LMFDB - Embedded newform 2240.2.b.h.1121.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(1121,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("2240.1121");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.b (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:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} + 35 x^{10} - 120 x^{9} + 362 x^{8} - 794 x^{7} + 1463 x^{6} - 2048 x^{5} + 2258 x^{4} + \cdots + 61 $ x^12 - 6x^11 + 35x^10 - 120x^9 + 362x^8 - 794x^7 + 1463x^6 - 2048x^5 + 2258x^4 - 1822x^3 + 1035x^2 - 364*x + 61
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1121.1
Root			$0.500000 + 0.234551i$ of defining polynomial
Character	$\chi$	$=$	2240.1121
Dual form			2240.2.b.h.1121.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.32646i q^{3} -1.00000i q^{5} +1.00000 q^{7} -8.06534 q^{9} +O(q^{10})$	$q-3.32646i q^{3} -1.00000i q^{5} +1.00000 q^{7} -8.06534 q^{9} +3.78873i q^{11} -4.49865i q^{13} -3.32646 q^{15} -2.58941 q^{17} -5.09856i q^{19} -3.32646i q^{21} -2.16036 q^{23} -1.00000 q^{25} +16.8497i q^{27} +0.164774i q^{29} -6.36151 q^{31} +12.6031 q^{33} -1.00000i q^{35} +7.95666i q^{37} -14.9646 q^{39} +3.23582 q^{41} +8.69371i q^{43} +8.06534i q^{45} -12.3012 q^{47} +1.00000 q^{49} +8.61357i q^{51} -6.26428i q^{53} +3.78873 q^{55} -16.9602 q^{57} -6.88394i q^{59} -8.08623i q^{61} -8.06534 q^{63} -4.49865 q^{65} -2.05793i q^{67} +7.18635i q^{69} -12.7699 q^{71} +7.80848 q^{73} +3.32646i q^{75} +3.78873i q^{77} -6.08065 q^{79} +31.8537 q^{81} +17.7985i q^{83} +2.58941i q^{85} +0.548115 q^{87} +4.16036 q^{89} -4.49865i q^{91} +21.1613i q^{93} -5.09856 q^{95} -0.338793 q^{97} -30.5574i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{7} - 20 q^{9}+O(q^{10}) $ 12 * q + 12 * q^7 - 20 * q^9	$ 12 q + 12 q^{7} - 20 q^{9} + 16 q^{17} + 8 q^{23} - 12 q^{25} - 8 q^{31} + 72 q^{33} - 32 q^{39} - 32 q^{47} + 12 q^{49} + 8 q^{55} + 8 q^{57} - 20 q^{63} + 8 q^{65} + 48 q^{71} + 8 q^{73} - 16 q^{79} + 92 q^{81} - 32 q^{87} + 16 q^{89} + 32 q^{97}+O(q^{100}) $ 12 * q + 12 * q^7 - 20 * q^9 + 16 * q^17 + 8 * q^23 - 12 * q^25 - 8 * q^31 + 72 * q^33 - 32 * q^39 - 32 * q^47 + 12 * q^49 + 8 * q^55 + 8 * q^57 - 20 * q^63 + 8 * q^65 + 48 * q^71 + 8 * q^73 - 16 * q^79 + 92 * q^81 - 32 * q^87 + 16 * q^89 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$.

$n$	$897$	$1471$	$1541$	$1921$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	− 3.32646i	− 1.92053i	−0.279085	−	0.960267i	$-0.590031\pi$
$3$	− 3.32646i	− 1.92053i	0.279085	−	0.960267i	$-0.409969\pi$
$4$	0	0
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−8.06534	−2.68845
$10$	0	0
$11$	3.78873i	1.14235i	0.820830	+	0.571173i	$0.193511\pi$
$11$	3.78873i	1.14235i	−0.820830	+	0.571173i	$0.806489\pi$
$12$	0	0
$13$	− 4.49865i	− 1.24770i	−0.781544	−	0.623850i	$-0.785567\pi$
$13$	− 4.49865i	− 1.24770i	0.781544	−	0.623850i	$-0.214433\pi$
$14$	0	0
$15$	−3.32646	−0.858888
$16$	0	0
$17$	−2.58941	−0.628024	−0.314012	−	0.949419i	$-0.601673\pi$
$17$	−2.58941	−0.628024	−0.314012	+	0.949419i	$0.601673\pi$
$18$	0	0
$19$	− 5.09856i	− 1.16969i	−0.811145	−	0.584845i	$-0.801155\pi$
$19$	− 5.09856i	− 1.16969i	0.811145	−	0.584845i	$-0.198845\pi$
$20$	0	0
$21$	− 3.32646i	− 0.725893i
$22$	0	0
$23$	−2.16036	−0.450466	−0.225233	−	0.974305i	$-0.572314\pi$
$23$	−2.16036	−0.450466	−0.225233	+	0.974305i	$0.572314\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	16.8497i	3.24272i
$28$	0	0
$29$	0.164774i	0.0305978i	0.999883	+	0.0152989i	$0.00486998\pi$
$29$	0.164774i	0.0305978i	−0.999883	+	0.0152989i	$0.995130\pi$
$30$	0	0
$31$	−6.36151	−1.14256	−0.571281	−	0.820755i	$-0.693553\pi$
$31$	−6.36151	−1.14256	−0.571281	+	0.820755i	$0.693553\pi$
$32$	0	0
$33$	12.6031	2.19391
$34$	0	0
$35$	− 1.00000i	− 0.169031i
$36$	0	0
$37$	7.95666i	1.30807i	0.756465	+	0.654034i	$0.226925\pi$
$37$	7.95666i	1.30807i	−0.756465	+	0.654034i	$0.773075\pi$
$38$	0	0
$39$	−14.9646	−2.39625
$40$	0	0
$41$	3.23582	0.505350	0.252675	−	0.967551i	$-0.418690\pi$
$41$	3.23582	0.505350	0.252675	+	0.967551i	$0.418690\pi$
$42$	0	0
$43$	8.69371i	1.32578i	0.748717	+	0.662889i	$0.230670\pi$
$43$	8.69371i	1.32578i	−0.748717	+	0.662889i	$0.769330\pi$
$44$	0	0
$45$	8.06534i	1.20231i
$46$	0	0
$47$	−12.3012	−1.79431	−0.897154	−	0.441718i	$-0.854369\pi$
$47$	−12.3012	−1.79431	−0.897154	+	0.441718i	$0.854369\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	8.61357i	1.20614i
$52$	0	0
$53$	− 6.26428i	− 0.860465i	−0.902718	−	0.430232i	$-0.858432\pi$
$53$	− 6.26428i	− 0.860465i	0.902718	−	0.430232i	$-0.141568\pi$
$54$	0	0
$55$	3.78873	0.510873
$56$	0	0
$57$	−16.9602	−2.24643
$58$	0	0
$59$	− 6.88394i	− 0.896212i	−0.893980	−	0.448106i	$-0.852099\pi$
$59$	− 6.88394i	− 0.896212i	0.893980	−	0.448106i	$-0.147901\pi$
$60$	0	0
$61$	− 8.08623i	− 1.03534i	−0.855582	−	0.517668i	$-0.826800\pi$
$61$	− 8.08623i	− 1.03534i	0.855582	−	0.517668i	$-0.173200\pi$
$62$	0	0
$63$	−8.06534	−1.01614
$64$	0	0
$65$	−4.49865	−0.557989
$66$	0	0
$67$	− 2.05793i	− 0.251416i	−0.992067	−	0.125708i	$-0.959880\pi$
$67$	− 2.05793i	− 0.251416i	0.992067	−	0.125708i	$-0.0401202\pi$
$68$	0	0
$69$	7.18635i	0.865135i
$70$	0	0
$71$	−12.7699	−1.51551	−0.757757	−	0.652537i	$-0.773705\pi$
$71$	−12.7699	−1.51551	−0.757757	+	0.652537i	$0.773705\pi$
$72$	0	0
$73$	7.80848	0.913913	0.456957	−	0.889489i	$-0.348939\pi$
$73$	7.80848	0.913913	0.456957	+	0.889489i	$0.348939\pi$
$74$	0	0
$75$	3.32646i	0.384107i
$76$	0	0
$77$	3.78873i	0.431766i
$78$	0	0
$79$	−6.08065	−0.684126	−0.342063	−	0.939677i	$-0.611126\pi$
$79$	−6.08065	−0.684126	−0.342063	+	0.939677i	$0.611126\pi$
$80$	0	0
$81$	31.8537	3.53930
$82$	0	0
$83$	17.7985i	1.95364i	0.214069	+	0.976818i	$0.431328\pi$
$83$	17.7985i	1.95364i	−0.214069	+	0.976818i	$0.568672\pi$
$84$	0	0
$85$	2.58941i	0.280861i
$86$	0	0
$87$	0.548115	0.0587641
$88$	0	0
$89$	4.16036	0.440997	0.220499	−	0.975387i	$-0.429232\pi$
$89$	4.16036	0.440997	0.220499	+	0.975387i	$0.429232\pi$
$90$	0	0
$91$	− 4.49865i	− 0.471587i
$92$	0	0
$93$	21.1613i	2.19433i
$94$	0	0
$95$	−5.09856	−0.523101
$96$	0	0
$97$	−0.338793	−0.0343992	−0.0171996	−	0.999852i	$-0.505475\pi$
$97$	−0.338793	−0.0343992	−0.0171996	+	0.999852i	$0.505475\pi$
$98$	0	0
$99$	− 30.5574i	− 3.07114i
$100$	0	0
$101$	− 12.9111i	− 1.28470i	−0.766412	−	0.642350i	$-0.777960\pi$
$101$	− 12.9111i	− 1.28470i	0.766412	−	0.642350i	$-0.222040\pi$
$102$	0	0
$103$	12.3012	1.21207	0.606035	−	0.795438i	$-0.292759\pi$
$103$	12.3012	1.21207	0.606035	+	0.795438i	$0.292759\pi$
$104$	0	0
$105$	−3.32646	−0.324629
$106$	0	0
$107$	13.7284i	1.32717i	0.748100	+	0.663586i	$0.230966\pi$
$107$	13.7284i	1.32717i	−0.748100	+	0.663586i	$0.769034\pi$
$108$	0	0
$109$	− 4.29180i	− 0.411080i	−0.978649	−	0.205540i	$-0.934105\pi$
$109$	− 4.29180i	− 0.411080i	0.978649	−	0.205540i	$-0.0658950\pi$
$110$	0	0
$111$	26.4675	2.51219
$112$	0	0
$113$	−13.2924	−1.25044	−0.625220	−	0.780448i	$-0.714991\pi$
$113$	−13.2924	−1.25044	−0.625220	+	0.780448i	$0.714991\pi$
$114$	0	0
$115$	2.16036i	0.201455i
$116$	0	0
$117$	36.2831i	3.35438i
$118$	0	0
$119$	−2.58941	−0.237371
$120$	0	0
$121$	−3.35449	−0.304954
$122$	0	0
$123$	− 10.7638i	− 0.970541i
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	20.7836	1.84425	0.922123	−	0.386896i	$-0.126453\pi$
$127$	20.7836	1.84425	0.922123	+	0.386896i	$0.126453\pi$
$128$	0	0
$129$	28.9193	2.54620
$130$	0	0
$131$	− 14.5517i	− 1.27138i	−0.771943	−	0.635692i	$-0.780715\pi$
$131$	− 14.5517i	− 1.27138i	0.771943	−	0.635692i	$-0.219285\pi$
$132$	0	0
$133$	− 5.09856i	− 0.442101i
$134$	0	0
$135$	16.8497	1.45019
$136$	0	0
$137$	−8.08623	−0.690853	−0.345427	−	0.938446i	$-0.612266\pi$
$137$	−8.08623	−0.690853	−0.345427	+	0.938446i	$0.612266\pi$
$138$	0	0
$139$	9.53686i	0.808906i	0.914559	+	0.404453i	$0.132538\pi$
$139$	9.53686i	0.808906i	−0.914559	+	0.404453i	$0.867462\pi$
$140$	0	0
$141$	40.9193i	3.44603i
$142$	0	0
$143$	17.0442	1.42531
$144$	0	0
$145$	0.164774	0.0136838
$146$	0	0
$147$	− 3.32646i	− 0.274362i
$148$	0	0
$149$	− 5.79482i	− 0.474730i	−0.971421	−	0.237365i	$-0.923716\pi$
$149$	− 5.79482i	− 0.474730i	0.971421	−	0.237365i	$-0.0762838\pi$
$150$	0	0
$151$	4.11648	0.334994	0.167497	−	0.985873i	$-0.446432\pi$
$151$	4.11648	0.334994	0.167497	+	0.985873i	$0.446432\pi$
$152$	0	0
$153$	20.8845	1.68841
$154$	0	0
$155$	6.36151i	0.510969i
$156$	0	0
$157$	− 17.8591i	− 1.42531i	−0.701516	−	0.712654i	$-0.747493\pi$
$157$	− 17.8591i	− 1.42531i	0.701516	−	0.712654i	$-0.252507\pi$
$158$	0	0
$159$	−20.8379	−1.65255
$160$	0	0
$161$	−2.16036	−0.170260
$162$	0	0
$163$	− 21.5269i	− 1.68611i	−0.537824	−	0.843057i	$-0.680754\pi$
$163$	− 21.5269i	− 1.68611i	0.537824	−	0.843057i	$-0.319246\pi$
$164$	0	0
$165$	− 12.6031i	− 0.981148i
$166$	0	0
$167$	4.62970	0.358257	0.179128	−	0.983826i	$-0.442672\pi$
$167$	4.62970	0.358257	0.179128	+	0.983826i	$0.442672\pi$
$168$	0	0
$169$	−7.23785	−0.556758
$170$	0	0
$171$	41.1216i	3.14465i
$172$	0	0
$173$	− 9.82472i	− 0.746960i	−0.927638	−	0.373480i	$-0.878164\pi$
$173$	− 9.82472i	− 0.746960i	0.927638	−	0.373480i	$-0.121836\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−22.8991	−1.72120
$178$	0	0
$179$	− 1.45410i	− 0.108685i	−0.998522	−	0.0543424i	$-0.982694\pi$
$179$	− 1.45410i	− 0.108685i	0.998522	−	0.0543424i	$-0.0173062\pi$
$180$	0	0
$181$	− 16.1639i	− 1.20146i	−0.799453	−	0.600728i	$-0.794877\pi$
$181$	− 16.1639i	− 1.20146i	0.799453	−	0.600728i	$-0.205123\pi$
$182$	0	0
$183$	−26.8985	−1.98840
$184$	0	0
$185$	7.95666	0.584986
$186$	0	0
$187$	− 9.81058i	− 0.717421i
$188$	0	0
$189$	16.8497i	1.22563i
$190$	0	0
$191$	−10.0537	−0.727460	−0.363730	−	0.931504i	$-0.618497\pi$
$191$	−10.0537	−0.727460	−0.363730	+	0.931504i	$0.618497\pi$
$192$	0	0
$193$	9.13803	0.657770	0.328885	−	0.944370i	$-0.393327\pi$
$193$	9.13803	0.657770	0.328885	+	0.944370i	$0.393327\pi$
$194$	0	0
$195$	14.9646i	1.07164i
$196$	0	0
$197$	20.1777i	1.43760i	0.695217	+	0.718800i	$0.255308\pi$
$197$	20.1777i	1.43760i	−0.695217	+	0.718800i	$0.744692\pi$
$198$	0	0
$199$	−13.8011	−0.978336	−0.489168	−	0.872189i	$-0.662700\pi$
$199$	−13.8011	−0.978336	−0.489168	+	0.872189i	$0.662700\pi$
$200$	0	0
$201$	−6.84561	−0.482852
$202$	0	0
$203$	0.164774i	0.0115649i
$204$	0	0
$205$	− 3.23582i	− 0.225999i
$206$	0	0
$207$	17.4240	1.21105
$208$	0	0
$209$	19.3171	1.33619
$210$	0	0
$211$	7.44801i	0.512742i	0.966578	+	0.256371i	$0.0825270\pi$
$211$	7.44801i	0.512742i	−0.966578	+	0.256371i	$0.917473\pi$
$212$	0	0
$213$	42.4787i	2.91059i
$214$	0	0
$215$	8.69371	0.592906
$216$	0	0
$217$	−6.36151	−0.431848
$218$	0	0
$219$	− 25.9746i	− 1.75520i
$220$	0	0
$221$	11.6488i	0.783586i
$222$	0	0
$223$	−28.5009	−1.90856	−0.954282	−	0.298908i	$-0.903378\pi$
$223$	−28.5009	−1.90856	−0.954282	+	0.298908i	$0.903378\pi$
$224$	0	0
$225$	8.06534	0.537689
$226$	0	0
$227$	8.74630i	0.580512i	0.956949	+	0.290256i	$0.0937405\pi$
$227$	8.74630i	0.580512i	−0.956949	+	0.290256i	$0.906260\pi$
$228$	0	0
$229$	− 1.51513i	− 0.100122i	−0.998746	−	0.0500612i	$-0.984058\pi$
$229$	− 1.51513i	− 0.100122i	0.998746	−	0.0500612i	$-0.0159416\pi$
$230$	0	0
$231$	12.6031	0.829221
$232$	0	0
$233$	6.33586	0.415076	0.207538	−	0.978227i	$-0.433455\pi$
$233$	6.33586	0.415076	0.207538	+	0.978227i	$0.433455\pi$
$234$	0	0
$235$	12.3012i	0.802439i
$236$	0	0
$237$	20.2270i	1.31389i
$238$	0	0
$239$	5.57864	0.360852	0.180426	−	0.983589i	$-0.442252\pi$
$239$	5.57864	0.360852	0.180426	+	0.983589i	$0.442252\pi$
$240$	0	0
$241$	−23.2762	−1.49935	−0.749675	−	0.661806i	$-0.769790\pi$
$241$	−23.2762	−1.49935	−0.749675	+	0.661806i	$0.769790\pi$
$242$	0	0
$243$	− 55.4111i	− 3.55463i
$244$	0	0
$245$	− 1.00000i	− 0.0638877i
$246$	0	0
$247$	−22.9366	−1.45942
$248$	0	0
$249$	59.2059	3.75202
$250$	0	0
$251$	15.3659i	0.969890i	0.874545	+	0.484945i	$0.161160\pi$
$251$	15.3659i	0.969890i	−0.874545	+	0.484945i	$0.838840\pi$
$252$	0	0
$253$	− 8.18502i	− 0.514588i
$254$	0	0
$255$	8.61357	0.539403
$256$	0	0
$257$	−4.48529	−0.279785	−0.139892	−	0.990167i	$-0.544676\pi$
$257$	−4.48529	−0.279785	−0.139892	+	0.990167i	$0.544676\pi$
$258$	0	0
$259$	7.95666i	0.494403i
$260$	0	0
$261$	− 1.32896i	− 0.0822606i
$262$	0	0
$263$	1.76164	0.108627	0.0543137	−	0.998524i	$-0.482703\pi$
$263$	1.76164	0.108627	0.0543137	+	0.998524i	$0.482703\pi$
$264$	0	0
$265$	−6.26428	−0.384811
$266$	0	0
$267$	− 13.8393i	− 0.846950i
$268$	0	0
$269$	4.09033i	0.249392i	0.992195	+	0.124696i	$0.0397956\pi$
$269$	4.09033i	0.249392i	−0.992195	+	0.124696i	$0.960204\pi$
$270$	0	0
$271$	−3.98287	−0.241942	−0.120971	−	0.992656i	$-0.538601\pi$
$271$	−3.98287	−0.241942	−0.120971	+	0.992656i	$0.538601\pi$
$272$	0	0
$273$	−14.9646	−0.905698
$274$	0	0
$275$	− 3.78873i	− 0.228469i
$276$	0	0
$277$	− 5.38618i	− 0.323624i	−0.986822	−	0.161812i	$-0.948266\pi$
$277$	− 5.38618i	− 0.323624i	0.986822	−	0.161812i	$-0.0517338\pi$
$278$	0	0
$279$	51.3078	3.07172
$280$	0	0
$281$	22.0969	1.31819	0.659096	−	0.752059i	$-0.270939\pi$
$281$	22.0969	1.31819	0.659096	+	0.752059i	$0.270939\pi$
$282$	0	0
$283$	− 16.1846i	− 0.962072i	−0.876701	−	0.481036i	$-0.840261\pi$
$283$	− 16.1846i	− 0.962072i	0.876701	−	0.481036i	$-0.159739\pi$
$284$	0	0
$285$	16.9602i	1.00463i
$286$	0	0
$287$	3.23582	0.191004
$288$	0	0
$289$	−10.2950	−0.605586
$290$	0	0
$291$	1.12698i	0.0660648i
$292$	0	0
$293$	− 4.07611i	− 0.238129i	−0.992887	−	0.119065i	$-0.962010\pi$
$293$	− 4.07611i	− 0.238129i	0.992887	−	0.119065i	$-0.0379896\pi$
$294$	0	0
$295$	−6.88394	−0.400798
$296$	0	0
$297$	−63.8388	−3.70431
$298$	0	0
$299$	9.71870i	0.562047i
$300$	0	0
$301$	8.69371i	0.501097i
$302$	0	0
$303$	−42.9482	−2.46731
$304$	0	0
$305$	−8.08623	−0.463016
$306$	0	0
$307$	11.3719i	0.649028i	0.945881	+	0.324514i	$0.105201\pi$
$307$	11.3719i	0.649028i	−0.945881	+	0.324514i	$0.894799\pi$
$308$	0	0
$309$	− 40.9193i	− 2.32782i
$310$	0	0
$311$	−29.6165	−1.67940	−0.839699	−	0.543052i	$-0.817269\pi$
$311$	−29.6165	−1.67940	−0.839699	+	0.543052i	$0.817269\pi$
$312$	0	0
$313$	−32.5113	−1.83764	−0.918822	−	0.394671i	$-0.870858\pi$
$313$	−32.5113	−1.83764	−0.918822	+	0.394671i	$0.870858\pi$
$314$	0	0
$315$	8.06534i	0.454430i
$316$	0	0
$317$	− 26.1898i	− 1.47096i	−0.677544	−	0.735482i	$-0.736956\pi$
$317$	− 26.1898i	− 1.47096i	0.677544	−	0.735482i	$-0.263044\pi$
$318$	0	0
$319$	−0.624285	−0.0349533
$320$	0	0
$321$	45.6669	2.54888
$322$	0	0
$323$	13.2023i	0.734594i
$324$	0	0
$325$	4.49865i	0.249540i
$326$	0	0
$327$	−14.2765	−0.789492
$328$	0	0
$329$	−12.3012	−0.678185
$330$	0	0
$331$	− 30.6714i	− 1.68585i	−0.538029	−	0.842927i	$-0.680831\pi$
$331$	− 30.6714i	− 1.68585i	0.538029	−	0.842927i	$-0.319169\pi$
$332$	0	0
$333$	− 64.1732i	− 3.51667i
$334$	0	0
$335$	−2.05793	−0.112437
$336$	0	0
$337$	−4.78095	−0.260435	−0.130217	−	0.991485i	$-0.541568\pi$
$337$	−4.78095	−0.260435	−0.130217	+	0.991485i	$0.541568\pi$
$338$	0	0
$339$	44.2165i	2.40151i
$340$	0	0
$341$	− 24.1021i	− 1.30520i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	7.18635	0.386900
$346$	0	0
$347$	− 7.22062i	− 0.387623i	−0.981039	−	0.193812i	$-0.937915\pi$
$347$	− 7.22062i	− 0.387623i	0.981039	−	0.193812i	$-0.0620850\pi$
$348$	0	0
$349$	− 29.0366i	− 1.55429i	−0.629320	−	0.777146i	$-0.716666\pi$
$349$	− 29.0366i	− 1.55429i	0.629320	−	0.777146i	$-0.283334\pi$
$350$	0	0
$351$	75.8007	4.04594
$352$	0	0
$353$	−7.03715	−0.374550	−0.187275	−	0.982308i	$-0.559966\pi$
$353$	−7.03715	−0.374550	−0.187275	+	0.982308i	$0.559966\pi$
$354$	0	0
$355$	12.7699i	0.677758i
$356$	0	0
$357$	8.61357i	0.455879i
$358$	0	0
$359$	−7.87446	−0.415598	−0.207799	−	0.978172i	$-0.566630\pi$
$359$	−7.87446	−0.415598	−0.207799	+	0.978172i	$0.566630\pi$
$360$	0	0
$361$	−6.99533	−0.368175
$362$	0	0
$363$	11.1586i	0.585674i
$364$	0	0
$365$	− 7.80848i	− 0.408714i
$366$	0	0
$367$	9.86379	0.514886	0.257443	−	0.966294i	$-0.417120\pi$
$367$	9.86379	0.514886	0.257443	+	0.966294i	$0.417120\pi$
$368$	0	0
$369$	−26.0980	−1.35861
$370$	0	0
$371$	− 6.26428i	− 0.325225i
$372$	0	0
$373$	28.7241i	1.48728i	0.668581	+	0.743639i	$0.266902\pi$
$373$	28.7241i	1.48728i	−0.668581	+	0.743639i	$0.733098\pi$
$374$	0	0
$375$	3.32646	0.171778
$376$	0	0
$377$	0.741261	0.0381769
$378$	0	0
$379$	− 1.32584i	− 0.0681039i	−0.999420	−	0.0340520i	$-0.989159\pi$
$379$	− 1.32584i	− 0.0681039i	0.999420	−	0.0340520i	$-0.0108412\pi$
$380$	0	0
$381$	− 69.1358i	− 3.54194i
$382$	0	0
$383$	−0.0542666	−0.00277290	−0.00138645	−	0.999999i	$-0.500441\pi$
$383$	−0.0542666	−0.00277290	−0.00138645	+	0.999999i	$0.500441\pi$
$384$	0	0
$385$	3.78873	0.193092
$386$	0	0
$387$	− 70.1178i	− 3.56429i
$388$	0	0
$389$	11.4865i	0.582386i	0.956664	+	0.291193i	$0.0940522\pi$
$389$	11.4865i	0.582386i	−0.956664	+	0.291193i	$0.905948\pi$
$390$	0	0
$391$	5.59406	0.282904
$392$	0	0
$393$	−48.4055	−2.44174
$394$	0	0
$395$	6.08065i	0.305950i
$396$	0	0
$397$	− 21.6782i	− 1.08800i	−0.839085	−	0.544000i	$-0.816909\pi$
$397$	− 21.6782i	− 1.08800i	0.839085	−	0.544000i	$-0.183091\pi$
$398$	0	0
$399$	−16.9602	−0.849070
$400$	0	0
$401$	23.6650	1.18177	0.590887	−	0.806754i	$-0.298778\pi$
$401$	23.6650	1.18177	0.590887	+	0.806754i	$0.298778\pi$
$402$	0	0
$403$	28.6182i	1.42557i
$404$	0	0
$405$	− 31.8537i	− 1.58282i
$406$	0	0
$407$	−30.1457	−1.49427
$408$	0	0
$409$	−30.9319	−1.52948	−0.764741	−	0.644338i	$-0.777133\pi$
$409$	−30.9319	−1.52948	−0.764741	+	0.644338i	$0.777133\pi$
$410$	0	0
$411$	26.8985i	1.32681i
$412$	0	0
$413$	− 6.88394i	− 0.338736i
$414$	0	0
$415$	17.7985	0.873693
$416$	0	0
$417$	31.7240	1.55353
$418$	0	0
$419$	− 12.4123i	− 0.606381i	−0.952930	−	0.303190i	$-0.901948\pi$
$419$	− 12.4123i	− 0.606381i	0.952930	−	0.303190i	$-0.0980518\pi$
$420$	0	0
$421$	− 29.8984i	− 1.45716i	−0.684962	−	0.728578i	$-0.740181\pi$
$421$	− 29.8984i	− 1.45716i	0.684962	−	0.728578i	$-0.259819\pi$
$422$	0	0
$423$	99.2130	4.82390
$424$	0	0
$425$	2.58941	0.125605
$426$	0	0
$427$	− 8.08623i	− 0.391320i
$428$	0	0
$429$	− 56.6968i	− 2.73735i
$430$	0	0
$431$	13.8670	0.667947	0.333974	−	0.942582i	$-0.391610\pi$
$431$	13.8670	0.667947	0.333974	+	0.942582i	$0.391610\pi$
$432$	0	0
$433$	−12.0695	−0.580022	−0.290011	−	0.957023i	$-0.593659\pi$
$433$	−12.0695	−0.580022	−0.290011	+	0.957023i	$0.593659\pi$
$434$	0	0
$435$	− 0.548115i	− 0.0262801i
$436$	0	0
$437$	11.0147i	0.526906i
$438$	0	0
$439$	27.6996	1.32203	0.661016	−	0.750372i	$-0.270126\pi$
$439$	27.6996	1.32203	0.661016	+	0.750372i	$0.270126\pi$
$440$	0	0
$441$	−8.06534	−0.384064
$442$	0	0
$443$	− 1.21001i	− 0.0574893i	−0.999587	−	0.0287446i	$-0.990849\pi$
$443$	− 1.21001i	− 0.0574893i	0.999587	−	0.0287446i	$-0.00915096\pi$
$444$	0	0
$445$	− 4.16036i	− 0.197220i
$446$	0	0
$447$	−19.2762	−0.911735
$448$	0	0
$449$	14.2432	0.672176	0.336088	−	0.941831i	$-0.390896\pi$
$449$	14.2432	0.672176	0.336088	+	0.941831i	$0.390896\pi$
$450$	0	0
$451$	12.2596i	0.577284i
$452$	0	0
$453$	− 13.6933i	− 0.643368i
$454$	0	0
$455$	−4.49865	−0.210900
$456$	0	0
$457$	−8.62385	−0.403406	−0.201703	−	0.979447i	$-0.564648\pi$
$457$	−8.62385	−0.403406	−0.201703	+	0.979447i	$0.564648\pi$
$458$	0	0
$459$	− 43.6307i	− 2.03651i
$460$	0	0
$461$	31.2313i	1.45459i	0.686326	+	0.727294i	$0.259222\pi$
$461$	31.2313i	1.45459i	−0.686326	+	0.727294i	$0.740778\pi$
$462$	0	0
$463$	−16.0612	−0.746427	−0.373214	−	0.927745i	$-0.621744\pi$
$463$	−16.0612	−0.746427	−0.373214	+	0.927745i	$0.621744\pi$
$464$	0	0
$465$	21.1613	0.981333
$466$	0	0
$467$	38.9918i	1.80433i	0.431395	+	0.902163i	$0.358022\pi$
$467$	38.9918i	1.80433i	−0.431395	+	0.902163i	$0.641978\pi$
$468$	0	0
$469$	− 2.05793i	− 0.0950262i
$470$	0	0
$471$	−59.4075	−2.73735
$472$	0	0
$473$	−32.9382	−1.51450
$474$	0	0
$475$	5.09856i	0.233938i
$476$	0	0
$477$	50.5235i	2.31331i
$478$	0	0
$479$	−37.4743	−1.71225	−0.856123	−	0.516772i	$-0.827134\pi$
$479$	−37.4743	−1.71225	−0.856123	+	0.516772i	$0.827134\pi$
$480$	0	0
$481$	35.7942	1.63208
$482$	0	0
$483$	7.18635i	0.326990i
$484$	0	0
$485$	0.338793i	0.0153838i
$486$	0	0
$487$	−16.8490	−0.763503	−0.381751	−	0.924265i	$-0.624679\pi$
$487$	−16.8490	−0.763503	−0.381751	+	0.924265i	$0.624679\pi$
$488$	0	0
$489$	−71.6082	−3.23824
$490$	0	0
$491$	− 6.18660i	− 0.279197i	−0.990208	−	0.139599i	$-0.955419\pi$
$491$	− 6.18660i	− 0.279197i	0.990208	−	0.139599i	$-0.0445813\pi$
$492$	0	0
$493$	− 0.426668i	− 0.0192162i
$494$	0	0
$495$	−30.5574	−1.37345
$496$	0	0
$497$	−12.7699	−0.572810
$498$	0	0
$499$	− 14.7169i	− 0.658821i	−0.944187	−	0.329410i	$-0.893150\pi$
$499$	− 14.7169i	− 0.658821i	0.944187	−	0.329410i	$-0.106850\pi$
$500$	0	0
$501$	− 15.4005i	− 0.688044i
$502$	0	0
$503$	17.9698	0.801234	0.400617	−	0.916246i	$-0.368796\pi$
$503$	17.9698	0.801234	0.400617	+	0.916246i	$0.368796\pi$
$504$	0	0
$505$	−12.9111	−0.574535
$506$	0	0
$507$	24.0764i	1.06927i
$508$	0	0
$509$	36.5442i	1.61979i	0.586574	+	0.809896i	$0.300476\pi$
$509$	36.5442i	1.61979i	−0.586574	+	0.809896i	$0.699524\pi$
$510$	0	0
$511$	7.80848	0.345427
$512$	0	0
$513$	85.9090	3.79298
$514$	0	0
$515$	− 12.3012i	− 0.542054i
$516$	0	0
$517$	− 46.6058i	− 2.04972i
$518$	0	0
$519$	−32.6816	−1.43456
$520$	0	0
$521$	−3.71262	−0.162653	−0.0813264	−	0.996688i	$-0.525916\pi$
$521$	−3.71262	−0.162653	−0.0813264	+	0.996688i	$0.525916\pi$
$522$	0	0
$523$	− 10.8921i	− 0.476280i	−0.971231	−	0.238140i	$-0.923462\pi$
$523$	− 10.8921i	− 0.476280i	0.971231	−	0.238140i	$-0.0765377\pi$
$524$	0	0
$525$	3.32646i	0.145179i
$526$	0	0
$527$	16.4726	0.717556
$528$	0	0
$529$	−18.3328	−0.797080
$530$	0	0
$531$	55.5213i	2.40942i
$532$	0	0
$533$	− 14.5568i	− 0.630525i
$534$	0	0
$535$	13.7284	0.593530
$536$	0	0
$537$	−4.83701	−0.208733
$538$	0	0
$539$	3.78873i	0.163192i
$540$	0	0
$541$	− 20.9421i	− 0.900370i	−0.892935	−	0.450185i	$-0.851358\pi$
$541$	− 20.9421i	− 0.900370i	0.892935	−	0.450185i	$-0.148642\pi$
$542$	0	0
$543$	−53.7687	−2.30744
$544$	0	0
$545$	−4.29180	−0.183840
$546$	0	0
$547$	− 3.72882i	− 0.159433i	−0.996818	−	0.0797164i	$-0.974599\pi$
$547$	− 3.72882i	− 0.159433i	0.996818	−	0.0797164i	$-0.0254015\pi$
$548$	0	0
$549$	65.2182i	2.78345i
$550$	0	0
$551$	0.840111	0.0357899
$552$	0	0
$553$	−6.08065	−0.258575
$554$	0	0
$555$	− 26.4675i	− 1.12348i
$556$	0	0
$557$	0.983640i	0.0416782i	0.999783	+	0.0208391i	$0.00663376\pi$
$557$	0.983640i	0.0416782i	−0.999783	+	0.0208391i	$0.993366\pi$
$558$	0	0
$559$	39.1100	1.65418
$560$	0	0
$561$	−32.6345	−1.37783
$562$	0	0
$563$	7.97634i	0.336163i	0.985773	+	0.168081i	$0.0537571\pi$
$563$	7.97634i	0.336163i	−0.985773	+	0.168081i	$0.946243\pi$
$564$	0	0
$565$	13.2924i	0.559214i
$566$	0	0
$567$	31.8537	1.33773
$568$	0	0
$569$	−39.7001	−1.66431	−0.832157	−	0.554540i	$-0.812894\pi$
$569$	−39.7001	−1.66431	−0.832157	+	0.554540i	$0.812894\pi$
$570$	0	0
$571$	− 45.8606i	− 1.91921i	−0.281356	−	0.959603i	$-0.590784\pi$
$571$	− 45.8606i	− 1.91921i	0.281356	−	0.959603i	$-0.409216\pi$
$572$	0	0
$573$	33.4432i	1.39711i
$574$	0	0
$575$	2.16036	0.0900932
$576$	0	0
$577$	24.0877	1.00278	0.501392	−	0.865220i	$-0.332821\pi$
$577$	24.0877	1.00278	0.501392	+	0.865220i	$0.332821\pi$
$578$	0	0
$579$	− 30.3973i	− 1.26327i
$580$	0	0
$581$	17.7985i	0.738405i
$582$	0	0
$583$	23.7337	0.982948
$584$	0	0
$585$	36.2831	1.50012
$586$	0	0
$587$	− 26.9561i	− 1.11260i	−0.830983	−	0.556298i	$-0.812221\pi$
$587$	− 26.9561i	− 1.11260i	0.830983	−	0.556298i	$-0.187779\pi$
$588$	0	0
$589$	32.4346i	1.33644i
$590$	0	0
$591$	67.1203	2.76096
$592$	0	0
$593$	6.13361	0.251877	0.125939	−	0.992038i	$-0.459806\pi$
$593$	6.13361	0.251877	0.125939	+	0.992038i	$0.459806\pi$
$594$	0	0
$595$	2.58941i	0.106155i
$596$	0	0
$597$	45.9089i	1.87893i
$598$	0	0
$599$	−14.6746	−0.599589	−0.299795	−	0.954004i	$-0.596918\pi$
$599$	−14.6746	−0.599589	−0.299795	+	0.954004i	$0.596918\pi$
$600$	0	0
$601$	2.84333	0.115982	0.0579909	−	0.998317i	$-0.481531\pi$
$601$	2.84333	0.115982	0.0579909	+	0.998317i	$0.481531\pi$
$602$	0	0
$603$	16.5979i	0.675918i
$604$	0	0
$605$	3.35449i	0.136380i
$606$	0	0
$607$	9.58677	0.389115	0.194557	−	0.980891i	$-0.437673\pi$
$607$	9.58677	0.389115	0.194557	+	0.980891i	$0.437673\pi$
$608$	0	0
$609$	0.548115	0.0222107
$610$	0	0
$611$	55.3386i	2.23876i
$612$	0	0
$613$	− 19.5376i	− 0.789117i	−0.918871	−	0.394559i	$-0.870897\pi$
$613$	− 19.5376i	− 0.789117i	0.918871	−	0.394559i	$-0.129103\pi$
$614$	0	0
$615$	−10.7638	−0.434039
$616$	0	0
$617$	34.6597	1.39535	0.697673	−	0.716416i	$-0.254219\pi$
$617$	34.6597	1.39535	0.697673	+	0.716416i	$0.254219\pi$
$618$	0	0
$619$	− 11.5053i	− 0.462436i	−0.972902	−	0.231218i	$-0.925729\pi$
$619$	− 11.5053i	− 0.462436i	0.972902	−	0.231218i	$-0.0742710\pi$
$620$	0	0
$621$	− 36.4013i	− 1.46073i
$622$	0	0
$623$	4.16036	0.166681
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	− 64.2575i	− 2.56620i
$628$	0	0
$629$	− 20.6031i	− 0.821498i
$630$	0	0
$631$	16.5523	0.658936	0.329468	−	0.944167i	$-0.393131\pi$
$631$	16.5523	0.658936	0.329468	+	0.944167i	$0.393131\pi$
$632$	0	0
$633$	24.7755	0.984739
$634$	0	0
$635$	− 20.7836i	− 0.824772i
$636$	0	0
$637$	− 4.49865i	− 0.178243i
$638$	0	0
$639$	102.994	4.07438
$640$	0	0
$641$	37.2781	1.47240	0.736198	−	0.676766i	$-0.236619\pi$
$641$	37.2781	1.47240	0.736198	+	0.676766i	$0.236619\pi$
$642$	0	0
$643$	− 1.89360i	− 0.0746761i	−0.999303	−	0.0373381i	$-0.988112\pi$
$643$	− 1.89360i	− 0.0746761i	0.999303	−	0.0373381i	$-0.0118878\pi$
$644$	0	0
$645$	− 28.9193i	− 1.13870i
$646$	0	0
$647$	−20.0019	−0.786354	−0.393177	−	0.919463i	$-0.628624\pi$
$647$	−20.0019	−0.786354	−0.393177	+	0.919463i	$0.628624\pi$
$648$	0	0
$649$	26.0814	1.02378
$650$	0	0
$651$	21.1613i	0.829377i
$652$	0	0
$653$	14.1109i	0.552203i	0.961128	+	0.276102i	$0.0890426\pi$
$653$	14.1109i	0.552203i	−0.961128	+	0.276102i	$0.910957\pi$
$654$	0	0
$655$	−14.5517	−0.568580
$656$	0	0
$657$	−62.9780	−2.45701
$658$	0	0
$659$	− 3.20106i	− 0.124695i	−0.998054	−	0.0623477i	$-0.980141\pi$
$659$	− 3.20106i	− 0.124695i	0.998054	−	0.0623477i	$-0.0198588\pi$
$660$	0	0
$661$	3.88158i	0.150976i	0.997147	+	0.0754879i	$0.0240514\pi$
$661$	3.88158i	0.150976i	−0.997147	+	0.0754879i	$0.975949\pi$
$662$	0	0
$663$	38.7494	1.50490
$664$	0	0
$665$	−5.09856	−0.197714
$666$	0	0
$667$	− 0.355971i	− 0.0137833i
$668$	0	0
$669$	94.8072i	3.66546i
$670$	0	0
$671$	30.6366	1.18271
$672$	0	0
$673$	−2.53366	−0.0976655	−0.0488327	−	0.998807i	$-0.515550\pi$
$673$	−2.53366	−0.0976655	−0.0488327	+	0.998807i	$0.515550\pi$
$674$	0	0
$675$	− 16.8497i	− 0.648544i
$676$	0	0
$677$	− 31.0982i	− 1.19520i	−0.801795	−	0.597600i	$-0.796121\pi$
$677$	− 31.0982i	− 1.19520i	0.801795	−	0.597600i	$-0.203879\pi$
$678$	0	0
$679$	−0.338793	−0.0130017
$680$	0	0
$681$	29.0942	1.11489
$682$	0	0
$683$	− 12.6342i	− 0.483435i	−0.970347	−	0.241718i	$-0.922289\pi$
$683$	− 12.6342i	− 0.483435i	0.970347	−	0.241718i	$-0.0777107\pi$
$684$	0	0
$685$	8.08623i	0.308959i
$686$	0	0
$687$	−5.04001	−0.192288
$688$	0	0
$689$	−28.1808	−1.07360
$690$	0	0
$691$	7.57863i	0.288305i	0.989555	+	0.144152i	$0.0460456\pi$
$691$	7.57863i	0.288305i	−0.989555	+	0.144152i	$0.953954\pi$
$692$	0	0
$693$	− 30.5574i	− 1.16078i
$694$	0	0
$695$	9.53686	0.361754
$696$	0	0
$697$	−8.37886	−0.317372
$698$	0	0
$699$	− 21.0760i	− 0.797167i
$700$	0	0
$701$	− 19.8028i	− 0.747942i	−0.927440	−	0.373971i	$-0.877996\pi$
$701$	− 19.8028i	− 0.747942i	0.927440	−	0.373971i	$-0.122004\pi$
$702$	0	0
$703$	40.5675	1.53003
$704$	0	0
$705$	40.9193	1.54111
$706$	0	0
$707$	− 12.9111i	− 0.485571i
$708$	0	0
$709$	− 38.8494i	− 1.45902i	−0.683971	−	0.729509i	$-0.739748\pi$
$709$	− 38.8494i	− 1.45902i	0.683971	−	0.729509i	$-0.260252\pi$
$710$	0	0
$711$	49.0425	1.83924
$712$	0	0
$713$	13.7432	0.514685
$714$	0	0
$715$	− 17.0442i	− 0.637416i
$716$	0	0
$717$	− 18.5571i	− 0.693028i
$718$	0	0
$719$	−40.5862	−1.51361	−0.756805	−	0.653641i	$-0.773241\pi$
$719$	−40.5862	−1.51361	−0.756805	+	0.653641i	$0.773241\pi$
$720$	0	0
$721$	12.3012	0.458119
$722$	0	0
$723$	77.4273i	2.87955i
$724$	0	0
$725$	− 0.164774i	− 0.00611956i
$726$	0	0
$727$	−9.75047	−0.361625	−0.180812	−	0.983518i	$-0.557873\pi$
$727$	−9.75047	−0.361625	−0.180812	+	0.983518i	$0.557873\pi$
$728$	0	0
$729$	−88.7618	−3.28747
$730$	0	0
$731$	− 22.5116i	− 0.832621i
$732$	0	0
$733$	− 42.2532i	− 1.56066i	−0.625369	−	0.780330i	$-0.715051\pi$
$733$	− 42.2532i	− 1.56066i	0.625369	−	0.780330i	$-0.284949\pi$
$734$	0	0
$735$	−3.32646	−0.122698
$736$	0	0
$737$	7.79693	0.287204
$738$	0	0
$739$	28.4031i	1.04482i	0.852693	+	0.522412i	$0.174968\pi$
$739$	28.4031i	1.04482i	−0.852693	+	0.522412i	$0.825032\pi$
$740$	0	0
$741$	76.2978i	2.80287i
$742$	0	0
$743$	15.5443	0.570265	0.285132	−	0.958488i	$-0.407962\pi$
$743$	15.5443	0.570265	0.285132	+	0.958488i	$0.407962\pi$
$744$	0	0
$745$	−5.79482	−0.212306
$746$	0	0
$747$	− 143.551i	− 5.25225i
$748$	0	0
$749$	13.7284i	0.501624i
$750$	0	0
$751$	−1.73572	−0.0633373	−0.0316687	−	0.999498i	$-0.510082\pi$
$751$	−1.73572	−0.0633373	−0.0316687	+	0.999498i	$0.510082\pi$
$752$	0	0
$753$	51.1142	1.86271
$754$	0	0
$755$	− 4.11648i	− 0.149814i
$756$	0	0
$757$	18.9267i	0.687903i	0.938987	+	0.343952i	$0.111766\pi$
$757$	18.9267i	0.687903i	−0.938987	+	0.343952i	$0.888234\pi$
$758$	0	0
$759$	−27.2272	−0.988283
$760$	0	0
$761$	12.2341	0.443487	0.221743	−	0.975105i	$-0.428825\pi$
$761$	12.2341	0.443487	0.221743	+	0.975105i	$0.428825\pi$
$762$	0	0
$763$	− 4.29180i	− 0.155373i
$764$	0	0
$765$	− 20.8845i	− 0.755080i
$766$	0	0
$767$	−30.9684	−1.11820
$768$	0	0
$769$	−11.1640	−0.402586	−0.201293	−	0.979531i	$-0.564514\pi$
$769$	−11.1640	−0.402586	−0.201293	+	0.979531i	$0.564514\pi$
$770$	0	0
$771$	14.9201i	0.537336i
$772$	0	0
$773$	47.4529i	1.70676i	0.521287	+	0.853381i	$0.325452\pi$
$773$	47.4529i	1.70676i	−0.521287	+	0.853381i	$0.674548\pi$
$774$	0	0
$775$	6.36151	0.228512
$776$	0	0
$777$	26.4675	0.949517
$778$	0	0
$779$	− 16.4980i	− 0.591103i
$780$	0	0
$781$	− 48.3819i	− 1.73124i
$782$	0	0
$783$	−2.77639	−0.0992200
$784$	0	0
$785$	−17.8591	−0.637417
$786$	0	0
$787$	− 26.6866i	− 0.951275i	−0.879641	−	0.475638i	$-0.842217\pi$
$787$	− 26.6866i	− 0.951275i	0.879641	−	0.475638i	$-0.157783\pi$
$788$	0	0
$789$	− 5.86003i	− 0.208623i
$790$	0	0
$791$	−13.2924	−0.472622
$792$	0	0
$793$	−36.3771	−1.29179
$794$	0	0
$795$	20.8379i	0.739043i
$796$	0	0
$797$	− 10.0143i	− 0.354726i	−0.984145	−	0.177363i	$-0.943243\pi$
$797$	− 10.0143i	− 0.354726i	0.984145	−	0.177363i	$-0.0567567\pi$
$798$	0	0
$799$	31.8527	1.12687
$800$	0	0
$801$	−33.5547	−1.18560
$802$	0	0
$803$	29.5842i	1.04401i
$804$	0	0
$805$	2.16036i	0.0761427i
$806$	0	0
$807$	13.6063	0.478966
$808$	0	0
$809$	19.6455	0.690699	0.345350	−	0.938474i	$-0.387760\pi$
$809$	19.6455	0.690699	0.345350	+	0.938474i	$0.387760\pi$
$810$	0	0
$811$	− 34.8354i	− 1.22324i	−0.791153	−	0.611618i	$-0.790519\pi$
$811$	− 34.8354i	− 1.22324i	0.791153	−	0.611618i	$-0.209481\pi$
$812$	0	0
$813$	13.2488i	0.464657i
$814$	0	0
$815$	−21.5269	−0.754053
$816$	0	0
$817$	44.3254	1.55075
$818$	0	0
$819$	36.2831i	1.26784i
$820$	0	0
$821$	42.6418i	1.48821i	0.668064	+	0.744104i	$0.267123\pi$
$821$	42.6418i	1.48821i	−0.668064	+	0.744104i	$0.732877\pi$
$822$	0	0
$823$	13.1802	0.459433	0.229717	−	0.973258i	$-0.426220\pi$
$823$	13.1802	0.459433	0.229717	+	0.973258i	$0.426220\pi$
$824$	0	0
$825$	−12.6031	−0.438783
$826$	0	0
$827$	− 27.2431i	− 0.947334i	−0.880704	−	0.473667i	$-0.842930\pi$
$827$	− 27.2431i	− 0.947334i	0.880704	−	0.473667i	$-0.157070\pi$
$828$	0	0
$829$	39.0178i	1.35514i	0.735457	+	0.677572i	$0.236968\pi$
$829$	39.0178i	1.35514i	−0.735457	+	0.677572i	$0.763032\pi$
$830$	0	0
$831$	−17.9169	−0.621531
$832$	0	0
$833$	−2.58941	−0.0897177
$834$	0	0
$835$	− 4.62970i	− 0.160217i
$836$	0	0
$837$	− 107.189i	− 3.70500i
$838$	0	0
$839$	−12.5202	−0.432247	−0.216123	−	0.976366i	$-0.569341\pi$
$839$	−12.5202	−0.432247	−0.216123	+	0.976366i	$0.569341\pi$
$840$	0	0
$841$	28.9728	0.999064
$842$	0	0
$843$	− 73.5045i	− 2.53163i
$844$	0	0
$845$	7.23785i	0.248990i
$846$	0	0
$847$	−3.35449	−0.115262
$848$	0	0
$849$	−53.8373	−1.84769
$850$	0	0
$851$	− 17.1893i	− 0.589240i
$852$	0	0
$853$	16.1155i	0.551786i	0.961188	+	0.275893i	$0.0889735\pi$
$853$	16.1155i	0.551786i	−0.961188	+	0.275893i	$0.911027\pi$
$854$	0	0
$855$	41.1216	1.40633
$856$	0	0
$857$	13.3966	0.457618	0.228809	−	0.973471i	$-0.426517\pi$
$857$	13.3966	0.457618	0.228809	+	0.973471i	$0.426517\pi$
$858$	0	0
$859$	51.5860i	1.76009i	0.474890	+	0.880045i	$0.342488\pi$
$859$	51.5860i	1.76009i	−0.474890	+	0.880045i	$0.657512\pi$
$860$	0	0
$861$	− 10.7638i	− 0.366830i
$862$	0	0
$863$	−20.9388	−0.712765	−0.356382	−	0.934340i	$-0.615990\pi$
$863$	−20.9388	−0.712765	−0.356382	+	0.934340i	$0.615990\pi$
$864$	0	0
$865$	−9.82472	−0.334051
$866$	0	0
$867$	34.2458i	1.16305i
$868$	0	0
$869$	− 23.0379i	− 0.781509i
$870$	0	0
$871$	−9.25789	−0.313692
$872$	0	0
$873$	2.73248	0.0924805
$874$	0	0
$875$	1.00000i	0.0338062i
$876$	0	0
$877$	5.78490i	0.195342i	0.995219	+	0.0976710i	$0.0311393\pi$
$877$	5.78490i	0.195342i	−0.995219	+	0.0976710i	$0.968861\pi$
$878$	0	0
$879$	−13.5590	−0.457335
$880$	0	0
$881$	−7.47470	−0.251829	−0.125915	−	0.992041i	$-0.540187\pi$
$881$	−7.47470	−0.251829	−0.125915	+	0.992041i	$0.540187\pi$
$882$	0	0
$883$	46.3587i	1.56009i	0.625721	+	0.780047i	$0.284805\pi$
$883$	46.3587i	1.56009i	−0.625721	+	0.780047i	$0.715195\pi$
$884$	0	0
$885$	22.8991i	0.769746i
$886$	0	0
$887$	3.43749	0.115420	0.0577099	−	0.998333i	$-0.481620\pi$
$887$	3.43749	0.115420	0.0577099	+	0.998333i	$0.481620\pi$
$888$	0	0
$889$	20.7836	0.697060
$890$	0	0
$891$	120.685i	4.04311i
$892$	0	0
$893$	62.7182i	2.09878i
$894$	0	0
$895$	−1.45410	−0.0486053
$896$	0	0
$897$	32.3289	1.07943
$898$	0	0
$899$	− 1.04821i	− 0.0349599i
$900$	0	0
$901$	16.2208i	0.540393i
$902$	0	0
$903$	28.9193	0.962374
$904$	0	0
$905$	−16.1639	−0.537308
$906$	0	0
$907$	− 37.3061i	− 1.23873i	−0.785104	−	0.619364i	$-0.787390\pi$
$907$	− 37.3061i	− 1.23873i	0.785104	−	0.619364i	$-0.212610\pi$
$908$	0	0
$909$	104.132i	3.45385i
$910$	0	0
$911$	−22.8886	−0.758333	−0.379167	−	0.925328i	$-0.623789\pi$
$911$	−22.8886	−0.758333	−0.379167	+	0.925328i	$0.623789\pi$
$912$	0	0
$913$	−67.4337	−2.23173
$914$	0	0
$915$	26.8985i	0.889238i
$916$	0	0
$917$	− 14.5517i	− 0.480538i
$918$	0	0
$919$	40.4410	1.33403	0.667013	−	0.745046i	$-0.267573\pi$
$919$	40.4410	1.33403	0.667013	+	0.745046i	$0.267573\pi$
$920$	0	0
$921$	37.8282	1.24648
$922$	0	0
$923$	57.4475i	1.89091i
$924$	0	0
$925$	− 7.95666i	− 0.261613i
$926$	0	0
$927$	−99.2130	−3.25858
$928$	0	0
$929$	−12.0701	−0.396007	−0.198004	−	0.980201i	$-0.563446\pi$
$929$	−12.0701	−0.396007	−0.198004	+	0.980201i	$0.563446\pi$
$930$	0	0
$931$	− 5.09856i	− 0.167099i
$932$	0	0
$933$	98.5182i	3.22534i
$934$	0	0
$935$	−9.81058	−0.320840
$936$	0	0
$937$	−16.3045	−0.532646	−0.266323	−	0.963884i	$-0.585809\pi$
$937$	−16.3045	−0.532646	−0.266323	+	0.963884i	$0.585809\pi$
$938$	0	0
$939$	108.147i	3.52926i
$940$	0	0
$941$	− 21.3378i	− 0.695593i	−0.937570	−	0.347796i	$-0.886930\pi$
$941$	− 21.3378i	− 0.695593i	0.937570	−	0.347796i	$-0.113070\pi$
$942$	0	0
$943$	−6.99053	−0.227643
$944$	0	0
$945$	16.8497	0.548119
$946$	0	0
$947$	− 20.3715i	− 0.661984i	−0.943634	−	0.330992i	$-0.892617\pi$
$947$	− 20.3715i	− 0.661984i	0.943634	−	0.330992i	$-0.107383\pi$
$948$	0	0
$949$	− 35.1276i	− 1.14029i
$950$	0	0
$951$	−87.1193	−2.82504
$952$	0	0
$953$	41.9301	1.35825	0.679124	−	0.734024i	$-0.262360\pi$
$953$	41.9301	1.35825	0.679124	+	0.734024i	$0.262360\pi$
$954$	0	0
$955$	10.0537i	0.325330i
$956$	0	0
$957$	2.07666i	0.0671289i
$958$	0	0
$959$	−8.08623	−0.261118
$960$	0	0
$961$	9.46883	0.305446
$962$	0	0
$963$	− 110.724i	− 3.56803i
$964$	0	0
$965$	− 9.13803i	− 0.294164i
$966$	0	0
$967$	37.1592	1.19496	0.597480	−	0.801884i	$-0.296169\pi$
$967$	37.1592	1.19496	0.597480	+	0.801884i	$0.296169\pi$
$968$	0	0
$969$	43.9168	1.41081
$970$	0	0
$971$	7.96541i	0.255622i	0.991799	+	0.127811i	$0.0407951\pi$
$971$	7.96541i	0.255622i	−0.991799	+	0.127811i	$0.959205\pi$
$972$	0	0
$973$	9.53686i	0.305738i
$974$	0	0
$975$	14.9646	0.479250
$976$	0	0
$977$	10.7944	0.345344	0.172672	−	0.984979i	$-0.444760\pi$
$977$	10.7944	0.345344	0.172672	+	0.984979i	$0.444760\pi$
$978$	0	0
$979$	15.7625i	0.503771i
$980$	0	0
$981$	34.6148i	1.10517i
$982$	0	0
$983$	4.44703	0.141838	0.0709192	−	0.997482i	$-0.477407\pi$
$983$	4.44703	0.141838	0.0709192	+	0.997482i	$0.477407\pi$
$984$	0	0
$985$	20.1777	0.642914
$986$	0	0
$987$	40.9193i	1.30248i
$988$	0	0
$989$	− 18.7815i	− 0.597218i
$990$	0	0
$991$	8.79041	0.279237	0.139618	−	0.990205i	$-0.455412\pi$
$991$	8.79041	0.279237	0.139618	+	0.990205i	$0.455412\pi$
$992$	0	0
$993$	−102.027	−3.23774
$994$	0	0
$995$	13.8011i	0.437525i
$996$	0	0
$997$	− 4.07611i	− 0.129092i	−0.997915	−	0.0645459i	$-0.979440\pi$
$997$	− 4.07611i	− 0.129092i	0.997915	−	0.0645459i	$-0.0205599\pi$
$998$	0	0
$999$	−134.067	−4.24169

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.b.h.1121.1	yes	12
4.3	odd	2		2240.2.b.g.1121.12	yes	12
8.3	odd	2		2240.2.b.g.1121.1	✓	12
8.5	even	2	inner	2240.2.b.h.1121.12	yes	12
16.3	odd	4		8960.2.a.cc.1.1		6
16.5	even	4		8960.2.a.ce.1.1		6
16.11	odd	4		8960.2.a.ch.1.6		6
16.13	even	4		8960.2.a.cb.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2240.2.b.g.1121.1	✓	12	8.3	odd	2
2240.2.b.g.1121.12	yes	12	4.3	odd	2
2240.2.b.h.1121.1	yes	12	1.1	even	1	trivial
2240.2.b.h.1121.12	yes	12	8.5	even	2	inner
8960.2.a.cb.1.6		6	16.13	even	4
8960.2.a.cc.1.1		6	16.3	odd	4
8960.2.a.ce.1.1		6	16.5	even	4
8960.2.a.ch.1.6		6	16.11	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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.32646i q^{3} -1.00000i q^{5} +1.00000 q^{7} -8.06534 q^{9} +O(q^{10})\)	\(q-3.32646i q^{3} -1.00000i q^{5} +1.00000 q^{7} -8.06534 q^{9} +3.78873i q^{11} -4.49865i q^{13} -3.32646 q^{15} -2.58941 q^{17} -5.09856i q^{19} -3.32646i q^{21} -2.16036 q^{23} -1.00000 q^{25} +16.8497i q^{27} +0.164774i q^{29} -6.36151 q^{31} +12.6031 q^{33} -1.00000i q^{35} +7.95666i q^{37} -14.9646 q^{39} +3.23582 q^{41} +8.69371i q^{43} +8.06534i q^{45} -12.3012 q^{47} +1.00000 q^{49} +8.61357i q^{51} -6.26428i q^{53} +3.78873 q^{55} -16.9602 q^{57} -6.88394i q^{59} -8.08623i q^{61} -8.06534 q^{63} -4.49865 q^{65} -2.05793i q^{67} +7.18635i q^{69} -12.7699 q^{71} +7.80848 q^{73} +3.32646i q^{75} +3.78873i q^{77} -6.08065 q^{79} +31.8537 q^{81} +17.7985i q^{83} +2.58941i q^{85} +0.548115 q^{87} +4.16036 q^{89} -4.49865i q^{91} +21.1613i q^{93} -5.09856 q^{95} -0.338793 q^{97} -30.5574i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{7} - 20 q^{9}+O(q^{10}) \) 12 * q + 12 * q^7 - 20 * q^9	\( 12 q + 12 q^{7} - 20 q^{9} + 16 q^{17} + 8 q^{23} - 12 q^{25} - 8 q^{31} + 72 q^{33} - 32 q^{39} - 32 q^{47} + 12 q^{49} + 8 q^{55} + 8 q^{57} - 20 q^{63} + 8 q^{65} + 48 q^{71} + 8 q^{73} - 16 q^{79} + 92 q^{81} - 32 q^{87} + 16 q^{89} + 32 q^{97}+O(q^{100}) \) 12 * q + 12 * q^7 - 20 * q^9 + 16 * q^17 + 8 * q^23 - 12 * q^25 - 8 * q^31 + 72 * q^33 - 32 * q^39 - 32 * q^47 + 12 * q^49 + 8 * q^55 + 8 * q^57 - 20 * q^63 + 8 * q^65 + 48 * q^71 + 8 * q^73 - 16 * q^79 + 92 * q^81 - 32 * q^87 + 16 * q^89 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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