LMFDB - Embedded newform 5520.2.be.c.1471.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5520,2,Mod(1471,5520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5520, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5520.1471");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5520.be (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$44.0774219157$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.19
Character	$\chi$	$=$	5520.1471
Dual form			5520.2.be.c.1471.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -1.00000i q^{5} -2.01693 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -1.00000i q^{5} -2.01693 q^{7} -1.00000 q^{9} -0.697775 q^{11} +0.195235 q^{13} -1.00000 q^{15} -0.430024i q^{17} +7.71006 q^{19} +2.01693i q^{21} +(-4.23049 - 2.25897i) q^{23} -1.00000 q^{25} +1.00000i q^{27} -9.64181 q^{29} +1.05072i q^{31} +0.697775i q^{33} +2.01693i q^{35} -4.89822i q^{37} -0.195235i q^{39} -3.71938 q^{41} +5.85669 q^{43} +1.00000i q^{45} +10.6764i q^{47} -2.93201 q^{49} -0.430024 q^{51} -0.464177i q^{53} +0.697775i q^{55} -7.71006i q^{57} -7.64910i q^{59} +4.18983i q^{61} +2.01693 q^{63} -0.195235i q^{65} -3.70657 q^{67} +(-2.25897 + 4.23049i) q^{69} +5.69569i q^{71} -3.51188 q^{73} +1.00000i q^{75} +1.40736 q^{77} -9.68233 q^{79} +1.00000 q^{81} +0.982199 q^{83} -0.430024 q^{85} +9.64181i q^{87} +5.48101i q^{89} -0.393776 q^{91} +1.05072 q^{93} -7.71006i q^{95} -13.5553i q^{97} +0.697775 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 8 q^{7} - 32 q^{9}+O(q^{10}) $ 32 * q - 8 * q^7 - 32 * q^9	$ 32 q - 8 q^{7} - 32 q^{9} + 8 q^{11} - 8 q^{13} - 32 q^{15} - 32 q^{25} + 4 q^{29} + 20 q^{41} + 52 q^{49} - 4 q^{51} + 8 q^{63} + 32 q^{67} - 40 q^{73} - 24 q^{77} + 32 q^{79} + 32 q^{81} - 4 q^{85} - 48 q^{91} - 8 q^{99}+O(q^{100}) $ 32 * q - 8 * q^7 - 32 * q^9 + 8 * q^11 - 8 * q^13 - 32 * q^15 - 32 * q^25 + 4 * q^29 + 20 * q^41 + 52 * q^49 - 4 * q^51 + 8 * q^63 + 32 * q^67 - 40 * q^73 - 24 * q^77 + 32 * q^79 + 32 * q^81 - 4 * q^85 - 48 * q^91 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5520\mathbb{Z}\right)^\times$.

$n$	$1201$	$1381$	$1841$	$4417$	$4831$
$\chi(n)$	$-1$	$1$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$	−2.01693			−0.762327			−0.381163	−	0.924508i	$-0.624476\pi$
$7$	−2.01693			−0.762327			−0.381163	+	0.924508i	$0.624476\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−0.697775			−0.210387			−0.105194	−	0.994452i	$-0.533546\pi$
$11$	−0.697775			−0.210387			−0.105194	+	0.994452i	$0.533546\pi$
$12$		0			0
$13$	0.195235			0.0541486			0.0270743	−	0.999633i	$-0.491381\pi$
$13$	0.195235			0.0541486			0.0270743	+	0.999633i	$0.491381\pi$
$14$		0			0
$15$	−1.00000			−0.258199
$16$		0			0
$17$		−	0.430024i		−	0.104296i	−0.998639	−	0.0521480i	$-0.983393\pi$
$17$		−	0.430024i		−	0.104296i	0.998639	−	0.0521480i	$-0.0166068\pi$
$18$		0			0
$19$	7.71006			1.76881			0.884405	−	0.466720i	$-0.154564\pi$
$19$	7.71006			1.76881			0.884405	+	0.466720i	$0.154564\pi$
$20$		0			0
$21$			2.01693i			0.440130i
$22$		0			0
$23$	−4.23049	−	2.25897i	−0.882119	−	0.471027i
$23$	−4.23049	−	2.25897i	−0.882119	−	0.471027i
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	−9.64181			−1.79044			−0.895220	−	0.445625i	$-0.852982\pi$
$29$	−9.64181			−1.79044			−0.895220	+	0.445625i	$0.852982\pi$
$30$		0			0
$31$			1.05072i			0.188714i	0.995538	+	0.0943570i	$0.0300795\pi$
$31$			1.05072i			0.188714i	−0.995538	+	0.0943570i	$0.969920\pi$
$32$		0			0
$33$			0.697775i			0.121467i
$34$		0			0
$35$			2.01693i			0.340923i
$36$		0			0
$37$		−	4.89822i		−	0.805262i	−0.915362	−	0.402631i	$-0.868096\pi$
$37$		−	4.89822i		−	0.805262i	0.915362	−	0.402631i	$-0.131904\pi$
$38$		0			0
$39$		−	0.195235i		−	0.0312627i
$40$		0			0
$41$	−3.71938			−0.580870			−0.290435	−	0.956895i	$-0.593800\pi$
$41$	−3.71938			−0.580870			−0.290435	+	0.956895i	$0.593800\pi$
$42$		0			0
$43$	5.85669			0.893136			0.446568	−	0.894750i	$-0.352646\pi$
$43$	5.85669			0.893136			0.446568	+	0.894750i	$0.352646\pi$
$44$		0			0
$45$			1.00000i			0.149071i
$46$		0			0
$47$			10.6764i			1.55731i	0.627452	+	0.778655i	$0.284098\pi$
$47$			10.6764i			1.55731i	−0.627452	+	0.778655i	$0.715902\pi$
$48$		0			0
$49$	−2.93201			−0.418858
$50$		0			0
$51$	−0.430024			−0.0602154
$52$		0			0
$53$		−	0.464177i		−	0.0637597i	−0.999492	−	0.0318798i	$-0.989851\pi$
$53$		−	0.464177i		−	0.0637597i	0.999492	−	0.0318798i	$-0.0101494\pi$
$54$		0			0
$55$			0.697775i			0.0940880i
$56$		0			0
$57$		−	7.71006i		−	1.02122i
$58$		0			0
$59$		−	7.64910i		−	0.995828i	−0.867226	−	0.497914i	$-0.834099\pi$
$59$		−	7.64910i		−	0.995828i	0.867226	−	0.497914i	$-0.165901\pi$
$60$		0			0
$61$			4.18983i			0.536452i	0.963356	+	0.268226i	$0.0864375\pi$
$61$			4.18983i			0.536452i	−0.963356	+	0.268226i	$0.913563\pi$
$62$		0			0
$63$	2.01693			0.254109
$64$		0			0
$65$		−	0.195235i		−	0.0242160i
$66$		0			0
$67$	−3.70657			−0.452830			−0.226415	−	0.974031i	$-0.572700\pi$
$67$	−3.70657			−0.452830			−0.226415	+	0.974031i	$0.572700\pi$
$68$		0			0
$69$	−2.25897	+	4.23049i	−0.271948	+	0.509292i
$70$		0			0
$71$			5.69569i			0.675954i	0.941154	+	0.337977i	$0.109743\pi$
$71$			5.69569i			0.675954i	−0.941154	+	0.337977i	$0.890257\pi$
$72$		0			0
$73$	−3.51188			−0.411034			−0.205517	−	0.978654i	$-0.565888\pi$
$73$	−3.51188			−0.411034			−0.205517	+	0.978654i	$0.565888\pi$
$74$		0			0
$75$			1.00000i			0.115470i
$76$		0			0
$77$	1.40736			0.160384
$78$		0			0
$79$	−9.68233			−1.08935			−0.544674	−	0.838648i	$-0.683346\pi$
$79$	−9.68233			−1.08935			−0.544674	+	0.838648i	$0.683346\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	0.982199			0.107810			0.0539052	−	0.998546i	$-0.482833\pi$
$83$	0.982199			0.107810			0.0539052	+	0.998546i	$0.482833\pi$
$84$		0			0
$85$	−0.430024			−0.0466426
$86$		0			0
$87$			9.64181i			1.03371i
$88$		0			0
$89$			5.48101i			0.580986i	0.956877	+	0.290493i	$0.0938192\pi$
$89$			5.48101i			0.580986i	−0.956877	+	0.290493i	$0.906181\pi$
$90$		0			0
$91$	−0.393776			−0.0412789
$92$		0			0
$93$	1.05072			0.108954
$94$		0			0
$95$		−	7.71006i		−	0.791036i
$96$		0			0
$97$		−	13.5553i		−	1.37634i	−0.725552	−	0.688168i	$-0.758415\pi$
$97$		−	13.5553i		−	1.37634i	0.725552	−	0.688168i	$-0.241585\pi$
$98$		0			0
$99$	0.697775			0.0701291
$100$		0			0
$101$	−9.87390			−0.982490			−0.491245	−	0.871021i	$-0.663458\pi$
$101$	−9.87390			−0.982490			−0.491245	+	0.871021i	$0.663458\pi$
$102$		0			0
$103$	7.65129			0.753904			0.376952	−	0.926233i	$-0.376972\pi$
$103$	7.65129			0.753904			0.376952	+	0.926233i	$0.376972\pi$
$104$		0			0
$105$	2.01693			0.196832
$106$		0			0
$107$	15.7693			1.52448			0.762238	−	0.647296i	$-0.224100\pi$
$107$	15.7693			1.52448			0.762238	+	0.647296i	$0.224100\pi$
$108$		0			0
$109$			7.29036i			0.698289i	0.937069	+	0.349145i	$0.113528\pi$
$109$			7.29036i			0.698289i	−0.937069	+	0.349145i	$0.886472\pi$
$110$		0			0
$111$	−4.89822			−0.464918
$112$		0			0
$113$			11.6243i			1.09352i	0.837288	+	0.546762i	$0.184140\pi$
$113$			11.6243i			1.09352i	−0.837288	+	0.546762i	$0.815860\pi$
$114$		0			0
$115$	−2.25897	+	4.23049i	−0.210650	+	0.394496i
$116$		0			0
$117$	−0.195235			−0.0180495
$118$		0			0
$119$			0.867327i			0.0795077i
$120$		0			0
$121$	−10.5131			−0.955737
$122$		0			0
$123$			3.71938i			0.335365i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$		−	5.20841i		−	0.462172i	−0.972933	−	0.231086i	$-0.925772\pi$
$127$		−	5.20841i		−	0.462172i	0.972933	−	0.231086i	$-0.0742279\pi$
$128$		0			0
$129$		−	5.85669i		−	0.515653i
$130$		0			0
$131$		−	2.04355i		−	0.178546i	−0.996007	−	0.0892728i	$-0.971546\pi$
$131$		−	2.04355i		−	0.178546i	0.996007	−	0.0892728i	$-0.0284543\pi$
$132$		0			0
$133$	−15.5506			−1.34841
$134$		0			0
$135$	1.00000			0.0860663
$136$		0			0
$137$			21.0004i			1.79419i	0.441841	+	0.897093i	$0.354325\pi$
$137$			21.0004i			1.79419i	−0.441841	+	0.897093i	$0.645675\pi$
$138$		0			0
$139$			2.56106i			0.217227i	0.994084	+	0.108613i	$0.0346410\pi$
$139$			2.56106i			0.217227i	−0.994084	+	0.108613i	$0.965359\pi$
$140$		0			0
$141$	10.6764			0.899113
$142$		0			0
$143$	−0.136231			−0.0113922
$144$		0			0
$145$			9.64181i			0.800709i
$146$		0			0
$147$			2.93201i			0.241828i
$148$		0			0
$149$			9.82277i			0.804712i	0.915483	+	0.402356i	$0.131809\pi$
$149$			9.82277i			0.804712i	−0.915483	+	0.402356i	$0.868191\pi$
$150$		0			0
$151$			14.8419i			1.20782i	0.797054	+	0.603909i	$0.206391\pi$
$151$			14.8419i			1.20782i	−0.797054	+	0.603909i	$0.793609\pi$
$152$		0			0
$153$			0.430024i			0.0347654i
$154$		0			0
$155$	1.05072			0.0843955
$156$		0			0
$157$			1.12666i			0.0899172i	0.998989	+	0.0449586i	$0.0143156\pi$
$157$			1.12666i			0.0899172i	−0.998989	+	0.0449586i	$0.985684\pi$
$158$		0			0
$159$	−0.464177			−0.0368117
$160$		0			0
$161$	8.53260	+	4.55617i	0.672463	+	0.359077i
$162$		0			0
$163$			17.1655i			1.34451i	0.740321	+	0.672254i	$0.234674\pi$
$163$			17.1655i			1.34451i	−0.740321	+	0.672254i	$0.765326\pi$
$164$		0			0
$165$	0.697775			0.0543217
$166$		0			0
$167$		−	0.809244i		−	0.0626212i	−0.999510	−	0.0313106i	$-0.990032\pi$
$167$		−	0.809244i		−	0.0626212i	0.999510	−	0.0313106i	$-0.00996809\pi$
$168$		0			0
$169$	−12.9619			−0.997068
$170$		0			0
$171$	−7.71006			−0.589603
$172$		0			0
$173$	0.192946			0.0146694			0.00733472	−	0.999973i	$-0.497665\pi$
$173$	0.192946			0.0146694			0.00733472	+	0.999973i	$0.497665\pi$
$174$		0			0
$175$	2.01693			0.152465
$176$		0			0
$177$	−7.64910			−0.574942
$178$		0			0
$179$			23.5203i			1.75799i	0.476832	+	0.878994i	$0.341785\pi$
$179$			23.5203i			1.75799i	−0.476832	+	0.878994i	$0.658215\pi$
$180$		0			0
$181$			9.77478i			0.726554i	0.931681	+	0.363277i	$0.118342\pi$
$181$			9.77478i			0.726554i	−0.931681	+	0.363277i	$0.881658\pi$
$182$		0			0
$183$	4.18983			0.309721
$184$		0			0
$185$	−4.89822			−0.360124
$186$		0			0
$187$			0.300060i			0.0219426i
$188$		0			0
$189$		−	2.01693i		−	0.146710i
$190$		0			0
$191$	7.21224			0.521859			0.260930	−	0.965358i	$-0.415971\pi$
$191$	7.21224			0.521859			0.260930	+	0.965358i	$0.415971\pi$
$192$		0			0
$193$	24.9860			1.79853			0.899266	−	0.437402i	$-0.144101\pi$
$193$	24.9860			1.79853			0.899266	+	0.437402i	$0.144101\pi$
$194$		0			0
$195$	−0.195235			−0.0139811
$196$		0			0
$197$	−6.96165			−0.495997			−0.247998	−	0.968760i	$-0.579773\pi$
$197$	−6.96165			−0.495997			−0.247998	+	0.968760i	$0.579773\pi$
$198$		0			0
$199$	2.94085			0.208471			0.104236	−	0.994553i	$-0.466760\pi$
$199$	2.94085			0.208471			0.104236	+	0.994553i	$0.466760\pi$
$200$		0			0
$201$			3.70657i			0.261441i
$202$		0			0
$203$	19.4468			1.36490
$204$		0			0
$205$			3.71938i			0.259773i
$206$		0			0
$207$	4.23049	+	2.25897i	0.294040	+	0.157009i
$208$		0			0
$209$	−5.37989			−0.372135
$210$		0			0
$211$			28.0259i			1.92938i	0.263381	+	0.964692i	$0.415162\pi$
$211$			28.0259i			1.92938i	−0.263381	+	0.964692i	$0.584838\pi$
$212$		0			0
$213$	5.69569			0.390262
$214$		0			0
$215$		−	5.85669i		−	0.399423i
$216$		0			0
$217$		−	2.11922i		−	0.143862i
$218$		0			0
$219$			3.51188i			0.237311i
$220$		0			0
$221$		−	0.0839559i		−	0.00564748i
$222$		0			0
$223$		−	1.81830i		−	0.121763i	−0.998145	−	0.0608813i	$-0.980609\pi$
$223$		−	1.81830i		−	0.121763i	0.998145	−	0.0608813i	$-0.0193911\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	−15.7824			−1.04752			−0.523758	−	0.851867i	$-0.675470\pi$
$227$	−15.7824			−1.04752			−0.523758	+	0.851867i	$0.675470\pi$
$228$		0			0
$229$		−	7.05803i		−	0.466408i	−0.972428	−	0.233204i	$-0.925079\pi$
$229$		−	7.05803i		−	0.466408i	0.972428	−	0.233204i	$-0.0749210\pi$
$230$		0			0
$231$		−	1.40736i		−	0.0925976i
$232$		0			0
$233$	−5.71879			−0.374650			−0.187325	−	0.982298i	$-0.559982\pi$
$233$	−5.71879			−0.374650			−0.187325	+	0.982298i	$0.559982\pi$
$234$		0			0
$235$	10.6764			0.696450
$236$		0			0
$237$			9.68233i			0.628935i
$238$		0			0
$239$		−	20.6125i		−	1.33331i	−0.745365	−	0.666657i	$-0.767725\pi$
$239$		−	20.6125i		−	1.33331i	0.745365	−	0.666657i	$-0.232275\pi$
$240$		0			0
$241$		−	23.5987i		−	1.52013i	−0.649849	−	0.760063i	$-0.725168\pi$
$241$		−	23.5987i		−	1.52013i	0.649849	−	0.760063i	$-0.274832\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$			2.93201i			0.187319i
$246$		0			0
$247$	1.50528			0.0957785
$248$		0			0
$249$		−	0.982199i		−	0.0622443i
$250$		0			0
$251$	1.45913			0.0920994			0.0460497	−	0.998939i	$-0.485337\pi$
$251$	1.45913			0.0920994			0.0460497	+	0.998939i	$0.485337\pi$
$252$		0			0
$253$	2.95193	+	1.57625i	0.185587	+	0.0990981i
$254$		0			0
$255$			0.430024i			0.0269291i
$256$		0			0
$257$	−20.3730			−1.27083			−0.635417	−	0.772169i	$-0.719172\pi$
$257$	−20.3730			−1.27083			−0.635417	+	0.772169i	$0.719172\pi$
$258$		0			0
$259$			9.87935i			0.613873i
$260$		0			0
$261$	9.64181			0.596813
$262$		0			0
$263$	6.84599			0.422142			0.211071	−	0.977471i	$-0.432305\pi$
$263$	6.84599			0.422142			0.211071	+	0.977471i	$0.432305\pi$
$264$		0			0
$265$	−0.464177			−0.0285142
$266$		0			0
$267$	5.48101			0.335432
$268$		0			0
$269$	−15.7765			−0.961912			−0.480956	−	0.876745i	$-0.659710\pi$
$269$	−15.7765			−0.961912			−0.480956	+	0.876745i	$0.659710\pi$
$270$		0			0
$271$			15.4121i			0.936216i	0.883671	+	0.468108i	$0.155064\pi$
$271$			15.4121i			0.936216i	−0.883671	+	0.468108i	$0.844936\pi$
$272$		0			0
$273$			0.393776i			0.0238324i
$274$		0			0
$275$	0.697775			0.0420774
$276$		0			0
$277$	7.24637			0.435392			0.217696	−	0.976017i	$-0.430146\pi$
$277$	7.24637			0.435392			0.217696	+	0.976017i	$0.430146\pi$
$278$		0			0
$279$		−	1.05072i		−	0.0629047i
$280$		0			0
$281$			0.321739i			0.0191933i	0.999954	+	0.00959666i	$0.00305476\pi$
$281$			0.321739i			0.0191933i	−0.999954	+	0.00959666i	$0.996945\pi$
$282$		0			0
$283$	16.0317			0.952988			0.476494	−	0.879178i	$-0.341907\pi$
$283$	16.0317			0.952988			0.476494	+	0.879178i	$0.341907\pi$
$284$		0			0
$285$	−7.71006			−0.456705
$286$		0			0
$287$	7.50172			0.442812
$288$		0			0
$289$	16.8151			0.989122
$290$		0			0
$291$	−13.5553			−0.794628
$292$		0			0
$293$			6.96765i			0.407055i	0.979069	+	0.203527i	$0.0652406\pi$
$293$			6.96765i			0.407055i	−0.979069	+	0.203527i	$0.934759\pi$
$294$		0			0
$295$	−7.64910			−0.445348
$296$		0			0
$297$		−	0.697775i		−	0.0404890i
$298$		0			0
$299$	−0.825942	−	0.441030i	−0.0477655	−	0.0255054i
$300$		0			0
$301$	−11.8125			−0.680862
$302$		0			0
$303$			9.87390i			0.567241i
$304$		0			0
$305$	4.18983			0.239909
$306$		0			0
$307$		−	23.8378i		−	1.36049i	−0.732983	−	0.680246i	$-0.761873\pi$
$307$		−	23.8378i		−	1.36049i	0.732983	−	0.680246i	$-0.238127\pi$
$308$		0			0
$309$		−	7.65129i		−	0.435266i
$310$		0			0
$311$		−	2.02594i		−	0.114880i	−0.998349	−	0.0574401i	$-0.981706\pi$
$311$		−	2.02594i		−	0.114880i	0.998349	−	0.0574401i	$-0.0182938\pi$
$312$		0			0
$313$		−	16.4162i		−	0.927900i	−0.885861	−	0.463950i	$-0.846432\pi$
$313$		−	16.4162i		−	0.927900i	0.885861	−	0.463950i	$-0.153568\pi$
$314$		0			0
$315$		−	2.01693i		−	0.113641i
$316$		0			0
$317$	−21.2081			−1.19117			−0.595583	−	0.803294i	$-0.703079\pi$
$317$	−21.2081			−1.19117			−0.595583	+	0.803294i	$0.703079\pi$
$318$		0			0
$319$	6.72782			0.376686
$320$		0			0
$321$		−	15.7693i		−	0.880157i
$322$		0			0
$323$		−	3.31551i		−	0.184480i
$324$		0			0
$325$	−0.195235			−0.0108297
$326$		0			0
$327$	7.29036			0.403158
$328$		0			0
$329$		−	21.5335i		−	1.18718i
$330$		0			0
$331$		−	2.15797i		−	0.118613i	−0.998240	−	0.0593063i	$-0.981111\pi$
$331$		−	2.15797i		−	0.118613i	0.998240	−	0.0593063i	$-0.0188889\pi$
$332$		0			0
$333$			4.89822i			0.268421i
$334$		0			0
$335$			3.70657i			0.202512i
$336$		0			0
$337$		−	3.24431i		−	0.176729i	−0.996088	−	0.0883645i	$-0.971836\pi$
$337$		−	3.24431i		−	0.176729i	0.996088	−	0.0883645i	$-0.0281640\pi$
$338$		0			0
$339$	11.6243			0.631346
$340$		0			0
$341$		−	0.733163i		−	0.0397030i
$342$		0			0
$343$	20.0321			1.08163
$344$		0			0
$345$	4.23049	+	2.25897i	0.227762	+	0.121619i
$346$		0			0
$347$			24.8303i			1.33296i	0.745524	+	0.666479i	$0.232200\pi$
$347$			24.8303i			1.33296i	−0.745524	+	0.666479i	$0.767800\pi$
$348$		0			0
$349$	2.69841			0.144442			0.0722212	−	0.997389i	$-0.476991\pi$
$349$	2.69841			0.144442			0.0722212	+	0.997389i	$0.476991\pi$
$350$		0			0
$351$			0.195235i			0.0104209i
$352$		0			0
$353$	6.10284			0.324821			0.162411	−	0.986723i	$-0.448073\pi$
$353$	6.10284			0.324821			0.162411	+	0.986723i	$0.448073\pi$
$354$		0			0
$355$	5.69569			0.302296
$356$		0			0
$357$	0.867327			0.0459038
$358$		0			0
$359$	13.7445			0.725406			0.362703	−	0.931905i	$-0.381854\pi$
$359$	13.7445			0.725406			0.362703	+	0.931905i	$0.381854\pi$
$360$		0			0
$361$	40.4451			2.12869
$362$		0			0
$363$			10.5131i			0.551795i
$364$		0			0
$365$			3.51188i			0.183820i
$366$		0			0
$367$	−2.83223			−0.147841			−0.0739206	−	0.997264i	$-0.523551\pi$
$367$	−2.83223			−0.147841			−0.0739206	+	0.997264i	$0.523551\pi$
$368$		0			0
$369$	3.71938			0.193623
$370$		0			0
$371$			0.936212i			0.0486057i
$372$		0			0
$373$			13.7736i			0.713168i	0.934263	+	0.356584i	$0.116059\pi$
$373$			13.7736i			0.713168i	−0.934263	+	0.356584i	$0.883941\pi$
$374$		0			0
$375$	1.00000			0.0516398
$376$		0			0
$377$	−1.88242			−0.0969498
$378$		0			0
$379$	12.0709			0.620040			0.310020	−	0.950730i	$-0.399664\pi$
$379$	12.0709			0.620040			0.310020	+	0.950730i	$0.399664\pi$
$380$		0			0
$381$	−5.20841			−0.266835
$382$		0			0
$383$	−10.1553			−0.518910			−0.259455	−	0.965755i	$-0.583543\pi$
$383$	−10.1553			−0.518910			−0.259455	+	0.965755i	$0.583543\pi$
$384$		0			0
$385$		−	1.40736i		−	0.0717258i
$386$		0			0
$387$	−5.85669			−0.297712
$388$		0			0
$389$			12.2653i			0.621877i	0.950430	+	0.310939i	$0.100643\pi$
$389$			12.2653i			0.621877i	−0.950430	+	0.310939i	$0.899357\pi$
$390$		0			0
$391$	−0.971409	+	1.81921i	−0.0491263	+	0.0920015i
$392$		0			0
$393$	−2.04355			−0.103083
$394$		0			0
$395$			9.68233i			0.487171i
$396$		0			0
$397$	−10.7383			−0.538942			−0.269471	−	0.963008i	$-0.586849\pi$
$397$	−10.7383			−0.538942			−0.269471	+	0.963008i	$0.586849\pi$
$398$		0			0
$399$			15.5506i			0.778506i
$400$		0			0
$401$			18.0035i			0.899049i	0.893268	+	0.449525i	$0.148407\pi$
$401$			18.0035i			0.899049i	−0.893268	+	0.449525i	$0.851593\pi$
$402$		0			0
$403$			0.205137i			0.0102186i
$404$		0			0
$405$		−	1.00000i		−	0.0496904i
$406$		0			0
$407$			3.41786i			0.169417i
$408$		0			0
$409$	−19.4255			−0.960527			−0.480264	−	0.877124i	$-0.659459\pi$
$409$	−19.4255			−0.960527			−0.480264	+	0.877124i	$0.659459\pi$
$410$		0			0
$411$	21.0004			1.03587
$412$		0			0
$413$			15.4277i			0.759147i
$414$		0			0
$415$		−	0.982199i		−	0.0482143i
$416$		0			0
$417$	2.56106			0.125416
$418$		0			0
$419$	−19.8285			−0.968685			−0.484343	−	0.874878i	$-0.660941\pi$
$419$	−19.8285			−0.968685			−0.484343	+	0.874878i	$0.660941\pi$
$420$		0			0
$421$			14.9197i			0.727140i	0.931567	+	0.363570i	$0.118442\pi$
$421$			14.9197i			0.727140i	−0.931567	+	0.363570i	$0.881558\pi$
$422$		0			0
$423$		−	10.6764i		−	0.519103i
$424$		0			0
$425$			0.430024i			0.0208592i
$426$		0			0
$427$		−	8.45058i		−	0.408952i
$428$		0			0
$429$			0.136231i			0.00657727i
$430$		0			0
$431$	−37.3682			−1.79996			−0.899982	−	0.435926i	$-0.856421\pi$
$431$	−37.3682			−1.79996			−0.899982	+	0.435926i	$0.856421\pi$
$432$		0			0
$433$		−	4.24734i		−	0.204114i	−0.994779	−	0.102057i	$-0.967458\pi$
$433$		−	4.24734i		−	0.204114i	0.994779	−	0.102057i	$-0.0325425\pi$
$434$		0			0
$435$	9.64181			0.462290
$436$		0			0
$437$	−32.6174	−	17.4168i	−1.56030	−	0.833157i
$438$		0			0
$439$			36.0964i			1.72278i	0.507940	+	0.861392i	$0.330407\pi$
$439$			36.0964i			1.72278i	−0.507940	+	0.861392i	$0.669593\pi$
$440$		0			0
$441$	2.93201			0.139619
$442$		0			0
$443$			8.68329i			0.412556i	0.978493	+	0.206278i	$0.0661350\pi$
$443$			8.68329i			0.412556i	−0.978493	+	0.206278i	$0.933865\pi$
$444$		0			0
$445$	5.48101			0.259825
$446$		0			0
$447$	9.82277			0.464601
$448$		0			0
$449$	13.7648			0.649599			0.324799	−	0.945783i	$-0.394703\pi$
$449$	13.7648			0.649599			0.324799	+	0.945783i	$0.394703\pi$
$450$		0			0
$451$	2.59529			0.122208
$452$		0			0
$453$	14.8419			0.697334
$454$		0			0
$455$			0.393776i			0.0184605i
$456$		0			0
$457$		−	13.9155i		−	0.650939i	−0.945553	−	0.325469i	$-0.894478\pi$
$457$		−	13.9155i		−	0.650939i	0.945553	−	0.325469i	$-0.105522\pi$
$458$		0			0
$459$	0.430024			0.0200718
$460$		0			0
$461$	−23.7166			−1.10459			−0.552296	−	0.833648i	$-0.686248\pi$
$461$	−23.7166			−1.10459			−0.552296	+	0.833648i	$0.686248\pi$
$462$		0			0
$463$		−	9.78160i		−	0.454589i	−0.973826	−	0.227295i	$-0.927012\pi$
$463$		−	9.78160i		−	0.454589i	0.973826	−	0.227295i	$-0.0729881\pi$
$464$		0			0
$465$		−	1.05072i		−	0.0487258i
$466$		0			0
$467$	−37.7537			−1.74703			−0.873516	−	0.486795i	$-0.838166\pi$
$467$	−37.7537			−1.74703			−0.873516	+	0.486795i	$0.838166\pi$
$468$		0			0
$469$	7.47588			0.345204
$470$		0			0
$471$	1.12666			0.0519137
$472$		0			0
$473$	−4.08665			−0.187904
$474$		0			0
$475$	−7.71006			−0.353762
$476$		0			0
$477$			0.464177i			0.0212532i
$478$		0			0
$479$	13.8613			0.633337			0.316668	−	0.948536i	$-0.397436\pi$
$479$	13.8613			0.633337			0.316668	+	0.948536i	$0.397436\pi$
$480$		0			0
$481$		−	0.956306i		−	0.0436038i
$482$		0			0
$483$	4.55617	−	8.53260i	0.207313	−	0.388247i
$484$		0			0
$485$	−13.5553			−0.615516
$486$		0			0
$487$			20.7362i			0.939646i	0.882761	+	0.469823i	$0.155682\pi$
$487$			20.7362i			0.939646i	−0.882761	+	0.469823i	$0.844318\pi$
$488$		0			0
$489$	17.1655			0.776252
$490$		0			0
$491$			32.8875i			1.48419i	0.670295	+	0.742095i	$0.266168\pi$
$491$			32.8875i			1.48419i	−0.670295	+	0.742095i	$0.733832\pi$
$492$		0			0
$493$			4.14621i			0.186736i
$494$		0			0
$495$		−	0.697775i		−	0.0313627i
$496$		0			0
$497$		−	11.4878i		−	0.515298i
$498$		0			0
$499$			0.150831i			0.00675210i	0.999994	+	0.00337605i	$0.00107463\pi$
$499$			0.150831i			0.00675210i	−0.999994	+	0.00337605i	$0.998925\pi$
$500$		0			0
$501$	−0.809244			−0.0361543
$502$		0			0
$503$	6.93423			0.309182			0.154591	−	0.987979i	$-0.450594\pi$
$503$	6.93423			0.309182			0.154591	+	0.987979i	$0.450594\pi$
$504$		0			0
$505$			9.87390i			0.439383i
$506$		0			0
$507$			12.9619i			0.575657i
$508$		0			0
$509$	−10.6312			−0.471220			−0.235610	−	0.971848i	$-0.575709\pi$
$509$	−10.6312			−0.471220			−0.235610	+	0.971848i	$0.575709\pi$
$510$		0			0
$511$	7.08320			0.313343
$512$		0			0
$513$			7.71006i			0.340408i
$514$		0			0
$515$		−	7.65129i		−	0.337156i
$516$		0			0
$517$		−	7.44971i		−	0.327638i
$518$		0			0
$519$		−	0.192946i		−	0.00846941i
$520$		0			0
$521$		−	12.6607i		−	0.554673i	−0.960773	−	0.277337i	$-0.910548\pi$
$521$		−	12.6607i		−	0.554673i	0.960773	−	0.277337i	$-0.0894517\pi$
$522$		0			0
$523$	35.9210			1.57072			0.785358	−	0.619042i	$-0.212479\pi$
$523$	35.9210			1.57072			0.785358	+	0.619042i	$0.212479\pi$
$524$		0			0
$525$		−	2.01693i		−	0.0880259i
$526$		0			0
$527$	0.451833			0.0196821
$528$		0			0
$529$	12.7941	+	19.1131i	0.556267	+	0.831004i
$530$		0			0
$531$			7.64910i			0.331943i
$532$		0			0
$533$	−0.726155			−0.0314533
$534$		0			0
$535$		−	15.7693i		−	0.681767i
$536$		0			0
$537$	23.5203			1.01498
$538$		0			0
$539$	2.04588			0.0881224
$540$		0			0
$541$	−15.5417			−0.668191			−0.334095	−	0.942539i	$-0.608431\pi$
$541$	−15.5417			−0.668191			−0.334095	+	0.942539i	$0.608431\pi$
$542$		0			0
$543$	9.77478			0.419476
$544$		0			0
$545$	7.29036			0.312285
$546$		0			0
$547$		−	12.4468i		−	0.532187i	−0.963947	−	0.266093i	$-0.914267\pi$
$547$		−	12.4468i		−	0.532187i	0.963947	−	0.266093i	$-0.0857330\pi$
$548$		0			0
$549$		−	4.18983i		−	0.178817i
$550$		0			0
$551$	−74.3390			−3.16695
$552$		0			0
$553$	19.5286			0.830438
$554$		0			0
$555$			4.89822i			0.207918i
$556$		0			0
$557$		−	24.8342i		−	1.05226i	−0.850404	−	0.526130i	$-0.823643\pi$
$557$		−	24.8342i		−	1.05226i	0.850404	−	0.526130i	$-0.176357\pi$
$558$		0			0
$559$	1.14343			0.0483621
$560$		0			0
$561$	0.300060			0.0126685
$562$		0			0
$563$	41.0729			1.73101			0.865507	−	0.500896i	$-0.166996\pi$
$563$	41.0729			1.73101			0.865507	+	0.500896i	$0.166996\pi$
$564$		0			0
$565$	11.6243			0.489038
$566$		0			0
$567$	−2.01693			−0.0847030
$568$		0			0
$569$		−	32.8724i		−	1.37808i	−0.724722	−	0.689041i	$-0.758032\pi$
$569$		−	32.8724i		−	1.37808i	0.724722	−	0.689041i	$-0.241968\pi$
$570$		0			0
$571$	−38.5411			−1.61290			−0.806448	−	0.591305i	$-0.798613\pi$
$571$	−38.5411			−1.61290			−0.806448	+	0.591305i	$0.798613\pi$
$572$		0			0
$573$		−	7.21224i		−	0.301296i
$574$		0			0
$575$	4.23049	+	2.25897i	0.176424	+	0.0942054i
$576$		0			0
$577$	33.2323			1.38348			0.691740	−	0.722147i	$-0.256844\pi$
$577$	33.2323			1.38348			0.691740	+	0.722147i	$0.256844\pi$
$578$		0			0
$579$		−	24.9860i		−	1.03838i
$580$		0			0
$581$	−1.98102			−0.0821867
$582$		0			0
$583$			0.323892i			0.0134142i
$584$		0			0
$585$			0.195235i			0.00807199i
$586$		0			0
$587$		−	10.1033i		−	0.417008i	−0.978021	−	0.208504i	$-0.933141\pi$
$587$		−	10.1033i		−	0.417008i	0.978021	−	0.208504i	$-0.0668595\pi$
$588$		0			0
$589$			8.10108i			0.333799i
$590$		0			0
$591$			6.96165i			0.286364i
$592$		0			0
$593$	−0.289921			−0.0119056			−0.00595281	−	0.999982i	$-0.501895\pi$
$593$	−0.289921			−0.0119056			−0.00595281	+	0.999982i	$0.501895\pi$
$594$		0			0
$595$	0.867327			0.0355569
$596$		0			0
$597$		−	2.94085i		−	0.120361i
$598$		0			0
$599$		−	37.0346i		−	1.51319i	−0.653883	−	0.756596i	$-0.726861\pi$
$599$		−	37.0346i		−	1.51319i	0.653883	−	0.756596i	$-0.273139\pi$
$600$		0			0
$601$	−38.1805			−1.55742			−0.778708	−	0.627387i	$-0.784125\pi$
$601$	−38.1805			−1.55742			−0.778708	+	0.627387i	$0.784125\pi$
$602$		0			0
$603$	3.70657			0.150943
$604$		0			0
$605$			10.5131i			0.427419i
$606$		0			0
$607$			39.4410i			1.60086i	0.599426	+	0.800430i	$0.295395\pi$
$607$			39.4410i			1.60086i	−0.599426	+	0.800430i	$0.704605\pi$
$608$		0			0
$609$		−	19.4468i		−	0.788025i
$610$		0			0
$611$			2.08441i			0.0843261i
$612$		0			0
$613$		−	12.2889i		−	0.496345i	−0.968716	−	0.248173i	$-0.920170\pi$
$613$		−	12.2889i		−	0.496345i	0.968716	−	0.248173i	$-0.0798300\pi$
$614$		0			0
$615$	3.71938			0.149980
$616$		0			0
$617$			10.2034i			0.410773i	0.978681	+	0.205386i	$0.0658451\pi$
$617$			10.2034i			0.410773i	−0.978681	+	0.205386i	$0.934155\pi$
$618$		0			0
$619$	−31.5655			−1.26873			−0.634363	−	0.773036i	$-0.718737\pi$
$619$	−31.5655			−1.26873			−0.634363	+	0.773036i	$0.718737\pi$
$620$		0			0
$621$	2.25897	−	4.23049i	0.0906492	−	0.169764i
$622$		0			0
$623$		−	11.0548i		−	0.442901i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			5.37989i			0.214852i
$628$		0			0
$629$	−2.10635			−0.0839857
$630$		0			0
$631$	18.4861			0.735919			0.367960	−	0.929842i	$-0.380057\pi$
$631$	18.4861			0.735919			0.367960	+	0.929842i	$0.380057\pi$
$632$		0			0
$633$	28.0259			1.11393
$634$		0			0
$635$	−5.20841			−0.206690
$636$		0			0
$637$	−0.572431			−0.0226806
$638$		0			0
$639$		−	5.69569i		−	0.225318i
$640$		0			0
$641$			31.0234i			1.22535i	0.790335	+	0.612674i	$0.209906\pi$
$641$			31.0234i			1.22535i	−0.790335	+	0.612674i	$0.790094\pi$
$642$		0			0
$643$	4.62461			0.182377			0.0911884	−	0.995834i	$-0.470933\pi$
$643$	4.62461			0.182377			0.0911884	+	0.995834i	$0.470933\pi$
$644$		0			0
$645$	−5.85669			−0.230607
$646$		0			0
$647$		−	20.6906i		−	0.813433i	−0.913554	−	0.406716i	$-0.866674\pi$
$647$		−	20.6906i		−	0.813433i	0.913554	−	0.406716i	$-0.133326\pi$
$648$		0			0
$649$			5.33736i			0.209510i
$650$		0			0
$651$	−2.11922			−0.0830586
$652$		0			0
$653$	−44.7001			−1.74925			−0.874626	−	0.484799i	$-0.838893\pi$
$653$	−44.7001			−1.74925			−0.874626	+	0.484799i	$0.838893\pi$
$654$		0			0
$655$	−2.04355			−0.0798480
$656$		0			0
$657$	3.51188			0.137011
$658$		0			0
$659$	−4.80658			−0.187238			−0.0936190	−	0.995608i	$-0.529844\pi$
$659$	−4.80658			−0.187238			−0.0936190	+	0.995608i	$0.529844\pi$
$660$		0			0
$661$		−	37.6532i		−	1.46454i	−0.681014	−	0.732270i	$-0.738461\pi$
$661$		−	37.6532i		−	1.46454i	0.681014	−	0.732270i	$-0.261539\pi$
$662$		0			0
$663$	−0.0839559			−0.00326058
$664$		0			0
$665$			15.5506i			0.603028i
$666$		0			0
$667$	40.7896	+	21.7805i	1.57938	+	0.843346i
$668$		0			0
$669$	−1.81830			−0.0702997
$670$		0			0
$671$		−	2.92356i		−	0.112863i
$672$		0			0
$673$	−29.9838			−1.15579			−0.577895	−	0.816111i	$-0.696126\pi$
$673$	−29.9838			−1.15579			−0.577895	+	0.816111i	$0.696126\pi$
$674$		0			0
$675$		−	1.00000i		−	0.0384900i
$676$		0			0
$677$		−	26.9203i		−	1.03463i	−0.855794	−	0.517316i	$-0.826931\pi$
$677$		−	26.9203i		−	1.03463i	0.855794	−	0.517316i	$-0.173069\pi$
$678$		0			0
$679$			27.3401i			1.04922i
$680$		0			0
$681$			15.7824i			0.604783i
$682$		0			0
$683$		−	34.8703i		−	1.33427i	−0.744935	−	0.667137i	$-0.767520\pi$
$683$		−	34.8703i		−	1.33427i	0.744935	−	0.667137i	$-0.232480\pi$
$684$		0			0
$685$	21.0004			0.802385
$686$		0			0
$687$	−7.05803			−0.269281
$688$		0			0
$689$		−	0.0906239i		−	0.00345250i
$690$		0			0
$691$		−	5.22615i		−	0.198812i	−0.995047	−	0.0994061i	$-0.968306\pi$
$691$		−	5.22615i		−	0.198812i	0.995047	−	0.0994061i	$-0.0316943\pi$
$692$		0			0
$693$	−1.40736			−0.0534613
$694$		0			0
$695$	2.56106			0.0971467
$696$		0			0
$697$			1.59942i			0.0605824i
$698$		0			0
$699$			5.71879i			0.216305i
$700$		0			0
$701$			15.5942i			0.588985i	0.955654	+	0.294493i	$0.0951507\pi$
$701$			15.5942i			0.588985i	−0.955654	+	0.294493i	$0.904849\pi$
$702$		0			0
$703$		−	37.7656i		−	1.42436i
$704$		0			0
$705$		−	10.6764i		−	0.402096i
$706$		0			0
$707$	19.9149			0.748978
$708$		0			0
$709$			31.2212i			1.17254i	0.810117	+	0.586268i	$0.199404\pi$
$709$			31.2212i			1.17254i	−0.810117	+	0.586268i	$0.800596\pi$
$710$		0			0
$711$	9.68233			0.363116
$712$		0			0
$713$	2.37353	−	4.44504i	0.0888894	−	0.166468i
$714$		0			0
$715$			0.136231i			0.00509473i
$716$		0			0
$717$	−20.6125			−0.769789
$718$		0			0
$719$			2.74815i			0.102489i	0.998686	+	0.0512443i	$0.0163187\pi$
$719$			2.74815i			0.102489i	−0.998686	+	0.0512443i	$0.983681\pi$
$720$		0			0
$721$	−15.4321			−0.574721
$722$		0			0
$723$	−23.5987			−0.877646
$724$		0			0
$725$	9.64181			0.358088
$726$		0			0
$727$	−28.4656			−1.05573			−0.527865	−	0.849328i	$-0.677007\pi$
$727$	−28.4656			−1.05573			−0.527865	+	0.849328i	$0.677007\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		−	2.51852i		−	0.0931506i
$732$		0			0
$733$			31.8684i			1.17709i	0.808466	+	0.588543i	$0.200298\pi$
$733$			31.8684i			1.17709i	−0.808466	+	0.588543i	$0.799702\pi$
$734$		0			0
$735$	2.93201			0.108149
$736$		0			0
$737$	2.58635			0.0952695
$738$		0			0
$739$			19.4238i			0.714515i	0.934006	+	0.357258i	$0.116288\pi$
$739$			19.4238i			0.714515i	−0.934006	+	0.357258i	$0.883712\pi$
$740$		0			0
$741$		−	1.50528i		−	0.0552978i
$742$		0			0
$743$	−18.2199			−0.668425			−0.334212	−	0.942498i	$-0.608470\pi$
$743$	−18.2199			−0.668425			−0.334212	+	0.942498i	$0.608470\pi$
$744$		0			0
$745$	9.82277			0.359878
$746$		0			0
$747$	−0.982199			−0.0359368
$748$		0			0
$749$	−31.8055			−1.16215
$750$		0			0
$751$	−50.1935			−1.83159			−0.915794	−	0.401649i	$-0.868437\pi$
$751$	−50.1935			−1.83159			−0.915794	+	0.401649i	$0.868437\pi$
$752$		0			0
$753$		−	1.45913i		−	0.0531736i
$754$		0			0
$755$	14.8419			0.540152
$756$		0			0
$757$		−	24.7967i		−	0.901250i	−0.892713	−	0.450625i	$-0.851201\pi$
$757$		−	24.7967i		−	0.901250i	0.892713	−	0.450625i	$-0.148799\pi$
$758$		0			0
$759$	1.57625	−	2.95193i	0.0572143	−	0.107148i
$760$		0			0
$761$	−30.6359			−1.11055			−0.555275	−	0.831667i	$-0.687387\pi$
$761$	−30.6359			−1.11055			−0.555275	+	0.831667i	$0.687387\pi$
$762$		0			0
$763$		−	14.7041i		−	0.532325i
$764$		0			0
$765$	0.430024			0.0155475
$766$		0			0
$767$		−	1.49338i		−	0.0539227i
$768$		0			0
$769$		−	35.2375i		−	1.27070i	−0.772226	−	0.635348i	$-0.780857\pi$
$769$		−	35.2375i		−	1.27070i	0.772226	−	0.635348i	$-0.219143\pi$
$770$		0			0
$771$			20.3730i			0.733716i
$772$		0			0
$773$			51.1400i			1.83938i	0.392650	+	0.919688i	$0.371559\pi$
$773$			51.1400i			1.83938i	−0.392650	+	0.919688i	$0.628441\pi$
$774$		0			0
$775$		−	1.05072i		−	0.0377428i
$776$		0			0
$777$	9.87935			0.354420
$778$		0			0
$779$	−28.6767			−1.02745
$780$		0			0
$781$		−	3.97431i		−	0.142212i
$782$		0			0
$783$		−	9.64181i		−	0.344570i
$784$		0			0
$785$	1.12666			0.0402122
$786$		0			0
$787$	−36.8424			−1.31329			−0.656645	−	0.754200i	$-0.728025\pi$
$787$	−36.8424			−1.31329			−0.656645	+	0.754200i	$0.728025\pi$
$788$		0			0
$789$		−	6.84599i		−	0.243724i
$790$		0			0
$791$		−	23.4454i		−	0.833622i
$792$		0			0
$793$			0.818003i			0.0290481i
$794$		0			0
$795$			0.464177i			0.0164627i
$796$		0			0
$797$			24.8874i			0.881556i	0.897616	+	0.440778i	$0.145297\pi$
$797$			24.8874i			0.881556i	−0.897616	+	0.440778i	$0.854703\pi$
$798$		0			0
$799$	4.59110			0.162421
$800$		0			0
$801$		−	5.48101i		−	0.193662i
$802$		0			0
$803$	2.45050			0.0864764
$804$		0			0
$805$	4.55617	−	8.53260i	0.160584	−	0.300734i
$806$		0			0
$807$			15.7765i			0.555360i
$808$		0			0
$809$	−39.4308			−1.38631			−0.693156	−	0.720787i	$-0.743780\pi$
$809$	−39.4308			−1.38631			−0.693156	+	0.720787i	$0.743780\pi$
$810$		0			0
$811$		−	23.8345i		−	0.836943i	−0.908230	−	0.418471i	$-0.862566\pi$
$811$		−	23.8345i		−	0.836943i	0.908230	−	0.418471i	$-0.137434\pi$
$812$		0			0
$813$	15.4121			0.540524
$814$		0			0
$815$	17.1655			0.601282
$816$		0			0
$817$	45.1554			1.57979
$818$		0			0
$819$	0.393776			0.0137596
$820$		0			0
$821$	0.770729			0.0268986			0.0134493	−	0.999910i	$-0.495719\pi$
$821$	0.770729			0.0268986			0.0134493	+	0.999910i	$0.495719\pi$
$822$		0			0
$823$			15.2384i			0.531177i	0.964086	+	0.265589i	$0.0855663\pi$
$823$			15.2384i			0.531177i	−0.964086	+	0.265589i	$0.914434\pi$
$824$		0			0
$825$		−	0.697775i		−	0.0242934i
$826$		0			0
$827$	24.9491			0.867567			0.433783	−	0.901017i	$-0.357178\pi$
$827$	24.9491			0.867567			0.433783	+	0.901017i	$0.357178\pi$
$828$		0			0
$829$	27.4879			0.954695			0.477348	−	0.878715i	$-0.341598\pi$
$829$	27.4879			0.954695			0.477348	+	0.878715i	$0.341598\pi$
$830$		0			0
$831$		−	7.24637i		−	0.251374i
$832$		0			0
$833$			1.26083i			0.0436853i
$834$		0			0
$835$	−0.809244			−0.0280050
$836$		0			0
$837$	−1.05072			−0.0363180
$838$		0			0
$839$	9.34476			0.322617			0.161309	−	0.986904i	$-0.448429\pi$
$839$	9.34476			0.322617			0.161309	+	0.986904i	$0.448429\pi$
$840$		0			0
$841$	63.9646			2.20567
$842$		0			0
$843$	0.321739			0.0110813
$844$		0			0
$845$			12.9619i			0.445902i
$846$		0			0
$847$	21.2042			0.728584
$848$		0			0
$849$		−	16.0317i		−	0.550208i
$850$		0			0
$851$	−11.0649	+	20.7219i	−0.379300	+	0.710337i
$852$		0			0
$853$	22.9627			0.786228			0.393114	−	0.919490i	$-0.371398\pi$
$853$	22.9627			0.786228			0.393114	+	0.919490i	$0.371398\pi$
$854$		0			0
$855$			7.71006i			0.263679i
$856$		0			0
$857$	22.9504			0.783972			0.391986	−	0.919971i	$-0.371788\pi$
$857$	22.9504			0.783972			0.391986	+	0.919971i	$0.371788\pi$
$858$		0			0
$859$		−	3.21344i		−	0.109641i	−0.998496	−	0.0548206i	$-0.982541\pi$
$859$		−	3.21344i		−	0.109641i	0.998496	−	0.0548206i	$-0.0174587\pi$
$860$		0			0
$861$		−	7.50172i		−	0.255658i
$862$		0			0
$863$		−	47.8566i		−	1.62906i	−0.580122	−	0.814529i	$-0.696995\pi$
$863$		−	47.8566i		−	1.62906i	0.580122	−	0.814529i	$-0.303005\pi$
$864$		0			0
$865$		−	0.192946i		−	0.00656038i
$866$		0			0
$867$		−	16.8151i		−	0.571070i
$868$		0			0
$869$	6.75609			0.229185
$870$		0			0
$871$	−0.723654			−0.0245201
$872$		0			0
$873$			13.5553i			0.458779i
$874$		0			0
$875$		−	2.01693i		−	0.0681846i
$876$		0			0
$877$	12.5419			0.423512			0.211756	−	0.977323i	$-0.432082\pi$
$877$	12.5419			0.423512			0.211756	+	0.977323i	$0.432082\pi$
$878$		0			0
$879$	6.96765			0.235013
$880$		0			0
$881$			14.1413i			0.476431i	0.971212	+	0.238216i	$0.0765625\pi$
$881$			14.1413i			0.476431i	−0.971212	+	0.238216i	$0.923437\pi$
$882$		0			0
$883$		−	31.3945i		−	1.05651i	−0.849086	−	0.528255i	$-0.822846\pi$
$883$		−	31.3945i		−	1.05651i	0.849086	−	0.528255i	$-0.177154\pi$
$884$		0			0
$885$			7.64910i			0.257122i
$886$		0			0
$887$		−	30.6407i		−	1.02881i	−0.857546	−	0.514407i	$-0.828012\pi$
$887$		−	30.6407i		−	1.02881i	0.857546	−	0.514407i	$-0.171988\pi$
$888$		0			0
$889$			10.5050i			0.352326i
$890$		0			0
$891$	−0.697775			−0.0233764
$892$		0			0
$893$			82.3156i			2.75459i
$894$		0			0
$895$	23.5203			0.786196
$896$		0			0
$897$	−0.441030	+	0.825942i	−0.0147256	+	0.0275774i
$898$		0			0
$899$		−	10.1308i		−	0.337881i
$900$		0			0
$901$	−0.199607			−0.00664988
$902$		0			0
$903$			11.8125i			0.393096i
$904$		0			0
$905$	9.77478			0.324925
$906$		0			0
$907$	−26.7078			−0.886818			−0.443409	−	0.896319i	$-0.646231\pi$
$907$	−26.7078			−0.886818			−0.443409	+	0.896319i	$0.646231\pi$
$908$		0			0
$909$	9.87390			0.327497
$910$		0			0
$911$	−10.5901			−0.350865			−0.175432	−	0.984492i	$-0.556132\pi$
$911$	−10.5901			−0.350865			−0.175432	+	0.984492i	$0.556132\pi$
$912$		0			0
$913$	−0.685354			−0.0226819
$914$		0			0
$915$		−	4.18983i		−	0.138511i
$916$		0			0
$917$			4.12169i			0.136110i
$918$		0			0
$919$	−32.8569			−1.08385			−0.541925	−	0.840427i	$-0.682304\pi$
$919$	−32.8569			−1.08385			−0.541925	+	0.840427i	$0.682304\pi$
$920$		0			0
$921$	−23.8378			−0.785481
$922$		0			0
$923$			1.11200i			0.0366019i
$924$		0			0
$925$			4.89822i			0.161052i
$926$		0			0
$927$	−7.65129			−0.251301
$928$		0			0
$929$	34.6352			1.13634			0.568172	−	0.822910i	$-0.307651\pi$
$929$	34.6352			1.13634			0.568172	+	0.822910i	$0.307651\pi$
$930$		0			0
$931$	−22.6060			−0.740880
$932$		0			0
$933$	−2.02594			−0.0663262
$934$		0			0
$935$	0.300060			0.00981301
$936$		0			0
$937$			39.0172i			1.27464i	0.770601	+	0.637318i	$0.219956\pi$
$937$			39.0172i			1.27464i	−0.770601	+	0.637318i	$0.780044\pi$
$938$		0			0
$939$	−16.4162			−0.535723
$940$		0			0
$941$		−	16.3857i		−	0.534159i	−0.963675	−	0.267079i	$-0.913941\pi$
$941$		−	16.3857i		−	0.534159i	0.963675	−	0.267079i	$-0.0860586\pi$
$942$		0			0
$943$	15.7348	+	8.40195i	0.512396	+	0.273605i
$944$		0			0
$945$	−2.01693			−0.0656106
$946$		0			0
$947$		−	48.1701i		−	1.56532i	−0.622451	−	0.782658i	$-0.713863\pi$
$947$		−	48.1701i		−	1.56532i	0.622451	−	0.782658i	$-0.286137\pi$
$948$		0			0
$949$	−0.685643			−0.0222569
$950$		0			0
$951$			21.2081i			0.687720i
$952$		0			0
$953$		−	37.1941i		−	1.20483i	−0.798181	−	0.602417i	$-0.794204\pi$
$953$		−	37.1941i		−	1.20483i	0.798181	−	0.602417i	$-0.205796\pi$
$954$		0			0
$955$		−	7.21224i		−	0.233383i
$956$		0			0
$957$		−	6.72782i		−	0.217480i
$958$		0			0
$959$		−	42.3563i		−	1.36776i
$960$		0			0
$961$	29.8960			0.964387
$962$		0			0
$963$	−15.7693			−0.508159
$964$		0			0
$965$		−	24.9860i		−	0.804328i
$966$		0			0
$967$		−	9.39440i		−	0.302103i	−0.988526	−	0.151052i	$-0.951734\pi$
$967$		−	9.39440i		−	0.302103i	0.988526	−	0.151052i	$-0.0482660\pi$
$968$		0			0
$969$	−3.31551			−0.106510
$970$		0			0
$971$	−50.5564			−1.62243			−0.811216	−	0.584747i	$-0.801194\pi$
$971$	−50.5564			−1.62243			−0.811216	+	0.584747i	$0.801194\pi$
$972$		0			0
$973$		−	5.16548i		−	0.165598i
$974$		0			0
$975$			0.195235i			0.00625254i
$976$		0			0
$977$			23.7373i			0.759424i	0.925105	+	0.379712i	$0.123977\pi$
$977$			23.7373i			0.759424i	−0.925105	+	0.379712i	$0.876023\pi$
$978$		0			0
$979$		−	3.82451i		−	0.122232i
$980$		0			0
$981$		−	7.29036i		−	0.232763i
$982$		0			0
$983$	8.88451			0.283372			0.141686	−	0.989912i	$-0.454748\pi$
$983$	8.88451			0.283372			0.141686	+	0.989912i	$0.454748\pi$
$984$		0			0
$985$			6.96165i			0.221816i
$986$		0			0
$987$	−21.5335			−0.685418
$988$		0			0
$989$	−24.7767	−	13.2301i	−0.787852	−	0.420691i
$990$		0			0
$991$		−	30.2770i		−	0.961781i	−0.876781	−	0.480890i	$-0.840314\pi$
$991$		−	30.2770i		−	0.961781i	0.876781	−	0.480890i	$-0.159686\pi$
$992$		0			0
$993$	−2.15797			−0.0684810
$994$		0			0
$995$		−	2.94085i		−	0.0932312i
$996$		0			0
$997$	−46.1127			−1.46040			−0.730202	−	0.683232i	$-0.760574\pi$
$997$	−46.1127			−1.46040			−0.730202	+	0.683232i	$0.760574\pi$
$998$		0			0
$999$	4.89822			0.154973

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.be.c.1471.19	✓	32
4.3	odd	2		5520.2.be.d.1471.20	yes	32
23.22	odd	2		5520.2.be.d.1471.19	yes	32
92.91	even	2	inner	5520.2.be.c.1471.20	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5520.2.be.c.1471.19	✓	32	1.1	even	1	trivial
5520.2.be.c.1471.20	yes	32	92.91	even	2	inner
5520.2.be.d.1471.19	yes	32	23.22	odd	2
5520.2.be.d.1471.20	yes	32	4.3	odd	2

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:	\( N \)	\(=\)	\( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5520.be (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -1.00000i q^{5} -2.01693 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -1.00000i q^{5} -2.01693 q^{7} -1.00000 q^{9} -0.697775 q^{11} +0.195235 q^{13} -1.00000 q^{15} -0.430024i q^{17} +7.71006 q^{19} +2.01693i q^{21} +(-4.23049 - 2.25897i) q^{23} -1.00000 q^{25} +1.00000i q^{27} -9.64181 q^{29} +1.05072i q^{31} +0.697775i q^{33} +2.01693i q^{35} -4.89822i q^{37} -0.195235i q^{39} -3.71938 q^{41} +5.85669 q^{43} +1.00000i q^{45} +10.6764i q^{47} -2.93201 q^{49} -0.430024 q^{51} -0.464177i q^{53} +0.697775i q^{55} -7.71006i q^{57} -7.64910i q^{59} +4.18983i q^{61} +2.01693 q^{63} -0.195235i q^{65} -3.70657 q^{67} +(-2.25897 + 4.23049i) q^{69} +5.69569i q^{71} -3.51188 q^{73} +1.00000i q^{75} +1.40736 q^{77} -9.68233 q^{79} +1.00000 q^{81} +0.982199 q^{83} -0.430024 q^{85} +9.64181i q^{87} +5.48101i q^{89} -0.393776 q^{91} +1.05072 q^{93} -7.71006i q^{95} -13.5553i q^{97} +0.697775 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 8 q^{7} - 32 q^{9}+O(q^{10}) \) 32 * q - 8 * q^7 - 32 * q^9	\( 32 q - 8 q^{7} - 32 q^{9} + 8 q^{11} - 8 q^{13} - 32 q^{15} - 32 q^{25} + 4 q^{29} + 20 q^{41} + 52 q^{49} - 4 q^{51} + 8 q^{63} + 32 q^{67} - 40 q^{73} - 24 q^{77} + 32 q^{79} + 32 q^{81} - 4 q^{85} - 48 q^{91} - 8 q^{99}+O(q^{100}) \) 32 * q - 8 * q^7 - 32 * q^9 + 8 * q^11 - 8 * q^13 - 32 * q^15 - 32 * q^25 + 4 * q^29 + 20 * q^41 + 52 * q^49 - 4 * q^51 + 8 * q^63 + 32 * q^67 - 40 * q^73 - 24 * q^77 + 32 * q^79 + 32 * q^81 - 4 * q^85 - 48 * q^91 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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