LMFDB - Embedded newform 2240.2.bd.a.561.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(561,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(4))

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("2240.561");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.bd (of order $4$, degree $2$, 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:	$44$
Relative dimension:	$22$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 560)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			561.20
Character	$\chi$	$=$	2240.561
Dual form			2240.2.bd.a.1681.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.67958 - 1.67958i) q^{3} +(0.707107 + 0.707107i) q^{5} -1.00000i q^{7} -2.64195i q^{9} +O(q^{10})$	\(q+(1.67958 - 1.67958i) q^{3} +(0.707107 + 0.707107i) q^{5} -1.00000i q^{7} -2.64195i q^{9} +(4.03337 + 4.03337i) q^{11} +(-4.75317 + 4.75317i) q^{13} +2.37528 q^{15} +4.01138 q^{17} +(-1.17869 + 1.17869i) q^{19} +(-1.67958 - 1.67958i) q^{21} -0.610935i q^{23} +1.00000i q^{25} +(0.601365 + 0.601365i) q^{27} +(2.42606 - 2.42606i) q^{29} +8.63834 q^{31} +13.5487 q^{33} +(0.707107 - 0.707107i) q^{35} +(-1.14755 - 1.14755i) q^{37} +15.9666i q^{39} +1.24783i q^{41} +(8.70723 + 8.70723i) q^{43} +(1.86814 - 1.86814i) q^{45} -7.92976 q^{47} -1.00000 q^{49} +(6.73743 - 6.73743i) q^{51} +(-5.64673 - 5.64673i) q^{53} +5.70405i q^{55} +3.95940i q^{57} +(1.23973 + 1.23973i) q^{59} +(5.71963 - 5.71963i) q^{61} -2.64195 q^{63} -6.72199 q^{65} +(-2.38301 + 2.38301i) q^{67} +(-1.02611 - 1.02611i) q^{69} -1.57667i q^{71} -1.85964i q^{73} +(1.67958 + 1.67958i) q^{75} +(4.03337 - 4.03337i) q^{77} -3.14874 q^{79} +9.94594 q^{81} +(6.01779 - 6.01779i) q^{83} +(2.83648 + 2.83648i) q^{85} -8.14951i q^{87} -18.0907i q^{89} +(4.75317 + 4.75317i) q^{91} +(14.5087 - 14.5087i) q^{93} -1.66692 q^{95} -10.6607 q^{97} +(10.6560 - 10.6560i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q+O(q^{10}) $ 44 * q	$ 44 q - 12 q^{11} - 8 q^{15} - 8 q^{19} + 24 q^{27} + 12 q^{29} + 28 q^{37} + 44 q^{43} - 44 q^{49} + 8 q^{51} - 12 q^{53} - 24 q^{59} - 16 q^{61} - 28 q^{63} - 40 q^{65} + 28 q^{67} + 40 q^{69} - 12 q^{77} + 16 q^{79} + 20 q^{81} + 16 q^{85} + 88 q^{93} + 32 q^{95} - 28 q^{99}+O(q^{100}) $ 44 * q - 12 * q^11 - 8 * q^15 - 8 * q^19 + 24 * q^27 + 12 * q^29 + 28 * q^37 + 44 * q^43 - 44 * q^49 + 8 * q^51 - 12 * q^53 - 24 * q^59 - 16 * q^61 - 28 * q^63 - 40 * q^65 + 28 * q^67 + 40 * q^69 - 12 * q^77 + 16 * q^79 + 20 * q^81 + 16 * q^85 + 88 * q^93 + 32 * q^95 - 28 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$897$	$1471$	$1541$	$1921$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$	$1$

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
$3$	1.67958	−	1.67958i	0.969704	−	0.969704i	−0.0298505	−	0.999554i	$-0.509503\pi$
$3$	1.67958	−	1.67958i	0.969704	−	0.969704i	0.999554	+	0.0298505i	$0.00950311\pi$
$4$		0			0
$5$	0.707107	+	0.707107i	0.316228	+	0.316228i
$5$	0.707107	+	0.707107i	0.316228	+	0.316228i
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$		−	2.64195i		−	0.880651i
$10$		0			0
$11$	4.03337	+	4.03337i	1.21611	+	1.21611i	0.968984	+	0.247122i	$0.0794850\pi$
$11$	4.03337	+	4.03337i	1.21611	+	1.21611i	0.247122	+	0.968984i	$0.420515\pi$
$12$		0			0
$13$	−4.75317	+	4.75317i	−1.31829	+	1.31829i	−0.403163	+	0.915128i	$0.632089\pi$
$13$	−4.75317	+	4.75317i	−1.31829	+	1.31829i	−0.915128	+	0.403163i	$0.867911\pi$
$14$		0			0
$15$	2.37528			0.613295
$16$		0			0
$17$	4.01138			0.972903			0.486452	−	0.873707i	$-0.338291\pi$
$17$	4.01138			0.972903			0.486452	+	0.873707i	$0.338291\pi$
$18$		0			0
$19$	−1.17869	+	1.17869i	−0.270410	+	0.270410i	−0.829265	−	0.558855i	$-0.811241\pi$
$19$	−1.17869	+	1.17869i	−0.270410	+	0.270410i	0.558855	+	0.829265i	$0.311241\pi$
$20$		0			0
$21$	−1.67958	−	1.67958i	−0.366514	−	0.366514i
$22$		0			0
$23$		−	0.610935i		−	0.127389i	−0.997969	−	0.0636944i	$-0.979712\pi$
$23$		−	0.610935i		−	0.127389i	0.997969	−	0.0636944i	$-0.0202883\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$	0.601365	+	0.601365i	0.115733	+	0.115733i
$28$		0			0
$29$	2.42606	−	2.42606i	0.450508	−	0.450508i	−0.445015	−	0.895523i	$-0.646802\pi$
$29$	2.42606	−	2.42606i	0.450508	−	0.450508i	0.895523	+	0.445015i	$0.146802\pi$
$30$		0			0
$31$	8.63834			1.55149			0.775746	−	0.631046i	$-0.217374\pi$
$31$	8.63834			1.55149			0.775746	+	0.631046i	$0.217374\pi$
$32$		0			0
$33$	13.5487			2.35853
$34$		0			0
$35$	0.707107	−	0.707107i	0.119523	−	0.119523i
$36$		0			0
$37$	−1.14755	−	1.14755i	−0.188655	−	0.188655i	0.606459	−	0.795115i	$-0.292589\pi$
$37$	−1.14755	−	1.14755i	−0.188655	−	0.188655i	−0.795115	+	0.606459i	$0.792589\pi$
$38$		0			0
$39$			15.9666i			2.55670i
$40$		0			0
$41$			1.24783i			0.194879i	0.995241	+	0.0974394i	$0.0310652\pi$
$41$			1.24783i			0.194879i	−0.995241	+	0.0974394i	$0.968935\pi$
$42$		0			0
$43$	8.70723	+	8.70723i	1.32784	+	1.32784i	0.907252	+	0.420588i	$0.138176\pi$
$43$	8.70723	+	8.70723i	1.32784	+	1.32784i	0.420588	+	0.907252i	$0.361824\pi$
$44$		0			0
$45$	1.86814	−	1.86814i	0.278486	−	0.278486i
$46$		0			0
$47$	−7.92976			−1.15667			−0.578337	−	0.815798i	$-0.696298\pi$
$47$	−7.92976			−1.15667			−0.578337	+	0.815798i	$0.696298\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	6.73743	−	6.73743i	0.943428	−	0.943428i
$52$		0			0
$53$	−5.64673	−	5.64673i	−0.775638	−	0.775638i	0.203447	−	0.979086i	$-0.434785\pi$
$53$	−5.64673	−	5.64673i	−0.775638	−	0.775638i	−0.979086	+	0.203447i	$0.934785\pi$
$54$		0			0
$55$			5.70405i			0.769133i
$56$		0			0
$57$			3.95940i			0.524436i
$58$		0			0
$59$	1.23973	+	1.23973i	0.161399	+	0.161399i	0.783186	−	0.621787i	$-0.213593\pi$
$59$	1.23973	+	1.23973i	0.161399	+	0.161399i	−0.621787	+	0.783186i	$0.713593\pi$
$60$		0			0
$61$	5.71963	−	5.71963i	0.732324	−	0.732324i	−0.238756	−	0.971080i	$-0.576740\pi$
$61$	5.71963	−	5.71963i	0.732324	−	0.732324i	0.971080	+	0.238756i	$0.0767396\pi$
$62$		0			0
$63$	−2.64195			−0.332855
$64$		0			0
$65$	−6.72199			−0.833761
$66$		0			0
$67$	−2.38301	+	2.38301i	−0.291132	+	0.291132i	−0.837527	−	0.546396i	$-0.816000\pi$
$67$	−2.38301	+	2.38301i	−0.291132	+	0.291132i	0.546396	+	0.837527i	$0.316000\pi$
$68$		0			0
$69$	−1.02611	−	1.02611i	−0.123529	−	0.123529i
$70$		0			0
$71$		−	1.57667i		−	0.187116i	−0.995614	−	0.0935580i	$-0.970176\pi$
$71$		−	1.57667i		−	0.187116i	0.995614	−	0.0935580i	$-0.0298240\pi$
$72$		0			0
$73$		−	1.85964i		−	0.217655i	−0.994061	−	0.108827i	$-0.965290\pi$
$73$		−	1.85964i		−	0.217655i	0.994061	−	0.108827i	$-0.0347096\pi$
$74$		0			0
$75$	1.67958	+	1.67958i	0.193941	+	0.193941i
$76$		0			0
$77$	4.03337	−	4.03337i	0.459645	−	0.459645i
$78$		0			0
$79$	−3.14874			−0.354261			−0.177130	−	0.984187i	$-0.556681\pi$
$79$	−3.14874			−0.354261			−0.177130	+	0.984187i	$0.556681\pi$
$80$		0			0
$81$	9.94594			1.10510
$82$		0			0
$83$	6.01779	−	6.01779i	0.660538	−	0.660538i	−0.294969	−	0.955507i	$-0.595309\pi$
$83$	6.01779	−	6.01779i	0.660538	−	0.660538i	0.955507	+	0.294969i	$0.0953092\pi$
$84$		0			0
$85$	2.83648	+	2.83648i	0.307659	+	0.307659i
$86$		0			0
$87$		−	8.14951i		−	0.873719i
$88$		0			0
$89$		−	18.0907i		−	1.91761i	−0.284060	−	0.958806i	$-0.591682\pi$
$89$		−	18.0907i		−	1.91761i	0.284060	−	0.958806i	$-0.408318\pi$
$90$		0			0
$91$	4.75317	+	4.75317i	0.498267	+	0.498267i
$92$		0			0
$93$	14.5087	−	14.5087i	1.50449	−	1.50449i
$94$		0			0
$95$	−1.66692			−0.171022
$96$		0			0
$97$	−10.6607			−1.08243			−0.541213	−	0.840886i	$-0.682035\pi$
$97$	−10.6607			−1.08243			−0.541213	+	0.840886i	$0.682035\pi$
$98$		0			0
$99$	10.6560	−	10.6560i	1.07097	−	1.07097i
$100$		0			0
$101$	3.13938	+	3.13938i	0.312380	+	0.312380i	0.845831	−	0.533451i	$-0.179105\pi$
$101$	3.13938	+	3.13938i	0.312380	+	0.312380i	−0.533451	+	0.845831i	$0.679105\pi$
$102$		0			0
$103$			3.04008i			0.299548i	0.988720	+	0.149774i	$0.0478546\pi$
$103$			3.04008i			0.299548i	−0.988720	+	0.149774i	$0.952145\pi$
$104$		0			0
$105$		−	2.37528i		−	0.231804i
$106$		0			0
$107$	−0.349930	−	0.349930i	−0.0338290	−	0.0338290i	0.689990	−	0.723819i	$-0.257615\pi$
$107$	−0.349930	−	0.349930i	−0.0338290	−	0.0338290i	−0.723819	+	0.689990i	$0.757615\pi$
$108$		0			0
$109$	−9.66324	+	9.66324i	−0.925571	+	0.925571i	−0.997416	−	0.0718452i	$-0.977111\pi$
$109$	−9.66324	+	9.66324i	−0.925571	+	0.925571i	0.0718452	+	0.997416i	$0.477111\pi$
$110$		0			0
$111$	−3.85478			−0.365880
$112$		0			0
$113$	−15.6991			−1.47685			−0.738425	−	0.674335i	$-0.764430\pi$
$113$	−15.6991			−1.47685			−0.738425	+	0.674335i	$0.764430\pi$
$114$		0			0
$115$	0.431996	−	0.431996i	0.0402839	−	0.0402839i
$116$		0			0
$117$	12.5576	+	12.5576i	1.16095	+	1.16095i
$118$		0			0
$119$		−	4.01138i		−	0.367723i
$120$		0			0
$121$			21.5361i			1.95783i
$122$		0			0
$123$	2.09583	+	2.09583i	0.188975	+	0.188975i
$124$		0			0
$125$	−0.707107	+	0.707107i	−0.0632456	+	0.0632456i
$126$		0			0
$127$	9.68222			0.859158			0.429579	−	0.903029i	$-0.358662\pi$
$127$	9.68222			0.859158			0.429579	+	0.903029i	$0.358662\pi$
$128$		0			0
$129$	29.2489			2.57522
$130$		0			0
$131$	10.3532	−	10.3532i	0.904562	−	0.904562i	−0.0912644	−	0.995827i	$-0.529091\pi$
$131$	10.3532	−	10.3532i	0.904562	−	0.904562i	0.995827	+	0.0912644i	$0.0290909\pi$
$132$		0			0
$133$	1.17869	+	1.17869i	0.102205	+	0.102205i
$134$		0			0
$135$			0.850459i			0.0731958i
$136$		0			0
$137$			7.80415i			0.666754i	0.942794	+	0.333377i	$0.108188\pi$
$137$			7.80415i			0.666754i	−0.942794	+	0.333377i	$0.891812\pi$
$138$		0			0
$139$	−12.8660	−	12.8660i	−1.09128	−	1.09128i	−0.995392	−	0.0958852i	$-0.969432\pi$
$139$	−12.8660	−	12.8660i	−1.09128	−	1.09128i	−0.0958852	−	0.995392i	$-0.530568\pi$
$140$		0			0
$141$	−13.3186	+	13.3186i	−1.12163	+	1.12163i
$142$		0			0
$143$	−38.3426			−3.20637
$144$		0			0
$145$	3.43097			0.284926
$146$		0			0
$147$	−1.67958	+	1.67958i	−0.138529	+	0.138529i
$148$		0			0
$149$	−7.43213	−	7.43213i	−0.608864	−	0.608864i	0.333785	−	0.942649i	$-0.391674\pi$
$149$	−7.43213	−	7.43213i	−0.608864	−	0.608864i	−0.942649	+	0.333785i	$0.891674\pi$
$150$		0			0
$151$		−	17.0059i		−	1.38392i	−0.721936	−	0.691960i	$-0.756747\pi$
$151$		−	17.0059i		−	1.38392i	0.721936	−	0.691960i	$-0.243253\pi$
$152$		0			0
$153$		−	10.5979i		−	0.856789i
$154$		0			0
$155$	6.10823	+	6.10823i	0.490625	+	0.490625i
$156$		0			0
$157$	−10.2852	+	10.2852i	−0.820847	+	0.820847i	−0.986229	−	0.165383i	$-0.947114\pi$
$157$	−10.2852	+	10.2852i	−0.820847	+	0.820847i	0.165383	+	0.986229i	$0.447114\pi$
$158$		0			0
$159$	−18.9682			−1.50428
$160$		0			0
$161$	−0.610935			−0.0481484
$162$		0			0
$163$	1.57991	−	1.57991i	0.123748	−	0.123748i	−0.642521	−	0.766268i	$-0.722111\pi$
$163$	1.57991	−	1.57991i	0.123748	−	0.123748i	0.766268	+	0.642521i	$0.222111\pi$
$164$		0			0
$165$	9.58038	+	9.58038i	0.745832	+	0.745832i
$166$		0			0
$167$			19.4076i			1.50180i	0.660414	+	0.750902i	$0.270381\pi$
$167$			19.4076i			1.50180i	−0.660414	+	0.750902i	$0.729619\pi$
$168$		0			0
$169$		−	32.1852i		−	2.47578i
$170$		0			0
$171$	3.11405	+	3.11405i	0.238137	+	0.238137i
$172$		0			0
$173$	−2.53665	+	2.53665i	−0.192858	+	0.192858i	−0.796930	−	0.604072i	$-0.793544\pi$
$173$	−2.53665	+	2.53665i	−0.192858	+	0.192858i	0.604072	+	0.796930i	$0.293544\pi$
$174$		0			0
$175$	1.00000			0.0755929
$176$		0			0
$177$	4.16445			0.313019
$178$		0			0
$179$	0.426744	−	0.426744i	0.0318964	−	0.0318964i	−0.690979	−	0.722875i	$-0.742820\pi$
$179$	0.426744	−	0.426744i	0.0318964	−	0.0318964i	0.722875	+	0.690979i	$0.242820\pi$
$180$		0			0
$181$	4.85919	+	4.85919i	0.361181	+	0.361181i	0.864248	−	0.503067i	$-0.167795\pi$
$181$	4.85919	+	4.85919i	0.361181	+	0.361181i	−0.503067	+	0.864248i	$0.667795\pi$
$182$		0			0
$183$		−	19.2131i		−	1.42027i
$184$		0			0
$185$		−	1.62288i		−	0.119316i
$186$		0			0
$187$	16.1794	+	16.1794i	1.18315	+	1.18315i
$188$		0			0
$189$	0.601365	−	0.601365i	0.0437429	−	0.0437429i
$190$		0			0
$191$	11.4717			0.830062			0.415031	−	0.909807i	$-0.363771\pi$
$191$	11.4717			0.830062			0.415031	+	0.909807i	$0.363771\pi$
$192$		0			0
$193$	13.4250			0.966355			0.483177	−	0.875522i	$-0.339483\pi$
$193$	13.4250			0.966355			0.483177	+	0.875522i	$0.339483\pi$
$194$		0			0
$195$	−11.2901	+	11.2901i	−0.808501	+	0.808501i
$196$		0			0
$197$	10.2883	+	10.2883i	0.733015	+	0.733015i	0.971216	−	0.238201i	$-0.0765578\pi$
$197$	10.2883	+	10.2883i	0.733015	+	0.733015i	−0.238201	+	0.971216i	$0.576558\pi$
$198$		0			0
$199$		−	25.7491i		−	1.82530i	−0.408737	−	0.912652i	$-0.634031\pi$
$199$		−	25.7491i		−	1.82530i	0.408737	−	0.912652i	$-0.365969\pi$
$200$		0			0
$201$			8.00491i			0.564623i
$202$		0			0
$203$	−2.42606	−	2.42606i	−0.170276	−	0.170276i
$204$		0			0
$205$	−0.882352	+	0.882352i	−0.0616261	+	0.0616261i
$206$		0			0
$207$	−1.61406			−0.112185
$208$		0			0
$209$	−9.50819			−0.657695
$210$		0			0
$211$	12.1272	−	12.1272i	0.834868	−	0.834868i	−0.153310	−	0.988178i	$-0.548993\pi$
$211$	12.1272	−	12.1272i	0.834868	−	0.834868i	0.988178	+	0.153310i	$0.0489933\pi$
$212$		0			0
$213$	−2.64813	−	2.64813i	−0.181447	−	0.181447i
$214$		0			0
$215$			12.3139i			0.839799i
$216$		0			0
$217$		−	8.63834i		−	0.586409i
$218$		0			0
$219$	−3.12342	−	3.12342i	−0.211061	−	0.211061i
$220$		0			0
$221$	−19.0668	+	19.0668i	−1.28257	+	1.28257i
$222$		0			0
$223$	−1.64904			−0.110428			−0.0552140	−	0.998475i	$-0.517584\pi$
$223$	−1.64904			−0.110428			−0.0552140	+	0.998475i	$0.517584\pi$
$224$		0			0
$225$	2.64195			0.176130
$226$		0			0
$227$	−0.582397	+	0.582397i	−0.0386551	+	0.0386551i	−0.726170	−	0.687515i	$-0.758702\pi$
$227$	−0.582397	+	0.582397i	−0.0386551	+	0.0386551i	0.687515	+	0.726170i	$0.258702\pi$
$228$		0			0
$229$	5.72794	+	5.72794i	0.378513	+	0.378513i	0.870565	−	0.492053i	$-0.163753\pi$
$229$	5.72794	+	5.72794i	0.378513	+	0.378513i	−0.492053	+	0.870565i	$0.663753\pi$
$230$		0			0
$231$		−	13.5487i		−	0.891439i
$232$		0			0
$233$			1.17346i			0.0768760i	0.999261	+	0.0384380i	$0.0122382\pi$
$233$			1.17346i			0.0768760i	−0.999261	+	0.0384380i	$0.987762\pi$
$234$		0			0
$235$	−5.60719	−	5.60719i	−0.365773	−	0.365773i
$236$		0			0
$237$	−5.28855	+	5.28855i	−0.343528	+	0.343528i
$238$		0			0
$239$	28.4618			1.84104			0.920520	−	0.390697i	$-0.127766\pi$
$239$	28.4618			1.84104			0.920520	+	0.390697i	$0.127766\pi$
$240$		0			0
$241$	−18.9114			−1.21819			−0.609096	−	0.793097i	$-0.708467\pi$
$241$	−18.9114			−1.21819			−0.609096	+	0.793097i	$0.708467\pi$
$242$		0			0
$243$	14.9009	−	14.9009i	0.955891	−	0.955891i
$244$		0			0
$245$	−0.707107	−	0.707107i	−0.0451754	−	0.0451754i
$246$		0			0
$247$		−	11.2050i		−	0.712959i
$248$		0			0
$249$		−	20.2147i		−	1.28105i
$250$		0			0
$251$	9.17726	+	9.17726i	0.579263	+	0.579263i	0.934700	−	0.355437i	$-0.115668\pi$
$251$	9.17726	+	9.17726i	0.579263	+	0.579263i	−0.355437	+	0.934700i	$0.615668\pi$
$252$		0			0
$253$	2.46413	−	2.46413i	0.154918	−	0.154918i
$254$		0			0
$255$	9.52816			0.596676
$256$		0			0
$257$	−17.4767			−1.09017			−0.545084	−	0.838381i	$-0.683502\pi$
$257$	−17.4767			−1.09017			−0.545084	+	0.838381i	$0.683502\pi$
$258$		0			0
$259$	−1.14755	+	1.14755i	−0.0713051	+	0.0713051i
$260$		0			0
$261$	−6.40954	−	6.40954i	−0.396741	−	0.396741i
$262$		0			0
$263$			14.7508i			0.909575i	0.890600	+	0.454787i	$0.150285\pi$
$263$			14.7508i			0.909575i	−0.890600	+	0.454787i	$0.849715\pi$
$264$		0			0
$265$		−	7.98569i		−	0.490557i
$266$		0			0
$267$	−30.3848	−	30.3848i	−1.85952	−	1.85952i
$268$		0			0
$269$	−7.14150	+	7.14150i	−0.435425	+	0.435425i	−0.890469	−	0.455044i	$-0.849624\pi$
$269$	−7.14150	+	7.14150i	−0.435425	+	0.435425i	0.455044	+	0.890469i	$0.349624\pi$
$270$		0			0
$271$	−24.2531			−1.47327			−0.736634	−	0.676292i	$-0.763586\pi$
$271$	−24.2531			−1.47327			−0.736634	+	0.676292i	$0.763586\pi$
$272$		0			0
$273$	15.9666			0.966343
$274$		0			0
$275$	−4.03337	+	4.03337i	−0.243221	+	0.243221i
$276$		0			0
$277$	−13.6965	−	13.6965i	−0.822945	−	0.822945i	0.163585	−	0.986529i	$-0.447694\pi$
$277$	−13.6965	−	13.6965i	−0.822945	−	0.822945i	−0.986529	+	0.163585i	$0.947694\pi$
$278$		0			0
$279$		−	22.8221i		−	1.36632i
$280$		0			0
$281$		−	16.1993i		−	0.966367i	−0.875519	−	0.483184i	$-0.839480\pi$
$281$		−	16.1993i		−	0.966367i	0.875519	−	0.483184i	$-0.160520\pi$
$282$		0			0
$283$	−10.7931	−	10.7931i	−0.641584	−	0.641584i	0.309361	−	0.950945i	$-0.399885\pi$
$283$	−10.7931	−	10.7931i	−0.641584	−	0.641584i	−0.950945	+	0.309361i	$0.899885\pi$
$284$		0			0
$285$	−2.79972	+	2.79972i	−0.165841	+	0.165841i
$286$		0			0
$287$	1.24783			0.0736573
$288$		0			0
$289$	−0.908803			−0.0534590
$290$		0			0
$291$	−17.9054	+	17.9054i	−1.04963	+	1.04963i
$292$		0			0
$293$	4.38233	+	4.38233i	0.256019	+	0.256019i	0.823433	−	0.567414i	$-0.192056\pi$
$293$	4.38233	+	4.38233i	0.256019	+	0.256019i	−0.567414	+	0.823433i	$0.692056\pi$
$294$		0			0
$295$			1.75325i			0.102078i
$296$		0			0
$297$			4.85106i			0.281487i
$298$		0			0
$299$	2.90388	+	2.90388i	0.167935	+	0.167935i
$300$		0			0
$301$	8.70723	−	8.70723i	0.501876	−	0.501876i
$302$		0			0
$303$	10.5457			0.605832
$304$		0			0
$305$	8.08878			0.463162
$306$		0			0
$307$	8.17429	−	8.17429i	0.466531	−	0.466531i	−0.434258	−	0.900789i	$-0.642989\pi$
$307$	8.17429	−	8.17429i	0.466531	−	0.466531i	0.900789	+	0.434258i	$0.142989\pi$
$308$		0			0
$309$	5.10604	+	5.10604i	0.290473	+	0.290473i
$310$		0			0
$311$			18.4682i			1.04724i	0.851953	+	0.523618i	$0.175418\pi$
$311$			18.4682i			1.04724i	−0.851953	+	0.523618i	$0.824582\pi$
$312$		0			0
$313$		−	8.02087i		−	0.453366i	−0.973969	−	0.226683i	$-0.927212\pi$
$313$		−	8.02087i		−	0.453366i	0.973969	−	0.226683i	$-0.0727882\pi$
$314$		0			0
$315$	−1.86814	−	1.86814i	−0.105258	−	0.105258i
$316$		0			0
$317$	9.62720	−	9.62720i	0.540717	−	0.540717i	−0.383022	−	0.923739i	$-0.625117\pi$
$317$	9.62720	−	9.62720i	0.540717	−	0.540717i	0.923739	+	0.383022i	$0.125117\pi$
$318$		0			0
$319$	19.5704			1.09573
$320$		0			0
$321$	−1.17547			−0.0656082
$322$		0			0
$323$	−4.72818	+	4.72818i	−0.263083	+	0.263083i
$324$		0			0
$325$	−4.75317	−	4.75317i	−0.263658	−	0.263658i
$326$		0			0
$327$			32.4603i			1.79506i
$328$		0			0
$329$			7.92976i			0.437182i
$330$		0			0
$331$	14.5030	+	14.5030i	0.797159	+	0.797159i	0.982647	−	0.185488i	$-0.0593865\pi$
$331$	14.5030	+	14.5030i	0.797159	+	0.797159i	−0.185488	+	0.982647i	$0.559386\pi$
$332$		0			0
$333$	−3.03177	+	3.03177i	−0.166140	+	0.166140i
$334$		0			0
$335$	−3.37009			−0.184128
$336$		0			0
$337$	−6.26474			−0.341262			−0.170631	−	0.985335i	$-0.554581\pi$
$337$	−6.26474			−0.341262			−0.170631	+	0.985335i	$0.554581\pi$
$338$		0			0
$339$	−26.3679	+	26.3679i	−1.43211	+	1.43211i
$340$		0			0
$341$	34.8416	+	34.8416i	1.88678	+	1.88678i
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$		−	1.45114i		−	0.0781268i
$346$		0			0
$347$	1.52476	+	1.52476i	0.0818535	+	0.0818535i	0.746848	−	0.664995i	$-0.231566\pi$
$347$	1.52476	+	1.52476i	0.0818535	+	0.0818535i	−0.664995	+	0.746848i	$0.731566\pi$
$348$		0			0
$349$	0.611349	−	0.611349i	0.0327248	−	0.0327248i	−0.690555	−	0.723280i	$-0.742634\pi$
$349$	0.611349	−	0.611349i	0.0327248	−	0.0327248i	0.723280	+	0.690555i	$0.242634\pi$
$350$		0			0
$351$	−5.71678			−0.305139
$352$		0			0
$353$	−24.7889			−1.31938			−0.659690	−	0.751538i	$-0.729312\pi$
$353$	−24.7889			−1.31938			−0.659690	+	0.751538i	$0.729312\pi$
$354$		0			0
$355$	1.11487	−	1.11487i	0.0591713	−	0.0591713i
$356$		0			0
$357$	−6.73743	−	6.73743i	−0.356582	−	0.356582i
$358$		0			0
$359$		−	17.3351i		−	0.914911i	−0.889233	−	0.457455i	$-0.848761\pi$
$359$		−	17.3351i		−	0.914911i	0.889233	−	0.457455i	$-0.151239\pi$
$360$		0			0
$361$			16.2214i			0.853757i
$362$		0			0
$363$	36.1716	+	36.1716i	1.89852	+	1.89852i
$364$		0			0
$365$	1.31497	−	1.31497i	0.0688285	−	0.0688285i
$366$		0			0
$367$	−6.87342			−0.358790			−0.179395	−	0.983777i	$-0.557414\pi$
$367$	−6.87342			−0.358790			−0.179395	+	0.983777i	$0.557414\pi$
$368$		0			0
$369$	3.29672			0.171620
$370$		0			0
$371$	−5.64673	+	5.64673i	−0.293164	+	0.293164i
$372$		0			0
$373$	20.1924	+	20.1924i	1.04552	+	1.04552i	0.998913	+	0.0466080i	$0.0148412\pi$
$373$	20.1924	+	20.1924i	1.04552	+	1.04552i	0.0466080	+	0.998913i	$0.485159\pi$
$374$		0			0
$375$			2.37528i			0.122659i
$376$		0			0
$377$			23.0629i			1.18780i
$378$		0			0
$379$	3.43766	+	3.43766i	0.176581	+	0.176581i	0.789864	−	0.613283i	$-0.210151\pi$
$379$	3.43766	+	3.43766i	0.176581	+	0.176581i	−0.613283	+	0.789864i	$0.710151\pi$
$380$		0			0
$381$	16.2620	−	16.2620i	0.833129	−	0.833129i
$382$		0			0
$383$	−17.7783			−0.908431			−0.454216	−	0.890892i	$-0.650080\pi$
$383$	−17.7783			−0.908431			−0.454216	+	0.890892i	$0.650080\pi$
$384$		0			0
$385$	5.70405			0.290705
$386$		0			0
$387$	23.0041	−	23.0041i	1.16936	−	1.16936i
$388$		0			0
$389$	17.7695	+	17.7695i	0.900951	+	0.900951i	0.995518	−	0.0945674i	$-0.0301468\pi$
$389$	17.7695	+	17.7695i	0.900951	+	0.900951i	−0.0945674	+	0.995518i	$0.530147\pi$
$390$		0			0
$391$		−	2.45069i		−	0.123937i
$392$		0			0
$393$		−	34.7779i		−	1.75432i
$394$		0			0
$395$	−2.22649	−	2.22649i	−0.112027	−	0.112027i
$396$		0			0
$397$	12.7634	−	12.7634i	0.640575	−	0.640575i	−0.310122	−	0.950697i	$-0.600370\pi$
$397$	12.7634	−	12.7634i	0.640575	−	0.640575i	0.950697	+	0.310122i	$0.100370\pi$
$398$		0			0
$399$	3.95940			0.198218
$400$		0			0
$401$	20.5555			1.02649			0.513245	−	0.858242i	$-0.328443\pi$
$401$	20.5555			1.02649			0.513245	+	0.858242i	$0.328443\pi$
$402$		0			0
$403$	−41.0595	+	41.0595i	−2.04532	+	2.04532i
$404$		0			0
$405$	7.03284	+	7.03284i	0.349465	+	0.349465i
$406$		0			0
$407$		−	9.25696i		−	0.458850i
$408$		0			0
$409$		−	20.7319i		−	1.02513i	−0.858649	−	0.512564i	$-0.828696\pi$
$409$		−	20.7319i		−	1.02513i	0.858649	−	0.512564i	$-0.171304\pi$
$410$		0			0
$411$	13.1077	+	13.1077i	0.646554	+	0.646554i
$412$		0			0
$413$	1.23973	−	1.23973i	0.0610033	−	0.0610033i
$414$		0			0
$415$	8.51044			0.417761
$416$		0			0
$417$	−43.2188			−2.11643
$418$		0			0
$419$	−3.85590	+	3.85590i	−0.188373	+	0.188373i	−0.794992	−	0.606620i	$-0.792525\pi$
$419$	−3.85590	+	3.85590i	−0.188373	+	0.188373i	0.606620	+	0.794992i	$0.292525\pi$
$420$		0			0
$421$	−0.480906	−	0.480906i	−0.0234379	−	0.0234379i	0.695291	−	0.718729i	$-0.255276\pi$
$421$	−0.480906	−	0.480906i	−0.0234379	−	0.0234379i	−0.718729	+	0.695291i	$0.755276\pi$
$422$		0			0
$423$			20.9501i			1.01863i
$424$		0			0
$425$			4.01138i			0.194581i
$426$		0			0
$427$	−5.71963	−	5.71963i	−0.276792	−	0.276792i
$428$		0			0
$429$	−64.3992	+	64.3992i	−3.10923	+	3.10923i
$430$		0			0
$431$	−15.8219			−0.762114			−0.381057	−	0.924551i	$-0.624440\pi$
$431$	−15.8219			−0.762114			−0.381057	+	0.924551i	$0.624440\pi$
$432$		0			0
$433$	−10.7353			−0.515906			−0.257953	−	0.966157i	$-0.583048\pi$
$433$	−10.7353			−0.515906			−0.257953	+	0.966157i	$0.583048\pi$
$434$		0			0
$435$	5.76257	−	5.76257i	0.276294	−	0.276294i
$436$		0			0
$437$	0.720104	+	0.720104i	0.0344472	+	0.0344472i
$438$		0			0
$439$		−	32.3977i		−	1.54626i	−0.634250	−	0.773128i	$-0.718691\pi$
$439$		−	32.3977i		−	1.54626i	0.634250	−	0.773128i	$-0.281309\pi$
$440$		0			0
$441$			2.64195i			0.125807i
$442$		0			0
$443$	19.5555	+	19.5555i	0.929110	+	0.929110i	0.997648	−	0.0685383i	$-0.0218335\pi$
$443$	19.5555	+	19.5555i	0.929110	+	0.929110i	−0.0685383	+	0.997648i	$0.521834\pi$
$444$		0			0
$445$	12.7921	−	12.7921i	0.606402	−	0.606402i
$446$		0			0
$447$	−24.9657			−1.18083
$448$		0			0
$449$	−41.8148			−1.97336			−0.986680	−	0.162673i	$-0.947989\pi$
$449$	−41.8148			−1.97336			−0.986680	+	0.162673i	$0.947989\pi$
$450$		0			0
$451$	−5.03297	+	5.03297i	−0.236993	+	0.236993i
$452$		0			0
$453$	−28.5627	−	28.5627i	−1.34199	−	1.34199i
$454$		0			0
$455$			6.72199i			0.315132i
$456$		0			0
$457$		−	18.4598i		−	0.863515i	−0.901990	−	0.431757i	$-0.857894\pi$
$457$		−	18.4598i		−	0.863515i	0.901990	−	0.431757i	$-0.142106\pi$
$458$		0			0
$459$	2.41231	+	2.41231i	0.112597	+	0.112597i
$460$		0			0
$461$	6.90312	−	6.90312i	0.321510	−	0.321510i	−0.527836	−	0.849346i	$-0.676996\pi$
$461$	6.90312	−	6.90312i	0.321510	−	0.321510i	0.849346	+	0.527836i	$0.176996\pi$
$462$		0			0
$463$	−23.8189			−1.10696			−0.553480	−	0.832862i	$-0.686700\pi$
$463$	−23.8189			−1.10696			−0.553480	+	0.832862i	$0.686700\pi$
$464$		0			0
$465$	20.5185			0.951521
$466$		0			0
$467$	−9.73816	+	9.73816i	−0.450628	+	0.450628i	−0.895563	−	0.444935i	$-0.853227\pi$
$467$	−9.73816	+	9.73816i	−0.450628	+	0.450628i	0.444935	+	0.895563i	$0.353227\pi$
$468$		0			0
$469$	2.38301	+	2.38301i	0.110037	+	0.110037i
$470$		0			0
$471$			34.5495i			1.59196i
$472$		0			0
$473$			70.2389i			3.22959i
$474$		0			0
$475$	−1.17869	−	1.17869i	−0.0540820	−	0.0540820i
$476$		0			0
$477$	−14.9184	+	14.9184i	−0.683067	+	0.683067i
$478$		0			0
$479$	26.2926			1.20134			0.600670	−	0.799497i	$-0.294900\pi$
$479$	26.2926			1.20134			0.600670	+	0.799497i	$0.294900\pi$
$480$		0			0
$481$	10.9090			0.497406
$482$		0			0
$483$	−1.02611	+	1.02611i	−0.0466897	+	0.0466897i
$484$		0			0
$485$	−7.53822	−	7.53822i	−0.342293	−	0.342293i
$486$		0			0
$487$			9.31040i			0.421895i	0.977497	+	0.210947i	$0.0676549\pi$
$487$			9.31040i			0.421895i	−0.977497	+	0.210947i	$0.932345\pi$
$488$		0			0
$489$		−	5.30715i		−	0.239998i
$490$		0			0
$491$	−17.8430	−	17.8430i	−0.805245	−	0.805245i	0.178665	−	0.983910i	$-0.442822\pi$
$491$	−17.8430	−	17.8430i	−0.805245	−	0.805245i	−0.983910	+	0.178665i	$0.942822\pi$
$492$		0			0
$493$	9.73186	−	9.73186i	0.438301	−	0.438301i
$494$		0			0
$495$	15.0698			0.677338
$496$		0			0
$497$	−1.57667			−0.0707232
$498$		0			0
$499$	−24.3976	+	24.3976i	−1.09219	+	1.09219i	−0.0968934	+	0.995295i	$0.530891\pi$
$499$	−24.3976	+	24.3976i	−1.09219	+	1.09219i	−0.995295	+	0.0968934i	$0.969109\pi$
$500$		0			0
$501$	32.5965	+	32.5965i	1.45631	+	1.45631i
$502$		0			0
$503$		−	41.9958i		−	1.87250i	−0.351336	−	0.936249i	$-0.614273\pi$
$503$		−	41.9958i		−	1.87250i	0.351336	−	0.936249i	$-0.385727\pi$
$504$		0			0
$505$			4.43975i			0.197566i
$506$		0			0
$507$	−54.0575	−	54.0575i	−2.40078	−	2.40078i
$508$		0			0
$509$	−7.57957	+	7.57957i	−0.335959	+	0.335959i	−0.854844	−	0.518885i	$-0.826347\pi$
$509$	−7.57957	+	7.57957i	−0.335959	+	0.335959i	0.518885	+	0.854844i	$0.326347\pi$
$510$		0			0
$511$	−1.85964			−0.0822658
$512$		0			0
$513$	−1.41765			−0.0625907
$514$		0			0
$515$	−2.14966	+	2.14966i	−0.0947253	+	0.0947253i
$516$		0			0
$517$	−31.9837	−	31.9837i	−1.40664	−	1.40664i
$518$		0			0
$519$			8.52099i			0.374030i
$520$		0			0
$521$		−	11.7275i		−	0.513793i	−0.966439	−	0.256897i	$-0.917300\pi$
$521$		−	11.7275i		−	0.513793i	0.966439	−	0.256897i	$-0.0827000\pi$
$522$		0			0
$523$	−18.3603	−	18.3603i	−0.802839	−	0.802839i	0.180699	−	0.983538i	$-0.442164\pi$
$523$	−18.3603	−	18.3603i	−0.802839	−	0.802839i	−0.983538	+	0.180699i	$0.942164\pi$
$524$		0			0
$525$	1.67958	−	1.67958i	0.0733027	−	0.0733027i
$526$		0			0
$527$	34.6517			1.50945
$528$		0			0
$529$	22.6268			0.983772
$530$		0			0
$531$	3.27532	−	3.27532i	0.142137	−	0.142137i
$532$		0			0
$533$	−5.93116	−	5.93116i	−0.256907	−	0.256907i
$534$		0			0
$535$		−	0.494875i		−	0.0213953i
$536$		0			0
$537$		−	1.43350i		−	0.0618600i
$538$		0			0
$539$	−4.03337	−	4.03337i	−0.173730	−	0.173730i
$540$		0			0
$541$	−8.83042	+	8.83042i	−0.379649	+	0.379649i	−0.870976	−	0.491326i	$-0.836512\pi$
$541$	−8.83042	+	8.83042i	−0.379649	+	0.379649i	0.491326	+	0.870976i	$0.336512\pi$
$542$		0			0
$543$	16.3228			0.700477
$544$		0			0
$545$	−13.6659			−0.585382
$546$		0			0
$547$	13.7417	−	13.7417i	0.587552	−	0.587552i	−0.349416	−	0.936968i	$-0.613620\pi$
$547$	13.7417	−	13.7417i	0.587552	−	0.587552i	0.936968	+	0.349416i	$0.113620\pi$
$548$		0			0
$549$	−15.1110	−	15.1110i	−0.644922	−	0.644922i
$550$		0			0
$551$			5.71915i			0.243644i
$552$		0			0
$553$			3.14874i			0.133898i
$554$		0			0
$555$	−2.72574	−	2.72574i	−0.115701	−	0.115701i
$556$		0			0
$557$	−2.00922	+	2.00922i	−0.0851335	+	0.0851335i	−0.748391	−	0.663258i	$-0.769173\pi$
$557$	−2.00922	+	2.00922i	−0.0851335	+	0.0851335i	0.663258	+	0.748391i	$0.269173\pi$
$558$		0			0
$559$	−82.7738			−3.50096
$560$		0			0
$561$	54.3490			2.29462
$562$		0			0
$563$	12.4610	−	12.4610i	0.525167	−	0.525167i	−0.393961	−	0.919127i	$-0.628895\pi$
$563$	12.4610	−	12.4610i	0.525167	−	0.525167i	0.919127	+	0.393961i	$0.128895\pi$
$564$		0			0
$565$	−11.1010	−	11.1010i	−0.467021	−	0.467021i
$566$		0			0
$567$		−	9.94594i		−	0.417690i
$568$		0			0
$569$			22.7065i			0.951905i	0.879471	+	0.475952i	$0.157896\pi$
$569$			22.7065i			0.951905i	−0.879471	+	0.475952i	$0.842104\pi$
$570$		0			0
$571$	10.2731	+	10.2731i	0.429915	+	0.429915i	0.888599	−	0.458684i	$-0.151679\pi$
$571$	10.2731	+	10.2731i	0.429915	+	0.429915i	−0.458684	+	0.888599i	$0.651679\pi$
$572$		0			0
$573$	19.2676	−	19.2676i	0.804915	−	0.804915i
$574$		0			0
$575$	0.610935			0.0254778
$576$		0			0
$577$	−18.1866			−0.757120			−0.378560	−	0.925577i	$-0.623581\pi$
$577$	−18.1866			−0.757120			−0.378560	+	0.925577i	$0.623581\pi$
$578$		0			0
$579$	22.5484	−	22.5484i	0.937078	−	0.937078i
$580$		0			0
$581$	−6.01779	−	6.01779i	−0.249660	−	0.249660i
$582$		0			0
$583$		−	45.5507i		−	1.88652i
$584$		0			0
$585$			17.7592i			0.734252i
$586$		0			0
$587$	−14.3795	−	14.3795i	−0.593504	−	0.593504i	0.345072	−	0.938576i	$-0.387855\pi$
$587$	−14.3795	−	14.3795i	−0.593504	−	0.593504i	−0.938576	+	0.345072i	$0.887855\pi$
$588$		0			0
$589$	−10.1819	+	10.1819i	−0.419539	+	0.419539i
$590$		0			0
$591$	34.5601			1.42161
$592$		0			0
$593$	−43.2423			−1.77575			−0.887875	−	0.460085i	$-0.847819\pi$
$593$	−43.2423			−1.77575			−0.887875	+	0.460085i	$0.847819\pi$
$594$		0			0
$595$	2.83648	−	2.83648i	0.116284	−	0.116284i
$596$		0			0
$597$	−43.2476	−	43.2476i	−1.77000	−	1.77000i
$598$		0			0
$599$			12.2633i			0.501067i	0.968108	+	0.250533i	$0.0806060\pi$
$599$			12.2633i			0.501067i	−0.968108	+	0.250533i	$0.919394\pi$
$600$		0			0
$601$			10.1758i			0.415078i	0.978227	+	0.207539i	$0.0665455\pi$
$601$			10.1758i			0.415078i	−0.978227	+	0.207539i	$0.933455\pi$
$602$		0			0
$603$	6.29582	+	6.29582i	0.256385	+	0.256385i
$604$		0			0
$605$	−15.2283	+	15.2283i	−0.619120	+	0.619120i
$606$		0			0
$607$	−34.7601			−1.41087			−0.705434	−	0.708776i	$-0.749248\pi$
$607$	−34.7601			−1.41087			−0.705434	+	0.708776i	$0.749248\pi$
$608$		0			0
$609$	−8.14951			−0.330235
$610$		0			0
$611$	37.6915	−	37.6915i	1.52483	−	1.52483i
$612$		0			0
$613$	−21.1023	−	21.1023i	−0.852314	−	0.852314i	0.138103	−	0.990418i	$-0.455899\pi$
$613$	−21.1023	−	21.1023i	−0.852314	−	0.852314i	−0.990418	+	0.138103i	$0.955899\pi$
$614$		0			0
$615$			2.96395i			0.119518i
$616$		0			0
$617$		−	29.6621i		−	1.19415i	−0.802185	−	0.597075i	$-0.796329\pi$
$617$		−	29.6621i		−	1.19415i	0.802185	−	0.597075i	$-0.203671\pi$
$618$		0			0
$619$	22.1895	+	22.1895i	0.891872	+	0.891872i	0.994699	−	0.102827i	$-0.0327888\pi$
$619$	22.1895	+	22.1895i	0.891872	+	0.891872i	−0.102827	+	0.994699i	$0.532789\pi$
$620$		0			0
$621$	0.367395	−	0.367395i	0.0147431	−	0.0147431i
$622$		0			0
$623$	−18.0907			−0.724790
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$	−15.9697	+	15.9697i	−0.637770	+	0.637770i
$628$		0			0
$629$	−4.60325	−	4.60325i	−0.183544	−	0.183544i
$630$		0			0
$631$			24.2301i			0.964583i	0.876011	+	0.482292i	$0.160195\pi$
$631$			24.2301i			0.964583i	−0.876011	+	0.482292i	$0.839805\pi$
$632$		0			0
$633$		−	40.7370i		−	1.61915i
$634$		0			0
$635$	6.84636	+	6.84636i	0.271690	+	0.271690i
$636$		0			0
$637$	4.75317	−	4.75317i	0.188327	−	0.188327i
$638$		0			0
$639$	−4.16548			−0.164784
$640$		0			0
$641$	−29.6060			−1.16937			−0.584683	−	0.811262i	$-0.698781\pi$
$641$	−29.6060			−1.16937			−0.584683	+	0.811262i	$0.698781\pi$
$642$		0			0
$643$	23.5440	−	23.5440i	0.928485	−	0.928485i	−0.0691234	−	0.997608i	$-0.522020\pi$
$643$	23.5440	−	23.5440i	0.928485	−	0.928485i	0.997608	+	0.0691234i	$0.0220202\pi$
$644$		0			0
$645$	20.6821	+	20.6821i	0.814357	+	0.814357i
$646$		0			0
$647$			0.629008i			0.0247288i	0.999924	+	0.0123644i	$0.00393582\pi$
$647$			0.629008i			0.0247288i	−0.999924	+	0.0123644i	$0.996064\pi$
$648$		0			0
$649$			10.0006i			0.392558i
$650$		0			0
$651$	−14.5087	−	14.5087i	−0.568643	−	0.568643i
$652$		0			0
$653$	16.0838	−	16.0838i	0.629409	−	0.629409i	−0.318510	−	0.947919i	$-0.603183\pi$
$653$	16.0838	−	16.0838i	0.629409	−	0.629409i	0.947919	+	0.318510i	$0.103183\pi$
$654$		0			0
$655$	14.6416			0.572095
$656$		0			0
$657$	−4.91310			−0.191678
$658$		0			0
$659$	−32.6711	+	32.6711i	−1.27268	+	1.27268i	−0.328011	+	0.944674i	$0.606378\pi$
$659$	−32.6711	+	32.6711i	−1.27268	+	1.27268i	−0.944674	+	0.328011i	$0.893622\pi$
$660$		0			0
$661$	−12.6881	−	12.6881i	−0.493512	−	0.493512i	0.415899	−	0.909411i	$-0.363467\pi$
$661$	−12.6881	−	12.6881i	−0.493512	−	0.493512i	−0.909411	+	0.415899i	$0.863467\pi$
$662$		0			0
$663$			64.0482i			2.48743i
$664$		0			0
$665$			1.66692i			0.0646404i
$666$		0			0
$667$	−1.48217	−	1.48217i	−0.0573897	−	0.0573897i
$668$		0			0
$669$	−2.76969	+	2.76969i	−0.107082	+	0.107082i
$670$		0			0
$671$	46.1388			1.78117
$672$		0			0
$673$	45.0856			1.73792			0.868961	−	0.494880i	$-0.164788\pi$
$673$	45.0856			1.73792			0.868961	+	0.494880i	$0.164788\pi$
$674$		0			0
$675$	−0.601365	+	0.601365i	−0.0231466	+	0.0231466i
$676$		0			0
$677$	6.31413	+	6.31413i	0.242672	+	0.242672i	0.817955	−	0.575283i	$-0.195108\pi$
$677$	6.31413	+	6.31413i	0.242672	+	0.242672i	−0.575283	+	0.817955i	$0.695108\pi$
$678$		0			0
$679$			10.6607i			0.409118i
$680$		0			0
$681$			1.95636i			0.0749679i
$682$		0			0
$683$	1.23835	+	1.23835i	0.0473841	+	0.0473841i	0.730402	−	0.683018i	$-0.239333\pi$
$683$	1.23835	+	1.23835i	0.0473841	+	0.0473841i	−0.683018	+	0.730402i	$0.739333\pi$
$684$		0			0
$685$	−5.51837	+	5.51837i	−0.210846	+	0.210846i
$686$		0			0
$687$	19.2410			0.734090
$688$		0			0
$689$	53.6797			2.04503
$690$		0			0
$691$	−4.67186	+	4.67186i	−0.177726	+	0.177726i	−0.790364	−	0.612638i	$-0.790108\pi$
$691$	−4.67186	+	4.67186i	−0.177726	+	0.177726i	0.612638	+	0.790364i	$0.290108\pi$
$692$		0			0
$693$	−10.6560	−	10.6560i	−0.404787	−	0.404787i
$694$		0			0
$695$		−	18.1952i		−	0.690185i
$696$		0			0
$697$			5.00554i			0.189598i
$698$		0			0
$699$	1.97092	+	1.97092i	0.0745469	+	0.0745469i
$700$		0			0
$701$	13.3471	−	13.3471i	0.504113	−	0.504113i	−0.408600	−	0.912713i	$-0.633983\pi$
$701$	13.3471	−	13.3471i	0.504113	−	0.504113i	0.912713	+	0.408600i	$0.133983\pi$
$702$		0			0
$703$	2.70521			0.102029
$704$		0			0
$705$	−18.8354			−0.709382
$706$		0			0
$707$	3.13938	−	3.13938i	0.118069	−	0.118069i
$708$		0			0
$709$	−28.0487	−	28.0487i	−1.05339	−	1.05339i	−0.998492	−	0.0548997i	$-0.982516\pi$
$709$	−28.0487	−	28.0487i	−1.05339	−	1.05339i	−0.0548997	−	0.998492i	$-0.517484\pi$
$710$		0			0
$711$			8.31883i			0.311980i
$712$		0			0
$713$		−	5.27746i		−	0.197643i
$714$		0			0
$715$	−27.1123	−	27.1123i	−1.01394	−	1.01394i
$716$		0			0
$717$	47.8037	−	47.8037i	1.78526	−	1.78526i
$718$		0			0
$719$	−1.45224			−0.0541594			−0.0270797	−	0.999633i	$-0.508621\pi$
$719$	−1.45224			−0.0541594			−0.0270797	+	0.999633i	$0.508621\pi$
$720$		0			0
$721$	3.04008			0.113218
$722$		0			0
$723$	−31.7632	+	31.7632i	−1.18128	+	1.18128i
$724$		0			0
$725$	2.42606	+	2.42606i	0.0901017	+	0.0901017i
$726$		0			0
$727$		−	1.90322i		−	0.0705866i	−0.999377	−	0.0352933i	$-0.988763\pi$
$727$		−	1.90322i		−	0.0705866i	0.999377	−	0.0352933i	$-0.0112365\pi$
$728$		0			0
$729$		−	20.2165i		−	0.748759i
$730$		0			0
$731$	34.9280	+	34.9280i	1.29186	+	1.29186i
$732$		0			0
$733$	−13.0224	+	13.0224i	−0.480993	+	0.480993i	−0.905449	−	0.424456i	$-0.860465\pi$
$733$	−13.0224	+	13.0224i	−0.480993	+	0.480993i	0.424456	+	0.905449i	$0.360465\pi$
$734$		0			0
$735$	−2.37528			−0.0876135
$736$		0			0
$737$	−19.2232			−0.708094
$738$		0			0
$739$	−0.749479	+	0.749479i	−0.0275700	+	0.0275700i	−0.720757	−	0.693187i	$-0.756206\pi$
$739$	−0.749479	+	0.749479i	−0.0275700	+	0.0275700i	0.693187	+	0.720757i	$0.256206\pi$
$740$		0			0
$741$	−18.8197	−	18.8197i	−0.691359	−	0.691359i
$742$		0			0
$743$			34.5259i			1.26663i	0.773893	+	0.633317i	$0.218307\pi$
$743$			34.5259i			1.26663i	−0.773893	+	0.633317i	$0.781693\pi$
$744$		0			0
$745$		−	10.5106i		−	0.385079i
$746$		0			0
$747$	−15.8987	−	15.8987i	−0.581704	−	0.581704i
$748$		0			0
$749$	−0.349930	+	0.349930i	−0.0127861	+	0.0127861i
$750$		0			0
$751$	1.69258			0.0617631			0.0308815	−	0.999523i	$-0.490169\pi$
$751$	1.69258			0.0617631			0.0308815	+	0.999523i	$0.490169\pi$
$752$		0			0
$753$	30.8278			1.12343
$754$		0			0
$755$	12.0250	−	12.0250i	0.437634	−	0.437634i
$756$		0			0
$757$	17.2025	+	17.2025i	0.625235	+	0.625235i	0.946865	−	0.321630i	$-0.104231\pi$
$757$	17.2025	+	17.2025i	0.625235	+	0.625235i	−0.321630	+	0.946865i	$0.604231\pi$
$758$		0			0
$759$		−	8.27738i		−	0.300450i
$760$		0			0
$761$			15.5249i			0.562778i	0.959594	+	0.281389i	$0.0907951\pi$
$761$			15.5249i			0.562778i	−0.959594	+	0.281389i	$0.909205\pi$
$762$		0			0
$763$	9.66324	+	9.66324i	0.349833	+	0.349833i
$764$		0			0
$765$	7.49384	−	7.49384i	0.270940	−	0.270940i
$766$		0			0
$767$	−11.7853			−0.425543
$768$		0			0
$769$	−3.03643			−0.109496			−0.0547482	−	0.998500i	$-0.517436\pi$
$769$	−3.03643			−0.109496			−0.0547482	+	0.998500i	$0.517436\pi$
$770$		0			0
$771$	−29.3535	+	29.3535i	−1.05714	+	1.05714i
$772$		0			0
$773$	−28.2372	−	28.2372i	−1.01562	−	1.01562i	−0.999876	−	0.0157441i	$-0.994988\pi$
$773$	−28.2372	−	28.2372i	−1.01562	−	1.01562i	−0.0157441	−	0.999876i	$-0.505012\pi$
$774$		0			0
$775$			8.63834i			0.310298i
$776$		0			0
$777$			3.85478i			0.138290i
$778$		0			0
$779$	−1.47081	−	1.47081i	−0.0526972	−	0.0526972i
$780$		0			0
$781$	6.35928	−	6.35928i	0.227553	−	0.227553i
$782$		0			0
$783$	2.91790			0.104277
$784$		0			0
$785$	−14.5454			−0.519149
$786$		0			0
$787$	8.21518	−	8.21518i	0.292839	−	0.292839i	−0.545362	−	0.838201i	$-0.683608\pi$
$787$	8.21518	−	8.21518i	0.292839	−	0.292839i	0.838201	+	0.545362i	$0.183608\pi$
$788$		0			0
$789$	24.7751	+	24.7751i	0.882018	+	0.882018i
$790$		0			0
$791$			15.6991i			0.558197i
$792$		0			0
$793$			54.3727i			1.93083i
$794$		0			0
$795$	−13.4126	−	13.4126i	−0.475695	−	0.475695i
$796$		0			0
$797$	21.3358	−	21.3358i	0.755754	−	0.755754i	−0.219793	−	0.975547i	$-0.570538\pi$
$797$	21.3358	−	21.3358i	0.755754	−	0.755754i	0.975547	+	0.219793i	$0.0705381\pi$
$798$		0			0
$799$	−31.8093			−1.12533
$800$		0			0
$801$	−47.7949			−1.68875
$802$		0			0
$803$	7.50063	−	7.50063i	0.264692	−	0.264692i
$804$		0			0
$805$	−0.431996	−	0.431996i	−0.0152259	−	0.0152259i
$806$		0			0
$807$			23.9894i			0.844467i
$808$		0			0
$809$			18.4079i			0.647186i	0.946196	+	0.323593i	$0.104891\pi$
$809$			18.4079i			0.647186i	−0.946196	+	0.323593i	$0.895109\pi$
$810$		0			0
$811$	−30.9453	−	30.9453i	−1.08664	−	1.08664i	−0.995872	−	0.0907642i	$-0.971069\pi$
$811$	−30.9453	−	30.9453i	−1.08664	−	1.08664i	−0.0907642	−	0.995872i	$-0.528931\pi$
$812$		0			0
$813$	−40.7349	+	40.7349i	−1.42863	+	1.42863i
$814$		0			0
$815$	2.23433			0.0782651
$816$		0			0
$817$	−20.5263			−0.718123
$818$		0			0
$819$	12.5576	−	12.5576i	0.438800	−	0.438800i
$820$		0			0
$821$	−26.7423	−	26.7423i	−0.933312	−	0.933312i	0.0645994	−	0.997911i	$-0.479423\pi$
$821$	−26.7423	−	26.7423i	−0.933312	−	0.933312i	−0.997911	+	0.0645994i	$0.979423\pi$
$822$		0			0
$823$		−	17.1926i		−	0.599297i	−0.954050	−	0.299649i	$-0.903131\pi$
$823$		−	17.1926i		−	0.599297i	0.954050	−	0.299649i	$-0.0968694\pi$
$824$		0			0
$825$			13.5487i			0.471705i
$826$		0			0
$827$	−20.7087	−	20.7087i	−0.720112	−	0.720112i	0.248516	−	0.968628i	$-0.420057\pi$
$827$	−20.7087	−	20.7087i	−0.720112	−	0.720112i	−0.968628	+	0.248516i	$0.920057\pi$
$828$		0			0
$829$	23.2314	−	23.2314i	0.806860	−	0.806860i	−0.177297	−	0.984157i	$-0.556735\pi$
$829$	23.2314	−	23.2314i	0.806860	−	0.806860i	0.984157	+	0.177297i	$0.0567354\pi$
$830$		0			0
$831$	−46.0087			−1.59603
$832$		0			0
$833$	−4.01138			−0.138986
$834$		0			0
$835$	−13.7232	+	13.7232i	−0.474912	+	0.474912i
$836$		0			0
$837$	5.19480	+	5.19480i	0.179558	+	0.179558i
$838$		0			0
$839$		−	3.91663i		−	0.135217i	−0.997712	−	0.0676085i	$-0.978463\pi$
$839$		−	3.91663i		−	0.135217i	0.997712	−	0.0676085i	$-0.0215369\pi$
$840$		0			0
$841$			17.2285i			0.594085i
$842$		0			0
$843$	−27.2079	−	27.2079i	−0.937090	−	0.937090i
$844$		0			0
$845$	22.7584	−	22.7584i	0.782911	−	0.782911i
$846$		0			0
$847$	21.5361			0.739990
$848$		0			0
$849$	−36.2557			−1.24429
$850$		0			0
$851$	−0.701076	+	0.701076i	−0.0240326	+	0.0240326i
$852$		0			0
$853$	−19.3540	−	19.3540i	−0.662667	−	0.662667i	0.293341	−	0.956008i	$-0.405233\pi$
$853$	−19.3540	−	19.3540i	−0.662667	−	0.662667i	−0.956008	+	0.293341i	$0.905233\pi$
$854$		0			0
$855$			4.40393i			0.150611i
$856$		0			0
$857$			57.3516i			1.95909i	0.201218	+	0.979546i	$0.435510\pi$
$857$			57.3516i			1.95909i	−0.201218	+	0.979546i	$0.564490\pi$
$858$		0			0
$859$	14.9811	+	14.9811i	0.511150	+	0.511150i	0.914879	−	0.403729i	$-0.132286\pi$
$859$	14.9811	+	14.9811i	0.511150	+	0.511150i	−0.403729	+	0.914879i	$0.632286\pi$
$860$		0			0
$861$	2.09583	−	2.09583i	0.0714258	−	0.0714258i
$862$		0			0
$863$	−28.9117			−0.984167			−0.492083	−	0.870548i	$-0.663765\pi$
$863$	−28.9117			−0.984167			−0.492083	+	0.870548i	$0.663765\pi$
$864$		0			0
$865$	−3.58736			−0.121974
$866$		0			0
$867$	−1.52640	+	1.52640i	−0.0518394	+	0.0518394i
$868$		0			0
$869$	−12.7000	−	12.7000i	−0.430819	−	0.430819i
$870$		0			0
$871$		−	22.6537i		−	0.767592i
$872$		0			0
$873$			28.1650i			0.953240i
$874$		0			0
$875$	0.707107	+	0.707107i	0.0239046	+	0.0239046i
$876$		0			0
$877$	17.3785	−	17.3785i	0.586829	−	0.586829i	−0.349942	−	0.936771i	$-0.613799\pi$
$877$	17.3785	−	17.3785i	0.586829	−	0.586829i	0.936771	+	0.349942i	$0.113799\pi$
$878$		0			0
$879$	14.7209			0.496525
$880$		0			0
$881$	10.8578			0.365807			0.182904	−	0.983131i	$-0.441450\pi$
$881$	10.8578			0.365807			0.182904	+	0.983131i	$0.441450\pi$
$882$		0			0
$883$	−16.5408	+	16.5408i	−0.556644	+	0.556644i	−0.928350	−	0.371707i	$-0.878773\pi$
$883$	−16.5408	+	16.5408i	−0.556644	+	0.556644i	0.371707	+	0.928350i	$0.378773\pi$
$884$		0			0
$885$	2.94471	+	2.94471i	0.0989854	+	0.0989854i
$886$		0			0
$887$		−	31.7134i		−	1.06483i	−0.846483	−	0.532416i	$-0.821284\pi$
$887$		−	31.7134i		−	1.06483i	0.846483	−	0.532416i	$-0.178716\pi$
$888$		0			0
$889$		−	9.68222i		−	0.324731i
$890$		0			0
$891$	40.1157	+	40.1157i	1.34392	+	1.34392i
$892$		0			0
$893$	9.34674	−	9.34674i	0.312777	−	0.312777i
$894$		0			0
$895$	0.603507			0.0201730
$896$		0			0
$897$	9.75456			0.325695
$898$		0			0
$899$	20.9571	−	20.9571i	0.698960	−	0.698960i
$900$		0			0
$901$	−22.6512	−	22.6512i	−0.754621	−	0.754621i
$902$		0			0
$903$		−	29.2489i		−	0.973343i
$904$		0			0
$905$			6.87194i			0.228431i
$906$		0			0
$907$	26.9647	+	26.9647i	0.895347	+	0.895347i	0.995020	−	0.0996733i	$-0.0317798\pi$
$907$	26.9647	+	26.9647i	0.895347	+	0.895347i	−0.0996733	+	0.995020i	$0.531780\pi$
$908$		0			0
$909$	8.29410	−	8.29410i	0.275098	−	0.275098i
$910$		0			0
$911$	−3.58613			−0.118814			−0.0594070	−	0.998234i	$-0.518921\pi$
$911$	−3.58613			−0.118814			−0.0594070	+	0.998234i	$0.518921\pi$
$912$		0			0
$913$	48.5439			1.60657
$914$		0			0
$915$	13.5857	−	13.5857i	0.449130	−	0.449130i
$916$		0			0
$917$	−10.3532	−	10.3532i	−0.341892	−	0.341892i
$918$		0			0
$919$			23.7442i			0.783249i	0.920125	+	0.391624i	$0.128087\pi$
$919$			23.7442i			0.783249i	−0.920125	+	0.391624i	$0.871913\pi$
$920$		0			0
$921$		−	27.4587i		−	0.904794i
$922$		0			0
$923$	7.49416	+	7.49416i	0.246673	+	0.246673i
$924$		0			0
$925$	1.14755	−	1.14755i	0.0377311	−	0.0377311i
$926$		0			0
$927$	8.03174			0.263797
$928$		0			0
$929$	5.63142			0.184761			0.0923805	−	0.995724i	$-0.470552\pi$
$929$	5.63142			0.184761			0.0923805	+	0.995724i	$0.470552\pi$
$930$		0			0
$931$	1.17869	−	1.17869i	0.0386300	−	0.0386300i
$932$		0			0
$933$	31.0188	+	31.0188i	1.01551	+	1.01551i
$934$		0			0
$935$			22.8811i			0.748292i
$936$		0			0
$937$		−	22.1711i		−	0.724299i	−0.932120	−	0.362150i	$-0.882043\pi$
$937$		−	22.1711i		−	0.724299i	0.932120	−	0.362150i	$-0.117957\pi$
$938$		0			0
$939$	−13.4717	−	13.4717i	−0.439631	−	0.439631i
$940$		0			0
$941$	−23.5907	+	23.5907i	−0.769036	+	0.769036i	−0.977937	−	0.208900i	$-0.933012\pi$
$941$	−23.5907	+	23.5907i	−0.769036	+	0.769036i	0.208900	+	0.977937i	$0.433012\pi$
$942$		0			0
$943$	0.762345			0.0248254
$944$		0			0
$945$	0.850459			0.0276654
$946$		0			0
$947$	−17.6049	+	17.6049i	−0.572084	+	0.572084i	−0.932710	−	0.360627i	$-0.882563\pi$
$947$	−17.6049	+	17.6049i	−0.572084	+	0.572084i	0.360627	+	0.932710i	$0.382563\pi$
$948$		0			0
$949$	8.83920	+	8.83920i	0.286933	+	0.286933i
$950$		0			0
$951$		−	32.3392i		−	1.04867i
$952$		0			0
$953$		−	27.9895i		−	0.906670i	−0.891340	−	0.453335i	$-0.850234\pi$
$953$		−	27.9895i		−	0.906670i	0.891340	−	0.453335i	$-0.149766\pi$
$954$		0			0
$955$	8.11171	+	8.11171i	0.262489	+	0.262489i
$956$		0			0
$957$	32.8700	−	32.8700i	1.06254	−	1.06254i
$958$		0			0
$959$	7.80415			0.252009
$960$		0			0
$961$	43.6209			1.40713
$962$		0			0
$963$	−0.924498	+	0.924498i	−0.0297915	+	0.0297915i
$964$		0			0
$965$	9.49293	+	9.49293i	0.305588	+	0.305588i
$966$		0			0
$967$		−	32.9438i		−	1.05940i	−0.848185	−	0.529700i	$-0.822304\pi$
$967$		−	32.9438i		−	1.05940i	0.848185	−	0.529700i	$-0.177696\pi$
$968$		0			0
$969$			15.8827i			0.510225i
$970$		0			0
$971$	−3.25667	−	3.25667i	−0.104512	−	0.104512i	0.652917	−	0.757429i	$-0.273545\pi$
$971$	−3.25667	−	3.25667i	−0.104512	−	0.104512i	−0.757429	+	0.652917i	$0.773545\pi$
$972$		0			0
$973$	−12.8660	+	12.8660i	−0.412464	+	0.412464i
$974$		0			0
$975$	−15.9666			−0.511341
$976$		0			0
$977$	14.8275			0.474373			0.237187	−	0.971464i	$-0.423775\pi$
$977$	14.8275			0.474373			0.237187	+	0.971464i	$0.423775\pi$
$978$		0			0
$979$	72.9666	−	72.9666i	2.33202	−	2.33202i
$980$		0			0
$981$	25.5298	+	25.5298i	0.815105	+	0.815105i
$982$		0			0
$983$			10.9665i			0.349778i	0.984588	+	0.174889i	$0.0559567\pi$
$983$			10.9665i			0.349778i	−0.984588	+	0.174889i	$0.944043\pi$
$984$		0			0
$985$			14.5499i			0.463599i
$986$		0			0
$987$	13.3186	+	13.3186i	0.423937	+	0.423937i
$988$		0			0
$989$	5.31955	−	5.31955i	0.169152	−	0.169152i
$990$		0			0
$991$	44.7006			1.41996			0.709981	−	0.704221i	$-0.248703\pi$
$991$	44.7006			1.41996			0.709981	+	0.704221i	$0.248703\pi$
$992$		0			0
$993$	48.7179			1.54602
$994$		0			0
$995$	18.2074	−	18.2074i	0.577212	−	0.577212i
$996$		0			0
$997$	29.8446	+	29.8446i	0.945187	+	0.945187i	0.998574	−	0.0533873i	$-0.0170018\pi$
$997$	29.8446	+	29.8446i	0.945187	+	0.945187i	−0.0533873	+	0.998574i	$0.517002\pi$
$998$		0			0
$999$		−	1.38019i		−	0.0436673i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.bd.a.561.20		44
4.3	odd	2		560.2.bd.a.421.4	yes	44
16.3	odd	4		560.2.bd.a.141.4	✓	44
16.13	even	4	inner	2240.2.bd.a.1681.20		44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.bd.a.141.4	✓	44	16.3	odd	4
560.2.bd.a.421.4	yes	44	4.3	odd	2
2240.2.bd.a.561.20		44	1.1	even	1	trivial
2240.2.bd.a.1681.20		44	16.13	even	4	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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.bd (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.67958 - 1.67958i) q^{3} +(0.707107 + 0.707107i) q^{5} -1.00000i q^{7} -2.64195i q^{9} +O(q^{10})\)	\(q+(1.67958 - 1.67958i) q^{3} +(0.707107 + 0.707107i) q^{5} -1.00000i q^{7} -2.64195i q^{9} +(4.03337 + 4.03337i) q^{11} +(-4.75317 + 4.75317i) q^{13} +2.37528 q^{15} +4.01138 q^{17} +(-1.17869 + 1.17869i) q^{19} +(-1.67958 - 1.67958i) q^{21} -0.610935i q^{23} +1.00000i q^{25} +(0.601365 + 0.601365i) q^{27} +(2.42606 - 2.42606i) q^{29} +8.63834 q^{31} +13.5487 q^{33} +(0.707107 - 0.707107i) q^{35} +(-1.14755 - 1.14755i) q^{37} +15.9666i q^{39} +1.24783i q^{41} +(8.70723 + 8.70723i) q^{43} +(1.86814 - 1.86814i) q^{45} -7.92976 q^{47} -1.00000 q^{49} +(6.73743 - 6.73743i) q^{51} +(-5.64673 - 5.64673i) q^{53} +5.70405i q^{55} +3.95940i q^{57} +(1.23973 + 1.23973i) q^{59} +(5.71963 - 5.71963i) q^{61} -2.64195 q^{63} -6.72199 q^{65} +(-2.38301 + 2.38301i) q^{67} +(-1.02611 - 1.02611i) q^{69} -1.57667i q^{71} -1.85964i q^{73} +(1.67958 + 1.67958i) q^{75} +(4.03337 - 4.03337i) q^{77} -3.14874 q^{79} +9.94594 q^{81} +(6.01779 - 6.01779i) q^{83} +(2.83648 + 2.83648i) q^{85} -8.14951i q^{87} -18.0907i q^{89} +(4.75317 + 4.75317i) q^{91} +(14.5087 - 14.5087i) q^{93} -1.66692 q^{95} -10.6607 q^{97} +(10.6560 - 10.6560i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q+O(q^{10}) \) 44 * q	\( 44 q - 12 q^{11} - 8 q^{15} - 8 q^{19} + 24 q^{27} + 12 q^{29} + 28 q^{37} + 44 q^{43} - 44 q^{49} + 8 q^{51} - 12 q^{53} - 24 q^{59} - 16 q^{61} - 28 q^{63} - 40 q^{65} + 28 q^{67} + 40 q^{69} - 12 q^{77} + 16 q^{79} + 20 q^{81} + 16 q^{85} + 88 q^{93} + 32 q^{95} - 28 q^{99}+O(q^{100}) \) 44 * q - 12 * q^11 - 8 * q^15 - 8 * q^19 + 24 * q^27 + 12 * q^29 + 28 * q^37 + 44 * q^43 - 44 * q^49 + 8 * q^51 - 12 * q^53 - 24 * q^59 - 16 * q^61 - 28 * q^63 - 40 * q^65 + 28 * q^67 + 40 * q^69 - 12 * q^77 + 16 * q^79 + 20 * q^81 + 16 * q^85 + 88 * q^93 + 32 * q^95 - 28 * q^99

Introduction

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