LMFDB - Embedded newform 1792.2.b.o.897.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1792,2,Mod(897,1792)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1792, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1792.897");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1792.b (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:	$14.3091920422$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.399424.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 $ x^6 - 2x^5 + 3x^4 - 6x^3 + 6x^2 - 8*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 896)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			897.3
Root			$-0.671462 - 1.24464i$ of defining polynomial
Character	$\chi$	$=$	1792.897
Dual form			1792.2.b.o.897.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.14637i q^{3} +3.83221i q^{5} -1.00000 q^{7} +1.68585 q^{9} +O(q^{10})$	$q-1.14637i q^{3} +3.83221i q^{5} -1.00000 q^{7} +1.68585 q^{9} +4.68585i q^{11} +5.53948i q^{13} +4.39312 q^{15} +0.292731 q^{17} -5.14637i q^{19} +1.14637i q^{21} -4.97858 q^{23} -9.68585 q^{25} -5.37169i q^{27} +4.29273i q^{29} -7.66442 q^{31} +5.37169 q^{33} -3.83221i q^{35} -9.66442i q^{37} +6.35027 q^{39} -3.70727 q^{41} -5.27131i q^{43} +6.46052i q^{45} +2.29273 q^{47} +1.00000 q^{49} -0.335577i q^{51} +2.00000i q^{53} -17.9572 q^{55} -5.89962 q^{57} +9.93260i q^{59} +4.16779i q^{61} -1.68585 q^{63} -21.2285 q^{65} +10.9786i q^{67} +5.70727i q^{69} -7.37169 q^{73} +11.1035i q^{75} -4.68585i q^{77} +13.9572 q^{79} -1.10038 q^{81} -4.81079i q^{83} +1.12181i q^{85} +4.92104 q^{87} +2.58546 q^{89} -5.53948i q^{91} +8.78623i q^{93} +19.7220 q^{95} -14.2499 q^{97} +7.89962i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{7} - 14 q^{9}+O(q^{10}) $ 6 * q - 6 * q^7 - 14 * q^9	$ 6 q - 6 q^{7} - 14 q^{9} + 8 q^{15} - 4 q^{17} - 34 q^{25} + 8 q^{31} - 16 q^{33} - 40 q^{39} - 28 q^{41} + 8 q^{47} + 6 q^{49} - 48 q^{55} - 48 q^{57} + 14 q^{63} - 32 q^{65} + 4 q^{73} + 24 q^{79} + 6 q^{81} + 72 q^{87} + 4 q^{89} - 8 q^{95} - 20 q^{97}+O(q^{100}) $ 6 * q - 6 * q^7 - 14 * q^9 + 8 * q^15 - 4 * q^17 - 34 * q^25 + 8 * q^31 - 16 * q^33 - 40 * q^39 - 28 * q^41 + 8 * q^47 + 6 * q^49 - 48 * q^55 - 48 * q^57 + 14 * q^63 - 32 * q^65 + 4 * q^73 + 24 * q^79 + 6 * q^81 + 72 * q^87 + 4 * q^89 - 8 * q^95 - 20 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 1.14637i	− 0.661854i	−0.943656	−	0.330927i	$-0.892639\pi$
$3$	− 1.14637i	− 0.661854i	0.943656	−	0.330927i	$-0.107361\pi$
$4$	0	0
$5$	3.83221i	1.71382i	0.515468	+	0.856909i	$0.327618\pi$
$5$	3.83221i	1.71382i	−0.515468	+	0.856909i	$0.672382\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.68585	0.561949
$10$	0	0
$11$	4.68585i	1.41284i	0.707795	+	0.706418i	$0.249690\pi$
$11$	4.68585i	1.41284i	−0.707795	+	0.706418i	$0.750310\pi$
$12$	0	0
$13$	5.53948i	1.53638i	0.640225	+	0.768188i	$0.278841\pi$
$13$	5.53948i	1.53638i	−0.640225	+	0.768188i	$0.721159\pi$
$14$	0	0
$15$	4.39312	1.13430
$16$	0	0
$17$	0.292731	0.0709977	0.0354988	−	0.999370i	$-0.488698\pi$
$17$	0.292731	0.0709977	0.0354988	+	0.999370i	$0.488698\pi$
$18$	0	0
$19$	− 5.14637i	− 1.18066i	−0.807163	−	0.590329i	$-0.798998\pi$
$19$	− 5.14637i	− 1.18066i	0.807163	−	0.590329i	$-0.201002\pi$
$20$	0	0
$21$	1.14637i	0.250157i
$22$	0	0
$23$	−4.97858	−1.03811	−0.519053	−	0.854742i	$-0.673715\pi$
$23$	−4.97858	−1.03811	−0.519053	+	0.854742i	$0.673715\pi$
$24$	0	0
$25$	−9.68585	−1.93717
$26$	0	0
$27$	− 5.37169i	− 1.03378i
$28$	0	0
$29$	4.29273i	0.797140i	0.917138	+	0.398570i	$0.130493\pi$
$29$	4.29273i	0.797140i	−0.917138	+	0.398570i	$0.869507\pi$
$30$	0	0
$31$	−7.66442	−1.37657	−0.688286	−	0.725440i	$-0.741636\pi$
$31$	−7.66442	−1.37657	−0.688286	+	0.725440i	$0.741636\pi$
$32$	0	0
$33$	5.37169	0.935092
$34$	0	0
$35$	− 3.83221i	− 0.647762i
$36$	0	0
$37$	− 9.66442i	− 1.58882i	−0.607381	−	0.794411i	$-0.707780\pi$
$37$	− 9.66442i	− 1.58882i	0.607381	−	0.794411i	$-0.292220\pi$
$38$	0	0
$39$	6.35027	1.01686
$40$	0	0
$41$	−3.70727	−0.578978	−0.289489	−	0.957181i	$-0.593485\pi$
$41$	−3.70727	−0.578978	−0.289489	+	0.957181i	$0.593485\pi$
$42$	0	0
$43$	− 5.27131i	− 0.803867i	−0.915669	−	0.401933i	$-0.868338\pi$
$43$	− 5.27131i	− 0.803867i	0.915669	−	0.401933i	$-0.131662\pi$
$44$	0	0
$45$	6.46052i	0.963077i
$46$	0	0
$47$	2.29273	0.334429	0.167215	−	0.985921i	$-0.446523\pi$
$47$	2.29273	0.334429	0.167215	+	0.985921i	$0.446523\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	− 0.335577i	− 0.0469901i
$52$	0	0
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	0	0
$55$	−17.9572	−2.42134
$56$	0	0
$57$	−5.89962	−0.781423
$58$	0	0
$59$	9.93260i	1.29311i	0.762866	+	0.646557i	$0.223792\pi$
$59$	9.93260i	1.29311i	−0.762866	+	0.646557i	$0.776208\pi$
$60$	0	0
$61$	4.16779i	0.533631i	0.963748	+	0.266815i	$0.0859714\pi$
$61$	4.16779i	0.533631i	−0.963748	+	0.266815i	$0.914029\pi$
$62$	0	0
$63$	−1.68585	−0.212397
$64$	0	0
$65$	−21.2285	−2.63307
$66$	0	0
$67$	10.9786i	1.34125i	0.741798	+	0.670623i	$0.233973\pi$
$67$	10.9786i	1.34125i	−0.741798	+	0.670623i	$0.766027\pi$
$68$	0	0
$69$	5.70727i	0.687074i
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−7.37169	−0.862791	−0.431396	−	0.902163i	$-0.641979\pi$
$73$	−7.37169	−0.862791	−0.431396	+	0.902163i	$0.641979\pi$
$74$	0	0
$75$	11.1035i	1.28212i
$76$	0	0
$77$	− 4.68585i	− 0.534002i
$78$	0	0
$79$	13.9572	1.57030	0.785151	−	0.619304i	$-0.212585\pi$
$79$	13.9572	1.57030	0.785151	+	0.619304i	$0.212585\pi$
$80$	0	0
$81$	−1.10038	−0.122265
$82$	0	0
$83$	− 4.81079i	− 0.528053i	−0.964515	−	0.264026i	$-0.914949\pi$
$83$	− 4.81079i	− 0.528053i	0.964515	−	0.264026i	$-0.0850506\pi$
$84$	0	0
$85$	1.12181i	0.121677i
$86$	0	0
$87$	4.92104	0.527591
$88$	0	0
$89$	2.58546	0.274058	0.137029	−	0.990567i	$-0.456245\pi$
$89$	2.58546	0.274058	0.137029	+	0.990567i	$0.456245\pi$
$90$	0	0
$91$	− 5.53948i	− 0.580695i
$92$	0	0
$93$	8.78623i	0.911090i
$94$	0	0
$95$	19.7220	2.02343
$96$	0	0
$97$	−14.2499	−1.44686	−0.723428	−	0.690399i	$-0.757435\pi$
$97$	−14.2499	−1.44686	−0.723428	+	0.690399i	$0.757435\pi$
$98$	0	0
$99$	7.89962i	0.793941i
$100$	0	0
$101$	− 1.87506i	− 0.186575i	−0.995639	−	0.0932876i	$-0.970262\pi$
$101$	− 1.87506i	− 0.186575i	0.995639	−	0.0932876i	$-0.0297376\pi$
$102$	0	0
$103$	7.66442	0.755198	0.377599	−	0.925969i	$-0.376750\pi$
$103$	7.66442	0.755198	0.377599	+	0.925969i	$0.376750\pi$
$104$	0	0
$105$	−4.39312	−0.428724
$106$	0	0
$107$	− 2.39312i	− 0.231351i	−0.993287	−	0.115676i	$-0.963097\pi$
$107$	− 2.39312i	− 0.231351i	0.993287	−	0.115676i	$-0.0369033\pi$
$108$	0	0
$109$	− 1.07896i	− 0.103346i	−0.998664	−	0.0516729i	$-0.983545\pi$
$109$	− 1.07896i	− 0.103346i	0.998664	−	0.0516729i	$-0.0164553\pi$
$110$	0	0
$111$	−11.0790	−1.05157
$112$	0	0
$113$	9.27131	0.872171	0.436086	−	0.899905i	$-0.356364\pi$
$113$	9.27131	0.872171	0.436086	+	0.899905i	$0.356364\pi$
$114$	0	0
$115$	− 19.0790i	− 1.77912i
$116$	0	0
$117$	9.33871i	0.863364i
$118$	0	0
$119$	−0.292731	−0.0268346
$120$	0	0
$121$	−10.9572	−0.996105
$122$	0	0
$123$	4.24989i	0.383199i
$124$	0	0
$125$	− 17.9572i	− 1.60614i
$126$	0	0
$127$	−6.35027	−0.563495	−0.281748	−	0.959489i	$-0.590914\pi$
$127$	−6.35027	−0.563495	−0.281748	+	0.959489i	$0.590914\pi$
$128$	0	0
$129$	−6.04285	−0.532043
$130$	0	0
$131$	10.5181i	0.918967i	0.888186	+	0.459483i	$0.151965\pi$
$131$	10.5181i	0.918967i	−0.888186	+	0.459483i	$0.848035\pi$
$132$	0	0
$133$	5.14637i	0.446246i
$134$	0	0
$135$	20.5855	1.77171
$136$	0	0
$137$	11.3717	0.971549	0.485775	−	0.874084i	$-0.338538\pi$
$137$	11.3717	0.971549	0.485775	+	0.874084i	$0.338538\pi$
$138$	0	0
$139$	10.8536i	0.920593i	0.887765	+	0.460297i	$0.152257\pi$
$139$	10.8536i	0.920593i	−0.887765	+	0.460297i	$0.847743\pi$
$140$	0	0
$141$	− 2.62831i	− 0.221343i
$142$	0	0
$143$	−25.9572	−2.17065
$144$	0	0
$145$	−16.4507	−1.36615
$146$	0	0
$147$	− 1.14637i	− 0.0945506i
$148$	0	0
$149$	2.78623i	0.228257i	0.993466	+	0.114128i	$0.0364075\pi$
$149$	2.78623i	0.228257i	−0.993466	+	0.114128i	$0.963592\pi$
$150$	0	0
$151$	1.56404	0.127280	0.0636398	−	0.997973i	$-0.479729\pi$
$151$	1.56404	0.127280	0.0636398	+	0.997973i	$0.479729\pi$
$152$	0	0
$153$	0.493499	0.0398971
$154$	0	0
$155$	− 29.3717i	− 2.35919i
$156$	0	0
$157$	14.7104i	1.17402i	0.809580	+	0.587009i	$0.199695\pi$
$157$	14.7104i	1.17402i	−0.809580	+	0.587009i	$0.800305\pi$
$158$	0	0
$159$	2.29273	0.181825
$160$	0	0
$161$	4.97858	0.392367
$162$	0	0
$163$	0.100384i	0.00786270i	0.999992	+	0.00393135i	$0.00125139\pi$
$163$	0.100384i	0.00786270i	−0.999992	+	0.00393135i	$0.998749\pi$
$164$	0	0
$165$	20.5855i	1.60258i
$166$	0	0
$167$	−5.70727	−0.441642	−0.220821	−	0.975314i	$-0.570874\pi$
$167$	−5.70727	−0.441642	−0.220821	+	0.975314i	$0.570874\pi$
$168$	0	0
$169$	−17.6858	−1.36045
$170$	0	0
$171$	− 8.67598i	− 0.663469i
$172$	0	0
$173$	12.8683i	0.978361i	0.872183	+	0.489180i	$0.162704\pi$
$173$	12.8683i	0.978361i	−0.872183	+	0.489180i	$0.837296\pi$
$174$	0	0
$175$	9.68585	0.732181
$176$	0	0
$177$	11.3864	0.855853
$178$	0	0
$179$	26.3074i	1.96631i	0.182776	+	0.983155i	$0.441492\pi$
$179$	26.3074i	1.96631i	−0.182776	+	0.983155i	$0.558508\pi$
$180$	0	0
$181$	− 3.24675i	− 0.241329i	−0.992693	−	0.120665i	$-0.961497\pi$
$181$	− 3.24675i	− 0.241329i	0.992693	−	0.120665i	$-0.0385025\pi$
$182$	0	0
$183$	4.77781	0.353186
$184$	0	0
$185$	37.0361	2.72295
$186$	0	0
$187$	1.37169i	0.100308i
$188$	0	0
$189$	5.37169i	0.390733i
$190$	0	0
$191$	−17.3717	−1.25697	−0.628486	−	0.777821i	$-0.716325\pi$
$191$	−17.3717	−1.25697	−0.628486	+	0.777821i	$0.716325\pi$
$192$	0	0
$193$	27.4292	1.97440	0.987200	−	0.159490i	$-0.0509848\pi$
$193$	27.4292	1.97440	0.987200	+	0.159490i	$0.0509848\pi$
$194$	0	0
$195$	24.3356i	1.74271i
$196$	0	0
$197$	10.0000i	0.712470i	0.934396	+	0.356235i	$0.115940\pi$
$197$	10.0000i	0.712470i	−0.934396	+	0.356235i	$0.884060\pi$
$198$	0	0
$199$	−4.24989	−0.301266	−0.150633	−	0.988590i	$-0.548131\pi$
$199$	−4.24989	−0.301266	−0.150633	+	0.988590i	$0.548131\pi$
$200$	0	0
$201$	12.5855	0.887710
$202$	0	0
$203$	− 4.29273i	− 0.301291i
$204$	0	0
$205$	− 14.2070i	− 0.992263i
$206$	0	0
$207$	−8.39312	−0.583362
$208$	0	0
$209$	24.1151	1.66807
$210$	0	0
$211$	− 20.3503i	− 1.40097i	−0.713667	−	0.700485i	$-0.752967\pi$
$211$	− 20.3503i	− 1.40097i	0.713667	−	0.700485i	$-0.247033\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	20.2008	1.37768
$216$	0	0
$217$	7.66442	0.520295
$218$	0	0
$219$	8.45065i	0.571042i
$220$	0	0
$221$	1.62158i	0.109079i
$222$	0	0
$223$	0.786230	0.0526499	0.0263249	−	0.999653i	$-0.491620\pi$
$223$	0.786230	0.0526499	0.0263249	+	0.999653i	$0.491620\pi$
$224$	0	0
$225$	−16.3288	−1.08859
$226$	0	0
$227$	10.8536i	0.720381i	0.932879	+	0.360191i	$0.117288\pi$
$227$	10.8536i	0.720381i	−0.932879	+	0.360191i	$0.882712\pi$
$228$	0	0
$229$	− 10.1249i	− 0.669075i	−0.942382	−	0.334538i	$-0.891420\pi$
$229$	− 10.1249i	− 0.669075i	0.942382	−	0.334538i	$-0.108580\pi$
$230$	0	0
$231$	−5.37169	−0.353431
$232$	0	0
$233$	18.7862	1.23073	0.615363	−	0.788244i	$-0.289009\pi$
$233$	18.7862	1.23073	0.615363	+	0.788244i	$0.289009\pi$
$234$	0	0
$235$	8.78623i	0.573150i
$236$	0	0
$237$	− 16.0000i	− 1.03931i
$238$	0	0
$239$	7.02142	0.454178	0.227089	−	0.973874i	$-0.427079\pi$
$239$	7.02142	0.454178	0.227089	+	0.973874i	$0.427079\pi$
$240$	0	0
$241$	−28.4078	−1.82991	−0.914954	−	0.403558i	$-0.867773\pi$
$241$	−28.4078	−1.82991	−0.914954	+	0.403558i	$0.867773\pi$
$242$	0	0
$243$	− 14.8536i	− 0.952861i
$244$	0	0
$245$	3.83221i	0.244831i
$246$	0	0
$247$	28.5082	1.81393
$248$	0	0
$249$	−5.51492	−0.349494
$250$	0	0
$251$	2.51806i	0.158938i	0.996837	+	0.0794692i	$0.0253225\pi$
$251$	2.51806i	0.158938i	−0.996837	+	0.0794692i	$0.974677\pi$
$252$	0	0
$253$	− 23.3288i	− 1.46667i
$254$	0	0
$255$	1.28600	0.0805325
$256$	0	0
$257$	−1.41454	−0.0882365	−0.0441182	−	0.999026i	$-0.514048\pi$
$257$	−1.41454	−0.0882365	−0.0441182	+	0.999026i	$0.514048\pi$
$258$	0	0
$259$	9.66442i	0.600518i
$260$	0	0
$261$	7.23688i	0.447952i
$262$	0	0
$263$	−27.9143	−1.72127	−0.860635	−	0.509222i	$-0.829933\pi$
$263$	−27.9143	−1.72127	−0.860635	+	0.509222i	$0.829933\pi$
$264$	0	0
$265$	−7.66442	−0.470822
$266$	0	0
$267$	− 2.96388i	− 0.181387i
$268$	0	0
$269$	18.1249i	1.10510i	0.833481	+	0.552549i	$0.186345\pi$
$269$	18.1249i	1.10510i	−0.833481	+	0.552549i	$0.813655\pi$
$270$	0	0
$271$	24.1151	1.46489	0.732443	−	0.680828i	$-0.238380\pi$
$271$	24.1151	1.46489	0.732443	+	0.680828i	$0.238380\pi$
$272$	0	0
$273$	−6.35027	−0.384336
$274$	0	0
$275$	− 45.3864i	− 2.73690i
$276$	0	0
$277$	− 0.628308i	− 0.0377513i	−0.999822	−	0.0188757i	$-0.993991\pi$
$277$	− 0.628308i	− 0.0377513i	0.999822	−	0.0188757i	$-0.00600867\pi$
$278$	0	0
$279$	−12.9210	−0.773562
$280$	0	0
$281$	−0.743385	−0.0443466	−0.0221733	−	0.999754i	$-0.507059\pi$
$281$	−0.743385	−0.0443466	−0.0221733	+	0.999754i	$0.507059\pi$
$282$	0	0
$283$	− 0.475212i	− 0.0282484i	−0.999900	−	0.0141242i	$-0.995504\pi$
$283$	− 0.475212i	− 0.0282484i	0.999900	−	0.0141242i	$-0.00449603\pi$
$284$	0	0
$285$	− 22.6086i	− 1.33922i
$286$	0	0
$287$	3.70727	0.218833
$288$	0	0
$289$	−16.9143	−0.994959
$290$	0	0
$291$	16.3356i	0.957608i
$292$	0	0
$293$	1.53948i	0.0899374i	0.998988	+	0.0449687i	$0.0143188\pi$
$293$	1.53948i	0.0899374i	−0.998988	+	0.0449687i	$0.985681\pi$
$294$	0	0
$295$	−38.0638	−2.21616
$296$	0	0
$297$	25.1709	1.46057
$298$	0	0
$299$	− 27.5787i	− 1.59492i
$300$	0	0
$301$	5.27131i	0.303833i
$302$	0	0
$303$	−2.14950	−0.123486
$304$	0	0
$305$	−15.9718	−0.914545
$306$	0	0
$307$	21.2614i	1.21345i	0.794910	+	0.606727i	$0.207518\pi$
$307$	21.2614i	1.21345i	−0.794910	+	0.606727i	$0.792482\pi$
$308$	0	0
$309$	− 8.78623i	− 0.499831i
$310$	0	0
$311$	15.3288	0.869219	0.434610	−	0.900619i	$-0.356886\pi$
$311$	15.3288	0.869219	0.434610	+	0.900619i	$0.356886\pi$
$312$	0	0
$313$	−3.70727	−0.209547	−0.104774	−	0.994496i	$-0.533412\pi$
$313$	−3.70727	−0.209547	−0.104774	+	0.994496i	$0.533412\pi$
$314$	0	0
$315$	− 6.46052i	− 0.364009i
$316$	0	0
$317$	− 22.5855i	− 1.26853i	−0.773117	−	0.634263i	$-0.781304\pi$
$317$	− 22.5855i	− 1.26853i	0.773117	−	0.634263i	$-0.218696\pi$
$318$	0	0
$319$	−20.1151	−1.12623
$320$	0	0
$321$	−2.74338	−0.153121
$322$	0	0
$323$	− 1.50650i	− 0.0838239i
$324$	0	0
$325$	− 53.6546i	− 2.97622i
$326$	0	0
$327$	−1.23688	−0.0683998
$328$	0	0
$329$	−2.29273	−0.126402
$330$	0	0
$331$	14.6430i	0.804852i	0.915452	+	0.402426i	$0.131833\pi$
$331$	14.6430i	0.804852i	−0.915452	+	0.402426i	$0.868167\pi$
$332$	0	0
$333$	− 16.2927i	− 0.892836i
$334$	0	0
$335$	−42.0722	−2.29865
$336$	0	0
$337$	21.4721	1.16966	0.584829	−	0.811156i	$-0.301162\pi$
$337$	21.4721	1.16966	0.584829	+	0.811156i	$0.301162\pi$
$338$	0	0
$339$	− 10.6283i	− 0.577250i
$340$	0	0
$341$	− 35.9143i	− 1.94487i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−21.8715	−1.17752
$346$	0	0
$347$	− 28.6002i	− 1.53534i	−0.640847	−	0.767668i	$-0.721417\pi$
$347$	− 28.6002i	− 1.53534i	0.640847	−	0.767668i	$-0.278583\pi$
$348$	0	0
$349$	23.8322i	1.27571i	0.770157	+	0.637855i	$0.220178\pi$
$349$	23.8322i	1.27571i	−0.770157	+	0.637855i	$0.779822\pi$
$350$	0	0
$351$	29.7564	1.58828
$352$	0	0
$353$	35.2860	1.87808	0.939042	−	0.343802i	$-0.111715\pi$
$353$	35.2860	1.87808	0.939042	+	0.343802i	$0.111715\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0.335577i	0.0177606i
$358$	0	0
$359$	−0.393115	−0.0207478	−0.0103739	−	0.999946i	$-0.503302\pi$
$359$	−0.393115	−0.0207478	−0.0103739	+	0.999946i	$0.503302\pi$
$360$	0	0
$361$	−7.48508	−0.393951
$362$	0	0
$363$	12.5609i	0.659276i
$364$	0	0
$365$	− 28.2499i	− 1.47867i
$366$	0	0
$367$	−2.62831	−0.137197	−0.0685983	−	0.997644i	$-0.521853\pi$
$367$	−2.62831	−0.137197	−0.0685983	+	0.997644i	$0.521853\pi$
$368$	0	0
$369$	−6.24989	−0.325356
$370$	0	0
$371$	− 2.00000i	− 0.103835i
$372$	0	0
$373$	− 14.7862i	− 0.765602i	−0.923831	−	0.382801i	$-0.874959\pi$
$373$	− 14.7862i	− 0.765602i	0.923831	−	0.382801i	$-0.125041\pi$
$374$	0	0
$375$	−20.5855	−1.06303
$376$	0	0
$377$	−23.7795	−1.22471
$378$	0	0
$379$	− 4.10038i	− 0.210623i	−0.994439	−	0.105311i	$-0.966416\pi$
$379$	− 4.10038i	− 0.210623i	0.994439	−	0.105311i	$-0.0335839\pi$
$380$	0	0
$381$	7.27973i	0.372952i
$382$	0	0
$383$	−11.4637	−0.585765	−0.292883	−	0.956148i	$-0.594615\pi$
$383$	−11.4637	−0.585765	−0.292883	+	0.956148i	$0.594615\pi$
$384$	0	0
$385$	17.9572	0.915181
$386$	0	0
$387$	− 8.88661i	− 0.451732i
$388$	0	0
$389$	− 24.9933i	− 1.26721i	−0.773657	−	0.633605i	$-0.781575\pi$
$389$	− 24.9933i	− 1.26721i	0.773657	−	0.633605i	$-0.218425\pi$
$390$	0	0
$391$	−1.45738	−0.0737031
$392$	0	0
$393$	12.0575	0.608222
$394$	0	0
$395$	53.4868i	2.69121i
$396$	0	0
$397$	7.83221i	0.393087i	0.980495	+	0.196544i	$0.0629718\pi$
$397$	7.83221i	0.393087i	−0.980495	+	0.196544i	$0.937028\pi$
$398$	0	0
$399$	5.89962	0.295350
$400$	0	0
$401$	37.3864	1.86699	0.933493	−	0.358594i	$-0.116744\pi$
$401$	37.3864	1.86699	0.933493	+	0.358594i	$0.116744\pi$
$402$	0	0
$403$	− 42.4569i	− 2.11493i
$404$	0	0
$405$	− 4.21691i	− 0.209540i
$406$	0	0
$407$	45.2860	2.24474
$408$	0	0
$409$	6.24989	0.309037	0.154518	−	0.987990i	$-0.450617\pi$
$409$	6.24989	0.309037	0.154518	+	0.987990i	$0.450617\pi$
$410$	0	0
$411$	− 13.0361i	− 0.643024i
$412$	0	0
$413$	− 9.93260i	− 0.488751i
$414$	0	0
$415$	18.4360	0.904986
$416$	0	0
$417$	12.4422	0.609299
$418$	0	0
$419$	19.4391i	0.949662i	0.880077	+	0.474831i	$0.157491\pi$
$419$	19.4391i	0.949662i	−0.880077	+	0.474831i	$0.842509\pi$
$420$	0	0
$421$	35.2860i	1.71973i	0.510518	+	0.859867i	$0.329454\pi$
$421$	35.2860i	1.71973i	−0.510518	+	0.859867i	$0.670546\pi$
$422$	0	0
$423$	3.86519	0.187932
$424$	0	0
$425$	−2.83535	−0.137535
$426$	0	0
$427$	− 4.16779i	− 0.201693i
$428$	0	0
$429$	29.7564i	1.43665i
$430$	0	0
$431$	7.80765	0.376081	0.188041	−	0.982161i	$-0.439786\pi$
$431$	7.80765	0.376081	0.188041	+	0.982161i	$0.439786\pi$
$432$	0	0
$433$	17.4637	0.839250	0.419625	−	0.907698i	$-0.362162\pi$
$433$	17.4637	0.839250	0.419625	+	0.907698i	$0.362162\pi$
$434$	0	0
$435$	18.8585i	0.904194i
$436$	0	0
$437$	25.6216i	1.22565i
$438$	0	0
$439$	−32.7862	−1.56480	−0.782401	−	0.622775i	$-0.786005\pi$
$439$	−32.7862	−1.56480	−0.782401	+	0.622775i	$0.786005\pi$
$440$	0	0
$441$	1.68585	0.0802784
$442$	0	0
$443$	− 34.5082i	− 1.63953i	−0.572697	−	0.819767i	$-0.694103\pi$
$443$	− 34.5082i	− 1.63953i	0.572697	−	0.819767i	$-0.305897\pi$
$444$	0	0
$445$	9.90804i	0.469686i
$446$	0	0
$447$	3.19404	0.151073
$448$	0	0
$449$	22.7862	1.07535	0.537674	−	0.843153i	$-0.319303\pi$
$449$	22.7862	1.07535	0.537674	+	0.843153i	$0.319303\pi$
$450$	0	0
$451$	− 17.3717i	− 0.818001i
$452$	0	0
$453$	− 1.79296i	− 0.0842406i
$454$	0	0
$455$	21.2285	0.995206
$456$	0	0
$457$	8.01469	0.374912	0.187456	−	0.982273i	$-0.439976\pi$
$457$	8.01469	0.374912	0.187456	+	0.982273i	$0.439976\pi$
$458$	0	0
$459$	− 1.57246i	− 0.0733962i
$460$	0	0
$461$	− 31.4966i	− 1.46694i	−0.679719	−	0.733472i	$-0.737898\pi$
$461$	− 31.4966i	− 1.46694i	0.679719	−	0.733472i	$-0.262102\pi$
$462$	0	0
$463$	28.7862	1.33781	0.668905	−	0.743348i	$-0.266763\pi$
$463$	28.7862	1.33781	0.668905	+	0.743348i	$0.266763\pi$
$464$	0	0
$465$	−33.6707	−1.56144
$466$	0	0
$467$	− 16.5609i	− 0.766347i	−0.923676	−	0.383174i	$-0.874831\pi$
$467$	− 16.5609i	− 0.766347i	0.923676	−	0.383174i	$-0.125169\pi$
$468$	0	0
$469$	− 10.9786i	− 0.506944i
$470$	0	0
$471$	16.8635	0.777029
$472$	0	0
$473$	24.7005	1.13573
$474$	0	0
$475$	49.8469i	2.28713i
$476$	0	0
$477$	3.37169i	0.154379i
$478$	0	0
$479$	15.6644	0.715726	0.357863	−	0.933774i	$-0.383506\pi$
$479$	15.6644	0.715726	0.357863	+	0.933774i	$0.383506\pi$
$480$	0	0
$481$	53.5359	2.44103
$482$	0	0
$483$	− 5.70727i	− 0.259690i
$484$	0	0
$485$	− 54.6086i	− 2.47965i
$486$	0	0
$487$	−31.7220	−1.43746	−0.718730	−	0.695290i	$-0.755276\pi$
$487$	−31.7220	−1.43746	−0.718730	+	0.695290i	$0.755276\pi$
$488$	0	0
$489$	0.115077	0.00520396
$490$	0	0
$491$	35.4783i	1.60112i	0.599256	+	0.800558i	$0.295463\pi$
$491$	35.4783i	1.60112i	−0.599256	+	0.800558i	$0.704537\pi$
$492$	0	0
$493$	1.25662i	0.0565951i
$494$	0	0
$495$	−30.2730	−1.36067
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 15.6497i	− 0.700578i	−0.936642	−	0.350289i	$-0.886083\pi$
$499$	− 15.6497i	− 0.700578i	0.936642	−	0.350289i	$-0.113917\pi$
$500$	0	0
$501$	6.54262i	0.292303i
$502$	0	0
$503$	−37.3717	−1.66632	−0.833161	−	0.553031i	$-0.813471\pi$
$503$	−37.3717	−1.66632	−0.833161	+	0.553031i	$0.813471\pi$
$504$	0	0
$505$	7.18562	0.319756
$506$	0	0
$507$	20.2744i	0.900420i
$508$	0	0
$509$	18.5756i	0.823349i	0.911331	+	0.411674i	$0.135056\pi$
$509$	18.5756i	0.823349i	−0.911331	+	0.411674i	$0.864944\pi$
$510$	0	0
$511$	7.37169	0.326104
$512$	0	0
$513$	−27.6447	−1.22054
$514$	0	0
$515$	29.3717i	1.29427i
$516$	0	0
$517$	10.7434i	0.472494i
$518$	0	0
$519$	14.7518	0.647532
$520$	0	0
$521$	−17.5787	−0.770138	−0.385069	−	0.922888i	$-0.625822\pi$
$521$	−17.5787	−0.770138	−0.385069	+	0.922888i	$0.625822\pi$
$522$	0	0
$523$	14.5181i	0.634830i	0.948287	+	0.317415i	$0.102815\pi$
$523$	14.5181i	0.634830i	−0.948287	+	0.317415i	$0.897185\pi$
$524$	0	0
$525$	− 11.1035i	− 0.484597i
$526$	0	0
$527$	−2.24361	−0.0977334
$528$	0	0
$529$	1.78623	0.0776622
$530$	0	0
$531$	16.7448i	0.726664i
$532$	0	0
$533$	− 20.5363i	− 0.889528i
$534$	0	0
$535$	9.17092	0.396494
$536$	0	0
$537$	30.1579	1.30141
$538$	0	0
$539$	4.68585i	0.201834i
$540$	0	0
$541$	12.5426i	0.539249i	0.962966	+	0.269625i	$0.0868996\pi$
$541$	12.5426i	0.539249i	−0.962966	+	0.269625i	$0.913100\pi$
$542$	0	0
$543$	−3.72196	−0.159725
$544$	0	0
$545$	4.13481	0.177116
$546$	0	0
$547$	0.771538i	0.0329886i	0.999864	+	0.0164943i	$0.00525053\pi$
$547$	0.771538i	0.0329886i	−0.999864	+	0.0164943i	$0.994749\pi$
$548$	0	0
$549$	7.02625i	0.299873i
$550$	0	0
$551$	22.0920	0.941149
$552$	0	0
$553$	−13.9572	−0.593519
$554$	0	0
$555$	− 42.4569i	− 1.80220i
$556$	0	0
$557$	− 28.7434i	− 1.21790i	−0.793210	−	0.608948i	$-0.791592\pi$
$557$	− 28.7434i	− 1.21790i	0.793210	−	0.608948i	$-0.208408\pi$
$558$	0	0
$559$	29.2003	1.23504
$560$	0	0
$561$	1.57246	0.0663893
$562$	0	0
$563$	5.34713i	0.225355i	0.993632	+	0.112677i	$0.0359427\pi$
$563$	5.34713i	0.225355i	−0.993632	+	0.112677i	$0.964057\pi$
$564$	0	0
$565$	35.5296i	1.49474i
$566$	0	0
$567$	1.10038	0.0462118
$568$	0	0
$569$	31.2285	1.30917	0.654583	−	0.755990i	$-0.272844\pi$
$569$	31.2285	1.30917	0.654583	+	0.755990i	$0.272844\pi$
$570$	0	0
$571$	18.7287i	0.783771i	0.920014	+	0.391886i	$0.128177\pi$
$571$	18.7287i	0.783771i	−0.920014	+	0.391886i	$0.871823\pi$
$572$	0	0
$573$	19.9143i	0.831932i
$574$	0	0
$575$	48.2217	2.01099
$576$	0	0
$577$	0.542616	0.0225894	0.0112947	−	0.999936i	$-0.496405\pi$
$577$	0.542616	0.0225894	0.0112947	+	0.999936i	$0.496405\pi$
$578$	0	0
$579$	− 31.4439i	− 1.30676i
$580$	0	0
$581$	4.81079i	0.199585i
$582$	0	0
$583$	−9.37169	−0.388136
$584$	0	0
$585$	−35.7879	−1.47965
$586$	0	0
$587$	7.43910i	0.307044i	0.988145	+	0.153522i	$0.0490617\pi$
$587$	7.43910i	0.307044i	−0.988145	+	0.153522i	$0.950938\pi$
$588$	0	0
$589$	39.4439i	1.62526i
$590$	0	0
$591$	11.4637	0.471552
$592$	0	0
$593$	−9.91431	−0.407132	−0.203566	−	0.979061i	$-0.565253\pi$
$593$	−9.91431	−0.407132	−0.203566	+	0.979061i	$0.565253\pi$
$594$	0	0
$595$	− 1.12181i	− 0.0459896i
$596$	0	0
$597$	4.87192i	0.199394i
$598$	0	0
$599$	−29.4868	−1.20480	−0.602398	−	0.798196i	$-0.705788\pi$
$599$	−29.4868	−1.20480	−0.602398	+	0.798196i	$0.705788\pi$
$600$	0	0
$601$	33.2432	1.35602	0.678008	−	0.735054i	$-0.262843\pi$
$601$	33.2432	1.35602	0.678008	+	0.735054i	$0.262843\pi$
$602$	0	0
$603$	18.5082i	0.753712i
$604$	0	0
$605$	− 41.9901i	− 1.70714i
$606$	0	0
$607$	−18.0722	−0.733529	−0.366765	−	0.930314i	$-0.619535\pi$
$607$	−18.0722	−0.733529	−0.366765	+	0.930314i	$0.619535\pi$
$608$	0	0
$609$	−4.92104	−0.199411
$610$	0	0
$611$	12.7005i	0.513809i
$612$	0	0
$613$	37.7795i	1.52590i	0.646458	+	0.762950i	$0.276250\pi$
$613$	37.7795i	1.52590i	−0.646458	+	0.762950i	$0.723750\pi$
$614$	0	0
$615$	−16.2865	−0.656733
$616$	0	0
$617$	16.4851	0.663664	0.331832	−	0.943338i	$-0.392333\pi$
$617$	16.4851	0.663664	0.331832	+	0.943338i	$0.392333\pi$
$618$	0	0
$619$	22.1825i	0.891589i	0.895135	+	0.445795i	$0.147079\pi$
$619$	22.1825i	0.891589i	−0.895135	+	0.445795i	$0.852921\pi$
$620$	0	0
$621$	26.7434i	1.07318i
$622$	0	0
$623$	−2.58546	−0.103584
$624$	0	0
$625$	20.3864	0.815455
$626$	0	0
$627$	− 27.6447i	− 1.10402i
$628$	0	0
$629$	− 2.82908i	− 0.112803i
$630$	0	0
$631$	−12.9870	−0.517004	−0.258502	−	0.966011i	$-0.583229\pi$
$631$	−12.9870	−0.517004	−0.258502	+	0.966011i	$0.583229\pi$
$632$	0	0
$633$	−23.3288	−0.927238
$634$	0	0
$635$	− 24.3356i	− 0.965728i
$636$	0	0
$637$	5.53948i	0.219482i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	6.72869	0.265767	0.132884	−	0.991132i	$-0.457576\pi$
$641$	6.72869	0.265767	0.132884	+	0.991132i	$0.457576\pi$
$642$	0	0
$643$	45.1758i	1.78156i	0.454437	+	0.890779i	$0.349840\pi$
$643$	45.1758i	1.78156i	−0.454437	+	0.890779i	$0.650160\pi$
$644$	0	0
$645$	− 23.1575i	− 0.911824i
$646$	0	0
$647$	34.4078	1.35271	0.676355	−	0.736576i	$-0.263558\pi$
$647$	34.4078	1.35271	0.676355	+	0.736576i	$0.263558\pi$
$648$	0	0
$649$	−46.5426	−1.82696
$650$	0	0
$651$	− 8.78623i	− 0.344360i
$652$	0	0
$653$	15.0361i	0.588409i	0.955743	+	0.294204i	$0.0950546\pi$
$653$	15.0361i	0.588409i	−0.955743	+	0.294204i	$0.904945\pi$
$654$	0	0
$655$	−40.3074	−1.57494
$656$	0	0
$657$	−12.4275	−0.484844
$658$	0	0
$659$	− 20.8866i	− 0.813627i	−0.913511	−	0.406813i	$-0.866640\pi$
$659$	− 20.8866i	− 0.813627i	0.913511	−	0.406813i	$-0.133360\pi$
$660$	0	0
$661$	6.79610i	0.264337i	0.991227	+	0.132169i	$0.0421941\pi$
$661$	6.79610i	0.264337i	−0.991227	+	0.132169i	$0.957806\pi$
$662$	0	0
$663$	1.85892	0.0721945
$664$	0	0
$665$	−19.7220	−0.764785
$666$	0	0
$667$	− 21.3717i	− 0.827515i
$668$	0	0
$669$	− 0.901307i	− 0.0348466i
$670$	0	0
$671$	−19.5296	−0.753932
$672$	0	0
$673$	−8.15792	−0.314465	−0.157232	−	0.987562i	$-0.550257\pi$
$673$	−8.15792	−0.314465	−0.157232	+	0.987562i	$0.550257\pi$
$674$	0	0
$675$	52.0294i	2.00261i
$676$	0	0
$677$	− 5.58860i	− 0.214787i	−0.994217	−	0.107394i	$-0.965749\pi$
$677$	− 5.58860i	− 0.214787i	0.994217	−	0.107394i	$-0.0342505\pi$
$678$	0	0
$679$	14.2499	0.546860
$680$	0	0
$681$	12.4422	0.476787
$682$	0	0
$683$	− 6.30742i	− 0.241347i	−0.992692	−	0.120673i	$-0.961495\pi$
$683$	− 6.30742i	− 0.241347i	0.992692	−	0.120673i	$-0.0385054\pi$
$684$	0	0
$685$	43.5787i	1.66506i
$686$	0	0
$687$	−11.6069	−0.442830
$688$	0	0
$689$	−11.0790	−0.422075
$690$	0	0
$691$	0.945597i	0.0359722i	0.999838	+	0.0179861i	$0.00572546\pi$
$691$	0.945597i	0.0359722i	−0.999838	+	0.0179861i	$0.994275\pi$
$692$	0	0
$693$	− 7.89962i	− 0.300082i
$694$	0	0
$695$	−41.5934	−1.57773
$696$	0	0
$697$	−1.08523	−0.0411061
$698$	0	0
$699$	− 21.5359i	− 0.814562i
$700$	0	0
$701$	− 12.2070i	− 0.461054i	−0.973066	−	0.230527i	$-0.925955\pi$
$701$	− 12.2070i	− 0.461054i	0.973066	−	0.230527i	$-0.0740449\pi$
$702$	0	0
$703$	−49.7367	−1.87585
$704$	0	0
$705$	10.0722	0.379342
$706$	0	0
$707$	1.87506i	0.0705188i
$708$	0	0
$709$	34.2499i	1.28628i	0.765748	+	0.643141i	$0.222369\pi$
$709$	34.2499i	1.28628i	−0.765748	+	0.643141i	$0.777631\pi$
$710$	0	0
$711$	23.5296	0.882430
$712$	0	0
$713$	38.1579	1.42903
$714$	0	0
$715$	− 99.4733i	− 3.72009i
$716$	0	0
$717$	− 8.04912i	− 0.300600i
$718$	0	0
$719$	25.1218	0.936885	0.468443	−	0.883494i	$-0.344815\pi$
$719$	25.1218	0.936885	0.468443	+	0.883494i	$0.344815\pi$
$720$	0	0
$721$	−7.66442	−0.285438
$722$	0	0
$723$	32.5657i	1.21113i
$724$	0	0
$725$	− 41.5787i	− 1.54420i
$726$	0	0
$727$	31.2797	1.16010	0.580050	−	0.814581i	$-0.303033\pi$
$727$	31.2797	1.16010	0.580050	+	0.814581i	$0.303033\pi$
$728$	0	0
$729$	−20.3288	−0.752920
$730$	0	0
$731$	− 1.54308i	− 0.0570727i
$732$	0	0
$733$	− 21.0031i	− 0.775769i	−0.921708	−	0.387884i	$-0.873206\pi$
$733$	− 21.0031i	− 0.775769i	0.921708	−	0.387884i	$-0.126794\pi$
$734$	0	0
$735$	4.39312	0.162042
$736$	0	0
$737$	−51.4439	−1.89496
$738$	0	0
$739$	− 37.3864i	− 1.37528i	−0.726052	−	0.687640i	$-0.758647\pi$
$739$	− 37.3864i	− 1.37528i	0.726052	−	0.687640i	$-0.241353\pi$
$740$	0	0
$741$	− 32.6808i	− 1.20056i
$742$	0	0
$743$	19.9227	0.730894	0.365447	−	0.930832i	$-0.380916\pi$
$743$	19.9227	0.730894	0.365447	+	0.930832i	$0.380916\pi$
$744$	0	0
$745$	−10.6774	−0.391191
$746$	0	0
$747$	− 8.11025i	− 0.296739i
$748$	0	0
$749$	2.39312i	0.0874425i
$750$	0	0
$751$	44.8929	1.63816	0.819082	−	0.573676i	$-0.194483\pi$
$751$	44.8929	1.63816	0.819082	+	0.573676i	$0.194483\pi$
$752$	0	0
$753$	2.88661	0.105194
$754$	0	0
$755$	5.99373i	0.218134i
$756$	0	0
$757$	9.46365i	0.343962i	0.985100	+	0.171981i	$0.0550168\pi$
$757$	9.46365i	0.343962i	−0.985100	+	0.171981i	$0.944983\pi$
$758$	0	0
$759$	−26.7434	−0.970723
$760$	0	0
$761$	25.7795	0.934506	0.467253	−	0.884124i	$-0.345244\pi$
$761$	25.7795	0.934506	0.467253	+	0.884124i	$0.345244\pi$
$762$	0	0
$763$	1.07896i	0.0390610i
$764$	0	0
$765$	1.89119i	0.0683763i
$766$	0	0
$767$	−55.0214	−1.98671
$768$	0	0
$769$	−19.5065	−0.703422	−0.351711	−	0.936109i	$-0.614400\pi$
$769$	−19.5065	−0.703422	−0.351711	+	0.936109i	$0.614400\pi$
$770$	0	0
$771$	1.62158i	0.0583997i
$772$	0	0
$773$	− 44.6676i	− 1.60658i	−0.595588	−	0.803290i	$-0.703081\pi$
$773$	− 44.6676i	− 1.60658i	0.595588	−	0.803290i	$-0.296919\pi$
$774$	0	0
$775$	74.2364	2.66665
$776$	0	0
$777$	11.0790	0.397456
$778$	0	0
$779$	19.0790i	0.683575i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	23.0592	0.824070
$784$	0	0
$785$	−56.3734	−2.01205
$786$	0	0
$787$	34.1825i	1.21847i	0.792988	+	0.609237i	$0.208524\pi$
$787$	34.1825i	1.21847i	−0.792988	+	0.609237i	$0.791476\pi$
$788$	0	0
$789$	32.0000i	1.13923i
$790$	0	0
$791$	−9.27131	−0.329650
$792$	0	0
$793$	−23.0874	−0.819857
$794$	0	0
$795$	8.78623i	0.311615i
$796$	0	0
$797$	− 24.6184i	− 0.872030i	−0.899939	−	0.436015i	$-0.856389\pi$
$797$	− 24.6184i	− 0.872030i	0.899939	−	0.436015i	$-0.143611\pi$
$798$	0	0
$799$	0.671153	0.0237437
$800$	0	0
$801$	4.35869	0.154007
$802$	0	0
$803$	− 34.5426i	− 1.21898i
$804$	0	0
$805$	19.0790i	0.672445i
$806$	0	0
$807$	20.7778	0.731414
$808$	0	0
$809$	−31.8139	−1.11852	−0.559259	−	0.828993i	$-0.688914\pi$
$809$	−31.8139	−1.11852	−0.559259	+	0.828993i	$0.688914\pi$
$810$	0	0
$811$	43.9389i	1.54290i	0.636289	+	0.771451i	$0.280469\pi$
$811$	43.9389i	1.54290i	−0.636289	+	0.771451i	$0.719531\pi$
$812$	0	0
$813$	− 27.6447i	− 0.969542i
$814$	0	0
$815$	−0.384694	−0.0134752
$816$	0	0
$817$	−27.1281	−0.949091
$818$	0	0
$819$	− 9.33871i	− 0.326321i
$820$	0	0
$821$	45.1281i	1.57498i	0.616327	+	0.787490i	$0.288620\pi$
$821$	45.1281i	1.57498i	−0.616327	+	0.787490i	$0.711380\pi$
$822$	0	0
$823$	14.1579	0.493514	0.246757	−	0.969077i	$-0.420635\pi$
$823$	14.1579	0.493514	0.246757	+	0.969077i	$0.420635\pi$
$824$	0	0
$825$	−52.0294	−1.81143
$826$	0	0
$827$	14.5082i	0.504499i	0.967662	+	0.252250i	$0.0811704\pi$
$827$	14.5082i	0.504499i	−0.967662	+	0.252250i	$0.918830\pi$
$828$	0	0
$829$	− 15.7465i	− 0.546899i	−0.961886	−	0.273450i	$-0.911835\pi$
$829$	− 15.7465i	− 0.546899i	0.961886	−	0.273450i	$-0.0881647\pi$
$830$	0	0
$831$	−0.720270	−0.0249859
$832$	0	0
$833$	0.292731	0.0101425
$834$	0	0
$835$	− 21.8715i	− 0.756893i
$836$	0	0
$837$	41.1709i	1.42308i
$838$	0	0
$839$	−7.66442	−0.264605	−0.132303	−	0.991209i	$-0.542237\pi$
$839$	−7.66442	−0.264605	−0.132303	+	0.991209i	$0.542237\pi$
$840$	0	0
$841$	10.5725	0.364568
$842$	0	0
$843$	0.852191i	0.0293510i
$844$	0	0
$845$	− 67.7759i	− 2.33156i
$846$	0	0
$847$	10.9572	0.376492
$848$	0	0
$849$	−0.544767	−0.0186963
$850$	0	0
$851$	48.1151i	1.64936i
$852$	0	0
$853$	− 16.9540i	− 0.580495i	−0.956952	−	0.290247i	$-0.906262\pi$
$853$	− 16.9540i	− 0.580495i	0.956952	−	0.290247i	$-0.0937376\pi$
$854$	0	0
$855$	33.2482	1.13706
$856$	0	0
$857$	−17.0790	−0.583406	−0.291703	−	0.956509i	$-0.594222\pi$
$857$	−17.0790	−0.583406	−0.291703	+	0.956509i	$0.594222\pi$
$858$	0	0
$859$	10.5181i	0.358872i	0.983770	+	0.179436i	$0.0574272\pi$
$859$	10.5181i	0.358872i	−0.983770	+	0.179436i	$0.942573\pi$
$860$	0	0
$861$	− 4.24989i	− 0.144836i
$862$	0	0
$863$	36.1151	1.22937	0.614686	−	0.788772i	$-0.289283\pi$
$863$	36.1151	1.22937	0.614686	+	0.788772i	$0.289283\pi$
$864$	0	0
$865$	−49.3142	−1.67673
$866$	0	0
$867$	19.3900i	0.658518i
$868$	0	0
$869$	65.4011i	2.21858i
$870$	0	0
$871$	−60.8156	−2.06066
$872$	0	0
$873$	−24.0231	−0.813059
$874$	0	0
$875$	17.9572i	0.607063i
$876$	0	0
$877$	− 0.177654i	− 0.00599895i	−0.999996	−	0.00299947i	$-0.999045\pi$
$877$	− 0.177654i	− 0.00599895i	0.999996	−	0.00299947i	$-0.000954764\pi$
$878$	0	0
$879$	1.76481	0.0595255
$880$	0	0
$881$	−1.52962	−0.0515340	−0.0257670	−	0.999668i	$-0.508203\pi$
$881$	−1.52962	−0.0515340	−0.0257670	+	0.999668i	$0.508203\pi$
$882$	0	0
$883$	− 43.7942i	− 1.47379i	−0.676006	−	0.736896i	$-0.736291\pi$
$883$	− 43.7942i	− 1.47379i	0.676006	−	0.736896i	$-0.263709\pi$
$884$	0	0
$885$	43.6350i	1.46678i
$886$	0	0
$887$	−35.6938	−1.19848	−0.599240	−	0.800569i	$-0.704531\pi$
$887$	−35.6938	−1.19848	−0.599240	+	0.800569i	$0.704531\pi$
$888$	0	0
$889$	6.35027	0.212981
$890$	0	0
$891$	− 5.15623i	− 0.172740i
$892$	0	0
$893$	− 11.7992i	− 0.394846i
$894$	0	0
$895$	−100.816	−3.36989
$896$	0	0
$897$	−31.6153	−1.05560
$898$	0	0
$899$	− 32.9013i	− 1.09732i
$900$	0	0
$901$	0.585462i	0.0195046i
$902$	0	0
$903$	6.04285	0.201093
$904$	0	0
$905$	12.4422	0.413594
$906$	0	0
$907$	− 33.3373i	− 1.10695i	−0.832867	−	0.553473i	$-0.813302\pi$
$907$	− 33.3373i	− 1.10695i	0.832867	−	0.553473i	$-0.186698\pi$
$908$	0	0
$909$	− 3.16106i	− 0.104846i
$910$	0	0
$911$	−45.5640	−1.50960	−0.754802	−	0.655953i	$-0.772267\pi$
$911$	−45.5640	−1.50960	−0.754802	+	0.655953i	$0.772267\pi$
$912$	0	0
$913$	22.5426	0.746052
$914$	0	0
$915$	18.3096i	0.605296i
$916$	0	0
$917$	− 10.5181i	− 0.347337i
$918$	0	0
$919$	6.15792	0.203131	0.101566	−	0.994829i	$-0.467615\pi$
$919$	6.15792	0.203131	0.101566	+	0.994829i	$0.467615\pi$
$920$	0	0
$921$	24.3734	0.803130
$922$	0	0
$923$	0	0
$924$	0	0
$925$	93.6081i	3.07782i
$926$	0	0
$927$	12.9210	0.424383
$928$	0	0
$929$	−42.9504	−1.40916	−0.704579	−	0.709626i	$-0.748864\pi$
$929$	−42.9504	−1.40916	−0.704579	+	0.709626i	$0.748864\pi$
$930$	0	0
$931$	− 5.14637i	− 0.168665i
$932$	0	0
$933$	− 17.5725i	− 0.575297i
$934$	0	0
$935$	−5.25662	−0.171910
$936$	0	0
$937$	41.1281	1.34360	0.671798	−	0.740735i	$-0.265522\pi$
$937$	41.1281	1.34360	0.671798	+	0.740735i	$0.265522\pi$
$938$	0	0
$939$	4.24989i	0.138690i
$940$	0	0
$941$	4.61844i	0.150557i	0.997163	+	0.0752785i	$0.0239846\pi$
$941$	4.61844i	0.150557i	−0.997163	+	0.0752785i	$0.976015\pi$
$942$	0	0
$943$	18.4569	0.601040
$944$	0	0
$945$	−20.5855	−0.669645
$946$	0	0
$947$	− 0.800923i	− 0.0260265i	−0.999915	−	0.0130133i	$-0.995858\pi$
$947$	− 0.800923i	− 0.0260265i	0.999915	−	0.0130133i	$-0.00414236\pi$
$948$	0	0
$949$	− 40.8353i	− 1.32557i
$950$	0	0
$951$	−25.8912	−0.839579
$952$	0	0
$953$	−13.9143	−0.450729	−0.225364	−	0.974275i	$-0.572357\pi$
$953$	−13.9143	−0.450729	−0.225364	+	0.974275i	$0.572357\pi$
$954$	0	0
$955$	− 66.5720i	− 2.15422i
$956$	0	0
$957$	23.0592i	0.745399i
$958$	0	0
$959$	−11.3717	−0.367211
$960$	0	0
$961$	27.7434	0.894948
$962$	0	0
$963$	− 4.03442i	− 0.130007i
$964$	0	0
$965$	105.115i	3.38376i
$966$	0	0
$967$	−27.6363	−0.888723	−0.444361	−	0.895848i	$-0.646569\pi$
$967$	−27.6363	−0.888723	−0.444361	+	0.895848i	$0.646569\pi$
$968$	0	0
$969$	−1.72700	−0.0554792
$970$	0	0
$971$	− 0.810789i	− 0.0260195i	−0.999915	−	0.0130097i	$-0.995859\pi$
$971$	− 0.810789i	− 0.0260195i	0.999915	−	0.0130097i	$-0.00414124\pi$
$972$	0	0
$973$	− 10.8536i	− 0.347952i
$974$	0	0
$975$	−61.5077	−1.96982
$976$	0	0
$977$	−4.62831	−0.148073	−0.0740363	−	0.997256i	$-0.523588\pi$
$977$	−4.62831	−0.148073	−0.0740363	+	0.997256i	$0.523588\pi$
$978$	0	0
$979$	12.1151i	0.387200i
$980$	0	0
$981$	− 1.81896i	− 0.0580750i
$982$	0	0
$983$	5.42081	0.172897	0.0864485	−	0.996256i	$-0.472448\pi$
$983$	5.42081	0.172897	0.0864485	+	0.996256i	$0.472448\pi$
$984$	0	0
$985$	−38.3221	−1.22104
$986$	0	0
$987$	2.62831i	0.0836600i
$988$	0	0
$989$	26.2436i	0.834498i
$990$	0	0
$991$	16.2008	0.514634	0.257317	−	0.966327i	$-0.417161\pi$
$991$	16.2008	0.514634	0.257317	+	0.966327i	$0.417161\pi$
$992$	0	0
$993$	16.7862	0.532695
$994$	0	0
$995$	− 16.2865i	− 0.516315i
$996$	0	0
$997$	17.9901i	0.569753i	0.958564	+	0.284877i	$0.0919526\pi$
$997$	17.9901i	0.569753i	−0.958564	+	0.284877i	$0.908047\pi$
$998$	0	0
$999$	−51.9143	−1.64250

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.2.b.o.897.3		6
4.3	odd	2		1792.2.b.p.897.4		6
8.3	odd	2		1792.2.b.p.897.3		6
8.5	even	2	inner	1792.2.b.o.897.4		6
16.3	odd	4		896.2.a.i.1.2	✓	3
16.5	even	4		896.2.a.j.1.2	yes	3
16.11	odd	4		896.2.a.l.1.2	yes	3
16.13	even	4		896.2.a.k.1.2	yes	3
48.5	odd	4		8064.2.a.cb.1.3		3
48.11	even	4		8064.2.a.bu.1.3		3
48.29	odd	4		8064.2.a.ch.1.1		3
48.35	even	4		8064.2.a.ce.1.1		3
112.13	odd	4		6272.2.a.v.1.2		3
112.27	even	4		6272.2.a.u.1.2		3
112.69	odd	4		6272.2.a.w.1.2		3
112.83	even	4		6272.2.a.x.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.a.i.1.2	✓	3	16.3	odd	4
896.2.a.j.1.2	yes	3	16.5	even	4
896.2.a.k.1.2	yes	3	16.13	even	4
896.2.a.l.1.2	yes	3	16.11	odd	4
1792.2.b.o.897.3		6	1.1	even	1	trivial
1792.2.b.o.897.4		6	8.5	even	2	inner
1792.2.b.p.897.3		6	8.3	odd	2
1792.2.b.p.897.4		6	4.3	odd	2
6272.2.a.u.1.2		3	112.27	even	4
6272.2.a.v.1.2		3	112.13	odd	4
6272.2.a.w.1.2		3	112.69	odd	4
6272.2.a.x.1.2		3	112.83	even	4
8064.2.a.bu.1.3		3	48.11	even	4
8064.2.a.cb.1.3		3	48.5	odd	4
8064.2.a.ce.1.1		3	48.35	even	4
8064.2.a.ch.1.1		3	48.29	odd	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:	\( N \)	\(=\)	\( 1792 = 2^{8} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1792.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.14637i q^{3} +3.83221i q^{5} -1.00000 q^{7} +1.68585 q^{9} +O(q^{10})\)	\(q-1.14637i q^{3} +3.83221i q^{5} -1.00000 q^{7} +1.68585 q^{9} +4.68585i q^{11} +5.53948i q^{13} +4.39312 q^{15} +0.292731 q^{17} -5.14637i q^{19} +1.14637i q^{21} -4.97858 q^{23} -9.68585 q^{25} -5.37169i q^{27} +4.29273i q^{29} -7.66442 q^{31} +5.37169 q^{33} -3.83221i q^{35} -9.66442i q^{37} +6.35027 q^{39} -3.70727 q^{41} -5.27131i q^{43} +6.46052i q^{45} +2.29273 q^{47} +1.00000 q^{49} -0.335577i q^{51} +2.00000i q^{53} -17.9572 q^{55} -5.89962 q^{57} +9.93260i q^{59} +4.16779i q^{61} -1.68585 q^{63} -21.2285 q^{65} +10.9786i q^{67} +5.70727i q^{69} -7.37169 q^{73} +11.1035i q^{75} -4.68585i q^{77} +13.9572 q^{79} -1.10038 q^{81} -4.81079i q^{83} +1.12181i q^{85} +4.92104 q^{87} +2.58546 q^{89} -5.53948i q^{91} +8.78623i q^{93} +19.7220 q^{95} -14.2499 q^{97} +7.89962i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{7} - 14 q^{9}+O(q^{10}) \) 6 * q - 6 * q^7 - 14 * q^9	\( 6 q - 6 q^{7} - 14 q^{9} + 8 q^{15} - 4 q^{17} - 34 q^{25} + 8 q^{31} - 16 q^{33} - 40 q^{39} - 28 q^{41} + 8 q^{47} + 6 q^{49} - 48 q^{55} - 48 q^{57} + 14 q^{63} - 32 q^{65} + 4 q^{73} + 24 q^{79} + 6 q^{81} + 72 q^{87} + 4 q^{89} - 8 q^{95} - 20 q^{97}+O(q^{100}) \) 6 * q - 6 * q^7 - 14 * q^9 + 8 * q^15 - 4 * q^17 - 34 * q^25 + 8 * q^31 - 16 * q^33 - 40 * q^39 - 28 * q^41 + 8 * q^47 + 6 * q^49 - 48 * q^55 - 48 * q^57 + 14 * q^63 - 32 * q^65 + 4 * q^73 + 24 * q^79 + 6 * q^81 + 72 * q^87 + 4 * q^89 - 8 * q^95 - 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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