LMFDB - Embedded newform 2816.2.g.d.1407.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2816,2,Mod(1407,2816)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2816, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("2816.1407");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2816 = 2^{8} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2816.g (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:	$22.4858732092$
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^{11} $
Twist minimal:	no (minimal twist has level 352)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1407.4
Root			$0.892524 - 1.09700i$ of defining polynomial
Character	$\chi$	$=$	2816.1407
Dual form			2816.2.g.d.1407.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.81361 q^{3} +0.289169i q^{5} +4.38799 q^{7} +4.91638 q^{9} +O(q^{10})$	$q-2.81361 q^{3} +0.289169i q^{5} +4.38799 q^{7} +4.91638 q^{9} +(2.10278 - 2.56483i) q^{11} +1.55956 q^{13} -0.813607i q^{15} -3.57009i q^{17} -6.39852i q^{19} -12.3461 q^{21} -3.39194i q^{23} +4.91638 q^{25} -5.39194 q^{27} -3.64632 q^{29} -9.01916i q^{31} +(-5.91638 + 7.21641i) q^{33} +1.26887i q^{35} +0.289169i q^{37} -4.38799 q^{39} +6.68921i q^{41} -5.12965i q^{43} +1.42166i q^{45} +7.04888i q^{47} +12.2544 q^{49} +10.0448i q^{51} +9.25443i q^{53} +(0.741667 + 0.608056i) q^{55} +18.0029i q^{57} -0.440820 q^{59} -10.7865 q^{61} +21.5730 q^{63} +0.450976i q^{65} -3.97028 q^{67} +9.54359i q^{69} +5.76473i q^{71} -6.68921i q^{73} -13.8328 q^{75} +(9.22695 - 11.2544i) q^{77} -11.3137 q^{79} +0.421663 q^{81} -13.9056i q^{83} +1.03236 q^{85} +10.2593 q^{87} -12.5925 q^{89} +6.84333 q^{91} +25.3764i q^{93} +1.85025 q^{95} -8.75971 q^{97} +(10.3380 - 12.6097i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 8 q^{3} + 4 q^{9}+O(q^{10}) $ 12 * q - 8 * q^3 + 4 * q^9	$ 12 q - 8 q^{3} + 4 q^{9} - 4 q^{11} + 4 q^{25} - 32 q^{27} - 16 q^{33} + 44 q^{49} + 72 q^{59} - 8 q^{67} - 56 q^{75} + 12 q^{81} + 96 q^{91} - 64 q^{97} + 76 q^{99}+O(q^{100}) $ 12 * q - 8 * q^3 + 4 * q^9 - 4 * q^11 + 4 * q^25 - 32 * q^27 - 16 * q^33 + 44 * q^49 + 72 * q^59 - 8 * q^67 - 56 * q^75 + 12 * q^81 + 96 * q^91 - 64 * q^97 + 76 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2816\mathbb{Z}\right)^\times$.

$n$	$1025$	$1541$	$2047$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$	−2.81361			−1.62444			−0.812218	−	0.583354i	$-0.801740\pi$
$3$	−2.81361			−1.62444			−0.812218	+	0.583354i	$0.801740\pi$
$4$		0			0
$5$			0.289169i			0.129320i	0.997907	+	0.0646601i	$0.0205963\pi$
$5$			0.289169i			0.129320i	−0.997907	+	0.0646601i	$0.979404\pi$
$6$		0			0
$7$	4.38799			1.65850			0.829252	−	0.558876i	$-0.188767\pi$
$7$	4.38799			1.65850			0.829252	+	0.558876i	$0.188767\pi$
$8$		0			0
$9$	4.91638			1.63879
$10$		0			0
$11$	2.10278	−	2.56483i	0.634011	−	0.773324i
$11$	2.10278	−	2.56483i	0.634011	−	0.773324i
$12$		0			0
$13$	1.55956			0.432544			0.216272	−	0.976333i	$-0.430610\pi$
$13$	1.55956			0.432544			0.216272	+	0.976333i	$0.430610\pi$
$14$		0			0
$15$		−	0.813607i		−	0.210072i
$16$		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
$19$		−	6.39852i		−	1.46792i	−0.679192	−	0.733961i	$-0.737670\pi$
$19$		−	6.39852i		−	1.46792i	0.679192	−	0.733961i	$-0.262330\pi$
$20$		0			0
$21$	−12.3461			−2.69413
$22$		0			0
$23$		−	3.39194i		−	0.707269i	−0.935384	−	0.353635i	$-0.884946\pi$
$23$		−	3.39194i		−	0.707269i	0.935384	−	0.353635i	$-0.115054\pi$
$24$		0			0
$25$	4.91638			0.983276
$26$		0			0
$27$	−5.39194			−1.03768
$28$		0			0
$29$	−3.64632			−0.677104			−0.338552	−	0.940948i	$-0.609937\pi$
$29$	−3.64632			−0.677104			−0.338552	+	0.940948i	$0.609937\pi$
$30$		0			0
$31$		−	9.01916i		−	1.61989i	−0.586507	−	0.809944i	$-0.699497\pi$
$31$		−	9.01916i		−	1.61989i	0.586507	−	0.809944i	$-0.300503\pi$
$32$		0			0
$33$	−5.91638	+	7.21641i	−1.02991	+	1.25622i
$34$		0			0
$35$			1.26887i			0.214478i
$36$		0			0
$37$			0.289169i			0.0475390i	0.999717	+	0.0237695i	$0.00756678\pi$
$37$			0.289169i			0.0475390i	−0.999717	+	0.0237695i	$0.992433\pi$
$38$		0			0
$39$	−4.38799			−0.702640
$40$		0			0
$41$			6.68921i			1.04468i	0.852737	+	0.522340i	$0.174941\pi$
$41$			6.68921i			1.04468i	−0.852737	+	0.522340i	$0.825059\pi$
$42$		0			0
$43$		−	5.12965i		−	0.782265i	−0.920334	−	0.391132i	$-0.872083\pi$
$43$		−	5.12965i		−	0.782265i	0.920334	−	0.391132i	$-0.127917\pi$
$44$		0			0
$45$			1.42166i			0.211929i
$46$		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
$51$			10.0448i			1.40656i
$52$		0			0
$53$			9.25443i			1.27119i	0.772021	+	0.635597i	$0.219246\pi$
$53$			9.25443i			1.27119i	−0.772021	+	0.635597i	$0.780754\pi$
$54$		0			0
$55$	0.741667	+	0.608056i	0.100006	+	0.0819903i
$56$		0			0
$57$			18.0029i			2.38455i
$58$		0			0
$59$	−0.440820			−0.0573898			−0.0286949	−	0.999588i	$-0.509135\pi$
$59$	−0.440820			−0.0573898			−0.0286949	+	0.999588i	$0.509135\pi$
$60$		0			0
$61$	−10.7865			−1.38107			−0.690535	−	0.723299i	$-0.742625\pi$
$61$	−10.7865			−1.38107			−0.690535	+	0.723299i	$0.742625\pi$
$62$		0			0
$63$	21.5730			2.71794
$64$		0			0
$65$			0.450976i			0.0559366i
$66$		0			0
$67$	−3.97028			−0.485047			−0.242523	−	0.970146i	$-0.577975\pi$
$67$	−3.97028			−0.485047			−0.242523	+	0.970146i	$0.577975\pi$
$68$		0			0
$69$			9.54359i			1.14891i
$70$		0			0
$71$			5.76473i			0.684148i	0.939673	+	0.342074i	$0.111129\pi$
$71$			5.76473i			0.684148i	−0.939673	+	0.342074i	$0.888871\pi$
$72$		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
$75$	−13.8328			−1.59727
$76$		0			0
$77$	9.22695	−	11.2544i	1.05151	−	1.28256i
$78$		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
$81$	0.421663			0.0468514
$82$		0			0
$83$		−	13.9056i		−	1.52634i	−0.646197	−	0.763170i	$-0.723642\pi$
$83$		−	13.9056i		−	1.52634i	0.646197	−	0.763170i	$-0.276358\pi$
$84$		0			0
$85$	1.03236			0.111975
$86$		0			0
$87$	10.2593			1.09991
$88$		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
$91$	6.84333			0.717375
$92$		0			0
$93$			25.3764i			2.63141i
$94$		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
$99$	10.3380	−	12.6097i	1.03901	−	1.26732i
$100$		0			0
$101$	12.8733			1.28094			0.640469	−	0.767984i	$-0.278740\pi$
$101$	12.8733			1.28094			0.640469	+	0.767984i	$0.278740\pi$
$102$		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
$105$		−	3.57009i		−	0.348406i
$106$		0			0
$107$			13.6912i			1.32357i	0.749692	+	0.661787i	$0.230202\pi$
$107$			13.6912i			1.32357i	−0.749692	+	0.661787i	$0.769798\pi$
$108$		0			0
$109$	0.527200			0.0504966			0.0252483	−	0.999681i	$-0.491962\pi$
$109$	0.527200			0.0504966			0.0252483	+	0.999681i	$0.491962\pi$
$110$		0			0
$111$		−	0.813607i		−	0.0772241i
$112$		0			0
$113$	2.49472			0.234683			0.117342	−	0.993092i	$-0.462563\pi$
$113$	2.49472			0.234683			0.117342	+	0.993092i	$0.462563\pi$
$114$		0			0
$115$	0.980843			0.0914641
$116$		0			0
$117$	7.66739			0.708850
$118$		0			0
$119$		−	15.6655i		−	1.43606i
$120$		0			0
$121$	−2.15667	−	10.7865i	−0.196061	−	0.980592i
$122$		0			0
$123$		−	18.8208i		−	1.69702i
$124$		0			0
$125$			2.86751i			0.256478i
$126$		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
$129$			14.4328i			1.27074i
$130$		0			0
$131$		−	6.39852i		−	0.559041i	−0.960140	−	0.279521i	$-0.909824\pi$
$131$		−	6.39852i		−	0.559041i	0.960140	−	0.279521i	$-0.0901756\pi$
$132$		0			0
$133$		−	28.0766i		−	2.43455i
$134$		0			0
$135$		−	1.55918i		−	0.134193i
$136$		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
$139$			2.37745i			0.201653i	0.994904	+	0.100826i	$0.0321487\pi$
$139$			2.37745i			0.201653i	−0.994904	+	0.100826i	$0.967851\pi$
$140$		0			0
$141$		−	19.8328i		−	1.67022i
$142$		0			0
$143$	3.27940	−	4.00000i	0.274237	−	0.334497i
$144$		0			0
$145$		−	1.05440i		−	0.0875632i
$146$		0			0
$147$	−34.4791			−2.84379
$148$		0			0
$149$	1.55956			0.127764			0.0638820	−	0.997957i	$-0.479652\pi$
$149$	1.55956			0.127764			0.0638820	+	0.997957i	$0.479652\pi$
$150$		0			0
$151$	7.72157			0.628373			0.314186	−	0.949361i	$-0.398268\pi$
$151$	7.72157			0.628373			0.314186	+	0.949361i	$0.398268\pi$
$152$		0			0
$153$		−	17.5519i		−	1.41899i
$154$		0			0
$155$	2.60806			0.209484
$156$		0			0
$157$			4.12193i			0.328966i	0.986380	+	0.164483i	$0.0525956\pi$
$157$			4.12193i			0.328966i	−0.986380	+	0.164483i	$0.947404\pi$
$158$		0			0
$159$		−	26.0383i		−	2.06497i
$160$		0			0
$161$		−	14.8838i		−	1.17301i
$162$		0			0
$163$	−18.7144			−1.46583			−0.732913	−	0.680323i	$-0.761840\pi$
$163$	−18.7144			−1.46583			−0.732913	+	0.680323i	$0.761840\pi$
$164$		0			0
$165$	−2.08676	−	1.71083i	−0.162454	−	0.133188i
$166$		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
$171$		−	31.4576i		−	2.40562i
$172$		0			0
$173$	18.0791			1.37453			0.687266	−	0.726406i	$-0.258811\pi$
$173$	18.0791			1.37453			0.687266	+	0.726406i	$0.258811\pi$
$174$		0			0
$175$	21.5730			1.63077
$176$		0			0
$177$	1.24029			0.0932261
$178$		0			0
$179$	11.6952			0.874144			0.437072	−	0.899427i	$-0.356016\pi$
$179$	11.6952			0.874144			0.437072	+	0.899427i	$0.356016\pi$
$180$		0			0
$181$		−	2.96526i		−	0.220406i	−0.993909	−	0.110203i	$-0.964850\pi$
$181$		−	2.96526i		−	0.220406i	0.993909	−	0.110203i	$-0.0351501\pi$
$182$		0			0
$183$	30.3490			2.24346
$184$		0			0
$185$	−0.0836184			−0.00614775
$186$		0			0
$187$	−9.15667	−	7.50711i	−0.669602	−	0.548974i
$188$		0			0
$189$	−23.6598			−1.72100
$190$		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	$-0.952006\pi$
$193$		−	4.17352i		−	0.300417i	0.988654	−	0.150208i	$-0.0479944\pi$
$194$		0			0
$195$		−	1.26887i		−	0.0908655i
$196$		0			0
$197$	13.9056			0.990735			0.495367	−	0.868684i	$-0.335033\pi$
$197$	13.9056			0.990735			0.495367	+	0.868684i	$0.335033\pi$
$198$		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
$201$	11.1708			0.787928
$202$		0			0
$203$	−16.0000			−1.12298
$204$		0			0
$205$	−1.93431			−0.135098
$206$		0			0
$207$		−	16.6761i		−	1.15907i
$208$		0			0
$209$	−16.4111	−	13.4547i	−1.13518	−	0.930678i
$210$		0			0
$211$			6.18405i			0.425728i	0.977082	+	0.212864i	$0.0682791\pi$
$211$			6.18405i			0.425728i	−0.977082	+	0.212864i	$0.931721\pi$
$212$		0			0
$213$		−	16.2197i		−	1.11135i
$214$		0			0
$215$	1.48333			0.101163
$216$		0			0
$217$		−	39.5759i		−	2.68659i
$218$		0			0
$219$			18.8208i			1.27179i
$220$		0			0
$221$		−	5.56777i		−	0.374529i
$222$		0			0
$223$			6.98084i			0.467472i	0.972300	+	0.233736i	$0.0750951\pi$
$223$			6.98084i			0.467472i	−0.972300	+	0.233736i	$0.924905\pi$
$224$		0			0
$225$	24.1708			1.61139
$226$		0			0
$227$			13.6912i			0.908714i	0.890820	+	0.454357i	$0.150131\pi$
$227$			13.6912i			0.908714i	−0.890820	+	0.454357i	$0.849869\pi$
$228$		0			0
$229$		−	21.2786i		−	1.40613i	−0.711126	−	0.703065i	$-0.751814\pi$
$229$		−	21.2786i		−	1.40613i	0.711126	−	0.703065i	$-0.248186\pi$
$230$		0			0
$231$	−25.9610	+	31.6655i	−1.70811	+	2.08344i
$232$		0			0
$233$		−	27.8113i		−	1.82198i	−0.412433	−	0.910988i	$-0.635321\pi$
$233$		−	27.8113i		−	1.82198i	0.412433	−	0.910988i	$-0.364679\pi$
$234$		0			0
$235$	−2.03831			−0.132965
$236$		0			0
$237$	31.8323			2.06773
$238$		0			0
$239$	−5.50440			−0.356050			−0.178025	−	0.984026i	$-0.556971\pi$
$239$	−5.50440			−0.356050			−0.178025	+	0.984026i	$0.556971\pi$
$240$		0			0
$241$			10.4118i			0.670680i	0.942097	+	0.335340i	$0.108851\pi$
$241$			10.4118i			0.670680i	−0.942097	+	0.335340i	$0.891149\pi$
$242$		0			0
$243$	14.9894			0.961573
$244$		0			0
$245$			3.54359i			0.226392i
$246$		0			0
$247$		−	9.97887i		−	0.634941i
$248$		0			0
$249$			39.1250i			2.47944i
$250$		0			0
$251$	−15.2841			−0.964727			−0.482363	−	0.875971i	$-0.660222\pi$
$251$	−15.2841			−0.964727			−0.482363	+	0.875971i	$0.660222\pi$
$252$		0			0
$253$	−8.69975	−	7.13249i	−0.546948	−	0.448416i
$254$		0			0
$255$	−2.90465			−0.181896
$256$		0			0
$257$	−11.5678			−0.721578			−0.360789	−	0.932647i	$-0.617493\pi$
$257$	−11.5678			−0.721578			−0.360789	+	0.932647i	$0.617493\pi$
$258$		0			0
$259$			1.26887i			0.0788436i
$260$		0			0
$261$	−17.9267			−1.10963
$262$		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
$267$	35.4303			2.16830
$268$		0			0
$269$		−	1.25443i		−	0.0764837i	−0.999269	−	0.0382419i	$-0.987824\pi$
$269$		−	1.25443i		−	0.0764837i	0.999269	−	0.0382419i	$-0.0121757\pi$
$270$		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
$273$	−19.2544			−1.16533
$274$		0			0
$275$	10.3380	−	12.6097i	0.623408	−	0.760392i
$276$		0			0
$277$	0.527200			0.0316764			0.0158382	−	0.999875i	$-0.494958\pi$
$277$	0.527200			0.0316764			0.0158382	+	0.999875i	$0.494958\pi$
$278$		0			0
$279$		−	44.3416i		−	2.65466i
$280$		0			0
$281$			10.2593i			0.612019i	0.952028	+	0.306009i	$0.0989939\pi$
$281$			10.2593i			0.612019i	−0.952028	+	0.306009i	$0.901006\pi$
$282$		0			0
$283$		−	2.59192i		−	0.154074i	−0.997028	−	0.0770368i	$-0.975454\pi$
$283$		−	2.59192i		−	0.154074i	0.997028	−	0.0770368i	$-0.0245459\pi$
$284$		0			0
$285$	−5.20588			−0.308370
$286$		0			0
$287$			29.3522i			1.73260i
$288$		0			0
$289$	4.25443			0.250260
$290$		0			0
$291$	24.6464			1.44480
$292$		0			0
$293$	5.73308			0.334930			0.167465	−	0.985878i	$-0.446442\pi$
$293$	5.73308			0.334930			0.167465	+	0.985878i	$0.446442\pi$
$294$		0			0
$295$		−	0.127471i		−	0.00742166i
$296$		0			0
$297$	−11.3380	+	13.8294i	−0.657900	+	0.802463i
$298$		0			0
$299$		−	5.28994i		−	0.305925i
$300$		0			0
$301$		−	22.5089i		−	1.29739i
$302$		0			0
$303$	−36.2203			−2.08080
$304$		0			0
$305$		−	3.11912i		−	0.178600i
$306$		0			0
$307$		−	15.1745i		−	0.866054i	−0.901381	−	0.433027i	$-0.857445\pi$
$307$		−	15.1745i		−	0.866054i	0.901381	−	0.433027i	$-0.142555\pi$
$308$		0			0
$309$			45.5960i			2.59387i
$310$		0			0
$311$			13.1466i			0.745477i	0.927936	+	0.372738i	$0.121581\pi$
$311$			13.1466i			0.745477i	−0.927936	+	0.372738i	$0.878419\pi$
$312$		0			0
$313$	15.6030			0.881936			0.440968	−	0.897523i	$-0.354635\pi$
$313$	15.6030			0.881936			0.440968	+	0.897523i	$0.354635\pi$
$314$		0			0
$315$			6.23824i			0.351485i
$316$		0			0
$317$		−	1.44584i		−	0.0812066i	−0.999175	−	0.0406033i	$-0.987072\pi$
$317$		−	1.44584i		−	0.0812066i	0.999175	−	0.0406033i	$-0.0129280\pi$
$318$		0			0
$319$	−7.66739	+	9.35218i	−0.429291	+	0.523621i
$320$		0			0
$321$		−	38.5215i		−	2.15006i
$322$		0			0
$323$	−22.8433			−1.27104
$324$		0			0
$325$	7.66739			0.425310
$326$		0			0
$327$	−1.48333			−0.0820286
$328$		0			0
$329$			30.9304i			1.70525i
$330$		0			0
$331$	14.0680			0.773249			0.386624	−	0.922237i	$-0.373641\pi$
$331$	14.0680			0.773249			0.386624	+	0.922237i	$0.373641\pi$
$332$		0			0
$333$			1.42166i			0.0779066i
$334$		0			0
$335$		−	1.14808i		−	0.0627263i
$336$		0			0
$337$		−	2.66814i		−	0.145343i	−0.997356	−	0.0726715i	$-0.976848\pi$
$337$		−	2.66814i		−	0.145343i	0.997356	−	0.0726715i	$-0.0231525\pi$
$338$		0			0
$339$	−7.01916			−0.381228
$340$		0			0
$341$	−23.1326	−	18.9653i	−1.25270	−	1.02703i
$342$		0			0
$343$	23.0564			1.24493
$344$		0			0
$345$	−2.75971			−0.148578
$346$		0			0
$347$			19.9294i			1.06987i	0.844894	+	0.534933i	$0.179663\pi$
$347$			19.9294i			1.06987i	−0.844894	+	0.534933i	$0.820337\pi$
$348$		0			0
$349$	9.60170			0.513967			0.256984	−	0.966416i	$-0.417271\pi$
$349$	9.60170			0.513967			0.256984	+	0.966416i	$0.417271\pi$
$350$		0			0
$351$	−8.40906			−0.448842
$352$		0			0
$353$	7.24029			0.385362			0.192681	−	0.981261i	$-0.438282\pi$
$353$	7.24029			0.385362			0.192681	+	0.981261i	$0.438282\pi$
$354$		0			0
$355$	−1.66698			−0.0884740
$356$		0			0
$357$			44.0766i			2.33278i
$358$		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
$363$	6.06803	+	30.3490i	0.318489	+	1.59291i
$364$		0			0
$365$	1.93431			0.101246
$366$		0			0
$367$		−	15.5280i		−	0.810555i	−0.914194	−	0.405278i	$-0.867175\pi$
$367$		−	15.5280i		−	0.810555i	0.914194	−	0.405278i	$-0.132825\pi$
$368$		0			0
$369$			32.8867i			1.71201i
$370$		0			0
$371$			40.6083i			2.10828i
$372$		0			0
$373$	33.3919			1.72897			0.864483	−	0.502662i	$-0.167646\pi$
$373$	33.3919			1.72897			0.864483	+	0.502662i	$0.167646\pi$
$374$		0			0
$375$		−	8.06803i		−	0.416631i
$376$		0			0
$377$	−5.68665			−0.292877
$378$		0			0
$379$	−14.1275			−0.725679			−0.362840	−	0.931852i	$-0.618193\pi$
$379$	−14.1275			−0.725679			−0.362840	+	0.931852i	$0.618193\pi$
$380$		0			0
$381$	8.17255			0.418692
$382$		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
$387$		−	25.2193i		−	1.28197i
$388$		0			0
$389$			28.3658i			1.43820i	0.694905	+	0.719101i	$0.255446\pi$
$389$			28.3658i			1.43820i	−0.694905	+	0.719101i	$0.744554\pi$
$390$		0			0
$391$	−12.1096			−0.612407
$392$		0			0
$393$			18.0029i			0.908127i
$394$		0			0
$395$		−	3.27157i		−	0.164610i
$396$		0			0
$397$			21.2544i			1.06673i	0.845885	+	0.533365i	$0.179073\pi$
$397$			21.2544i			1.06673i	−0.845885	+	0.533365i	$0.820927\pi$
$398$		0			0
$399$			78.9966i			3.95478i
$400$		0			0
$401$	−18.0766			−0.902704			−0.451352	−	0.892346i	$-0.649058\pi$
$401$	−18.0766			−0.902704			−0.451352	+	0.892346i	$0.649058\pi$
$402$		0			0
$403$		−	14.0659i		−	0.700673i
$404$		0			0
$405$			0.121932i			0.00605883i
$406$		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
$411$	3.76473			0.185700
$412$		0			0
$413$	−1.93431			−0.0951812
$414$		0			0
$415$	4.02107			0.197387
$416$		0			0
$417$		−	6.68921i		−	0.327572i
$418$		0			0
$419$	4.61665			0.225538			0.112769	−	0.993621i	$-0.464028\pi$
$419$	4.61665			0.225538			0.112769	+	0.993621i	$0.464028\pi$
$420$		0			0
$421$		−	13.2544i		−	0.645981i	−0.946402	−	0.322991i	$-0.895312\pi$
$421$		−	13.2544i		−	0.645981i	0.946402	−	0.322991i	$-0.104688\pi$
$422$		0			0
$423$			34.6550i			1.68498i
$424$		0			0
$425$		−	17.5519i		−	0.851394i
$426$		0			0
$427$	−47.3311			−2.29051
$428$		0			0
$429$	−9.22695	+	11.2544i	−0.445481	+	0.543369i
$430$		0			0
$431$	−3.33359			−0.160573			−0.0802866	−	0.996772i	$-0.525584\pi$
$431$	−3.33359			−0.160573			−0.0802866	+	0.996772i	$0.525584\pi$
$432$		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
$435$			2.96667i			0.142241i
$436$		0			0
$437$	−21.7034			−1.03822
$438$		0			0
$439$	10.2593			0.489650			0.244825	−	0.969567i	$-0.421269\pi$
$439$	10.2593			0.489650			0.244825	+	0.969567i	$0.421269\pi$
$440$		0			0
$441$	60.2474			2.86893
$442$		0			0
$443$	29.7336			1.41268			0.706342	−	0.707871i	$-0.250344\pi$
$443$	29.7336			1.41268			0.706342	+	0.707871i	$0.250344\pi$
$444$		0			0
$445$		−	3.64135i		−	0.172616i
$446$		0			0
$447$	−4.38799			−0.207545
$448$		0			0
$449$	17.0036			0.802448			0.401224	−	0.915980i	$-0.368585\pi$
$449$	17.0036			0.802448			0.401224	+	0.915980i	$0.368585\pi$
$450$		0			0
$451$	17.1567	+	14.0659i	0.807876	+	0.662338i
$452$		0			0
$453$	−21.7255			−1.02075
$454$		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
$459$			19.2497i			0.898501i
$460$		0			0
$461$	27.2841			1.27075			0.635373	−	0.772206i	$-0.280847\pi$
$461$	27.2841			1.27075			0.635373	+	0.772206i	$0.280847\pi$
$462$		0			0
$463$		−	15.8625i		−	0.737192i	−0.929590	−	0.368596i	$-0.879839\pi$
$463$		−	15.8625i		−	0.737192i	0.929590	−	0.368596i	$-0.120161\pi$
$464$		0			0
$465$	−7.33804			−0.340294
$466$		0			0
$467$	−1.32246			−0.0611961			−0.0305980	−	0.999532i	$-0.509741\pi$
$467$	−1.32246			−0.0611961			−0.0305980	+	0.999532i	$0.509741\pi$
$468$		0			0
$469$	−17.4215			−0.804452
$470$		0			0
$471$		−	11.5975i		−	0.534384i
$472$		0			0
$473$	−13.1567	−	10.7865i	−0.604945	−	0.495964i
$474$		0			0
$475$		−	31.4576i		−	1.44337i
$476$		0			0
$477$			45.4983i			2.08322i
$478$		0			0
$479$	37.2747			1.70313			0.851563	−	0.524253i	$-0.175655\pi$
$479$	37.2747			1.70313			0.851563	+	0.524253i	$0.175655\pi$
$480$		0			0
$481$			0.450976i			0.0205627i
$482$		0			0
$483$			41.8772i			1.90548i
$484$		0			0
$485$		−	2.53303i		−	0.115019i
$486$		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
$489$	52.6550			2.38114
$490$		0			0
$491$			0.374751i			0.0169123i	0.999964	+	0.00845613i	$0.00269170\pi$
$491$			0.374751i			0.0169123i	−0.999964	+	0.00845613i	$0.997308\pi$
$492$		0			0
$493$			13.0177i			0.586288i
$494$		0			0
$495$	3.64632	+	2.98944i	0.163890	+	0.134365i
$496$		0			0
$497$			25.2956i			1.13466i
$498$		0			0
$499$	4.61665			0.206670			0.103335	−	0.994647i	$-0.467049\pi$
$499$	4.61665			0.206670			0.103335	+	0.994647i	$0.467049\pi$
$500$		0			0
$501$	−19.4863			−0.870582
$502$		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
$507$	29.7336			1.32051
$508$		0			0
$509$		−	17.4458i		−	0.773273i	−0.922232	−	0.386637i	$-0.873637\pi$
$509$		−	17.4458i		−	0.773273i	0.922232	−	0.386637i	$-0.126363\pi$
$510$		0			0
$511$		−	29.3522i		−	1.29846i
$512$		0			0
$513$			34.5005i			1.52323i
$514$		0			0
$515$	4.68614			0.206496
$516$		0			0
$517$	18.0791	+	14.8222i	0.795120	+	0.651880i
$518$		0			0
$519$	−50.8676			−2.23284
$520$		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
$523$		−	20.9838i		−	0.917557i	−0.888551	−	0.458779i	$-0.848287\pi$
$523$		−	20.9838i		−	0.917557i	0.888551	−	0.458779i	$-0.151713\pi$
$524$		0			0
$525$	−60.6980			−2.64908
$526$		0			0
$527$	−32.1992			−1.40262
$528$		0			0
$529$	11.4947			0.499770
$530$		0			0
$531$	−2.16724			−0.0940501
$532$		0			0
$533$			10.4322i			0.451870i
$534$		0			0
$535$	−3.95905			−0.171165
$536$		0			0
$537$	−32.9058			−1.41999
$538$		0			0
$539$	25.7683	−	31.4305i	1.10992	−	1.35381i
$540$		0			0
$541$	23.2850			1.00110			0.500551	−	0.865707i	$-0.333131\pi$
$541$	23.2850			1.00110			0.500551	+	0.865707i	$0.333131\pi$
$542$		0			0
$543$			8.34307i			0.358035i
$544$		0			0
$545$			0.152450i			0.00653023i
$546$		0			0
$547$		−	22.6816i		−	0.969795i	−0.874571	−	0.484898i	$-0.838857\pi$
$547$		−	22.6816i		−	0.969795i	0.874571	−	0.484898i	$-0.161143\pi$
$548$		0			0
$549$	−53.0306			−2.26329
$550$		0			0
$551$			23.3311i			0.993936i
$552$		0			0
$553$	−49.6444			−2.11109
$554$		0			0
$555$	0.235269			0.00998663
$556$		0			0
$557$	−40.6845			−1.72386			−0.861929	−	0.507029i	$-0.830744\pi$
$557$	−40.6845			−1.72386			−0.861929	+	0.507029i	$0.830744\pi$
$558$		0			0
$559$		−	8.00000i		−	0.338364i
$560$		0			0
$561$	25.7633	+	21.1220i	1.08773	+	0.891773i
$562$		0			0
$563$			19.0895i			0.804525i	0.915524	+	0.402262i	$0.131776\pi$
$563$			19.0895i			0.804525i	−0.915524	+	0.402262i	$0.868224\pi$
$564$		0			0
$565$			0.721394i			0.0303493i
$566$		0			0
$567$	1.85025			0.0777032
$568$		0			0
$569$			26.6485i			1.11716i	0.829450	+	0.558581i	$0.188654\pi$
$569$			26.6485i			1.11716i	−0.829450	+	0.558581i	$0.811346\pi$
$570$		0			0
$571$			18.6605i			0.780919i	0.920620	+	0.390459i	$0.127684\pi$
$571$			18.6605i			0.780919i	−0.920620	+	0.390459i	$0.872316\pi$
$572$		0			0
$573$		−	19.6413i		−	0.820529i
$574$		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
$579$			11.7426i			0.488008i
$580$		0			0
$581$		−	61.0177i		−	2.53144i
$582$		0			0
$583$	23.7360	+	19.4600i	0.983045	+	0.805950i
$584$		0			0
$585$			2.21717i			0.0916686i
$586$		0			0
$587$	13.3622			0.551518			0.275759	−	0.961227i	$-0.411071\pi$
$587$	13.3622			0.551518			0.275759	+	0.961227i	$0.411071\pi$
$588$		0			0
$589$	−57.7093			−2.37787
$590$		0			0
$591$	−39.1250			−1.60939
$592$		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
$595$	4.52998			0.185711
$596$		0			0
$597$		−	43.4983i		−	1.78027i
$598$		0			0
$599$			33.7733i			1.37994i	0.723838	+	0.689970i	$0.242376\pi$
$599$			33.7733i			1.37994i	−0.723838	+	0.689970i	$0.757624\pi$
$600$		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
$603$	−19.5194			−0.794892
$604$		0			0
$605$	3.11912	−	0.623642i	0.126810	−	0.0253547i
$606$		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
$609$	45.0177			1.82421
$610$		0			0
$611$			10.9931i			0.444735i
$612$		0			0
$613$	35.3262			1.42681			0.713406	−	0.700751i	$-0.247152\pi$
$613$	35.3262			1.42681			0.713406	+	0.700751i	$0.247152\pi$
$614$		0			0
$615$	5.44239			0.219458
$616$		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
$619$	19.7547			0.794008			0.397004	−	0.917817i	$-0.370050\pi$
$619$	19.7547			0.794008			0.397004	+	0.917817i	$0.370050\pi$
$620$		0			0
$621$			18.2892i			0.733919i
$622$		0			0
$623$	−55.2556			−2.21377
$624$		0			0
$625$	23.7527			0.950109
$626$		0			0
$627$	46.1744	+	37.8561i	1.84403	+	1.51183i
$628$		0			0
$629$	1.03236			0.0411628
$630$		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
$633$		−	17.3995i		−	0.691568i
$634$		0			0
$635$		−	0.839934i		−	0.0333318i
$636$		0			0
$637$	19.1115			0.757225
$638$		0			0
$639$			28.3416i			1.12118i
$640$		0			0
$641$	−22.3275			−0.881883			−0.440941	−	0.897536i	$-0.645355\pi$
$641$	−22.3275			−0.881883			−0.440941	+	0.897536i	$0.645355\pi$
$642$		0			0
$643$	−11.9703			−0.472062			−0.236031	−	0.971746i	$-0.575847\pi$
$643$	−11.9703			−0.472062			−0.236031	+	0.971746i	$0.575847\pi$
$644$		0			0
$645$	−4.17352			−0.164332
$646$		0			0
$647$		−	10.2353i		−	0.402390i	−0.979551	−	0.201195i	$-0.935518\pi$
$647$		−	10.2353i		−	0.402390i	0.979551	−	0.201195i	$-0.0644825\pi$
$648$		0			0
$649$	−0.926944	+	1.13063i	−0.0363857	+	0.0443809i
$650$		0			0
$651$			111.351i			4.36419i
$652$		0			0
$653$		−	15.1325i		−	0.592180i	−0.955160	−	0.296090i	$-0.904317\pi$
$653$		−	15.1325i		−	0.592180i	0.955160	−	0.296090i	$-0.0956829\pi$
$654$		0			0
$655$	1.85025			0.0722953
$656$		0			0
$657$		−	32.8867i		−	1.28303i
$658$		0			0
$659$			0.374751i			0.0145982i	0.999973	+	0.00729911i	$0.00232340\pi$
$659$			0.374751i			0.0145982i	−0.999973	+	0.00729911i	$0.997677\pi$
$660$		0			0
$661$			47.6202i			1.85221i	0.377263	+	0.926106i	$0.376865\pi$
$661$			47.6202i			1.85221i	−0.377263	+	0.926106i	$0.623135\pi$
$662$		0			0
$663$			15.6655i			0.608399i
$664$		0			0
$665$	8.11888			0.314837
$666$		0			0
$667$			12.3681i			0.478895i
$668$		0			0
$669$		−	19.6413i		−	0.759378i
$670$		0			0
$671$	−22.6816	+	27.6655i	−0.875613	+	1.06802i
$672$		0			0
$673$			39.8745i			1.53705i	0.639821	+	0.768524i	$0.279008\pi$
$673$			39.8745i			1.53705i	−0.639821	+	0.768524i	$0.720992\pi$
$674$		0			0
$675$	−26.5089			−1.02033
$676$		0			0
$677$	14.9380			0.574113			0.287057	−	0.957914i	$-0.407323\pi$
$677$	14.9380			0.574113			0.287057	+	0.957914i	$0.407323\pi$
$678$		0			0
$679$	−38.4375			−1.47510
$680$		0			0
$681$		−	38.5215i		−	1.47615i
$682$		0			0
$683$	12.2056			0.467032			0.233516	−	0.972353i	$-0.424977\pi$
$683$	12.2056			0.467032			0.233516	+	0.972353i	$0.424977\pi$
$684$		0			0
$685$		−	0.386920i		−	0.0147835i
$686$		0			0
$687$			59.8696i			2.28417i
$688$		0			0
$689$			14.4328i			0.549847i
$690$		0			0
$691$	−32.1658			−1.22364			−0.611822	−	0.790995i	$-0.709563\pi$
$691$	−32.1658			−1.22364			−0.611822	+	0.790995i	$0.709563\pi$
$692$		0			0
$693$	45.3632	−	55.3311i	1.72321	−	2.10185i
$694$		0			0
$695$	−0.687484			−0.0260778
$696$		0			0
$697$	23.8811			0.904562
$698$		0			0
$699$			78.2499i			2.95968i
$700$		0			0
$701$	0.657608			0.0248375			0.0124188	−	0.999923i	$-0.496047\pi$
$701$	0.657608			0.0248375			0.0124188	+	0.999923i	$0.496047\pi$
$702$		0			0
$703$	1.85025			0.0697835
$704$		0			0
$705$	5.73501			0.215993
$706$		0			0
$707$	56.4877			2.12444
$708$		0			0
$709$			33.2302i			1.24799i	0.781429	+	0.623994i	$0.214491\pi$
$709$			33.2302i			1.24799i	−0.781429	+	0.623994i	$0.785509\pi$
$710$		0			0
$711$	−55.6225			−2.08601
$712$		0			0
$713$	−30.5925			−1.14570
$714$		0			0
$715$	1.15667	+	0.948300i	0.0432572	+	0.0354644i
$716$		0			0
$717$	15.4872			0.578381
$718$		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
$723$		−	29.2946i		−	1.08948i
$724$		0			0
$725$	−17.9267			−0.665781
$726$		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
$729$	−43.4394			−1.60887
$730$		0			0
$731$	−18.3133			−0.677344
$732$		0			0
$733$	34.4683			1.27312			0.636558	−	0.771229i	$-0.280357\pi$
$733$	34.4683			1.27312			0.636558	+	0.771229i	$0.280357\pi$
$734$		0			0
$735$		−	9.97028i		−	0.367759i
$736$		0			0
$737$	−8.34861	+	10.1831i	−0.307525	+	0.375099i
$738$		0			0
$739$			9.15072i			0.336615i	0.985735	+	0.168307i	$0.0538301\pi$
$739$			9.15072i			0.336615i	−0.985735	+	0.168307i	$0.946170\pi$
$740$		0			0
$741$			28.0766i			1.03142i
$742$		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
$747$		−	68.3654i		−	2.50136i
$748$		0			0
$749$			60.0766i			2.19515i
$750$		0			0
$751$			40.5285i			1.47891i	0.673208	+	0.739453i	$0.264916\pi$
$751$			40.5285i			1.47891i	−0.673208	+	0.739453i	$0.735084\pi$
$752$		0			0
$753$	43.0036			1.56714
$754$		0			0
$755$			2.23284i			0.0812612i
$756$		0			0
$757$		−	6.74557i		−	0.245172i	−0.992458	−	0.122586i	$-0.960881\pi$
$757$		−	6.74557i		−	0.245172i	0.992458	−	0.122586i	$-0.0391187\pi$
$758$		0			0
$759$	24.4777	+	20.0680i	0.888483	+	0.728423i
$760$		0			0
$761$		−	15.0363i		−	0.545064i	−0.962147	−	0.272532i	$-0.912139\pi$
$761$		−	15.0363i		−	0.545064i	0.962147	−	0.272532i	$-0.0878610\pi$
$762$		0			0
$763$	2.31335			0.0837488
$764$		0			0
$765$	5.07547			0.183504
$766$		0			0
$767$	−0.687484			−0.0248236
$768$		0			0
$769$		−	2.66814i		−	0.0962157i	−0.998842	−	0.0481079i	$-0.984681\pi$
$769$		−	2.66814i		−	0.0962157i	0.998842	−	0.0481079i	$-0.0153191\pi$
$770$		0			0
$771$	32.5472			1.17216
$772$		0			0
$773$		−	41.3311i		−	1.48657i	−0.668972	−	0.743287i	$-0.733266\pi$
$773$		−	41.3311i		−	1.48657i	0.668972	−	0.743287i	$-0.266734\pi$
$774$		0			0
$775$		−	44.3416i		−	1.59280i
$776$		0			0
$777$		−	3.57009i		−	0.128076i
$778$		0			0
$779$	42.8011			1.53351
$780$		0			0
$781$	14.7855	+	12.1219i	0.529068	+	0.433757i
$782$		0			0
$783$	19.6607			0.702618
$784$		0			0
$785$	−1.19193			−0.0425419
$786$		0			0
$787$		−	14.3346i		−	0.510972i	−0.966813	−	0.255486i	$-0.917765\pi$
$787$		−	14.3346i		−	0.510972i	0.966813	−	0.255486i	$-0.0822355\pi$
$788$		0			0
$789$	77.2176			2.74902
$790$		0			0
$791$	10.9468			0.389223
$792$		0			0
$793$	−16.8222			−0.597374
$794$		0			0
$795$	7.52946			0.267042
$796$		0			0
$797$		−	36.0313i		−	1.27629i	−0.769914	−	0.638147i	$-0.779701\pi$
$797$		−	36.0313i		−	1.27629i	0.769914	−	0.638147i	$-0.220299\pi$
$798$		0			0
$799$	25.1652			0.890279
$800$		0			0
$801$	−61.9094			−2.18746
$802$		0			0
$803$	−17.1567	−	14.0659i	−0.605446	−	0.496375i
$804$		0			0
$805$	4.30393			0.151694
$806$		0			0
$807$			3.52946i			0.124243i
$808$		0			0
$809$		−	47.7705i		−	1.67952i	−0.542956	−	0.839761i	$-0.682695\pi$
$809$		−	47.7705i		−	1.67952i	0.542956	−	0.839761i	$-0.317305\pi$
$810$		0			0
$811$		−	36.9620i		−	1.29791i	−0.760827	−	0.648955i	$-0.775206\pi$
$811$		−	36.9620i		−	1.29791i	0.760827	−	0.648955i	$-0.224794\pi$
$812$		0			0
$813$	−73.0440			−2.56177
$814$		0			0
$815$		−	5.41162i		−	0.189561i
$816$		0			0
$817$	−32.8222			−1.14830
$818$		0			0
$819$	33.6444			1.17563
$820$		0			0
$821$	14.8076			0.516788			0.258394	−	0.966040i	$-0.416807\pi$
$821$	14.8076			0.516788			0.258394	+	0.966040i	$0.416807\pi$
$822$		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
$825$	−29.0872	+	35.4786i	−1.01269	+	1.23521i
$826$		0			0
$827$			12.9573i			0.450570i	0.974293	+	0.225285i	$0.0723314\pi$
$827$			12.9573i			0.450570i	−0.974293	+	0.225285i	$0.927669\pi$
$828$		0			0
$829$			14.5542i			0.505487i	0.967533	+	0.252743i	$0.0813328\pi$
$829$			14.5542i			0.505487i	−0.967533	+	0.252743i	$0.918667\pi$
$830$		0			0
$831$	−1.48333			−0.0514563
$832$		0			0
$833$		−	43.7495i		−	1.51583i
$834$		0			0
$835$			2.00270i			0.0693063i
$836$		0			0
$837$			48.6308i			1.68093i
$838$		0			0
$839$			39.7436i			1.37210i	0.727554	+	0.686051i	$0.240657\pi$
$839$			39.7436i			1.37210i	−0.727554	+	0.686051i	$0.759343\pi$
$840$		0			0
$841$	−15.7044			−0.541530
$842$		0			0
$843$		−	28.8657i		−	0.994186i
$844$		0			0
$845$		−	3.05587i		−	0.105125i
$846$		0			0
$847$	−9.46346	−	47.3311i	−0.325168	−	1.62631i
$848$		0			0
$849$			7.29264i			0.250283i
$850$		0			0
$851$	0.980843			0.0336229
$852$		0			0
$853$	12.7429			0.436307			0.218154	−	0.975914i	$-0.429997\pi$
$853$	12.7429			0.436307			0.218154	+	0.975914i	$0.429997\pi$
$854$		0			0
$855$	9.09654			0.311095
$856$		0			0
$857$			11.1613i			0.381261i	0.981662	+	0.190631i	$0.0610533\pi$
$857$			11.1613i			0.381261i	−0.981662	+	0.190631i	$0.938947\pi$
$858$		0			0
$859$	−54.6152			−1.86345			−0.931723	−	0.363169i	$-0.881695\pi$
$859$	−54.6152			−1.86345			−0.931723	+	0.363169i	$0.881695\pi$
$860$		0			0
$861$		−	82.5855i		−	2.81451i
$862$		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
$867$	−11.9703			−0.406532
$868$		0			0
$869$	−23.7902	+	29.0177i	−0.807027	+	0.984358i
$870$		0			0
$871$	−6.19189			−0.209804
$872$		0			0
$873$	−43.0661			−1.45757
$874$		0			0
$875$			12.5826i			0.425369i
$876$		0			0
$877$	33.3919			1.12756			0.563782	−	0.825924i	$-0.309346\pi$
$877$	33.3919			1.12756			0.563782	+	0.825924i	$0.309346\pi$
$878$		0			0
$879$	−16.1306			−0.544073
$880$		0			0
$881$	43.4358			1.46339			0.731695	−	0.681633i	$-0.238730\pi$
$881$	43.4358			1.46339			0.731695	+	0.681633i	$0.238730\pi$
$882$		0			0
$883$	13.7733			0.463509			0.231755	−	0.972774i	$-0.425553\pi$
$883$	13.7733			0.463509			0.231755	+	0.972774i	$0.425553\pi$
$884$		0			0
$885$			0.358654i			0.0120560i
$886$		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
$891$	0.886662	−	1.08149i	0.0297043	−	0.0362314i
$892$		0			0
$893$	45.1024			1.50929
$894$		0			0
$895$			3.38190i			0.113044i
$896$		0			0
$897$			14.8838i			0.496956i
$898$		0			0
$899$			32.8867i			1.09683i
$900$		0			0
$901$	33.0392			1.10069
$902$		0			0
$903$			63.3311i			2.10753i
$904$		0			0
$905$	0.857459			0.0285029
$906$		0			0
$907$	12.2056			0.405279			0.202639	−	0.979253i	$-0.435048\pi$
$907$	12.2056			0.405279			0.202639	+	0.979253i	$0.435048\pi$
$908$		0			0
$909$	63.2899			2.09919
$910$		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
$915$			8.77597i			0.290125i
$916$		0			0
$917$		−	28.0766i		−	0.927172i
$918$		0			0
$919$	27.8113			0.917409			0.458704	−	0.888589i	$-0.348314\pi$
$919$	27.8113			0.917409			0.458704	+	0.888589i	$0.348314\pi$
$920$		0			0
$921$			42.6951i			1.40685i
$922$		0			0
$923$			8.99044i			0.295924i
$924$		0			0
$925$			1.42166i			0.0467440i
$926$		0			0
$927$		−	79.6727i		−	2.61679i
$928$		0			0
$929$	−7.76328			−0.254705			−0.127352	−	0.991858i	$-0.540648\pi$
$929$	−7.76328			−0.254705			−0.127352	+	0.991858i	$0.540648\pi$
$930$		0			0
$931$		−	78.4102i		−	2.56979i
$932$		0			0
$933$		−	36.9894i		−	1.21098i
$934$		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
$939$	−43.9008			−1.43265
$940$		0			0
$941$	20.1439			0.656671			0.328336	−	0.944561i	$-0.393512\pi$
$941$	20.1439			0.656671			0.328336	+	0.944561i	$0.393512\pi$
$942$		0			0
$943$	22.6894			0.738870
$944$		0			0
$945$		−	6.84166i		−	0.222559i
$946$		0			0
$947$	−41.3225			−1.34280			−0.671400	−	0.741095i	$-0.734307\pi$
$947$	−41.3225			−1.34280			−0.671400	+	0.741095i	$0.734307\pi$
$948$		0			0
$949$		−	10.4322i		−	0.338644i
$950$		0			0
$951$			4.06803i			0.131915i
$952$		0			0
$953$		−	28.1098i		−	0.910565i	−0.890347	−	0.455283i	$-0.849538\pi$
$953$		−	28.1098i		−	0.910565i	0.890347	−	0.455283i	$-0.150462\pi$
$954$		0			0
$955$	−2.01864			−0.0653217
$956$		0			0
$957$	21.5730	−	26.3133i	0.697357	−	0.850590i
$958$		0			0
$959$	−5.87132			−0.189595
$960$		0			0
$961$	−50.3452			−1.62404
$962$		0			0
$963$			67.3110i			2.16907i
$964$		0			0
$965$	1.20685			0.0388499
$966$		0			0
$967$	21.5730			0.693741			0.346871	−	0.937913i	$-0.387244\pi$
$967$	21.5730			0.693741			0.346871	+	0.937913i	$0.387244\pi$
$968$		0			0
$969$	64.2721			2.06472
$970$		0			0
$971$	−17.9320			−0.575464			−0.287732	−	0.957711i	$-0.592901\pi$
$971$	−17.9320			−0.575464			−0.287732	+	0.957711i	$0.592901\pi$
$972$		0			0
$973$			10.4322i			0.334442i
$974$		0			0
$975$	−21.5730			−0.690889
$976$		0			0
$977$	−43.8953			−1.40433			−0.702167	−	0.712012i	$-0.747784\pi$
$977$	−43.8953			−1.40433			−0.702167	+	0.712012i	$0.747784\pi$
$978$		0			0
$979$	−26.4791	+	32.2975i	−0.846277	+	1.03223i
$980$		0			0
$981$	2.59192			0.0827536
$982$		0			0
$983$		−	21.7053i		−	0.692291i	−0.938181	−	0.346146i	$-0.887490\pi$
$983$		−	21.7053i		−	0.692291i	0.938181	−	0.346146i	$-0.112510\pi$
$984$		0			0
$985$			4.02107i			0.128122i
$986$		0			0
$987$		−	87.0259i		−	2.77006i
$988$		0			0
$989$	−17.3995			−0.553272
$990$		0			0
$991$			54.3799i			1.72744i	0.503976	+	0.863718i	$0.331870\pi$
$991$			54.3799i			1.72744i	−0.503976	+	0.863718i	$0.668130\pi$
$992$		0			0
$993$	−39.5819			−1.25609
$994$		0			0
$995$	−4.47054			−0.141726
$996$		0			0
$997$	−59.1384			−1.87293			−0.936466	−	0.350758i	$-0.885924\pi$
$997$	−59.1384			−1.87293			−0.936466	+	0.350758i	$0.885924\pi$
$998$		0			0
$999$		−	1.55918i		−	0.0493303i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2816.2.g.d.1407.4		12
4.3	odd	2		2816.2.g.i.1407.11		12
8.3	odd	2	inner	2816.2.g.d.1407.1		12
8.5	even	2		2816.2.g.i.1407.10		12
11.10	odd	2	inner	2816.2.g.d.1407.3		12
16.3	odd	4		352.2.e.a.351.12	yes	12
16.5	even	4		704.2.e.d.703.11		12
16.11	odd	4		704.2.e.d.703.2		12
16.13	even	4		352.2.e.a.351.1	✓	12
44.43	even	2		2816.2.g.i.1407.12		12
48.29	odd	4		3168.2.o.e.703.6		12
48.35	even	4		3168.2.o.e.703.7		12
88.21	odd	2		2816.2.g.i.1407.9		12
88.43	even	2	inner	2816.2.g.d.1407.2		12
176.21	odd	4		704.2.e.d.703.12		12
176.43	even	4		704.2.e.d.703.1		12
176.109	odd	4		352.2.e.a.351.2	yes	12
176.131	even	4		352.2.e.a.351.11	yes	12
528.131	odd	4		3168.2.o.e.703.5		12
528.461	even	4		3168.2.o.e.703.8		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
352.2.e.a.351.1	✓	12	16.13	even	4
352.2.e.a.351.2	yes	12	176.109	odd	4
352.2.e.a.351.11	yes	12	176.131	even	4
352.2.e.a.351.12	yes	12	16.3	odd	4
704.2.e.d.703.1		12	176.43	even	4
704.2.e.d.703.2		12	16.11	odd	4
704.2.e.d.703.11		12	16.5	even	4
704.2.e.d.703.12		12	176.21	odd	4
2816.2.g.d.1407.1		12	8.3	odd	2	inner
2816.2.g.d.1407.2		12	88.43	even	2	inner
2816.2.g.d.1407.3		12	11.10	odd	2	inner
2816.2.g.d.1407.4		12	1.1	even	1	trivial
2816.2.g.i.1407.9		12	88.21	odd	2
2816.2.g.i.1407.10		12	8.5	even	2
2816.2.g.i.1407.11		12	4.3	odd	2
2816.2.g.i.1407.12		12	44.43	even	2
3168.2.o.e.703.5		12	528.131	odd	4
3168.2.o.e.703.6		12	48.29	odd	4
3168.2.o.e.703.7		12	48.35	even	4
3168.2.o.e.703.8		12	528.461	even	4

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:	\( N \)	\(=\)	\( 2816 = 2^{8} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2816.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.81361 q^{3} +0.289169i q^{5} +4.38799 q^{7} +4.91638 q^{9} +O(q^{10})\)	\(q-2.81361 q^{3} +0.289169i q^{5} +4.38799 q^{7} +4.91638 q^{9} +(2.10278 - 2.56483i) q^{11} +1.55956 q^{13} -0.813607i q^{15} -3.57009i q^{17} -6.39852i q^{19} -12.3461 q^{21} -3.39194i q^{23} +4.91638 q^{25} -5.39194 q^{27} -3.64632 q^{29} -9.01916i q^{31} +(-5.91638 + 7.21641i) q^{33} +1.26887i q^{35} +0.289169i q^{37} -4.38799 q^{39} +6.68921i q^{41} -5.12965i q^{43} +1.42166i q^{45} +7.04888i q^{47} +12.2544 q^{49} +10.0448i q^{51} +9.25443i q^{53} +(0.741667 + 0.608056i) q^{55} +18.0029i q^{57} -0.440820 q^{59} -10.7865 q^{61} +21.5730 q^{63} +0.450976i q^{65} -3.97028 q^{67} +9.54359i q^{69} +5.76473i q^{71} -6.68921i q^{73} -13.8328 q^{75} +(9.22695 - 11.2544i) q^{77} -11.3137 q^{79} +0.421663 q^{81} -13.9056i q^{83} +1.03236 q^{85} +10.2593 q^{87} -12.5925 q^{89} +6.84333 q^{91} +25.3764i q^{93} +1.85025 q^{95} -8.75971 q^{97} +(10.3380 - 12.6097i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{3} + 4 q^{9}+O(q^{10}) \) 12 * q - 8 * q^3 + 4 * q^9	\( 12 q - 8 q^{3} + 4 q^{9} - 4 q^{11} + 4 q^{25} - 32 q^{27} - 16 q^{33} + 44 q^{49} + 72 q^{59} - 8 q^{67} - 56 q^{75} + 12 q^{81} + 96 q^{91} - 64 q^{97} + 76 q^{99}+O(q^{100}) \) 12 * q - 8 * q^3 + 4 * q^9 - 4 * q^11 + 4 * q^25 - 32 * q^27 - 16 * q^33 + 44 * q^49 + 72 * q^59 - 8 * q^67 - 56 * q^75 + 12 * q^81 + 96 * q^91 - 64 * q^97 + 76 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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