LMFDB - Embedded newform 2240.2.bd.b.561.8

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:	$52$
Relative dimension:	$26$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 560)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			561.8
Character	$\chi$	$=$	2240.561
Dual form			2240.2.bd.b.1681.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.975138 + 0.975138i) q^{3} +(-0.707107 - 0.707107i) q^{5} +1.00000i q^{7} +1.09821i q^{9} +O(q^{10})$	\(q+(-0.975138 + 0.975138i) q^{3} +(-0.707107 - 0.707107i) q^{5} +1.00000i q^{7} +1.09821i q^{9} +(3.17705 + 3.17705i) q^{11} +(1.49224 - 1.49224i) q^{13} +1.37905 q^{15} -3.05153 q^{17} +(-2.31880 + 2.31880i) q^{19} +(-0.975138 - 0.975138i) q^{21} +3.15707i q^{23} +1.00000i q^{25} +(-3.99632 - 3.99632i) q^{27} +(3.98915 - 3.98915i) q^{29} +2.24393 q^{31} -6.19612 q^{33} +(0.707107 - 0.707107i) q^{35} +(6.11938 + 6.11938i) q^{37} +2.91029i q^{39} -1.53335i q^{41} +(1.42149 + 1.42149i) q^{43} +(0.776554 - 0.776554i) q^{45} -4.45086 q^{47} -1.00000 q^{49} +(2.97566 - 2.97566i) q^{51} +(-1.87531 - 1.87531i) q^{53} -4.49303i q^{55} -4.52229i q^{57} +(-1.74077 - 1.74077i) q^{59} +(3.83294 - 3.83294i) q^{61} -1.09821 q^{63} -2.11035 q^{65} +(-4.56660 + 4.56660i) q^{67} +(-3.07858 - 3.07858i) q^{69} +16.1135i q^{71} +6.07011i q^{73} +(-0.975138 - 0.975138i) q^{75} +(-3.17705 + 3.17705i) q^{77} -17.0083 q^{79} +4.49929 q^{81} +(4.44738 - 4.44738i) q^{83} +(2.15776 + 2.15776i) q^{85} +7.77994i q^{87} +15.1211i q^{89} +(1.49224 + 1.49224i) q^{91} +(-2.18814 + 2.18814i) q^{93} +3.27927 q^{95} -18.8094 q^{97} +(-3.48908 + 3.48908i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 52 q+O(q^{10}) $ 52 * q	$ 52 q + 4 q^{11} - 8 q^{15} - 8 q^{19} + 24 q^{27} + 4 q^{29} - 12 q^{37} - 36 q^{43} - 52 q^{49} + 8 q^{51} - 4 q^{53} - 24 q^{59} - 16 q^{61} + 68 q^{63} + 40 q^{65} + 12 q^{67} - 72 q^{69} - 4 q^{77} + 16 q^{79} - 116 q^{81} + 16 q^{85} + 8 q^{93} + 32 q^{95} + 52 q^{99}+O(q^{100}) $ 52 * q + 4 * q^11 - 8 * q^15 - 8 * q^19 + 24 * q^27 + 4 * q^29 - 12 * q^37 - 36 * q^43 - 52 * q^49 + 8 * q^51 - 4 * q^53 - 24 * q^59 - 16 * q^61 + 68 * q^63 + 40 * q^65 + 12 * q^67 - 72 * q^69 - 4 * q^77 + 16 * q^79 - 116 * q^81 + 16 * q^85 + 8 * q^93 + 32 * q^95 + 52 * 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
$3$	−0.975138	+	0.975138i	−0.562996	+	0.562996i	−0.930157	−	0.367161i	$-0.880330\pi$
$3$	−0.975138	+	0.975138i	−0.562996	+	0.562996i	0.367161	+	0.930157i	$0.380330\pi$
$4$		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
$9$			1.09821i			0.366071i
$10$		0			0
$11$	3.17705	+	3.17705i	0.957916	+	0.957916i	0.999150	−	0.0412333i	$-0.0131287\pi$
$11$	3.17705	+	3.17705i	0.957916	+	0.957916i	−0.0412333	+	0.999150i	$0.513129\pi$
$12$		0			0
$13$	1.49224	−	1.49224i	0.413874	−	0.413874i	−0.469212	−	0.883086i	$-0.655462\pi$
$13$	1.49224	−	1.49224i	0.413874	−	0.413874i	0.883086	+	0.469212i	$0.155462\pi$
$14$		0			0
$15$	1.37905			0.356070
$16$		0			0
$17$	−3.05153			−0.740105			−0.370052	−	0.929011i	$-0.620660\pi$
$17$	−3.05153			−0.740105			−0.370052	+	0.929011i	$0.620660\pi$
$18$		0			0
$19$	−2.31880	+	2.31880i	−0.531968	+	0.531968i	−0.921158	−	0.389189i	$-0.872755\pi$
$19$	−2.31880	+	2.31880i	−0.531968	+	0.531968i	0.389189	+	0.921158i	$0.372755\pi$
$20$		0			0
$21$	−0.975138	−	0.975138i	−0.212792	−	0.212792i
$22$		0			0
$23$			3.15707i			0.658294i	0.944279	+	0.329147i	$0.106761\pi$
$23$			3.15707i			0.658294i	−0.944279	+	0.329147i	$0.893239\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$	−3.99632	−	3.99632i	−0.769093	−	0.769093i
$28$		0			0
$29$	3.98915	−	3.98915i	0.740767	−	0.740767i	−0.231959	−	0.972726i	$-0.574514\pi$
$29$	3.98915	−	3.98915i	0.740767	−	0.740767i	0.972726	+	0.231959i	$0.0745135\pi$
$30$		0			0
$31$	2.24393			0.403021			0.201511	−	0.979486i	$-0.435415\pi$
$31$	2.24393			0.403021			0.201511	+	0.979486i	$0.435415\pi$
$32$		0			0
$33$	−6.19612			−1.07861
$34$		0			0
$35$	0.707107	−	0.707107i	0.119523	−	0.119523i
$36$		0			0
$37$	6.11938	+	6.11938i	1.00602	+	1.00602i	0.999982	+	0.00603771i	$0.00192187\pi$
$37$	6.11938	+	6.11938i	1.00602	+	1.00602i	0.00603771	+	0.999982i	$0.498078\pi$
$38$		0			0
$39$			2.91029i			0.466019i
$40$		0			0
$41$		−	1.53335i		−	0.239469i	−0.992806	−	0.119734i	$-0.961796\pi$
$41$		−	1.53335i		−	0.239469i	0.992806	−	0.119734i	$-0.0382043\pi$
$42$		0			0
$43$	1.42149	+	1.42149i	0.216776	+	0.216776i	0.807138	−	0.590362i	$-0.201015\pi$
$43$	1.42149	+	1.42149i	0.216776	+	0.216776i	−0.590362	+	0.807138i	$0.701015\pi$
$44$		0			0
$45$	0.776554	−	0.776554i	0.115762	−	0.115762i
$46$		0			0
$47$	−4.45086			−0.649225			−0.324612	−	0.945847i	$-0.605234\pi$
$47$	−4.45086			−0.649225			−0.324612	+	0.945847i	$0.605234\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	2.97566	−	2.97566i	0.416676	−	0.416676i
$52$		0			0
$53$	−1.87531	−	1.87531i	−0.257594	−	0.257594i	0.566481	−	0.824075i	$-0.308305\pi$
$53$	−1.87531	−	1.87531i	−0.257594	−	0.257594i	−0.824075	+	0.566481i	$0.808305\pi$
$54$		0			0
$55$		−	4.49303i		−	0.605839i
$56$		0			0
$57$		−	4.52229i		−	0.598992i
$58$		0			0
$59$	−1.74077	−	1.74077i	−0.226629	−	0.226629i	0.584654	−	0.811283i	$-0.301230\pi$
$59$	−1.74077	−	1.74077i	−0.226629	−	0.226629i	−0.811283	+	0.584654i	$0.801230\pi$
$60$		0			0
$61$	3.83294	−	3.83294i	0.490757	−	0.490757i	−0.417788	−	0.908545i	$-0.637194\pi$
$61$	3.83294	−	3.83294i	0.490757	−	0.490757i	0.908545	+	0.417788i	$0.137194\pi$
$62$		0			0
$63$	−1.09821			−0.138362
$64$		0			0
$65$	−2.11035			−0.261757
$66$		0			0
$67$	−4.56660	+	4.56660i	−0.557899	+	0.557899i	−0.928709	−	0.370810i	$-0.879080\pi$
$67$	−4.56660	+	4.56660i	−0.557899	+	0.557899i	0.370810	+	0.928709i	$0.379080\pi$
$68$		0			0
$69$	−3.07858	−	3.07858i	−0.370617	−	0.370617i
$70$		0			0
$71$			16.1135i			1.91232i	0.292844	+	0.956160i	$0.405398\pi$
$71$			16.1135i			1.91232i	−0.292844	+	0.956160i	$0.594602\pi$
$72$		0			0
$73$			6.07011i			0.710453i	0.934780	+	0.355227i	$0.115596\pi$
$73$			6.07011i			0.710453i	−0.934780	+	0.355227i	$0.884404\pi$
$74$		0			0
$75$	−0.975138	−	0.975138i	−0.112599	−	0.112599i
$76$		0			0
$77$	−3.17705	+	3.17705i	−0.362058	+	0.362058i
$78$		0			0
$79$	−17.0083			−1.91358			−0.956792	−	0.290772i	$-0.906088\pi$
$79$	−17.0083			−1.91358			−0.956792	+	0.290772i	$0.906088\pi$
$80$		0			0
$81$	4.49929			0.499921
$82$		0			0
$83$	4.44738	−	4.44738i	0.488164	−	0.488164i	−0.419563	−	0.907726i	$-0.637817\pi$
$83$	4.44738	−	4.44738i	0.488164	−	0.488164i	0.907726	+	0.419563i	$0.137817\pi$
$84$		0			0
$85$	2.15776	+	2.15776i	0.234042	+	0.234042i
$86$		0			0
$87$			7.77994i			0.834097i
$88$		0			0
$89$			15.1211i			1.60283i	0.598109	+	0.801415i	$0.295919\pi$
$89$			15.1211i			1.60283i	−0.598109	+	0.801415i	$0.704081\pi$
$90$		0			0
$91$	1.49224	+	1.49224i	0.156430	+	0.156430i
$92$		0			0
$93$	−2.18814	+	2.18814i	−0.226899	+	0.226899i
$94$		0			0
$95$	3.27927			0.336446
$96$		0			0
$97$	−18.8094			−1.90980			−0.954901	−	0.296924i	$-0.904039\pi$
$97$	−18.8094			−1.90980			−0.954901	+	0.296924i	$0.904039\pi$
$98$		0			0
$99$	−3.48908	+	3.48908i	−0.350665	+	0.350665i
$100$		0			0
$101$	−3.85350	−	3.85350i	−0.383437	−	0.383437i	0.488902	−	0.872339i	$-0.337398\pi$
$101$	−3.85350	−	3.85350i	−0.383437	−	0.383437i	−0.872339	+	0.488902i	$0.837398\pi$
$102$		0			0
$103$			6.69505i			0.659683i	0.944036	+	0.329841i	$0.106995\pi$
$103$			6.69505i			0.659683i	−0.944036	+	0.329841i	$0.893005\pi$
$104$		0			0
$105$			1.37905i			0.134582i
$106$		0			0
$107$	−8.22368	−	8.22368i	−0.795013	−	0.795013i	0.187291	−	0.982304i	$-0.440029\pi$
$107$	−8.22368	−	8.22368i	−0.795013	−	0.795013i	−0.982304	+	0.187291i	$0.940029\pi$
$108$		0			0
$109$	2.99972	−	2.99972i	0.287321	−	0.287321i	−0.548699	−	0.836020i	$-0.684877\pi$
$109$	2.99972	−	2.99972i	0.287321	−	0.287321i	0.836020	+	0.548699i	$0.184877\pi$
$110$		0			0
$111$	−11.9345			−1.13277
$112$		0			0
$113$	−11.2367			−1.05706			−0.528528	−	0.848916i	$-0.677256\pi$
$113$	−11.2367			−1.05706			−0.528528	+	0.848916i	$0.677256\pi$
$114$		0			0
$115$	2.23238	−	2.23238i	0.208171	−	0.208171i
$116$		0			0
$117$	1.63880	+	1.63880i	0.151507	+	0.151507i
$118$		0			0
$119$		−	3.05153i		−	0.279733i
$120$		0			0
$121$			9.18728i			0.835207i
$122$		0			0
$123$	1.49523	+	1.49523i	0.134820	+	0.134820i
$124$		0			0
$125$	0.707107	−	0.707107i	0.0632456	−	0.0632456i
$126$		0			0
$127$	−15.1910			−1.34799			−0.673994	−	0.738737i	$-0.735422\pi$
$127$	−15.1910			−1.34799			−0.673994	+	0.738737i	$0.735422\pi$
$128$		0			0
$129$	−2.77231			−0.244088
$130$		0			0
$131$	−8.79814	+	8.79814i	−0.768697	+	0.768697i	−0.977877	−	0.209180i	$-0.932921\pi$
$131$	−8.79814	+	8.79814i	−0.768697	+	0.768697i	0.209180	+	0.977877i	$0.432921\pi$
$132$		0			0
$133$	−2.31880	−	2.31880i	−0.201065	−	0.201065i
$134$		0			0
$135$			5.65165i			0.486417i
$136$		0			0
$137$		−	11.7852i		−	1.00688i	−0.864030	−	0.503440i	$-0.832067\pi$
$137$		−	11.7852i		−	1.00688i	0.864030	−	0.503440i	$-0.167933\pi$
$138$		0			0
$139$	3.24773	+	3.24773i	0.275469	+	0.275469i	0.831297	−	0.555828i	$-0.187599\pi$
$139$	3.24773	+	3.24773i	0.275469	+	0.275469i	−0.555828	+	0.831297i	$0.687599\pi$
$140$		0			0
$141$	4.34020	−	4.34020i	0.365511	−	0.365511i
$142$		0			0
$143$	9.48186			0.792913
$144$		0			0
$145$	−5.64151			−0.468502
$146$		0			0
$147$	0.975138	−	0.975138i	0.0804280	−	0.0804280i
$148$		0			0
$149$	15.8516	+	15.8516i	1.29861	+	1.29861i	0.929309	+	0.369302i	$0.120403\pi$
$149$	15.8516	+	15.8516i	1.29861	+	1.29861i	0.369302	+	0.929309i	$0.379597\pi$
$150$		0			0
$151$			4.55022i			0.370292i	0.982711	+	0.185146i	$0.0592757\pi$
$151$			4.55022i			0.370292i	−0.982711	+	0.185146i	$0.940724\pi$
$152$		0			0
$153$		−	3.35123i		−	0.270931i
$154$		0			0
$155$	−1.58670	−	1.58670i	−0.127446	−	0.127446i
$156$		0			0
$157$	−4.18859	+	4.18859i	−0.334286	+	0.334286i	−0.854211	−	0.519926i	$-0.825960\pi$
$157$	−4.18859	+	4.18859i	−0.334286	+	0.334286i	0.519926	+	0.854211i	$0.325960\pi$
$158$		0			0
$159$	3.65738			0.290049
$160$		0			0
$161$	−3.15707			−0.248812
$162$		0			0
$163$	3.28289	−	3.28289i	0.257136	−	0.257136i	−0.566752	−	0.823888i	$-0.691800\pi$
$163$	3.28289	−	3.28289i	0.257136	−	0.257136i	0.823888	+	0.566752i	$0.191800\pi$
$164$		0			0
$165$	4.38132	+	4.38132i	0.341085	+	0.341085i
$166$		0			0
$167$		−	13.7112i		−	1.06100i	−0.847685	−	0.530500i	$-0.822004\pi$
$167$		−	13.7112i		−	1.06100i	0.847685	−	0.530500i	$-0.177996\pi$
$168$		0			0
$169$			8.54642i			0.657417i
$170$		0			0
$171$	−2.54653	−	2.54653i	−0.194738	−	0.194738i
$172$		0			0
$173$	−4.88909	+	4.88909i	−0.371710	+	0.371710i	−0.868100	−	0.496390i	$-0.834659\pi$
$173$	−4.88909	+	4.88909i	−0.371710	+	0.371710i	0.496390	+	0.868100i	$0.334659\pi$
$174$		0			0
$175$	−1.00000			−0.0755929
$176$		0			0
$177$	3.39499			0.255183
$178$		0			0
$179$	−4.90158	+	4.90158i	−0.366361	+	0.366361i	−0.866148	−	0.499787i	$-0.833412\pi$
$179$	−4.90158	+	4.90158i	−0.366361	+	0.366361i	0.499787	+	0.866148i	$0.333412\pi$
$180$		0			0
$181$	15.4760	+	15.4760i	1.15032	+	1.15032i	0.986488	+	0.163835i	$0.0523864\pi$
$181$	15.4760	+	15.4760i	1.15032	+	1.15032i	0.163835	+	0.986488i	$0.447614\pi$
$182$		0			0
$183$			7.47528i			0.552589i
$184$		0			0
$185$		−	8.65411i		−	0.636263i
$186$		0			0
$187$	−9.69486	−	9.69486i	−0.708959	−	0.708959i
$188$		0			0
$189$	3.99632	−	3.99632i	0.290690	−	0.290690i
$190$		0			0
$191$	0.996679			0.0721171			0.0360586	−	0.999350i	$-0.488520\pi$
$191$	0.996679			0.0721171			0.0360586	+	0.999350i	$0.488520\pi$
$192$		0			0
$193$	2.00192			0.144101			0.0720507	−	0.997401i	$-0.477046\pi$
$193$	2.00192			0.144101			0.0720507	+	0.997401i	$0.477046\pi$
$194$		0			0
$195$	2.05788	−	2.05788i	0.147368	−	0.147368i
$196$		0			0
$197$	−16.6710	−	16.6710i	−1.18776	−	1.18776i	−0.977685	−	0.210079i	$-0.932628\pi$
$197$	−16.6710	−	16.6710i	−1.18776	−	1.18776i	−0.210079	−	0.977685i	$-0.567372\pi$
$198$		0			0
$199$		−	14.7492i		−	1.04554i	−0.852473	−	0.522771i	$-0.824898\pi$
$199$		−	14.7492i		−	1.04554i	0.852473	−	0.522771i	$-0.175102\pi$
$200$		0			0
$201$		−	8.90613i		−	0.628190i
$202$		0			0
$203$	3.98915	+	3.98915i	0.279983	+	0.279983i
$204$		0			0
$205$	−1.08424	+	1.08424i	−0.0757267	+	0.0757267i
$206$		0			0
$207$	−3.46713			−0.240982
$208$		0			0
$209$	−14.7339			−1.01916
$210$		0			0
$211$	3.35667	−	3.35667i	0.231083	−	0.231083i	−0.582062	−	0.813145i	$-0.697754\pi$
$211$	3.35667	−	3.35667i	0.231083	−	0.231083i	0.813145	+	0.582062i	$0.197754\pi$
$212$		0			0
$213$	−15.7129	−	15.7129i	−1.07663	−	1.07663i
$214$		0			0
$215$		−	2.01030i		−	0.137101i
$216$		0			0
$217$			2.24393i			0.152328i
$218$		0			0
$219$	−5.91920	−	5.91920i	−0.399982	−	0.399982i
$220$		0			0
$221$	−4.55363	+	4.55363i	−0.306310	+	0.306310i
$222$		0			0
$223$	5.84435			0.391367			0.195683	−	0.980667i	$-0.437308\pi$
$223$	5.84435			0.391367			0.195683	+	0.980667i	$0.437308\pi$
$224$		0			0
$225$	−1.09821			−0.0732142
$226$		0			0
$227$	4.08793	−	4.08793i	0.271325	−	0.271325i	−0.558308	−	0.829634i	$-0.688549\pi$
$227$	4.08793	−	4.08793i	0.271325	−	0.271325i	0.829634	+	0.558308i	$0.188549\pi$
$228$		0			0
$229$	6.50824	+	6.50824i	0.430076	+	0.430076i	0.888654	−	0.458578i	$-0.151641\pi$
$229$	6.50824	+	6.50824i	0.430076	+	0.430076i	−0.458578	+	0.888654i	$0.651641\pi$
$230$		0			0
$231$		−	6.19612i		−	0.407675i
$232$		0			0
$233$		−	7.26919i		−	0.476221i	−0.971238	−	0.238110i	$-0.923472\pi$
$233$		−	7.26919i		−	0.476221i	0.971238	−	0.238110i	$-0.0765280\pi$
$234$		0			0
$235$	3.14723	+	3.14723i	0.205303	+	0.205303i
$236$		0			0
$237$	16.5854	−	16.5854i	1.07734	−	1.07734i
$238$		0			0
$239$	7.52221			0.486571			0.243286	−	0.969955i	$-0.421775\pi$
$239$	7.52221			0.486571			0.243286	+	0.969955i	$0.421775\pi$
$240$		0			0
$241$	8.86584			0.571099			0.285549	−	0.958364i	$-0.407824\pi$
$241$	8.86584			0.571099			0.285549	+	0.958364i	$0.407824\pi$
$242$		0			0
$243$	7.60154	−	7.60154i	0.487639	−	0.487639i
$244$		0			0
$245$	0.707107	+	0.707107i	0.0451754	+	0.0451754i
$246$		0			0
$247$			6.92042i			0.440336i
$248$		0			0
$249$			8.67362i			0.549668i
$250$		0			0
$251$	−18.4049	−	18.4049i	−1.16171	−	1.16171i	−0.984102	−	0.177607i	$-0.943164\pi$
$251$	−18.4049	−	18.4049i	−1.16171	−	1.16171i	−0.177607	−	0.984102i	$-0.556836\pi$
$252$		0			0
$253$	−10.0302	+	10.0302i	−0.630591	+	0.630591i
$254$		0			0
$255$	−4.20822			−0.263529
$256$		0			0
$257$	−27.3671			−1.70711			−0.853556	−	0.521000i	$-0.825559\pi$
$257$	−27.3671			−1.70711			−0.853556	+	0.521000i	$0.825559\pi$
$258$		0			0
$259$	−6.11938	+	6.11938i	−0.380240	+	0.380240i
$260$		0			0
$261$	4.38094	+	4.38094i	0.271173	+	0.271173i
$262$		0			0
$263$			19.3216i			1.19142i	0.803199	+	0.595711i	$0.203129\pi$
$263$			19.3216i			1.19142i	−0.803199	+	0.595711i	$0.796871\pi$
$264$		0			0
$265$			2.65209i			0.162917i
$266$		0			0
$267$	−14.7451	−	14.7451i	−0.902386	−	0.902386i
$268$		0			0
$269$	−11.3620	+	11.3620i	−0.692752	+	0.692752i	−0.962837	−	0.270085i	$-0.912948\pi$
$269$	−11.3620	+	11.3620i	−0.692752	+	0.692752i	0.270085	+	0.962837i	$0.412948\pi$
$270$		0			0
$271$	30.7777			1.86961			0.934806	−	0.355158i	$-0.115573\pi$
$271$	30.7777			1.86961			0.934806	+	0.355158i	$0.115573\pi$
$272$		0			0
$273$	−2.91029			−0.176139
$274$		0			0
$275$	−3.17705	+	3.17705i	−0.191583	+	0.191583i
$276$		0			0
$277$	17.5499	+	17.5499i	1.05447	+	1.05447i	0.998428	+	0.0560432i	$0.0178484\pi$
$277$	17.5499	+	17.5499i	1.05447	+	1.05447i	0.0560432	+	0.998428i	$0.482152\pi$
$278$		0			0
$279$			2.46431i			0.147534i
$280$		0			0
$281$			6.90649i			0.412007i	0.978551	+	0.206003i	$0.0660458\pi$
$281$			6.90649i			0.412007i	−0.978551	+	0.206003i	$0.933954\pi$
$282$		0			0
$283$	−13.4481	−	13.4481i	−0.799405	−	0.799405i	0.183597	−	0.983002i	$-0.441226\pi$
$283$	−13.4481	−	13.4481i	−0.799405	−	0.799405i	−0.983002	+	0.183597i	$0.941226\pi$
$284$		0			0
$285$	−3.19774	+	3.19774i	−0.189418	+	0.189418i
$286$		0			0
$287$	1.53335			0.0905107
$288$		0			0
$289$	−7.68816			−0.452245
$290$		0			0
$291$	18.3417	−	18.3417i	1.07521	−	1.07521i
$292$		0			0
$293$	−0.506804	−	0.506804i	−0.0296078	−	0.0296078i	0.692148	−	0.721756i	$-0.256665\pi$
$293$	−0.506804	−	0.506804i	−0.0296078	−	0.0296078i	−0.721756	+	0.692148i	$0.756665\pi$
$294$		0			0
$295$			2.46183i			0.143333i
$296$		0			0
$297$		−	25.3930i		−	1.47345i
$298$		0			0
$299$	4.71111	+	4.71111i	0.272451	+	0.272451i
$300$		0			0
$301$	−1.42149	+	1.42149i	−0.0819336	+	0.0819336i
$302$		0			0
$303$	7.51538			0.431747
$304$		0			0
$305$	−5.42059			−0.310382
$306$		0			0
$307$	19.1376	−	19.1376i	1.09224	−	1.09224i	0.0969530	−	0.995289i	$-0.469090\pi$
$307$	19.1376	−	19.1376i	1.09224	−	1.09224i	0.995289	−	0.0969530i	$-0.0309097\pi$
$308$		0			0
$309$	−6.52860	−	6.52860i	−0.371399	−	0.371399i
$310$		0			0
$311$			25.4164i			1.44123i	0.693335	+	0.720616i	$0.256141\pi$
$311$			25.4164i			1.44123i	−0.693335	+	0.720616i	$0.743859\pi$
$312$		0			0
$313$			33.8929i			1.91574i	0.287205	+	0.957869i	$0.407274\pi$
$313$			33.8929i			1.91574i	−0.287205	+	0.957869i	$0.592726\pi$
$314$		0			0
$315$	0.776554	+	0.776554i	0.0437539	+	0.0437539i
$316$		0			0
$317$	−3.61007	+	3.61007i	−0.202761	+	0.202761i	−0.801182	−	0.598421i	$-0.795795\pi$
$317$	−3.61007	+	3.61007i	−0.202761	+	0.202761i	0.598421	+	0.801182i	$0.295795\pi$
$318$		0			0
$319$	25.3474			1.41918
$320$		0			0
$321$	16.0384			0.895178
$322$		0			0
$323$	7.07588	−	7.07588i	0.393713	−	0.393713i
$324$		0			0
$325$	1.49224	+	1.49224i	0.0827748	+	0.0827748i
$326$		0			0
$327$			5.85029i			0.323522i
$328$		0			0
$329$		−	4.45086i		−	0.245384i
$330$		0			0
$331$	19.5373	+	19.5373i	1.07387	+	1.07387i	0.997045	+	0.0768218i	$0.0244773\pi$
$331$	19.5373	+	19.5373i	1.07387	+	1.07387i	0.0768218	+	0.997045i	$0.475523\pi$
$332$		0			0
$333$	−6.72038	+	6.72038i	−0.368275	+	0.368275i
$334$		0			0
$335$	6.45815			0.352846
$336$		0			0
$337$	7.09280			0.386370			0.193185	−	0.981162i	$-0.438118\pi$
$337$	7.09280			0.386370			0.193185	+	0.981162i	$0.438118\pi$
$338$		0			0
$339$	10.9573	−	10.9573i	0.595118	−	0.595118i
$340$		0			0
$341$	7.12907	+	7.12907i	0.386061	+	0.386061i
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$			4.35376i			0.234399i
$346$		0			0
$347$	14.7293	+	14.7293i	0.790709	+	0.790709i	0.981609	−	0.190900i	$-0.0611407\pi$
$347$	14.7293	+	14.7293i	0.790709	+	0.790709i	−0.190900	+	0.981609i	$0.561141\pi$
$348$		0			0
$349$	2.94167	−	2.94167i	0.157464	−	0.157464i	−0.623978	−	0.781442i	$-0.714485\pi$
$349$	2.94167	−	2.94167i	0.157464	−	0.157464i	0.781442	+	0.623978i	$0.214485\pi$
$350$		0			0
$351$	−11.9270			−0.636615
$352$		0			0
$353$	1.21700			0.0647743			0.0323872	−	0.999475i	$-0.489689\pi$
$353$	1.21700			0.0647743			0.0323872	+	0.999475i	$0.489689\pi$
$354$		0			0
$355$	11.3940	−	11.3940i	0.604729	−	0.604729i
$356$		0			0
$357$	2.97566	+	2.97566i	0.157489	+	0.157489i
$358$		0			0
$359$		−	29.6037i		−	1.56243i	−0.624265	−	0.781213i	$-0.714601\pi$
$359$		−	29.6037i		−	1.56243i	0.624265	−	0.781213i	$-0.285399\pi$
$360$		0			0
$361$			8.24636i			0.434019i
$362$		0			0
$363$	−8.95886	−	8.95886i	−0.470218	−	0.470218i
$364$		0			0
$365$	4.29222	−	4.29222i	0.224665	−	0.224665i
$366$		0			0
$367$	−16.7272			−0.873153			−0.436577	−	0.899667i	$-0.643809\pi$
$367$	−16.7272			−0.873153			−0.436577	+	0.899667i	$0.643809\pi$
$368$		0			0
$369$	1.68394			0.0876626
$370$		0			0
$371$	1.87531	−	1.87531i	0.0973614	−	0.0973614i
$372$		0			0
$373$	6.20835	+	6.20835i	0.321456	+	0.321456i	0.849326	−	0.527869i	$-0.177009\pi$
$373$	6.20835	+	6.20835i	0.321456	+	0.321456i	−0.527869	+	0.849326i	$0.677009\pi$
$374$		0			0
$375$			1.37905i			0.0712140i
$376$		0			0
$377$		−	11.9056i		−	0.613168i
$378$		0			0
$379$	−11.1519	−	11.1519i	−0.572835	−	0.572835i	0.360084	−	0.932920i	$-0.382748\pi$
$379$	−11.1519	−	11.1519i	−0.572835	−	0.572835i	−0.932920	+	0.360084i	$0.882748\pi$
$380$		0			0
$381$	14.8134	−	14.8134i	0.758911	−	0.758911i
$382$		0			0
$383$	33.5067			1.71211			0.856056	−	0.516884i	$-0.172908\pi$
$383$	33.5067			1.71211			0.856056	+	0.516884i	$0.172908\pi$
$384$		0			0
$385$	4.49303			0.228986
$386$		0			0
$387$	−1.56110	+	1.56110i	−0.0793553	+	0.0793553i
$388$		0			0
$389$	−17.9185	−	17.9185i	−0.908504	−	0.908504i	0.0876477	−	0.996152i	$-0.472065\pi$
$389$	−17.9185	−	17.9185i	−0.908504	−	0.908504i	−0.996152	+	0.0876477i	$0.972065\pi$
$390$		0			0
$391$		−	9.63389i		−	0.487207i
$392$		0			0
$393$		−	17.1588i		−	0.865547i
$394$		0			0
$395$	12.0267	+	12.0267i	0.605129	+	0.605129i
$396$		0			0
$397$	9.36255	−	9.36255i	0.469893	−	0.469893i	−0.431987	−	0.901880i	$-0.642187\pi$
$397$	9.36255	−	9.36255i	0.469893	−	0.469893i	0.901880	+	0.431987i	$0.142187\pi$
$398$		0			0
$399$	4.52229			0.226398
$400$		0			0
$401$	29.0099			1.44868			0.724342	−	0.689440i	$-0.242144\pi$
$401$	29.0099			1.44868			0.724342	+	0.689440i	$0.242144\pi$
$402$		0			0
$403$	3.34849	−	3.34849i	0.166800	−	0.166800i
$404$		0			0
$405$	−3.18148	−	3.18148i	−0.158089	−	0.158089i
$406$		0			0
$407$			38.8831i			1.92736i
$408$		0			0
$409$		−	32.8724i		−	1.62543i	−0.582659	−	0.812717i	$-0.697987\pi$
$409$		−	32.8724i		−	1.62543i	0.582659	−	0.812717i	$-0.302013\pi$
$410$		0			0
$411$	11.4922	+	11.4922i	0.566870	+	0.566870i
$412$		0			0
$413$	1.74077	−	1.74077i	0.0856579	−	0.0856579i
$414$		0			0
$415$	−6.28955			−0.308742
$416$		0			0
$417$	−6.33397			−0.310176
$418$		0			0
$419$	−20.0843	+	20.0843i	−0.981181	+	0.981181i	−0.999826	−	0.0186451i	$-0.994065\pi$
$419$	−20.0843	+	20.0843i	−0.981181	+	0.981181i	0.0186451	+	0.999826i	$0.494065\pi$
$420$		0			0
$421$	8.71466	+	8.71466i	0.424726	+	0.424726i	0.886827	−	0.462101i	$-0.152904\pi$
$421$	8.71466	+	8.71466i	0.424726	+	0.424726i	−0.462101	+	0.886827i	$0.652904\pi$
$422$		0			0
$423$		−	4.88799i		−	0.237662i
$424$		0			0
$425$		−	3.05153i		−	0.148021i
$426$		0			0
$427$	3.83294	+	3.83294i	0.185489	+	0.185489i
$428$		0			0
$429$	−9.24612	+	9.24612i	−0.446407	+	0.446407i
$430$		0			0
$431$	12.0221			0.579083			0.289541	−	0.957165i	$-0.406497\pi$
$431$	12.0221			0.579083			0.289541	+	0.957165i	$0.406497\pi$
$432$		0			0
$433$	24.3294			1.16919			0.584597	−	0.811324i	$-0.301253\pi$
$433$	24.3294			1.16919			0.584597	+	0.811324i	$0.301253\pi$
$434$		0			0
$435$	5.50125	−	5.50125i	0.263765	−	0.263765i
$436$		0			0
$437$	−7.32060	−	7.32060i	−0.350192	−	0.350192i
$438$		0			0
$439$			21.6658i			1.03405i	0.855970	+	0.517025i	$0.172961\pi$
$439$			21.6658i			1.03405i	−0.855970	+	0.517025i	$0.827039\pi$
$440$		0			0
$441$		−	1.09821i		−	0.0522959i
$442$		0			0
$443$	−10.2237	−	10.2237i	−0.485744	−	0.485744i	0.421216	−	0.906960i	$-0.361603\pi$
$443$	−10.2237	−	10.2237i	−0.485744	−	0.485744i	−0.906960	+	0.421216i	$0.861603\pi$
$444$		0			0
$445$	10.6922	−	10.6922i	0.506859	−	0.506859i
$446$		0			0
$447$	−30.9149			−1.46223
$448$		0			0
$449$	12.7472			0.601580			0.300790	−	0.953690i	$-0.402750\pi$
$449$	12.7472			0.601580			0.300790	+	0.953690i	$0.402750\pi$
$450$		0			0
$451$	4.87153	−	4.87153i	0.229391	−	0.229391i
$452$		0			0
$453$	−4.43709	−	4.43709i	−0.208473	−	0.208473i
$454$		0			0
$455$		−	2.11035i		−	0.0989348i
$456$		0			0
$457$			29.6985i			1.38924i	0.719378	+	0.694619i	$0.244427\pi$
$457$			29.6985i			1.38924i	−0.719378	+	0.694619i	$0.755573\pi$
$458$		0			0
$459$	12.1949	+	12.1949i	0.569209	+	0.569209i
$460$		0			0
$461$	24.5157	−	24.5157i	1.14181	−	1.14181i	0.153689	−	0.988119i	$-0.450885\pi$
$461$	24.5157	−	24.5157i	1.14181	−	1.14181i	0.988119	−	0.153689i	$-0.0491154\pi$
$462$		0			0
$463$	24.8873			1.15661			0.578305	−	0.815821i	$-0.303714\pi$
$463$	24.8873			1.15661			0.578305	+	0.815821i	$0.303714\pi$
$464$		0			0
$465$	3.09449			0.143504
$466$		0			0
$467$	2.29426	−	2.29426i	0.106166	−	0.106166i	−0.652029	−	0.758194i	$-0.726082\pi$
$467$	2.29426	−	2.29426i	0.106166	−	0.106166i	0.758194	+	0.652029i	$0.226082\pi$
$468$		0			0
$469$	−4.56660	−	4.56660i	−0.210866	−	0.210866i
$470$		0			0
$471$		−	8.16890i		−	0.376403i
$472$		0			0
$473$			9.03231i			0.415306i
$474$		0			0
$475$	−2.31880	−	2.31880i	−0.106394	−	0.106394i
$476$		0			0
$477$	2.05949	−	2.05949i	0.0942977	−	0.0942977i
$478$		0			0
$479$	3.87812			0.177196			0.0885979	−	0.996067i	$-0.471761\pi$
$479$	3.87812			0.177196			0.0885979	+	0.996067i	$0.471761\pi$
$480$		0			0
$481$	18.2632			0.832731
$482$		0			0
$483$	3.07858	−	3.07858i	0.140080	−	0.140080i
$484$		0			0
$485$	13.3002	+	13.3002i	0.603933	+	0.603933i
$486$		0			0
$487$			12.9846i			0.588387i	0.955746	+	0.294194i	$0.0950510\pi$
$487$			12.9846i			0.588387i	−0.955746	+	0.294194i	$0.904949\pi$
$488$		0			0
$489$			6.40255i			0.289533i
$490$		0			0
$491$	−26.3265	−	26.3265i	−1.18810	−	1.18810i	−0.977593	−	0.210503i	$-0.932490\pi$
$491$	−26.3265	−	26.3265i	−1.18810	−	1.18810i	−0.210503	−	0.977593i	$-0.567510\pi$
$492$		0			0
$493$	−12.1730	+	12.1730i	−0.548245	+	0.548245i
$494$		0			0
$495$	4.93430			0.221780
$496$		0			0
$497$	−16.1135			−0.722789
$498$		0			0
$499$	−25.1426	+	25.1426i	−1.12554	+	1.12554i	−0.134644	+	0.990894i	$0.542989\pi$
$499$	−25.1426	+	25.1426i	−1.12554	+	1.12554i	−0.990894	+	0.134644i	$0.957011\pi$
$500$		0			0
$501$	13.3703	+	13.3703i	0.597339	+	0.597339i
$502$		0			0
$503$		−	40.1668i		−	1.79095i	−0.445113	−	0.895474i	$-0.646837\pi$
$503$		−	40.1668i		−	1.79095i	0.445113	−	0.895474i	$-0.353163\pi$
$504$		0			0
$505$			5.44967i			0.242507i
$506$		0			0
$507$	−8.33393	−	8.33393i	−0.370123	−	0.370123i
$508$		0			0
$509$	28.7339	−	28.7339i	1.27361	−	1.27361i	0.329427	−	0.944181i	$-0.393144\pi$
$509$	28.7339	−	28.7339i	1.27361	−	1.27361i	0.944181	−	0.329427i	$-0.106856\pi$
$510$		0			0
$511$	−6.07011			−0.268526
$512$		0			0
$513$	18.5333			0.818266
$514$		0			0
$515$	4.73412	−	4.73412i	0.208610	−	0.208610i
$516$		0			0
$517$	−14.1406	−	14.1406i	−0.621903	−	0.621903i
$518$		0			0
$519$		−	9.53506i		−	0.418543i
$520$		0			0
$521$		−	19.1131i		−	0.837361i	−0.908134	−	0.418680i	$-0.862493\pi$
$521$		−	19.1131i		−	0.837361i	0.908134	−	0.418680i	$-0.137507\pi$
$522$		0			0
$523$	8.30451	+	8.30451i	0.363131	+	0.363131i	0.864964	−	0.501833i	$-0.167341\pi$
$523$	8.30451	+	8.30451i	0.363131	+	0.363131i	−0.501833	+	0.864964i	$0.667341\pi$
$524$		0			0
$525$	0.975138	−	0.975138i	0.0425585	−	0.0425585i
$526$		0			0
$527$	−6.84741			−0.298278
$528$		0			0
$529$	13.0329			0.566649
$530$		0			0
$531$	1.91174	−	1.91174i	0.0829624	−	0.0829624i
$532$		0			0
$533$	−2.28813	−	2.28813i	−0.0991099	−	0.0991099i
$534$		0			0
$535$			11.6300i			0.502810i
$536$		0			0
$537$		−	9.55943i		−	0.412520i
$538$		0			0
$539$	−3.17705	−	3.17705i	−0.136845	−	0.136845i
$540$		0			0
$541$	11.4933	−	11.4933i	0.494135	−	0.494135i	−0.415471	−	0.909606i	$-0.636383\pi$
$541$	11.4933	−	11.4933i	0.494135	−	0.494135i	0.909606	+	0.415471i	$0.136383\pi$
$542$		0			0
$543$	−30.1825			−1.29525
$544$		0			0
$545$	−4.24225			−0.181718
$546$		0			0
$547$	−3.27497	+	3.27497i	−0.140028	+	0.140028i	−0.773646	−	0.633618i	$-0.781569\pi$
$547$	−3.27497	+	3.27497i	−0.140028	+	0.140028i	0.633618	+	0.773646i	$0.281569\pi$
$548$		0			0
$549$	4.20938	+	4.20938i	0.179652	+	0.179652i
$550$		0			0
$551$			18.5001i			0.788129i
$552$		0			0
$553$		−	17.0083i		−	0.723267i
$554$		0			0
$555$	8.43895	+	8.43895i	0.358213	+	0.358213i
$556$		0			0
$557$	−11.6257	+	11.6257i	−0.492599	+	0.492599i	−0.909124	−	0.416526i	$-0.863248\pi$
$557$	−11.6257	+	11.6257i	−0.492599	+	0.492599i	0.416526	+	0.909124i	$0.363248\pi$
$558$		0			0
$559$	4.24243			0.179436
$560$		0			0
$561$	18.9077			0.798282
$562$		0			0
$563$	−16.1051	+	16.1051i	−0.678748	+	0.678748i	−0.959717	−	0.280969i	$-0.909344\pi$
$563$	−16.1051	+	16.1051i	−0.678748	+	0.678748i	0.280969	+	0.959717i	$0.409344\pi$
$564$		0			0
$565$	7.94552	+	7.94552i	0.334271	+	0.334271i
$566$		0			0
$567$			4.49929i			0.188952i
$568$		0			0
$569$		−	18.4164i		−	0.772055i	−0.922487	−	0.386027i	$-0.873847\pi$
$569$		−	18.4164i		−	0.772055i	0.922487	−	0.386027i	$-0.126153\pi$
$570$		0			0
$571$	13.2864	+	13.2864i	0.556017	+	0.556017i	0.928171	−	0.372154i	$-0.121381\pi$
$571$	13.2864	+	13.2864i	0.556017	+	0.556017i	−0.372154	+	0.928171i	$0.621381\pi$
$572$		0			0
$573$	−0.971899	+	0.971899i	−0.0406017	+	0.0406017i
$574$		0			0
$575$	−3.15707			−0.131659
$576$		0			0
$577$	−3.47382			−0.144617			−0.0723086	−	0.997382i	$-0.523037\pi$
$577$	−3.47382			−0.144617			−0.0723086	+	0.997382i	$0.523037\pi$
$578$		0			0
$579$	−1.95215	+	1.95215i	−0.0811285	+	0.0811285i
$580$		0			0
$581$	4.44738	+	4.44738i	0.184508	+	0.184508i
$582$		0			0
$583$		−	11.9159i		−	0.493507i
$584$		0			0
$585$		−	2.31762i		−	0.0958216i
$586$		0			0
$587$	−17.5866	−	17.5866i	−0.725875	−	0.725875i	0.243921	−	0.969795i	$-0.421566\pi$
$587$	−17.5866	−	17.5866i	−0.725875	−	0.725875i	−0.969795	+	0.243921i	$0.921566\pi$
$588$		0			0
$589$	−5.20321	+	5.20321i	−0.214395	+	0.214395i
$590$		0			0
$591$	32.5131			1.33741
$592$		0			0
$593$	−41.4630			−1.70268			−0.851341	−	0.524613i	$-0.824210\pi$
$593$	−41.4630			−1.70268			−0.851341	+	0.524613i	$0.824210\pi$
$594$		0			0
$595$	−2.15776	+	2.15776i	−0.0884595	+	0.0884595i
$596$		0			0
$597$	14.3825	+	14.3825i	0.588636	+	0.588636i
$598$		0			0
$599$		−	13.8641i		−	0.566473i	−0.959050	−	0.283237i	$-0.908592\pi$
$599$		−	13.8641i		−	0.566473i	0.959050	−	0.283237i	$-0.0914082\pi$
$600$		0			0
$601$		−	2.65189i		−	0.108173i	−0.998536	−	0.0540864i	$-0.982775\pi$
$601$		−	2.65189i		−	0.108173i	0.998536	−	0.0540864i	$-0.0172246\pi$
$602$		0			0
$603$	−5.01510	−	5.01510i	−0.204231	−	0.204231i
$604$		0			0
$605$	6.49639	−	6.49639i	0.264116	−	0.264116i
$606$		0			0
$607$	−13.3324			−0.541147			−0.270573	−	0.962699i	$-0.587213\pi$
$607$	−13.3324			−0.541147			−0.270573	+	0.962699i	$0.587213\pi$
$608$		0			0
$609$	−7.77994			−0.315259
$610$		0			0
$611$	−6.64177	+	6.64177i	−0.268697	+	0.268697i
$612$		0			0
$613$	12.3726	+	12.3726i	0.499725	+	0.499725i	0.911352	−	0.411627i	$-0.135039\pi$
$613$	12.3726	+	12.3726i	0.499725	+	0.499725i	−0.411627	+	0.911352i	$0.635039\pi$
$614$		0			0
$615$		−	2.11457i		−	0.0852677i
$616$		0			0
$617$		−	17.9671i		−	0.723330i	−0.932308	−	0.361665i	$-0.882208\pi$
$617$		−	17.9671i		−	0.723330i	0.932308	−	0.361665i	$-0.117792\pi$
$618$		0			0
$619$	12.8788	+	12.8788i	0.517644	+	0.517644i	0.916858	−	0.399214i	$-0.130717\pi$
$619$	12.8788	+	12.8788i	0.517644	+	0.517644i	−0.399214	+	0.916858i	$0.630717\pi$
$620$		0			0
$621$	12.6167	−	12.6167i	0.506289	−	0.506289i
$622$		0			0
$623$	−15.1211			−0.605812
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$	14.3675	−	14.3675i	0.573784	−	0.573784i
$628$		0			0
$629$	−18.6735	−	18.6735i	−0.744560	−	0.744560i
$630$		0			0
$631$		−	25.0441i		−	0.996991i	−0.866892	−	0.498496i	$-0.833886\pi$
$631$		−	25.0441i		−	0.996991i	0.866892	−	0.498496i	$-0.166114\pi$
$632$		0			0
$633$			6.54643i			0.260197i
$634$		0			0
$635$	10.7417	+	10.7417i	0.426271	+	0.426271i
$636$		0			0
$637$	−1.49224	+	1.49224i	−0.0591249	+	0.0591249i
$638$		0			0
$639$	−17.6961			−0.700045
$640$		0			0
$641$	31.2728			1.23520			0.617601	−	0.786492i	$-0.288105\pi$
$641$	31.2728			1.23520			0.617601	+	0.786492i	$0.288105\pi$
$642$		0			0
$643$	−25.1751	+	25.1751i	−0.992807	+	0.992807i	−0.999974	−	0.00716702i	$-0.997719\pi$
$643$	−25.1751	+	25.1751i	−0.992807	+	0.992807i	0.00716702	+	0.999974i	$0.497719\pi$
$644$		0			0
$645$	1.96032	+	1.96032i	0.0771874	+	0.0771874i
$646$		0			0
$647$		−	2.88325i		−	0.113352i	−0.998393	−	0.0566762i	$-0.981950\pi$
$647$		−	2.88325i		−	0.113352i	0.998393	−	0.0566762i	$-0.0180503\pi$
$648$		0			0
$649$		−	11.0610i		−	0.434184i
$650$		0			0
$651$	−2.18814	−	2.18814i	−0.0857599	−	0.0857599i
$652$		0			0
$653$	25.2650	−	25.2650i	0.988695	−	0.988695i	−0.0112415	−	0.999937i	$-0.503578\pi$
$653$	25.2650	−	25.2650i	0.988695	−	0.988695i	0.999937	+	0.0112415i	$0.00357836\pi$
$654$		0			0
$655$	12.4424			0.486167
$656$		0			0
$657$	−6.66628			−0.260076
$658$		0			0
$659$	3.37439	−	3.37439i	0.131448	−	0.131448i	−0.638322	−	0.769770i	$-0.720371\pi$
$659$	3.37439	−	3.37439i	0.131448	−	0.131448i	0.769770	+	0.638322i	$0.220371\pi$
$660$		0			0
$661$	−8.16542	−	8.16542i	−0.317598	−	0.317598i	0.530246	−	0.847844i	$-0.322100\pi$
$661$	−8.16542	−	8.16542i	−0.317598	−	0.317598i	−0.847844	+	0.530246i	$0.822100\pi$
$662$		0			0
$663$		−	8.88083i		−	0.344903i
$664$		0			0
$665$			3.27927i			0.127165i
$666$		0			0
$667$	12.5940	+	12.5940i	0.487642	+	0.487642i
$668$		0			0
$669$	−5.69905	+	5.69905i	−0.220338	+	0.220338i
$670$		0			0
$671$	24.3548			0.940208
$672$		0			0
$673$	39.5505			1.52456			0.762280	−	0.647248i	$-0.224080\pi$
$673$	39.5505			1.52456			0.762280	+	0.647248i	$0.224080\pi$
$674$		0			0
$675$	3.99632	−	3.99632i	0.153819	−	0.153819i
$676$		0			0
$677$	16.8401	+	16.8401i	0.647217	+	0.647217i	0.952319	−	0.305103i	$-0.0986908\pi$
$677$	16.8401	+	16.8401i	0.647217	+	0.647217i	−0.305103	+	0.952319i	$0.598691\pi$
$678$		0			0
$679$		−	18.8094i		−	0.721838i
$680$		0			0
$681$			7.97258i			0.305510i
$682$		0			0
$683$	16.0509	+	16.0509i	0.614172	+	0.614172i	0.944030	−	0.329859i	$-0.107001\pi$
$683$	16.0509	+	16.0509i	0.614172	+	0.614172i	−0.329859	+	0.944030i	$0.607001\pi$
$684$		0			0
$685$	−8.33342	+	8.33342i	−0.318404	+	0.318404i
$686$		0			0
$687$	−12.6929			−0.484263
$688$		0			0
$689$	−5.59685			−0.213223
$690$		0			0
$691$	−31.0492	+	31.0492i	−1.18117	+	1.18117i	−0.201726	+	0.979442i	$0.564655\pi$
$691$	−31.0492	+	31.0492i	−1.18117	+	1.18117i	−0.979442	+	0.201726i	$0.935345\pi$
$692$		0			0
$693$	−3.48908	−	3.48908i	−0.132539	−	0.132539i
$694$		0			0
$695$		−	4.59299i		−	0.174222i
$696$		0			0
$697$			4.67906i			0.177232i
$698$		0			0
$699$	7.08846	+	7.08846i	0.268110	+	0.268110i
$700$		0			0
$701$	2.76649	−	2.76649i	0.104489	−	0.104489i	−0.652930	−	0.757419i	$-0.726460\pi$
$701$	2.76649	−	2.76649i	0.104489	−	0.104489i	0.757419	+	0.652930i	$0.226460\pi$
$702$		0			0
$703$	−28.3792			−1.07034
$704$		0			0
$705$	−6.13797			−0.231169
$706$		0			0
$707$	3.85350	−	3.85350i	0.144926	−	0.144926i
$708$		0			0
$709$	−29.2877	−	29.2877i	−1.09992	−	1.09992i	−0.994419	−	0.105502i	$-0.966355\pi$
$709$	−29.2877	−	29.2877i	−1.09992	−	1.09992i	−0.105502	−	0.994419i	$-0.533645\pi$
$710$		0			0
$711$		−	18.6787i		−	0.700508i
$712$		0			0
$713$			7.08423i			0.265306i
$714$		0			0
$715$	−6.70469	−	6.70469i	−0.250741	−	0.250741i
$716$		0			0
$717$	−7.33519	+	7.33519i	−0.273938	+	0.273938i
$718$		0			0
$719$	−1.60830			−0.0599796			−0.0299898	−	0.999550i	$-0.509547\pi$
$719$	−1.60830			−0.0599796			−0.0299898	+	0.999550i	$0.509547\pi$
$720$		0			0
$721$	−6.69505			−0.249337
$722$		0			0
$723$	−8.64541	+	8.64541i	−0.321526	+	0.321526i
$724$		0			0
$725$	3.98915	+	3.98915i	0.148153	+	0.148153i
$726$		0			0
$727$		−	4.75004i		−	0.176169i	−0.996113	−	0.0880845i	$-0.971925\pi$
$727$		−	4.75004i		−	0.176169i	0.996113	−	0.0880845i	$-0.0280746\pi$
$728$		0			0
$729$			28.3230i			1.04900i
$730$		0			0
$731$	−4.33773	−	4.33773i	−0.160437	−	0.160437i
$732$		0			0
$733$	15.2960	−	15.2960i	0.564970	−	0.564970i	−0.365745	−	0.930715i	$-0.619186\pi$
$733$	15.2960	−	15.2960i	0.564970	−	0.564970i	0.930715	+	0.365745i	$0.119186\pi$
$734$		0			0
$735$	−1.37905			−0.0508671
$736$		0			0
$737$	−29.0166			−1.06884
$738$		0			0
$739$	−15.3503	+	15.3503i	−0.564669	+	0.564669i	−0.930630	−	0.365961i	$-0.880740\pi$
$739$	−15.3503	+	15.3503i	−0.564669	+	0.564669i	0.365961	+	0.930630i	$0.380740\pi$
$740$		0			0
$741$	−6.74836	−	6.74836i	−0.247907	−	0.247907i
$742$		0			0
$743$		−	34.8262i		−	1.27765i	−0.769352	−	0.638825i	$-0.779421\pi$
$743$		−	34.8262i		−	1.27765i	0.769352	−	0.638825i	$-0.220579\pi$
$744$		0			0
$745$		−	22.4175i		−	0.821314i
$746$		0			0
$747$	4.88417	+	4.88417i	0.178703	+	0.178703i
$748$		0			0
$749$	8.22368	−	8.22368i	0.300487	−	0.300487i
$750$		0			0
$751$	17.1957			0.627480			0.313740	−	0.949509i	$-0.398418\pi$
$751$	17.1957			0.627480			0.313740	+	0.949509i	$0.398418\pi$
$752$		0			0
$753$	35.8947			1.30807
$754$		0			0
$755$	3.21749	−	3.21749i	0.117096	−	0.117096i
$756$		0			0
$757$	15.4913	+	15.4913i	0.563040	+	0.563040i	0.930170	−	0.367129i	$-0.119659\pi$
$757$	15.4913	+	15.4913i	0.563040	+	0.563040i	−0.367129	+	0.930170i	$0.619659\pi$
$758$		0			0
$759$		−	19.5616i		−	0.710040i
$760$		0			0
$761$			20.2388i			0.733657i	0.930289	+	0.366828i	$0.119556\pi$
$761$			20.2388i			0.733657i	−0.930289	+	0.366828i	$0.880444\pi$
$762$		0			0
$763$	2.99972	+	2.99972i	0.108597	+	0.108597i
$764$		0			0
$765$	−2.36968	+	2.36968i	−0.0856759	+	0.0856759i
$766$		0			0
$767$	−5.19532			−0.187592
$768$		0			0
$769$	−39.9415			−1.44033			−0.720164	−	0.693804i	$-0.755933\pi$
$769$	−39.9415			−1.44033			−0.720164	+	0.693804i	$0.755933\pi$
$770$		0			0
$771$	26.6867	−	26.6867i	0.961098	−	0.961098i
$772$		0			0
$773$	−37.4959	−	37.4959i	−1.34863	−	1.34863i	−0.887148	−	0.461486i	$-0.847317\pi$
$773$	−37.4959	−	37.4959i	−1.34863	−	1.34863i	−0.461486	−	0.887148i	$-0.652683\pi$
$774$		0			0
$775$			2.24393i			0.0806042i
$776$		0			0
$777$		−	11.9345i		−	0.428147i
$778$		0			0
$779$	3.55552	+	3.55552i	0.127390	+	0.127390i
$780$		0			0
$781$	−51.1934	+	51.1934i	−1.83184	+	1.83184i
$782$		0			0
$783$	−31.8839			−1.13944
$784$		0			0
$785$	5.92356			0.211421
$786$		0			0
$787$	38.5192	−	38.5192i	1.37306	−	1.37306i	0.517194	−	0.855868i	$-0.326977\pi$
$787$	38.5192	−	38.5192i	1.37306	−	1.37306i	0.855868	−	0.517194i	$-0.173023\pi$
$788$		0			0
$789$	−18.8412	−	18.8412i	−0.670765	−	0.670765i
$790$		0			0
$791$		−	11.2367i		−	0.399530i
$792$		0			0
$793$		−	11.4393i		−	0.406223i
$794$		0			0
$795$	−2.58616	−	2.58616i	−0.0917215	−	0.0917215i
$796$		0			0
$797$	11.0170	−	11.0170i	0.390244	−	0.390244i	−0.484531	−	0.874774i	$-0.661010\pi$
$797$	11.0170	−	11.0170i	0.390244	−	0.390244i	0.874774	+	0.484531i	$0.161010\pi$
$798$		0			0
$799$	13.5819			0.480495
$800$		0			0
$801$	−16.6061			−0.586749
$802$		0			0
$803$	−19.2850	+	19.2850i	−0.680555	+	0.680555i
$804$		0			0
$805$	2.23238	+	2.23238i	0.0786812	+	0.0786812i
$806$		0			0
$807$		−	22.1590i		−	0.780033i
$808$		0			0
$809$			18.8890i			0.664101i	0.943262	+	0.332051i	$0.107741\pi$
$809$			18.8890i			0.664101i	−0.943262	+	0.332051i	$0.892259\pi$
$810$		0			0
$811$	33.4961	+	33.4961i	1.17621	+	1.17621i	0.980703	+	0.195506i	$0.0626348\pi$
$811$	33.4961	+	33.4961i	1.17621	+	1.17621i	0.195506	+	0.980703i	$0.437365\pi$
$812$		0			0
$813$	−30.0125	+	30.0125i	−1.05258	+	1.05258i
$814$		0			0
$815$	−4.64271			−0.162627
$816$		0			0
$817$	−6.59231			−0.230636
$818$		0			0
$819$	−1.63880	+	1.63880i	−0.0572644	+	0.0572644i
$820$		0			0
$821$	−24.1873	−	24.1873i	−0.844141	−	0.844141i	0.145253	−	0.989395i	$-0.453600\pi$
$821$	−24.1873	−	24.1873i	−0.844141	−	0.844141i	−0.989395	+	0.145253i	$0.953600\pi$
$822$		0			0
$823$			45.2276i			1.57653i	0.615333	+	0.788267i	$0.289022\pi$
$823$			45.2276i			1.57653i	−0.615333	+	0.788267i	$0.710978\pi$
$824$		0			0
$825$		−	6.19612i		−	0.215721i
$826$		0			0
$827$	17.3160	+	17.3160i	0.602136	+	0.602136i	0.940879	−	0.338743i	$-0.110002\pi$
$827$	17.3160	+	17.3160i	0.602136	+	0.602136i	−0.338743	+	0.940879i	$0.610002\pi$
$828$		0			0
$829$	−7.37463	+	7.37463i	−0.256131	+	0.256131i	−0.823479	−	0.567347i	$-0.807970\pi$
$829$	−7.37463	+	7.37463i	−0.256131	+	0.256131i	0.567347	+	0.823479i	$0.307970\pi$
$830$		0			0
$831$	−34.2271			−1.18733
$832$		0			0
$833$	3.05153			0.105729
$834$		0			0
$835$	−9.69525	+	9.69525i	−0.335518	+	0.335518i
$836$		0			0
$837$	−8.96745	−	8.96745i	−0.309961	−	0.309961i
$838$		0			0
$839$		−	11.3271i		−	0.391053i	−0.980698	−	0.195527i	$-0.937358\pi$
$839$		−	11.3271i		−	0.391053i	0.980698	−	0.195527i	$-0.0626416\pi$
$840$		0			0
$841$		−	2.82664i		−	0.0974702i
$842$		0			0
$843$	−6.73478	−	6.73478i	−0.231958	−	0.231958i
$844$		0			0
$845$	6.04323	−	6.04323i	0.207893	−	0.207893i
$846$		0			0
$847$	−9.18728			−0.315679
$848$		0			0
$849$	26.2274			0.900123
$850$		0			0
$851$	−19.3193	+	19.3193i	−0.662257	+	0.662257i
$852$		0			0
$853$	14.9559	+	14.9559i	0.512080	+	0.512080i	0.915163	−	0.403084i	$-0.132062\pi$
$853$	14.9559	+	14.9559i	0.512080	+	0.512080i	−0.403084	+	0.915163i	$0.632062\pi$
$854$		0			0
$855$			3.60134i			0.123163i
$856$		0			0
$857$			53.2542i			1.81913i	0.415564	+	0.909564i	$0.363584\pi$
$857$			53.2542i			1.81913i	−0.415564	+	0.909564i	$0.636416\pi$
$858$		0			0
$859$	9.59294	+	9.59294i	0.327307	+	0.327307i	0.851562	−	0.524255i	$-0.175656\pi$
$859$	9.59294	+	9.59294i	0.327307	+	0.327307i	−0.524255	+	0.851562i	$0.675656\pi$
$860$		0			0
$861$	−1.49523	+	1.49523i	−0.0509572	+	0.0509572i
$862$		0			0
$863$	13.4327			0.457253			0.228627	−	0.973514i	$-0.426577\pi$
$863$	13.4327			0.457253			0.228627	+	0.973514i	$0.426577\pi$
$864$		0			0
$865$	6.91421			0.235090
$866$		0			0
$867$	7.49701	−	7.49701i	0.254612	−	0.254612i
$868$		0			0
$869$	−54.0362	−	54.0362i	−1.83305	−	1.83305i
$870$		0			0
$871$			13.6290i			0.461800i
$872$		0			0
$873$		−	20.6567i		−	0.699123i
$874$		0			0
$875$	0.707107	+	0.707107i	0.0239046	+	0.0239046i
$876$		0			0
$877$	−30.0789	+	30.0789i	−1.01569	+	1.01569i	−0.0158176	+	0.999875i	$0.505035\pi$
$877$	−30.0789	+	30.0789i	−1.01569	+	1.01569i	−0.999875	+	0.0158176i	$0.994965\pi$
$878$		0			0
$879$	0.988408			0.0333382
$880$		0			0
$881$	−12.3932			−0.417538			−0.208769	−	0.977965i	$-0.566946\pi$
$881$	−12.3932			−0.417538			−0.208769	+	0.977965i	$0.566946\pi$
$882$		0			0
$883$	38.5305	−	38.5305i	1.29666	−	1.29666i	0.366067	−	0.930589i	$-0.380704\pi$
$883$	38.5305	−	38.5305i	1.29666	−	1.29666i	0.930589	−	0.366067i	$-0.119296\pi$
$884$		0			0
$885$	−2.40062	−	2.40062i	−0.0806959	−	0.0806959i
$886$		0			0
$887$		−	4.60507i		−	0.154623i	−0.997007	−	0.0773115i	$-0.975366\pi$
$887$		−	4.60507i		−	0.154623i	0.997007	−	0.0773115i	$-0.0246336\pi$
$888$		0			0
$889$		−	15.1910i		−	0.509491i
$890$		0			0
$891$	14.2945	+	14.2945i	0.478882	+	0.478882i
$892$		0			0
$893$	10.3206	−	10.3206i	0.345367	−	0.345367i
$894$		0			0
$895$	6.93188			0.231707
$896$		0			0
$897$	−9.18797			−0.306777
$898$		0			0
$899$	8.95136	−	8.95136i	0.298545	−	0.298545i
$900$		0			0
$901$	5.72258	+	5.72258i	0.190647	+	0.190647i
$902$		0			0
$903$		−	2.77231i		−	0.0922565i
$904$		0			0
$905$		−	21.8864i		−	0.727528i
$906$		0			0
$907$	4.85880	+	4.85880i	0.161334	+	0.161334i	0.783157	−	0.621824i	$-0.213608\pi$
$907$	4.85880	+	4.85880i	0.161334	+	0.161334i	−0.621824	+	0.783157i	$0.713608\pi$
$908$		0			0
$909$	4.23196	−	4.23196i	0.140365	−	0.140365i
$910$		0			0
$911$	25.5468			0.846404			0.423202	−	0.906035i	$-0.360906\pi$
$911$	25.5468			0.846404			0.423202	+	0.906035i	$0.360906\pi$
$912$		0			0
$913$	28.2591			0.935240
$914$		0			0
$915$	5.28582	−	5.28582i	0.174744	−	0.174744i
$916$		0			0
$917$	−8.79814	−	8.79814i	−0.290540	−	0.290540i
$918$		0			0
$919$			6.90027i			0.227619i	0.993503	+	0.113809i	$0.0363053\pi$
$919$			6.90027i			0.227619i	−0.993503	+	0.113809i	$0.963695\pi$
$920$		0			0
$921$			37.3236i			1.22986i
$922$		0			0
$923$	24.0453	+	24.0453i	0.791460	+	0.791460i
$924$		0			0
$925$	−6.11938	+	6.11938i	−0.201204	+	0.201204i
$926$		0			0
$927$	−7.35259			−0.241491
$928$		0			0
$929$	45.2739			1.48539			0.742694	−	0.669631i	$-0.233548\pi$
$929$	45.2739			1.48539			0.742694	+	0.669631i	$0.233548\pi$
$930$		0			0
$931$	2.31880	−	2.31880i	0.0759955	−	0.0759955i
$932$		0			0
$933$	−24.7845	−	24.7845i	−0.811408	−	0.811408i
$934$		0			0
$935$			13.7106i			0.448385i
$936$		0			0
$937$		−	40.5121i		−	1.32347i	−0.749737	−	0.661736i	$-0.769820\pi$
$937$		−	40.5121i		−	1.32347i	0.749737	−	0.661736i	$-0.230180\pi$
$938$		0			0
$939$	−33.0502	−	33.0502i	−1.07855	−	1.07855i
$940$		0			0
$941$	−36.3523	+	36.3523i	−1.18505	+	1.18505i	−0.206634	+	0.978418i	$0.566251\pi$
$941$	−36.3523	+	36.3523i	−1.18505	+	1.18505i	−0.978418	+	0.206634i	$0.933749\pi$
$942$		0			0
$943$	4.84089			0.157641
$944$		0			0
$945$	−5.65165			−0.183848
$946$		0			0
$947$	4.34342	−	4.34342i	0.141142	−	0.141142i	−0.633005	−	0.774148i	$-0.718179\pi$
$947$	4.34342	−	4.34342i	0.141142	−	0.141142i	0.774148	+	0.633005i	$0.218179\pi$
$948$		0			0
$949$	9.05809	+	9.05809i	0.294038	+	0.294038i
$950$		0			0
$951$		−	7.04062i		−	0.228308i
$952$		0			0
$953$			20.2806i			0.656954i	0.944512	+	0.328477i	$0.106535\pi$
$953$			20.2806i			0.656954i	−0.944512	+	0.328477i	$0.893465\pi$
$954$		0			0
$955$	−0.704758	−	0.704758i	−0.0228054	−	0.0228054i
$956$		0			0
$957$	−24.7173	+	24.7173i	−0.798995	+	0.798995i
$958$		0			0
$959$	11.7852			0.380565
$960$		0			0
$961$	−25.9648			−0.837574
$962$		0			0
$963$	9.03135	−	9.03135i	0.291031	−	0.291031i
$964$		0			0
$965$	−1.41557	−	1.41557i	−0.0455688	−	0.0455688i
$966$		0			0
$967$			48.2182i			1.55059i	0.631598	+	0.775296i	$0.282399\pi$
$967$			48.2182i			1.55059i	−0.631598	+	0.775296i	$0.717601\pi$
$968$		0			0
$969$			13.7999i			0.443317i
$970$		0			0
$971$	32.9747	+	32.9747i	1.05821	+	1.05821i	0.998198	+	0.0600094i	$0.0191131\pi$
$971$	32.9747	+	32.9747i	1.05821	+	1.05821i	0.0600094	+	0.998198i	$0.480887\pi$
$972$		0			0
$973$	−3.24773	+	3.24773i	−0.104118	+	0.104118i
$974$		0			0
$975$	−2.91029			−0.0932038
$976$		0			0
$977$	19.4943			0.623677			0.311838	−	0.950135i	$-0.399055\pi$
$977$	19.4943			0.623677			0.311838	+	0.950135i	$0.399055\pi$
$978$		0			0
$979$	−48.0403	+	48.0403i	−1.53538	+	1.53538i
$980$		0			0
$981$	3.29433	+	3.29433i	0.105180	+	0.105180i
$982$		0			0
$983$		−	0.206086i		−	0.00657311i	−0.999995	−	0.00328656i	$-0.998954\pi$
$983$		−	0.206086i		−	0.00657311i	0.999995	−	0.00328656i	$-0.00104614\pi$
$984$		0			0
$985$			23.5764i			0.751207i
$986$		0			0
$987$	4.34020	+	4.34020i	0.138150	+	0.138150i
$988$		0			0
$989$	−4.48775	+	4.48775i	−0.142702	+	0.142702i
$990$		0			0
$991$	−2.67226			−0.0848872			−0.0424436	−	0.999099i	$-0.513514\pi$
$991$	−2.67226			−0.0848872			−0.0424436	+	0.999099i	$0.513514\pi$
$992$		0			0
$993$	−38.1031			−1.20917
$994$		0			0
$995$	−10.4292	+	10.4292i	−0.330629	+	0.330629i
$996$		0			0
$997$	30.6277	+	30.6277i	0.969988	+	0.969988i	0.999563	−	0.0295743i	$-0.00941516\pi$
$997$	30.6277	+	30.6277i	0.969988	+	0.969988i	−0.0295743	+	0.999563i	$0.509415\pi$
$998$		0			0
$999$		−	48.9100i		−	1.54744i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.bd.b.561.8		52
4.3	odd	2		560.2.bd.b.421.17	yes	52
16.3	odd	4		560.2.bd.b.141.17	✓	52
16.13	even	4	inner	2240.2.bd.b.1681.8		52

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.bd.b.141.17	✓	52	16.3	odd	4
560.2.bd.b.421.17	yes	52	4.3	odd	2
2240.2.bd.b.561.8		52	1.1	even	1	trivial
2240.2.bd.b.1681.8		52	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)

\(f(q)\)	\(=\)	\(q+(-0.975138 + 0.975138i) q^{3} +(-0.707107 - 0.707107i) q^{5} +1.00000i q^{7} +1.09821i q^{9} +O(q^{10})\)	\(q+(-0.975138 + 0.975138i) q^{3} +(-0.707107 - 0.707107i) q^{5} +1.00000i q^{7} +1.09821i q^{9} +(3.17705 + 3.17705i) q^{11} +(1.49224 - 1.49224i) q^{13} +1.37905 q^{15} -3.05153 q^{17} +(-2.31880 + 2.31880i) q^{19} +(-0.975138 - 0.975138i) q^{21} +3.15707i q^{23} +1.00000i q^{25} +(-3.99632 - 3.99632i) q^{27} +(3.98915 - 3.98915i) q^{29} +2.24393 q^{31} -6.19612 q^{33} +(0.707107 - 0.707107i) q^{35} +(6.11938 + 6.11938i) q^{37} +2.91029i q^{39} -1.53335i q^{41} +(1.42149 + 1.42149i) q^{43} +(0.776554 - 0.776554i) q^{45} -4.45086 q^{47} -1.00000 q^{49} +(2.97566 - 2.97566i) q^{51} +(-1.87531 - 1.87531i) q^{53} -4.49303i q^{55} -4.52229i q^{57} +(-1.74077 - 1.74077i) q^{59} +(3.83294 - 3.83294i) q^{61} -1.09821 q^{63} -2.11035 q^{65} +(-4.56660 + 4.56660i) q^{67} +(-3.07858 - 3.07858i) q^{69} +16.1135i q^{71} +6.07011i q^{73} +(-0.975138 - 0.975138i) q^{75} +(-3.17705 + 3.17705i) q^{77} -17.0083 q^{79} +4.49929 q^{81} +(4.44738 - 4.44738i) q^{83} +(2.15776 + 2.15776i) q^{85} +7.77994i q^{87} +15.1211i q^{89} +(1.49224 + 1.49224i) q^{91} +(-2.18814 + 2.18814i) q^{93} +3.27927 q^{95} -18.8094 q^{97} +(-3.48908 + 3.48908i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q+O(q^{10}) \) 52 * q	\( 52 q + 4 q^{11} - 8 q^{15} - 8 q^{19} + 24 q^{27} + 4 q^{29} - 12 q^{37} - 36 q^{43} - 52 q^{49} + 8 q^{51} - 4 q^{53} - 24 q^{59} - 16 q^{61} + 68 q^{63} + 40 q^{65} + 12 q^{67} - 72 q^{69} - 4 q^{77} + 16 q^{79} - 116 q^{81} + 16 q^{85} + 8 q^{93} + 32 q^{95} + 52 q^{99}+O(q^{100}) \) 52 * q + 4 * q^11 - 8 * q^15 - 8 * q^19 + 24 * q^27 + 4 * q^29 - 12 * q^37 - 36 * q^43 - 52 * q^49 + 8 * q^51 - 4 * q^53 - 24 * q^59 - 16 * q^61 + 68 * q^63 + 40 * q^65 + 12 * q^67 - 72 * q^69 - 4 * q^77 + 16 * q^79 - 116 * q^81 + 16 * q^85 + 8 * q^93 + 32 * q^95 + 52 * q^99

Introduction

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