LMFDB - Embedded newform 2320.2.d.f.929.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2320,2,Mod(929,2320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2320, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2320.929");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2320 = 2^{4} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2320.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$18.5252932689$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 145)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			929.4
Root			$1.93185i$ of defining polynomial
Character	$\chi$	$=$	2320.929
Dual form			2320.2.d.f.929.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{3} +(1.73205 - 1.41421i) q^{5} -2.44949i q^{7} +1.00000 q^{9} +4.73205 q^{11} +6.69213i q^{13} +(2.00000 + 2.44949i) q^{15} +1.41421i q^{17} -6.73205 q^{19} +3.46410 q^{21} +3.48477i q^{23} +(1.00000 - 4.89898i) q^{25} +5.65685i q^{27} -1.00000 q^{29} +5.26795 q^{31} +6.69213i q^{33} +(-3.46410 - 4.24264i) q^{35} -0.656339i q^{37} -9.46410 q^{39} +6.92820 q^{41} -0.656339i q^{43} +(1.73205 - 1.41421i) q^{45} -1.41421i q^{47} +1.00000 q^{49} -2.00000 q^{51} +8.76268i q^{53} +(8.19615 - 6.69213i) q^{55} -9.52056i q^{57} +10.3923 q^{59} -10.9282 q^{61} -2.44949i q^{63} +(9.46410 + 11.5911i) q^{65} -4.24264i q^{67} -4.92820 q^{69} +3.46410 q^{71} -7.34847i q^{73} +(6.92820 + 1.41421i) q^{75} -11.5911i q^{77} +6.19615 q^{79} -5.00000 q^{81} -4.52004i q^{83} +(2.00000 + 2.44949i) q^{85} -1.41421i q^{87} +10.3923 q^{89} +16.3923 q^{91} +7.45001i q^{93} +(-11.6603 + 9.52056i) q^{95} -6.03579i q^{97} +4.73205 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{9} + 12 q^{11} + 8 q^{15} - 20 q^{19} + 4 q^{25} - 4 q^{29} + 28 q^{31} - 24 q^{39} + 4 q^{49} - 8 q^{51} + 12 q^{55} - 16 q^{61} + 24 q^{65} + 8 q^{69} + 4 q^{79} - 20 q^{81} + 8 q^{85} + 24 q^{91}+ \cdots + 12 q^{99}+O(q^{100}) $ 4 * q + 4 * q^9 + 12 * q^11 + 8 * q^15 - 20 * q^19 + 4 * q^25 - 4 * q^29 + 28 * q^31 - 24 * q^39 + 4 * q^49 - 8 * q^51 + 12 * q^55 - 16 * q^61 + 24 * q^65 + 8 * q^69 + 4 * q^79 - 20 * q^81 + 8 * q^85 + 24 * q^91 - 12 * q^95 + 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2320\mathbb{Z}\right)^\times$.

$n$	$321$	$581$	$1857$	$2031$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$			1.41421i			0.816497i	0.912871	+	0.408248i	$0.133860\pi$
$3$			1.41421i			0.816497i	−0.912871	+	0.408248i	$0.866140\pi$
$4$		0			0
$5$	1.73205	−	1.41421i	0.774597	−	0.632456i
$5$	1.73205	−	1.41421i	0.774597	−	0.632456i
$6$		0			0
$7$		−	2.44949i		−	0.925820i	−0.886405	−	0.462910i	$-0.846805\pi$
$7$		−	2.44949i		−	0.925820i	0.886405	−	0.462910i	$-0.153195\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$	4.73205			1.42677			0.713384	−	0.700774i	$-0.247162\pi$
$11$	4.73205			1.42677			0.713384	+	0.700774i	$0.247162\pi$
$12$		0			0
$13$			6.69213i			1.85606i	0.372502	+	0.928032i	$0.378500\pi$
$13$			6.69213i			1.85606i	−0.372502	+	0.928032i	$0.621500\pi$
$14$		0			0
$15$	2.00000	+	2.44949i	0.516398	+	0.632456i
$16$		0			0
$17$			1.41421i			0.342997i	0.985184	+	0.171499i	$0.0548609\pi$
$17$			1.41421i			0.342997i	−0.985184	+	0.171499i	$0.945139\pi$
$18$		0			0
$19$	−6.73205			−1.54444			−0.772219	−	0.635356i	$-0.780853\pi$
$19$	−6.73205			−1.54444			−0.772219	+	0.635356i	$0.780853\pi$
$20$		0			0
$21$	3.46410			0.755929
$22$		0			0
$23$			3.48477i			0.726624i	0.931668	+	0.363312i	$0.118354\pi$
$23$			3.48477i			0.726624i	−0.931668	+	0.363312i	$0.881646\pi$
$24$		0			0
$25$	1.00000	−	4.89898i	0.200000	−	0.979796i
$26$		0			0
$27$			5.65685i			1.08866i
$28$		0			0
$29$	−1.00000			−0.185695
$29$	−1.00000			−0.185695
$30$		0			0
$31$	5.26795			0.946152			0.473076	−	0.881022i	$-0.343144\pi$
$31$	5.26795			0.946152			0.473076	+	0.881022i	$0.343144\pi$
$32$		0			0
$33$			6.69213i			1.16495i
$34$		0			0
$35$	−3.46410	−	4.24264i	−0.585540	−	0.717137i
$36$		0			0
$37$		−	0.656339i		−	0.107901i	−0.998544	−	0.0539507i	$-0.982819\pi$
$37$		−	0.656339i		−	0.107901i	0.998544	−	0.0539507i	$-0.0171814\pi$
$38$		0			0
$39$	−9.46410			−1.51547
$40$		0			0
$41$	6.92820			1.08200			0.541002	−	0.841021i	$-0.318045\pi$
$41$	6.92820			1.08200			0.541002	+	0.841021i	$0.318045\pi$
$42$		0			0
$43$		−	0.656339i		−	0.100091i	−0.998747	−	0.0500454i	$-0.984063\pi$
$43$		−	0.656339i		−	0.100091i	0.998747	−	0.0500454i	$-0.0159366\pi$
$44$		0			0
$45$	1.73205	−	1.41421i	0.258199	−	0.210819i
$46$		0			0
$47$		−	1.41421i		−	0.206284i	−0.994667	−	0.103142i	$-0.967110\pi$
$47$		−	1.41421i		−	0.206284i	0.994667	−	0.103142i	$-0.0328896\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$	−2.00000			−0.280056
$52$		0			0
$53$			8.76268i			1.20365i	0.798629	+	0.601824i	$0.205559\pi$
$53$			8.76268i			1.20365i	−0.798629	+	0.601824i	$0.794441\pi$
$54$		0			0
$55$	8.19615	−	6.69213i	1.10517	−	0.902367i
$56$		0			0
$57$		−	9.52056i		−	1.26103i
$58$		0			0
$59$	10.3923			1.35296			0.676481	−	0.736460i	$-0.263504\pi$
$59$	10.3923			1.35296			0.676481	+	0.736460i	$0.263504\pi$
$60$		0			0
$61$	−10.9282			−1.39921			−0.699607	−	0.714528i	$-0.746641\pi$
$61$	−10.9282			−1.39921			−0.699607	+	0.714528i	$0.746641\pi$
$62$		0			0
$63$		−	2.44949i		−	0.308607i
$64$		0			0
$65$	9.46410	+	11.5911i	1.17388	+	1.43770i
$66$		0			0
$67$		−	4.24264i		−	0.518321i	−0.965834	−	0.259161i	$-0.916554\pi$
$67$		−	4.24264i		−	0.518321i	0.965834	−	0.259161i	$-0.0834459\pi$
$68$		0			0
$69$	−4.92820			−0.593286
$70$		0			0
$71$	3.46410			0.411113			0.205557	−	0.978645i	$-0.434100\pi$
$71$	3.46410			0.411113			0.205557	+	0.978645i	$0.434100\pi$
$72$		0			0
$73$		−	7.34847i		−	0.860073i	−0.902811	−	0.430037i	$-0.858501\pi$
$73$		−	7.34847i		−	0.860073i	0.902811	−	0.430037i	$-0.141499\pi$
$74$		0			0
$75$	6.92820	+	1.41421i	0.800000	+	0.163299i
$76$		0			0
$77$		−	11.5911i		−	1.32093i
$78$		0			0
$79$	6.19615			0.697122			0.348561	−	0.937286i	$-0.386670\pi$
$79$	6.19615			0.697122			0.348561	+	0.937286i	$0.386670\pi$
$80$		0			0
$81$	−5.00000			−0.555556
$82$		0			0
$83$		−	4.52004i		−	0.496139i	−0.968742	−	0.248070i	$-0.920204\pi$
$83$		−	4.52004i		−	0.496139i	0.968742	−	0.248070i	$-0.0797961\pi$
$84$		0			0
$85$	2.00000	+	2.44949i	0.216930	+	0.265684i
$86$		0			0
$87$		−	1.41421i		−	0.151620i
$88$		0			0
$89$	10.3923			1.10158			0.550791	−	0.834643i	$-0.314326\pi$
$89$	10.3923			1.10158			0.550791	+	0.834643i	$0.314326\pi$
$90$		0			0
$91$	16.3923			1.71838
$92$		0			0
$93$			7.45001i			0.772530i
$94$		0			0
$95$	−11.6603	+	9.52056i	−1.19632	+	0.976789i
$96$		0			0
$97$		−	6.03579i		−	0.612842i	−0.951896	−	0.306421i	$-0.900869\pi$
$97$		−	6.03579i		−	0.612842i	0.951896	−	0.306421i	$-0.0991315\pi$
$98$		0			0
$99$	4.73205			0.475589
$100$		0			0
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$		0			0
$103$		−	0.656339i		−	0.0646710i	−0.999477	−	0.0323355i	$-0.989705\pi$
$103$		−	0.656339i		−	0.0646710i	0.999477	−	0.0323355i	$-0.0102945\pi$
$104$		0			0
$105$	6.00000	−	4.89898i	0.585540	−	0.478091i
$106$		0			0
$107$			3.96524i			0.383334i	0.981460	+	0.191667i	$0.0613894\pi$
$107$			3.96524i			0.383334i	−0.981460	+	0.191667i	$0.938611\pi$
$108$		0			0
$109$	9.85641			0.944073			0.472036	−	0.881579i	$-0.343519\pi$
$109$	9.85641			0.944073			0.472036	+	0.881579i	$0.343519\pi$
$110$		0			0
$111$	0.928203			0.0881012
$112$		0			0
$113$		−	1.69161i		−	0.159134i	−0.996830	−	0.0795669i	$-0.974646\pi$
$113$		−	1.69161i		−	0.159134i	0.996830	−	0.0795669i	$-0.0253537\pi$
$114$		0			0
$115$	4.92820	+	6.03579i	0.459557	+	0.562840i
$116$		0			0
$117$			6.69213i			0.618688i
$118$		0			0
$119$	3.46410			0.317554
$120$		0			0
$121$	11.3923			1.03566
$122$		0			0
$123$			9.79796i			0.883452i
$124$		0			0
$125$	−5.19615	−	9.89949i	−0.464758	−	0.885438i
$126$		0			0
$127$		−	4.24264i		−	0.376473i	−0.982124	−	0.188237i	$-0.939723\pi$
$127$		−	4.24264i		−	0.376473i	0.982124	−	0.188237i	$-0.0602772\pi$
$128$		0			0
$129$	0.928203			0.0817237
$130$		0			0
$131$	1.26795			0.110781			0.0553906	−	0.998465i	$-0.482360\pi$
$131$	1.26795			0.110781			0.0553906	+	0.998465i	$0.482360\pi$
$132$		0			0
$133$			16.4901i			1.42987i
$134$		0			0
$135$	8.00000	+	9.79796i	0.688530	+	0.843274i
$136$		0			0
$137$			5.00052i			0.427223i	0.976919	+	0.213611i	$0.0685226\pi$
$137$			5.00052i			0.427223i	−0.976919	+	0.213611i	$0.931477\pi$
$138$		0			0
$139$	−8.00000			−0.678551			−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000			−0.678551			−0.339276	+	0.940687i	$0.610182\pi$
$140$		0			0
$141$	2.00000			0.168430
$142$		0			0
$143$			31.6675i			2.64817i
$144$		0			0
$145$	−1.73205	+	1.41421i	−0.143839	+	0.117444i
$146$		0			0
$147$			1.41421i			0.116642i
$148$		0			0
$149$	−12.9282			−1.05912			−0.529560	−	0.848273i	$-0.677643\pi$
$149$	−12.9282			−1.05912			−0.529560	+	0.848273i	$0.677643\pi$
$150$		0			0
$151$	−22.7846			−1.85419			−0.927093	−	0.374832i	$-0.877700\pi$
$151$	−22.7846			−1.85419			−0.927093	+	0.374832i	$0.877700\pi$
$152$		0			0
$153$			1.41421i			0.114332i
$154$		0			0
$155$	9.12436	−	7.45001i	0.732886	−	0.598399i
$156$		0			0
$157$			6.03579i			0.481709i	0.970561	+	0.240854i	$0.0774276\pi$
$157$			6.03579i			0.481709i	−0.970561	+	0.240854i	$0.922572\pi$
$158$		0			0
$159$	−12.3923			−0.982774
$160$		0			0
$161$	8.53590			0.672723
$162$		0			0
$163$		−	5.55532i		−	0.435126i	−0.976046	−	0.217563i	$-0.930189\pi$
$163$		−	5.55532i		−	0.435126i	0.976046	−	0.217563i	$-0.0698108\pi$
$164$		0			0
$165$	9.46410	+	11.5911i	0.736779	+	0.902367i
$166$		0			0
$167$			8.38375i			0.648754i	0.945928	+	0.324377i	$0.105155\pi$
$167$			8.38375i			0.648754i	−0.945928	+	0.324377i	$0.894845\pi$
$168$		0			0
$169$	−31.7846			−2.44497
$170$		0			0
$171$	−6.73205			−0.514813
$172$		0			0
$173$		−	9.52056i		−	0.723835i	−0.932210	−	0.361917i	$-0.882122\pi$
$173$		−	9.52056i		−	0.723835i	0.932210	−	0.361917i	$-0.117878\pi$
$174$		0			0
$175$	−12.0000	−	2.44949i	−0.907115	−	0.185164i
$176$		0			0
$177$			14.6969i			1.10469i
$178$		0			0
$179$	9.46410			0.707380			0.353690	−	0.935363i	$-0.384927\pi$
$179$	9.46410			0.707380			0.353690	+	0.935363i	$0.384927\pi$
$180$		0			0
$181$	−16.0000			−1.18927			−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000			−1.18927			−0.594635	+	0.803996i	$0.702704\pi$
$182$		0			0
$183$		−	15.4548i		−	1.14245i
$184$		0			0
$185$	−0.928203	−	1.13681i	−0.0682429	−	0.0835801i
$186$		0			0
$187$			6.69213i			0.489377i
$188$		0			0
$189$	13.8564			1.00791
$190$		0			0
$191$	9.12436			0.660215			0.330108	−	0.943943i	$-0.392915\pi$
$191$	9.12436			0.660215			0.330108	+	0.943943i	$0.392915\pi$
$192$		0			0
$193$		−	9.14162i		−	0.658028i	−0.944325	−	0.329014i	$-0.893284\pi$
$193$		−	9.14162i		−	0.658028i	0.944325	−	0.329014i	$-0.106716\pi$
$194$		0			0
$195$	−16.3923	+	13.3843i	−1.17388	+	0.958467i
$196$		0			0
$197$			16.7675i			1.19463i	0.802005	+	0.597317i	$0.203767\pi$
$197$			16.7675i			1.19463i	−0.802005	+	0.597317i	$0.796233\pi$
$198$		0			0
$199$	−14.0000			−0.992434			−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000			−0.992434			−0.496217	+	0.868199i	$0.665278\pi$
$200$		0			0
$201$	6.00000			0.423207
$202$		0			0
$203$			2.44949i			0.171920i
$204$		0			0
$205$	12.0000	−	9.79796i	0.838116	−	0.684319i
$206$		0			0
$207$			3.48477i			0.242208i
$208$		0			0
$209$	−31.8564			−2.20355
$210$		0			0
$211$	11.2679			0.775718			0.387859	−	0.921719i	$-0.373215\pi$
$211$	11.2679			0.775718			0.387859	+	0.921719i	$0.373215\pi$
$212$		0			0
$213$			4.89898i			0.335673i
$214$		0			0
$215$	−0.928203	−	1.13681i	−0.0633029	−	0.0775299i
$216$		0			0
$217$		−	12.9038i		−	0.875966i
$218$		0			0
$219$	10.3923			0.702247
$220$		0			0
$221$	−9.46410			−0.636624
$222$		0			0
$223$			23.8386i			1.59635i	0.602427	+	0.798174i	$0.294200\pi$
$223$			23.8386i			1.59635i	−0.602427	+	0.798174i	$0.705800\pi$
$224$		0			0
$225$	1.00000	−	4.89898i	0.0666667	−	0.326599i
$226$		0			0
$227$			10.6574i			0.707354i	0.935368	+	0.353677i	$0.115069\pi$
$227$			10.6574i			0.707354i	−0.935368	+	0.353677i	$0.884931\pi$
$228$		0			0
$229$	22.7846			1.50565			0.752825	−	0.658221i	$-0.228691\pi$
$229$	22.7846			1.50565			0.752825	+	0.658221i	$0.228691\pi$
$230$		0			0
$231$	16.3923			1.07853
$232$		0			0
$233$		−	26.0106i		−	1.70401i	−0.523530	−	0.852007i	$-0.675385\pi$
$233$		−	26.0106i		−	1.70401i	0.523530	−	0.852007i	$-0.324615\pi$
$234$		0			0
$235$	−2.00000	−	2.44949i	−0.130466	−	0.159787i
$236$		0			0
$237$			8.76268i			0.569197i
$238$		0			0
$239$	−10.3923			−0.672222			−0.336111	−	0.941822i	$-0.609112\pi$
$239$	−10.3923			−0.672222			−0.336111	+	0.941822i	$0.609112\pi$
$240$		0			0
$241$	4.53590			0.292183			0.146091	−	0.989271i	$-0.453331\pi$
$241$	4.53590			0.292183			0.146091	+	0.989271i	$0.453331\pi$
$242$		0			0
$243$			9.89949i			0.635053i
$244$		0			0
$245$	1.73205	−	1.41421i	0.110657	−	0.0903508i
$246$		0			0
$247$		−	45.0518i		−	2.86657i
$248$		0			0
$249$	6.39230			0.405096
$250$		0			0
$251$	9.12436			0.575924			0.287962	−	0.957642i	$-0.407022\pi$
$251$	9.12436			0.575924			0.287962	+	0.957642i	$0.407022\pi$
$252$		0			0
$253$			16.4901i			1.03672i
$254$		0			0
$255$	−3.46410	+	2.82843i	−0.216930	+	0.177123i
$256$		0			0
$257$		−	4.62158i		−	0.288286i	−0.989557	−	0.144143i	$-0.953957\pi$
$257$		−	4.62158i		−	0.288286i	0.989557	−	0.144143i	$-0.0460425\pi$
$258$		0			0
$259$	−1.60770			−0.0998973
$260$		0			0
$261$	−1.00000			−0.0618984
$262$		0			0
$263$		−	24.5964i		−	1.51668i	−0.651859	−	0.758341i	$-0.726010\pi$
$263$		−	24.5964i		−	1.51668i	0.651859	−	0.758341i	$-0.273990\pi$
$264$		0			0
$265$	12.3923	+	15.1774i	0.761253	+	0.932341i
$266$		0			0
$267$			14.6969i			0.899438i
$268$		0			0
$269$	12.0000			0.731653			0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000			0.731653			0.365826	+	0.930683i	$0.380786\pi$
$270$		0			0
$271$	23.2679			1.41343			0.706714	−	0.707500i	$-0.250177\pi$
$271$	23.2679			1.41343			0.706714	+	0.707500i	$0.250177\pi$
$272$		0			0
$273$			23.1822i			1.40305i
$274$		0			0
$275$	4.73205	−	23.1822i	0.285353	−	1.39794i
$276$		0			0
$277$			23.6627i			1.42175i	0.703317	+	0.710877i	$0.251702\pi$
$277$			23.6627i			1.42175i	−0.703317	+	0.710877i	$0.748298\pi$
$278$		0			0
$279$	5.26795			0.315384
$280$		0			0
$281$	−8.78461			−0.524046			−0.262023	−	0.965062i	$-0.584390\pi$
$281$	−8.78461			−0.524046			−0.262023	+	0.965062i	$0.584390\pi$
$282$		0			0
$283$			1.96902i			0.117046i	0.998286	+	0.0585229i	$0.0186391\pi$
$283$			1.96902i			0.117046i	−0.998286	+	0.0585229i	$0.981361\pi$
$284$		0			0
$285$	−13.4641	−	16.4901i	−0.797545	−	0.976789i
$286$		0			0
$287$		−	16.9706i		−	1.00174i
$288$		0			0
$289$	15.0000			0.882353
$290$		0			0
$291$	8.53590			0.500383
$292$		0			0
$293$		−	25.8348i		−	1.50928i	−0.656137	−	0.754642i	$-0.727811\pi$
$293$		−	25.8348i		−	1.50928i	0.656137	−	0.754642i	$-0.272189\pi$
$294$		0			0
$295$	18.0000	−	14.6969i	1.04800	−	0.855689i
$296$		0			0
$297$			26.7685i			1.55327i
$298$		0			0
$299$	−23.3205			−1.34866
$300$		0			0
$301$	−1.60770			−0.0926660
$302$		0			0
$303$		−	8.48528i		−	0.487467i
$304$		0			0
$305$	−18.9282	+	15.4548i	−1.08383	+	0.884940i
$306$		0			0
$307$			18.9396i			1.08094i	0.841364	+	0.540469i	$0.181753\pi$
$307$			18.9396i			1.08094i	−0.841364	+	0.540469i	$0.818247\pi$
$308$		0			0
$309$	0.928203			0.0528036
$310$		0			0
$311$	−24.5885			−1.39428			−0.697142	−	0.716933i	$-0.745545\pi$
$311$	−24.5885			−1.39428			−0.697142	+	0.716933i	$0.745545\pi$
$312$		0			0
$313$			18.7637i			1.06059i	0.847814	+	0.530294i	$0.177918\pi$
$313$			18.7637i			1.06059i	−0.847814	+	0.530294i	$0.822082\pi$
$314$		0			0
$315$	−3.46410	−	4.24264i	−0.195180	−	0.239046i
$316$		0			0
$317$		−	25.3543i		−	1.42404i	−0.702159	−	0.712020i	$-0.747781\pi$
$317$		−	25.3543i		−	1.42404i	0.702159	−	0.712020i	$-0.252219\pi$
$318$		0			0
$319$	−4.73205			−0.264944
$320$		0			0
$321$	−5.60770			−0.312991
$322$		0			0
$323$		−	9.52056i		−	0.529738i
$324$		0			0
$325$	32.7846	+	6.69213i	1.81856	+	0.371213i
$326$		0			0
$327$			13.9391i			0.770832i
$328$		0			0
$329$	−3.46410			−0.190982
$330$		0			0
$331$	7.80385			0.428938			0.214469	−	0.976731i	$-0.431198\pi$
$331$	7.80385			0.428938			0.214469	+	0.976731i	$0.431198\pi$
$332$		0			0
$333$		−	0.656339i		−	0.0359671i
$334$		0			0
$335$	−6.00000	−	7.34847i	−0.327815	−	0.401490i
$336$		0			0
$337$		−	23.0064i		−	1.25324i	−0.779327	−	0.626618i	$-0.784439\pi$
$337$		−	23.0064i		−	1.25324i	0.779327	−	0.626618i	$-0.215561\pi$
$338$		0			0
$339$	2.39230			0.129932
$340$		0			0
$341$	24.9282			1.34994
$342$		0			0
$343$		−	19.5959i		−	1.05808i
$344$		0			0
$345$	−8.53590	+	6.96953i	−0.459557	+	0.375227i
$346$		0			0
$347$		−	6.79367i		−	0.364703i	−0.983233	−	0.182352i	$-0.941629\pi$
$347$		−	6.79367i		−	0.364703i	0.983233	−	0.182352i	$-0.0583709\pi$
$348$		0			0
$349$	−18.7846			−1.00552			−0.502759	−	0.864427i	$-0.667682\pi$
$349$	−18.7846			−1.00552			−0.502759	+	0.864427i	$0.667682\pi$
$350$		0			0
$351$	−37.8564			−2.02063
$352$		0			0
$353$		−	12.1459i		−	0.646462i	−0.946320	−	0.323231i	$-0.895231\pi$
$353$		−	12.1459i		−	0.646462i	0.946320	−	0.323231i	$-0.104769\pi$
$354$		0			0
$355$	6.00000	−	4.89898i	0.318447	−	0.260011i
$356$		0			0
$357$			4.89898i			0.259281i
$358$		0			0
$359$	−10.7321			−0.566416			−0.283208	−	0.959059i	$-0.591399\pi$
$359$	−10.7321			−0.566416			−0.283208	+	0.959059i	$0.591399\pi$
$360$		0			0
$361$	26.3205			1.38529
$362$		0			0
$363$			16.1112i			0.845616i
$364$		0			0
$365$	−10.3923	−	12.7279i	−0.543958	−	0.666210i
$366$		0			0
$367$		−	25.1512i		−	1.31288i	−0.754377	−	0.656442i	$-0.772061\pi$
$367$		−	25.1512i		−	1.31288i	0.754377	−	0.656442i	$-0.227939\pi$
$368$		0			0
$369$	6.92820			0.360668
$370$		0			0
$371$	21.4641			1.11436
$372$		0			0
$373$		−	19.5959i		−	1.01464i	−0.861758	−	0.507319i	$-0.830637\pi$
$373$		−	19.5959i		−	1.01464i	0.861758	−	0.507319i	$-0.169363\pi$
$374$		0			0
$375$	14.0000	−	7.34847i	0.722957	−	0.379473i
$376$		0			0
$377$		−	6.69213i		−	0.344662i
$378$		0			0
$379$	−5.80385			−0.298124			−0.149062	−	0.988828i	$-0.547625\pi$
$379$	−5.80385			−0.298124			−0.149062	+	0.988828i	$0.547625\pi$
$380$		0			0
$381$	6.00000			0.307389
$382$		0			0
$383$			23.5612i			1.20392i	0.798527	+	0.601959i	$0.205613\pi$
$383$			23.5612i			1.20392i	−0.798527	+	0.601959i	$0.794387\pi$
$384$		0			0
$385$	−16.3923	−	20.0764i	−0.835429	−	1.02319i
$386$		0			0
$387$		−	0.656339i		−	0.0333636i
$388$		0			0
$389$	−20.7846			−1.05382			−0.526911	−	0.849921i	$-0.676650\pi$
$389$	−20.7846			−1.05382			−0.526911	+	0.849921i	$0.676650\pi$
$390$		0			0
$391$	−4.92820			−0.249230
$392$		0			0
$393$			1.79315i			0.0904525i
$394$		0			0
$395$	10.7321	−	8.76268i	0.539988	−	0.440898i
$396$		0			0
$397$		−	4.89898i		−	0.245873i	−0.992415	−	0.122936i	$-0.960769\pi$
$397$		−	4.89898i		−	0.245873i	0.992415	−	0.122936i	$-0.0392311\pi$
$398$		0			0
$399$	−23.3205			−1.16749
$400$		0			0
$401$	10.1436			0.506547			0.253273	−	0.967395i	$-0.418493\pi$
$401$	10.1436			0.506547			0.253273	+	0.967395i	$0.418493\pi$
$402$		0			0
$403$			35.2538i			1.75612i
$404$		0			0
$405$	−8.66025	+	7.07107i	−0.430331	+	0.351364i
$406$		0			0
$407$		−	3.10583i		−	0.153950i
$408$		0			0
$409$	−22.9282			−1.13373			−0.566863	−	0.823812i	$-0.691843\pi$
$409$	−22.9282			−1.13373			−0.566863	+	0.823812i	$0.691843\pi$
$410$		0			0
$411$	−7.07180			−0.348826
$412$		0			0
$413$		−	25.4558i		−	1.25260i
$414$		0			0
$415$	−6.39230	−	7.82894i	−0.313786	−	0.384308i
$416$		0			0
$417$		−	11.3137i		−	0.554035i
$418$		0			0
$419$	−6.00000			−0.293119			−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000			−0.293119			−0.146560	+	0.989202i	$0.546820\pi$
$420$		0			0
$421$	−1.46410			−0.0713559			−0.0356780	−	0.999363i	$-0.511359\pi$
$421$	−1.46410			−0.0713559			−0.0356780	+	0.999363i	$0.511359\pi$
$422$		0			0
$423$		−	1.41421i		−	0.0687614i
$424$		0			0
$425$	6.92820	+	1.41421i	0.336067	+	0.0685994i
$426$		0			0
$427$			26.7685i			1.29542i
$428$		0			0
$429$	−44.7846			−2.16222
$430$		0			0
$431$	−25.1769			−1.21273			−0.606365	−	0.795187i	$-0.707373\pi$
$431$	−25.1769			−1.21273			−0.606365	+	0.795187i	$0.707373\pi$
$432$		0			0
$433$			22.5259i			1.08252i	0.840854	+	0.541262i	$0.182053\pi$
$433$			22.5259i			1.08252i	−0.840854	+	0.541262i	$0.817947\pi$
$434$		0			0
$435$	−2.00000	−	2.44949i	−0.0958927	−	0.117444i
$436$		0			0
$437$		−	23.4596i		−	1.12223i
$438$		0			0
$439$	−20.9282			−0.998849			−0.499424	−	0.866358i	$-0.666455\pi$
$439$	−20.9282			−0.998849			−0.499424	+	0.866358i	$0.666455\pi$
$440$		0			0
$441$	1.00000			0.0476190
$442$		0			0
$443$		−	21.0101i		−	0.998221i	−0.866538	−	0.499111i	$-0.833660\pi$
$443$		−	21.0101i		−	0.998221i	0.866538	−	0.499111i	$-0.166340\pi$
$444$		0			0
$445$	18.0000	−	14.6969i	0.853282	−	0.696702i
$446$		0			0
$447$		−	18.2832i		−	0.864768i
$448$		0			0
$449$	−1.85641			−0.0876092			−0.0438046	−	0.999040i	$-0.513948\pi$
$449$	−1.85641			−0.0876092			−0.0438046	+	0.999040i	$0.513948\pi$
$450$		0			0
$451$	32.7846			1.54377
$452$		0			0
$453$		−	32.2223i		−	1.51394i
$454$		0			0
$455$	28.3923	−	23.1822i	1.33105	−	1.08680i
$456$		0			0
$457$		−	18.7637i		−	0.877730i	−0.898553	−	0.438865i	$-0.855381\pi$
$457$		−	18.7637i		−	0.877730i	0.898553	−	0.438865i	$-0.144619\pi$
$458$		0			0
$459$	−8.00000			−0.373408
$460$		0			0
$461$	11.0718			0.515665			0.257832	−	0.966190i	$-0.416992\pi$
$461$	11.0718			0.515665			0.257832	+	0.966190i	$0.416992\pi$
$462$		0			0
$463$		−	32.3238i		−	1.50222i	−0.660179	−	0.751108i	$-0.729520\pi$
$463$		−	32.3238i		−	1.50222i	0.660179	−	0.751108i	$-0.270480\pi$
$464$		0			0
$465$	10.5359	+	12.9038i	0.488591	+	0.598399i
$466$		0			0
$467$			18.1817i			0.841349i	0.907212	+	0.420674i	$0.138207\pi$
$467$			18.1817i			0.841349i	−0.907212	+	0.420674i	$0.861793\pi$
$468$		0			0
$469$	−10.3923			−0.479872
$470$		0			0
$471$	−8.53590			−0.393313
$472$		0			0
$473$		−	3.10583i		−	0.142806i
$474$		0			0
$475$	−6.73205	+	32.9802i	−0.308888	+	1.51323i
$476$		0			0
$477$			8.76268i			0.401216i
$478$		0			0
$479$	−11.6603			−0.532771			−0.266385	−	0.963867i	$-0.585829\pi$
$479$	−11.6603			−0.532771			−0.266385	+	0.963867i	$0.585829\pi$
$480$		0			0
$481$	4.39230			0.200272
$482$		0			0
$483$			12.0716i			0.549276i
$484$		0			0
$485$	−8.53590	−	10.4543i	−0.387595	−	0.474705i
$486$		0			0
$487$			6.03579i			0.273508i	0.990605	+	0.136754i	$0.0436670\pi$
$487$			6.03579i			0.273508i	−0.990605	+	0.136754i	$0.956333\pi$
$488$		0			0
$489$	7.85641			0.355279
$490$		0			0
$491$	−10.7321			−0.484331			−0.242165	−	0.970235i	$-0.577858\pi$
$491$	−10.7321			−0.484331			−0.242165	+	0.970235i	$0.577858\pi$
$492$		0			0
$493$		−	1.41421i		−	0.0636930i
$494$		0			0
$495$	8.19615	−	6.69213i	0.368390	−	0.300789i
$496$		0			0
$497$		−	8.48528i		−	0.380617i
$498$		0			0
$499$	−14.9282			−0.668278			−0.334139	−	0.942524i	$-0.608446\pi$
$499$	−14.9282			−0.668278			−0.334139	+	0.942524i	$0.608446\pi$
$500$		0			0
$501$	−11.8564			−0.529705
$502$		0			0
$503$		−	41.5670i		−	1.85338i	−0.375826	−	0.926690i	$-0.622641\pi$
$503$		−	41.5670i		−	1.85338i	0.375826	−	0.926690i	$-0.377359\pi$
$504$		0			0
$505$	−10.3923	+	8.48528i	−0.462451	+	0.377590i
$506$		0			0
$507$		−	44.9502i		−	1.99631i
$508$		0			0
$509$	−26.5359			−1.17618			−0.588092	−	0.808794i	$-0.700121\pi$
$509$	−26.5359			−1.17618			−0.588092	+	0.808794i	$0.700121\pi$
$510$		0			0
$511$	−18.0000			−0.796273
$512$		0			0
$513$		−	38.0822i		−	1.68137i
$514$		0			0
$515$	−0.928203	−	1.13681i	−0.0409015	−	0.0500939i
$516$		0			0
$517$		−	6.69213i		−	0.294320i
$518$		0			0
$519$	13.4641			0.591008
$520$		0			0
$521$	−16.3923			−0.718160			−0.359080	−	0.933307i	$-0.616909\pi$
$521$	−16.3923			−0.718160			−0.359080	+	0.933307i	$0.616909\pi$
$522$		0			0
$523$			40.8091i			1.78446i	0.451583	+	0.892229i	$0.350860\pi$
$523$			40.8091i			1.78446i	−0.451583	+	0.892229i	$0.649140\pi$
$524$		0			0
$525$	3.46410	−	16.9706i	0.151186	−	0.740656i
$526$		0			0
$527$			7.45001i			0.324527i
$528$		0			0
$529$	10.8564			0.472018
$530$		0			0
$531$	10.3923			0.450988
$532$		0			0
$533$			46.3644i			2.00827i
$534$		0			0
$535$	5.60770	+	6.86800i	0.242442	+	0.296929i
$536$		0			0
$537$			13.3843i			0.577573i
$538$		0			0
$539$	4.73205			0.203824
$540$		0			0
$541$	21.6077			0.928987			0.464494	−	0.885576i	$-0.346236\pi$
$541$	21.6077			0.928987			0.464494	+	0.885576i	$0.346236\pi$
$542$		0			0
$543$		−	22.6274i		−	0.971035i
$544$		0			0
$545$	17.0718	−	13.9391i	0.731275	−	0.597084i
$546$		0			0
$547$		−	2.44949i		−	0.104733i	−0.998628	−	0.0523663i	$-0.983324\pi$
$547$		−	2.44949i		−	0.104733i	0.998628	−	0.0523663i	$-0.0166763\pi$
$548$		0			0
$549$	−10.9282			−0.466404
$550$		0			0
$551$	6.73205			0.286795
$552$		0			0
$553$		−	15.1774i		−	0.645409i
$554$		0			0
$555$	1.60770	−	1.31268i	0.0682429	−	0.0557201i
$556$		0			0
$557$			28.8391i			1.22195i	0.791650	+	0.610975i	$0.209223\pi$
$557$			28.8391i			1.22195i	−0.791650	+	0.610975i	$0.790777\pi$
$558$		0			0
$559$	4.39230			0.185775
$560$		0			0
$561$	−9.46410			−0.399575
$562$		0			0
$563$		−	16.1112i		−	0.679004i	−0.940605	−	0.339502i	$-0.889741\pi$
$563$		−	16.1112i		−	0.679004i	0.940605	−	0.339502i	$-0.110259\pi$
$564$		0			0
$565$	−2.39230	−	2.92996i	−0.100645	−	0.123264i
$566$		0			0
$567$			12.2474i			0.514344i
$568$		0			0
$569$	−14.5359			−0.609377			−0.304688	−	0.952452i	$-0.598552\pi$
$569$	−14.5359			−0.609377			−0.304688	+	0.952452i	$0.598552\pi$
$570$		0			0
$571$	4.67949			0.195831			0.0979153	−	0.995195i	$-0.468783\pi$
$571$	4.67949			0.195831			0.0979153	+	0.995195i	$0.468783\pi$
$572$		0			0
$573$			12.9038i			0.539063i
$574$		0			0
$575$	17.0718	+	3.48477i	0.711943	+	0.145325i
$576$		0			0
$577$		−	0.656339i		−	0.0273237i	−0.999907	−	0.0136619i	$-0.995651\pi$
$577$		−	0.656339i		−	0.0273237i	0.999907	−	0.0136619i	$-0.00434884\pi$
$578$		0			0
$579$	12.9282			0.537278
$580$		0			0
$581$	−11.0718			−0.459336
$582$		0			0
$583$			41.4655i			1.71732i
$584$		0			0
$585$	9.46410	+	11.5911i	0.391292	+	0.479233i
$586$		0			0
$587$			15.5563i			0.642079i	0.947066	+	0.321040i	$0.104032\pi$
$587$			15.5563i			0.642079i	−0.947066	+	0.321040i	$0.895968\pi$
$588$		0			0
$589$	−35.4641			−1.46127
$590$		0			0
$591$	−23.7128			−0.975415
$592$		0			0
$593$			28.3586i			1.16455i	0.812993	+	0.582274i	$0.197837\pi$
$593$			28.3586i			1.16455i	−0.812993	+	0.582274i	$0.802163\pi$
$594$		0			0
$595$	6.00000	−	4.89898i	0.245976	−	0.200839i
$596$		0			0
$597$		−	19.7990i		−	0.810319i
$598$		0			0
$599$	4.73205			0.193346			0.0966732	−	0.995316i	$-0.469180\pi$
$599$	4.73205			0.193346			0.0966732	+	0.995316i	$0.469180\pi$
$600$		0			0
$601$	−24.7846			−1.01099			−0.505493	−	0.862831i	$-0.668689\pi$
$601$	−24.7846			−1.01099			−0.505493	+	0.862831i	$0.668689\pi$
$602$		0			0
$603$		−	4.24264i		−	0.172774i
$604$		0			0
$605$	19.7321	−	16.1112i	0.802222	−	0.655011i
$606$		0			0
$607$		−	26.1122i		−	1.05986i	−0.848041	−	0.529930i	$-0.822218\pi$
$607$		−	26.1122i		−	1.05986i	0.848041	−	0.529930i	$-0.177782\pi$
$608$		0			0
$609$	−3.46410			−0.140372
$610$		0			0
$611$	9.46410			0.382877
$612$		0			0
$613$			1.31268i			0.0530185i	0.999649	+	0.0265093i	$0.00843915\pi$
$613$			1.31268i			0.0530185i	−0.999649	+	0.0265093i	$0.991561\pi$
$614$		0			0
$615$	13.8564	+	16.9706i	0.558744	+	0.684319i
$616$		0			0
$617$		−	7.55154i		−	0.304014i	−0.988379	−	0.152007i	$-0.951426\pi$
$617$		−	7.55154i		−	0.304014i	0.988379	−	0.152007i	$-0.0485736\pi$
$618$		0			0
$619$	−34.1962			−1.37446			−0.687230	−	0.726440i	$-0.741173\pi$
$619$	−34.1962			−1.37446			−0.687230	+	0.726440i	$0.741173\pi$
$620$		0			0
$621$	−19.7128			−0.791048
$622$		0			0
$623$		−	25.4558i		−	1.01987i
$624$		0			0
$625$	−23.0000	−	9.79796i	−0.920000	−	0.391918i
$626$		0			0
$627$		−	45.0518i		−	1.79919i
$628$		0			0
$629$	0.928203			0.0370099
$630$		0			0
$631$	−36.3923			−1.44875			−0.724377	−	0.689404i	$-0.757873\pi$
$631$	−36.3923			−1.44875			−0.724377	+	0.689404i	$0.757873\pi$
$632$		0			0
$633$			15.9353i			0.633371i
$634$		0			0
$635$	−6.00000	−	7.34847i	−0.238103	−	0.291615i
$636$		0			0
$637$			6.69213i			0.265152i
$638$		0			0
$639$	3.46410			0.137038
$640$		0			0
$641$	−34.6410			−1.36824			−0.684119	−	0.729370i	$-0.739813\pi$
$641$	−34.6410			−1.36824			−0.684119	+	0.729370i	$0.739813\pi$
$642$		0			0
$643$			18.9396i			0.746904i	0.927649	+	0.373452i	$0.121826\pi$
$643$			18.9396i			0.746904i	−0.927649	+	0.373452i	$0.878174\pi$
$644$		0			0
$645$	1.60770	−	1.31268i	0.0633029	−	0.0516866i
$646$		0			0
$647$			19.9749i			0.785293i	0.919690	+	0.392646i	$0.128440\pi$
$647$			19.9749i			0.785293i	−0.919690	+	0.392646i	$0.871560\pi$
$648$		0			0
$649$	49.1769			1.93036
$650$		0			0
$651$	18.2487			0.715223
$652$		0			0
$653$		−	1.69161i		−	0.0661980i	−0.999452	−	0.0330990i	$-0.989462\pi$
$653$		−	1.69161i		−	0.0661980i	0.999452	−	0.0330990i	$-0.0105377\pi$
$654$		0			0
$655$	2.19615	−	1.79315i	0.0858108	−	0.0700642i
$656$		0			0
$657$		−	7.34847i		−	0.286691i
$658$		0			0
$659$	−26.4449			−1.03015			−0.515073	−	0.857146i	$-0.672235\pi$
$659$	−26.4449			−1.03015			−0.515073	+	0.857146i	$0.672235\pi$
$660$		0			0
$661$	40.7846			1.58634			0.793169	−	0.609002i	$-0.208430\pi$
$661$	40.7846			1.58634			0.793169	+	0.609002i	$0.208430\pi$
$662$		0			0
$663$		−	13.3843i		−	0.519802i
$664$		0			0
$665$	23.3205	+	28.5617i	0.904331	+	1.10757i
$666$		0			0
$667$		−	3.48477i		−	0.134931i
$668$		0			0
$669$	−33.7128			−1.30341
$670$		0			0
$671$	−51.7128			−1.99635
$672$		0			0
$673$			12.4233i			0.478884i	0.970911	+	0.239442i	$0.0769644\pi$
$673$			12.4233i			0.478884i	−0.970911	+	0.239442i	$0.923036\pi$
$674$		0			0
$675$	27.7128	+	5.65685i	1.06667	+	0.217732i
$676$		0			0
$677$			46.9464i			1.80430i	0.431424	+	0.902149i	$0.358011\pi$
$677$			46.9464i			1.80430i	−0.431424	+	0.902149i	$0.641989\pi$
$678$		0			0
$679$	−14.7846			−0.567381
$680$		0			0
$681$	−15.0718			−0.577553
$682$		0			0
$683$		−	31.7690i		−	1.21561i	−0.794087	−	0.607804i	$-0.792051\pi$
$683$		−	31.7690i		−	1.21561i	0.794087	−	0.607804i	$-0.207949\pi$
$684$		0			0
$685$	7.07180	+	8.66115i	0.270199	+	0.330925i
$686$		0			0
$687$			32.2223i			1.22936i
$688$		0			0
$689$	−58.6410			−2.23404
$690$		0			0
$691$	13.7128			0.521660			0.260830	−	0.965385i	$-0.416004\pi$
$691$	13.7128			0.521660			0.260830	+	0.965385i	$0.416004\pi$
$692$		0			0
$693$		−	11.5911i		−	0.440310i
$694$		0			0
$695$	−13.8564	+	11.3137i	−0.525603	+	0.429153i
$696$		0			0
$697$			9.79796i			0.371124i
$698$		0			0
$699$	36.7846			1.39132
$700$		0			0
$701$	−27.7128			−1.04670			−0.523349	−	0.852118i	$-0.675318\pi$
$701$	−27.7128			−1.04670			−0.523349	+	0.852118i	$0.675318\pi$
$702$		0			0
$703$			4.41851i			0.166647i
$704$		0			0
$705$	3.46410	−	2.82843i	0.130466	−	0.106525i
$706$		0			0
$707$			14.6969i			0.552735i
$708$		0			0
$709$	0.392305			0.0147333			0.00736666	−	0.999973i	$-0.497655\pi$
$709$	0.392305			0.0147333			0.00736666	+	0.999973i	$0.497655\pi$
$710$		0			0
$711$	6.19615			0.232374
$712$		0			0
$713$			18.3576i			0.687496i
$714$		0			0
$715$	44.7846	+	54.8497i	1.67485	+	2.05126i
$716$		0			0
$717$		−	14.6969i		−	0.548867i
$718$		0			0
$719$	7.85641			0.292995			0.146497	−	0.989211i	$-0.453200\pi$
$719$	7.85641			0.292995			0.146497	+	0.989211i	$0.453200\pi$
$720$		0			0
$721$	−1.60770			−0.0598737
$722$		0			0
$723$			6.41473i			0.238566i
$724$		0			0
$725$	−1.00000	+	4.89898i	−0.0371391	+	0.181944i
$726$		0			0
$727$			46.6690i			1.73086i	0.501031	+	0.865430i	$0.332954\pi$
$727$			46.6690i			1.73086i	−0.501031	+	0.865430i	$0.667046\pi$
$728$		0			0
$729$	−29.0000			−1.07407
$730$		0			0
$731$	0.928203			0.0343308
$732$		0			0
$733$		−	12.7279i		−	0.470117i	−0.971981	−	0.235058i	$-0.924472\pi$
$733$		−	12.7279i		−	0.470117i	0.971981	−	0.235058i	$-0.0755281\pi$
$734$		0			0
$735$	2.00000	+	2.44949i	0.0737711	+	0.0903508i
$736$		0			0
$737$		−	20.0764i		−	0.739523i
$738$		0			0
$739$	32.9808			1.21322			0.606608	−	0.795001i	$-0.292530\pi$
$739$	32.9808			1.21322			0.606608	+	0.795001i	$0.292530\pi$
$740$		0			0
$741$	63.7128			2.34055
$742$		0			0
$743$		−	9.89949i		−	0.363177i	−0.983375	−	0.181589i	$-0.941876\pi$
$743$		−	9.89949i		−	0.363177i	0.983375	−	0.181589i	$-0.0581239\pi$
$744$		0			0
$745$	−22.3923	+	18.2832i	−0.820391	+	0.669846i
$746$		0			0
$747$		−	4.52004i		−	0.165380i
$748$		0			0
$749$	9.71281			0.354898
$750$		0			0
$751$	15.9090			0.580526			0.290263	−	0.956947i	$-0.406257\pi$
$751$	15.9090			0.580526			0.290263	+	0.956947i	$0.406257\pi$
$752$		0			0
$753$			12.9038i			0.470240i
$754$		0			0
$755$	−39.4641	+	32.2223i	−1.43625	+	1.17269i
$756$		0			0
$757$			6.03579i			0.219375i	0.993966	+	0.109687i	$0.0349849\pi$
$757$			6.03579i			0.219375i	−0.993966	+	0.109687i	$0.965015\pi$
$758$		0			0
$759$	−23.3205			−0.846481
$760$		0			0
$761$	47.5692			1.72438			0.862191	−	0.506583i	$-0.169091\pi$
$761$	47.5692			1.72438			0.862191	+	0.506583i	$0.169091\pi$
$762$		0			0
$763$		−	24.1432i		−	0.874041i
$764$		0			0
$765$	2.00000	+	2.44949i	0.0723102	+	0.0885615i
$766$		0			0
$767$			69.5467i			2.51118i
$768$		0			0
$769$	32.9282			1.18742			0.593711	−	0.804679i	$-0.297662\pi$
$769$	32.9282			1.18742			0.593711	+	0.804679i	$0.297662\pi$
$770$		0			0
$771$	6.53590			0.235385
$772$		0			0
$773$		−	8.38375i		−	0.301542i	−0.988569	−	0.150771i	$-0.951824\pi$
$773$		−	8.38375i		−	0.301542i	0.988569	−	0.150771i	$-0.0481757\pi$
$774$		0			0
$775$	5.26795	−	25.8076i	0.189230	−	0.927035i
$776$		0			0
$777$		−	2.27362i		−	0.0815658i
$778$		0			0
$779$	−46.6410			−1.67109
$780$		0			0
$781$	16.3923			0.586563
$782$		0			0
$783$		−	5.65685i		−	0.202159i
$784$		0			0
$785$	8.53590	+	10.4543i	0.304659	+	0.373130i
$786$		0			0
$787$		−	49.7749i		−	1.77428i	−0.461498	−	0.887141i	$-0.652688\pi$
$787$		−	49.7749i		−	1.77428i	0.461498	−	0.887141i	$-0.347312\pi$
$788$		0			0
$789$	34.7846			1.23836
$790$		0			0
$791$	−4.14359			−0.147329
$792$		0			0
$793$		−	73.1330i		−	2.59703i
$794$		0			0
$795$	−21.4641	+	17.5254i	−0.761253	+	0.621561i
$796$		0			0
$797$		−	34.6718i		−	1.22814i	−0.789252	−	0.614069i	$-0.789532\pi$
$797$		−	34.6718i		−	1.22814i	0.789252	−	0.614069i	$-0.210468\pi$
$798$		0			0
$799$	2.00000			0.0707549
$800$		0			0
$801$	10.3923			0.367194
$802$		0			0
$803$		−	34.7733i		−	1.22712i
$804$		0			0
$805$	14.7846	−	12.0716i	0.521089	−	0.425467i
$806$		0			0
$807$			16.9706i			0.597392i
$808$		0			0
$809$	−34.3923			−1.20917			−0.604585	−	0.796541i	$-0.706661\pi$
$809$	−34.3923			−1.20917			−0.604585	+	0.796541i	$0.706661\pi$
$810$		0			0
$811$	−4.78461			−0.168010			−0.0840052	−	0.996465i	$-0.526771\pi$
$811$	−4.78461			−0.168010			−0.0840052	+	0.996465i	$0.526771\pi$
$812$		0			0
$813$			32.9058i			1.15406i
$814$		0			0
$815$	−7.85641	−	9.62209i	−0.275198	−	0.337047i
$816$		0			0
$817$			4.41851i			0.154584i
$818$		0			0
$819$	16.3923			0.572793
$820$		0			0
$821$	50.7846			1.77240			0.886198	−	0.463308i	$-0.153337\pi$
$821$	50.7846			1.77240			0.886198	+	0.463308i	$0.153337\pi$
$822$		0			0
$823$		−	1.61729i		−	0.0563750i	−0.999603	−	0.0281875i	$-0.991026\pi$
$823$		−	1.61729i		−	0.0563750i	0.999603	−	0.0281875i	$-0.00897355\pi$
$824$		0			0
$825$	32.7846	+	6.69213i	1.14141	+	0.232990i
$826$		0			0
$827$		−	33.0817i		−	1.15036i	−0.818025	−	0.575182i	$-0.804931\pi$
$827$		−	33.0817i		−	1.15036i	0.818025	−	0.575182i	$-0.195069\pi$
$828$		0			0
$829$	−24.7846			−0.860805			−0.430403	−	0.902637i	$-0.641628\pi$
$829$	−24.7846			−0.860805			−0.430403	+	0.902637i	$0.641628\pi$
$830$		0			0
$831$	−33.4641			−1.16086
$832$		0			0
$833$			1.41421i			0.0489996i
$834$		0			0
$835$	11.8564	+	14.5211i	0.410308	+	0.502522i
$836$		0			0
$837$			29.8000i			1.03004i
$838$		0			0
$839$	−10.9808			−0.379098			−0.189549	−	0.981871i	$-0.560703\pi$
$839$	−10.9808			−0.379098			−0.189549	+	0.981871i	$0.560703\pi$
$840$		0			0
$841$	1.00000			0.0344828
$842$		0			0
$843$		−	12.4233i		−	0.427882i
$844$		0			0
$845$	−55.0526	+	44.9502i	−1.89387	+	1.54633i
$846$		0			0
$847$		−	27.9053i		−	0.958839i
$848$		0			0
$849$	−2.78461			−0.0955676
$850$		0			0
$851$	2.28719			0.0784038
$852$		0			0
$853$			6.86800i			0.235156i	0.993064	+	0.117578i	$0.0375130\pi$
$853$			6.86800i			0.235156i	−0.993064	+	0.117578i	$0.962487\pi$
$854$		0			0
$855$	−11.6603	+	9.52056i	−0.398772	+	0.325596i
$856$		0			0
$857$		−	30.4292i		−	1.03944i	−0.854337	−	0.519720i	$-0.826036\pi$
$857$		−	30.4292i		−	1.03944i	0.854337	−	0.519720i	$-0.173964\pi$
$858$		0			0
$859$	−15.2679			−0.520936			−0.260468	−	0.965483i	$-0.583877\pi$
$859$	−15.2679			−0.520936			−0.260468	+	0.965483i	$0.583877\pi$
$860$		0			0
$861$	24.0000			0.817918
$862$		0			0
$863$		−	13.0053i		−	0.442706i	−0.975194	−	0.221353i	$-0.928953\pi$
$863$		−	13.0053i		−	0.442706i	0.975194	−	0.221353i	$-0.0710474\pi$
$864$		0			0
$865$	−13.4641	−	16.4901i	−0.457793	−	0.560680i
$866$		0			0
$867$			21.2132i			0.720438i
$868$		0			0
$869$	29.3205			0.994630
$870$		0			0
$871$	28.3923			0.962037
$872$		0			0
$873$		−	6.03579i		−	0.204281i
$874$		0			0
$875$	−24.2487	+	12.7279i	−0.819756	+	0.430282i
$876$		0			0
$877$			0.480473i			0.0162244i	0.999967	+	0.00811222i	$0.00258223\pi$
$877$			0.480473i			0.0162244i	−0.999967	+	0.00811222i	$0.997418\pi$
$878$		0			0
$879$	36.5359			1.23233
$880$		0			0
$881$	−57.4641			−1.93601			−0.968007	−	0.250922i	$-0.919266\pi$
$881$	−57.4641			−1.93601			−0.968007	+	0.250922i	$0.919266\pi$
$882$		0			0
$883$		−	52.4002i		−	1.76341i	−0.471803	−	0.881704i	$-0.656397\pi$
$883$		−	52.4002i		−	1.76341i	0.471803	−	0.881704i	$-0.343603\pi$
$884$		0			0
$885$	20.7846	+	25.4558i	0.698667	+	0.855689i
$886$		0			0
$887$			18.1817i			0.610482i	0.952275	+	0.305241i	$0.0987370\pi$
$887$			18.1817i			0.610482i	−0.952275	+	0.305241i	$0.901263\pi$
$888$		0			0
$889$	−10.3923			−0.348547
$890$		0			0
$891$	−23.6603			−0.792648
$892$		0			0
$893$			9.52056i			0.318593i
$894$		0			0
$895$	16.3923	−	13.3843i	0.547934	−	0.447386i
$896$		0			0
$897$		−	32.9802i		−	1.10118i
$898$		0			0
$899$	−5.26795			−0.175696
$900$		0			0
$901$	−12.3923			−0.412848
$902$		0			0
$903$		−	2.27362i		−	0.0756615i
$904$		0			0
$905$	−27.7128	+	22.6274i	−0.921205	+	0.752161i
$906$		0			0
$907$		−	3.28169i		−	0.108967i	−0.998515	−	0.0544834i	$-0.982649\pi$
$907$		−	3.28169i		−	0.108967i	0.998515	−	0.0544834i	$-0.0173512\pi$
$908$		0			0
$909$	−6.00000			−0.199007
$910$		0			0
$911$	20.1962			0.669128			0.334564	−	0.942373i	$-0.391411\pi$
$911$	20.1962			0.669128			0.334564	+	0.942373i	$0.391411\pi$
$912$		0			0
$913$		−	21.3891i		−	0.707875i
$914$		0			0
$915$	−21.8564	−	26.7685i	−0.722551	−	0.884940i
$916$		0			0
$917$		−	3.10583i		−	0.102563i
$918$		0			0
$919$	−59.9615			−1.97795			−0.988974	−	0.148089i	$-0.952688\pi$
$919$	−59.9615			−1.97795			−0.988974	+	0.148089i	$0.952688\pi$
$920$		0			0
$921$	−26.7846			−0.882583
$922$		0			0
$923$			23.1822i			0.763052i
$924$		0			0
$925$	−3.21539	−	0.656339i	−0.105721	−	0.0215803i
$926$		0			0
$927$		−	0.656339i		−	0.0215570i
$928$		0			0
$929$	−28.1436			−0.923361			−0.461681	−	0.887046i	$-0.652753\pi$
$929$	−28.1436			−0.923361			−0.461681	+	0.887046i	$0.652753\pi$
$930$		0			0
$931$	−6.73205			−0.220634
$932$		0			0
$933$		−	34.7733i		−	1.13843i
$934$		0			0
$935$	9.46410	+	11.5911i	0.309509	+	0.379070i
$936$		0			0
$937$			56.6429i			1.85044i	0.379429	+	0.925221i	$0.376121\pi$
$937$			56.6429i			1.85044i	−0.379429	+	0.925221i	$0.623879\pi$
$938$		0			0
$939$	−26.5359			−0.865966
$940$		0			0
$941$	19.8564			0.647300			0.323650	−	0.946177i	$-0.395090\pi$
$941$	19.8564			0.647300			0.323650	+	0.946177i	$0.395090\pi$
$942$		0			0
$943$			24.1432i			0.786210i
$944$		0			0
$945$	24.0000	−	19.5959i	0.780720	−	0.637455i
$946$		0			0
$947$		−	30.8081i		−	1.00113i	−0.865699	−	0.500564i	$-0.833126\pi$
$947$		−	30.8081i		−	1.00113i	0.865699	−	0.500564i	$-0.166874\pi$
$948$		0			0
$949$	49.1769			1.59635
$950$		0			0
$951$	35.8564			1.16272
$952$		0			0
$953$			5.65685i			0.183243i	0.995794	+	0.0916217i	$0.0292051\pi$
$953$			5.65685i			0.183243i	−0.995794	+	0.0916217i	$0.970795\pi$
$954$		0			0
$955$	15.8038	−	12.9038i	0.511400	−	0.417557i
$956$		0			0
$957$		−	6.69213i		−	0.216326i
$958$		0			0
$959$	12.2487			0.395532
$960$		0			0
$961$	−3.24871			−0.104797
$962$		0			0
$963$			3.96524i			0.127778i
$964$		0			0
$965$	−12.9282	−	15.8338i	−0.416174	−	0.509706i
$966$		0			0
$967$		−	27.4249i		−	0.881924i	−0.897526	−	0.440962i	$-0.854637\pi$
$967$		−	27.4249i		−	0.881924i	0.897526	−	0.440962i	$-0.145363\pi$
$968$		0			0
$969$	13.4641			0.432529
$970$		0			0
$971$	4.48334			0.143877			0.0719386	−	0.997409i	$-0.477081\pi$
$971$	4.48334			0.143877			0.0719386	+	0.997409i	$0.477081\pi$
$972$		0			0
$973$			19.5959i			0.628216i
$974$		0			0
$975$	−9.46410	+	46.3644i	−0.303094	+	1.48485i
$976$		0			0
$977$		−	8.20788i		−	0.262593i	−0.991343	−	0.131297i	$-0.958086\pi$
$977$		−	8.20788i		−	0.262593i	0.991343	−	0.131297i	$-0.0419140\pi$
$978$		0			0
$979$	49.1769			1.57170
$980$		0			0
$981$	9.85641			0.314691
$982$		0			0
$983$			0.859411i			0.0274109i	0.999906	+	0.0137055i	$0.00436272\pi$
$983$			0.859411i			0.0274109i	−0.999906	+	0.0137055i	$0.995637\pi$
$984$		0			0
$985$	23.7128	+	29.0421i	0.755553	+	0.925360i
$986$		0			0
$987$		−	4.89898i		−	0.155936i
$988$		0			0
$989$	2.28719			0.0727283
$990$		0			0
$991$	6.78461			0.215520			0.107760	−	0.994177i	$-0.465632\pi$
$991$	6.78461			0.215520			0.107760	+	0.994177i	$0.465632\pi$
$992$		0			0
$993$			11.0363i			0.350227i
$994$		0			0
$995$	−24.2487	+	19.7990i	−0.768736	+	0.627670i
$996$		0			0
$997$			41.2896i			1.30765i	0.756644	+	0.653827i	$0.226838\pi$
$997$			41.2896i			1.30765i	−0.756644	+	0.653827i	$0.773162\pi$
$998$		0			0
$999$	3.71281			0.117468

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2320.2.d.f.929.4		4
4.3	odd	2		145.2.b.b.59.1	✓	4
5.4	even	2	inner	2320.2.d.f.929.2		4
12.11	even	2		1305.2.c.f.784.4		4
20.3	even	4		725.2.a.f.1.1		4
20.7	even	4		725.2.a.f.1.4		4
20.19	odd	2		145.2.b.b.59.4	yes	4
60.23	odd	4		6525.2.a.bj.1.4		4
60.47	odd	4		6525.2.a.bj.1.1		4
60.59	even	2		1305.2.c.f.784.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.b.b.59.1	✓	4	4.3	odd	2
145.2.b.b.59.4	yes	4	20.19	odd	2
725.2.a.f.1.1		4	20.3	even	4
725.2.a.f.1.4		4	20.7	even	4
1305.2.c.f.784.1		4	60.59	even	2
1305.2.c.f.784.4		4	12.11	even	2
2320.2.d.f.929.2		4	5.4	even	2	inner
2320.2.d.f.929.4		4	1.1	even	1	trivial
6525.2.a.bj.1.1		4	60.47	odd	4
6525.2.a.bj.1.4		4	60.23	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2320.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{3} +(1.73205 - 1.41421i) q^{5} -2.44949i q^{7} +1.00000 q^{9} +4.73205 q^{11} +6.69213i q^{13} +(2.00000 + 2.44949i) q^{15} +1.41421i q^{17} -6.73205 q^{19} +3.46410 q^{21} +3.48477i q^{23} +(1.00000 - 4.89898i) q^{25} +5.65685i q^{27} -1.00000 q^{29} +5.26795 q^{31} +6.69213i q^{33} +(-3.46410 - 4.24264i) q^{35} -0.656339i q^{37} -9.46410 q^{39} +6.92820 q^{41} -0.656339i q^{43} +(1.73205 - 1.41421i) q^{45} -1.41421i q^{47} +1.00000 q^{49} -2.00000 q^{51} +8.76268i q^{53} +(8.19615 - 6.69213i) q^{55} -9.52056i q^{57} +10.3923 q^{59} -10.9282 q^{61} -2.44949i q^{63} +(9.46410 + 11.5911i) q^{65} -4.24264i q^{67} -4.92820 q^{69} +3.46410 q^{71} -7.34847i q^{73} +(6.92820 + 1.41421i) q^{75} -11.5911i q^{77} +6.19615 q^{79} -5.00000 q^{81} -4.52004i q^{83} +(2.00000 + 2.44949i) q^{85} -1.41421i q^{87} +10.3923 q^{89} +16.3923 q^{91} +7.45001i q^{93} +(-11.6603 + 9.52056i) q^{95} -6.03579i q^{97} +4.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{9} + 12 q^{11} + 8 q^{15} - 20 q^{19} + 4 q^{25} - 4 q^{29} + 28 q^{31} - 24 q^{39} + 4 q^{49} - 8 q^{51} + 12 q^{55} - 16 q^{61} + 24 q^{65} + 8 q^{69} + 4 q^{79} - 20 q^{81} + 8 q^{85} + 24 q^{91}+ \cdots + 12 q^{99}+O(q^{100}) \) 4 * q + 4 * q^9 + 12 * q^11 + 8 * q^15 - 20 * q^19 + 4 * q^25 - 4 * q^29 + 28 * q^31 - 24 * q^39 + 4 * q^49 - 8 * q^51 + 12 * q^55 - 16 * q^61 + 24 * q^65 + 8 * q^69 + 4 * q^79 - 20 * q^81 + 8 * q^85 + 24 * q^91 - 12 * q^95 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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