LMFDB - Embedded newform 896.2.f.b.895.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [896,2,Mod(895,896)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(896, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("896.895");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 896 = 2^{7} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	896.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$7.15459602111$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			895.1
Root			$0.437016 + 0.437016i$ of defining polynomial
Character	$\chi$	$=$	896.895
Dual form			896.2.f.b.895.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.28825 q^{3} -0.540182i q^{5} +(-1.41421 + 2.23607i) q^{7} +2.23607 q^{9} +O(q^{10})$	$q-2.28825 q^{3} -0.540182i q^{5} +(-1.41421 + 2.23607i) q^{7} +2.23607 q^{9} -1.23607i q^{11} -2.28825i q^{13} +1.23607i q^{15} +1.74806i q^{17} -5.11667 q^{19} +(3.23607 - 5.11667i) q^{21} +3.23607i q^{23} +4.70820 q^{25} +1.74806 q^{27} -2.00000 q^{29} +3.90879 q^{31} +2.82843i q^{33} +(1.20788 + 0.763932i) q^{35} +6.94427 q^{37} +5.23607i q^{39} -10.2333i q^{41} -11.7082i q^{43} -1.20788i q^{45} +10.9010 q^{47} +(-3.00000 - 6.32456i) q^{49} -4.00000i q^{51} +8.47214 q^{53} -0.667701 q^{55} +11.7082 q^{57} +1.62054 q^{59} +13.1893i q^{61} +(-3.16228 + 5.00000i) q^{63} -1.23607 q^{65} -3.70820i q^{67} -7.40492i q^{69} -10.0000i q^{71} -14.1421i q^{73} -10.7735 q^{75} +(2.76393 + 1.74806i) q^{77} -3.52786i q^{79} -10.7082 q^{81} +9.02546 q^{83} +0.944272 q^{85} +4.57649 q^{87} +15.4775i q^{89} +(5.11667 + 3.23607i) q^{91} -8.94427 q^{93} +2.76393i q^{95} -1.74806i q^{97} -2.76393i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 8 q^{21} - 16 q^{25} - 16 q^{29} - 16 q^{37} - 24 q^{49} + 32 q^{53} + 40 q^{57} + 8 q^{65} + 40 q^{77} - 32 q^{81} - 64 q^{85}+O(q^{100}) $ 8 * q + 8 * q^21 - 16 * q^25 - 16 * q^29 - 16 * q^37 - 24 * q^49 + 32 * q^53 + 40 * q^57 + 8 * q^65 + 40 * q^77 - 32 * q^81 - 64 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/896\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$645$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	−2.28825			−1.32112			−0.660560	−	0.750774i	$-0.729681\pi$
$3$	−2.28825			−1.32112			−0.660560	+	0.750774i	$0.729681\pi$
$4$		0			0
$5$		−	0.540182i		−	0.241577i	−0.992678	−	0.120788i	$-0.961458\pi$
$5$		−	0.540182i		−	0.241577i	0.992678	−	0.120788i	$-0.0385422\pi$
$6$		0			0
$7$	−1.41421	+	2.23607i	−0.534522	+	0.845154i
$7$	−1.41421	+	2.23607i	−0.534522	+	0.845154i
$8$		0			0
$9$	2.23607			0.745356
$10$		0			0
$11$		−	1.23607i		−	0.372689i	−0.982485	−	0.186344i	$-0.940336\pi$
$11$		−	1.23607i		−	0.372689i	0.982485	−	0.186344i	$-0.0596640\pi$
$12$		0			0
$13$		−	2.28825i		−	0.634645i	−0.948318	−	0.317323i	$-0.897216\pi$
$13$		−	2.28825i		−	0.634645i	0.948318	−	0.317323i	$-0.102784\pi$
$14$		0			0
$15$			1.23607i			0.319151i
$16$		0			0
$17$			1.74806i			0.423968i	0.977273	+	0.211984i	$0.0679924\pi$
$17$			1.74806i			0.423968i	−0.977273	+	0.211984i	$0.932008\pi$
$18$		0			0
$19$	−5.11667			−1.17385			−0.586923	−	0.809643i	$-0.699661\pi$
$19$	−5.11667			−1.17385			−0.586923	+	0.809643i	$0.699661\pi$
$20$		0			0
$21$	3.23607	−	5.11667i	0.706168	−	1.11655i
$22$		0			0
$23$			3.23607i			0.674767i	0.941367	+	0.337383i	$0.109542\pi$
$23$			3.23607i			0.674767i	−0.941367	+	0.337383i	$0.890458\pi$
$24$		0			0
$25$	4.70820			0.941641
$26$		0			0
$27$	1.74806			0.336415
$28$		0			0
$29$	−2.00000			−0.371391			−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000			−0.371391			−0.185695	+	0.982607i	$0.559454\pi$
$30$		0			0
$31$	3.90879			0.702039			0.351020	−	0.936368i	$-0.385835\pi$
$31$	3.90879			0.702039			0.351020	+	0.936368i	$0.385835\pi$
$32$		0			0
$33$			2.82843i			0.492366i
$34$		0			0
$35$	1.20788	+	0.763932i	0.204169	+	0.129128i
$36$		0			0
$37$	6.94427			1.14163			0.570816	−	0.821078i	$-0.306627\pi$
$37$	6.94427			1.14163			0.570816	+	0.821078i	$0.306627\pi$
$38$		0			0
$39$			5.23607i			0.838442i
$40$		0			0
$41$		−	10.2333i		−	1.59818i	−0.601211	−	0.799090i	$-0.705315\pi$
$41$		−	10.2333i		−	1.59818i	0.601211	−	0.799090i	$-0.294685\pi$
$42$		0			0
$43$		−	11.7082i		−	1.78548i	−0.450568	−	0.892742i	$-0.648778\pi$
$43$		−	11.7082i		−	1.78548i	0.450568	−	0.892742i	$-0.351222\pi$
$44$		0			0
$45$		−	1.20788i		−	0.180061i
$46$		0			0
$47$	10.9010			1.59008			0.795041	−	0.606556i	$-0.207450\pi$
$47$	10.9010			1.59008			0.795041	+	0.606556i	$0.207450\pi$
$48$		0			0
$49$	−3.00000	−	6.32456i	−0.428571	−	0.903508i
$50$		0			0
$51$		−	4.00000i		−	0.560112i
$52$		0			0
$53$	8.47214			1.16374			0.581869	−	0.813283i	$-0.302322\pi$
$53$	8.47214			1.16374			0.581869	+	0.813283i	$0.302322\pi$
$54$		0			0
$55$	−0.667701			−0.0900328
$56$		0			0
$57$	11.7082			1.55079
$58$		0			0
$59$	1.62054			0.210977			0.105488	−	0.994421i	$-0.466359\pi$
$59$	1.62054			0.210977			0.105488	+	0.994421i	$0.466359\pi$
$60$		0			0
$61$			13.1893i			1.68872i	0.535780	+	0.844358i	$0.320018\pi$
$61$			13.1893i			1.68872i	−0.535780	+	0.844358i	$0.679982\pi$
$62$		0			0
$63$	−3.16228	+	5.00000i	−0.398410	+	0.629941i
$64$		0			0
$65$	−1.23607			−0.153315
$66$		0			0
$67$		−	3.70820i		−	0.453029i	−0.974008	−	0.226515i	$-0.927267\pi$
$67$		−	3.70820i		−	0.453029i	0.974008	−	0.226515i	$-0.0727331\pi$
$68$		0			0
$69$		−	7.40492i		−	0.891447i
$70$		0			0
$71$		−	10.0000i		−	1.18678i	−0.804914	−	0.593391i	$-0.797789\pi$
$71$		−	10.0000i		−	1.18678i	0.804914	−	0.593391i	$-0.202211\pi$
$72$		0			0
$73$		−	14.1421i		−	1.65521i	−0.561310	−	0.827606i	$-0.689702\pi$
$73$		−	14.1421i		−	1.65521i	0.561310	−	0.827606i	$-0.310298\pi$
$74$		0			0
$75$	−10.7735			−1.24402
$76$		0			0
$77$	2.76393	+	1.74806i	0.314979	+	0.199210i
$78$		0			0
$79$		−	3.52786i		−	0.396916i	−0.980109	−	0.198458i	$-0.936407\pi$
$79$		−	3.52786i		−	0.396916i	0.980109	−	0.198458i	$-0.0635933\pi$
$80$		0			0
$81$	−10.7082			−1.18980
$82$		0			0
$83$	9.02546			0.990673			0.495337	−	0.868701i	$-0.335045\pi$
$83$	9.02546			0.990673			0.495337	+	0.868701i	$0.335045\pi$
$84$		0			0
$85$	0.944272			0.102421
$86$		0			0
$87$	4.57649			0.490651
$88$		0			0
$89$			15.4775i			1.64062i	0.571922	+	0.820308i	$0.306198\pi$
$89$			15.4775i			1.64062i	−0.571922	+	0.820308i	$0.693802\pi$
$90$		0			0
$91$	5.11667	+	3.23607i	0.536373	+	0.339232i
$92$		0			0
$93$	−8.94427			−0.927478
$94$		0			0
$95$			2.76393i			0.283573i
$96$		0			0
$97$		−	1.74806i		−	0.177489i	−0.996054	−	0.0887445i	$-0.971715\pi$
$97$		−	1.74806i		−	0.177489i	0.996054	−	0.0887445i	$-0.0282855\pi$
$98$		0			0
$99$		−	2.76393i		−	0.277786i
$100$		0			0
$101$			11.4412i			1.13844i	0.822184	+	0.569222i	$0.192756\pi$
$101$			11.4412i			1.13844i	−0.822184	+	0.569222i	$0.807244\pi$
$102$		0			0
$103$	13.7295			1.35281			0.676403	−	0.736532i	$-0.263538\pi$
$103$	13.7295			1.35281			0.676403	+	0.736532i	$0.263538\pi$
$104$		0			0
$105$	−2.76393	−	1.74806i	−0.269732	−	0.170594i
$106$		0			0
$107$			0.291796i			0.0282090i	0.999901	+	0.0141045i	$0.00448975\pi$
$107$			0.291796i			0.0282090i	−0.999901	+	0.0141045i	$0.995510\pi$
$108$		0			0
$109$	3.52786			0.337908			0.168954	−	0.985624i	$-0.445961\pi$
$109$	3.52786			0.337908			0.168954	+	0.985624i	$0.445961\pi$
$110$		0			0
$111$	−15.8902			−1.50823
$112$		0			0
$113$	2.29180			0.215594			0.107797	−	0.994173i	$-0.465620\pi$
$113$	2.29180			0.215594			0.107797	+	0.994173i	$0.465620\pi$
$114$		0			0
$115$	1.74806			0.163008
$116$		0			0
$117$		−	5.11667i		−	0.473037i
$118$		0			0
$119$	−3.90879	−	2.47214i	−0.358318	−	0.226620i
$120$		0			0
$121$	9.47214			0.861103
$122$		0			0
$123$			23.4164i			2.11139i
$124$		0			0
$125$		−	5.24419i		−	0.469055i
$126$		0			0
$127$		−	0.763932i		−	0.0677880i	−0.999425	−	0.0338940i	$-0.989209\pi$
$127$		−	0.763932i		−	0.0677880i	0.999425	−	0.0338940i	$-0.0107909\pi$
$128$		0			0
$129$			26.7912i			2.35884i
$130$		0			0
$131$	−5.78437			−0.505383			−0.252692	−	0.967547i	$-0.581316\pi$
$131$	−5.78437			−0.505383			−0.252692	+	0.967547i	$0.581316\pi$
$132$		0			0
$133$	7.23607	−	11.4412i	0.627447	−	0.992080i
$134$		0			0
$135$		−	0.944272i		−	0.0812700i
$136$		0			0
$137$	−12.4721			−1.06557			−0.532783	−	0.846252i	$-0.678854\pi$
$137$	−12.4721			−1.06557			−0.532783	+	0.846252i	$0.678854\pi$
$138$		0			0
$139$	−2.70091			−0.229088			−0.114544	−	0.993418i	$-0.536541\pi$
$139$	−2.70091			−0.229088			−0.114544	+	0.993418i	$0.536541\pi$
$140$		0			0
$141$	−24.9443			−2.10069
$142$		0			0
$143$	−2.82843			−0.236525
$144$		0			0
$145$			1.08036i			0.0897193i
$146$		0			0
$147$	6.86474	+	14.4721i	0.566194	+	1.19364i
$148$		0			0
$149$	−13.4164			−1.09911			−0.549557	−	0.835456i	$-0.685204\pi$
$149$	−13.4164			−1.09911			−0.549557	+	0.835456i	$0.685204\pi$
$150$		0			0
$151$			11.2361i			0.914378i	0.889369	+	0.457189i	$0.151144\pi$
$151$			11.2361i			0.914378i	−0.889369	+	0.457189i	$0.848856\pi$
$152$		0			0
$153$			3.90879i			0.316007i
$154$		0			0
$155$		−	2.11146i		−	0.169596i
$156$		0			0
$157$		−	11.4412i		−	0.913109i	−0.889695	−	0.456555i	$-0.849083\pi$
$157$		−	11.4412i		−	0.913109i	0.889695	−	0.456555i	$-0.150917\pi$
$158$		0			0
$159$	−19.3863			−1.53744
$160$		0			0
$161$	−7.23607	−	4.57649i	−0.570282	−	0.360678i
$162$		0			0
$163$		−	17.2361i		−	1.35003i	−0.737803	−	0.675017i	$-0.764136\pi$
$163$		−	17.2361i		−	1.35003i	0.737803	−	0.675017i	$-0.235864\pi$
$164$		0			0
$165$	1.52786			0.118944
$166$		0			0
$167$	−8.07262			−0.624678			−0.312339	−	0.949971i	$-0.601112\pi$
$167$	−8.07262			−0.624678			−0.312339	+	0.949971i	$0.601112\pi$
$168$		0			0
$169$	7.76393			0.597226
$170$		0			0
$171$	−11.4412			−0.874933
$172$		0			0
$173$			18.1784i			1.38208i	0.722816	+	0.691041i	$0.242848\pi$
$173$			18.1784i			1.38208i	−0.722816	+	0.691041i	$0.757152\pi$
$174$		0			0
$175$	−6.65841	+	10.5279i	−0.503328	+	0.795832i
$176$		0			0
$177$	−3.70820			−0.278726
$178$		0			0
$179$			20.6525i			1.54364i	0.635842	+	0.771819i	$0.280653\pi$
$179$			20.6525i			1.54364i	−0.635842	+	0.771819i	$0.719347\pi$
$180$		0			0
$181$		−	0.540182i		−	0.0401514i	−0.999798	−	0.0200757i	$-0.993609\pi$
$181$		−	0.540182i		−	0.0401514i	0.999798	−	0.0200757i	$-0.00639072\pi$
$182$		0			0
$183$		−	30.1803i		−	2.23099i
$184$		0			0
$185$		−	3.75117i		−	0.275791i
$186$		0			0
$187$	2.16073			0.158008
$188$		0			0
$189$	−2.47214	+	3.90879i	−0.179821	+	0.284323i
$190$		0			0
$191$		−	22.9443i		−	1.66019i	−0.557623	−	0.830095i	$-0.688286\pi$
$191$		−	22.9443i		−	1.66019i	0.557623	−	0.830095i	$-0.311714\pi$
$192$		0			0
$193$	−11.2361			−0.808790			−0.404395	−	0.914584i	$-0.632518\pi$
$193$	−11.2361			−0.808790			−0.404395	+	0.914584i	$0.632518\pi$
$194$		0			0
$195$	2.82843			0.202548
$196$		0			0
$197$	−18.9443			−1.34972			−0.674862	−	0.737944i	$-0.735797\pi$
$197$	−18.9443			−1.34972			−0.674862	+	0.737944i	$0.735797\pi$
$198$		0			0
$199$	−6.73722			−0.477589			−0.238794	−	0.971070i	$-0.576752\pi$
$199$	−6.73722			−0.477589			−0.238794	+	0.971070i	$0.576752\pi$
$200$		0			0
$201$			8.48528i			0.598506i
$202$		0			0
$203$	2.82843	−	4.47214i	0.198517	−	0.313882i
$204$		0			0
$205$	−5.52786			−0.386083
$206$		0			0
$207$			7.23607i			0.502941i
$208$		0			0
$209$			6.32456i			0.437479i
$210$		0			0
$211$			1.23607i			0.0850944i	0.999094	+	0.0425472i	$0.0135473\pi$
$211$			1.23607i			0.0850944i	−0.999094	+	0.0425472i	$0.986453\pi$
$212$		0			0
$213$			22.8825i			1.56788i
$214$		0			0
$215$	−6.32456			−0.431331
$216$		0			0
$217$	−5.52786	+	8.74032i	−0.375256	+	0.593332i
$218$		0			0
$219$			32.3607i			2.18673i
$220$		0			0
$221$	4.00000			0.269069
$222$		0			0
$223$	16.1452			1.08117			0.540583	−	0.841291i	$-0.318204\pi$
$223$	16.1452			1.08117			0.540583	+	0.841291i	$0.318204\pi$
$224$		0			0
$225$	10.5279			0.701858
$226$		0			0
$227$	1.87558			0.124487			0.0622434	−	0.998061i	$-0.480174\pi$
$227$	1.87558			0.124487			0.0622434	+	0.998061i	$0.480174\pi$
$228$		0			0
$229$		−	1.20788i		−	0.0798191i	−0.999203	−	0.0399096i	$-0.987293\pi$
$229$		−	1.20788i		−	0.0798191i	0.999203	−	0.0399096i	$-0.0127070\pi$
$230$		0			0
$231$	−6.32456	−	4.00000i	−0.416125	−	0.263181i
$232$		0			0
$233$	−13.4164			−0.878938			−0.439469	−	0.898258i	$-0.644833\pi$
$233$	−13.4164			−0.878938			−0.439469	+	0.898258i	$0.644833\pi$
$234$		0			0
$235$		−	5.88854i		−	0.384126i
$236$		0			0
$237$			8.07262i			0.524373i
$238$		0			0
$239$		−	18.6525i		−	1.20653i	−0.797541	−	0.603264i	$-0.793866\pi$
$239$		−	18.6525i		−	1.20653i	0.797541	−	0.603264i	$-0.206134\pi$
$240$		0			0
$241$			1.74806i			0.112603i	0.998414	+	0.0563014i	$0.0179308\pi$
$241$			1.74806i			0.112603i	−0.998414	+	0.0563014i	$0.982069\pi$
$242$		0			0
$243$	19.2588			1.23545
$244$		0			0
$245$	−3.41641	+	1.62054i	−0.218266	+	0.103533i
$246$		0			0
$247$			11.7082i			0.744975i
$248$		0			0
$249$	−20.6525			−1.30880
$250$		0			0
$251$	19.2588			1.21561			0.607803	−	0.794088i	$-0.292051\pi$
$251$	19.2588			1.21561			0.607803	+	0.794088i	$0.292051\pi$
$252$		0			0
$253$	4.00000			0.251478
$254$		0			0
$255$	−2.16073			−0.135310
$256$		0			0
$257$		−	16.9706i		−	1.05859i	−0.848436	−	0.529297i	$-0.822456\pi$
$257$		−	16.9706i		−	1.05859i	0.848436	−	0.529297i	$-0.177544\pi$
$258$		0			0
$259$	−9.82068	+	15.5279i	−0.610228	+	0.964855i
$260$		0			0
$261$	−4.47214			−0.276818
$262$		0			0
$263$		−	3.88854i		−	0.239778i	−0.992787	−	0.119889i	$-0.961746\pi$
$263$		−	3.88854i		−	0.239778i	0.992787	−	0.119889i	$-0.0382539\pi$
$264$		0			0
$265$		−	4.57649i		−	0.281132i
$266$		0			0
$267$		−	35.4164i		−	2.16745i
$268$		0			0
$269$		−	19.2588i		−	1.17423i	−0.809503	−	0.587115i	$-0.800264\pi$
$269$		−	19.2588i		−	1.17423i	0.809503	−	0.587115i	$-0.199736\pi$
$270$		0			0
$271$	26.1235			1.58689			0.793446	−	0.608640i	$-0.208285\pi$
$271$	26.1235			1.58689			0.793446	+	0.608640i	$0.208285\pi$
$272$		0			0
$273$	−11.7082	−	7.40492i	−0.708613	−	0.448166i
$274$		0			0
$275$		−	5.81966i		−	0.350939i
$276$		0			0
$277$	18.9443			1.13825			0.569125	−	0.822251i	$-0.307282\pi$
$277$	18.9443			1.13825			0.569125	+	0.822251i	$0.307282\pi$
$278$		0			0
$279$	8.74032			0.523269
$280$		0			0
$281$	−3.88854			−0.231971			−0.115986	−	0.993251i	$-0.537003\pi$
$281$	−3.88854			−0.231971			−0.115986	+	0.993251i	$0.537003\pi$
$282$		0			0
$283$	20.5942			1.22420			0.612099	−	0.790781i	$-0.290325\pi$
$283$	20.5942			1.22420			0.612099	+	0.790781i	$0.290325\pi$
$284$		0			0
$285$		−	6.32456i		−	0.374634i
$286$		0			0
$287$	22.8825	+	14.4721i	1.35071	+	0.854263i
$288$		0			0
$289$	13.9443			0.820251
$290$		0			0
$291$			4.00000i			0.234484i
$292$		0			0
$293$			28.4118i			1.65983i	0.557886	+	0.829917i	$0.311612\pi$
$293$			28.4118i			1.65983i	−0.557886	+	0.829917i	$0.688388\pi$
$294$		0			0
$295$		−	0.875388i		−	0.0509671i
$296$		0			0
$297$		−	2.16073i		−	0.125378i
$298$		0			0
$299$	7.40492			0.428237
$300$		0			0
$301$	26.1803	+	16.5579i	1.50901	+	0.954382i
$302$		0			0
$303$		−	26.1803i		−	1.50402i
$304$		0			0
$305$	7.12461			0.407954
$306$		0			0
$307$	−13.6020			−0.776305			−0.388152	−	0.921595i	$-0.626887\pi$
$307$	−13.6020			−0.776305			−0.388152	+	0.921595i	$0.626887\pi$
$308$		0			0
$309$	−31.4164			−1.78722
$310$		0			0
$311$	−14.1421			−0.801927			−0.400963	−	0.916094i	$-0.631325\pi$
$311$	−14.1421			−0.801927			−0.400963	+	0.916094i	$0.631325\pi$
$312$		0			0
$313$		−	8.89794i		−	0.502941i	−0.967865	−	0.251471i	$-0.919086\pi$
$313$		−	8.89794i		−	0.502941i	0.967865	−	0.251471i	$-0.0809142\pi$
$314$		0			0
$315$	2.70091	+	1.70820i	0.152179	+	0.0962464i
$316$		0			0
$317$	19.8885			1.11705			0.558526	−	0.829487i	$-0.311367\pi$
$317$	19.8885			1.11705			0.558526	+	0.829487i	$0.311367\pi$
$318$		0			0
$319$			2.47214i			0.138413i
$320$		0			0
$321$		−	0.667701i		−	0.0372674i
$322$		0			0
$323$		−	8.94427i		−	0.497673i
$324$		0			0
$325$		−	10.7735i		−	0.597608i
$326$		0			0
$327$	−8.07262			−0.446417
$328$		0			0
$329$	−15.4164	+	24.3755i	−0.849934	+	1.34386i
$330$		0			0
$331$		−	15.1246i		−	0.831324i	−0.909519	−	0.415662i	$-0.863550\pi$
$331$		−	15.1246i		−	0.831324i	0.909519	−	0.415662i	$-0.136450\pi$
$332$		0			0
$333$	15.5279			0.850922
$334$		0			0
$335$	−2.00310			−0.109441
$336$		0			0
$337$	25.1246			1.36862			0.684312	−	0.729189i	$-0.260102\pi$
$337$	25.1246			1.36862			0.684312	+	0.729189i	$0.260102\pi$
$338$		0			0
$339$	−5.24419			−0.284825
$340$		0			0
$341$		−	4.83153i		−	0.261642i
$342$		0			0
$343$	18.3848	+	2.23607i	0.992685	+	0.120736i
$344$		0			0
$345$	−4.00000			−0.215353
$346$		0			0
$347$			9.23607i			0.495818i	0.968783	+	0.247909i	$0.0797434\pi$
$347$			9.23607i			0.495818i	−0.968783	+	0.247909i	$0.920257\pi$
$348$		0			0
$349$		−	12.9343i		−	0.692355i	−0.938169	−	0.346177i	$-0.887480\pi$
$349$		−	12.9343i		−	0.692355i	0.938169	−	0.346177i	$-0.112520\pi$
$350$		0			0
$351$		−	4.00000i		−	0.213504i
$352$		0			0
$353$		−	16.1452i		−	0.859324i	−0.902990	−	0.429662i	$-0.858633\pi$
$353$		−	16.1452i		−	0.859324i	0.902990	−	0.429662i	$-0.141367\pi$
$354$		0			0
$355$	−5.40182			−0.286699
$356$		0			0
$357$	8.94427	+	5.65685i	0.473381	+	0.299392i
$358$		0			0
$359$			27.2361i			1.43746i	0.695287	+	0.718732i	$0.255277\pi$
$359$			27.2361i			1.43746i	−0.695287	+	0.718732i	$0.744723\pi$
$360$		0			0
$361$	7.18034			0.377913
$362$		0			0
$363$	−21.6746			−1.13762
$364$		0			0
$365$	−7.63932			−0.399860
$366$		0			0
$367$	5.65685			0.295285			0.147643	−	0.989041i	$-0.452831\pi$
$367$	5.65685			0.295285			0.147643	+	0.989041i	$0.452831\pi$
$368$		0			0
$369$		−	22.8825i		−	1.19121i
$370$		0			0
$371$	−11.9814	+	18.9443i	−0.622044	+	0.983538i
$372$		0			0
$373$	−5.05573			−0.261776			−0.130888	−	0.991397i	$-0.541783\pi$
$373$	−5.05573			−0.261776			−0.130888	+	0.991397i	$0.541783\pi$
$374$		0			0
$375$			12.0000i			0.619677i
$376$		0			0
$377$			4.57649i			0.235701i
$378$		0			0
$379$		−	8.29180i		−	0.425921i	−0.977061	−	0.212960i	$-0.931689\pi$
$379$		−	8.29180i		−	0.425921i	0.977061	−	0.212960i	$-0.0683106\pi$
$380$		0			0
$381$			1.74806i			0.0895560i
$382$		0			0
$383$	−29.2070			−1.49241			−0.746204	−	0.665717i	$-0.768126\pi$
$383$	−29.2070			−1.49241			−0.746204	+	0.665717i	$0.768126\pi$
$384$		0			0
$385$	0.944272	−	1.49302i	0.0481246	−	0.0760916i
$386$		0			0
$387$		−	26.1803i		−	1.33082i
$388$		0			0
$389$	−28.8328			−1.46188			−0.730941	−	0.682441i	$-0.760919\pi$
$389$	−28.8328			−1.46188			−0.730941	+	0.682441i	$0.760919\pi$
$390$		0			0
$391$	−5.65685			−0.286079
$392$		0			0
$393$	13.2361			0.667671
$394$		0			0
$395$	−1.90569			−0.0958855
$396$		0			0
$397$		−	31.2402i		−	1.56790i	−0.620823	−	0.783951i	$-0.713201\pi$
$397$		−	31.2402i		−	1.56790i	0.620823	−	0.783951i	$-0.286799\pi$
$398$		0			0
$399$	−16.5579	+	26.1803i	−0.828932	+	1.31066i
$400$		0			0
$401$	14.6525			0.731710			0.365855	−	0.930672i	$-0.380777\pi$
$401$	14.6525			0.731710			0.365855	+	0.930672i	$0.380777\pi$
$402$		0			0
$403$		−	8.94427i		−	0.445546i
$404$		0			0
$405$			5.78437i			0.287428i
$406$		0			0
$407$		−	8.58359i		−	0.425473i
$408$		0			0
$409$		−	5.91189i		−	0.292324i	−0.989261	−	0.146162i	$-0.953308\pi$
$409$		−	5.91189i		−	0.292324i	0.989261	−	0.146162i	$-0.0466921\pi$
$410$		0			0
$411$	28.5393			1.40774
$412$		0			0
$413$	−2.29180	+	3.62365i	−0.112772	+	0.178308i
$414$		0			0
$415$		−	4.87539i		−	0.239323i
$416$		0			0
$417$	6.18034			0.302653
$418$		0			0
$419$	−22.7549			−1.11165			−0.555826	−	0.831299i	$-0.687598\pi$
$419$	−22.7549			−1.11165			−0.555826	+	0.831299i	$0.687598\pi$
$420$		0			0
$421$	17.4164			0.848824			0.424412	−	0.905469i	$-0.360481\pi$
$421$	17.4164			0.848824			0.424412	+	0.905469i	$0.360481\pi$
$422$		0			0
$423$	24.3755			1.18518
$424$		0			0
$425$			8.23024i			0.399225i
$426$		0			0
$427$	−29.4922	−	18.6525i	−1.42723	−	0.902657i
$428$		0			0
$429$	6.47214			0.312478
$430$		0			0
$431$			2.29180i			0.110392i	0.998476	+	0.0551960i	$0.0175784\pi$
$431$			2.29180i			0.110392i	−0.998476	+	0.0551960i	$0.982422\pi$
$432$		0			0
$433$			14.3972i			0.691884i	0.938256	+	0.345942i	$0.112441\pi$
$433$			14.3972i			0.691884i	−0.938256	+	0.345942i	$0.887559\pi$
$434$		0			0
$435$		−	2.47214i		−	0.118530i
$436$		0			0
$437$		−	16.5579i		−	0.792072i
$438$		0			0
$439$	−26.7912			−1.27868			−0.639338	−	0.768926i	$-0.720792\pi$
$439$	−26.7912			−1.27868			−0.639338	+	0.768926i	$0.720792\pi$
$440$		0			0
$441$	−6.70820	−	14.1421i	−0.319438	−	0.673435i
$442$		0			0
$443$			31.7082i			1.50650i	0.657733	+	0.753251i	$0.271515\pi$
$443$			31.7082i			1.50650i	−0.657733	+	0.753251i	$0.728485\pi$
$444$		0			0
$445$	8.36068			0.396334
$446$		0			0
$447$	30.7000			1.45206
$448$		0			0
$449$	30.3607			1.43281			0.716405	−	0.697685i	$-0.245787\pi$
$449$	30.3607			1.43281			0.716405	+	0.697685i	$0.245787\pi$
$450$		0			0
$451$	−12.6491			−0.595623
$452$		0			0
$453$		−	25.7109i		−	1.20800i
$454$		0			0
$455$	1.74806	−	2.76393i	0.0819505	−	0.129575i
$456$		0			0
$457$	30.6525			1.43386			0.716931	−	0.697144i	$-0.245546\pi$
$457$	30.6525			1.43386			0.716931	+	0.697144i	$0.245546\pi$
$458$		0			0
$459$			3.05573i			0.142629i
$460$		0			0
$461$		−	17.7658i		−	0.827435i	−0.910405	−	0.413718i	$-0.864230\pi$
$461$		−	17.7658i		−	0.827435i	0.910405	−	0.413718i	$-0.135770\pi$
$462$		0			0
$463$			31.8885i			1.48199i	0.671513	+	0.740993i	$0.265645\pi$
$463$			31.8885i			1.48199i	−0.671513	+	0.740993i	$0.734355\pi$
$464$		0			0
$465$			4.83153i			0.224057i
$466$		0			0
$467$	0.540182			0.0249966			0.0124983	−	0.999922i	$-0.496022\pi$
$467$	0.540182			0.0249966			0.0124983	+	0.999922i	$0.496022\pi$
$468$		0			0
$469$	8.29180	+	5.24419i	0.382880	+	0.242154i
$470$		0			0
$471$			26.1803i			1.20633i
$472$		0			0
$473$	−14.4721			−0.665430
$474$		0			0
$475$	−24.0903			−1.10534
$476$		0			0
$477$	18.9443			0.867399
$478$		0			0
$479$	−27.8716			−1.27349			−0.636743	−	0.771076i	$-0.719719\pi$
$479$	−27.8716			−1.27349			−0.636743	+	0.771076i	$0.719719\pi$
$480$		0			0
$481$		−	15.8902i		−	0.724531i
$482$		0			0
$483$	16.5579	+	10.4721i	0.753411	+	0.476499i
$484$		0			0
$485$	−0.944272			−0.0428772
$486$		0			0
$487$			34.0689i			1.54381i	0.635739	+	0.771904i	$0.280696\pi$
$487$			34.0689i			1.54381i	−0.635739	+	0.771904i	$0.719304\pi$
$488$		0			0
$489$			39.4404i			1.78355i
$490$		0			0
$491$		−	27.7082i		−	1.25045i	−0.780443	−	0.625227i	$-0.785006\pi$
$491$		−	27.7082i		−	1.25045i	0.780443	−	0.625227i	$-0.214994\pi$
$492$		0			0
$493$		−	3.49613i		−	0.157458i
$494$		0			0
$495$	−1.49302			−0.0671065
$496$		0			0
$497$	22.3607	+	14.1421i	1.00301	+	0.634361i
$498$		0			0
$499$		−	25.2361i		−	1.12972i	−0.825186	−	0.564861i	$-0.808930\pi$
$499$		−	25.2361i		−	1.12972i	0.825186	−	0.564861i	$-0.191070\pi$
$500$		0			0
$501$	18.4721			0.825274
$502$		0			0
$503$	28.1266			1.25411			0.627053	−	0.778977i	$-0.284261\pi$
$503$	28.1266			1.25411			0.627053	+	0.778977i	$0.284261\pi$
$504$		0			0
$505$	6.18034			0.275022
$506$		0			0
$507$	−17.7658			−0.789006
$508$		0			0
$509$		−	25.5834i		−	1.13396i	−0.823731	−	0.566981i	$-0.808111\pi$
$509$		−	25.5834i		−	1.13396i	0.823731	−	0.566981i	$-0.191889\pi$
$510$		0			0
$511$	31.6228	+	20.0000i	1.39891	+	0.884748i
$512$		0			0
$513$	−8.94427			−0.394899
$514$		0			0
$515$		−	7.41641i		−	0.326806i
$516$		0			0
$517$		−	13.4744i		−	0.592605i
$518$		0			0
$519$		−	41.5967i		−	1.82589i
$520$		0			0
$521$			15.0649i			0.660004i	0.943980	+	0.330002i	$0.107049\pi$
$521$			15.0649i			0.660004i	−0.943980	+	0.330002i	$0.892951\pi$
$522$		0			0
$523$	−15.3500			−0.671209			−0.335605	−	0.942003i	$-0.608941\pi$
$523$	−15.3500			−0.671209			−0.335605	+	0.942003i	$0.608941\pi$
$524$		0			0
$525$	15.2361	−	24.0903i	0.664957	−	1.05139i
$526$		0			0
$527$			6.83282i			0.297642i
$528$		0			0
$529$	12.5279			0.544690
$530$		0			0
$531$	3.62365			0.157253
$532$		0			0
$533$	−23.4164			−1.01428
$534$		0			0
$535$	0.157623			0.00681463
$536$		0			0
$537$		−	47.2579i		−	2.03933i
$538$		0			0
$539$	−7.81758	+	3.70820i	−0.336727	+	0.159724i
$540$		0			0
$541$	−8.47214			−0.364246			−0.182123	−	0.983276i	$-0.558297\pi$
$541$	−8.47214			−0.364246			−0.182123	+	0.983276i	$0.558297\pi$
$542$		0			0
$543$			1.23607i			0.0530448i
$544$		0			0
$545$		−	1.90569i		−	0.0816307i
$546$		0			0
$547$		−	3.12461i		−	0.133599i	−0.997766	−	0.0667994i	$-0.978721\pi$
$547$		−	3.12461i		−	0.133599i	0.997766	−	0.0667994i	$-0.0212787\pi$
$548$		0			0
$549$			29.4922i			1.25869i
$550$		0			0
$551$	10.2333			0.435955
$552$		0			0
$553$	7.88854	+	4.98915i	0.335455	+	0.212160i
$554$		0			0
$555$			8.58359i			0.364353i
$556$		0			0
$557$	42.3607			1.79488			0.897440	−	0.441137i	$-0.145425\pi$
$557$	42.3607			1.79488			0.897440	+	0.441137i	$0.145425\pi$
$558$		0			0
$559$	−26.7912			−1.13315
$560$		0			0
$561$	−4.94427			−0.208747
$562$		0			0
$563$	35.8167			1.50949			0.754747	−	0.656016i	$-0.227760\pi$
$563$	35.8167			1.50949			0.754747	+	0.656016i	$0.227760\pi$
$564$		0			0
$565$		−	1.23799i		−	0.0520825i
$566$		0			0
$567$	15.1437	−	23.9443i	0.635975	−	1.00556i
$568$		0			0
$569$	−34.0689			−1.42824			−0.714121	−	0.700022i	$-0.753173\pi$
$569$	−34.0689			−1.42824			−0.714121	+	0.700022i	$0.753173\pi$
$570$		0			0
$571$			14.1803i			0.593429i	0.954966	+	0.296714i	$0.0958909\pi$
$571$			14.1803i			0.593429i	−0.954966	+	0.296714i	$0.904109\pi$
$572$		0			0
$573$			52.5021i			2.19331i
$574$		0			0
$575$			15.2361i			0.635388i
$576$		0			0
$577$		−	35.2765i		−	1.46858i	−0.678835	−	0.734291i	$-0.737515\pi$
$577$		−	35.2765i		−	1.46858i	0.678835	−	0.734291i	$-0.262485\pi$
$578$		0			0
$579$	25.7109			1.06851
$580$		0			0
$581$	−12.7639	+	20.1815i	−0.529537	+	0.837272i
$582$		0			0
$583$		−	10.4721i		−	0.433712i
$584$		0			0
$585$	−2.76393			−0.114275
$586$		0			0
$587$	−8.86784			−0.366015			−0.183007	−	0.983112i	$-0.558583\pi$
$587$	−8.86784			−0.366015			−0.183007	+	0.983112i	$0.558583\pi$
$588$		0			0
$589$	−20.0000			−0.824086
$590$		0			0
$591$	43.3491			1.78315
$592$		0			0
$593$			11.3137i			0.464598i	0.972644	+	0.232299i	$0.0746248\pi$
$593$			11.3137i			0.464598i	−0.972644	+	0.232299i	$0.925375\pi$
$594$		0			0
$595$	−1.33540	+	2.11146i	−0.0547462	+	0.0865613i
$596$		0			0
$597$	15.4164			0.630952
$598$		0			0
$599$		−	3.52786i		−	0.144145i	−0.997399	−	0.0720723i	$-0.977039\pi$
$599$		−	3.52786i		−	0.144145i	0.997399	−	0.0720723i	$-0.0229612\pi$
$600$		0			0
$601$		−	24.6305i		−	1.00470i	−0.864664	−	0.502350i	$-0.832469\pi$
$601$		−	24.6305i		−	1.00470i	0.864664	−	0.502350i	$-0.167531\pi$
$602$		0			0
$603$		−	8.29180i		−	0.337668i
$604$		0			0
$605$		−	5.11667i		−	0.208022i
$606$		0			0
$607$	−23.1375			−0.939122			−0.469561	−	0.882900i	$-0.655588\pi$
$607$	−23.1375			−0.939122			−0.469561	+	0.882900i	$0.655588\pi$
$608$		0			0
$609$	−6.47214	+	10.2333i	−0.262264	+	0.414676i
$610$		0			0
$611$		−	24.9443i		−	1.00914i
$612$		0			0
$613$	9.41641			0.380325			0.190163	−	0.981753i	$-0.439098\pi$
$613$	9.41641			0.380325			0.190163	+	0.981753i	$0.439098\pi$
$614$		0			0
$615$	12.6491			0.510061
$616$		0			0
$617$	7.23607			0.291313			0.145657	−	0.989335i	$-0.453471\pi$
$617$	7.23607			0.291313			0.145657	+	0.989335i	$0.453471\pi$
$618$		0			0
$619$	−4.70401			−0.189070			−0.0945351	−	0.995522i	$-0.530136\pi$
$619$	−4.70401			−0.189070			−0.0945351	+	0.995522i	$0.530136\pi$
$620$		0			0
$621$			5.65685i			0.227002i
$622$		0			0
$623$	−34.6088	−	21.8885i	−1.38657	−	0.876946i
$624$		0			0
$625$	20.7082			0.828328
$626$		0			0
$627$		−	14.4721i		−	0.577961i
$628$		0			0
$629$			12.1390i			0.484015i
$630$		0			0
$631$		−	0.472136i		−	0.0187954i	−0.999956	−	0.00939772i	$-0.997009\pi$
$631$		−	0.472136i		−	0.0187954i	0.999956	−	0.00939772i	$-0.00299143\pi$
$632$		0			0
$633$		−	2.82843i		−	0.112420i
$634$		0			0
$635$	−0.412662			−0.0163760
$636$		0			0
$637$	−14.4721	+	6.86474i	−0.573407	+	0.271991i
$638$		0			0
$639$		−	22.3607i		−	0.884575i
$640$		0			0
$641$	−16.1803			−0.639085			−0.319543	−	0.947572i	$-0.603529\pi$
$641$	−16.1803			−0.639085			−0.319543	+	0.947572i	$0.603529\pi$
$642$		0			0
$643$	16.6854			0.658009			0.329004	−	0.944328i	$-0.393287\pi$
$643$	16.6854			0.658009			0.329004	+	0.944328i	$0.393287\pi$
$644$		0			0
$645$	14.4721			0.569840
$646$		0			0
$647$	−2.41577			−0.0949735			−0.0474868	−	0.998872i	$-0.515121\pi$
$647$	−2.41577			−0.0949735			−0.0474868	+	0.998872i	$0.515121\pi$
$648$		0			0
$649$		−	2.00310i		−	0.0786287i
$650$		0			0
$651$	12.6491	−	20.0000i	0.495758	−	0.783862i
$652$		0			0
$653$	−13.4164			−0.525025			−0.262512	−	0.964929i	$-0.584551\pi$
$653$	−13.4164			−0.525025			−0.262512	+	0.964929i	$0.584551\pi$
$654$		0			0
$655$			3.12461i			0.122089i
$656$		0			0
$657$		−	31.6228i		−	1.23372i
$658$		0			0
$659$			41.0132i			1.59765i	0.601566	+	0.798823i	$0.294544\pi$
$659$			41.0132i			1.59765i	−0.601566	+	0.798823i	$0.705456\pi$
$660$		0			0
$661$		−	26.5061i		−	1.03097i	−0.856899	−	0.515484i	$-0.827612\pi$
$661$		−	26.5061i		−	1.03097i	0.856899	−	0.515484i	$-0.172388\pi$
$662$		0			0
$663$	−9.15298			−0.355472
$664$		0			0
$665$	−6.18034	−	3.90879i	−0.239663	−	0.151576i
$666$		0			0
$667$		−	6.47214i		−	0.250602i
$668$		0			0
$669$	−36.9443			−1.42835
$670$		0			0
$671$	16.3029			0.629365
$672$		0			0
$673$	−40.8328			−1.57399			−0.786995	−	0.616960i	$-0.788364\pi$
$673$	−40.8328			−1.57399			−0.786995	+	0.616960i	$0.788364\pi$
$674$		0			0
$675$	8.23024			0.316782
$676$		0			0
$677$			32.5756i			1.25198i	0.779830	+	0.625991i	$0.215306\pi$
$677$			32.5756i			1.25198i	−0.779830	+	0.625991i	$0.784694\pi$
$678$		0			0
$679$	3.90879	+	2.47214i	0.150006	+	0.0948719i
$680$		0			0
$681$	−4.29180			−0.164462
$682$		0			0
$683$			10.1803i			0.389540i	0.980849	+	0.194770i	$0.0623960\pi$
$683$			10.1803i			0.389540i	−0.980849	+	0.194770i	$0.937604\pi$
$684$		0			0
$685$			6.73722i			0.257416i
$686$		0			0
$687$			2.76393i			0.105451i
$688$		0			0
$689$		−	19.3863i		−	0.738560i
$690$		0			0
$691$	−6.60970			−0.251445			−0.125722	−	0.992065i	$-0.540125\pi$
$691$	−6.60970			−0.251445			−0.125722	+	0.992065i	$0.540125\pi$
$692$		0			0
$693$	6.18034	+	3.90879i	0.234772	+	0.148483i
$694$		0			0
$695$			1.45898i			0.0553423i
$696$		0			0
$697$	17.8885			0.677577
$698$		0			0
$699$	30.7000			1.16118
$700$		0			0
$701$	31.5279			1.19079			0.595395	−	0.803433i	$-0.296995\pi$
$701$	31.5279			1.19079			0.595395	+	0.803433i	$0.296995\pi$
$702$		0			0
$703$	−35.5316			−1.34010
$704$		0			0
$705$			13.4744i			0.507477i
$706$		0			0
$707$	−25.5834	−	16.1803i	−0.962161	−	0.608524i
$708$		0			0
$709$	−12.4721			−0.468401			−0.234200	−	0.972188i	$-0.575247\pi$
$709$	−12.4721			−0.468401			−0.234200	+	0.972188i	$0.575247\pi$
$710$		0			0
$711$		−	7.88854i		−	0.295844i
$712$		0			0
$713$			12.6491i			0.473713i
$714$		0			0
$715$			1.52786i			0.0571389i
$716$		0			0
$717$			42.6814i			1.59397i
$718$		0			0
$719$	45.3523			1.69135			0.845677	−	0.533695i	$-0.179197\pi$
$719$	45.3523			1.69135			0.845677	+	0.533695i	$0.179197\pi$
$720$		0			0
$721$	−19.4164	+	30.7000i	−0.723105	+	1.14333i
$722$		0			0
$723$		−	4.00000i		−	0.148762i
$724$		0			0
$725$	−9.41641			−0.349717
$726$		0			0
$727$	−2.41577			−0.0895958			−0.0447979	−	0.998996i	$-0.514264\pi$
$727$	−2.41577			−0.0895958			−0.0447979	+	0.998996i	$0.514264\pi$
$728$		0			0
$729$	−11.9443			−0.442380
$730$		0			0
$731$	20.4667			0.756988
$732$		0			0
$733$			7.02236i			0.259377i	0.991555	+	0.129688i	$0.0413977\pi$
$733$			7.02236i			0.259377i	−0.991555	+	0.129688i	$0.958602\pi$
$734$		0			0
$735$	7.81758	−	3.70820i	0.288356	−	0.136779i
$736$		0			0
$737$	−4.58359			−0.168839
$738$		0			0
$739$		−	2.40325i		−	0.0884051i	−0.999023	−	0.0442025i	$-0.985925\pi$
$739$		−	2.40325i		−	0.0884051i	0.999023	−	0.0442025i	$-0.0140747\pi$
$740$		0			0
$741$		−	26.7912i		−	0.984201i
$742$		0			0
$743$		−	17.7082i		−	0.649651i	−0.945774	−	0.324825i	$-0.894694\pi$
$743$		−	17.7082i		−	0.649651i	0.945774	−	0.324825i	$-0.105306\pi$
$744$		0			0
$745$			7.24730i			0.265520i
$746$		0			0
$747$	20.1815			0.738404
$748$		0			0
$749$	−0.652476	−	0.412662i	−0.0238409	−	0.0150783i
$750$		0			0
$751$		−	52.5410i		−	1.91725i	−0.284674	−	0.958625i	$-0.591885\pi$
$751$		−	52.5410i		−	1.91725i	0.284674	−	0.958625i	$-0.408115\pi$
$752$		0			0
$753$	−44.0689			−1.60596
$754$		0			0
$755$	6.06952			0.220892
$756$		0			0
$757$	22.3607			0.812713			0.406356	−	0.913715i	$-0.366799\pi$
$757$	22.3607			0.812713			0.406356	+	0.913715i	$0.366799\pi$
$758$		0			0
$759$	−9.15298			−0.332232
$760$		0			0
$761$			8.07262i			0.292632i	0.989238	+	0.146316i	$0.0467417\pi$
$761$			8.07262i			0.292632i	−0.989238	+	0.146316i	$0.953258\pi$
$762$		0			0
$763$	−4.98915	+	7.88854i	−0.180619	+	0.285584i
$764$		0			0
$765$	2.11146			0.0763399
$766$		0			0
$767$		−	3.70820i		−	0.133895i
$768$		0			0
$769$			8.23024i			0.296790i	0.988928	+	0.148395i	$0.0474107\pi$
$769$			8.23024i			0.296790i	−0.988928	+	0.148395i	$0.952589\pi$
$770$		0			0
$771$			38.8328i			1.39853i
$772$		0			0
$773$		−	49.8012i		−	1.79123i	−0.444835	−	0.895613i	$-0.646738\pi$
$773$		−	49.8012i		−	1.79123i	0.444835	−	0.895613i	$-0.353262\pi$
$774$		0			0
$775$	18.4034			0.661069
$776$		0			0
$777$	22.4721	−	35.5316i	0.806183	−	1.27469i
$778$		0			0
$779$			52.3607i			1.87602i
$780$		0			0
$781$	−12.3607			−0.442300
$782$		0			0
$783$	−3.49613			−0.124941
$784$		0			0
$785$	−6.18034			−0.220586
$786$		0			0
$787$	−10.7735			−0.384035			−0.192017	−	0.981392i	$-0.561503\pi$
$787$	−10.7735			−0.384035			−0.192017	+	0.981392i	$0.561503\pi$
$788$		0			0
$789$			8.89794i			0.316775i
$790$		0			0
$791$	−3.24109	+	5.12461i	−0.115240	+	0.182210i
$792$		0			0
$793$	30.1803			1.07174
$794$		0			0
$795$			10.4721i			0.371408i
$796$		0			0
$797$		−	13.6020i		−	0.481806i	−0.970549	−	0.240903i	$-0.922556\pi$
$797$		−	13.6020i		−	0.481806i	0.970549	−	0.240903i	$-0.0774436\pi$
$798$		0			0
$799$			19.0557i			0.674143i
$800$		0			0
$801$			34.6088i			1.22284i
$802$		0			0
$803$	−17.4806			−0.616878
$804$		0			0
$805$	−2.47214	+	3.90879i	−0.0871313	+	0.137767i
$806$		0			0
$807$			44.0689i			1.55130i
$808$		0			0
$809$	−2.06888			−0.0727381			−0.0363690	−	0.999338i	$-0.511579\pi$
$809$	−2.06888			−0.0727381			−0.0363690	+	0.999338i	$0.511579\pi$
$810$		0			0
$811$	36.7394			1.29010			0.645048	−	0.764142i	$-0.276837\pi$
$811$	36.7394			1.29010			0.645048	+	0.764142i	$0.276837\pi$
$812$		0			0
$813$	−59.7771			−2.09647
$814$		0			0
$815$	−9.31061			−0.326136
$816$		0			0
$817$			59.9070i			2.09588i
$818$		0			0
$819$	11.4412	+	7.23607i	0.399789	+	0.252849i
$820$		0			0
$821$	−11.8885			−0.414913			−0.207457	−	0.978244i	$-0.566519\pi$
$821$	−11.8885			−0.414913			−0.207457	+	0.978244i	$0.566519\pi$
$822$		0			0
$823$		−	16.8328i		−	0.586755i	−0.955997	−	0.293378i	$-0.905221\pi$
$823$		−	16.8328i		−	0.586755i	0.955997	−	0.293378i	$-0.0947793\pi$
$824$		0			0
$825$			13.3168i			0.463632i
$826$		0			0
$827$		−	24.0689i		−	0.836957i	−0.908227	−	0.418479i	$-0.862564\pi$
$827$		−	24.0689i		−	0.836957i	0.908227	−	0.418479i	$-0.137436\pi$
$828$		0			0
$829$			35.8167i			1.24397i	0.783031	+	0.621983i	$0.213673\pi$
$829$			35.8167i			1.24397i	−0.783031	+	0.621983i	$0.786327\pi$
$830$		0			0
$831$	−43.3491			−1.50377
$832$		0			0
$833$	11.0557	−	5.24419i	0.383058	−	0.181700i
$834$		0			0
$835$			4.36068i			0.150908i
$836$		0			0
$837$	6.83282			0.236177
$838$		0			0
$839$	−54.6629			−1.88717			−0.943586	−	0.331128i	$-0.892571\pi$
$839$	−54.6629			−1.88717			−0.943586	+	0.331128i	$0.892571\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$	8.89794			0.306461
$844$		0			0
$845$		−	4.19393i		−	0.144276i
$846$		0			0
$847$	−13.3956	+	21.1803i	−0.460279	+	0.727765i
$848$		0			0
$849$	−47.1246			−1.61731
$850$		0			0
$851$			22.4721i			0.770335i
$852$		0			0
$853$		−	4.19393i		−	0.143598i	−0.997419	−	0.0717988i	$-0.977126\pi$
$853$		−	4.19393i		−	0.143598i	0.997419	−	0.0717988i	$-0.0228739\pi$
$854$		0			0
$855$			6.18034i			0.211363i
$856$		0			0
$857$			43.3491i			1.48078i	0.672178	+	0.740389i	$0.265359\pi$
$857$			43.3491i			1.48078i	−0.672178	+	0.740389i	$0.734641\pi$
$858$		0			0
$859$	−18.6885			−0.637644			−0.318822	−	0.947815i	$-0.603287\pi$
$859$	−18.6885			−0.637644			−0.318822	+	0.947815i	$0.603287\pi$
$860$		0			0
$861$	−52.3607	−	33.1158i	−1.78445	−	1.12858i
$862$		0			0
$863$		−	10.0000i		−	0.340404i	−0.985409	−	0.170202i	$-0.945558\pi$
$863$		−	10.0000i		−	0.340404i	0.985409	−	0.170202i	$-0.0544420\pi$
$864$		0			0
$865$	9.81966			0.333878
$866$		0			0
$867$	−31.9079			−1.08365
$868$		0			0
$869$	−4.36068			−0.147926
$870$		0			0
$871$	−8.48528			−0.287513
$872$		0			0
$873$		−	3.90879i		−	0.132293i
$874$		0			0
$875$	11.7264	+	7.41641i	0.396424	+	0.250720i
$876$		0			0
$877$	−30.0000			−1.01303			−0.506514	−	0.862232i	$-0.669066\pi$
$877$	−30.0000			−1.01303			−0.506514	+	0.862232i	$0.669066\pi$
$878$		0			0
$879$		−	65.0132i		−	2.19284i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$			32.2918i			1.08671i	0.839505	+	0.543353i	$0.182845\pi$
$883$			32.2918i			1.08671i	−0.839505	+	0.543353i	$0.817155\pi$
$884$		0			0
$885$			2.00310i			0.0673336i
$886$		0			0
$887$	−2.41577			−0.0811135			−0.0405567	−	0.999177i	$-0.512913\pi$
$887$	−2.41577			−0.0811135			−0.0405567	+	0.999177i	$0.512913\pi$
$888$		0			0
$889$	1.70820	+	1.08036i	0.0572913	+	0.0362342i
$890$		0			0
$891$			13.2361i			0.443425i
$892$		0			0
$893$	−55.7771			−1.86651
$894$		0			0
$895$	11.1561			0.372907
$896$		0			0
$897$	−16.9443			−0.565753
$898$		0			0
$899$	−7.81758			−0.260731
$900$		0			0
$901$			14.8098i			0.493387i
$902$		0			0
$903$	−59.9070	−	37.8885i	−1.99358	−	1.26085i
$904$		0			0
$905$	−0.291796			−0.00969963
$906$		0			0
$907$		−	0.291796i		−	0.00968893i	−0.999988	−	0.00484446i	$-0.998458\pi$
$907$		−	0.291796i		−	0.00968893i	0.999988	−	0.00484446i	$-0.00154205\pi$
$908$		0			0
$909$			25.5834i			0.848547i
$910$		0			0
$911$			25.1246i			0.832416i	0.909270	+	0.416208i	$0.136641\pi$
$911$			25.1246i			0.832416i	−0.909270	+	0.416208i	$0.863359\pi$
$912$		0			0
$913$		−	11.1561i		−	0.369213i
$914$		0			0
$915$	−16.3029			−0.538956
$916$		0			0
$917$	8.18034	−	12.9343i	0.270139	−	0.427127i
$918$		0			0
$919$		−	6.94427i		−	0.229070i	−0.993419	−	0.114535i	$-0.963462\pi$
$919$		−	6.94427i		−	0.229070i	0.993419	−	0.114535i	$-0.0365379\pi$
$920$		0			0
$921$	31.1246			1.02559
$922$		0			0
$923$	−22.8825			−0.753185
$924$		0			0
$925$	32.6950			1.07501
$926$		0			0
$927$	30.7000			1.00832
$928$		0			0
$929$			45.3523i			1.48796i	0.668202	+	0.743980i	$0.267064\pi$
$929$			45.3523i			1.48796i	−0.668202	+	0.743980i	$0.732936\pi$
$930$		0			0
$931$	15.3500	+	32.3607i	0.503077	+	1.06058i
$932$		0			0
$933$	32.3607			1.05944
$934$		0			0
$935$		−	1.16718i		−	0.0381710i
$936$		0			0
$937$		−	45.0972i		−	1.47326i	−0.676295	−	0.736631i	$-0.736416\pi$
$937$		−	45.0972i		−	1.47326i	0.676295	−	0.736631i	$-0.263584\pi$
$938$		0			0
$939$			20.3607i			0.664446i
$940$		0			0
$941$			39.9805i			1.30333i	0.758508	+	0.651664i	$0.225929\pi$
$941$			39.9805i			1.30333i	−0.758508	+	0.651664i	$0.774071\pi$
$942$		0			0
$943$	33.1158			1.07840
$944$		0			0
$945$	2.11146	+	1.33540i	0.0686857	+	0.0434406i
$946$		0			0
$947$			15.1246i			0.491484i	0.969335	+	0.245742i	$0.0790316\pi$
$947$			15.1246i			0.491484i	−0.969335	+	0.245742i	$0.920968\pi$
$948$		0			0
$949$	−32.3607			−1.05047
$950$		0			0
$951$	−45.5099			−1.47576
$952$		0			0
$953$	−1.05573			−0.0341984			−0.0170992	−	0.999854i	$-0.505443\pi$
$953$	−1.05573			−0.0341984			−0.0170992	+	0.999854i	$0.505443\pi$
$954$		0			0
$955$	−12.3941			−0.401063
$956$		0			0
$957$		−	5.65685i		−	0.182860i
$958$		0			0
$959$	17.6383	−	27.8885i	0.569569	−	0.900568i
$960$		0			0
$961$	−15.7214			−0.507141
$962$		0			0
$963$			0.652476i			0.0210257i
$964$		0			0
$965$			6.06952i			0.195385i
$966$		0			0
$967$			17.3475i			0.557859i	0.960312	+	0.278929i	$0.0899795\pi$
$967$			17.3475i			0.557859i	−0.960312	+	0.278929i	$0.910020\pi$
$968$		0			0
$969$			20.4667i			0.657485i
$970$		0			0
$971$	−20.1815			−0.647657			−0.323828	−	0.946116i	$-0.604970\pi$
$971$	−20.1815			−0.647657			−0.323828	+	0.946116i	$0.604970\pi$
$972$		0			0
$973$	3.81966	−	6.03941i	0.122453	−	0.193615i
$974$		0			0
$975$			24.6525i			0.789511i
$976$		0			0
$977$	33.4164			1.06909			0.534543	−	0.845141i	$-0.320484\pi$
$977$	33.4164			1.06909			0.534543	+	0.845141i	$0.320484\pi$
$978$		0			0
$979$	19.1313			0.611439
$980$		0			0
$981$	7.88854			0.251862
$982$		0			0
$983$	−1.90569			−0.0607820			−0.0303910	−	0.999538i	$-0.509675\pi$
$983$	−1.90569			−0.0607820			−0.0303910	+	0.999538i	$0.509675\pi$
$984$		0			0
$985$			10.2333i			0.326061i
$986$		0			0
$987$	35.2765	−	55.7771i	1.12286	−	1.77540i
$988$		0			0
$989$	37.8885			1.20479
$990$		0			0
$991$		−	34.3607i		−	1.09150i	−0.837947	−	0.545751i	$-0.816244\pi$
$991$		−	34.3607i		−	1.09150i	0.837947	−	0.545751i	$-0.183756\pi$
$992$		0			0
$993$			34.6088i			1.09828i
$994$		0			0
$995$			3.63932i			0.115374i
$996$		0			0
$997$		−	37.9774i		−	1.20276i	−0.798964	−	0.601379i	$-0.794618\pi$
$997$		−	37.9774i		−	1.20276i	0.798964	−	0.601379i	$-0.205382\pi$
$998$		0			0
$999$	12.1390			0.384062

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.f.b.895.1	yes	8
4.3	odd	2	inner	896.2.f.b.895.7	yes	8
7.6	odd	2	inner	896.2.f.b.895.8	yes	8
8.3	odd	2		896.2.f.a.895.2	yes	8
8.5	even	2		896.2.f.a.895.8	yes	8
16.3	odd	4		1792.2.e.h.895.7		8
16.5	even	4		1792.2.e.h.895.8		8
16.11	odd	4		1792.2.e.i.895.2		8
16.13	even	4		1792.2.e.i.895.1		8
28.27	even	2	inner	896.2.f.b.895.2	yes	8
56.13	odd	2		896.2.f.a.895.1	✓	8
56.27	even	2		896.2.f.a.895.7	yes	8
112.13	odd	4		1792.2.e.i.895.8		8
112.27	even	4		1792.2.e.i.895.7		8
112.69	odd	4		1792.2.e.h.895.1		8
112.83	even	4		1792.2.e.h.895.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.f.a.895.1	✓	8	56.13	odd	2
896.2.f.a.895.2	yes	8	8.3	odd	2
896.2.f.a.895.7	yes	8	56.27	even	2
896.2.f.a.895.8	yes	8	8.5	even	2
896.2.f.b.895.1	yes	8	1.1	even	1	trivial
896.2.f.b.895.2	yes	8	28.27	even	2	inner
896.2.f.b.895.7	yes	8	4.3	odd	2	inner
896.2.f.b.895.8	yes	8	7.6	odd	2	inner
1792.2.e.h.895.1		8	112.69	odd	4
1792.2.e.h.895.2		8	112.83	even	4
1792.2.e.h.895.7		8	16.3	odd	4
1792.2.e.h.895.8		8	16.5	even	4
1792.2.e.i.895.1		8	16.13	even	4
1792.2.e.i.895.2		8	16.11	odd	4
1792.2.e.i.895.7		8	112.27	even	4
1792.2.e.i.895.8		8	112.13	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:	\( N \)	\(=\)	\( 896 = 2^{7} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	896.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.28825 q^{3} -0.540182i q^{5} +(-1.41421 + 2.23607i) q^{7} +2.23607 q^{9} +O(q^{10})\)	\(q-2.28825 q^{3} -0.540182i q^{5} +(-1.41421 + 2.23607i) q^{7} +2.23607 q^{9} -1.23607i q^{11} -2.28825i q^{13} +1.23607i q^{15} +1.74806i q^{17} -5.11667 q^{19} +(3.23607 - 5.11667i) q^{21} +3.23607i q^{23} +4.70820 q^{25} +1.74806 q^{27} -2.00000 q^{29} +3.90879 q^{31} +2.82843i q^{33} +(1.20788 + 0.763932i) q^{35} +6.94427 q^{37} +5.23607i q^{39} -10.2333i q^{41} -11.7082i q^{43} -1.20788i q^{45} +10.9010 q^{47} +(-3.00000 - 6.32456i) q^{49} -4.00000i q^{51} +8.47214 q^{53} -0.667701 q^{55} +11.7082 q^{57} +1.62054 q^{59} +13.1893i q^{61} +(-3.16228 + 5.00000i) q^{63} -1.23607 q^{65} -3.70820i q^{67} -7.40492i q^{69} -10.0000i q^{71} -14.1421i q^{73} -10.7735 q^{75} +(2.76393 + 1.74806i) q^{77} -3.52786i q^{79} -10.7082 q^{81} +9.02546 q^{83} +0.944272 q^{85} +4.57649 q^{87} +15.4775i q^{89} +(5.11667 + 3.23607i) q^{91} -8.94427 q^{93} +2.76393i q^{95} -1.74806i q^{97} -2.76393i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 8 q^{21} - 16 q^{25} - 16 q^{29} - 16 q^{37} - 24 q^{49} + 32 q^{53} + 40 q^{57} + 8 q^{65} + 40 q^{77} - 32 q^{81} - 64 q^{85}+O(q^{100}) \) 8 * q + 8 * q^21 - 16 * q^25 - 16 * q^29 - 16 * q^37 - 24 * q^49 + 32 * q^53 + 40 * q^57 + 8 * q^65 + 40 * q^77 - 32 * q^81 - 64 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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