LMFDB - Embedded newform 1575.2.d.i.1324.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,2,Mod(1324,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1575.1324");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1575.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:	$12.5764383184$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1324.2
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1575.1324
Dual form			1575.2.d.i.1324.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.73205i q^{8} +O(q^{10})$	$q-1.73205i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.73205i q^{8} -3.46410 q^{11} -2.00000i q^{13} +1.73205 q^{14} -5.00000 q^{16} -3.46410i q^{17} +4.00000 q^{19} +6.00000i q^{22} -3.46410i q^{23} -3.46410 q^{26} -1.00000i q^{28} -4.00000 q^{31} +5.19615i q^{32} -6.00000 q^{34} +2.00000i q^{37} -6.92820i q^{38} -10.3923 q^{41} +4.00000i q^{43} +3.46410 q^{44} -6.00000 q^{46} -6.92820i q^{47} -1.00000 q^{49} +2.00000i q^{52} -6.92820i q^{53} +1.73205 q^{56} -6.92820 q^{59} -10.0000 q^{61} +6.92820i q^{62} -1.00000 q^{64} -4.00000i q^{67} +3.46410i q^{68} +10.3923 q^{71} -14.0000i q^{73} +3.46410 q^{74} -4.00000 q^{76} -3.46410i q^{77} -8.00000 q^{79} +18.0000i q^{82} +6.92820 q^{86} +6.00000i q^{88} -3.46410 q^{89} +2.00000 q^{91} +3.46410i q^{92} -12.0000 q^{94} +14.0000i q^{97} +1.73205i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} - 20 q^{16} + 16 q^{19} - 16 q^{31} - 24 q^{34} - 24 q^{46} - 4 q^{49} - 40 q^{61} - 4 q^{64} - 16 q^{76} - 32 q^{79} + 8 q^{91} - 48 q^{94}+O(q^{100}) $ 4 * q - 4 * q^4 - 20 * q^16 + 16 * q^19 - 16 * q^31 - 24 * q^34 - 24 * q^46 - 4 * q^49 - 40 * q^61 - 4 * q^64 - 16 * q^76 - 32 * q^79 + 8 * q^91 - 48 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$1226$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	− 1.73205i	− 1.22474i	−0.790569	−	0.612372i	$-0.790215\pi$
$2$	− 1.73205i	− 1.22474i	0.790569	−	0.612372i	$-0.209785\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	− 1.73205i	− 0.612372i
$9$	0	0
$10$	0	0
$11$	−3.46410	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$12$	0	0
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	1.73205	0.462910
$15$	0	0
$16$	−5.00000	−1.25000
$17$	− 3.46410i	− 0.840168i	−0.907485	−	0.420084i	$-0.862001\pi$
$17$	− 3.46410i	− 0.840168i	0.907485	−	0.420084i	$-0.137999\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	6.00000i	1.27920i
$23$	− 3.46410i	− 0.722315i	−0.932505	−	0.361158i	$-0.882382\pi$
$23$	− 3.46410i	− 0.722315i	0.932505	−	0.361158i	$-0.117618\pi$
$24$	0	0
$25$	0	0
$26$	−3.46410	−0.679366
$27$	0	0
$28$	− 1.00000i	− 0.188982i
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	5.19615i	0.918559i
$33$	0	0
$34$	−6.00000	−1.02899
$35$	0	0
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	− 6.92820i	− 1.12390i
$39$	0	0
$40$	0	0
$41$	−10.3923	−1.62301	−0.811503	−	0.584349i	$-0.801350\pi$
$41$	−10.3923	−1.62301	−0.811503	+	0.584349i	$0.801350\pi$
$42$	0	0
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	3.46410	0.522233
$45$	0	0
$46$	−6.00000	−0.884652
$47$	− 6.92820i	− 1.01058i	−0.862949	−	0.505291i	$-0.831385\pi$
$47$	− 6.92820i	− 1.01058i	0.862949	−	0.505291i	$-0.168615\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	2.00000i	0.277350i
$53$	− 6.92820i	− 0.951662i	−0.879537	−	0.475831i	$-0.842147\pi$
$53$	− 6.92820i	− 0.951662i	0.879537	−	0.475831i	$-0.157853\pi$
$54$	0	0
$55$	0	0
$56$	1.73205	0.231455
$57$	0	0
$58$	0	0
$59$	−6.92820	−0.901975	−0.450988	−	0.892530i	$-0.648928\pi$
$59$	−6.92820	−0.901975	−0.450988	+	0.892530i	$0.648928\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	6.92820i	0.879883i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	3.46410i	0.420084i
$69$	0	0
$70$	0	0
$71$	10.3923	1.23334	0.616670	−	0.787222i	$-0.288481\pi$
$71$	10.3923	1.23334	0.616670	+	0.787222i	$0.288481\pi$
$72$	0	0
$73$	− 14.0000i	− 1.63858i	−0.573382	−	0.819288i	$-0.694369\pi$
$73$	− 14.0000i	− 1.63858i	0.573382	−	0.819288i	$-0.305631\pi$
$74$	3.46410	0.402694
$75$	0	0
$76$	−4.00000	−0.458831
$77$	− 3.46410i	− 0.394771i
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	0	0
$82$	18.0000i	1.98777i
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	6.92820	0.747087
$87$	0	0
$88$	6.00000i	0.639602i
$89$	−3.46410	−0.367194	−0.183597	−	0.983002i	$-0.558774\pi$
$89$	−3.46410	−0.367194	−0.183597	+	0.983002i	$0.558774\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	3.46410i	0.361158i
$93$	0	0
$94$	−12.0000	−1.23771
$95$	0	0
$96$	0	0
$97$	14.0000i	1.42148i	0.703452	+	0.710742i	$0.251641\pi$
$97$	14.0000i	1.42148i	−0.703452	+	0.710742i	$0.748359\pi$
$98$	1.73205i	0.174964i
$99$	0	0
$100$	0	0
$101$	−3.46410	−0.344691	−0.172345	−	0.985037i	$-0.555135\pi$
$101$	−3.46410	−0.344691	−0.172345	+	0.985037i	$0.555135\pi$
$102$	0	0
$103$	4.00000i	0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$	4.00000i	0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	−3.46410	−0.339683
$105$	0	0
$106$	−12.0000	−1.16554
$107$	− 17.3205i	− 1.67444i	−0.546869	−	0.837218i	$-0.684180\pi$
$107$	− 17.3205i	− 1.67444i	0.546869	−	0.837218i	$-0.315820\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	0	0
$112$	− 5.00000i	− 0.472456i
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	12.0000i	1.10469i
$119$	3.46410	0.317554
$120$	0	0
$121$	1.00000	0.0909091
$122$	17.3205i	1.56813i
$123$	0	0
$124$	4.00000	0.359211
$125$	0	0
$126$	0	0
$127$	8.00000i	0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$	8.00000i	0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$	12.1244i	1.07165i
$129$	0	0
$130$	0	0
$131$	13.8564	1.21064	0.605320	−	0.795982i	$-0.293045\pi$
$131$	13.8564	1.21064	0.605320	+	0.795982i	$0.293045\pi$
$132$	0	0
$133$	4.00000i	0.346844i
$134$	−6.92820	−0.598506
$135$	0	0
$136$	−6.00000	−0.514496
$137$	6.92820i	0.591916i	0.955201	+	0.295958i	$0.0956389\pi$
$137$	6.92820i	0.591916i	−0.955201	+	0.295958i	$0.904361\pi$
$138$	0	0
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	0	0
$142$	− 18.0000i	− 1.51053i
$143$	6.92820i	0.579365i
$144$	0	0
$145$	0	0
$146$	−24.2487	−2.00684
$147$	0	0
$148$	− 2.00000i	− 0.164399i
$149$	6.92820	0.567581	0.283790	−	0.958886i	$-0.408408\pi$
$149$	6.92820	0.567581	0.283790	+	0.958886i	$0.408408\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	− 6.92820i	− 0.561951i
$153$	0	0
$154$	−6.00000	−0.483494
$155$	0	0
$156$	0	0
$157$	− 10.0000i	− 0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$	− 10.0000i	− 0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$	13.8564i	1.10236i
$159$	0	0
$160$	0	0
$161$	3.46410	0.273009
$162$	0	0
$163$	− 20.0000i	− 1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$	− 20.0000i	− 1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$	10.3923	0.811503
$165$	0	0
$166$	0	0
$167$	20.7846i	1.60836i	0.594385	+	0.804181i	$0.297396\pi$
$167$	20.7846i	1.60836i	−0.594385	+	0.804181i	$0.702604\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	− 4.00000i	− 0.304997i
$173$	17.3205i	1.31685i	0.752645	+	0.658427i	$0.228778\pi$
$173$	17.3205i	1.31685i	−0.752645	+	0.658427i	$0.771222\pi$
$174$	0	0
$175$	0	0
$176$	17.3205	1.30558
$177$	0	0
$178$	6.00000i	0.449719i
$179$	−17.3205	−1.29460	−0.647298	−	0.762237i	$-0.724101\pi$
$179$	−17.3205	−1.29460	−0.647298	+	0.762237i	$0.724101\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	− 3.46410i	− 0.256776i
$183$	0	0
$184$	−6.00000	−0.442326
$185$	0	0
$186$	0	0
$187$	12.0000i	0.877527i
$188$	6.92820i	0.505291i
$189$	0	0
$190$	0	0
$191$	24.2487	1.75458	0.877288	−	0.479965i	$-0.159351\pi$
$191$	24.2487	1.75458	0.877288	+	0.479965i	$0.159351\pi$
$192$	0	0
$193$	− 14.0000i	− 1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	− 14.0000i	− 1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	24.2487	1.74096
$195$	0	0
$196$	1.00000	0.0714286
$197$	− 20.7846i	− 1.48084i	−0.672143	−	0.740421i	$-0.734626\pi$
$197$	− 20.7846i	− 1.48084i	0.672143	−	0.740421i	$-0.265374\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	0	0
$202$	6.00000i	0.422159i
$203$	0	0
$204$	0	0
$205$	0	0
$206$	6.92820	0.482711
$207$	0	0
$208$	10.0000i	0.693375i
$209$	−13.8564	−0.958468
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	6.92820i	0.475831i
$213$	0	0
$214$	−30.0000	−2.05076
$215$	0	0
$216$	0	0
$217$	− 4.00000i	− 0.271538i
$218$	3.46410i	0.234619i
$219$	0	0
$220$	0	0
$221$	−6.92820	−0.466041
$222$	0	0
$223$	− 8.00000i	− 0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$	− 8.00000i	− 0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$	−5.19615	−0.347183
$225$	0	0
$226$	0	0
$227$	6.92820i	0.459841i	0.973209	+	0.229920i	$0.0738466\pi$
$227$	6.92820i	0.459841i	−0.973209	+	0.229920i	$0.926153\pi$
$228$	0	0
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.92820i	0.453882i	0.973909	+	0.226941i	$0.0728724\pi$
$233$	6.92820i	0.453882i	−0.973909	+	0.226941i	$0.927128\pi$
$234$	0	0
$235$	0	0
$236$	6.92820	0.450988
$237$	0	0
$238$	− 6.00000i	− 0.388922i
$239$	10.3923	0.672222	0.336111	−	0.941822i	$-0.390888\pi$
$239$	10.3923	0.672222	0.336111	+	0.941822i	$0.390888\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	− 1.73205i	− 0.111340i
$243$	0	0
$244$	10.0000	0.640184
$245$	0	0
$246$	0	0
$247$	− 8.00000i	− 0.509028i
$248$	6.92820i	0.439941i
$249$	0	0
$250$	0	0
$251$	−20.7846	−1.31191	−0.655956	−	0.754799i	$-0.727735\pi$
$251$	−20.7846	−1.31191	−0.655956	+	0.754799i	$0.727735\pi$
$252$	0	0
$253$	12.0000i	0.754434i
$254$	13.8564	0.869428
$255$	0	0
$256$	19.0000	1.18750
$257$	3.46410i	0.216085i	0.994146	+	0.108042i	$0.0344582\pi$
$257$	3.46410i	0.216085i	−0.994146	+	0.108042i	$0.965542\pi$
$258$	0	0
$259$	−2.00000	−0.124274
$260$	0	0
$261$	0	0
$262$	− 24.0000i	− 1.48272i
$263$	− 17.3205i	− 1.06803i	−0.845476	−	0.534014i	$-0.820683\pi$
$263$	− 17.3205i	− 1.06803i	0.845476	−	0.534014i	$-0.179317\pi$
$264$	0	0
$265$	0	0
$266$	6.92820	0.424795
$267$	0	0
$268$	4.00000i	0.244339i
$269$	−17.3205	−1.05605	−0.528025	−	0.849229i	$-0.677067\pi$
$269$	−17.3205	−1.05605	−0.528025	+	0.849229i	$0.677067\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	17.3205i	1.05021i
$273$	0	0
$274$	12.0000	0.724947
$275$	0	0
$276$	0	0
$277$	− 10.0000i	− 0.600842i	−0.953807	−	0.300421i	$-0.902873\pi$
$277$	− 10.0000i	− 0.600842i	0.953807	−	0.300421i	$-0.0971271\pi$
$278$	− 27.7128i	− 1.66210i
$279$	0	0
$280$	0	0
$281$	20.7846	1.23991	0.619953	−	0.784639i	$-0.287152\pi$
$281$	20.7846	1.23991	0.619953	+	0.784639i	$0.287152\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	−10.3923	−0.616670
$285$	0	0
$286$	12.0000	0.709575
$287$	− 10.3923i	− 0.613438i
$288$	0	0
$289$	5.00000	0.294118
$290$	0	0
$291$	0	0
$292$	14.0000i	0.819288i
$293$	− 10.3923i	− 0.607125i	−0.952812	−	0.303562i	$-0.901824\pi$
$293$	− 10.3923i	− 0.607125i	0.952812	−	0.303562i	$-0.0981761\pi$
$294$	0	0
$295$	0	0
$296$	3.46410	0.201347
$297$	0	0
$298$	− 12.0000i	− 0.695141i
$299$	−6.92820	−0.400668
$300$	0	0
$301$	−4.00000	−0.230556
$302$	− 13.8564i	− 0.797347i
$303$	0	0
$304$	−20.0000	−1.14708
$305$	0	0
$306$	0	0
$307$	− 28.0000i	− 1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$	− 28.0000i	− 1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$	3.46410i	0.197386i
$309$	0	0
$310$	0	0
$311$	−34.6410	−1.96431	−0.982156	−	0.188069i	$-0.939777\pi$
$311$	−34.6410	−1.96431	−0.982156	+	0.188069i	$0.939777\pi$
$312$	0	0
$313$	− 2.00000i	− 0.113047i	−0.998401	−	0.0565233i	$-0.981998\pi$
$313$	− 2.00000i	− 0.113047i	0.998401	−	0.0565233i	$-0.0180015\pi$
$314$	−17.3205	−0.977453
$315$	0	0
$316$	8.00000	0.450035
$317$	− 6.92820i	− 0.389127i	−0.980890	−	0.194563i	$-0.937671\pi$
$317$	− 6.92820i	− 0.389127i	0.980890	−	0.194563i	$-0.0623290\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	− 6.00000i	− 0.334367i
$323$	− 13.8564i	− 0.770991i
$324$	0	0
$325$	0	0
$326$	−34.6410	−1.91859
$327$	0	0
$328$	18.0000i	0.993884i
$329$	6.92820	0.381964
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	0	0
$333$	0	0
$334$	36.0000	1.96983
$335$	0	0
$336$	0	0
$337$	14.0000i	0.762629i	0.924445	+	0.381314i	$0.124528\pi$
$337$	14.0000i	0.762629i	−0.924445	+	0.381314i	$0.875472\pi$
$338$	− 15.5885i	− 0.847900i
$339$	0	0
$340$	0	0
$341$	13.8564	0.750366
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	6.92820	0.373544
$345$	0	0
$346$	30.0000	1.61281
$347$	17.3205i	0.929814i	0.885360	+	0.464907i	$0.153912\pi$
$347$	17.3205i	0.929814i	−0.885360	+	0.464907i	$0.846088\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	0	0
$352$	− 18.0000i	− 0.959403i
$353$	3.46410i	0.184376i	0.995742	+	0.0921878i	$0.0293860\pi$
$353$	3.46410i	0.184376i	−0.995742	+	0.0921878i	$0.970614\pi$
$354$	0	0
$355$	0	0
$356$	3.46410	0.183597
$357$	0	0
$358$	30.0000i	1.58555i
$359$	−24.2487	−1.27980	−0.639899	−	0.768459i	$-0.721024\pi$
$359$	−24.2487	−1.27980	−0.639899	+	0.768459i	$0.721024\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 3.46410i	− 0.182069i
$363$	0	0
$364$	−2.00000	−0.104828
$365$	0	0
$366$	0	0
$367$	− 16.0000i	− 0.835193i	−0.908633	−	0.417597i	$-0.862873\pi$
$367$	− 16.0000i	− 0.835193i	0.908633	−	0.417597i	$-0.137127\pi$
$368$	17.3205i	0.902894i
$369$	0	0
$370$	0	0
$371$	6.92820	0.359694
$372$	0	0
$373$	10.0000i	0.517780i	0.965907	+	0.258890i	$0.0833568\pi$
$373$	10.0000i	0.517780i	−0.965907	+	0.258890i	$0.916643\pi$
$374$	20.7846	1.07475
$375$	0	0
$376$	−12.0000	−0.618853
$377$	0	0
$378$	0	0
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	0	0
$381$	0	0
$382$	− 42.0000i	− 2.14891i
$383$	− 13.8564i	− 0.708029i	−0.935240	−	0.354015i	$-0.884816\pi$
$383$	− 13.8564i	− 0.708029i	0.935240	−	0.354015i	$-0.115184\pi$
$384$	0	0
$385$	0	0
$386$	−24.2487	−1.23423
$387$	0	0
$388$	− 14.0000i	− 0.710742i
$389$	13.8564	0.702548	0.351274	−	0.936273i	$-0.385749\pi$
$389$	13.8564	0.702548	0.351274	+	0.936273i	$0.385749\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	1.73205i	0.0874818i
$393$	0	0
$394$	−36.0000	−1.81365
$395$	0	0
$396$	0	0
$397$	38.0000i	1.90717i	0.301131	+	0.953583i	$0.402636\pi$
$397$	38.0000i	1.90717i	−0.301131	+	0.953583i	$0.597364\pi$
$398$	− 27.7128i	− 1.38912i
$399$	0	0
$400$	0	0
$401$	−6.92820	−0.345978	−0.172989	−	0.984924i	$-0.555343\pi$
$401$	−6.92820	−0.345978	−0.172989	+	0.984924i	$0.555343\pi$
$402$	0	0
$403$	8.00000i	0.398508i
$404$	3.46410	0.172345
$405$	0	0
$406$	0	0
$407$	− 6.92820i	− 0.343418i
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	0	0
$412$	− 4.00000i	− 0.197066i
$413$	− 6.92820i	− 0.340915i
$414$	0	0
$415$	0	0
$416$	10.3923	0.509525
$417$	0	0
$418$	24.0000i	1.17388i
$419$	20.7846	1.01539	0.507697	−	0.861536i	$-0.330497\pi$
$419$	20.7846	1.01539	0.507697	+	0.861536i	$0.330497\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	− 34.6410i	− 1.68630i
$423$	0	0
$424$	−12.0000	−0.582772
$425$	0	0
$426$	0	0
$427$	− 10.0000i	− 0.483934i
$428$	17.3205i	0.837218i
$429$	0	0
$430$	0	0
$431$	−3.46410	−0.166860	−0.0834300	−	0.996514i	$-0.526587\pi$
$431$	−3.46410	−0.166860	−0.0834300	+	0.996514i	$0.526587\pi$
$432$	0	0
$433$	− 26.0000i	− 1.24948i	−0.780833	−	0.624740i	$-0.785205\pi$
$433$	− 26.0000i	− 1.24948i	0.780833	−	0.624740i	$-0.214795\pi$
$434$	−6.92820	−0.332564
$435$	0	0
$436$	2.00000	0.0957826
$437$	− 13.8564i	− 0.662842i
$438$	0	0
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	0	0
$441$	0	0
$442$	12.0000i	0.570782i
$443$	− 3.46410i	− 0.164584i	−0.996608	−	0.0822922i	$-0.973776\pi$
$443$	− 3.46410i	− 0.164584i	0.996608	−	0.0822922i	$-0.0262241\pi$
$444$	0	0
$445$	0	0
$446$	−13.8564	−0.656120
$447$	0	0
$448$	− 1.00000i	− 0.0472456i
$449$	41.5692	1.96177	0.980886	−	0.194581i	$-0.0623348\pi$
$449$	41.5692	1.96177	0.980886	+	0.194581i	$0.0623348\pi$
$450$	0	0
$451$	36.0000	1.69517
$452$	0	0
$453$	0	0
$454$	12.0000	0.563188
$455$	0	0
$456$	0	0
$457$	− 10.0000i	− 0.467780i	−0.972263	−	0.233890i	$-0.924854\pi$
$457$	− 10.0000i	− 0.467780i	0.972263	−	0.233890i	$-0.0751456\pi$
$458$	− 38.1051i	− 1.78054i
$459$	0	0
$460$	0	0
$461$	−31.1769	−1.45205	−0.726027	−	0.687666i	$-0.758635\pi$
$461$	−31.1769	−1.45205	−0.726027	+	0.687666i	$0.758635\pi$
$462$	0	0
$463$	− 32.0000i	− 1.48717i	−0.668644	−	0.743583i	$-0.733125\pi$
$463$	− 32.0000i	− 1.48717i	0.668644	−	0.743583i	$-0.266875\pi$
$464$	0	0
$465$	0	0
$466$	12.0000	0.555889
$467$	− 6.92820i	− 0.320599i	−0.987068	−	0.160300i	$-0.948754\pi$
$467$	− 6.92820i	− 0.320599i	0.987068	−	0.160300i	$-0.0512460\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	0	0
$471$	0	0
$472$	12.0000i	0.552345i
$473$	− 13.8564i	− 0.637118i
$474$	0	0
$475$	0	0
$476$	−3.46410	−0.158777
$477$	0	0
$478$	− 18.0000i	− 0.823301i
$479$	−6.92820	−0.316558	−0.158279	−	0.987394i	$-0.550594\pi$
$479$	−6.92820	−0.316558	−0.158279	+	0.987394i	$0.550594\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	17.3205i	0.788928i
$483$	0	0
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	0	0
$487$	− 40.0000i	− 1.81257i	−0.422664	−	0.906287i	$-0.638905\pi$
$487$	− 40.0000i	− 1.81257i	0.422664	−	0.906287i	$-0.361095\pi$
$488$	17.3205i	0.784063i
$489$	0	0
$490$	0	0
$491$	−10.3923	−0.468998	−0.234499	−	0.972116i	$-0.575345\pi$
$491$	−10.3923	−0.468998	−0.234499	+	0.972116i	$0.575345\pi$
$492$	0	0
$493$	0	0
$494$	−13.8564	−0.623429
$495$	0	0
$496$	20.0000	0.898027
$497$	10.3923i	0.466159i
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	0	0
$502$	36.0000i	1.60676i
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	20.7846	0.923989
$507$	0	0
$508$	− 8.00000i	− 0.354943i
$509$	−3.46410	−0.153544	−0.0767718	−	0.997049i	$-0.524461\pi$
$509$	−3.46410	−0.153544	−0.0767718	+	0.997049i	$0.524461\pi$
$510$	0	0
$511$	14.0000	0.619324
$512$	− 8.66025i	− 0.382733i
$513$	0	0
$514$	6.00000	0.264649
$515$	0	0
$516$	0	0
$517$	24.0000i	1.05552i
$518$	3.46410i	0.152204i
$519$	0	0
$520$	0	0
$521$	−3.46410	−0.151765	−0.0758825	−	0.997117i	$-0.524177\pi$
$521$	−3.46410	−0.151765	−0.0758825	+	0.997117i	$0.524177\pi$
$522$	0	0
$523$	16.0000i	0.699631i	0.936819	+	0.349816i	$0.113756\pi$
$523$	16.0000i	0.699631i	−0.936819	+	0.349816i	$0.886244\pi$
$524$	−13.8564	−0.605320
$525$	0	0
$526$	−30.0000	−1.30806
$527$	13.8564i	0.603595i
$528$	0	0
$529$	11.0000	0.478261
$530$	0	0
$531$	0	0
$532$	− 4.00000i	− 0.173422i
$533$	20.7846i	0.900281i
$534$	0	0
$535$	0	0
$536$	−6.92820	−0.299253
$537$	0	0
$538$	30.0000i	1.29339i
$539$	3.46410	0.149209
$540$	0	0
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	− 34.6410i	− 1.48796i
$543$	0	0
$544$	18.0000	0.771744
$545$	0	0
$546$	0	0
$547$	− 4.00000i	− 0.171028i	−0.996337	−	0.0855138i	$-0.972747\pi$
$547$	− 4.00000i	− 0.171028i	0.996337	−	0.0855138i	$-0.0272532\pi$
$548$	− 6.92820i	− 0.295958i
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	− 8.00000i	− 0.340195i
$554$	−17.3205	−0.735878
$555$	0	0
$556$	−16.0000	−0.678551
$557$	6.92820i	0.293557i	0.989169	+	0.146779i	$0.0468905\pi$
$557$	6.92820i	0.293557i	−0.989169	+	0.146779i	$0.953109\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	− 36.0000i	− 1.51857i
$563$	34.6410i	1.45994i	0.683477	+	0.729972i	$0.260467\pi$
$563$	34.6410i	1.45994i	−0.683477	+	0.729972i	$0.739533\pi$
$564$	0	0
$565$	0	0
$566$	6.92820	0.291214
$567$	0	0
$568$	− 18.0000i	− 0.755263i
$569$	6.92820	0.290445	0.145223	−	0.989399i	$-0.453610\pi$
$569$	6.92820	0.290445	0.145223	+	0.989399i	$0.453610\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	− 6.92820i	− 0.289683i
$573$	0	0
$574$	−18.0000	−0.751305
$575$	0	0
$576$	0	0
$577$	2.00000i	0.0832611i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$	2.00000i	0.0832611i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$	− 8.66025i	− 0.360219i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	24.0000i	0.993978i
$584$	−24.2487	−1.00342
$585$	0	0
$586$	−18.0000	−0.743573
$587$	20.7846i	0.857873i	0.903335	+	0.428936i	$0.141112\pi$
$587$	20.7846i	0.857873i	−0.903335	+	0.428936i	$0.858888\pi$
$588$	0	0
$589$	−16.0000	−0.659269
$590$	0	0
$591$	0	0
$592$	− 10.0000i	− 0.410997i
$593$	− 24.2487i	− 0.995775i	−0.867242	−	0.497888i	$-0.834109\pi$
$593$	− 24.2487i	− 0.995775i	0.867242	−	0.497888i	$-0.165891\pi$
$594$	0	0
$595$	0	0
$596$	−6.92820	−0.283790
$597$	0	0
$598$	12.0000i	0.490716i
$599$	45.0333	1.84001	0.920006	−	0.391905i	$-0.128184\pi$
$599$	45.0333	1.84001	0.920006	+	0.391905i	$0.128184\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	6.92820i	0.282372i
$603$	0	0
$604$	−8.00000	−0.325515
$605$	0	0
$606$	0	0
$607$	8.00000i	0.324710i	0.986732	+	0.162355i	$0.0519090\pi$
$607$	8.00000i	0.324710i	−0.986732	+	0.162355i	$0.948091\pi$
$608$	20.7846i	0.842927i
$609$	0	0
$610$	0	0
$611$	−13.8564	−0.560570
$612$	0	0
$613$	− 38.0000i	− 1.53481i	−0.641165	−	0.767403i	$-0.721549\pi$
$613$	− 38.0000i	− 1.53481i	0.641165	−	0.767403i	$-0.278451\pi$
$614$	−48.4974	−1.95720
$615$	0	0
$616$	−6.00000	−0.241747
$617$	20.7846i	0.836757i	0.908273	+	0.418378i	$0.137401\pi$
$617$	20.7846i	0.836757i	−0.908273	+	0.418378i	$0.862599\pi$
$618$	0	0
$619$	−8.00000	−0.321547	−0.160774	−	0.986991i	$-0.551399\pi$
$619$	−8.00000	−0.321547	−0.160774	+	0.986991i	$0.551399\pi$
$620$	0	0
$621$	0	0
$622$	60.0000i	2.40578i
$623$	− 3.46410i	− 0.138786i
$624$	0	0
$625$	0	0
$626$	−3.46410	−0.138453
$627$	0	0
$628$	10.0000i	0.399043i
$629$	6.92820	0.276246
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	13.8564i	0.551178i
$633$	0	0
$634$	−12.0000	−0.476581
$635$	0	0
$636$	0	0
$637$	2.00000i	0.0792429i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	48.4974	1.91553	0.957767	−	0.287547i	$-0.0928398\pi$
$641$	48.4974	1.91553	0.957767	+	0.287547i	$0.0928398\pi$
$642$	0	0
$643$	− 20.0000i	− 0.788723i	−0.918955	−	0.394362i	$-0.870966\pi$
$643$	− 20.0000i	− 0.788723i	0.918955	−	0.394362i	$-0.129034\pi$
$644$	−3.46410	−0.136505
$645$	0	0
$646$	−24.0000	−0.944267
$647$	6.92820i	0.272376i	0.990683	+	0.136188i	$0.0434851\pi$
$647$	6.92820i	0.272376i	−0.990683	+	0.136188i	$0.956515\pi$
$648$	0	0
$649$	24.0000	0.942082
$650$	0	0
$651$	0	0
$652$	20.0000i	0.783260i
$653$	27.7128i	1.08449i	0.840222	+	0.542243i	$0.182425\pi$
$653$	27.7128i	1.08449i	−0.840222	+	0.542243i	$0.817575\pi$
$654$	0	0
$655$	0	0
$656$	51.9615	2.02876
$657$	0	0
$658$	− 12.0000i	− 0.467809i
$659$	−10.3923	−0.404827	−0.202413	−	0.979300i	$-0.564878\pi$
$659$	−10.3923	−0.404827	−0.202413	+	0.979300i	$0.564878\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	− 34.6410i	− 1.34636i
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	− 20.7846i	− 0.804181i
$669$	0	0
$670$	0	0
$671$	34.6410	1.33730
$672$	0	0
$673$	10.0000i	0.385472i	0.981251	+	0.192736i	$0.0617360\pi$
$673$	10.0000i	0.385472i	−0.981251	+	0.192736i	$0.938264\pi$
$674$	24.2487	0.934025
$675$	0	0
$676$	−9.00000	−0.346154
$677$	24.2487i	0.931954i	0.884797	+	0.465977i	$0.154297\pi$
$677$	24.2487i	0.931954i	−0.884797	+	0.465977i	$0.845703\pi$
$678$	0	0
$679$	−14.0000	−0.537271
$680$	0	0
$681$	0	0
$682$	− 24.0000i	− 0.919007i
$683$	24.2487i	0.927851i	0.885874	+	0.463926i	$0.153559\pi$
$683$	24.2487i	0.927851i	−0.885874	+	0.463926i	$0.846441\pi$
$684$	0	0
$685$	0	0
$686$	−1.73205	−0.0661300
$687$	0	0
$688$	− 20.0000i	− 0.762493i
$689$	−13.8564	−0.527887
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	− 17.3205i	− 0.658427i
$693$	0	0
$694$	30.0000	1.13878
$695$	0	0
$696$	0	0
$697$	36.0000i	1.36360i
$698$	24.2487i	0.917827i
$699$	0	0
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	8.00000i	0.301726i
$704$	3.46410	0.130558
$705$	0	0
$706$	6.00000	0.225813
$707$	− 3.46410i	− 0.130281i
$708$	0	0
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	0	0
$711$	0	0
$712$	6.00000i	0.224860i
$713$	13.8564i	0.518927i
$714$	0	0
$715$	0	0
$716$	17.3205	0.647298
$717$	0	0
$718$	42.0000i	1.56743i
$719$	27.7128	1.03351	0.516757	−	0.856132i	$-0.327139\pi$
$719$	27.7128	1.03351	0.516757	+	0.856132i	$0.327139\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	5.19615i	0.193381i
$723$	0	0
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	0	0
$727$	− 4.00000i	− 0.148352i	−0.997245	−	0.0741759i	$-0.976367\pi$
$727$	− 4.00000i	− 0.148352i	0.997245	−	0.0741759i	$-0.0236326\pi$
$728$	− 3.46410i	− 0.128388i
$729$	0	0
$730$	0	0
$731$	13.8564	0.512498
$732$	0	0
$733$	22.0000i	0.812589i	0.913742	+	0.406294i	$0.133179\pi$
$733$	22.0000i	0.812589i	−0.913742	+	0.406294i	$0.866821\pi$
$734$	−27.7128	−1.02290
$735$	0	0
$736$	18.0000	0.663489
$737$	13.8564i	0.510407i
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	0	0
$742$	− 12.0000i	− 0.440534i
$743$	− 10.3923i	− 0.381257i	−0.981662	−	0.190628i	$-0.938947\pi$
$743$	− 10.3923i	− 0.381257i	0.981662	−	0.190628i	$-0.0610525\pi$
$744$	0	0
$745$	0	0
$746$	17.3205	0.634149
$747$	0	0
$748$	− 12.0000i	− 0.438763i
$749$	17.3205	0.632878
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	34.6410i	1.26323i
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	26.0000i	0.944986i	0.881334	+	0.472493i	$0.156646\pi$
$757$	26.0000i	0.944986i	−0.881334	+	0.472493i	$0.843354\pi$
$758$	− 48.4974i	− 1.76151i
$759$	0	0
$760$	0	0
$761$	−38.1051	−1.38131	−0.690655	−	0.723185i	$-0.742678\pi$
$761$	−38.1051	−1.38131	−0.690655	+	0.723185i	$0.742678\pi$
$762$	0	0
$763$	− 2.00000i	− 0.0724049i
$764$	−24.2487	−0.877288
$765$	0	0
$766$	−24.0000	−0.867155
$767$	13.8564i	0.500326i
$768$	0	0
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	0	0
$771$	0	0
$772$	14.0000i	0.503871i
$773$	45.0333i	1.61974i	0.586612	+	0.809868i	$0.300461\pi$
$773$	45.0333i	1.61974i	−0.586612	+	0.809868i	$0.699539\pi$
$774$	0	0
$775$	0	0
$776$	24.2487	0.870478
$777$	0	0
$778$	− 24.0000i	− 0.860442i
$779$	−41.5692	−1.48937
$780$	0	0
$781$	−36.0000	−1.28818
$782$	20.7846i	0.743256i
$783$	0	0
$784$	5.00000	0.178571
$785$	0	0
$786$	0	0
$787$	32.0000i	1.14068i	0.821410	+	0.570338i	$0.193188\pi$
$787$	32.0000i	1.14068i	−0.821410	+	0.570338i	$0.806812\pi$
$788$	20.7846i	0.740421i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	20.0000i	0.710221i
$794$	65.8179	2.33579
$795$	0	0
$796$	−16.0000	−0.567105
$797$	− 31.1769i	− 1.10434i	−0.833731	−	0.552171i	$-0.813799\pi$
$797$	− 31.1769i	− 1.10434i	0.833731	−	0.552171i	$-0.186201\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	0	0
$801$	0	0
$802$	12.0000i	0.423735i
$803$	48.4974i	1.71144i
$804$	0	0
$805$	0	0
$806$	13.8564	0.488071
$807$	0	0
$808$	6.00000i	0.211079i
$809$	−27.7128	−0.974331	−0.487165	−	0.873310i	$-0.661969\pi$
$809$	−27.7128	−0.974331	−0.487165	+	0.873310i	$0.661969\pi$
$810$	0	0
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	0	0
$813$	0	0
$814$	−12.0000	−0.420600
$815$	0	0
$816$	0	0
$817$	16.0000i	0.559769i
$818$	24.2487i	0.847836i
$819$	0	0
$820$	0	0
$821$	−6.92820	−0.241796	−0.120898	−	0.992665i	$-0.538577\pi$
$821$	−6.92820	−0.241796	−0.120898	+	0.992665i	$0.538577\pi$
$822$	0	0
$823$	− 32.0000i	− 1.11545i	−0.830026	−	0.557725i	$-0.811674\pi$
$823$	− 32.0000i	− 1.11545i	0.830026	−	0.557725i	$-0.188326\pi$
$824$	6.92820	0.241355
$825$	0	0
$826$	−12.0000	−0.417533
$827$	10.3923i	0.361376i	0.983540	+	0.180688i	$0.0578324\pi$
$827$	10.3923i	0.361376i	−0.983540	+	0.180688i	$0.942168\pi$
$828$	0	0
$829$	−50.0000	−1.73657	−0.868286	−	0.496064i	$-0.834778\pi$
$829$	−50.0000	−1.73657	−0.868286	+	0.496064i	$0.834778\pi$
$830$	0	0
$831$	0	0
$832$	2.00000i	0.0693375i
$833$	3.46410i	0.120024i
$834$	0	0
$835$	0	0
$836$	13.8564	0.479234
$837$	0	0
$838$	− 36.0000i	− 1.24360i
$839$	20.7846	0.717564	0.358782	−	0.933421i	$-0.383192\pi$
$839$	20.7846	0.717564	0.358782	+	0.933421i	$0.383192\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	17.3205i	0.596904i
$843$	0	0
$844$	−20.0000	−0.688428
$845$	0	0
$846$	0	0
$847$	1.00000i	0.0343604i
$848$	34.6410i	1.18958i
$849$	0	0
$850$	0	0
$851$	6.92820	0.237496
$852$	0	0
$853$	10.0000i	0.342393i	0.985237	+	0.171197i	$0.0547634\pi$
$853$	10.0000i	0.342393i	−0.985237	+	0.171197i	$0.945237\pi$
$854$	−17.3205	−0.592696
$855$	0	0
$856$	−30.0000	−1.02538
$857$	17.3205i	0.591657i	0.955241	+	0.295829i	$0.0955957\pi$
$857$	17.3205i	0.591657i	−0.955241	+	0.295829i	$0.904404\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	0	0
$862$	6.00000i	0.204361i
$863$	38.1051i	1.29711i	0.761166	+	0.648557i	$0.224627\pi$
$863$	38.1051i	1.29711i	−0.761166	+	0.648557i	$0.775373\pi$
$864$	0	0
$865$	0	0
$866$	−45.0333	−1.53029
$867$	0	0
$868$	4.00000i	0.135769i
$869$	27.7128	0.940093
$870$	0	0
$871$	−8.00000	−0.271070
$872$	3.46410i	0.117309i
$873$	0	0
$874$	−24.0000	−0.811812
$875$	0	0
$876$	0	0
$877$	14.0000i	0.472746i	0.971662	+	0.236373i	$0.0759588\pi$
$877$	14.0000i	0.472746i	−0.971662	+	0.236373i	$0.924041\pi$
$878$	− 27.7128i	− 0.935262i
$879$	0	0
$880$	0	0
$881$	51.9615	1.75063	0.875314	−	0.483555i	$-0.160655\pi$
$881$	51.9615	1.75063	0.875314	+	0.483555i	$0.160655\pi$
$882$	0	0
$883$	52.0000i	1.74994i	0.484178	+	0.874970i	$0.339119\pi$
$883$	52.0000i	1.74994i	−0.484178	+	0.874970i	$0.660881\pi$
$884$	6.92820	0.233021
$885$	0	0
$886$	−6.00000	−0.201574
$887$	− 6.92820i	− 0.232626i	−0.993213	−	0.116313i	$-0.962892\pi$
$887$	− 6.92820i	− 0.232626i	0.993213	−	0.116313i	$-0.0371076\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	0	0
$892$	8.00000i	0.267860i
$893$	− 27.7128i	− 0.927374i
$894$	0	0
$895$	0	0
$896$	−12.1244	−0.405046
$897$	0	0
$898$	− 72.0000i	− 2.40267i
$899$	0	0
$900$	0	0
$901$	−24.0000	−0.799556
$902$	− 62.3538i	− 2.07616i
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 28.0000i	− 0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$	− 28.0000i	− 0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$	− 6.92820i	− 0.229920i
$909$	0	0
$910$	0	0
$911$	−31.1769	−1.03294	−0.516469	−	0.856306i	$-0.672754\pi$
$911$	−31.1769	−1.03294	−0.516469	+	0.856306i	$0.672754\pi$
$912$	0	0
$913$	0	0
$914$	−17.3205	−0.572911
$915$	0	0
$916$	−22.0000	−0.726900
$917$	13.8564i	0.457579i
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	0	0
$922$	54.0000i	1.77840i
$923$	− 20.7846i	− 0.684134i
$924$	0	0
$925$	0	0
$926$	−55.4256	−1.82140
$927$	0	0
$928$	0	0
$929$	−45.0333	−1.47750	−0.738748	−	0.673982i	$-0.764582\pi$
$929$	−45.0333	−1.47750	−0.738748	+	0.673982i	$0.764582\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	− 6.92820i	− 0.226941i
$933$	0	0
$934$	−12.0000	−0.392652
$935$	0	0
$936$	0	0
$937$	− 46.0000i	− 1.50275i	−0.659873	−	0.751377i	$-0.729390\pi$
$937$	− 46.0000i	− 1.50275i	0.659873	−	0.751377i	$-0.270610\pi$
$938$	− 6.92820i	− 0.226214i
$939$	0	0
$940$	0	0
$941$	17.3205	0.564632	0.282316	−	0.959321i	$-0.408897\pi$
$941$	17.3205	0.564632	0.282316	+	0.959321i	$0.408897\pi$
$942$	0	0
$943$	36.0000i	1.17232i
$944$	34.6410	1.12747
$945$	0	0
$946$	−24.0000	−0.780307
$947$	24.2487i	0.787977i	0.919115	+	0.393989i	$0.128905\pi$
$947$	24.2487i	0.787977i	−0.919115	+	0.393989i	$0.871095\pi$
$948$	0	0
$949$	−28.0000	−0.908918
$950$	0	0
$951$	0	0
$952$	− 6.00000i	− 0.194461i
$953$	− 41.5692i	− 1.34656i	−0.739388	−	0.673280i	$-0.764885\pi$
$953$	− 41.5692i	− 1.34656i	0.739388	−	0.673280i	$-0.235115\pi$
$954$	0	0
$955$	0	0
$956$	−10.3923	−0.336111
$957$	0	0
$958$	12.0000i	0.387702i
$959$	−6.92820	−0.223723
$960$	0	0
$961$	−15.0000	−0.483871
$962$	− 6.92820i	− 0.223374i
$963$	0	0
$964$	10.0000	0.322078
$965$	0	0
$966$	0	0
$967$	− 16.0000i	− 0.514525i	−0.966342	−	0.257263i	$-0.917179\pi$
$967$	− 16.0000i	− 0.514525i	0.966342	−	0.257263i	$-0.0828206\pi$
$968$	− 1.73205i	− 0.0556702i
$969$	0	0
$970$	0	0
$971$	−27.7128	−0.889346	−0.444673	−	0.895693i	$-0.646680\pi$
$971$	−27.7128	−0.889346	−0.444673	+	0.895693i	$0.646680\pi$
$972$	0	0
$973$	16.0000i	0.512936i
$974$	−69.2820	−2.21994
$975$	0	0
$976$	50.0000	1.60046
$977$	− 34.6410i	− 1.10826i	−0.832429	−	0.554132i	$-0.813050\pi$
$977$	− 34.6410i	− 1.10826i	0.832429	−	0.554132i	$-0.186950\pi$
$978$	0	0
$979$	12.0000	0.383522
$980$	0	0
$981$	0	0
$982$	18.0000i	0.574403i
$983$	13.8564i	0.441951i	0.975279	+	0.220975i	$0.0709240\pi$
$983$	13.8564i	0.441951i	−0.975279	+	0.220975i	$0.929076\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	8.00000i	0.254514i
$989$	13.8564	0.440608
$990$	0	0
$991$	56.0000	1.77890	0.889449	−	0.457034i	$-0.151088\pi$
$991$	56.0000	1.77890	0.889449	+	0.457034i	$0.151088\pi$
$992$	− 20.7846i	− 0.659912i
$993$	0	0
$994$	18.0000	0.570925
$995$	0	0
$996$	0	0
$997$	− 10.0000i	− 0.316703i	−0.987383	−	0.158352i	$-0.949382\pi$
$997$	− 10.0000i	− 0.316703i	0.987383	−	0.158352i	$-0.0506179\pi$
$998$	− 6.92820i	− 0.219308i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.2.d.i.1324.2		4
3.2	odd	2	inner	1575.2.d.i.1324.4		4
5.2	odd	4		1575.2.a.q.1.2		2
5.3	odd	4		63.2.a.b.1.1	✓	2
5.4	even	2	inner	1575.2.d.i.1324.3		4
15.2	even	4		1575.2.a.q.1.1		2
15.8	even	4		63.2.a.b.1.2	yes	2
15.14	odd	2	inner	1575.2.d.i.1324.1		4
20.3	even	4		1008.2.a.n.1.2		2
35.3	even	12		441.2.e.i.226.2		4
35.13	even	4		441.2.a.g.1.1		2
35.18	odd	12		441.2.e.j.226.2		4
35.23	odd	12		441.2.e.j.361.2		4
35.33	even	12		441.2.e.i.361.2		4
40.3	even	4		4032.2.a.bq.1.1		2
40.13	odd	4		4032.2.a.bt.1.1		2
45.13	odd	12		567.2.f.j.379.2		4
45.23	even	12		567.2.f.j.379.1		4
45.38	even	12		567.2.f.j.190.1		4
45.43	odd	12		567.2.f.j.190.2		4
55.43	even	4		7623.2.a.bi.1.2		2
60.23	odd	4		1008.2.a.n.1.1		2
105.23	even	12		441.2.e.j.361.1		4
105.38	odd	12		441.2.e.i.226.1		4
105.53	even	12		441.2.e.j.226.1		4
105.68	odd	12		441.2.e.i.361.1		4
105.83	odd	4		441.2.a.g.1.2		2
120.53	even	4		4032.2.a.bt.1.2		2
120.83	odd	4		4032.2.a.bq.1.2		2
140.83	odd	4		7056.2.a.cm.1.1		2
165.98	odd	4		7623.2.a.bi.1.1		2
420.83	even	4		7056.2.a.cm.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.2.a.b.1.1	✓	2	5.3	odd	4
63.2.a.b.1.2	yes	2	15.8	even	4
441.2.a.g.1.1		2	35.13	even	4
441.2.a.g.1.2		2	105.83	odd	4
441.2.e.i.226.1		4	105.38	odd	12
441.2.e.i.226.2		4	35.3	even	12
441.2.e.i.361.1		4	105.68	odd	12
441.2.e.i.361.2		4	35.33	even	12
441.2.e.j.226.1		4	105.53	even	12
441.2.e.j.226.2		4	35.18	odd	12
441.2.e.j.361.1		4	105.23	even	12
441.2.e.j.361.2		4	35.23	odd	12
567.2.f.j.190.1		4	45.38	even	12
567.2.f.j.190.2		4	45.43	odd	12
567.2.f.j.379.1		4	45.23	even	12
567.2.f.j.379.2		4	45.13	odd	12
1008.2.a.n.1.1		2	60.23	odd	4
1008.2.a.n.1.2		2	20.3	even	4
1575.2.a.q.1.1		2	15.2	even	4
1575.2.a.q.1.2		2	5.2	odd	4
1575.2.d.i.1324.1		4	15.14	odd	2	inner
1575.2.d.i.1324.2		4	1.1	even	1	trivial
1575.2.d.i.1324.3		4	5.4	even	2	inner
1575.2.d.i.1324.4		4	3.2	odd	2	inner
4032.2.a.bq.1.1		2	40.3	even	4
4032.2.a.bq.1.2		2	120.83	odd	4
4032.2.a.bt.1.1		2	40.13	odd	4
4032.2.a.bt.1.2		2	120.53	even	4
7056.2.a.cm.1.1		2	140.83	odd	4
7056.2.a.cm.1.2		2	420.83	even	4
7623.2.a.bi.1.1		2	165.98	odd	4
7623.2.a.bi.1.2		2	55.43	even	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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1575.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.73205i q^{8} +O(q^{10})\)	\(q-1.73205i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.73205i q^{8} -3.46410 q^{11} -2.00000i q^{13} +1.73205 q^{14} -5.00000 q^{16} -3.46410i q^{17} +4.00000 q^{19} +6.00000i q^{22} -3.46410i q^{23} -3.46410 q^{26} -1.00000i q^{28} -4.00000 q^{31} +5.19615i q^{32} -6.00000 q^{34} +2.00000i q^{37} -6.92820i q^{38} -10.3923 q^{41} +4.00000i q^{43} +3.46410 q^{44} -6.00000 q^{46} -6.92820i q^{47} -1.00000 q^{49} +2.00000i q^{52} -6.92820i q^{53} +1.73205 q^{56} -6.92820 q^{59} -10.0000 q^{61} +6.92820i q^{62} -1.00000 q^{64} -4.00000i q^{67} +3.46410i q^{68} +10.3923 q^{71} -14.0000i q^{73} +3.46410 q^{74} -4.00000 q^{76} -3.46410i q^{77} -8.00000 q^{79} +18.0000i q^{82} +6.92820 q^{86} +6.00000i q^{88} -3.46410 q^{89} +2.00000 q^{91} +3.46410i q^{92} -12.0000 q^{94} +14.0000i q^{97} +1.73205i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} - 20 q^{16} + 16 q^{19} - 16 q^{31} - 24 q^{34} - 24 q^{46} - 4 q^{49} - 40 q^{61} - 4 q^{64} - 16 q^{76} - 32 q^{79} + 8 q^{91} - 48 q^{94}+O(q^{100}) \) 4 * q - 4 * q^4 - 20 * q^16 + 16 * q^19 - 16 * q^31 - 24 * q^34 - 24 * q^46 - 4 * q^49 - 40 * q^61 - 4 * q^64 - 16 * q^76 - 32 * q^79 + 8 * q^91 - 48 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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