LMFDB - Embedded newform 3344.2.a.ba.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1,3344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.a (trivial)

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:	yes
Analytic conductor:	$26.7019744359$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 14x^{5} + 10x^{4} + 59x^{3} - 27x^{2} - 66x + 30 $ x^7 - x^6 - 14x^5 + 10x^4 + 59x^3 - 27x^2 - 66*x + 30
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 209)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.03821$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.87275 q^{3} -3.24760 q^{5} -1.92338 q^{7} +0.507178 q^{9} +O(q^{10})$	$q-1.87275 q^{3} -3.24760 q^{5} -1.92338 q^{7} +0.507178 q^{9} +1.00000 q^{11} +2.85122 q^{13} +6.08193 q^{15} -2.33033 q^{17} -1.00000 q^{19} +3.60199 q^{21} +2.74653 q^{23} +5.54689 q^{25} +4.66842 q^{27} -0.972965 q^{29} +0.00551178 q^{31} -1.87275 q^{33} +6.24635 q^{35} +9.67124 q^{37} -5.33962 q^{39} +6.65137 q^{41} -7.99413 q^{43} -1.64711 q^{45} -3.46982 q^{47} -3.30063 q^{49} +4.36412 q^{51} +10.5493 q^{53} -3.24760 q^{55} +1.87275 q^{57} +13.7814 q^{59} +3.74608 q^{61} -0.975494 q^{63} -9.25963 q^{65} +3.97172 q^{67} -5.14356 q^{69} -14.2688 q^{71} -13.2263 q^{73} -10.3879 q^{75} -1.92338 q^{77} +1.87656 q^{79} -10.2643 q^{81} +10.9619 q^{83} +7.56799 q^{85} +1.82212 q^{87} +15.0195 q^{89} -5.48397 q^{91} -0.0103222 q^{93} +3.24760 q^{95} -7.57248 q^{97} +0.507178 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 2 q^{3} + 2 q^{5} - 10 q^{7} + 11 q^{9}+O(q^{10}) $ 7 * q - 2 * q^3 + 2 * q^5 - 10 * q^7 + 11 * q^9	$ 7 q - 2 q^{3} + 2 q^{5} - 10 q^{7} + 11 q^{9} + 7 q^{11} - 4 q^{13} - 12 q^{15} + 2 q^{17} - 7 q^{19} - 14 q^{21} - 10 q^{23} + 9 q^{25} + 4 q^{27} - 18 q^{29} - 24 q^{31} - 2 q^{33} - 8 q^{35} - 24 q^{39} - 12 q^{41} - 2 q^{43} - 4 q^{45} - 8 q^{47} + 17 q^{49} + 24 q^{51} + 2 q^{53} + 2 q^{55} + 2 q^{57} + 10 q^{59} + 14 q^{61} - 14 q^{65} - 8 q^{67} - 6 q^{69} - 10 q^{71} - 6 q^{73} - 26 q^{75} - 10 q^{77} - 52 q^{79} - q^{81} + 10 q^{83} - 12 q^{85} - 6 q^{87} - 12 q^{91} + 2 q^{93} - 2 q^{95} - 24 q^{97} + 11 q^{99}+O(q^{100}) $ 7 * q - 2 * q^3 + 2 * q^5 - 10 * q^7 + 11 * q^9 + 7 * q^11 - 4 * q^13 - 12 * q^15 + 2 * q^17 - 7 * q^19 - 14 * q^21 - 10 * q^23 + 9 * q^25 + 4 * q^27 - 18 * q^29 - 24 * q^31 - 2 * q^33 - 8 * q^35 - 24 * q^39 - 12 * q^41 - 2 * q^43 - 4 * q^45 - 8 * q^47 + 17 * q^49 + 24 * q^51 + 2 * q^53 + 2 * q^55 + 2 * q^57 + 10 * q^59 + 14 * q^61 - 14 * q^65 - 8 * q^67 - 6 * q^69 - 10 * q^71 - 6 * q^73 - 26 * q^75 - 10 * q^77 - 52 * q^79 - q^81 + 10 * q^83 - 12 * q^85 - 6 * q^87 - 12 * q^91 + 2 * q^93 - 2 * q^95 - 24 * q^97 + 11 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−1.87275	−1.08123	−0.540615	−	0.841270i	$-0.681809\pi$
$3$	−1.87275	−1.08123	−0.540615	+	0.841270i	$0.681809\pi$
$4$	0	0
$5$	−3.24760	−1.45237	−0.726185	−	0.687499i	$-0.758708\pi$
$5$	−3.24760	−1.45237	−0.726185	+	0.687499i	$0.758708\pi$
$6$	0	0
$7$	−1.92338	−0.726967	−0.363484	−	0.931601i	$-0.618413\pi$
$7$	−1.92338	−0.726967	−0.363484	+	0.931601i	$0.618413\pi$
$8$	0	0
$9$	0.507178	0.169059
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	2.85122	0.790787	0.395394	−	0.918512i	$-0.370608\pi$
$13$	2.85122	0.790787	0.395394	+	0.918512i	$0.370608\pi$
$14$	0	0
$15$	6.08193	1.57035
$16$	0	0
$17$	−2.33033	−0.565189	−0.282594	−	0.959239i	$-0.591195\pi$
$17$	−2.33033	−0.565189	−0.282594	+	0.959239i	$0.591195\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	3.60199	0.786019
$22$	0	0
$23$	2.74653	0.572691	0.286346	−	0.958126i	$-0.407559\pi$
$23$	2.74653	0.572691	0.286346	+	0.958126i	$0.407559\pi$
$24$	0	0
$25$	5.54689	1.10938
$26$	0	0
$27$	4.66842	0.898438
$28$	0	0
$29$	−0.972965	−0.180675	−0.0903376	−	0.995911i	$-0.528795\pi$
$29$	−0.972965	−0.180675	−0.0903376	+	0.995911i	$0.528795\pi$
$30$	0	0
$31$	0.00551178	0.000989945 0	0.000494973	−	1.00000i	$-0.499842\pi$
$31$	0.00551178	0.000989945 0	0.000494973		1.00000i	$0.499842\pi$
$32$	0	0
$33$	−1.87275	−0.326003
$34$	0	0
$35$	6.24635	1.05583
$36$	0	0
$37$	9.67124	1.58994	0.794971	−	0.606647i	$-0.207486\pi$
$37$	9.67124	1.58994	0.794971	+	0.606647i	$0.207486\pi$
$38$	0	0
$39$	−5.33962	−0.855023
$40$	0	0
$41$	6.65137	1.03877	0.519385	−	0.854540i	$-0.326161\pi$
$41$	6.65137	1.03877	0.519385	+	0.854540i	$0.326161\pi$
$42$	0	0
$43$	−7.99413	−1.21909	−0.609547	−	0.792750i	$-0.708648\pi$
$43$	−7.99413	−1.21909	−0.609547	+	0.792750i	$0.708648\pi$
$44$	0	0
$45$	−1.64711	−0.245537
$46$	0	0
$47$	−3.46982	−0.506125	−0.253062	−	0.967450i	$-0.581438\pi$
$47$	−3.46982	−0.506125	−0.253062	+	0.967450i	$0.581438\pi$
$48$	0	0
$49$	−3.30063	−0.471518
$50$	0	0
$51$	4.36412	0.611100
$52$	0	0
$53$	10.5493	1.44905	0.724526	−	0.689247i	$-0.242058\pi$
$53$	10.5493	1.44905	0.724526	+	0.689247i	$0.242058\pi$
$54$	0	0
$55$	−3.24760	−0.437906
$56$	0	0
$57$	1.87275	0.248051
$58$	0	0
$59$	13.7814	1.79419	0.897096	−	0.441836i	$-0.145673\pi$
$59$	13.7814	1.79419	0.897096	+	0.441836i	$0.145673\pi$
$60$	0	0
$61$	3.74608	0.479636	0.239818	−	0.970818i	$-0.422912\pi$
$61$	3.74608	0.479636	0.239818	+	0.970818i	$0.422912\pi$
$62$	0	0
$63$	−0.975494	−0.122901
$64$	0	0
$65$	−9.25963	−1.14852
$66$	0	0
$67$	3.97172	0.485223	0.242612	−	0.970124i	$-0.421996\pi$
$67$	3.97172	0.485223	0.242612	+	0.970124i	$0.421996\pi$
$68$	0	0
$69$	−5.14356	−0.619211
$70$	0	0
$71$	−14.2688	−1.69339	−0.846695	−	0.532078i	$-0.821411\pi$
$71$	−14.2688	−1.69339	−0.846695	+	0.532078i	$0.821411\pi$
$72$	0	0
$73$	−13.2263	−1.54803	−0.774013	−	0.633170i	$-0.781753\pi$
$73$	−13.2263	−1.54803	−0.774013	+	0.633170i	$0.781753\pi$
$74$	0	0
$75$	−10.3879	−1.19949
$76$	0	0
$77$	−1.92338	−0.219189
$78$	0	0
$79$	1.87656	0.211130	0.105565	−	0.994412i	$-0.466335\pi$
$79$	1.87656	0.211130	0.105565	+	0.994412i	$0.466335\pi$
$80$	0	0
$81$	−10.2643	−1.14048
$82$	0	0
$83$	10.9619	1.20322	0.601612	−	0.798789i	$-0.294526\pi$
$83$	10.9619	1.20322	0.601612	+	0.798789i	$0.294526\pi$
$84$	0	0
$85$	7.56799	0.820863
$86$	0	0
$87$	1.82212	0.195351
$88$	0	0
$89$	15.0195	1.59207	0.796034	−	0.605253i	$-0.206928\pi$
$89$	15.0195	1.59207	0.796034	+	0.605253i	$0.206928\pi$
$90$	0	0
$91$	−5.48397	−0.574876
$92$	0	0
$93$	−0.0103222	−0.00107036
$94$	0	0
$95$	3.24760	0.333196
$96$	0	0
$97$	−7.57248	−0.768869	−0.384434	−	0.923152i	$-0.625604\pi$
$97$	−7.57248	−0.768869	−0.384434	+	0.923152i	$0.625604\pi$
$98$	0	0
$99$	0.507178	0.0509733
$100$	0	0
$101$	−15.0513	−1.49766	−0.748831	−	0.662761i	$-0.769385\pi$
$101$	−15.0513	−1.49766	−0.748831	+	0.662761i	$0.769385\pi$
$102$	0	0
$103$	0.543451	0.0535478	0.0267739	−	0.999642i	$-0.491477\pi$
$103$	0.543451	0.0535478	0.0267739	+	0.999642i	$0.491477\pi$
$104$	0	0
$105$	−11.6978	−1.14159
$106$	0	0
$107$	−14.7371	−1.42469	−0.712344	−	0.701831i	$-0.752366\pi$
$107$	−14.7371	−1.42469	−0.712344	+	0.701831i	$0.752366\pi$
$108$	0	0
$109$	−17.3711	−1.66385	−0.831925	−	0.554888i	$-0.812761\pi$
$109$	−17.3711	−1.66385	−0.831925	+	0.554888i	$0.812761\pi$
$110$	0	0
$111$	−18.1118	−1.71909
$112$	0	0
$113$	12.8865	1.21226	0.606132	−	0.795364i	$-0.292720\pi$
$113$	12.8865	1.21226	0.606132	+	0.795364i	$0.292720\pi$
$114$	0	0
$115$	−8.91963	−0.831760
$116$	0	0
$117$	1.44608	0.133690
$118$	0	0
$119$	4.48211	0.410874
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−12.4563	−1.12315
$124$	0	0
$125$	−1.77608	−0.158858
$126$	0	0
$127$	−15.4342	−1.36957	−0.684784	−	0.728746i	$-0.740103\pi$
$127$	−15.4342	−1.36957	−0.684784	+	0.728746i	$0.740103\pi$
$128$	0	0
$129$	14.9710	1.31812
$130$	0	0
$131$	12.0655	1.05417	0.527083	−	0.849814i	$-0.323286\pi$
$131$	12.0655	1.05417	0.527083	+	0.849814i	$0.323286\pi$
$132$	0	0
$133$	1.92338	0.166778
$134$	0	0
$135$	−15.1612	−1.30486
$136$	0	0
$137$	−5.53253	−0.472676	−0.236338	−	0.971671i	$-0.575947\pi$
$137$	−5.53253	−0.472676	−0.236338	+	0.971671i	$0.575947\pi$
$138$	0	0
$139$	8.66764	0.735180	0.367590	−	0.929988i	$-0.380183\pi$
$139$	8.66764	0.735180	0.367590	+	0.929988i	$0.380183\pi$
$140$	0	0
$141$	6.49808	0.547237
$142$	0	0
$143$	2.85122	0.238431
$144$	0	0
$145$	3.15980	0.262407
$146$	0	0
$147$	6.18124	0.509820
$148$	0	0
$149$	−19.3027	−1.58134	−0.790671	−	0.612241i	$-0.790268\pi$
$149$	−19.3027	−1.58134	−0.790671	+	0.612241i	$0.790268\pi$
$150$	0	0
$151$	−8.71384	−0.709122	−0.354561	−	0.935033i	$-0.615370\pi$
$151$	−8.71384	−0.709122	−0.354561	+	0.935033i	$0.615370\pi$
$152$	0	0
$153$	−1.18189	−0.0955505
$154$	0	0
$155$	−0.0179000	−0.00143777
$156$	0	0
$157$	−5.86640	−0.468189	−0.234095	−	0.972214i	$-0.575213\pi$
$157$	−5.86640	−0.468189	−0.234095	+	0.972214i	$0.575213\pi$
$158$	0	0
$159$	−19.7561	−1.56676
$160$	0	0
$161$	−5.28261	−0.416328
$162$	0	0
$163$	−14.8802	−1.16551	−0.582753	−	0.812649i	$-0.698025\pi$
$163$	−14.8802	−1.16551	−0.582753	+	0.812649i	$0.698025\pi$
$164$	0	0
$165$	6.08193	0.473477
$166$	0	0
$167$	−4.18971	−0.324209	−0.162105	−	0.986774i	$-0.551828\pi$
$167$	−4.18971	−0.324209	−0.162105	+	0.986774i	$0.551828\pi$
$168$	0	0
$169$	−4.87053	−0.374656
$170$	0	0
$171$	−0.507178	−0.0387849
$172$	0	0
$173$	0.707136	0.0537626	0.0268813	−	0.999639i	$-0.491442\pi$
$173$	0.707136	0.0537626	0.0268813	+	0.999639i	$0.491442\pi$
$174$	0	0
$175$	−10.6688	−0.806482
$176$	0	0
$177$	−25.8091	−1.93993
$178$	0	0
$179$	21.7962	1.62913	0.814563	−	0.580076i	$-0.196977\pi$
$179$	21.7962	1.62913	0.814563	+	0.580076i	$0.196977\pi$
$180$	0	0
$181$	−2.93416	−0.218094	−0.109047	−	0.994037i	$-0.534780\pi$
$181$	−2.93416	−0.218094	−0.109047	+	0.994037i	$0.534780\pi$
$182$	0	0
$183$	−7.01546	−0.518598
$184$	0	0
$185$	−31.4083	−2.30918
$186$	0	0
$187$	−2.33033	−0.170411
$188$	0	0
$189$	−8.97913	−0.653135
$190$	0	0
$191$	−22.4018	−1.62094	−0.810468	−	0.585783i	$-0.800787\pi$
$191$	−22.4018	−1.62094	−0.810468	+	0.585783i	$0.800787\pi$
$192$	0	0
$193$	2.55447	0.183875	0.0919375	−	0.995765i	$-0.470694\pi$
$193$	2.55447	0.183875	0.0919375	+	0.995765i	$0.470694\pi$
$194$	0	0
$195$	17.3409	1.24181
$196$	0	0
$197$	−12.6968	−0.904607	−0.452303	−	0.891864i	$-0.649398\pi$
$197$	−12.6968	−0.904607	−0.452303	+	0.891864i	$0.649398\pi$
$198$	0	0
$199$	−10.1553	−0.719892	−0.359946	−	0.932973i	$-0.617205\pi$
$199$	−10.1553	−0.719892	−0.359946	+	0.932973i	$0.617205\pi$
$200$	0	0
$201$	−7.43803	−0.524638
$202$	0	0
$203$	1.87138	0.131345
$204$	0	0
$205$	−21.6010	−1.50868
$206$	0	0
$207$	1.39298	0.0968188
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	8.25858	0.568544	0.284272	−	0.958744i	$-0.408248\pi$
$211$	8.25858	0.568544	0.284272	+	0.958744i	$0.408248\pi$
$212$	0	0
$213$	26.7218	1.83095
$214$	0	0
$215$	25.9617	1.77057
$216$	0	0
$217$	−0.0106012	−0.000719658 0
$218$	0	0
$219$	24.7696	1.67377
$220$	0	0
$221$	−6.64430	−0.446944
$222$	0	0
$223$	21.2289	1.42159	0.710797	−	0.703398i	$-0.248335\pi$
$223$	21.2289	1.42159	0.710797	+	0.703398i	$0.248335\pi$
$224$	0	0
$225$	2.81326	0.187551
$226$	0	0
$227$	9.06652	0.601766	0.300883	−	0.953661i	$-0.402719\pi$
$227$	9.06652	0.601766	0.300883	+	0.953661i	$0.402719\pi$
$228$	0	0
$229$	−5.25556	−0.347297	−0.173649	−	0.984808i	$-0.555556\pi$
$229$	−5.25556	−0.347297	−0.173649	+	0.984808i	$0.555556\pi$
$230$	0	0
$231$	3.60199	0.236994
$232$	0	0
$233$	−5.68870	−0.372679	−0.186340	−	0.982485i	$-0.559662\pi$
$233$	−5.68870	−0.372679	−0.186340	+	0.982485i	$0.559662\pi$
$234$	0	0
$235$	11.2686	0.735080
$236$	0	0
$237$	−3.51433	−0.228280
$238$	0	0
$239$	20.3787	1.31819	0.659094	−	0.752060i	$-0.270940\pi$
$239$	20.3787	1.31819	0.659094	+	0.752060i	$0.270940\pi$
$240$	0	0
$241$	17.6930	1.13971	0.569854	−	0.821746i	$-0.307000\pi$
$241$	17.6930	1.13971	0.569854	+	0.821746i	$0.307000\pi$
$242$	0	0
$243$	5.21717	0.334682
$244$	0	0
$245$	10.7191	0.684819
$246$	0	0
$247$	−2.85122	−0.181419
$248$	0	0
$249$	−20.5288	−1.30096
$250$	0	0
$251$	0.776543	0.0490149	0.0245075	−	0.999700i	$-0.492198\pi$
$251$	0.776543	0.0490149	0.0245075	+	0.999700i	$0.492198\pi$
$252$	0	0
$253$	2.74653	0.172673
$254$	0	0
$255$	−14.1729	−0.887543
$256$	0	0
$257$	29.2762	1.82620	0.913100	−	0.407736i	$-0.133682\pi$
$257$	29.2762	1.82620	0.913100	+	0.407736i	$0.133682\pi$
$258$	0	0
$259$	−18.6014	−1.15584
$260$	0	0
$261$	−0.493467	−0.0305448
$262$	0	0
$263$	5.90041	0.363835	0.181918	−	0.983314i	$-0.441770\pi$
$263$	5.90041	0.363835	0.181918	+	0.983314i	$0.441770\pi$
$264$	0	0
$265$	−34.2598	−2.10456
$266$	0	0
$267$	−28.1278	−1.72139
$268$	0	0
$269$	−10.7278	−0.654087	−0.327044	−	0.945009i	$-0.606052\pi$
$269$	−10.7278	−0.654087	−0.327044	+	0.945009i	$0.606052\pi$
$270$	0	0
$271$	−16.2707	−0.988375	−0.494188	−	0.869355i	$-0.664534\pi$
$271$	−16.2707	−0.988375	−0.494188	+	0.869355i	$0.664534\pi$
$272$	0	0
$273$	10.2701	0.621574
$274$	0	0
$275$	5.54689	0.334490
$276$	0	0
$277$	6.77040	0.406794	0.203397	−	0.979096i	$-0.434802\pi$
$277$	6.77040	0.406794	0.203397	+	0.979096i	$0.434802\pi$
$278$	0	0
$279$	0.00279545	0.000167359 0
$280$	0	0
$281$	−17.1455	−1.02281	−0.511407	−	0.859339i	$-0.670876\pi$
$281$	−17.1455	−1.02281	−0.511407	+	0.859339i	$0.670876\pi$
$282$	0	0
$283$	2.94787	0.175232	0.0876162	−	0.996154i	$-0.472075\pi$
$283$	2.94787	0.175232	0.0876162	+	0.996154i	$0.472075\pi$
$284$	0	0
$285$	−6.08193	−0.360262
$286$	0	0
$287$	−12.7931	−0.755152
$288$	0	0
$289$	−11.5695	−0.680561
$290$	0	0
$291$	14.1813	0.831325
$292$	0	0
$293$	2.57851	0.150638	0.0753192	−	0.997159i	$-0.476002\pi$
$293$	2.57851	0.150638	0.0753192	+	0.997159i	$0.476002\pi$
$294$	0	0
$295$	−44.7566	−2.60583
$296$	0	0
$297$	4.66842	0.270889
$298$	0	0
$299$	7.83097	0.452877
$300$	0	0
$301$	15.3757	0.886241
$302$	0	0
$303$	28.1873	1.61932
$304$	0	0
$305$	−12.1658	−0.696610
$306$	0	0
$307$	19.7888	1.12941	0.564704	−	0.825293i	$-0.308990\pi$
$307$	19.7888	1.12941	0.564704	+	0.825293i	$0.308990\pi$
$308$	0	0
$309$	−1.01775	−0.0578975
$310$	0	0
$311$	−19.7979	−1.12264	−0.561319	−	0.827599i	$-0.689706\pi$
$311$	−19.7979	−1.12264	−0.561319	+	0.827599i	$0.689706\pi$
$312$	0	0
$313$	10.9847	0.620890	0.310445	−	0.950591i	$-0.399522\pi$
$313$	10.9847	0.620890	0.310445	+	0.950591i	$0.399522\pi$
$314$	0	0
$315$	3.16801	0.178497
$316$	0	0
$317$	9.88351	0.555113	0.277557	−	0.960709i	$-0.410475\pi$
$317$	9.88351	0.555113	0.277557	+	0.960709i	$0.410475\pi$
$318$	0	0
$319$	−0.972965	−0.0544756
$320$	0	0
$321$	27.5988	1.54042
$322$	0	0
$323$	2.33033	0.129663
$324$	0	0
$325$	15.8154	0.877282
$326$	0	0
$327$	32.5317	1.79901
$328$	0	0
$329$	6.67376	0.367936
$330$	0	0
$331$	−25.7597	−1.41588	−0.707942	−	0.706271i	$-0.750376\pi$
$331$	−25.7597	−1.41588	−0.707942	+	0.706271i	$0.750376\pi$
$332$	0	0
$333$	4.90504	0.268795
$334$	0	0
$335$	−12.8986	−0.704723
$336$	0	0
$337$	21.8924	1.19256	0.596278	−	0.802778i	$-0.296646\pi$
$337$	21.8924	1.19256	0.596278	+	0.802778i	$0.296646\pi$
$338$	0	0
$339$	−24.1332	−1.31074
$340$	0	0
$341$	0.00551178	0.000298480 0
$342$	0	0
$343$	19.8120	1.06975
$344$	0	0
$345$	16.7042	0.899324
$346$	0	0
$347$	−19.2857	−1.03531	−0.517656	−	0.855589i	$-0.673195\pi$
$347$	−19.2857	−1.03531	−0.517656	+	0.855589i	$0.673195\pi$
$348$	0	0
$349$	15.5220	0.830872	0.415436	−	0.909622i	$-0.363629\pi$
$349$	15.5220	0.830872	0.415436	+	0.909622i	$0.363629\pi$
$350$	0	0
$351$	13.3107	0.710473
$352$	0	0
$353$	−8.92859	−0.475221	−0.237610	−	0.971361i	$-0.576364\pi$
$353$	−8.92859	−0.475221	−0.237610	+	0.971361i	$0.576364\pi$
$354$	0	0
$355$	46.3392	2.45943
$356$	0	0
$357$	−8.39385	−0.444249
$358$	0	0
$359$	−26.2672	−1.38633	−0.693165	−	0.720779i	$-0.743784\pi$
$359$	−26.2672	−1.38633	−0.693165	+	0.720779i	$0.743784\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−1.87275	−0.0982937
$364$	0	0
$365$	42.9538	2.24830
$366$	0	0
$367$	−10.2560	−0.535358	−0.267679	−	0.963508i	$-0.586257\pi$
$367$	−10.2560	−0.535358	−0.267679	+	0.963508i	$0.586257\pi$
$368$	0	0
$369$	3.37343	0.175614
$370$	0	0
$371$	−20.2902	−1.05341
$372$	0	0
$373$	−20.2242	−1.04717	−0.523584	−	0.851974i	$-0.675405\pi$
$373$	−20.2242	−1.04717	−0.523584	+	0.851974i	$0.675405\pi$
$374$	0	0
$375$	3.32615	0.171762
$376$	0	0
$377$	−2.77414	−0.142876
$378$	0	0
$379$	−14.1534	−0.727011	−0.363505	−	0.931592i	$-0.618420\pi$
$379$	−14.1534	−0.727011	−0.363505	+	0.931592i	$0.618420\pi$
$380$	0	0
$381$	28.9044	1.48082
$382$	0	0
$383$	−12.3217	−0.629611	−0.314806	−	0.949156i	$-0.601939\pi$
$383$	−12.3217	−0.629611	−0.314806	+	0.949156i	$0.601939\pi$
$384$	0	0
$385$	6.24635	0.318343
$386$	0	0
$387$	−4.05445	−0.206099
$388$	0	0
$389$	33.9248	1.72006	0.860029	−	0.510245i	$-0.170445\pi$
$389$	33.9248	1.72006	0.860029	+	0.510245i	$0.170445\pi$
$390$	0	0
$391$	−6.40033	−0.323679
$392$	0	0
$393$	−22.5956	−1.13980
$394$	0	0
$395$	−6.09432	−0.306639
$396$	0	0
$397$	−20.3357	−1.02062	−0.510309	−	0.859991i	$-0.670469\pi$
$397$	−20.3357	−1.02062	−0.510309	+	0.859991i	$0.670469\pi$
$398$	0	0
$399$	−3.60199	−0.180325
$400$	0	0
$401$	−30.6815	−1.53216	−0.766080	−	0.642746i	$-0.777795\pi$
$401$	−30.6815	−1.53216	−0.766080	+	0.642746i	$0.777795\pi$
$402$	0	0
$403$	0.0157153	0.000782836 0
$404$	0	0
$405$	33.3343	1.65640
$406$	0	0
$407$	9.67124	0.479386
$408$	0	0
$409$	−34.8086	−1.72117	−0.860586	−	0.509305i	$-0.829903\pi$
$409$	−34.8086	−1.72117	−0.860586	+	0.509305i	$0.829903\pi$
$410$	0	0
$411$	10.3610	0.511072
$412$	0	0
$413$	−26.5069	−1.30432
$414$	0	0
$415$	−35.5998	−1.74752
$416$	0	0
$417$	−16.2323	−0.794899
$418$	0	0
$419$	−9.04478	−0.441866	−0.220933	−	0.975289i	$-0.570910\pi$
$419$	−9.04478	−0.441866	−0.220933	+	0.975289i	$0.570910\pi$
$420$	0	0
$421$	29.9089	1.45767	0.728836	−	0.684688i	$-0.240062\pi$
$421$	29.9089	1.45767	0.728836	+	0.684688i	$0.240062\pi$
$422$	0	0
$423$	−1.75981	−0.0855651
$424$	0	0
$425$	−12.9261	−0.627008
$426$	0	0
$427$	−7.20512	−0.348680
$428$	0	0
$429$	−5.33962	−0.257799
$430$	0	0
$431$	13.7402	0.661840	0.330920	−	0.943659i	$-0.392641\pi$
$431$	13.7402	0.661840	0.330920	+	0.943659i	$0.392641\pi$
$432$	0	0
$433$	3.28875	0.158047	0.0790236	−	0.996873i	$-0.474820\pi$
$433$	3.28875	0.158047	0.0790236	+	0.996873i	$0.474820\pi$
$434$	0	0
$435$	−5.91750	−0.283723
$436$	0	0
$437$	−2.74653	−0.131384
$438$	0	0
$439$	−13.1729	−0.628707	−0.314353	−	0.949306i	$-0.601788\pi$
$439$	−13.1729	−0.628707	−0.314353	+	0.949306i	$0.601788\pi$
$440$	0	0
$441$	−1.67401	−0.0797146
$442$	0	0
$443$	24.9484	1.18533	0.592666	−	0.805448i	$-0.298075\pi$
$443$	24.9484	1.18533	0.592666	+	0.805448i	$0.298075\pi$
$444$	0	0
$445$	−48.7774	−2.31227
$446$	0	0
$447$	36.1491	1.70980
$448$	0	0
$449$	15.5530	0.733993	0.366996	−	0.930222i	$-0.380386\pi$
$449$	15.5530	0.733993	0.366996	+	0.930222i	$0.380386\pi$
$450$	0	0
$451$	6.65137	0.313201
$452$	0	0
$453$	16.3188	0.766724
$454$	0	0
$455$	17.8097	0.834933
$456$	0	0
$457$	−19.1451	−0.895571	−0.447786	−	0.894141i	$-0.647787\pi$
$457$	−19.1451	−0.895571	−0.447786	+	0.894141i	$0.647787\pi$
$458$	0	0
$459$	−10.8790	−0.507787
$460$	0	0
$461$	33.5045	1.56046	0.780229	−	0.625493i	$-0.215102\pi$
$461$	33.5045	1.56046	0.780229	+	0.625493i	$0.215102\pi$
$462$	0	0
$463$	−2.83101	−0.131568	−0.0657842	−	0.997834i	$-0.520955\pi$
$463$	−2.83101	−0.131568	−0.0657842	+	0.997834i	$0.520955\pi$
$464$	0	0
$465$	0.0335222	0.00155456
$466$	0	0
$467$	20.3717	0.942689	0.471344	−	0.881949i	$-0.343769\pi$
$467$	20.3717	0.942689	0.471344	+	0.881949i	$0.343769\pi$
$468$	0	0
$469$	−7.63911	−0.352741
$470$	0	0
$471$	10.9863	0.506221
$472$	0	0
$473$	−7.99413	−0.367570
$474$	0	0
$475$	−5.54689	−0.254509
$476$	0	0
$477$	5.35036	0.244976
$478$	0	0
$479$	7.81572	0.357109	0.178555	−	0.983930i	$-0.442858\pi$
$479$	7.81572	0.357109	0.178555	+	0.983930i	$0.442858\pi$
$480$	0	0
$481$	27.5749	1.25731
$482$	0	0
$483$	9.89299	0.450146
$484$	0	0
$485$	24.5924	1.11668
$486$	0	0
$487$	9.10523	0.412597	0.206299	−	0.978489i	$-0.433858\pi$
$487$	9.10523	0.412597	0.206299	+	0.978489i	$0.433858\pi$
$488$	0	0
$489$	27.8668	1.26018
$490$	0	0
$491$	−34.4175	−1.55324	−0.776619	−	0.629970i	$-0.783067\pi$
$491$	−34.4175	−1.55324	−0.776619	+	0.629970i	$0.783067\pi$
$492$	0	0
$493$	2.26733	0.102116
$494$	0	0
$495$	−1.64711	−0.0740321
$496$	0	0
$497$	27.4442	1.23104
$498$	0	0
$499$	−7.80798	−0.349533	−0.174767	−	0.984610i	$-0.555917\pi$
$499$	−7.80798	−0.349533	−0.174767	+	0.984610i	$0.555917\pi$
$500$	0	0
$501$	7.84625	0.350545
$502$	0	0
$503$	−34.9580	−1.55870	−0.779350	−	0.626589i	$-0.784450\pi$
$503$	−34.9580	−1.55870	−0.779350	+	0.626589i	$0.784450\pi$
$504$	0	0
$505$	48.8806	2.17516
$506$	0	0
$507$	9.12126	0.405089
$508$	0	0
$509$	−11.3952	−0.505082	−0.252541	−	0.967586i	$-0.581266\pi$
$509$	−11.3952	−0.505082	−0.252541	+	0.967586i	$0.581266\pi$
$510$	0	0
$511$	25.4392	1.12536
$512$	0	0
$513$	−4.66842	−0.206116
$514$	0	0
$515$	−1.76491	−0.0777712
$516$	0	0
$517$	−3.46982	−0.152602
$518$	0	0
$519$	−1.32429	−0.0581297
$520$	0	0
$521$	−28.0648	−1.22954	−0.614770	−	0.788707i	$-0.710751\pi$
$521$	−28.0648	−1.22954	−0.614770	+	0.788707i	$0.710751\pi$
$522$	0	0
$523$	−20.5683	−0.899389	−0.449694	−	0.893182i	$-0.648467\pi$
$523$	−20.5683	−0.899389	−0.449694	+	0.893182i	$0.648467\pi$
$524$	0	0
$525$	19.9799	0.871993
$526$	0	0
$527$	−0.0128443	−0.000559506 0
$528$	0	0
$529$	−15.4566	−0.672025
$530$	0	0
$531$	6.98965	0.303325
$532$	0	0
$533$	18.9645	0.821446
$534$	0	0
$535$	47.8601	2.06917
$536$	0	0
$537$	−40.8188	−1.76146
$538$	0	0
$539$	−3.30063	−0.142168
$540$	0	0
$541$	−4.13908	−0.177953	−0.0889766	−	0.996034i	$-0.528360\pi$
$541$	−4.13908	−0.177953	−0.0889766	+	0.996034i	$0.528360\pi$
$542$	0	0
$543$	5.49493	0.235810
$544$	0	0
$545$	56.4144	2.41653
$546$	0	0
$547$	30.7624	1.31530	0.657652	−	0.753322i	$-0.271550\pi$
$547$	30.7624	1.31530	0.657652	+	0.753322i	$0.271550\pi$
$548$	0	0
$549$	1.89993	0.0810870
$550$	0	0
$551$	0.972965	0.0414497
$552$	0	0
$553$	−3.60934	−0.153485
$554$	0	0
$555$	58.8198	2.49676
$556$	0	0
$557$	8.84004	0.374565	0.187282	−	0.982306i	$-0.440032\pi$
$557$	8.84004	0.374565	0.187282	+	0.982306i	$0.440032\pi$
$558$	0	0
$559$	−22.7930	−0.964043
$560$	0	0
$561$	4.36412	0.184253
$562$	0	0
$563$	3.15807	0.133097	0.0665484	−	0.997783i	$-0.478801\pi$
$563$	3.15807	0.133097	0.0665484	+	0.997783i	$0.478801\pi$
$564$	0	0
$565$	−41.8503	−1.76066
$566$	0	0
$567$	19.7421	0.829091
$568$	0	0
$569$	18.7192	0.784749	0.392375	−	0.919806i	$-0.371654\pi$
$569$	18.7192	0.784749	0.392375	+	0.919806i	$0.371654\pi$
$570$	0	0
$571$	−37.6834	−1.57700	−0.788500	−	0.615034i	$-0.789142\pi$
$571$	−37.6834	−1.57700	−0.788500	+	0.615034i	$0.789142\pi$
$572$	0	0
$573$	41.9529	1.75261
$574$	0	0
$575$	15.2347	0.635331
$576$	0	0
$577$	1.77272	0.0737993	0.0368997	−	0.999319i	$-0.488252\pi$
$577$	1.77272	0.0737993	0.0368997	+	0.999319i	$0.488252\pi$
$578$	0	0
$579$	−4.78388	−0.198811
$580$	0	0
$581$	−21.0838	−0.874704
$582$	0	0
$583$	10.5493	0.436906
$584$	0	0
$585$	−4.69628	−0.194167
$586$	0	0
$587$	32.1703	1.32781	0.663906	−	0.747816i	$-0.268897\pi$
$587$	32.1703	1.32781	0.663906	+	0.747816i	$0.268897\pi$
$588$	0	0
$589$	−0.00551178	−0.000227109 0
$590$	0	0
$591$	23.7778	0.978088
$592$	0	0
$593$	12.2719	0.503946	0.251973	−	0.967734i	$-0.418921\pi$
$593$	12.2719	0.503946	0.251973	+	0.967734i	$0.418921\pi$
$594$	0	0
$595$	−14.5561	−0.596741
$596$	0	0
$597$	19.0183	0.778369
$598$	0	0
$599$	−16.7005	−0.682366	−0.341183	−	0.939997i	$-0.610828\pi$
$599$	−16.7005	−0.682366	−0.341183	+	0.939997i	$0.610828\pi$
$600$	0	0
$601$	14.3030	0.583432	0.291716	−	0.956505i	$-0.405774\pi$
$601$	14.3030	0.583432	0.291716	+	0.956505i	$0.405774\pi$
$602$	0	0
$603$	2.01437	0.0820315
$604$	0	0
$605$	−3.24760	−0.132034
$606$	0	0
$607$	9.04370	0.367073	0.183536	−	0.983013i	$-0.441246\pi$
$607$	9.04370	0.367073	0.183536	+	0.983013i	$0.441246\pi$
$608$	0	0
$609$	−3.50461	−0.142014
$610$	0	0
$611$	−9.89322	−0.400237
$612$	0	0
$613$	−24.0096	−0.969738	−0.484869	−	0.874587i	$-0.661133\pi$
$613$	−24.0096	−0.969738	−0.484869	+	0.874587i	$0.661133\pi$
$614$	0	0
$615$	40.4532	1.63123
$616$	0	0
$617$	1.30852	0.0526791	0.0263395	−	0.999653i	$-0.491615\pi$
$617$	1.30852	0.0526791	0.0263395	+	0.999653i	$0.491615\pi$
$618$	0	0
$619$	−32.2747	−1.29723	−0.648616	−	0.761116i	$-0.724652\pi$
$619$	−32.2747	−1.29723	−0.648616	+	0.761116i	$0.724652\pi$
$620$	0	0
$621$	12.8220	0.514528
$622$	0	0
$623$	−28.8882	−1.15738
$624$	0	0
$625$	−21.9665	−0.878658
$626$	0	0
$627$	1.87275	0.0747903
$628$	0	0
$629$	−22.5372	−0.898618
$630$	0	0
$631$	−29.6689	−1.18110	−0.590551	−	0.807000i	$-0.701089\pi$
$631$	−29.6689	−1.18110	−0.590551	+	0.807000i	$0.701089\pi$
$632$	0	0
$633$	−15.4662	−0.614727
$634$	0	0
$635$	50.1242	1.98912
$636$	0	0
$637$	−9.41083	−0.372871
$638$	0	0
$639$	−7.23680	−0.286283
$640$	0	0
$641$	22.2716	0.879675	0.439838	−	0.898077i	$-0.355036\pi$
$641$	22.2716	0.879675	0.439838	+	0.898077i	$0.355036\pi$
$642$	0	0
$643$	16.6461	0.656456	0.328228	−	0.944599i	$-0.393549\pi$
$643$	16.6461	0.656456	0.328228	+	0.944599i	$0.393549\pi$
$644$	0	0
$645$	−48.6197	−1.91440
$646$	0	0
$647$	15.9531	0.627180	0.313590	−	0.949559i	$-0.398468\pi$
$647$	15.9531	0.627180	0.313590	+	0.949559i	$0.398468\pi$
$648$	0	0
$649$	13.7814	0.540969
$650$	0	0
$651$	0.0198534	0.000778116 0
$652$	0	0
$653$	14.7613	0.577655	0.288828	−	0.957381i	$-0.406735\pi$
$653$	14.7613	0.577655	0.288828	+	0.957381i	$0.406735\pi$
$654$	0	0
$655$	−39.1838	−1.53104
$656$	0	0
$657$	−6.70811	−0.261708
$658$	0	0
$659$	19.6143	0.764065	0.382032	−	0.924149i	$-0.375224\pi$
$659$	19.6143	0.764065	0.382032	+	0.924149i	$0.375224\pi$
$660$	0	0
$661$	−31.5523	−1.22724	−0.613621	−	0.789601i	$-0.710288\pi$
$661$	−31.5523	−1.22724	−0.613621	+	0.789601i	$0.710288\pi$
$662$	0	0
$663$	12.4431	0.483250
$664$	0	0
$665$	−6.24635	−0.242223
$666$	0	0
$667$	−2.67228	−0.103471
$668$	0	0
$669$	−39.7564	−1.53707
$670$	0	0
$671$	3.74608	0.144616
$672$	0	0
$673$	41.3876	1.59537	0.797687	−	0.603071i	$-0.206057\pi$
$673$	41.3876	1.59537	0.797687	+	0.603071i	$0.206057\pi$
$674$	0	0
$675$	25.8952	0.996708
$676$	0	0
$677$	−20.6981	−0.795491	−0.397746	−	0.917496i	$-0.630207\pi$
$677$	−20.6981	−0.795491	−0.397746	+	0.917496i	$0.630207\pi$
$678$	0	0
$679$	14.5647	0.558943
$680$	0	0
$681$	−16.9793	−0.650647
$682$	0	0
$683$	0.658543	0.0251985	0.0125992	−	0.999921i	$-0.495989\pi$
$683$	0.658543	0.0251985	0.0125992	+	0.999921i	$0.495989\pi$
$684$	0	0
$685$	17.9674	0.686501
$686$	0	0
$687$	9.84233	0.375508
$688$	0	0
$689$	30.0783	1.14589
$690$	0	0
$691$	−34.2462	−1.30279	−0.651393	−	0.758741i	$-0.725815\pi$
$691$	−34.2462	−1.30279	−0.651393	+	0.758741i	$0.725815\pi$
$692$	0	0
$693$	−0.975494	−0.0370559
$694$	0	0
$695$	−28.1490	−1.06775
$696$	0	0
$697$	−15.4999	−0.587101
$698$	0	0
$699$	10.6535	0.402952
$700$	0	0
$701$	22.4638	0.848445	0.424223	−	0.905558i	$-0.360547\pi$
$701$	22.4638	0.848445	0.424223	+	0.905558i	$0.360547\pi$
$702$	0	0
$703$	−9.67124	−0.364758
$704$	0	0
$705$	−21.1032	−0.794791
$706$	0	0
$707$	28.9493	1.08875
$708$	0	0
$709$	−0.410520	−0.0154174	−0.00770870	−	0.999970i	$-0.502454\pi$
$709$	−0.410520	−0.0154174	−0.00770870	+	0.999970i	$0.502454\pi$
$710$	0	0
$711$	0.951752	0.0356935
$712$	0	0
$713$	0.0151383	0.000566933 0
$714$	0	0
$715$	−9.25963	−0.346290
$716$	0	0
$717$	−38.1641	−1.42527
$718$	0	0
$719$	−36.5145	−1.36176	−0.680880	−	0.732395i	$-0.738402\pi$
$719$	−36.5145	−1.36176	−0.680880	+	0.732395i	$0.738402\pi$
$720$	0	0
$721$	−1.04526	−0.0389275
$722$	0	0
$723$	−33.1346	−1.23229
$724$	0	0
$725$	−5.39693	−0.200437
$726$	0	0
$727$	39.2587	1.45602	0.728012	−	0.685564i	$-0.240444\pi$
$727$	39.2587	1.45602	0.728012	+	0.685564i	$0.240444\pi$
$728$	0	0
$729$	21.0225	0.778610
$730$	0	0
$731$	18.6290	0.689018
$732$	0	0
$733$	−34.3259	−1.26786	−0.633928	−	0.773392i	$-0.718558\pi$
$733$	−34.3259	−1.26786	−0.633928	+	0.773392i	$0.718558\pi$
$734$	0	0
$735$	−20.0742	−0.740447
$736$	0	0
$737$	3.97172	0.146300
$738$	0	0
$739$	26.7732	0.984870	0.492435	−	0.870349i	$-0.336107\pi$
$739$	26.7732	0.984870	0.492435	+	0.870349i	$0.336107\pi$
$740$	0	0
$741$	5.33962	0.196156
$742$	0	0
$743$	9.38431	0.344277	0.172139	−	0.985073i	$-0.444932\pi$
$743$	9.38431	0.344277	0.172139	+	0.985073i	$0.444932\pi$
$744$	0	0
$745$	62.6876	2.29669
$746$	0	0
$747$	5.55963	0.203416
$748$	0	0
$749$	28.3449	1.03570
$750$	0	0
$751$	−7.66846	−0.279826	−0.139913	−	0.990164i	$-0.544682\pi$
$751$	−7.66846	−0.279826	−0.139913	+	0.990164i	$0.544682\pi$
$752$	0	0
$753$	−1.45427	−0.0529965
$754$	0	0
$755$	28.2990	1.02991
$756$	0	0
$757$	−6.75931	−0.245671	−0.122836	−	0.992427i	$-0.539199\pi$
$757$	−6.75931	−0.245671	−0.122836	+	0.992427i	$0.539199\pi$
$758$	0	0
$759$	−5.14356	−0.186699
$760$	0	0
$761$	−34.8774	−1.26430	−0.632152	−	0.774844i	$-0.717828\pi$
$761$	−34.8774	−1.26430	−0.632152	+	0.774844i	$0.717828\pi$
$762$	0	0
$763$	33.4111	1.20956
$764$	0	0
$765$	3.83832	0.138775
$766$	0	0
$767$	39.2940	1.41882
$768$	0	0
$769$	−0.00622027	−0.000224309 0	−0.000112154	−	1.00000i	$-0.500036\pi$
$769$	−0.00622027	−0.000224309 0	−0.000112154		1.00000i	$0.500036\pi$
$770$	0	0
$771$	−54.8269	−1.97454
$772$	0	0
$773$	10.8548	0.390418	0.195209	−	0.980762i	$-0.437461\pi$
$773$	10.8548	0.390418	0.195209	+	0.980762i	$0.437461\pi$
$774$	0	0
$775$	0.0305733	0.00109822
$776$	0	0
$777$	34.8357	1.24973
$778$	0	0
$779$	−6.65137	−0.238310
$780$	0	0
$781$	−14.2688	−0.510576
$782$	0	0
$783$	−4.54221	−0.162325
$784$	0	0
$785$	19.0517	0.679984
$786$	0	0
$787$	−37.8221	−1.34821	−0.674106	−	0.738635i	$-0.735471\pi$
$787$	−37.8221	−1.34821	−0.674106	+	0.738635i	$0.735471\pi$
$788$	0	0
$789$	−11.0500	−0.393390
$790$	0	0
$791$	−24.7857	−0.881277
$792$	0	0
$793$	10.6809	0.379290
$794$	0	0
$795$	64.1598	2.27552
$796$	0	0
$797$	−49.2853	−1.74577	−0.872887	−	0.487922i	$-0.837755\pi$
$797$	−49.2853	−1.74577	−0.872887	+	0.487922i	$0.837755\pi$
$798$	0	0
$799$	8.08583	0.286056
$800$	0	0
$801$	7.61758	0.269154
$802$	0	0
$803$	−13.2263	−0.466747
$804$	0	0
$805$	17.1558	0.604662
$806$	0	0
$807$	20.0905	0.707219
$808$	0	0
$809$	34.0416	1.19684	0.598420	−	0.801183i	$-0.295796\pi$
$809$	34.0416	1.19684	0.598420	+	0.801183i	$0.295796\pi$
$810$	0	0
$811$	44.7037	1.56976	0.784880	−	0.619648i	$-0.212725\pi$
$811$	44.7037	1.56976	0.784880	+	0.619648i	$0.212725\pi$
$812$	0	0
$813$	30.4709	1.06866
$814$	0	0
$815$	48.3249	1.69275
$816$	0	0
$817$	7.99413	0.279679
$818$	0	0
$819$	−2.78135	−0.0971882
$820$	0	0
$821$	−48.0778	−1.67793	−0.838963	−	0.544188i	$-0.816838\pi$
$821$	−48.0778	−1.67793	−0.838963	+	0.544188i	$0.816838\pi$
$822$	0	0
$823$	−18.6246	−0.649211	−0.324606	−	0.945849i	$-0.605232\pi$
$823$	−18.6246	−0.649211	−0.324606	+	0.945849i	$0.605232\pi$
$824$	0	0
$825$	−10.3879	−0.361661
$826$	0	0
$827$	−6.99744	−0.243325	−0.121662	−	0.992572i	$-0.538823\pi$
$827$	−6.99744	−0.243325	−0.121662	+	0.992572i	$0.538823\pi$
$828$	0	0
$829$	24.9441	0.866344	0.433172	−	0.901311i	$-0.357394\pi$
$829$	24.9441	0.866344	0.433172	+	0.901311i	$0.357394\pi$
$830$	0	0
$831$	−12.6792	−0.439838
$832$	0	0
$833$	7.69157	0.266497
$834$	0	0
$835$	13.6065	0.470872
$836$	0	0
$837$	0.0257313	0.000889405 0
$838$	0	0
$839$	−10.2122	−0.352566	−0.176283	−	0.984340i	$-0.556407\pi$
$839$	−10.2122	−0.352566	−0.176283	+	0.984340i	$0.556407\pi$
$840$	0	0
$841$	−28.0533	−0.967356
$842$	0	0
$843$	32.1091	1.10590
$844$	0	0
$845$	15.8175	0.544139
$846$	0	0
$847$	−1.92338	−0.0660879
$848$	0	0
$849$	−5.52060	−0.189467
$850$	0	0
$851$	26.5624	0.910546
$852$	0	0
$853$	33.5562	1.14894	0.574472	−	0.818524i	$-0.305207\pi$
$853$	33.5562	1.14894	0.574472	+	0.818524i	$0.305207\pi$
$854$	0	0
$855$	1.64711	0.0563300
$856$	0	0
$857$	34.8775	1.19139	0.595696	−	0.803210i	$-0.296876\pi$
$857$	34.8775	1.19139	0.595696	+	0.803210i	$0.296876\pi$
$858$	0	0
$859$	−13.1320	−0.448059	−0.224029	−	0.974582i	$-0.571921\pi$
$859$	−13.1320	−0.448059	−0.224029	+	0.974582i	$0.571921\pi$
$860$	0	0
$861$	23.9582	0.816493
$862$	0	0
$863$	27.5223	0.936871	0.468435	−	0.883498i	$-0.344818\pi$
$863$	27.5223	0.936871	0.468435	+	0.883498i	$0.344818\pi$
$864$	0	0
$865$	−2.29649	−0.0780831
$866$	0	0
$867$	21.6668	0.735844
$868$	0	0
$869$	1.87656	0.0636581
$870$	0	0
$871$	11.3243	0.383708
$872$	0	0
$873$	−3.84060	−0.129985
$874$	0	0
$875$	3.41607	0.115484
$876$	0	0
$877$	−18.2679	−0.616865	−0.308432	−	0.951246i	$-0.599804\pi$
$877$	−18.2679	−0.616865	−0.308432	+	0.951246i	$0.599804\pi$
$878$	0	0
$879$	−4.82890	−0.162875
$880$	0	0
$881$	−47.0363	−1.58469	−0.792346	−	0.610072i	$-0.791141\pi$
$881$	−47.0363	−1.58469	−0.792346	+	0.610072i	$0.791141\pi$
$882$	0	0
$883$	24.7005	0.831238	0.415619	−	0.909539i	$-0.363565\pi$
$883$	24.7005	0.831238	0.415619	+	0.909539i	$0.363565\pi$
$884$	0	0
$885$	83.8177	2.81750
$886$	0	0
$887$	−23.3946	−0.785515	−0.392758	−	0.919642i	$-0.628479\pi$
$887$	−23.3946	−0.785515	−0.392758	+	0.919642i	$0.628479\pi$
$888$	0	0
$889$	29.6858	0.995631
$890$	0	0
$891$	−10.2643	−0.343867
$892$	0	0
$893$	3.46982	0.116113
$894$	0	0
$895$	−70.7853	−2.36609
$896$	0	0
$897$	−14.6654	−0.489664
$898$	0	0
$899$	−0.00536277	−0.000178858 0
$900$	0	0
$901$	−24.5833	−0.818989
$902$	0	0
$903$	−28.7948	−0.958231
$904$	0	0
$905$	9.52897	0.316754
$906$	0	0
$907$	−19.2411	−0.638892	−0.319446	−	0.947605i	$-0.603497\pi$
$907$	−19.2411	−0.638892	−0.319446	+	0.947605i	$0.603497\pi$
$908$	0	0
$909$	−7.63370	−0.253194
$910$	0	0
$911$	−48.5854	−1.60971	−0.804853	−	0.593475i	$-0.797756\pi$
$911$	−48.5854	−1.60971	−0.804853	+	0.593475i	$0.797756\pi$
$912$	0	0
$913$	10.9619	0.362785
$914$	0	0
$915$	22.7834	0.753195
$916$	0	0
$917$	−23.2064	−0.766344
$918$	0	0
$919$	−21.2754	−0.701812	−0.350906	−	0.936411i	$-0.614126\pi$
$919$	−21.2754	−0.701812	−0.350906	+	0.936411i	$0.614126\pi$
$920$	0	0
$921$	−37.0595	−1.22115
$922$	0	0
$923$	−40.6834	−1.33911
$924$	0	0
$925$	53.6453	1.76385
$926$	0	0
$927$	0.275626	0.00905276
$928$	0	0
$929$	49.9319	1.63821	0.819106	−	0.573643i	$-0.194470\pi$
$929$	49.9319	1.63821	0.819106	+	0.573643i	$0.194470\pi$
$930$	0	0
$931$	3.30063	0.108174
$932$	0	0
$933$	37.0765	1.21383
$934$	0	0
$935$	7.56799	0.247500
$936$	0	0
$937$	11.2142	0.366353	0.183176	−	0.983080i	$-0.441362\pi$
$937$	11.2142	0.366353	0.183176	+	0.983080i	$0.441362\pi$
$938$	0	0
$939$	−20.5715	−0.671325
$940$	0	0
$941$	−47.9390	−1.56277	−0.781383	−	0.624052i	$-0.785485\pi$
$941$	−47.9390	−1.56277	−0.781383	+	0.624052i	$0.785485\pi$
$942$	0	0
$943$	18.2682	0.594894
$944$	0	0
$945$	29.1606	0.948594
$946$	0	0
$947$	−14.4633	−0.469993	−0.234997	−	0.971996i	$-0.575508\pi$
$947$	−14.4633	−0.469993	−0.234997	+	0.971996i	$0.575508\pi$
$948$	0	0
$949$	−37.7112	−1.22416
$950$	0	0
$951$	−18.5093	−0.600206
$952$	0	0
$953$	19.3671	0.627363	0.313682	−	0.949528i	$-0.398438\pi$
$953$	19.3671	0.627363	0.313682	+	0.949528i	$0.398438\pi$
$954$	0	0
$955$	72.7520	2.35420
$956$	0	0
$957$	1.82212	0.0589007
$958$	0	0
$959$	10.6411	0.343620
$960$	0	0
$961$	−31.0000	−0.999999
$962$	0	0
$963$	−7.47432	−0.240857
$964$	0	0
$965$	−8.29591	−0.267055
$966$	0	0
$967$	29.6452	0.953326	0.476663	−	0.879086i	$-0.341846\pi$
$967$	29.6452	0.953326	0.476663	+	0.879086i	$0.341846\pi$
$968$	0	0
$969$	−4.36412	−0.140196
$970$	0	0
$971$	27.9572	0.897190	0.448595	−	0.893735i	$-0.351925\pi$
$971$	27.9572	0.897190	0.448595	+	0.893735i	$0.351925\pi$
$972$	0	0
$973$	−16.6711	−0.534452
$974$	0	0
$975$	−29.6183	−0.948544
$976$	0	0
$977$	−25.4860	−0.815369	−0.407684	−	0.913123i	$-0.633664\pi$
$977$	−25.4860	−0.815369	−0.407684	+	0.913123i	$0.633664\pi$
$978$	0	0
$979$	15.0195	0.480026
$980$	0	0
$981$	−8.81024	−0.281289
$982$	0	0
$983$	23.7523	0.757582	0.378791	−	0.925482i	$-0.376340\pi$
$983$	23.7523	0.757582	0.378791	+	0.925482i	$0.376340\pi$
$984$	0	0
$985$	41.2340	1.31382
$986$	0	0
$987$	−12.4983	−0.397824
$988$	0	0
$989$	−21.9561	−0.698164
$990$	0	0
$991$	−10.8187	−0.343668	−0.171834	−	0.985126i	$-0.554969\pi$
$991$	−10.8187	−0.343668	−0.171834	+	0.985126i	$0.554969\pi$
$992$	0	0
$993$	48.2415	1.53090
$994$	0	0
$995$	32.9804	1.04555
$996$	0	0
$997$	−55.0261	−1.74270	−0.871348	−	0.490666i	$-0.836753\pi$
$997$	−55.0261	−1.74270	−0.871348	+	0.490666i	$0.836753\pi$
$998$	0	0
$999$	45.1494	1.42847

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.ba.1.2		7
4.3	odd	2		209.2.a.d.1.6	✓	7
12.11	even	2		1881.2.a.p.1.2		7
20.19	odd	2		5225.2.a.n.1.2		7
44.43	even	2		2299.2.a.q.1.2		7
76.75	even	2		3971.2.a.i.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.a.d.1.6	✓	7	4.3	odd	2
1881.2.a.p.1.2		7	12.11	even	2
2299.2.a.q.1.2		7	44.43	even	2
3344.2.a.ba.1.2		7	1.1	even	1	trivial
3971.2.a.i.1.2		7	76.75	even	2
5225.2.a.n.1.2		7	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q-1.87275 q^{3} -3.24760 q^{5} -1.92338 q^{7} +0.507178 q^{9} +O(q^{10})\)	\(q-1.87275 q^{3} -3.24760 q^{5} -1.92338 q^{7} +0.507178 q^{9} +1.00000 q^{11} +2.85122 q^{13} +6.08193 q^{15} -2.33033 q^{17} -1.00000 q^{19} +3.60199 q^{21} +2.74653 q^{23} +5.54689 q^{25} +4.66842 q^{27} -0.972965 q^{29} +0.00551178 q^{31} -1.87275 q^{33} +6.24635 q^{35} +9.67124 q^{37} -5.33962 q^{39} +6.65137 q^{41} -7.99413 q^{43} -1.64711 q^{45} -3.46982 q^{47} -3.30063 q^{49} +4.36412 q^{51} +10.5493 q^{53} -3.24760 q^{55} +1.87275 q^{57} +13.7814 q^{59} +3.74608 q^{61} -0.975494 q^{63} -9.25963 q^{65} +3.97172 q^{67} -5.14356 q^{69} -14.2688 q^{71} -13.2263 q^{73} -10.3879 q^{75} -1.92338 q^{77} +1.87656 q^{79} -10.2643 q^{81} +10.9619 q^{83} +7.56799 q^{85} +1.82212 q^{87} +15.0195 q^{89} -5.48397 q^{91} -0.0103222 q^{93} +3.24760 q^{95} -7.57248 q^{97} +0.507178 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 2 q^{3} + 2 q^{5} - 10 q^{7} + 11 q^{9}+O(q^{10}) \) 7 * q - 2 * q^3 + 2 * q^5 - 10 * q^7 + 11 * q^9	\( 7 q - 2 q^{3} + 2 q^{5} - 10 q^{7} + 11 q^{9} + 7 q^{11} - 4 q^{13} - 12 q^{15} + 2 q^{17} - 7 q^{19} - 14 q^{21} - 10 q^{23} + 9 q^{25} + 4 q^{27} - 18 q^{29} - 24 q^{31} - 2 q^{33} - 8 q^{35} - 24 q^{39} - 12 q^{41} - 2 q^{43} - 4 q^{45} - 8 q^{47} + 17 q^{49} + 24 q^{51} + 2 q^{53} + 2 q^{55} + 2 q^{57} + 10 q^{59} + 14 q^{61} - 14 q^{65} - 8 q^{67} - 6 q^{69} - 10 q^{71} - 6 q^{73} - 26 q^{75} - 10 q^{77} - 52 q^{79} - q^{81} + 10 q^{83} - 12 q^{85} - 6 q^{87} - 12 q^{91} + 2 q^{93} - 2 q^{95} - 24 q^{97} + 11 q^{99}+O(q^{100}) \) 7 * q - 2 * q^3 + 2 * q^5 - 10 * q^7 + 11 * q^9 + 7 * q^11 - 4 * q^13 - 12 * q^15 + 2 * q^17 - 7 * q^19 - 14 * q^21 - 10 * q^23 + 9 * q^25 + 4 * q^27 - 18 * q^29 - 24 * q^31 - 2 * q^33 - 8 * q^35 - 24 * q^39 - 12 * q^41 - 2 * q^43 - 4 * q^45 - 8 * q^47 + 17 * q^49 + 24 * q^51 + 2 * q^53 + 2 * q^55 + 2 * q^57 + 10 * q^59 + 14 * q^61 - 14 * q^65 - 8 * q^67 - 6 * q^69 - 10 * q^71 - 6 * q^73 - 26 * q^75 - 10 * q^77 - 52 * q^79 - q^81 + 10 * q^83 - 12 * q^85 - 6 * q^87 - 12 * q^91 + 2 * q^93 - 2 * q^95 - 24 * q^97 + 11 * q^99

Introduction

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