745i		−	1.53705i	0.639821	−	0.768524i	$-0.279008\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	14.9380i		−	0.574113i	−0.957914	−	0.287057i	$-0.907323\pi$
$677$		−	14.9380i		−	0.574113i	0.957914	−	0.287057i	$-0.0926768\pi$
$678$		0			0
$679$	38.4375			1.47510
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	12.2056i		−	0.467032i	−0.972353	−	0.233516i	$-0.924977\pi$
$683$		−	12.2056i		−	0.467032i	0.972353	−	0.233516i	$-0.0750232\pi$
$684$		0			0
$685$	0.386920			0.0147835
$686$		0			0
$687$		0			0
$688$		0			0
$689$			14.4328i			0.549847i
$690$		0			0
$691$			32.1658i			1.22364i	0.790995	+	0.611822i	$0.209563\pi$
$691$			32.1658i			1.22364i	−0.790995	+	0.611822i	$0.790437\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−0.687484			−0.0260778
$696$		0			0
$697$	−23.8811			−0.904562
$698$		0			0
$699$		0			0
$700$		0			0
$701$			0.657608i			0.0248375i	0.999923	+	0.0124188i	$0.00395312\pi$
$701$			0.657608i			0.0248375i	−0.999923	+	0.0124188i	$0.996047\pi$
$702$		0			0
$703$	1.85025			0.0697835
$704$		0			0
$705$		0			0
$706$		0			0
$707$			56.4877i			2.12444i
$708$		0			0
$709$	33.2302			1.24799			0.623994	−	0.781429i	$-0.285509\pi$
$709$	33.2302			1.24799			0.623994	+	0.781429i	$0.285509\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−30.5925			−1.14570
$714$		0			0
$715$	0.948300	+	1.15667i	0.0354644	+	0.0432572i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			18.1164i			0.675627i	0.941213	+	0.337814i	$0.109687\pi$
$719$			18.1164i			0.675627i	−0.941213	+	0.337814i	$0.890313\pi$
$720$		0			0
$721$			71.1097i			2.64826i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			17.9267i			0.665781i
$726$		0			0
$727$		−	21.3708i		−	0.792600i	−0.918121	−	0.396300i	$-0.870294\pi$
$727$		−	21.3708i		−	0.792600i	0.918121	−	0.396300i	$-0.129706\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			18.3133i			0.677344i
$732$		0			0
$733$		−	34.4683i		−	1.27312i	−0.771229	−	0.636558i	$-0.780357\pi$
$733$		−	34.4683i		−	1.27312i	0.771229	−	0.636558i	$-0.219643\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	8.34861	+	10.1831i	0.307525	+	0.375099i
$738$		0			0
$739$	−9.15072			−0.336615			−0.168307	−	0.985735i	$-0.553830\pi$
$739$	−9.15072			−0.336615			−0.168307	+	0.985735i	$0.553830\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−6.98774			−0.256355			−0.128178	−	0.991751i	$-0.540913\pi$
$743$	−6.98774			−0.256355			−0.128178	+	0.991751i	$0.540913\pi$
$744$		0			0
$745$			0.450976i			0.0165225i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	60.0766			2.19515
$750$		0			0
$751$		−	40.5285i		−	1.47891i	−0.673208	−	0.739453i	$-0.735084\pi$
$751$		−	40.5285i		−	1.47891i	0.673208	−	0.739453i	$-0.264916\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	2.23284			0.0812612
$756$		0			0
$757$	−6.74557			−0.245172			−0.122586	−	0.992458i	$-0.539119\pi$
$757$	−6.74557			−0.245172			−0.122586	+	0.992458i	$0.539119\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			15.0363i			0.545064i	0.962147	+	0.272532i	$0.0878610\pi$
$761$			15.0363i			0.545064i	−0.962147	+	0.272532i	$0.912139\pi$
$762$		0			0
$763$			2.31335i			0.0837488i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	0.687484			0.0248236
$768$		0			0
$769$			2.66814i			0.0962157i	0.998842	+	0.0481079i	$0.0153191\pi$
$769$			2.66814i			0.0962157i	−0.998842	+	0.0481079i	$0.984681\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	41.3311			1.48657			0.743287	−	0.668972i	$-0.233266\pi$
$773$	41.3311			1.48657			0.743287	+	0.668972i	$0.233266\pi$
$774$		0			0
$775$		−	44.3416i		−	1.59280i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	42.8011i		−	1.53351i
$780$		0			0
$781$	−12.1219	−	14.7855i	−0.433757	−	0.529068i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1.19193			0.0425419
$786$		0			0
$787$	14.3346			0.510972			0.255486	−	0.966813i	$-0.417765\pi$
$787$	14.3346			0.510972			0.255486	+	0.966813i	$0.417765\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	10.9468			0.389223
$792$		0			0
$793$	16.8222			0.597374
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−36.0313			−1.27629			−0.638147	−	0.769914i	$-0.720299\pi$
$797$	−36.0313			−1.27629			−0.638147	+	0.769914i	$0.720299\pi$
$798$		0			0
$799$	25.1652			0.890279
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−14.0659	−	17.1567i	−0.496375	−	0.605446i
$804$		0			0
$805$			4.30393i			0.151694i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			47.7705i			1.67952i	0.542956	+	0.839761i	$0.317305\pi$
$809$			47.7705i			1.67952i	−0.542956	+	0.839761i	$0.682695\pi$
$810$		0			0
$811$	−36.9620			−1.29791			−0.648955	−	0.760827i	$-0.724794\pi$
$811$	−36.9620			−1.29791			−0.648955	+	0.760827i	$0.724794\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	5.41162i		−	0.189561i
$816$		0			0
$817$	−32.8222			−1.14830
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	14.8076i		−	0.516788i	−0.966040	−	0.258394i	$-0.916807\pi$
$821$		−	14.8076i		−	0.516788i	0.966040	−	0.258394i	$-0.0831933\pi$
$822$		0			0
$823$		−	32.4096i		−	1.12973i	−0.825184	−	0.564865i	$-0.808928\pi$
$823$		−	32.4096i		−	1.12973i	0.825184	−	0.564865i	$-0.191072\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−12.9573			−0.450570			−0.225285	−	0.974293i	$-0.572331\pi$
$827$	−12.9573			−0.450570			−0.225285	+	0.974293i	$0.572331\pi$
$828$		0			0
$829$	−14.5542			−0.505487			−0.252743	−	0.967533i	$-0.581333\pi$
$829$	−14.5542			−0.505487			−0.252743	+	0.967533i	$0.581333\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	43.7495i		−	1.51583i
$834$		0			0
$835$	−2.00270			−0.0693063
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	39.7436i		−	1.37210i	−0.727554	−	0.686051i	$-0.759343\pi$
$839$		−	39.7436i		−	1.37210i	0.727554	−	0.686051i	$-0.240657\pi$
$840$		0			0
$841$	15.7044			0.541530
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3.05587			−0.105125
$846$		0			0
$847$	−9.46346	+	47.3311i	−0.325168	+	1.62631i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			0.980843i			0.0336229i
$852$		0			0
$853$			12.7429i			0.436307i	0.975914	+	0.218154i	$0.0700034\pi$
$853$			12.7429i			0.436307i	−0.975914	+	0.218154i	$0.929997\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	11.1613i		−	0.381261i	−0.981662	−	0.190631i	$-0.938947\pi$
$857$		−	11.1613i		−	0.381261i	0.981662	−	0.190631i	$-0.0610533\pi$
$858$		0			0
$859$		−	54.6152i		−	1.86345i	−0.363169	−	0.931723i	$-0.618305\pi$
$859$		−	54.6152i		−	1.86345i	0.363169	−	0.931723i	$-0.381695\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	37.9688i		−	1.29247i	−0.763137	−	0.646237i	$-0.776342\pi$
$863$		−	37.9688i		−	1.29247i	0.763137	−	0.646237i	$-0.223658\pi$
$864$		0			0
$865$		−	5.22792i		−	0.177755i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−29.0177	+	23.7902i	−0.984358	+	0.807027i
$870$		0			0
$871$	6.19189			0.209804
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−12.5826			−0.425369
$876$		0			0
$877$		−	33.3919i		−	1.12756i	−0.825924	−	0.563782i	$-0.809346\pi$
$877$		−	33.3919i		−	1.12756i	0.825924	−	0.563782i	$-0.190654\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−43.4358			−1.46339			−0.731695	−	0.681633i	$-0.761270\pi$
$881$	−43.4358			−1.46339			−0.731695	+	0.681633i	$0.761270\pi$
$882$		0			0
$883$		−	13.7733i		−	0.463509i	−0.972774	−	0.231755i	$-0.925553\pi$
$883$		−	13.7733i		−	0.463509i	0.972774	−	0.231755i	$-0.0744466\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−41.6627			−1.39890			−0.699448	−	0.714683i	$-0.746571\pi$
$887$	−41.6627			−1.39890			−0.699448	+	0.714683i	$0.746571\pi$
$888$		0			0
$889$	12.7456			0.427473
$890$		0			0
$891$		0			0
$892$		0			0
$893$			45.1024i			1.50929i
$894$		0			0
$895$		−	3.38190i		−	0.113044i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	32.8867			1.09683
$900$		0			0
$901$			33.0392i			1.10069i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	0.857459			0.0285029
$906$		0			0
$907$			12.2056i			0.405279i	0.979253	+	0.202639i	$0.0649519\pi$
$907$			12.2056i			0.405279i	−0.979253	+	0.202639i	$0.935048\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			9.36222i			0.310184i	0.987900	+	0.155092i	$0.0495675\pi$
$911$			9.36222i			0.310184i	−0.987900	+	0.155092i	$0.950433\pi$
$912$		0			0
$913$	−35.6655	+	29.2404i	−1.18036	+	0.967716i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	28.0766			0.927172
$918$		0			0
$919$	−27.8113			−0.917409			−0.458704	−	0.888589i	$-0.651686\pi$
$919$	−27.8113			−0.917409			−0.458704	+	0.888589i	$0.651686\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−8.99044			−0.295924
$924$		0			0
$925$	−1.42166			−0.0467440
$926$		0			0
$927$		0			0
$928$		0			0
$929$	7.76328			0.254705			0.127352	−	0.991858i	$-0.459352\pi$
$929$	7.76328			0.254705			0.127352	+	0.991858i	$0.459352\pi$
$930$		0			0
$931$	78.4102			2.56979
$932$		0			0
$933$		0			0
$934$		0			0
$935$	2.17082	+	2.64782i	0.0709934	+	0.0865930i
$936$		0			0
$937$			41.4882i			1.35536i	0.735357	+	0.677680i	$0.237015\pi$
$937$			41.4882i			1.35536i	−0.735357	+	0.677680i	$0.762985\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			20.1439i			0.656671i	0.944561	+	0.328336i	$0.106488\pi$
$941$			20.1439i			0.656671i	−0.944561	+	0.328336i	$0.893512\pi$
$942$		0			0
$943$	22.6894			0.738870
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	41.3225i		−	1.34280i	−0.741095	−	0.671400i	$-0.765693\pi$
$947$		−	41.3225i		−	1.34280i	0.741095	−	0.671400i	$-0.234307\pi$
$948$		0			0
$949$	−10.4322			−0.338644
$950$		0			0
$951$		0			0
$952$		0			0
$953$			28.1098i			0.910565i	0.890347	+	0.455283i	$0.150462\pi$
$953$			28.1098i			0.910565i	−0.890347	+	0.455283i	$0.849538\pi$
$954$		0			0
$955$		−	2.01864i		−	0.0653217i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	5.87132			0.189595
$960$		0			0
$961$	−50.3452			−1.62404
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	1.20685i		−	0.0388499i
$966$		0			0
$967$	−21.5730			−0.693741			−0.346871	−	0.937913i	$-0.612756\pi$
$967$	−21.5730			−0.693741			−0.346871	+	0.937913i	$0.612756\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			17.9320i			0.575464i	0.957711	+	0.287732i	$0.0929013\pi$
$971$			17.9320i			0.575464i	−0.957711	+	0.287732i	$0.907099\pi$
$972$		0			0
$973$	−10.4322			−0.334442
$974$		0			0
$975$		0			0
$976$		0			0
$977$	43.8953			1.40433			0.702167	−	0.712012i	$-0.252216\pi$
$977$	43.8953			1.40433			0.702167	+	0.712012i	$0.252216\pi$
$978$		0			0
$979$	−32.2975	+	26.4791i	−1.03223	+	0.846277i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			21.7053i			0.692291i	0.938181	+	0.346146i	$0.112510\pi$
$983$			21.7053i			0.692291i	−0.938181	+	0.346146i	$0.887490\pi$
$984$		0			0
$985$			4.02107i			0.128122i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	17.3995i		−	0.553272i
$990$		0			0
$991$		−	54.3799i		−	1.72744i	−0.503976	−	0.863718i	$-0.668130\pi$
$991$		−	54.3799i		−	1.72744i	0.503976	−	0.863718i	$-0.331870\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	4.47054i		−	0.141726i
$996$		0			0
$997$		−	59.1384i		−	1.87293i	−0.350758	−	0.936466i	$-0.614076\pi$
$997$		−	59.1384i		−	1.87293i	0.350758	−	0.936466i	$-0.385924\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3168.2.o.e.703.5		12
3.2	odd	2		352.2.e.a.351.11	yes	12
4.3	odd	2	inner	3168.2.o.e.703.8		12
11.10	odd	2	inner	3168.2.o.e.703.7		12
12.11	even	2		352.2.e.a.351.2	yes	12
24.5	odd	2		704.2.e.d.703.1		12
24.11	even	2		704.2.e.d.703.12		12
33.32	even	2		352.2.e.a.351.12	yes	12
44.43	even	2	inner	3168.2.o.e.703.6		12
48.5	odd	4		2816.2.g.i.1407.12		12
48.11	even	4		2816.2.g.d.1407.3		12
48.29	odd	4		2816.2.g.d.1407.2		12
48.35	even	4		2816.2.g.i.1407.9		12
132.131	odd	2		352.2.e.a.351.1	✓	12
264.131	odd	2		704.2.e.d.703.11		12
264.197	even	2		704.2.e.d.703.2		12
528.131	odd	4		2816.2.g.i.1407.10		12
528.197	even	4		2816.2.g.i.1407.11		12
528.395	odd	4		2816.2.g.d.1407.4		12
528.461	even	4		2816.2.g.d.1407.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
352.2.e.a.351.1	✓	12	132.131	odd	2
352.2.e.a.351.2	yes	12	12.11	even	2
352.2.e.a.351.11	yes	12	3.2	odd	2
352.2.e.a.351.12	yes	12	33.32	even	2
704.2.e.d.703.1		12	24.5	odd	2
704.2.e.d.703.2		12	264.197	even	2
704.2.e.d.703.11		12	264.131	odd	2
704.2.e.d.703.12		12	24.11	even	2
2816.2.g.d.1407.1		12	528.461	even	4
2816.2.g.d.1407.2		12	48.29	odd	4
2816.2.g.d.1407.3		12	48.11	even	4
2816.2.g.d.1407.4		12	528.395	odd	4
2816.2.g.i.1407.9		12	48.35	even	4
2816.2.g.i.1407.10		12	528.131	odd	4
2816.2.g.i.1407.11		12	528.197	even	4
2816.2.g.i.1407.12		12	48.5	odd	4
3168.2.o.e.703.5		12	1.1	even	1	trivial
3168.2.o.e.703.6		12	44.43	even	2	inner
3168.2.o.e.703.7		12	11.10	odd	2	inner
3168.2.o.e.703.8		12	4.3	odd	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:	\( N \)	\(=\)	\( 3168 = 2^{5} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3168.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.289169 q^{5} -4.38799 q^{7} +O(q^{10})\)	\(q-0.289169 q^{5} -4.38799 q^{7} +(2.56483 - 2.10278i) q^{11} -1.55956i q^{13} -3.57009i q^{17} +6.39852 q^{19} +3.39194i q^{23} -4.91638 q^{25} -3.64632i q^{29} +9.01916i q^{31} +1.26887 q^{35} +0.289169 q^{37} -6.68921i q^{41} -5.12965 q^{43} +7.04888i q^{47} +12.2544 q^{49} -9.25443 q^{53} +(-0.741667 + 0.608056i) q^{55} +0.440820i q^{59} +10.7865i q^{61} +0.450976i q^{65} +3.97028i q^{67} -5.76473i q^{71} -6.68921i q^{73} +(-11.2544 + 9.22695i) q^{77} -11.3137 q^{79} -13.9056 q^{83} +1.03236i q^{85} -12.5925 q^{89} +6.84333i q^{91} -1.85025 q^{95} -8.75971 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q - 4 q^{25} + 44 q^{49} - 8 q^{53} - 32 q^{77} - 64 q^{97}+O(q^{100}) \) 12 * q - 4 * q^25 + 44 * q^49 - 8 * q^53 - 32 * q^77 - 64 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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