LMFDB - Embedded newform 1148.2.d.a.1065.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1148,2,Mod(1065,1148)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1148, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1148.1065");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1148 = 2^{2} \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1148.d (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:	$9.16682615204$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 39 x^{18} + 531 x^{16} + 3488 x^{14} + 12661 x^{12} + 27027 x^{10} + 34540 x^{8} + 25909 x^{6} + \cdots + 144 $ x^20 + 39x^18 + 531x^16 + 3488x^14 + 12661x^12 + 27027x^10 + 34540x^8 + 25909x^6 + 10677x^4 + 2092*x^2 + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1065.5
Root			$1.20144i$ of defining polynomial
Character	$\chi$	$=$	1148.1065
Dual form			1148.2.d.a.1065.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23543i q^{3} +0.468853 q^{5} +1.00000i q^{7} -1.99714 q^{9} +O(q^{10})$	$q-2.23543i q^{3} +0.468853 q^{5} +1.00000i q^{7} -1.99714 q^{9} +4.98364i q^{11} +3.78611i q^{13} -1.04809i q^{15} +1.22201i q^{17} +7.89875i q^{19} +2.23543 q^{21} -0.249482 q^{23} -4.78018 q^{25} -2.24183i q^{27} +8.31531i q^{29} +3.17006 q^{31} +11.1406 q^{33} +0.468853i q^{35} -7.74950 q^{37} +8.46357 q^{39} +(-3.82365 - 5.13612i) q^{41} +4.90962 q^{43} -0.936364 q^{45} +1.80770i q^{47} -1.00000 q^{49} +2.73172 q^{51} -1.74173i q^{53} +2.33660i q^{55} +17.6571 q^{57} +2.25456 q^{59} -1.09904 q^{61} -1.99714i q^{63} +1.77513i q^{65} -11.8286i q^{67} +0.557699i q^{69} -12.3547i q^{71} -2.91035 q^{73} +10.6857i q^{75} -4.98364 q^{77} -3.79118i q^{79} -11.0029 q^{81} +8.75742 q^{83} +0.572943i q^{85} +18.5883 q^{87} +15.9289i q^{89} -3.78611 q^{91} -7.08645i q^{93} +3.70335i q^{95} +4.29519i q^{97} -9.95302i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 4 q^{5} - 20 q^{9}+O(q^{10}) $ 20 * q + 4 * q^5 - 20 * q^9	$ 20 q + 4 q^{5} - 20 q^{9} + 4 q^{21} + 8 q^{31} + 20 q^{37} + 4 q^{39} - 16 q^{41} + 20 q^{43} - 4 q^{45} - 20 q^{49} + 52 q^{51} - 36 q^{57} + 20 q^{59} - 4 q^{61} - 12 q^{73} + 8 q^{77} + 20 q^{81} - 48 q^{83} + 44 q^{87} - 4 q^{91}+O(q^{100}) $ 20 * q + 4 * q^5 - 20 * q^9 + 4 * q^21 + 8 * q^31 + 20 * q^37 + 4 * q^39 - 16 * q^41 + 20 * q^43 - 4 * q^45 - 20 * q^49 + 52 * q^51 - 36 * q^57 + 20 * q^59 - 4 * q^61 - 12 * q^73 + 8 * q^77 + 20 * q^81 - 48 * q^83 + 44 * q^87 - 4 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1148\mathbb{Z}\right)^\times$.

$n$	$493$	$575$	$785$
$\chi(n)$	$1$	$1$	$-1$

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
$3$		−	2.23543i		−	1.29062i	−0.763919	−	0.645312i	$-0.776727\pi$
$3$		−	2.23543i		−	1.29062i	0.763919	−	0.645312i	$-0.223273\pi$
$4$		0			0
$5$	0.468853			0.209677			0.104839	−	0.994489i	$-0.466567\pi$
$5$	0.468853			0.209677			0.104839	+	0.994489i	$0.466567\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	−1.99714			−0.665713
$10$		0			0
$11$			4.98364i			1.50263i	0.659946	+	0.751313i	$0.270579\pi$
$11$			4.98364i			1.50263i	−0.659946	+	0.751313i	$0.729421\pi$
$12$		0			0
$13$			3.78611i			1.05008i	0.851078	+	0.525039i	$0.175949\pi$
$13$			3.78611i			1.05008i	−0.851078	+	0.525039i	$0.824051\pi$
$14$		0			0
$15$		−	1.04809i		−	0.270615i
$16$		0			0
$17$			1.22201i			0.296381i	0.988959	+	0.148191i	$0.0473449\pi$
$17$			1.22201i			0.296381i	−0.988959	+	0.148191i	$0.952655\pi$
$18$		0			0
$19$			7.89875i			1.81210i	0.423173	+	0.906049i	$0.360916\pi$
$19$			7.89875i			1.81210i	−0.423173	+	0.906049i	$0.639084\pi$
$20$		0			0
$21$	2.23543			0.487810
$22$		0			0
$23$	−0.249482			−0.0520206			−0.0260103	−	0.999662i	$-0.508280\pi$
$23$	−0.249482			−0.0520206			−0.0260103	+	0.999662i	$0.508280\pi$
$24$		0			0
$25$	−4.78018			−0.956035
$26$		0			0
$27$		−	2.24183i		−	0.431440i
$28$		0			0
$29$			8.31531i			1.54411i	0.635553	+	0.772057i	$0.280772\pi$
$29$			8.31531i			1.54411i	−0.635553	+	0.772057i	$0.719228\pi$
$30$		0			0
$31$	3.17006			0.569360			0.284680	−	0.958623i	$-0.408113\pi$
$31$	3.17006			0.569360			0.284680	+	0.958623i	$0.408113\pi$
$32$		0			0
$33$	11.1406			1.93933
$34$		0			0
$35$			0.468853i			0.0792506i
$36$		0			0
$37$	−7.74950			−1.27401			−0.637005	−	0.770860i	$-0.719827\pi$
$37$	−7.74950			−1.27401			−0.637005	+	0.770860i	$0.719827\pi$
$38$		0			0
$39$	8.46357			1.35526
$40$		0			0
$41$	−3.82365	−	5.13612i	−0.597153	−	0.802127i
$41$	−3.82365	−	5.13612i	−0.597153	−	0.802127i
$42$		0			0
$43$	4.90962			0.748711			0.374355	−	0.927285i	$-0.377864\pi$
$43$	4.90962			0.748711			0.374355	+	0.927285i	$0.377864\pi$
$44$		0			0
$45$	−0.936364			−0.139585
$46$		0			0
$47$			1.80770i			0.263680i	0.991271	+	0.131840i	$0.0420885\pi$
$47$			1.80770i			0.263680i	−0.991271	+	0.131840i	$0.957911\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	2.73172			0.382517
$52$		0			0
$53$		−	1.74173i		−	0.239244i	−0.992819	−	0.119622i	$-0.961832\pi$
$53$		−	1.74173i		−	0.239244i	0.992819	−	0.119622i	$-0.0381683\pi$
$54$		0			0
$55$			2.33660i			0.315066i
$56$		0			0
$57$	17.6571			2.33874
$58$		0			0
$59$	2.25456			0.293519			0.146760	−	0.989172i	$-0.453116\pi$
$59$	2.25456			0.293519			0.146760	+	0.989172i	$0.453116\pi$
$60$		0			0
$61$	−1.09904			−0.140718			−0.0703589	−	0.997522i	$-0.522414\pi$
$61$	−1.09904			−0.140718			−0.0703589	+	0.997522i	$0.522414\pi$
$62$		0			0
$63$		−	1.99714i		−	0.251616i
$64$		0			0
$65$			1.77513i			0.220177i
$66$		0			0
$67$		−	11.8286i		−	1.44509i	−0.691323	−	0.722545i	$-0.742972\pi$
$67$		−	11.8286i		−	1.44509i	0.691323	−	0.722545i	$-0.257028\pi$
$68$		0			0
$69$			0.557699i			0.0671391i
$70$		0			0
$71$		−	12.3547i		−	1.46624i	−0.680100	−	0.733119i	$-0.738064\pi$
$71$		−	12.3547i		−	1.46624i	0.680100	−	0.733119i	$-0.261936\pi$
$72$		0			0
$73$	−2.91035			−0.340630			−0.170315	−	0.985390i	$-0.554479\pi$
$73$	−2.91035			−0.340630			−0.170315	+	0.985390i	$0.554479\pi$
$74$		0			0
$75$			10.6857i			1.23388i
$76$		0			0
$77$	−4.98364			−0.567939
$78$		0			0
$79$		−	3.79118i		−	0.426542i	−0.976993	−	0.213271i	$-0.931588\pi$
$79$		−	3.79118i		−	0.426542i	0.976993	−	0.213271i	$-0.0684117\pi$
$80$		0			0
$81$	−11.0029			−1.22254
$82$		0			0
$83$	8.75742			0.961252			0.480626	−	0.876926i	$-0.340410\pi$
$83$	8.75742			0.961252			0.480626	+	0.876926i	$0.340410\pi$
$84$		0			0
$85$			0.572943i			0.0621444i
$86$		0			0
$87$	18.5883			1.99287
$88$		0			0
$89$			15.9289i			1.68846i	0.535982	+	0.844229i	$0.319941\pi$
$89$			15.9289i			1.68846i	−0.535982	+	0.844229i	$0.680059\pi$
$90$		0			0
$91$	−3.78611			−0.396892
$92$		0			0
$93$		−	7.08645i		−	0.734831i
$94$		0			0
$95$			3.70335i			0.379956i
$96$		0			0
$97$			4.29519i			0.436111i	0.975936	+	0.218055i	$0.0699713\pi$
$97$			4.29519i			0.436111i	−0.975936	+	0.218055i	$0.930029\pi$
$98$		0			0
$99$		−	9.95302i		−	1.00032i
$100$		0			0
$101$		−	4.42421i		−	0.440225i	−0.975474	−	0.220113i	$-0.929358\pi$
$101$		−	4.42421i		−	0.440225i	0.975474	−	0.220113i	$-0.0706425\pi$
$102$		0			0
$103$	14.9664			1.47469			0.737343	−	0.675519i	$-0.236080\pi$
$103$	14.9664			1.47469			0.737343	+	0.675519i	$0.236080\pi$
$104$		0			0
$105$	1.04809			0.102283
$106$		0			0
$107$	17.4446			1.68644			0.843218	−	0.537572i	$-0.180658\pi$
$107$	17.4446			1.68644			0.843218	+	0.537572i	$0.180658\pi$
$108$		0			0
$109$		−	5.31886i		−	0.509455i	−0.967013	−	0.254727i	$-0.918014\pi$
$109$		−	5.31886i		−	0.509455i	0.967013	−	0.254727i	$-0.0819857\pi$
$110$		0			0
$111$			17.3234i			1.64427i
$112$		0			0
$113$	0.843987			0.0793957			0.0396978	−	0.999212i	$-0.487360\pi$
$113$	0.843987			0.0793957			0.0396978	+	0.999212i	$0.487360\pi$
$114$		0			0
$115$	−0.116970			−0.0109075
$116$		0			0
$117$		−	7.56138i		−	0.699050i
$118$		0			0
$119$	−1.22201			−0.112022
$120$		0			0
$121$	−13.8367			−1.25788
$122$		0			0
$123$	−11.4814	+	8.54748i	−1.03525	+	0.770701i
$124$		0			0
$125$	−4.58546			−0.410136
$126$		0			0
$127$	−8.18810			−0.726576			−0.363288	−	0.931677i	$-0.618346\pi$
$127$	−8.18810			−0.726576			−0.363288	+	0.931677i	$0.618346\pi$
$128$		0			0
$129$		−	10.9751i		−	0.966304i
$130$		0			0
$131$	5.65452			0.494038			0.247019	−	0.969011i	$-0.420549\pi$
$131$	5.65452			0.494038			0.247019	+	0.969011i	$0.420549\pi$
$132$		0			0
$133$	−7.89875			−0.684909
$134$		0			0
$135$		−	1.05109i		−	0.0904631i
$136$		0			0
$137$		−	3.57141i		−	0.305126i	−0.988294	−	0.152563i	$-0.951247\pi$
$137$		−	3.57141i		−	0.305126i	0.988294	−	0.152563i	$-0.0487528\pi$
$138$		0			0
$139$	20.9095			1.77352			0.886759	−	0.462232i	$-0.152951\pi$
$139$	20.9095			1.77352			0.886759	+	0.462232i	$0.152951\pi$
$140$		0			0
$141$	4.04098			0.340312
$142$		0			0
$143$	−18.8686			−1.57787
$144$		0			0
$145$			3.89866i			0.323766i
$146$		0			0
$147$			2.23543i			0.184375i
$148$		0			0
$149$			14.6546i			1.20055i	0.799792	+	0.600277i	$0.204943\pi$
$149$			14.6546i			1.20055i	−0.799792	+	0.600277i	$0.795057\pi$
$150$		0			0
$151$		−	1.70827i		−	0.139017i	−0.997581	−	0.0695087i	$-0.977857\pi$
$151$		−	1.70827i		−	0.139017i	0.997581	−	0.0695087i	$-0.0221431\pi$
$152$		0			0
$153$		−	2.44052i		−	0.197305i
$154$		0			0
$155$	1.48629			0.119382
$156$		0			0
$157$		−	14.1603i		−	1.13011i	−0.825052	−	0.565057i	$-0.808854\pi$
$157$		−	14.1603i		−	1.13011i	0.825052	−	0.565057i	$-0.191146\pi$
$158$		0			0
$159$	−3.89350			−0.308775
$160$		0			0
$161$		−	0.249482i		−	0.0196619i
$162$		0			0
$163$	−20.0614			−1.57133			−0.785664	−	0.618654i	$-0.787678\pi$
$163$	−20.0614			−1.57133			−0.785664	+	0.618654i	$0.787678\pi$
$164$		0			0
$165$	5.22329			0.406633
$166$		0			0
$167$			12.5991i			0.974946i	0.873138	+	0.487473i	$0.162081\pi$
$167$			12.5991i			0.974946i	−0.873138	+	0.487473i	$0.837919\pi$
$168$		0			0
$169$	−1.33461			−0.102662
$170$		0			0
$171$		−	15.7749i		−	1.20634i
$172$		0			0
$173$	0.134256			0.0102073			0.00510364	−	0.999987i	$-0.498375\pi$
$173$	0.134256			0.0102073			0.00510364	+	0.999987i	$0.498375\pi$
$174$		0			0
$175$		−	4.78018i		−	0.361347i
$176$		0			0
$177$		−	5.03992i		−	0.378823i
$178$		0			0
$179$		−	14.9327i		−	1.11613i	−0.829799	−	0.558063i	$-0.811545\pi$
$179$		−	14.9327i		−	1.11613i	0.829799	−	0.558063i	$-0.188455\pi$
$180$		0			0
$181$			22.6490i			1.68349i	0.539878	+	0.841744i	$0.318471\pi$
$181$			22.6490i			1.68349i	−0.539878	+	0.841744i	$0.681529\pi$
$182$		0			0
$183$			2.45683i			0.181614i
$184$		0			0
$185$	−3.63337			−0.267131
$186$		0			0
$187$	−6.09007			−0.445350
$188$		0			0
$189$	2.24183			0.163069
$190$		0			0
$191$			19.7885i			1.43185i	0.698178	+	0.715924i	$0.253994\pi$
$191$			19.7885i			1.43185i	−0.698178	+	0.715924i	$0.746006\pi$
$192$		0			0
$193$		−	23.4346i		−	1.68686i	−0.537237	−	0.843431i	$-0.680532\pi$
$193$		−	23.4346i		−	1.68686i	0.537237	−	0.843431i	$-0.319468\pi$
$194$		0			0
$195$	3.96817			0.284166
$196$		0			0
$197$	15.6808			1.11721			0.558606	−	0.829433i	$-0.311336\pi$
$197$	15.6808			1.11721			0.558606	+	0.829433i	$0.311336\pi$
$198$		0			0
$199$			18.6879i			1.32475i	0.749171	+	0.662376i	$0.230452\pi$
$199$			18.6879i			1.32475i	−0.749171	+	0.662376i	$0.769548\pi$
$200$		0			0
$201$	−26.4419			−1.86507
$202$		0			0
$203$	−8.31531			−0.583621
$204$		0			0
$205$	−1.79273	−	2.40808i	−0.125209	−	0.168188i
$206$		0			0
$207$	0.498250			0.0346308
$208$		0			0
$209$	−39.3646			−2.72290
$210$		0			0
$211$		−	17.8016i		−	1.22551i	−0.790272	−	0.612756i	$-0.790061\pi$
$211$		−	17.8016i		−	1.22551i	0.790272	−	0.612756i	$-0.209939\pi$
$212$		0			0
$213$	−27.6181			−1.89236
$214$		0			0
$215$	2.30189			0.156988
$216$		0			0
$217$			3.17006i			0.215198i
$218$		0			0
$219$			6.50587i			0.439626i
$220$		0			0
$221$	−4.62666			−0.311223
$222$		0			0
$223$	−18.4729			−1.23704			−0.618518	−	0.785771i	$-0.712267\pi$
$223$	−18.4729			−1.23704			−0.618518	+	0.785771i	$0.712267\pi$
$224$		0			0
$225$	9.54667			0.636445
$226$		0			0
$227$		−	11.1615i		−	0.740816i	−0.928869	−	0.370408i	$-0.879218\pi$
$227$		−	11.1615i		−	0.740816i	0.928869	−	0.370408i	$-0.120782\pi$
$228$		0			0
$229$			12.2692i			0.810774i	0.914145	+	0.405387i	$0.132863\pi$
$229$			12.2692i			0.810774i	−0.914145	+	0.405387i	$0.867137\pi$
$230$		0			0
$231$			11.1406i			0.732996i
$232$		0			0
$233$			5.38249i			0.352619i	0.984335	+	0.176309i	$0.0564159\pi$
$233$			5.38249i			0.352619i	−0.984335	+	0.176309i	$0.943584\pi$
$234$		0			0
$235$			0.847545i			0.0552878i
$236$		0			0
$237$	−8.47492			−0.550505
$238$		0			0
$239$			1.79783i			0.116292i	0.998308	+	0.0581461i	$0.0185189\pi$
$239$			1.79783i			0.116292i	−0.998308	+	0.0581461i	$0.981481\pi$
$240$		0			0
$241$	−16.4358			−1.05873			−0.529363	−	0.848396i	$-0.677569\pi$
$241$	−16.4358			−1.05873			−0.529363	+	0.848396i	$0.677569\pi$
$242$		0			0
$243$			17.8706i			1.14640i
$244$		0			0
$245$	−0.468853			−0.0299539
$246$		0			0
$247$	−29.9055			−1.90284
$248$		0			0
$249$		−	19.5766i		−	1.24062i
$250$		0			0
$251$	20.8489			1.31597			0.657986	−	0.753031i	$-0.271409\pi$
$251$	20.8489			1.31597			0.657986	+	0.753031i	$0.271409\pi$
$252$		0			0
$253$		−	1.24333i		−	0.0781675i
$254$		0			0
$255$	1.28077			0.0802051
$256$		0			0
$257$		−	11.7446i		−	0.732606i	−0.930496	−	0.366303i	$-0.880623\pi$
$257$		−	11.7446i		−	0.732606i	0.930496	−	0.366303i	$-0.119377\pi$
$258$		0			0
$259$		−	7.74950i		−	0.481530i
$260$		0			0
$261$		−	16.6068i		−	1.02794i
$262$		0			0
$263$			21.4258i			1.32117i	0.750750	+	0.660586i	$0.229692\pi$
$263$			21.4258i			1.32117i	−0.750750	+	0.660586i	$0.770308\pi$
$264$		0			0
$265$		−	0.816613i		−	0.0501641i
$266$		0			0
$267$	35.6079			2.17917
$268$		0			0
$269$	15.9515			0.972580			0.486290	−	0.873797i	$-0.338350\pi$
$269$	15.9515			0.972580			0.486290	+	0.873797i	$0.338350\pi$
$270$		0			0
$271$	−24.5899			−1.49373			−0.746865	−	0.664976i	$-0.768442\pi$
$271$	−24.5899			−1.49373			−0.746865	+	0.664976i	$0.768442\pi$
$272$		0			0
$273$			8.46357i			0.512239i
$274$		0			0
$275$		−	23.8227i		−	1.43656i
$276$		0			0
$277$	31.9459			1.91944			0.959721	−	0.280956i	$-0.0906514\pi$
$277$	31.9459			1.91944			0.959721	+	0.280956i	$0.0906514\pi$
$278$		0			0
$279$	−6.33106			−0.379030
$280$		0			0
$281$		−	14.0011i		−	0.835235i	−0.908623	−	0.417618i	$-0.862865\pi$
$281$		−	14.0011i		−	0.835235i	0.908623	−	0.417618i	$-0.137135\pi$
$282$		0			0
$283$	1.43345			0.0852097			0.0426049	−	0.999092i	$-0.486434\pi$
$283$	1.43345			0.0852097			0.0426049	+	0.999092i	$0.486434\pi$
$284$		0			0
$285$	8.27858			0.490381
$286$		0			0
$287$	5.13612	−	3.82365i	0.303176	−	0.225703i
$288$		0			0
$289$	15.5067			0.912158
$290$		0			0
$291$	9.60159			0.562855
$292$		0			0
$293$		−	14.0309i		−	0.819695i	−0.912154	−	0.409847i	$-0.865582\pi$
$293$		−	14.0309i		−	0.819695i	0.912154	−	0.409847i	$-0.134418\pi$
$294$		0			0
$295$	1.05706			0.0615444
$296$		0			0
$297$	11.1725			0.648292
$298$		0			0
$299$		−	0.944566i		−	0.0546257i
$300$		0			0
$301$			4.90962i			0.282986i
$302$		0			0
$303$	−9.89000			−0.568166
$304$		0			0
$305$	−0.515289			−0.0295053
$306$		0			0
$307$	−6.11435			−0.348964			−0.174482	−	0.984660i	$-0.555825\pi$
$307$	−6.11435			−0.348964			−0.174482	+	0.984660i	$0.555825\pi$
$308$		0			0
$309$		−	33.4564i		−	1.90327i
$310$		0			0
$311$			2.22194i			0.125995i	0.998014	+	0.0629973i	$0.0200660\pi$
$311$			2.22194i			0.125995i	−0.998014	+	0.0629973i	$0.979934\pi$
$312$		0			0
$313$		−	4.57533i		−	0.258613i	−0.991605	−	0.129307i	$-0.958725\pi$
$313$		−	4.57533i		−	0.258613i	0.991605	−	0.129307i	$-0.0412751\pi$
$314$		0			0
$315$		−	0.936364i		−	0.0527581i
$316$		0			0
$317$			7.39003i			0.415066i	0.978228	+	0.207533i	$0.0665434\pi$
$317$			7.39003i			0.415066i	−0.978228	+	0.207533i	$0.933457\pi$
$318$		0			0
$319$	−41.4406			−2.32023
$320$		0			0
$321$		−	38.9962i		−	2.17656i
$322$		0			0
$323$	−9.65236			−0.537072
$324$		0			0
$325$		−	18.0983i		−	1.00391i
$326$		0			0
$327$	−11.8899			−0.657515
$328$		0			0
$329$	−1.80770			−0.0996617
$330$		0			0
$331$		−	4.35261i		−	0.239241i	−0.992820	−	0.119620i	$-0.961832\pi$
$331$		−	4.35261i		−	0.239241i	0.992820	−	0.119620i	$-0.0381678\pi$
$332$		0			0
$333$	15.4768			0.848124
$334$		0			0
$335$		−	5.54586i		−	0.303003i
$336$		0			0
$337$	0.788204			0.0429362			0.0214681	−	0.999770i	$-0.493166\pi$
$337$	0.788204			0.0429362			0.0214681	+	0.999770i	$0.493166\pi$
$338$		0			0
$339$		−	1.88667i		−	0.102470i
$340$		0			0
$341$			15.7985i			0.855535i
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$			0.261479i			0.0140775i
$346$		0			0
$347$			4.37888i			0.235070i	0.993069	+	0.117535i	$0.0374993\pi$
$347$			4.37888i			0.235070i	−0.993069	+	0.117535i	$0.962501\pi$
$348$		0			0
$349$	−17.5805			−0.941061			−0.470530	−	0.882384i	$-0.655937\pi$
$349$	−17.5805			−0.941061			−0.470530	+	0.882384i	$0.655937\pi$
$350$		0			0
$351$	8.48779			0.453045
$352$		0			0
$353$	32.6489			1.73773			0.868864	−	0.495052i	$-0.164851\pi$
$353$	32.6489			1.73773			0.868864	+	0.495052i	$0.164851\pi$
$354$		0			0
$355$		−	5.79255i		−	0.307437i
$356$		0			0
$357$			2.73172i			0.144578i
$358$		0			0
$359$	28.0591			1.48090			0.740450	−	0.672111i	$-0.234612\pi$
$359$	28.0591			1.48090			0.740450	+	0.672111i	$0.234612\pi$
$360$		0			0
$361$	−43.3903			−2.28370
$362$		0			0
$363$			30.9310i			1.62345i
$364$		0			0
$365$	−1.36452			−0.0714224
$366$		0			0
$367$	19.8918			1.03834			0.519172	−	0.854670i	$-0.326240\pi$
$367$	19.8918			1.03834			0.519172	+	0.854670i	$0.326240\pi$
$368$		0			0
$369$	7.63635	+	10.2575i	0.397532	+	0.533986i
$370$		0			0
$371$	1.74173			0.0904259
$372$		0			0
$373$	2.48402			0.128617			0.0643087	−	0.997930i	$-0.479516\pi$
$373$	2.48402			0.128617			0.0643087	+	0.997930i	$0.479516\pi$
$374$		0			0
$375$			10.2505i			0.529332i
$376$		0			0
$377$	−31.4827			−1.62144
$378$		0			0
$379$	−11.5135			−0.591409			−0.295705	−	0.955279i	$-0.595554\pi$
$379$	−11.5135			−0.591409			−0.295705	+	0.955279i	$0.595554\pi$
$380$		0			0
$381$			18.3039i			0.937738i
$382$		0			0
$383$			36.6688i			1.87369i	0.349745	+	0.936845i	$0.386268\pi$
$383$			36.6688i			1.87369i	−0.349745	+	0.936845i	$0.613732\pi$
$384$		0			0
$385$	−2.33660			−0.119084
$386$		0			0
$387$	−9.80519			−0.498426
$388$		0			0
$389$	23.4388			1.18839			0.594197	−	0.804319i	$-0.297470\pi$
$389$	23.4388			1.18839			0.594197	+	0.804319i	$0.297470\pi$
$390$		0			0
$391$		−	0.304870i		−	0.0154179i
$392$		0			0
$393$		−	12.6403i		−	0.637618i
$394$		0			0
$395$		−	1.77751i		−	0.0894361i
$396$		0			0
$397$		−	32.4793i		−	1.63009i	−0.579397	−	0.815046i	$-0.696712\pi$
$397$		−	32.4793i		−	1.63009i	0.579397	−	0.815046i	$-0.303288\pi$
$398$		0			0
$399$			17.6571i			0.883960i
$400$		0			0
$401$	−4.98971			−0.249174			−0.124587	−	0.992209i	$-0.539761\pi$
$401$	−4.98971			−0.249174			−0.124587	+	0.992209i	$0.539761\pi$
$402$		0			0
$403$			12.0022i			0.597873i
$404$		0			0
$405$	−5.15872			−0.256339
$406$		0			0
$407$		−	38.6207i		−	1.91436i
$408$		0			0
$409$	21.2544			1.05096			0.525481	−	0.850805i	$-0.323885\pi$
$409$	21.2544			1.05096			0.525481	+	0.850805i	$0.323885\pi$
$410$		0			0
$411$	−7.98364			−0.393804
$412$		0			0
$413$			2.25456i			0.110940i
$414$		0			0
$415$	4.10594			0.201553
$416$		0			0
$417$		−	46.7416i		−	2.28895i
$418$		0			0
$419$	0.415097			0.0202788			0.0101394	−	0.999949i	$-0.496772\pi$
$419$	0.415097			0.0202788			0.0101394	+	0.999949i	$0.496772\pi$
$420$		0			0
$421$		−	24.4243i		−	1.19037i	−0.803590	−	0.595183i	$-0.797079\pi$
$421$		−	24.4243i		−	1.19037i	0.803590	−	0.595183i	$-0.202921\pi$
$422$		0			0
$423$		−	3.61023i		−	0.175535i
$424$		0			0
$425$		−	5.84143i		−	0.283351i
$426$		0			0
$427$		−	1.09904i		−	0.0531863i
$428$		0			0
$429$			42.1794i			2.03644i
$430$		0			0
$431$	33.4445			1.61096			0.805482	−	0.592620i	$-0.201907\pi$
$431$	33.4445			1.61096			0.805482	+	0.592620i	$0.201907\pi$
$432$		0			0
$433$	5.74246			0.275965			0.137983	−	0.990435i	$-0.455938\pi$
$433$	5.74246			0.275965			0.137983	+	0.990435i	$0.455938\pi$
$434$		0			0
$435$	8.71517			0.417860
$436$		0			0
$437$		−	1.97060i		−	0.0942664i
$438$		0			0
$439$			17.9479i			0.856606i	0.903635	+	0.428303i	$0.140888\pi$
$439$			17.9479i			0.856606i	−0.903635	+	0.428303i	$0.859112\pi$
$440$		0			0
$441$	1.99714			0.0951018
$442$		0			0
$443$	−12.7389			−0.605245			−0.302622	−	0.953111i	$-0.597862\pi$
$443$	−12.7389			−0.605245			−0.302622	+	0.953111i	$0.597862\pi$
$444$		0			0
$445$			7.46830i			0.354032i
$446$		0			0
$447$	32.7593			1.54946
$448$		0			0
$449$	−33.6046			−1.58590			−0.792949	−	0.609289i	$-0.791455\pi$
$449$	−33.6046			−1.58590			−0.792949	+	0.609289i	$0.791455\pi$
$450$		0			0
$451$	25.5966	−	19.0557i	1.20530	−	0.897297i
$452$		0			0
$453$	−3.81872			−0.179419
$454$		0			0
$455$	−1.77513			−0.0832193
$456$		0			0
$457$			30.3766i			1.42096i	0.703718	+	0.710480i	$0.251522\pi$
$457$			30.3766i			1.42096i	−0.703718	+	0.710480i	$0.748478\pi$
$458$		0			0
$459$	2.73954			0.127871
$460$		0			0
$461$	−10.4830			−0.488244			−0.244122	−	0.969745i	$-0.578500\pi$
$461$	−10.4830			−0.488244			−0.244122	+	0.969745i	$0.578500\pi$
$462$		0			0
$463$		−	16.9928i		−	0.789723i	−0.918741	−	0.394862i	$-0.870792\pi$
$463$		−	16.9928i		−	0.789723i	0.918741	−	0.394862i	$-0.129208\pi$
$464$		0			0
$465$		−	3.32250i		−	0.154077i
$466$		0			0
$467$	25.3701			1.17399			0.586994	−	0.809591i	$-0.300311\pi$
$467$	25.3701			1.17399			0.586994	+	0.809591i	$0.300311\pi$
$468$		0			0
$469$	11.8286			0.546193
$470$		0			0
$471$	−31.6543			−1.45855
$472$		0			0
$473$			24.4678i			1.12503i
$474$		0			0
$475$		−	37.7574i		−	1.73243i
$476$		0			0
$477$			3.47847i			0.159268i
$478$		0			0
$479$			5.92313i			0.270635i	0.990802	+	0.135317i	$0.0432054\pi$
$479$			5.92313i			0.270635i	−0.990802	+	0.135317i	$0.956795\pi$
$480$		0			0
$481$		−	29.3404i		−	1.33781i
$482$		0			0
$483$	−0.557699			−0.0253762
$484$		0			0
$485$			2.01381i			0.0914425i
$486$		0			0
$487$	5.12441			0.232209			0.116105	−	0.993237i	$-0.462959\pi$
$487$	5.12441			0.232209			0.116105	+	0.993237i	$0.462959\pi$
$488$		0			0
$489$			44.8457i			2.02799i
$490$		0			0
$491$	18.8111			0.848933			0.424467	−	0.905444i	$-0.360462\pi$
$491$	18.8111			0.848933			0.424467	+	0.905444i	$0.360462\pi$
$492$		0			0
$493$	−10.1614			−0.457646
$494$		0			0
$495$		−	4.66650i		−	0.209744i
$496$		0			0
$497$	12.3547			0.554186
$498$		0			0
$499$			2.07851i			0.0930470i	0.998917	+	0.0465235i	$0.0148142\pi$
$499$			2.07851i			0.0930470i	−0.998917	+	0.0465235i	$0.985186\pi$
$500$		0			0
$501$	28.1643			1.25829
$502$		0			0
$503$			26.1912i			1.16781i	0.811822	+	0.583905i	$0.198476\pi$
$503$			26.1912i			1.16781i	−0.811822	+	0.583905i	$0.801524\pi$
$504$		0			0
$505$		−	2.07430i		−	0.0923053i
$506$		0			0
$507$			2.98343i			0.132499i
$508$		0			0
$509$			31.6951i			1.40486i	0.711752	+	0.702431i	$0.247902\pi$
$509$			31.6951i			1.40486i	−0.711752	+	0.702431i	$0.752098\pi$
$510$		0			0
$511$		−	2.91035i		−	0.128746i
$512$		0			0
$513$	17.7076			0.781811
$514$		0			0
$515$	7.01705			0.309208
$516$		0			0
$517$	−9.00893			−0.396212
$518$		0			0
$519$		−	0.300119i		−	0.0131738i
$520$		0			0
$521$			37.0562i			1.62346i	0.584033	+	0.811730i	$0.301474\pi$
$521$			37.0562i			1.62346i	−0.584033	+	0.811730i	$0.698526\pi$
$522$		0			0
$523$	3.75364			0.164135			0.0820676	−	0.996627i	$-0.473848\pi$
$523$	3.75364			0.164135			0.0820676	+	0.996627i	$0.473848\pi$
$524$		0			0
$525$	−10.6857			−0.466364
$526$		0			0
$527$			3.87385i			0.168748i
$528$		0			0
$529$	−22.9378			−0.997294
$530$		0			0
$531$	−4.50268			−0.195400
$532$		0			0
$533$	19.4459	−	14.4767i	0.842296	−	0.627057i
$534$		0			0
$535$	8.17896			0.353607
$536$		0			0
$537$	−33.3811			−1.44050
$538$		0			0
$539$		−	4.98364i		−	0.214661i
$540$		0			0
$541$	12.0339			0.517380			0.258690	−	0.965960i	$-0.416709\pi$
$541$	12.0339			0.517380			0.258690	+	0.965960i	$0.416709\pi$
$542$		0			0
$543$	50.6302			2.17275
$544$		0			0
$545$		−	2.49376i		−	0.106821i
$546$		0			0
$547$			5.14839i			0.220129i	0.993924	+	0.110065i	$0.0351058\pi$
$547$			5.14839i			0.220129i	−0.993924	+	0.110065i	$0.964894\pi$
$548$		0			0
$549$	2.19494			0.0936776
$550$		0			0
$551$	−65.6806			−2.79809
$552$		0			0
$553$	3.79118			0.161218
$554$		0			0
$555$			8.12214i			0.344766i
$556$		0			0
$557$			46.7199i			1.97959i	0.142508	+	0.989794i	$0.454483\pi$
$557$			46.7199i			1.97959i	−0.142508	+	0.989794i	$0.545517\pi$
$558$		0			0
$559$			18.5884i			0.786204i
$560$		0			0
$561$			13.6139i			0.574779i
$562$		0			0
$563$			19.4210i			0.818496i	0.912423	+	0.409248i	$0.134209\pi$
$563$			19.4210i			0.818496i	−0.912423	+	0.409248i	$0.865791\pi$
$564$		0			0
$565$	0.395706			0.0166475
$566$		0			0
$567$		−	11.0029i		−	0.462076i
$568$		0			0
$569$	−11.9853			−0.502451			−0.251225	−	0.967929i	$-0.580833\pi$
$569$	−11.9853			−0.502451			−0.251225	+	0.967929i	$0.580833\pi$
$570$		0			0
$571$		−	28.2952i		−	1.18412i	−0.805895	−	0.592059i	$-0.798315\pi$
$571$		−	28.2952i		−	1.18412i	0.805895	−	0.592059i	$-0.201685\pi$
$572$		0			0
$573$	44.2359			1.84798
$574$		0			0
$575$	1.19257			0.0497335
$576$		0			0
$577$			10.4014i			0.433016i	0.976281	+	0.216508i	$0.0694667\pi$
$577$			10.4014i			0.433016i	−0.976281	+	0.216508i	$0.930533\pi$
$578$		0			0
$579$	−52.3864			−2.17711
$580$		0			0
$581$			8.75742i			0.363319i
$582$		0			0
$583$	8.68014			0.359495
$584$		0			0
$585$		−	3.54517i		−	0.146575i
$586$		0			0
$587$			21.5780i			0.890619i	0.895377	+	0.445310i	$0.146906\pi$
$587$			21.5780i			0.890619i	−0.895377	+	0.445310i	$0.853094\pi$
$588$		0			0
$589$			25.0396i			1.03174i
$590$		0			0
$591$		−	35.0534i		−	1.44190i
$592$		0			0
$593$		−	14.5336i		−	0.596823i	−0.954437	−	0.298412i	$-0.903543\pi$
$593$		−	14.5336i		−	0.596823i	0.954437	−	0.298412i	$-0.0964568\pi$
$594$		0			0
$595$	−0.572943			−0.0234884
$596$		0			0
$597$	41.7755			1.70976
$598$		0			0
$599$	5.12899			0.209565			0.104782	−	0.994495i	$-0.466585\pi$
$599$	5.12899			0.209565			0.104782	+	0.994495i	$0.466585\pi$
$600$		0			0
$601$			4.80002i			0.195797i	0.995196	+	0.0978984i	$0.0312120\pi$
$601$			4.80002i			0.195797i	−0.995196	+	0.0978984i	$0.968788\pi$
$602$		0			0
$603$			23.6233i			0.962015i
$604$		0			0
$605$	−6.48738			−0.263750
$606$		0			0
$607$	−17.8802			−0.725735			−0.362868	−	0.931841i	$-0.618202\pi$
$607$	−17.8802			−0.725735			−0.362868	+	0.931841i	$0.618202\pi$
$608$		0			0
$609$			18.5883i			0.753235i
$610$		0			0
$611$	−6.84415			−0.276885
$612$		0			0
$613$	−9.92497			−0.400866			−0.200433	−	0.979707i	$-0.564235\pi$
$613$	−9.92497			−0.400866			−0.200433	+	0.979707i	$0.564235\pi$
$614$		0			0
$615$	−5.38310	+	4.00751i	−0.217068	+	0.161598i
$616$		0			0
$617$	27.0895			1.09058			0.545291	−	0.838247i	$-0.316419\pi$
$617$	27.0895			1.09058			0.545291	+	0.838247i	$0.316419\pi$
$618$		0			0
$619$	−18.1664			−0.730170			−0.365085	−	0.930974i	$-0.618960\pi$
$619$	−18.1664			−0.730170			−0.365085	+	0.930974i	$0.618960\pi$
$620$		0			0
$621$			0.559295i			0.0224437i
$622$		0			0
$623$	−15.9289			−0.638177
$624$		0			0
$625$	21.7510			0.870039
$626$		0			0
$627$			87.9966i			3.51425i
$628$		0			0
$629$		−	9.46997i		−	0.377592i
$630$		0			0
$631$	11.1736			0.444812			0.222406	−	0.974954i	$-0.428609\pi$
$631$	11.1736			0.444812			0.222406	+	0.974954i	$0.428609\pi$
$632$		0			0
$633$	−39.7942			−1.58168
$634$		0			0
$635$	−3.83901			−0.152347
$636$		0			0
$637$		−	3.78611i		−	0.150011i
$638$		0			0
$639$			24.6741i			0.976093i
$640$		0			0
$641$		−	23.0981i		−	0.912320i	−0.889898	−	0.456160i	$-0.849224\pi$
$641$		−	23.0981i		−	0.912320i	0.889898	−	0.456160i	$-0.150776\pi$
$642$		0			0
$643$			8.28421i			0.326697i	0.986568	+	0.163349i	$0.0522295\pi$
$643$			8.28421i			0.326697i	−0.986568	+	0.163349i	$0.947770\pi$
$644$		0			0
$645$		−	5.14571i		−	0.202612i
$646$		0			0
$647$	49.0736			1.92928			0.964642	−	0.263565i	$-0.0848985\pi$
$647$	49.0736			1.92928			0.964642	+	0.263565i	$0.0848985\pi$
$648$		0			0
$649$			11.2359i			0.441050i
$650$		0			0
$651$	7.08645			0.277740
$652$		0			0
$653$		−	33.5708i		−	1.31373i	−0.754010	−	0.656863i	$-0.771883\pi$
$653$		−	33.5708i		−	1.31373i	0.754010	−	0.656863i	$-0.228117\pi$
$654$		0			0
$655$	2.65114			0.103589
$656$		0			0
$657$	5.81236			0.226762
$658$		0			0
$659$		−	28.9895i		−	1.12927i	−0.825341	−	0.564635i	$-0.809017\pi$
$659$		−	28.9895i		−	1.12927i	0.825341	−	0.564635i	$-0.190983\pi$
$660$		0			0
$661$	−1.64974			−0.0641673			−0.0320836	−	0.999485i	$-0.510214\pi$
$661$	−1.64974			−0.0641673			−0.0320836	+	0.999485i	$0.510214\pi$
$662$		0			0
$663$			10.3426i			0.401672i
$664$		0			0
$665$	−3.70335			−0.143610
$666$		0			0
$667$		−	2.07452i		−	0.0803258i
$668$		0			0
$669$			41.2948i			1.59655i
$670$		0			0
$671$		−	5.47723i		−	0.211446i
$672$		0			0
$673$		−	38.9087i		−	1.49982i	−0.661540	−	0.749910i	$-0.730097\pi$
$673$		−	38.9087i		−	1.49982i	0.661540	−	0.749910i	$-0.269903\pi$
$674$		0			0
$675$			10.7163i			0.412472i
$676$		0			0
$677$	16.5432			0.635805			0.317903	−	0.948123i	$-0.397021\pi$
$677$	16.5432			0.635805			0.317903	+	0.948123i	$0.397021\pi$
$678$		0			0
$679$	−4.29519			−0.164834
$680$		0			0
$681$	−24.9508			−0.956116
$682$		0			0
$683$			21.1382i			0.808830i	0.914576	+	0.404415i	$0.132525\pi$
$683$			21.1382i			0.808830i	−0.914576	+	0.404415i	$0.867475\pi$
$684$		0			0
$685$		−	1.67447i		−	0.0639781i
$686$		0			0
$687$	27.4270			1.04641
$688$		0			0
$689$	6.59436			0.251225
$690$		0			0
$691$		−	39.5315i		−	1.50385i	−0.659250	−	0.751924i	$-0.729126\pi$
$691$		−	39.5315i		−	1.50385i	0.659250	−	0.751924i	$-0.270874\pi$
$692$		0			0
$693$	9.95302			0.378084
$694$		0			0
$695$	9.80346			0.371867
$696$		0			0
$697$	6.27639	−	4.67253i	0.237735	−	0.176985i
$698$		0			0
$699$	12.0322			0.455099
$700$		0			0
$701$	−25.9383			−0.979675			−0.489838	−	0.871814i	$-0.662944\pi$
$701$	−25.9383			−0.979675			−0.489838	+	0.871814i	$0.662944\pi$
$702$		0			0
$703$		−	61.2113i		−	2.30863i
$704$		0			0
$705$	1.89463			0.0713557
$706$		0			0
$707$	4.42421			0.166389
$708$		0			0
$709$			29.6216i			1.11246i	0.831028	+	0.556231i	$0.187753\pi$
$709$			29.6216i			1.11246i	−0.831028	+	0.556231i	$0.812247\pi$
$710$		0			0
$711$			7.57152i			0.283954i
$712$		0			0
$713$	−0.790874			−0.0296185
$714$		0			0
$715$	−8.84660			−0.330844
$716$		0			0
$717$	4.01893			0.150090
$718$		0			0
$719$		−	27.4212i		−	1.02264i	−0.859391	−	0.511318i	$-0.829157\pi$
$719$		−	27.4212i		−	1.02264i	0.859391	−	0.511318i	$-0.170843\pi$
$720$		0			0
$721$			14.9664i			0.557379i
$722$		0			0
$723$			36.7411i			1.36642i
$724$		0			0
$725$		−	39.7487i		−	1.47623i
$726$		0			0
$727$		−	45.9715i		−	1.70499i	−0.522738	−	0.852493i	$-0.675089\pi$
$727$		−	45.9715i		−	1.70499i	0.522738	−	0.852493i	$-0.324911\pi$
$728$		0			0
$729$	6.93989			0.257033
$730$		0			0
$731$			5.99961i			0.221904i
$732$		0			0
$733$	−29.9746			−1.10714			−0.553569	−	0.832803i	$-0.686734\pi$
$733$	−29.9746			−1.10714			−0.553569	+	0.832803i	$0.686734\pi$
$734$		0			0
$735$			1.04809i			0.0386593i
$736$		0			0
$737$	58.9494			2.17143
$738$		0			0
$739$	−13.1355			−0.483196			−0.241598	−	0.970376i	$-0.577672\pi$
$739$	−13.1355			−0.483196			−0.241598	+	0.970376i	$0.577672\pi$
$740$		0			0
$741$			66.8516i			2.45586i
$742$		0			0
$743$	4.72772			0.173443			0.0867216	−	0.996233i	$-0.472361\pi$
$743$	4.72772			0.173443			0.0867216	+	0.996233i	$0.472361\pi$
$744$		0			0
$745$			6.87086i			0.251729i
$746$		0			0
$747$	−17.4898			−0.639917
$748$		0			0
$749$			17.4446i			0.637413i
$750$		0			0
$751$			35.5122i			1.29586i	0.761701	+	0.647929i	$0.224364\pi$
$751$			35.5122i			1.29586i	−0.761701	+	0.647929i	$0.775636\pi$
$752$		0			0
$753$		−	46.6062i		−	1.69843i
$754$		0			0
$755$		−	0.800929i		−	0.0291488i
$756$		0			0
$757$			45.9485i			1.67003i	0.550230	+	0.835013i	$0.314540\pi$
$757$			45.9485i			1.67003i	−0.550230	+	0.835013i	$0.685460\pi$
$758$		0			0
$759$	−2.77937			−0.100885
$760$		0			0
$761$	32.9598			1.19479			0.597396	−	0.801947i	$-0.296202\pi$
$761$	32.9598			1.19479			0.597396	+	0.801947i	$0.296202\pi$
$762$		0			0
$763$	5.31886			0.192556
$764$		0			0
$765$		−	1.14425i		−	0.0413703i
$766$		0			0
$767$			8.53603i			0.308218i
$768$		0			0
$769$	−20.4944			−0.739046			−0.369523	−	0.929222i	$-0.620479\pi$
$769$	−20.4944			−0.739046			−0.369523	+	0.929222i	$0.620479\pi$
$770$		0			0
$771$	−26.2541			−0.945520
$772$		0			0
$773$			6.62253i			0.238196i	0.992883	+	0.119098i	$0.0380002\pi$
$773$			6.62253i			0.238196i	−0.992883	+	0.119098i	$0.962000\pi$
$774$		0			0
$775$	−15.1535			−0.544329
$776$		0			0
$777$	−17.3234			−0.621475
$778$		0			0
$779$	40.5689	−	30.2020i	1.45353	−	1.08210i
$780$		0			0
$781$	61.5716			2.20321
$782$		0			0
$783$	18.6415			0.666192
$784$		0			0
$785$		−	6.63909i		−	0.236959i
$786$		0			0
$787$	30.6191			1.09145			0.545726	−	0.837964i	$-0.316254\pi$
$787$	30.6191			1.09145			0.545726	+	0.837964i	$0.316254\pi$
$788$		0			0
$789$	47.8959			1.70514
$790$		0			0
$791$			0.843987i			0.0300087i
$792$		0			0
$793$		−	4.16109i		−	0.147765i
$794$		0			0
$795$	−1.82548			−0.0647431
$796$		0			0
$797$	28.0187			0.992475			0.496237	−	0.868187i	$-0.334715\pi$
$797$	28.0187			0.992475			0.496237	+	0.868187i	$0.334715\pi$
$798$		0			0
$799$	−2.20903			−0.0781498
$800$		0			0
$801$		−	31.8122i		−	1.12403i
$802$		0			0
$803$		−	14.5041i		−	0.511840i
$804$		0			0
$805$		−	0.116970i		−	0.00412266i
$806$		0			0
$807$		−	35.6584i		−	1.25524i
$808$		0			0
$809$		−	47.4452i		−	1.66808i	−0.551701	−	0.834042i	$-0.686021\pi$
$809$		−	47.4452i		−	1.66808i	0.551701	−	0.834042i	$-0.313979\pi$
$810$		0			0
$811$	−32.0681			−1.12606			−0.563031	−	0.826435i	$-0.690365\pi$
$811$	−32.0681			−1.12606			−0.563031	+	0.826435i	$0.690365\pi$
$812$		0			0
$813$			54.9689i			1.92784i
$814$		0			0
$815$	−9.40583			−0.329472
$816$		0			0
$817$			38.7799i			1.35674i
$818$		0			0
$819$	7.56138			0.264216
$820$		0			0
$821$	29.7166			1.03711			0.518557	−	0.855043i	$-0.326469\pi$
$821$	29.7166			1.03711			0.518557	+	0.855043i	$0.326469\pi$
$822$		0			0
$823$		−	21.9764i		−	0.766048i	−0.923738	−	0.383024i	$-0.874883\pi$
$823$		−	21.9764i		−	0.766048i	0.923738	−	0.383024i	$-0.125117\pi$
$824$		0			0
$825$	−53.2539			−1.85406
$826$		0			0
$827$		−	11.1796i		−	0.388754i	−0.980927	−	0.194377i	$-0.937731\pi$
$827$		−	11.1796i		−	0.388754i	0.980927	−	0.194377i	$-0.0622686\pi$
$828$		0			0
$829$	−5.82399			−0.202276			−0.101138	−	0.994872i	$-0.532248\pi$
$829$	−5.82399			−0.202276			−0.101138	+	0.994872i	$0.532248\pi$
$830$		0			0
$831$		−	71.4127i		−	2.47728i
$832$		0			0
$833$		−	1.22201i		−	0.0423402i
$834$		0			0
$835$			5.90711i			0.204424i
$836$		0			0
$837$		−	7.10673i		−	0.245645i
$838$		0			0
$839$		−	20.5442i		−	0.709265i	−0.935006	−	0.354632i	$-0.884606\pi$
$839$		−	20.5442i		−	0.709265i	0.935006	−	0.354632i	$-0.115394\pi$
$840$		0			0
$841$	−40.1444			−1.38429
$842$		0			0
$843$	−31.2984			−1.07798
$844$		0			0
$845$	−0.625737			−0.0215260
$846$		0			0
$847$		−	13.8367i		−	0.475435i
$848$		0			0
$849$		−	3.20437i		−	0.109974i
$850$		0			0
$851$	1.93336			0.0662747
$852$		0			0
$853$	1.64795			0.0564246			0.0282123	−	0.999602i	$-0.491019\pi$
$853$	1.64795			0.0564246			0.0282123	+	0.999602i	$0.491019\pi$
$854$		0			0
$855$		−	7.39610i		−	0.252941i
$856$		0			0
$857$	13.3334			0.455460			0.227730	−	0.973724i	$-0.426870\pi$
$857$	13.3334			0.455460			0.227730	+	0.973724i	$0.426870\pi$
$858$		0			0
$859$	26.6900			0.910652			0.455326	−	0.890325i	$-0.349523\pi$
$859$	26.6900			0.910652			0.455326	+	0.890325i	$0.349523\pi$
$860$		0			0
$861$	−8.54748	−	11.4814i	−0.291297	−	0.391286i
$862$		0			0
$863$	−39.3025			−1.33787			−0.668936	−	0.743320i	$-0.733250\pi$
$863$	−39.3025			−1.33787			−0.668936	+	0.743320i	$0.733250\pi$
$864$		0			0
$865$	0.0629462			0.00214023
$866$		0			0
$867$		−	34.6641i		−	1.17725i
$868$		0			0
$869$	18.8939			0.640932
$870$		0			0
$871$	44.7843			1.51746
$872$		0			0
$873$		−	8.57809i		−	0.290324i
$874$		0			0
$875$		−	4.58546i		−	0.155017i
$876$		0			0
$877$	2.86125			0.0966176			0.0483088	−	0.998832i	$-0.484617\pi$
$877$	2.86125			0.0966176			0.0483088	+	0.998832i	$0.484617\pi$
$878$		0			0
$879$	−31.3651			−1.05792
$880$		0			0
$881$	−10.9699			−0.369587			−0.184793	−	0.982777i	$-0.559162\pi$
$881$	−10.9699			−0.369587			−0.184793	+	0.982777i	$0.559162\pi$
$882$		0			0
$883$		−	50.6462i		−	1.70438i	−0.523232	−	0.852190i	$-0.675274\pi$
$883$		−	50.6462i		−	1.70438i	0.523232	−	0.852190i	$-0.324726\pi$
$884$		0			0
$885$		−	2.36298i		−	0.0794307i
$886$		0			0
$887$		−	2.98749i		−	0.100310i	−0.998741	−	0.0501550i	$-0.984028\pi$
$887$		−	2.98749i		−	0.100310i	0.998741	−	0.0501550i	$-0.0159715\pi$
$888$		0			0
$889$		−	8.18810i		−	0.274620i
$890$		0			0
$891$		−	54.8343i		−	1.83702i
$892$		0			0
$893$	−14.2786			−0.477814
$894$		0			0
$895$		−	7.00126i		−	0.234026i
$896$		0			0
$897$	−2.11151			−0.0705012
$898$		0			0
$899$			26.3601i			0.879158i
$900$		0			0
$901$	2.12841			0.0709075
$902$		0			0
$903$	10.9751			0.365229
$904$		0			0
$905$			10.6190i			0.352989i
$906$		0			0
$907$	2.38010			0.0790300			0.0395150	−	0.999219i	$-0.487419\pi$
$907$	2.38010			0.0790300			0.0395150	+	0.999219i	$0.487419\pi$
$908$		0			0
$909$			8.83575i			0.293063i
$910$		0			0
$911$	−38.6679			−1.28112			−0.640562	−	0.767907i	$-0.721298\pi$
$911$	−38.6679			−1.28112			−0.640562	+	0.767907i	$0.721298\pi$
$912$		0			0
$913$			43.6439i			1.44440i
$914$		0			0
$915$			1.15189i			0.0380803i
$916$		0			0
$917$			5.65452i			0.186729i
$918$		0			0
$919$			45.1173i			1.48828i	0.668023	+	0.744141i	$0.267141\pi$
$919$			45.1173i			1.48828i	−0.668023	+	0.744141i	$0.732859\pi$
$920$		0			0
$921$			13.6682i			0.450382i
$922$		0			0
$923$	46.7764			1.53966
$924$		0			0
$925$	37.0440			1.21800
$926$		0			0
$927$	−29.8900			−0.981717
$928$		0			0
$929$		−	27.6620i		−	0.907561i	−0.891113	−	0.453781i	$-0.850075\pi$
$929$		−	27.6620i		−	0.907561i	0.891113	−	0.453781i	$-0.149925\pi$
$930$		0			0
$931$		−	7.89875i		−	0.258871i
$932$		0			0
$933$	4.96698			0.162612
$934$		0			0
$935$	−2.85534			−0.0933798
$936$		0			0
$937$			5.64793i			0.184510i	0.995735	+	0.0922549i	$0.0294075\pi$
$937$			5.64793i			0.184510i	−0.995735	+	0.0922549i	$0.970593\pi$
$938$		0			0
$939$	−10.2278			−0.333773
$940$		0			0
$941$	−40.3099			−1.31406			−0.657032	−	0.753862i	$-0.728188\pi$
$941$	−40.3099			−1.31406			−0.657032	+	0.753862i	$0.728188\pi$
$942$		0			0
$943$	0.953931	+	1.28137i	0.0310643	+	0.0417271i
$944$		0			0
$945$	1.05109			0.0341918
$946$		0			0
$947$	−17.4019			−0.565487			−0.282743	−	0.959196i	$-0.591245\pi$
$947$	−17.4019			−0.565487			−0.282743	+	0.959196i	$0.591245\pi$
$948$		0			0
$949$		−	11.0189i		−	0.357688i
$950$		0			0
$951$	16.5199			0.535694
$952$		0			0
$953$	−15.3681			−0.497823			−0.248911	−	0.968526i	$-0.580073\pi$
$953$	−15.3681			−0.497823			−0.248911	+	0.968526i	$0.580073\pi$
$954$		0			0
$955$			9.27791i			0.300226i
$956$		0			0
$957$			92.6374i			2.99454i
$958$		0			0
$959$	3.57141			0.115327
$960$		0			0
$961$	−20.9507			−0.675829
$962$		0			0
$963$	−34.8393			−1.12268
$964$		0			0
$965$		−	10.9874i		−	0.353697i
$966$		0			0
$967$		−	46.0434i		−	1.48065i	−0.672246	−	0.740327i	$-0.734670\pi$
$967$		−	46.0434i		−	1.48065i	0.672246	−	0.740327i	$-0.265330\pi$
$968$		0			0
$969$			21.5771i			0.693158i
$970$		0			0
$971$		−	30.8722i		−	0.990736i	−0.868683	−	0.495368i	$-0.835033\pi$
$971$		−	30.8722i		−	0.990736i	0.868683	−	0.495368i	$-0.164967\pi$
$972$		0			0
$973$			20.9095i			0.670327i
$974$		0			0
$975$	−40.4574			−1.29567
$976$		0			0
$977$		−	4.31882i		−	0.138171i	−0.997611	−	0.0690857i	$-0.977992\pi$
$977$		−	4.31882i		−	0.138171i	0.997611	−	0.0690857i	$-0.0220082\pi$
$978$		0			0
$979$	−79.3839			−2.53712
$980$		0			0
$981$			10.6225i			0.339150i
$982$		0			0
$983$	27.3025			0.870814			0.435407	−	0.900234i	$-0.356605\pi$
$983$	27.3025			0.870814			0.435407	+	0.900234i	$0.356605\pi$
$984$		0			0
$985$	7.35200			0.234254
$986$		0			0
$987$			4.04098i			0.128626i
$988$		0			0
$989$	−1.22486			−0.0389484
$990$		0			0
$991$		−	50.4305i		−	1.60198i	−0.598678	−	0.800989i	$-0.704307\pi$
$991$		−	50.4305i		−	1.60198i	0.598678	−	0.800989i	$-0.295693\pi$
$992$		0			0
$993$	−9.72994			−0.308770
$994$		0			0
$995$			8.76189i			0.277771i
$996$		0			0
$997$			4.64486i			0.147104i	0.997291	+	0.0735521i	$0.0234335\pi$
$997$			4.64486i			0.147104i	−0.997291	+	0.0735521i	$0.976566\pi$
$998$		0			0
$999$			17.3730i			0.549658i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1148.2.d.a.1065.5	✓	20
41.40	even	2	inner	1148.2.d.a.1065.16	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.d.a.1065.5	✓	20	1.1	even	1	trivial
1148.2.d.a.1065.16	yes	20	41.40	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1148.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.23543i q^{3} +0.468853 q^{5} +1.00000i q^{7} -1.99714 q^{9} +O(q^{10})\)	\(q-2.23543i q^{3} +0.468853 q^{5} +1.00000i q^{7} -1.99714 q^{9} +4.98364i q^{11} +3.78611i q^{13} -1.04809i q^{15} +1.22201i q^{17} +7.89875i q^{19} +2.23543 q^{21} -0.249482 q^{23} -4.78018 q^{25} -2.24183i q^{27} +8.31531i q^{29} +3.17006 q^{31} +11.1406 q^{33} +0.468853i q^{35} -7.74950 q^{37} +8.46357 q^{39} +(-3.82365 - 5.13612i) q^{41} +4.90962 q^{43} -0.936364 q^{45} +1.80770i q^{47} -1.00000 q^{49} +2.73172 q^{51} -1.74173i q^{53} +2.33660i q^{55} +17.6571 q^{57} +2.25456 q^{59} -1.09904 q^{61} -1.99714i q^{63} +1.77513i q^{65} -11.8286i q^{67} +0.557699i q^{69} -12.3547i q^{71} -2.91035 q^{73} +10.6857i q^{75} -4.98364 q^{77} -3.79118i q^{79} -11.0029 q^{81} +8.75742 q^{83} +0.572943i q^{85} +18.5883 q^{87} +15.9289i q^{89} -3.78611 q^{91} -7.08645i q^{93} +3.70335i q^{95} +4.29519i q^{97} -9.95302i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{5} - 20 q^{9}+O(q^{10}) \) 20 * q + 4 * q^5 - 20 * q^9	\( 20 q + 4 q^{5} - 20 q^{9} + 4 q^{21} + 8 q^{31} + 20 q^{37} + 4 q^{39} - 16 q^{41} + 20 q^{43} - 4 q^{45} - 20 q^{49} + 52 q^{51} - 36 q^{57} + 20 q^{59} - 4 q^{61} - 12 q^{73} + 8 q^{77} + 20 q^{81} - 48 q^{83} + 44 q^{87} - 4 q^{91}+O(q^{100}) \) 20 * q + 4 * q^5 - 20 * q^9 + 4 * q^21 + 8 * q^31 + 20 * q^37 + 4 * q^39 - 16 * q^41 + 20 * q^43 - 4 * q^45 - 20 * q^49 + 52 * q^51 - 36 * q^57 + 20 * q^59 - 4 * q^61 - 12 * q^73 + 8 * q^77 + 20 * q^81 - 48 * q^83 + 44 * q^87 - 4 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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