LMFDB - Embedded newform 2240.2.g.k.449.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(449,2240)]

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, 0, 1, 0]))

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

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

chi := DirichletCharacter("2240.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.1
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	2240.449
Dual form			2240.2.g.k.449.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -2.23607i q^{5} -1.00000i q^{7} +2.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -2.23607i q^{5} -1.00000i q^{7} +2.00000 q^{9} -2.23607 q^{11} -2.23607i q^{13} -2.23607 q^{15} +2.23607i q^{17} -4.47214 q^{19} -1.00000 q^{21} -4.00000i q^{23} -5.00000 q^{25} -5.00000i q^{27} -1.00000 q^{29} +2.23607i q^{33} -2.23607 q^{35} -8.94427i q^{37} -2.23607 q^{39} +6.00000i q^{43} -4.47214i q^{45} +3.00000i q^{47} -1.00000 q^{49} +2.23607 q^{51} -4.47214i q^{53} +5.00000i q^{55} +4.47214i q^{57} -4.47214 q^{59} +10.0000 q^{61} -2.00000i q^{63} -5.00000 q^{65} -2.00000i q^{67} -4.00000 q^{69} -8.94427 q^{71} +13.4164i q^{73} +5.00000i q^{75} +2.23607i q^{77} -15.6525 q^{79} +1.00000 q^{81} -4.00000i q^{83} +5.00000 q^{85} +1.00000i q^{87} +14.0000 q^{89} -2.23607 q^{91} +10.0000i q^{95} +2.23607i q^{97} -4.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{9}+O(q^{10}) $ 4 * q + 8 * q^9	$ 4 q + 8 q^{9} - 4 q^{21} - 20 q^{25} - 4 q^{29} - 4 q^{49} + 40 q^{61} - 20 q^{65} - 16 q^{69} + 4 q^{81} + 20 q^{85} + 56 q^{89}+O(q^{100}) $ 4 * q + 8 * q^9 - 4 * q^21 - 20 * q^25 - 4 * q^29 - 4 * q^49 + 40 * q^61 - 20 * q^65 - 16 * q^69 + 4 * q^81 + 20 * q^85 + 56 * q^89

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
$3$	− 1.00000i	− 0.577350i	−0.957427	−	0.288675i	$-0.906785\pi$
$3$	− 1.00000i	− 0.577350i	0.957427	−	0.288675i	$-0.0932147\pi$
$4$	0	0
$5$	− 2.23607i	− 1.00000i
$5$	− 2.23607i	− 1.00000i
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	−2.23607	−0.674200	−0.337100	−	0.941469i	$-0.609446\pi$
$11$	−2.23607	−0.674200	−0.337100	+	0.941469i	$0.609446\pi$
$12$	0	0
$13$	− 2.23607i	− 0.620174i	−0.950708	−	0.310087i	$-0.899642\pi$
$13$	− 2.23607i	− 0.620174i	0.950708	−	0.310087i	$-0.100358\pi$
$14$	0	0
$15$	−2.23607	−0.577350
$16$	0	0
$17$	2.23607i	0.542326i	0.962533	+	0.271163i	$0.0874083\pi$
$17$	2.23607i	0.542326i	−0.962533	+	0.271163i	$0.912592\pi$
$18$	0	0
$19$	−4.47214	−1.02598	−0.512989	−	0.858395i	$-0.671462\pi$
$19$	−4.47214	−1.02598	−0.512989	+	0.858395i	$0.671462\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	− 5.00000i	− 0.962250i
$28$	0	0
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	2.23607i	0.389249i
$34$	0	0
$35$	−2.23607	−0.377964
$36$	0	0
$37$	− 8.94427i	− 1.47043i	−0.677834	−	0.735215i	$-0.737081\pi$
$37$	− 8.94427i	− 1.47043i	0.677834	−	0.735215i	$-0.262919\pi$
$38$	0	0
$39$	−2.23607	−0.358057
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	6.00000i	0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$	6.00000i	0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$	0	0
$45$	− 4.47214i	− 0.666667i
$46$	0	0
$47$	3.00000i	0.437595i	0.975770	+	0.218797i	$0.0702134\pi$
$47$	3.00000i	0.437595i	−0.975770	+	0.218797i	$0.929787\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	2.23607	0.313112
$52$	0	0
$53$	− 4.47214i	− 0.614295i	−0.951662	−	0.307148i	$-0.900625\pi$
$53$	− 4.47214i	− 0.614295i	0.951662	−	0.307148i	$-0.0993745\pi$
$54$	0	0
$55$	5.00000i	0.674200i
$56$	0	0
$57$	4.47214i	0.592349i
$58$	0	0
$59$	−4.47214	−0.582223	−0.291111	−	0.956689i	$-0.594025\pi$
$59$	−4.47214	−0.582223	−0.291111	+	0.956689i	$0.594025\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	− 2.00000i	− 0.251976i
$64$	0	0
$65$	−5.00000	−0.620174
$66$	0	0
$67$	− 2.00000i	− 0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$	− 2.00000i	− 0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−8.94427	−1.06149	−0.530745	−	0.847532i	$-0.678088\pi$
$71$	−8.94427	−1.06149	−0.530745	+	0.847532i	$0.678088\pi$
$72$	0	0
$73$	13.4164i	1.57027i	0.619324	+	0.785136i	$0.287407\pi$
$73$	13.4164i	1.57027i	−0.619324	+	0.785136i	$0.712593\pi$
$74$	0	0
$75$	5.00000i	0.577350i
$76$	0	0
$77$	2.23607i	0.254824i
$78$	0	0
$79$	−15.6525	−1.76104	−0.880521	−	0.474008i	$-0.842807\pi$
$79$	−15.6525	−1.76104	−0.880521	+	0.474008i	$0.842807\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 4.00000i	− 0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$	− 4.00000i	− 0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$	0	0
$85$	5.00000	0.542326
$86$	0	0
$87$	1.00000i	0.107211i
$88$	0	0
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	−2.23607	−0.234404
$92$	0	0
$93$	0	0
$94$	0	0
$95$	10.0000i	1.02598i
$96$	0	0
$97$	2.23607i	0.227038i	0.993536	+	0.113519i	$0.0362123\pi$
$97$	2.23607i	0.227038i	−0.993536	+	0.113519i	$0.963788\pi$
$98$	0	0
$99$	−4.47214	−0.449467
$100$	0	0
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	− 11.0000i	− 1.08386i	−0.840423	−	0.541931i	$-0.817693\pi$
$103$	− 11.0000i	− 1.08386i	0.840423	−	0.541931i	$-0.182307\pi$
$104$	0	0
$105$	2.23607i	0.218218i
$106$	0	0
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	0	0
$109$	−15.0000	−1.43674	−0.718370	−	0.695662i	$-0.755111\pi$
$109$	−15.0000	−1.43674	−0.718370	+	0.695662i	$0.755111\pi$
$110$	0	0
$111$	−8.94427	−0.848953
$112$	0	0
$113$	− 8.94427i	− 0.841406i	−0.907198	−	0.420703i	$-0.861783\pi$
$113$	− 8.94427i	− 0.841406i	0.907198	−	0.420703i	$-0.138217\pi$
$114$	0	0
$115$	−8.94427	−0.834058
$116$	0	0
$117$	− 4.47214i	− 0.413449i
$118$	0	0
$119$	2.23607	0.204980
$120$	0	0
$121$	−6.00000	−0.545455
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.1803i	1.00000i
$126$	0	0
$127$	2.00000i	0.177471i	0.996055	+	0.0887357i	$0.0282826\pi$
$127$	2.00000i	0.177471i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$	0	0
$129$	6.00000	0.528271
$130$	0	0
$131$	−17.8885	−1.56293	−0.781465	−	0.623949i	$-0.785527\pi$
$131$	−17.8885	−1.56293	−0.781465	+	0.623949i	$0.785527\pi$
$132$	0	0
$133$	4.47214i	0.387783i
$134$	0	0
$135$	−11.1803	−0.962250
$136$	0	0
$137$	− 13.4164i	− 1.14624i	−0.819471	−	0.573121i	$-0.805733\pi$
$137$	− 13.4164i	− 1.14624i	0.819471	−	0.573121i	$-0.194267\pi$
$138$	0	0
$139$	13.4164	1.13796	0.568982	−	0.822350i	$-0.307337\pi$
$139$	13.4164	1.13796	0.568982	+	0.822350i	$0.307337\pi$
$140$	0	0
$141$	3.00000	0.252646
$142$	0	0
$143$	5.00000i	0.418121i
$144$	0	0
$145$	2.23607i	0.185695i
$146$	0	0
$147$	1.00000i	0.0824786i
$148$	0	0
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	15.6525	1.27378	0.636890	−	0.770955i	$-0.280220\pi$
$151$	15.6525	1.27378	0.636890	+	0.770955i	$0.280220\pi$
$152$	0	0
$153$	4.47214i	0.361551i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.47214i	0.356915i	0.983948	+	0.178458i	$0.0571108\pi$
$157$	4.47214i	0.356915i	−0.983948	+	0.178458i	$0.942889\pi$
$158$	0	0
$159$	−4.47214	−0.354663
$160$	0	0
$161$	−4.00000	−0.315244
$162$	0	0
$163$	− 14.0000i	− 1.09656i	−0.836293	−	0.548282i	$-0.815282\pi$
$163$	− 14.0000i	− 1.09656i	0.836293	−	0.548282i	$-0.184718\pi$
$164$	0	0
$165$	5.00000	0.389249
$166$	0	0
$167$	7.00000i	0.541676i	0.962625	+	0.270838i	$0.0873008\pi$
$167$	7.00000i	0.541676i	−0.962625	+	0.270838i	$0.912699\pi$
$168$	0	0
$169$	8.00000	0.615385
$170$	0	0
$171$	−8.94427	−0.683986
$172$	0	0
$173$	2.23607i	0.170005i	0.996381	+	0.0850026i	$0.0270898\pi$
$173$	2.23607i	0.170005i	−0.996381	+	0.0850026i	$0.972910\pi$
$174$	0	0
$175$	5.00000i	0.377964i
$176$	0	0
$177$	4.47214i	0.336146i
$178$	0	0
$179$	−17.8885	−1.33705	−0.668526	−	0.743689i	$-0.733075\pi$
$179$	−17.8885	−1.33705	−0.668526	+	0.743689i	$0.733075\pi$
$180$	0	0
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	− 10.0000i	− 0.739221i
$184$	0	0
$185$	−20.0000	−1.47043
$186$	0	0
$187$	− 5.00000i	− 0.365636i
$188$	0	0
$189$	−5.00000	−0.363696
$190$	0	0
$191$	−24.5967	−1.77976	−0.889879	−	0.456196i	$-0.849211\pi$
$191$	−24.5967	−1.77976	−0.889879	+	0.456196i	$0.849211\pi$
$192$	0	0
$193$	− 13.4164i	− 0.965734i	−0.875694	−	0.482867i	$-0.839595\pi$
$193$	− 13.4164i	− 0.965734i	0.875694	−	0.482867i	$-0.160405\pi$
$194$	0	0
$195$	5.00000i	0.358057i
$196$	0	0
$197$	17.8885i	1.27451i	0.770655	+	0.637253i	$0.219929\pi$
$197$	17.8885i	1.27451i	−0.770655	+	0.637253i	$0.780071\pi$
$198$	0	0
$199$	22.3607	1.58511	0.792553	−	0.609803i	$-0.208751\pi$
$199$	22.3607	1.58511	0.792553	+	0.609803i	$0.208751\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	0	0
$203$	1.00000i	0.0701862i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 8.00000i	− 0.556038i
$208$	0	0
$209$	10.0000	0.691714
$210$	0	0
$211$	−6.70820	−0.461812	−0.230906	−	0.972976i	$-0.574169\pi$
$211$	−6.70820	−0.461812	−0.230906	+	0.972976i	$0.574169\pi$
$212$	0	0
$213$	8.94427i	0.612851i
$214$	0	0
$215$	13.4164	0.914991
$216$	0	0
$217$	0	0
$218$	0	0
$219$	13.4164	0.906597
$220$	0	0
$221$	5.00000	0.336336
$222$	0	0
$223$	− 19.0000i	− 1.27233i	−0.771551	−	0.636167i	$-0.780519\pi$
$223$	− 19.0000i	− 1.27233i	0.771551	−	0.636167i	$-0.219481\pi$
$224$	0	0
$225$	−10.0000	−0.666667
$226$	0	0
$227$	17.0000i	1.12833i	0.825662	+	0.564165i	$0.190802\pi$
$227$	17.0000i	1.12833i	−0.825662	+	0.564165i	$0.809198\pi$
$228$	0	0
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	0	0
$231$	2.23607	0.147122
$232$	0	0
$233$	− 17.8885i	− 1.17192i	−0.810341	−	0.585959i	$-0.800718\pi$
$233$	− 17.8885i	− 1.17192i	0.810341	−	0.585959i	$-0.199282\pi$
$234$	0	0
$235$	6.70820	0.437595
$236$	0	0
$237$	15.6525i	1.01674i
$238$	0	0
$239$	−6.70820	−0.433918	−0.216959	−	0.976181i	$-0.569614\pi$
$239$	−6.70820	−0.433918	−0.216959	+	0.976181i	$0.569614\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	− 16.0000i	− 1.02640i
$244$	0	0
$245$	2.23607i	0.142857i
$246$	0	0
$247$	10.0000i	0.636285i
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	22.3607	1.41139	0.705697	−	0.708514i	$-0.250634\pi$
$251$	22.3607	1.41139	0.705697	+	0.708514i	$0.250634\pi$
$252$	0	0
$253$	8.94427i	0.562322i
$254$	0	0
$255$	− 5.00000i	− 0.313112i
$256$	0	0
$257$	− 22.3607i	− 1.39482i	−0.716672	−	0.697410i	$-0.754335\pi$
$257$	− 22.3607i	− 1.39482i	0.716672	−	0.697410i	$-0.245665\pi$
$258$	0	0
$259$	−8.94427	−0.555770
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	− 16.0000i	− 0.986602i	−0.869859	−	0.493301i	$-0.835790\pi$
$263$	− 16.0000i	− 0.986602i	0.869859	−	0.493301i	$-0.164210\pi$
$264$	0	0
$265$	−10.0000	−0.614295
$266$	0	0
$267$	− 14.0000i	− 0.856786i
$268$	0	0
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	4.47214	0.271663	0.135831	−	0.990732i	$-0.456629\pi$
$271$	4.47214	0.271663	0.135831	+	0.990732i	$0.456629\pi$
$272$	0	0
$273$	2.23607i	0.135333i
$274$	0	0
$275$	11.1803	0.674200
$276$	0	0
$277$	− 13.4164i	− 0.806114i	−0.915175	−	0.403057i	$-0.867948\pi$
$277$	− 13.4164i	− 0.806114i	0.915175	−	0.403057i	$-0.132052\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	15.0000	0.894825	0.447412	−	0.894328i	$-0.352346\pi$
$281$	15.0000	0.894825	0.447412	+	0.894328i	$0.352346\pi$
$282$	0	0
$283$	− 31.0000i	− 1.84276i	−0.388664	−	0.921379i	$-0.627063\pi$
$283$	− 31.0000i	− 1.84276i	0.388664	−	0.921379i	$-0.372937\pi$
$284$	0	0
$285$	10.0000	0.592349
$286$	0	0
$287$	0	0
$288$	0	0
$289$	12.0000	0.705882
$290$	0	0
$291$	2.23607	0.131081
$292$	0	0
$293$	11.1803i	0.653162i	0.945169	+	0.326581i	$0.105897\pi$
$293$	11.1803i	0.653162i	−0.945169	+	0.326581i	$0.894103\pi$
$294$	0	0
$295$	10.0000i	0.582223i
$296$	0	0
$297$	11.1803i	0.648749i
$298$	0	0
$299$	−8.94427	−0.517261
$300$	0	0
$301$	6.00000	0.345834
$302$	0	0
$303$	2.00000i	0.114897i
$304$	0	0
$305$	− 22.3607i	− 1.28037i
$306$	0	0
$307$	23.0000i	1.31268i	0.754466	+	0.656340i	$0.227896\pi$
$307$	23.0000i	1.31268i	−0.754466	+	0.656340i	$0.772104\pi$
$308$	0	0
$309$	−11.0000	−0.625768
$310$	0	0
$311$	22.3607	1.26796	0.633979	−	0.773350i	$-0.281421\pi$
$311$	22.3607	1.26796	0.633979	+	0.773350i	$0.281421\pi$
$312$	0	0
$313$	− 15.6525i	− 0.884730i	−0.896835	−	0.442365i	$-0.854140\pi$
$313$	− 15.6525i	− 0.884730i	0.896835	−	0.442365i	$-0.145860\pi$
$314$	0	0
$315$	−4.47214	−0.251976
$316$	0	0
$317$	17.8885i	1.00472i	0.864658	+	0.502360i	$0.167535\pi$
$317$	17.8885i	1.00472i	−0.864658	+	0.502360i	$0.832465\pi$
$318$	0	0
$319$	2.23607	0.125196
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	− 10.0000i	− 0.556415i
$324$	0	0
$325$	11.1803i	0.620174i
$326$	0	0
$327$	15.0000i	0.829502i
$328$	0	0
$329$	3.00000	0.165395
$330$	0	0
$331$	35.7771	1.96649	0.983243	−	0.182298i	$-0.0583536\pi$
$331$	35.7771	1.96649	0.983243	+	0.182298i	$0.0583536\pi$
$332$	0	0
$333$	− 17.8885i	− 0.980286i
$334$	0	0
$335$	−4.47214	−0.244339
$336$	0	0
$337$	13.4164i	0.730838i	0.930843	+	0.365419i	$0.119074\pi$
$337$	13.4164i	0.730838i	−0.930843	+	0.365419i	$0.880926\pi$
$338$	0	0
$339$	−8.94427	−0.485786
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	8.94427i	0.481543i
$346$	0	0
$347$	8.00000i	0.429463i	0.976673	+	0.214731i	$0.0688876\pi$
$347$	8.00000i	0.429463i	−0.976673	+	0.214731i	$0.931112\pi$
$348$	0	0
$349$	−34.0000	−1.81998	−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000	−1.81998	−0.909989	+	0.414632i	$0.863910\pi$
$350$	0	0
$351$	−11.1803	−0.596762
$352$	0	0
$353$	− 11.1803i	− 0.595069i	−0.954711	−	0.297535i	$-0.903836\pi$
$353$	− 11.1803i	− 0.595069i	0.954711	−	0.297535i	$-0.0961644\pi$
$354$	0	0
$355$	20.0000i	1.06149i
$356$	0	0
$357$	− 2.23607i	− 0.118345i
$358$	0	0
$359$	−17.8885	−0.944121	−0.472061	−	0.881566i	$-0.656490\pi$
$359$	−17.8885	−0.944121	−0.472061	+	0.881566i	$0.656490\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	6.00000i	0.314918i
$364$	0	0
$365$	30.0000	1.57027
$366$	0	0
$367$	− 17.0000i	− 0.887393i	−0.896177	−	0.443696i	$-0.853667\pi$
$367$	− 17.0000i	− 0.887393i	0.896177	−	0.443696i	$-0.146333\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−4.47214	−0.232182
$372$	0	0
$373$	− 13.4164i	− 0.694675i	−0.937740	−	0.347338i	$-0.887086\pi$
$373$	− 13.4164i	− 0.694675i	0.937740	−	0.347338i	$-0.112914\pi$
$374$	0	0
$375$	11.1803	0.577350
$376$	0	0
$377$	2.23607i	0.115163i
$378$	0	0
$379$	26.8328	1.37831	0.689155	−	0.724614i	$-0.257982\pi$
$379$	26.8328	1.37831	0.689155	+	0.724614i	$0.257982\pi$
$380$	0	0
$381$	2.00000	0.102463
$382$	0	0
$383$	16.0000i	0.817562i	0.912633	+	0.408781i	$0.134046\pi$
$383$	16.0000i	0.817562i	−0.912633	+	0.408781i	$0.865954\pi$
$384$	0	0
$385$	5.00000	0.254824
$386$	0	0
$387$	12.0000i	0.609994i
$388$	0	0
$389$	5.00000	0.253510	0.126755	−	0.991934i	$-0.459544\pi$
$389$	5.00000	0.253510	0.126755	+	0.991934i	$0.459544\pi$
$390$	0	0
$391$	8.94427	0.452331
$392$	0	0
$393$	17.8885i	0.902358i
$394$	0	0
$395$	35.0000i	1.76104i
$396$	0	0
$397$	− 29.0689i	− 1.45893i	−0.684021	−	0.729463i	$-0.739770\pi$
$397$	− 29.0689i	− 1.45893i	0.684021	−	0.729463i	$-0.260230\pi$
$398$	0	0
$399$	4.47214	0.223887
$400$	0	0
$401$	−27.0000	−1.34832	−0.674158	−	0.738587i	$-0.735493\pi$
$401$	−27.0000	−1.34832	−0.674158	+	0.738587i	$0.735493\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 2.23607i	− 0.111111i
$406$	0	0
$407$	20.0000i	0.991363i
$408$	0	0
$409$	20.0000	0.988936	0.494468	−	0.869196i	$-0.335363\pi$
$409$	20.0000	0.988936	0.494468	+	0.869196i	$0.335363\pi$
$410$	0	0
$411$	−13.4164	−0.661783
$412$	0	0
$413$	4.47214i	0.220059i
$414$	0	0
$415$	−8.94427	−0.439057
$416$	0	0
$417$	− 13.4164i	− 0.657004i
$418$	0	0
$419$	22.3607	1.09239	0.546195	−	0.837658i	$-0.316076\pi$
$419$	22.3607	1.09239	0.546195	+	0.837658i	$0.316076\pi$
$420$	0	0
$421$	−25.0000	−1.21843	−0.609213	−	0.793007i	$-0.708514\pi$
$421$	−25.0000	−1.21843	−0.609213	+	0.793007i	$0.708514\pi$
$422$	0	0
$423$	6.00000i	0.291730i
$424$	0	0
$425$	− 11.1803i	− 0.542326i
$426$	0	0
$427$	− 10.0000i	− 0.483934i
$428$	0	0
$429$	5.00000	0.241402
$430$	0	0
$431$	20.1246	0.969368	0.484684	−	0.874689i	$-0.338935\pi$
$431$	20.1246	0.969368	0.484684	+	0.874689i	$0.338935\pi$
$432$	0	0
$433$	31.3050i	1.50442i	0.658923	+	0.752210i	$0.271012\pi$
$433$	31.3050i	1.50442i	−0.658923	+	0.752210i	$0.728988\pi$
$434$	0	0
$435$	2.23607	0.107211
$436$	0	0
$437$	17.8885i	0.855725i
$438$	0	0
$439$	40.2492	1.92099	0.960495	−	0.278296i	$-0.0897697\pi$
$439$	40.2492	1.92099	0.960495	+	0.278296i	$0.0897697\pi$
$440$	0	0
$441$	−2.00000	−0.0952381
$442$	0	0
$443$	− 34.0000i	− 1.61539i	−0.589601	−	0.807694i	$-0.700715\pi$
$443$	− 34.0000i	− 1.61539i	0.589601	−	0.807694i	$-0.299285\pi$
$444$	0	0
$445$	− 31.3050i	− 1.48400i
$446$	0	0
$447$	10.0000i	0.472984i
$448$	0	0
$449$	−15.0000	−0.707894	−0.353947	−	0.935266i	$-0.615161\pi$
$449$	−15.0000	−0.707894	−0.353947	+	0.935266i	$0.615161\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	− 15.6525i	− 0.735417i
$454$	0	0
$455$	5.00000i	0.234404i
$456$	0	0
$457$	− 26.8328i	− 1.25519i	−0.778542	−	0.627593i	$-0.784040\pi$
$457$	− 26.8328i	− 1.25519i	0.778542	−	0.627593i	$-0.215960\pi$
$458$	0	0
$459$	11.1803	0.521854
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	0	0
$463$	− 26.0000i	− 1.20832i	−0.796862	−	0.604161i	$-0.793508\pi$
$463$	− 26.0000i	− 1.20832i	0.796862	−	0.604161i	$-0.206492\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.00000i	0.138823i	0.997588	+	0.0694117i	$0.0221122\pi$
$467$	3.00000i	0.138823i	−0.997588	+	0.0694117i	$0.977888\pi$
$468$	0	0
$469$	−2.00000	−0.0923514
$470$	0	0
$471$	4.47214	0.206065
$472$	0	0
$473$	− 13.4164i	− 0.616887i
$474$	0	0
$475$	22.3607	1.02598
$476$	0	0
$477$	− 8.94427i	− 0.409530i
$478$	0	0
$479$	−8.94427	−0.408674	−0.204337	−	0.978901i	$-0.565504\pi$
$479$	−8.94427	−0.408674	−0.204337	+	0.978901i	$0.565504\pi$
$480$	0	0
$481$	−20.0000	−0.911922
$482$	0	0
$483$	4.00000i	0.182006i
$484$	0	0
$485$	5.00000	0.227038
$486$	0	0
$487$	18.0000i	0.815658i	0.913058	+	0.407829i	$0.133714\pi$
$487$	18.0000i	0.815658i	−0.913058	+	0.407829i	$0.866286\pi$
$488$	0	0
$489$	−14.0000	−0.633102
$490$	0	0
$491$	20.1246	0.908211	0.454106	−	0.890948i	$-0.349959\pi$
$491$	20.1246	0.908211	0.454106	+	0.890948i	$0.349959\pi$
$492$	0	0
$493$	− 2.23607i	− 0.100707i
$494$	0	0
$495$	10.0000i	0.449467i
$496$	0	0
$497$	8.94427i	0.401205i
$498$	0	0
$499$	20.1246	0.900901	0.450451	−	0.892801i	$-0.351263\pi$
$499$	20.1246	0.900901	0.450451	+	0.892801i	$0.351263\pi$
$500$	0	0
$501$	7.00000	0.312737
$502$	0	0
$503$	− 21.0000i	− 0.936344i	−0.883637	−	0.468172i	$-0.844913\pi$
$503$	− 21.0000i	− 0.936344i	0.883637	−	0.468172i	$-0.155087\pi$
$504$	0	0
$505$	4.47214i	0.199007i
$506$	0	0
$507$	− 8.00000i	− 0.355292i
$508$	0	0
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	13.4164	0.593507
$512$	0	0
$513$	22.3607i	0.987248i
$514$	0	0
$515$	−24.5967	−1.08386
$516$	0	0
$517$	− 6.70820i	− 0.295026i
$518$	0	0
$519$	2.23607	0.0981525
$520$	0	0
$521$	2.00000	0.0876216	0.0438108	−	0.999040i	$-0.486050\pi$
$521$	2.00000	0.0876216	0.0438108	+	0.999040i	$0.486050\pi$
$522$	0	0
$523$	4.00000i	0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$	4.00000i	0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$	0	0
$525$	5.00000	0.218218
$526$	0	0
$527$	0	0
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	−8.94427	−0.388148
$532$	0	0
$533$	0	0
$534$	0	0
$535$	26.8328	1.16008
$536$	0	0
$537$	17.8885i	0.771948i
$538$	0	0
$539$	2.23607	0.0963143
$540$	0	0
$541$	17.0000	0.730887	0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000	0.730887	0.365444	+	0.930834i	$0.380917\pi$
$542$	0	0
$543$	− 22.0000i	− 0.944110i
$544$	0	0
$545$	33.5410i	1.43674i
$546$	0	0
$547$	22.0000i	0.940652i	0.882493	+	0.470326i	$0.155864\pi$
$547$	22.0000i	0.940652i	−0.882493	+	0.470326i	$0.844136\pi$
$548$	0	0
$549$	20.0000	0.853579
$550$	0	0
$551$	4.47214	0.190519
$552$	0	0
$553$	15.6525i	0.665611i
$554$	0	0
$555$	20.0000i	0.848953i
$556$	0	0
$557$	− 44.7214i	− 1.89490i	−0.319897	−	0.947452i	$-0.603648\pi$
$557$	− 44.7214i	− 1.89490i	0.319897	−	0.947452i	$-0.396352\pi$
$558$	0	0
$559$	13.4164	0.567454
$560$	0	0
$561$	−5.00000	−0.211100
$562$	0	0
$563$	− 36.0000i	− 1.51722i	−0.651546	−	0.758610i	$-0.725879\pi$
$563$	− 36.0000i	− 1.51722i	0.651546	−	0.758610i	$-0.274121\pi$
$564$	0	0
$565$	−20.0000	−0.841406
$566$	0	0
$567$	− 1.00000i	− 0.0419961i
$568$	0	0
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	17.8885	0.748612	0.374306	−	0.927305i	$-0.377881\pi$
$571$	17.8885	0.748612	0.374306	+	0.927305i	$0.377881\pi$
$572$	0	0
$573$	24.5967i	1.02754i
$574$	0	0
$575$	20.0000i	0.834058i
$576$	0	0
$577$	15.6525i	0.651621i	0.945435	+	0.325811i	$0.105637\pi$
$577$	15.6525i	0.651621i	−0.945435	+	0.325811i	$0.894363\pi$
$578$	0	0
$579$	−13.4164	−0.557567
$580$	0	0
$581$	−4.00000	−0.165948
$582$	0	0
$583$	10.0000i	0.414158i
$584$	0	0
$585$	−10.0000	−0.413449
$586$	0	0
$587$	− 8.00000i	− 0.330195i	−0.986277	−	0.165098i	$-0.947206\pi$
$587$	− 8.00000i	− 0.330195i	0.986277	−	0.165098i	$-0.0527939\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	17.8885	0.735836
$592$	0	0
$593$	− 6.70820i	− 0.275473i	−0.990469	−	0.137736i	$-0.956017\pi$
$593$	− 6.70820i	− 0.275473i	0.990469	−	0.137736i	$-0.0439827\pi$
$594$	0	0
$595$	− 5.00000i	− 0.204980i
$596$	0	0
$597$	− 22.3607i	− 0.915162i
$598$	0	0
$599$	24.5967	1.00500	0.502498	−	0.864578i	$-0.332414\pi$
$599$	24.5967	1.00500	0.502498	+	0.864578i	$0.332414\pi$
$600$	0	0
$601$	−20.0000	−0.815817	−0.407909	−	0.913023i	$-0.633742\pi$
$601$	−20.0000	−0.815817	−0.407909	+	0.913023i	$0.633742\pi$
$602$	0	0
$603$	− 4.00000i	− 0.162893i
$604$	0	0
$605$	13.4164i	0.545455i
$606$	0	0
$607$	− 47.0000i	− 1.90767i	−0.300329	−	0.953836i	$-0.597097\pi$
$607$	− 47.0000i	− 1.90767i	0.300329	−	0.953836i	$-0.402903\pi$
$608$	0	0
$609$	1.00000	0.0405220
$610$	0	0
$611$	6.70820	0.271385
$612$	0	0
$613$	− 26.8328i	− 1.08377i	−0.840454	−	0.541884i	$-0.817711\pi$
$613$	− 26.8328i	− 1.08377i	0.840454	−	0.541884i	$-0.182289\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4.47214i	0.180041i	0.995940	+	0.0900207i	$0.0286933\pi$
$617$	4.47214i	0.180041i	−0.995940	+	0.0900207i	$0.971307\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	−20.0000	−0.802572
$622$	0	0
$623$	− 14.0000i	− 0.560898i
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	− 10.0000i	− 0.399362i
$628$	0	0
$629$	20.0000	0.797452
$630$	0	0
$631$	20.1246	0.801148	0.400574	−	0.916264i	$-0.368811\pi$
$631$	20.1246	0.801148	0.400574	+	0.916264i	$0.368811\pi$
$632$	0	0
$633$	6.70820i	0.266627i
$634$	0	0
$635$	4.47214	0.177471
$636$	0	0
$637$	2.23607i	0.0885962i
$638$	0	0
$639$	−17.8885	−0.707660
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	− 21.0000i	− 0.828159i	−0.910241	−	0.414080i	$-0.864104\pi$
$643$	− 21.0000i	− 0.828159i	0.910241	−	0.414080i	$-0.135896\pi$
$644$	0	0
$645$	− 13.4164i	− 0.528271i
$646$	0	0
$647$	− 12.0000i	− 0.471769i	−0.971781	−	0.235884i	$-0.924201\pi$
$647$	− 12.0000i	− 0.471769i	0.971781	−	0.235884i	$-0.0757987\pi$
$648$	0	0
$649$	10.0000	0.392534
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.47214i	0.175008i	0.996164	+	0.0875041i	$0.0278891\pi$
$653$	4.47214i	0.175008i	−0.996164	+	0.0875041i	$0.972111\pi$
$654$	0	0
$655$	40.0000i	1.56293i
$656$	0	0
$657$	26.8328i	1.04685i
$658$	0	0
$659$	−6.70820	−0.261315	−0.130657	−	0.991428i	$-0.541709\pi$
$659$	−6.70820	−0.261315	−0.130657	+	0.991428i	$0.541709\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	0	0
$663$	− 5.00000i	− 0.194184i
$664$	0	0
$665$	10.0000	0.387783
$666$	0	0
$667$	4.00000i	0.154881i
$668$	0	0
$669$	−19.0000	−0.734582
$670$	0	0
$671$	−22.3607	−0.863224
$672$	0	0
$673$	40.2492i	1.55149i	0.631044	+	0.775747i	$0.282627\pi$
$673$	40.2492i	1.55149i	−0.631044	+	0.775747i	$0.717373\pi$
$674$	0	0
$675$	25.0000i	0.962250i
$676$	0	0
$677$	20.1246i	0.773452i	0.922195	+	0.386726i	$0.126394\pi$
$677$	20.1246i	0.773452i	−0.922195	+	0.386726i	$0.873606\pi$
$678$	0	0
$679$	2.23607	0.0858124
$680$	0	0
$681$	17.0000	0.651441
$682$	0	0
$683$	6.00000i	0.229584i	0.993390	+	0.114792i	$0.0366201\pi$
$683$	6.00000i	0.229584i	−0.993390	+	0.114792i	$0.963380\pi$
$684$	0	0
$685$	−30.0000	−1.14624
$686$	0	0
$687$	4.00000i	0.152610i
$688$	0	0
$689$	−10.0000	−0.380970
$690$	0	0
$691$	26.8328	1.02077	0.510384	−	0.859946i	$-0.329503\pi$
$691$	26.8328	1.02077	0.510384	+	0.859946i	$0.329503\pi$
$692$	0	0
$693$	4.47214i	0.169882i
$694$	0	0
$695$	− 30.0000i	− 1.13796i
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−17.8885	−0.676607
$700$	0	0
$701$	−35.0000	−1.32193	−0.660966	−	0.750416i	$-0.729853\pi$
$701$	−35.0000	−1.32193	−0.660966	+	0.750416i	$0.729853\pi$
$702$	0	0
$703$	40.0000i	1.50863i
$704$	0	0
$705$	− 6.70820i	− 0.252646i
$706$	0	0
$707$	2.00000i	0.0752177i
$708$	0	0
$709$	19.0000	0.713560	0.356780	−	0.934188i	$-0.383875\pi$
$709$	19.0000	0.713560	0.356780	+	0.934188i	$0.383875\pi$
$710$	0	0
$711$	−31.3050	−1.17403
$712$	0	0
$713$	0	0
$714$	0	0
$715$	11.1803	0.418121
$716$	0	0
$717$	6.70820i	0.250522i
$718$	0	0
$719$	8.94427	0.333565	0.166783	−	0.985994i	$-0.446662\pi$
$719$	8.94427	0.333565	0.166783	+	0.985994i	$0.446662\pi$
$720$	0	0
$721$	−11.0000	−0.409661
$722$	0	0
$723$	0	0
$724$	0	0
$725$	5.00000	0.185695
$726$	0	0
$727$	12.0000i	0.445055i	0.974926	+	0.222528i	$0.0714308\pi$
$727$	12.0000i	0.445055i	−0.974926	+	0.222528i	$0.928569\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−13.4164	−0.496224
$732$	0	0
$733$	− 20.1246i	− 0.743319i	−0.928369	−	0.371660i	$-0.878789\pi$
$733$	− 20.1246i	− 0.743319i	0.928369	−	0.371660i	$-0.121211\pi$
$734$	0	0
$735$	2.23607	0.0824786
$736$	0	0
$737$	4.47214i	0.164733i
$738$	0	0
$739$	11.1803	0.411275	0.205638	−	0.978628i	$-0.434073\pi$
$739$	11.1803	0.411275	0.205638	+	0.978628i	$0.434073\pi$
$740$	0	0
$741$	10.0000	0.367359
$742$	0	0
$743$	− 54.0000i	− 1.98107i	−0.137268	−	0.990534i	$-0.543832\pi$
$743$	− 54.0000i	− 1.98107i	0.137268	−	0.990534i	$-0.456168\pi$
$744$	0	0
$745$	22.3607i	0.819232i
$746$	0	0
$747$	− 8.00000i	− 0.292705i
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	−11.1803	−0.407976	−0.203988	−	0.978973i	$-0.565390\pi$
$751$	−11.1803	−0.407976	−0.203988	+	0.978973i	$0.565390\pi$
$752$	0	0
$753$	− 22.3607i	− 0.814868i
$754$	0	0
$755$	− 35.0000i	− 1.27378i
$756$	0	0
$757$	8.94427i	0.325085i	0.986702	+	0.162543i	$0.0519695\pi$
$757$	8.94427i	0.325085i	−0.986702	+	0.162543i	$0.948031\pi$
$758$	0	0
$759$	8.94427	0.324657
$760$	0	0
$761$	−48.0000	−1.74000	−0.869999	−	0.493053i	$-0.835881\pi$
$761$	−48.0000	−1.74000	−0.869999	+	0.493053i	$0.835881\pi$
$762$	0	0
$763$	15.0000i	0.543036i
$764$	0	0
$765$	10.0000	0.361551
$766$	0	0
$767$	10.0000i	0.361079i
$768$	0	0
$769$	−4.00000	−0.144244	−0.0721218	−	0.997396i	$-0.522977\pi$
$769$	−4.00000	−0.144244	−0.0721218	+	0.997396i	$0.522977\pi$
$770$	0	0
$771$	−22.3607	−0.805300
$772$	0	0
$773$	42.4853i	1.52809i	0.645163	+	0.764045i	$0.276789\pi$
$773$	42.4853i	1.52809i	−0.645163	+	0.764045i	$0.723211\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	8.94427i	0.320874i
$778$	0	0
$779$	0	0
$780$	0	0
$781$	20.0000	0.715656
$782$	0	0
$783$	5.00000i	0.178685i
$784$	0	0
$785$	10.0000	0.356915
$786$	0	0
$787$	− 3.00000i	− 0.106938i	−0.998569	−	0.0534692i	$-0.982972\pi$
$787$	− 3.00000i	− 0.106938i	0.998569	−	0.0534692i	$-0.0170279\pi$
$788$	0	0
$789$	−16.0000	−0.569615
$790$	0	0
$791$	−8.94427	−0.318022
$792$	0	0
$793$	− 22.3607i	− 0.794051i
$794$	0	0
$795$	10.0000i	0.354663i
$796$	0	0
$797$	15.6525i	0.554439i	0.960807	+	0.277220i	$0.0894129\pi$
$797$	15.6525i	0.554439i	−0.960807	+	0.277220i	$0.910587\pi$
$798$	0	0
$799$	−6.70820	−0.237319
$800$	0	0
$801$	28.0000	0.989331
$802$	0	0
$803$	− 30.0000i	− 1.05868i
$804$	0	0
$805$	8.94427i	0.315244i
$806$	0	0
$807$	10.0000i	0.352017i
$808$	0	0
$809$	−9.00000	−0.316423	−0.158212	−	0.987405i	$-0.550573\pi$
$809$	−9.00000	−0.316423	−0.158212	+	0.987405i	$0.550573\pi$
$810$	0	0
$811$	4.47214	0.157038	0.0785190	−	0.996913i	$-0.474981\pi$
$811$	4.47214	0.157038	0.0785190	+	0.996913i	$0.474981\pi$
$812$	0	0
$813$	− 4.47214i	− 0.156845i
$814$	0	0
$815$	−31.3050	−1.09656
$816$	0	0
$817$	− 26.8328i	− 0.938761i
$818$	0	0
$819$	−4.47214	−0.156269
$820$	0	0
$821$	−45.0000	−1.57051	−0.785255	−	0.619172i	$-0.787468\pi$
$821$	−45.0000	−1.57051	−0.785255	+	0.619172i	$0.787468\pi$
$822$	0	0
$823$	− 24.0000i	− 0.836587i	−0.908312	−	0.418294i	$-0.862628\pi$
$823$	− 24.0000i	− 0.836587i	0.908312	−	0.418294i	$-0.137372\pi$
$824$	0	0
$825$	− 11.1803i	− 0.389249i
$826$	0	0
$827$	18.0000i	0.625921i	0.949766	+	0.312961i	$0.101321\pi$
$827$	18.0000i	0.625921i	−0.949766	+	0.312961i	$0.898679\pi$
$828$	0	0
$829$	40.0000	1.38926	0.694629	−	0.719368i	$-0.255569\pi$
$829$	40.0000	1.38926	0.694629	+	0.719368i	$0.255569\pi$
$830$	0	0
$831$	−13.4164	−0.465410
$832$	0	0
$833$	− 2.23607i	− 0.0774752i
$834$	0	0
$835$	15.6525	0.541676
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−53.6656	−1.85274	−0.926372	−	0.376611i	$-0.877089\pi$
$839$	−53.6656	−1.85274	−0.926372	+	0.376611i	$0.877089\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	0	0
$843$	− 15.0000i	− 0.516627i
$844$	0	0
$845$	− 17.8885i	− 0.615385i
$846$	0	0
$847$	6.00000i	0.206162i
$848$	0	0
$849$	−31.0000	−1.06392
$850$	0	0
$851$	−35.7771	−1.22642
$852$	0	0
$853$	4.47214i	0.153123i	0.997065	+	0.0765615i	$0.0243942\pi$
$853$	4.47214i	0.153123i	−0.997065	+	0.0765615i	$0.975606\pi$
$854$	0	0
$855$	20.0000i	0.683986i
$856$	0	0
$857$	4.47214i	0.152765i	0.997079	+	0.0763826i	$0.0243370\pi$
$857$	4.47214i	0.152765i	−0.997079	+	0.0763826i	$0.975663\pi$
$858$	0	0
$859$	35.7771	1.22070	0.610349	−	0.792132i	$-0.291029\pi$
$859$	35.7771	1.22070	0.610349	+	0.792132i	$0.291029\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 24.0000i	− 0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$	− 24.0000i	− 0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$	0	0
$865$	5.00000	0.170005
$866$	0	0
$867$	− 12.0000i	− 0.407541i
$868$	0	0
$869$	35.0000	1.18729
$870$	0	0
$871$	−4.47214	−0.151533
$872$	0	0
$873$	4.47214i	0.151359i
$874$	0	0
$875$	11.1803	0.377964
$876$	0	0
$877$	4.47214i	0.151013i	0.997145	+	0.0755067i	$0.0240574\pi$
$877$	4.47214i	0.151013i	−0.997145	+	0.0755067i	$0.975943\pi$
$878$	0	0
$879$	11.1803	0.377104
$880$	0	0
$881$	40.0000	1.34763	0.673817	−	0.738898i	$-0.264654\pi$
$881$	40.0000	1.34763	0.673817	+	0.738898i	$0.264654\pi$
$882$	0	0
$883$	34.0000i	1.14419i	0.820187	+	0.572096i	$0.193869\pi$
$883$	34.0000i	1.14419i	−0.820187	+	0.572096i	$0.806131\pi$
$884$	0	0
$885$	10.0000	0.336146
$886$	0	0
$887$	− 8.00000i	− 0.268614i	−0.990940	−	0.134307i	$-0.957119\pi$
$887$	− 8.00000i	− 0.268614i	0.990940	−	0.134307i	$-0.0428808\pi$
$888$	0	0
$889$	2.00000	0.0670778
$890$	0	0
$891$	−2.23607	−0.0749111
$892$	0	0
$893$	− 13.4164i	− 0.448963i
$894$	0	0
$895$	40.0000i	1.33705i
$896$	0	0
$897$	8.94427i	0.298641i
$898$	0	0
$899$	0	0
$900$	0	0
$901$	10.0000	0.333148
$902$	0	0
$903$	− 6.00000i	− 0.199667i
$904$	0	0
$905$	− 49.1935i	− 1.63525i
$906$	0	0
$907$	− 32.0000i	− 1.06254i	−0.847202	−	0.531271i	$-0.821714\pi$
$907$	− 32.0000i	− 1.06254i	0.847202	−	0.531271i	$-0.178286\pi$
$908$	0	0
$909$	−4.00000	−0.132672
$910$	0	0
$911$	8.94427	0.296337	0.148168	−	0.988962i	$-0.452662\pi$
$911$	8.94427	0.296337	0.148168	+	0.988962i	$0.452662\pi$
$912$	0	0
$913$	8.94427i	0.296012i
$914$	0	0
$915$	−22.3607	−0.739221
$916$	0	0
$917$	17.8885i	0.590732i
$918$	0	0
$919$	46.9574	1.54898	0.774491	−	0.632585i	$-0.218006\pi$
$919$	46.9574	1.54898	0.774491	+	0.632585i	$0.218006\pi$
$920$	0	0
$921$	23.0000	0.757876
$922$	0	0
$923$	20.0000i	0.658308i
$924$	0	0
$925$	44.7214i	1.47043i
$926$	0	0
$927$	− 22.0000i	− 0.722575i
$928$	0	0
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	4.47214	0.146568
$932$	0	0
$933$	− 22.3607i	− 0.732056i
$934$	0	0
$935$	−11.1803	−0.365636
$936$	0	0
$937$	20.1246i	0.657442i	0.944427	+	0.328721i	$0.106618\pi$
$937$	20.1246i	0.657442i	−0.944427	+	0.328721i	$0.893382\pi$
$938$	0	0
$939$	−15.6525	−0.510799
$940$	0	0
$941$	−38.0000	−1.23876	−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000	−1.23876	−0.619382	+	0.785090i	$0.712617\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	11.1803i	0.363696i
$946$	0	0
$947$	48.0000i	1.55979i	0.625910	+	0.779895i	$0.284728\pi$
$947$	48.0000i	1.55979i	−0.625910	+	0.779895i	$0.715272\pi$
$948$	0	0
$949$	30.0000	0.973841
$950$	0	0
$951$	17.8885	0.580076
$952$	0	0
$953$	− 26.8328i	− 0.869200i	−0.900624	−	0.434600i	$-0.856890\pi$
$953$	− 26.8328i	− 0.869200i	0.900624	−	0.434600i	$-0.143110\pi$
$954$	0	0
$955$	55.0000i	1.77976i
$956$	0	0
$957$	− 2.23607i	− 0.0722818i
$958$	0	0
$959$	−13.4164	−0.433238
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	24.0000i	0.773389i
$964$	0	0
$965$	−30.0000	−0.965734
$966$	0	0
$967$	28.0000i	0.900419i	0.892923	+	0.450210i	$0.148651\pi$
$967$	28.0000i	0.900419i	−0.892923	+	0.450210i	$0.851349\pi$
$968$	0	0
$969$	−10.0000	−0.321246
$970$	0	0
$971$	−26.8328	−0.861106	−0.430553	−	0.902565i	$-0.641681\pi$
$971$	−26.8328	−0.861106	−0.430553	+	0.902565i	$0.641681\pi$
$972$	0	0
$973$	− 13.4164i	− 0.430110i
$974$	0	0
$975$	11.1803	0.358057
$976$	0	0
$977$	− 49.1935i	− 1.57384i	−0.617055	−	0.786920i	$-0.711675\pi$
$977$	− 49.1935i	− 1.57384i	0.617055	−	0.786920i	$-0.288325\pi$
$978$	0	0
$979$	−31.3050	−1.00051
$980$	0	0
$981$	−30.0000	−0.957826
$982$	0	0
$983$	39.0000i	1.24391i	0.783054	+	0.621953i	$0.213661\pi$
$983$	39.0000i	1.24391i	−0.783054	+	0.621953i	$0.786339\pi$
$984$	0	0
$985$	40.0000	1.27451
$986$	0	0
$987$	− 3.00000i	− 0.0954911i
$988$	0	0
$989$	24.0000	0.763156
$990$	0	0
$991$	−26.8328	−0.852372	−0.426186	−	0.904635i	$-0.640143\pi$
$991$	−26.8328	−0.852372	−0.426186	+	0.904635i	$0.640143\pi$
$992$	0	0
$993$	− 35.7771i	− 1.13535i
$994$	0	0
$995$	− 50.0000i	− 1.58511i
$996$	0	0
$997$	33.5410i	1.06225i	0.847292	+	0.531127i	$0.178231\pi$
$997$	33.5410i	1.06225i	−0.847292	+	0.531127i	$0.821769\pi$
$998$	0	0
$999$	−44.7214	−1.41492

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.g.k.449.1		4
4.3	odd	2	inner	2240.2.g.k.449.3		4
5.4	even	2	inner	2240.2.g.k.449.4		4
8.3	odd	2		1120.2.g.a.449.2	yes	4
8.5	even	2		1120.2.g.a.449.4	yes	4
20.19	odd	2	inner	2240.2.g.k.449.2		4
40.3	even	4		5600.2.a.bl.1.1		2
40.13	odd	4		5600.2.a.w.1.2		2
40.19	odd	2		1120.2.g.a.449.3	yes	4
40.27	even	4		5600.2.a.w.1.1		2
40.29	even	2		1120.2.g.a.449.1	✓	4
40.37	odd	4		5600.2.a.bl.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.2.g.a.449.1	✓	4	40.29	even	2
1120.2.g.a.449.2	yes	4	8.3	odd	2
1120.2.g.a.449.3	yes	4	40.19	odd	2
1120.2.g.a.449.4	yes	4	8.5	even	2
2240.2.g.k.449.1		4	1.1	even	1	trivial
2240.2.g.k.449.2		4	20.19	odd	2	inner
2240.2.g.k.449.3		4	4.3	odd	2	inner
2240.2.g.k.449.4		4	5.4	even	2	inner
5600.2.a.w.1.1		2	40.27	even	4
5600.2.a.w.1.2		2	40.13	odd	4
5600.2.a.bl.1.1		2	40.3	even	4
5600.2.a.bl.1.2		2	40.37	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.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -2.23607i q^{5} -1.00000i q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -2.23607i q^{5} -1.00000i q^{7} +2.00000 q^{9} -2.23607 q^{11} -2.23607i q^{13} -2.23607 q^{15} +2.23607i q^{17} -4.47214 q^{19} -1.00000 q^{21} -4.00000i q^{23} -5.00000 q^{25} -5.00000i q^{27} -1.00000 q^{29} +2.23607i q^{33} -2.23607 q^{35} -8.94427i q^{37} -2.23607 q^{39} +6.00000i q^{43} -4.47214i q^{45} +3.00000i q^{47} -1.00000 q^{49} +2.23607 q^{51} -4.47214i q^{53} +5.00000i q^{55} +4.47214i q^{57} -4.47214 q^{59} +10.0000 q^{61} -2.00000i q^{63} -5.00000 q^{65} -2.00000i q^{67} -4.00000 q^{69} -8.94427 q^{71} +13.4164i q^{73} +5.00000i q^{75} +2.23607i q^{77} -15.6525 q^{79} +1.00000 q^{81} -4.00000i q^{83} +5.00000 q^{85} +1.00000i q^{87} +14.0000 q^{89} -2.23607 q^{91} +10.0000i q^{95} +2.23607i q^{97} -4.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{9}+O(q^{10}) \) 4 * q + 8 * q^9	\( 4 q + 8 q^{9} - 4 q^{21} - 20 q^{25} - 4 q^{29} - 4 q^{49} + 40 q^{61} - 20 q^{65} - 16 q^{69} + 4 q^{81} + 20 q^{85} + 56 q^{89}+O(q^{100}) \) 4 * q + 8 * q^9 - 4 * q^21 - 20 * q^25 - 4 * q^29 - 4 * q^49 + 40 * q^61 - 20 * q^65 - 16 * q^69 + 4 * q^81 + 20 * q^85 + 56 * q^89

Introduction

L-functions

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