LMFDB - Embedded newform 9450.2.a.ex.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9450,2,Mod(1,9450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9450, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("9450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9450 = 2 \cdot 3^{3} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9450.a (trivial)

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:	yes
Analytic conductor:	$75.4586299101$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{10}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 10 $ x^2 - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.16228$ of defining polynomial
Character	$\chi$	$=$	9450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} +1.00000 q^{8} +2.00000 q^{11} -2.16228 q^{13} +1.00000 q^{14} +1.00000 q^{16} -5.32456 q^{17} +3.00000 q^{19} +2.00000 q^{22} -5.16228 q^{23} -2.16228 q^{26} +1.00000 q^{28} +8.16228 q^{29} +6.32456 q^{31} +1.00000 q^{32} -5.32456 q^{34} -11.4868 q^{37} +3.00000 q^{38} +4.00000 q^{41} -3.16228 q^{43} +2.00000 q^{44} -5.16228 q^{46} +8.48683 q^{47} +1.00000 q^{49} -2.16228 q^{52} +6.16228 q^{53} +1.00000 q^{56} +8.16228 q^{58} +12.3246 q^{59} -10.1623 q^{61} +6.32456 q^{62} +1.00000 q^{64} +15.4868 q^{67} -5.32456 q^{68} +2.32456 q^{71} +8.32456 q^{73} -11.4868 q^{74} +3.00000 q^{76} +2.00000 q^{77} +4.48683 q^{79} +4.00000 q^{82} -3.16228 q^{83} -3.16228 q^{86} +2.00000 q^{88} +5.00000 q^{89} -2.16228 q^{91} -5.16228 q^{92} +8.48683 q^{94} +7.48683 q^{97} +1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8} + 4 q^{11} + 2 q^{13} + 2 q^{14} + 2 q^{16} + 2 q^{17} + 6 q^{19} + 4 q^{22} - 4 q^{23} + 2 q^{26} + 2 q^{28} + 10 q^{29} + 2 q^{32} + 2 q^{34} - 4 q^{37} + 6 q^{38} + 8 q^{41} + 4 q^{44} - 4 q^{46} - 2 q^{47} + 2 q^{49} + 2 q^{52} + 6 q^{53} + 2 q^{56} + 10 q^{58} + 12 q^{59} - 14 q^{61} + 2 q^{64} + 12 q^{67} + 2 q^{68} - 8 q^{71} + 4 q^{73} - 4 q^{74} + 6 q^{76} + 4 q^{77} - 10 q^{79} + 8 q^{82} + 4 q^{88} + 10 q^{89} + 2 q^{91} - 4 q^{92} - 2 q^{94} - 4 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^7 + 2 * q^8 + 4 * q^11 + 2 * q^13 + 2 * q^14 + 2 * q^16 + 2 * q^17 + 6 * q^19 + 4 * q^22 - 4 * q^23 + 2 * q^26 + 2 * q^28 + 10 * q^29 + 2 * q^32 + 2 * q^34 - 4 * q^37 + 6 * q^38 + 8 * q^41 + 4 * q^44 - 4 * q^46 - 2 * q^47 + 2 * q^49 + 2 * q^52 + 6 * q^53 + 2 * q^56 + 10 * q^58 + 12 * q^59 - 14 * q^61 + 2 * q^64 + 12 * q^67 + 2 * q^68 - 8 * q^71 + 4 * q^73 - 4 * q^74 + 6 * q^76 + 4 * q^77 - 10 * q^79 + 8 * q^82 + 4 * q^88 + 10 * q^89 + 2 * q^91 - 4 * q^92 - 2 * q^94 - 4 * q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−2.16228	−0.599708	−0.299854	−	0.953985i	$-0.596938\pi$
$13$	−2.16228	−0.599708	−0.299854	+	0.953985i	$0.596938\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.32456	−1.29139	−0.645697	−	0.763594i	$-0.723433\pi$
$17$	−5.32456	−1.29139	−0.645697	+	0.763594i	$0.723433\pi$
$18$	0	0
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	0	0
$21$	0	0
$22$	2.00000	0.426401
$23$	−5.16228	−1.07641	−0.538205	−	0.842814i	$-0.680897\pi$
$23$	−5.16228	−1.07641	−0.538205	+	0.842814i	$0.680897\pi$
$24$	0	0
$25$	0	0
$26$	−2.16228	−0.424058
$27$	0	0
$28$	1.00000	0.188982
$29$	8.16228	1.51570	0.757848	−	0.652431i	$-0.226251\pi$
$29$	8.16228	1.51570	0.757848	+	0.652431i	$0.226251\pi$
$30$	0	0
$31$	6.32456	1.13592	0.567962	−	0.823055i	$-0.307732\pi$
$31$	6.32456	1.13592	0.567962	+	0.823055i	$0.307732\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−5.32456	−0.913154
$35$	0	0
$36$	0	0
$37$	−11.4868	−1.88842	−0.944212	−	0.329339i	$-0.893174\pi$
$37$	−11.4868	−1.88842	−0.944212	+	0.329339i	$0.893174\pi$
$38$	3.00000	0.486664
$39$	0	0
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	−3.16228	−0.482243	−0.241121	−	0.970495i	$-0.577515\pi$
$43$	−3.16228	−0.482243	−0.241121	+	0.970495i	$0.577515\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	−5.16228	−0.761136
$47$	8.48683	1.23793	0.618966	−	0.785418i	$-0.287552\pi$
$47$	8.48683	1.23793	0.618966	+	0.785418i	$0.287552\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	−2.16228	−0.299854
$53$	6.16228	0.846454	0.423227	−	0.906024i	$-0.360897\pi$
$53$	6.16228	0.846454	0.423227	+	0.906024i	$0.360897\pi$
$54$	0	0
$55$	0	0
$56$	1.00000	0.133631
$57$	0	0
$58$	8.16228	1.07176
$59$	12.3246	1.60452	0.802260	−	0.596974i	$-0.203631\pi$
$59$	12.3246	1.60452	0.802260	+	0.596974i	$0.203631\pi$
$60$	0	0
$61$	−10.1623	−1.30115	−0.650573	−	0.759444i	$-0.725471\pi$
$61$	−10.1623	−1.30115	−0.650573	+	0.759444i	$0.725471\pi$
$62$	6.32456	0.803219
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	15.4868	1.89202	0.946009	−	0.324141i	$-0.105075\pi$
$67$	15.4868	1.89202	0.946009	+	0.324141i	$0.105075\pi$
$68$	−5.32456	−0.645697
$69$	0	0
$70$	0	0
$71$	2.32456	0.275874	0.137937	−	0.990441i	$-0.455953\pi$
$71$	2.32456	0.275874	0.137937	+	0.990441i	$0.455953\pi$
$72$	0	0
$73$	8.32456	0.974316	0.487158	−	0.873314i	$-0.338034\pi$
$73$	8.32456	0.974316	0.487158	+	0.873314i	$0.338034\pi$
$74$	−11.4868	−1.33532
$75$	0	0
$76$	3.00000	0.344124
$77$	2.00000	0.227921
$78$	0	0
$79$	4.48683	0.504808	0.252404	−	0.967622i	$-0.418779\pi$
$79$	4.48683	0.504808	0.252404	+	0.967622i	$0.418779\pi$
$80$	0	0
$81$	0	0
$82$	4.00000	0.441726
$83$	−3.16228	−0.347105	−0.173553	−	0.984825i	$-0.555525\pi$
$83$	−3.16228	−0.347105	−0.173553	+	0.984825i	$0.555525\pi$
$84$	0	0
$85$	0	0
$86$	−3.16228	−0.340997
$87$	0	0
$88$	2.00000	0.213201
$89$	5.00000	0.529999	0.264999	−	0.964249i	$-0.414628\pi$
$89$	5.00000	0.529999	0.264999	+	0.964249i	$0.414628\pi$
$90$	0	0
$91$	−2.16228	−0.226668
$92$	−5.16228	−0.538205
$93$	0	0
$94$	8.48683	0.875350
$95$	0	0
$96$	0	0
$97$	7.48683	0.760173	0.380086	−	0.924951i	$-0.375894\pi$
$97$	7.48683	0.760173	0.380086	+	0.924951i	$0.375894\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	0	0
$101$	−3.48683	−0.346953	−0.173476	−	0.984838i	$-0.555500\pi$
$101$	−3.48683	−0.346953	−0.173476	+	0.984838i	$0.555500\pi$
$102$	0	0
$103$	6.32456	0.623177	0.311588	−	0.950217i	$-0.399139\pi$
$103$	6.32456	0.623177	0.311588	+	0.950217i	$0.399139\pi$
$104$	−2.16228	−0.212029
$105$	0	0
$106$	6.16228	0.598533
$107$	−4.32456	−0.418071	−0.209035	−	0.977908i	$-0.567032\pi$
$107$	−4.32456	−0.418071	−0.209035	+	0.977908i	$0.567032\pi$
$108$	0	0
$109$	−3.48683	−0.333978	−0.166989	−	0.985959i	$-0.553404\pi$
$109$	−3.48683	−0.333978	−0.166989	+	0.985959i	$0.553404\pi$
$110$	0	0
$111$	0	0
$112$	1.00000	0.0944911
$113$	10.6491	1.00178	0.500892	−	0.865510i	$-0.333005\pi$
$113$	10.6491	1.00178	0.500892	+	0.865510i	$0.333005\pi$
$114$	0	0
$115$	0	0
$116$	8.16228	0.757848
$117$	0	0
$118$	12.3246	1.13457
$119$	−5.32456	−0.488101
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−10.1623	−0.920049
$123$	0	0
$124$	6.32456	0.567962
$125$	0	0
$126$	0	0
$127$	14.4868	1.28550	0.642749	−	0.766077i	$-0.277794\pi$
$127$	14.4868	1.28550	0.642749	+	0.766077i	$0.277794\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	13.4868	1.17835	0.589175	−	0.808005i	$-0.299453\pi$
$131$	13.4868	1.17835	0.589175	+	0.808005i	$0.299453\pi$
$132$	0	0
$133$	3.00000	0.260133
$134$	15.4868	1.33786
$135$	0	0
$136$	−5.32456	−0.456577
$137$	−21.4868	−1.83574	−0.917872	−	0.396877i	$-0.870094\pi$
$137$	−21.4868	−1.83574	−0.917872	+	0.396877i	$0.870094\pi$
$138$	0	0
$139$	−0.324555	−0.0275284	−0.0137642	−	0.999905i	$-0.504381\pi$
$139$	−0.324555	−0.0275284	−0.0137642	+	0.999905i	$0.504381\pi$
$140$	0	0
$141$	0	0
$142$	2.32456	0.195072
$143$	−4.32456	−0.361637
$144$	0	0
$145$	0	0
$146$	8.32456	0.688945
$147$	0	0
$148$	−11.4868	−0.944212
$149$	−0.486833	−0.0398829	−0.0199415	−	0.999801i	$-0.506348\pi$
$149$	−0.486833	−0.0398829	−0.0199415	+	0.999801i	$0.506348\pi$
$150$	0	0
$151$	−1.83772	−0.149552	−0.0747759	−	0.997200i	$-0.523824\pi$
$151$	−1.83772	−0.149552	−0.0747759	+	0.997200i	$0.523824\pi$
$152$	3.00000	0.243332
$153$	0	0
$154$	2.00000	0.161165
$155$	0	0
$156$	0	0
$157$	17.8377	1.42361	0.711803	−	0.702380i	$-0.247879\pi$
$157$	17.8377	1.42361	0.711803	+	0.702380i	$0.247879\pi$
$158$	4.48683	0.356953
$159$	0	0
$160$	0	0
$161$	−5.16228	−0.406844
$162$	0	0
$163$	−12.3246	−0.965334	−0.482667	−	0.875804i	$-0.660332\pi$
$163$	−12.3246	−0.965334	−0.482667	+	0.875804i	$0.660332\pi$
$164$	4.00000	0.312348
$165$	0	0
$166$	−3.16228	−0.245440
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	−8.32456	−0.640350
$170$	0	0
$171$	0	0
$172$	−3.16228	−0.241121
$173$	19.1623	1.45688	0.728440	−	0.685109i	$-0.240245\pi$
$173$	19.1623	1.45688	0.728440	+	0.685109i	$0.240245\pi$
$174$	0	0
$175$	0	0
$176$	2.00000	0.150756
$177$	0	0
$178$	5.00000	0.374766
$179$	25.3246	1.89285	0.946423	−	0.322929i	$-0.104668\pi$
$179$	25.3246	1.89285	0.946423	+	0.322929i	$0.104668\pi$
$180$	0	0
$181$	−16.1623	−1.20133	−0.600666	−	0.799500i	$-0.705098\pi$
$181$	−16.1623	−1.20133	−0.600666	+	0.799500i	$0.705098\pi$
$182$	−2.16228	−0.160279
$183$	0	0
$184$	−5.16228	−0.380568
$185$	0	0
$186$	0	0
$187$	−10.6491	−0.778740
$188$	8.48683	0.618966
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−27.3246	−1.96686	−0.983432	−	0.181276i	$-0.941977\pi$
$193$	−27.3246	−1.96686	−0.983432	+	0.181276i	$0.941977\pi$
$194$	7.48683	0.537523
$195$	0	0
$196$	1.00000	0.0714286
$197$	18.8114	1.34026	0.670128	−	0.742246i	$-0.266239\pi$
$197$	18.8114	1.34026	0.670128	+	0.742246i	$0.266239\pi$
$198$	0	0
$199$	20.1359	1.42740	0.713700	−	0.700452i	$-0.247018\pi$
$199$	20.1359	1.42740	0.713700	+	0.700452i	$0.247018\pi$
$200$	0	0
$201$	0	0
$202$	−3.48683	−0.245333
$203$	8.16228	0.572880
$204$	0	0
$205$	0	0
$206$	6.32456	0.440653
$207$	0	0
$208$	−2.16228	−0.149927
$209$	6.00000	0.415029
$210$	0	0
$211$	−12.3246	−0.848457	−0.424229	−	0.905555i	$-0.639455\pi$
$211$	−12.3246	−0.848457	−0.424229	+	0.905555i	$0.639455\pi$
$212$	6.16228	0.423227
$213$	0	0
$214$	−4.32456	−0.295621
$215$	0	0
$216$	0	0
$217$	6.32456	0.429339
$218$	−3.48683	−0.236158
$219$	0	0
$220$	0	0
$221$	11.5132	0.774459
$222$	0	0
$223$	7.16228	0.479622	0.239811	−	0.970820i	$-0.422915\pi$
$223$	7.16228	0.479622	0.239811	+	0.970820i	$0.422915\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	10.6491	0.708368
$227$	21.8114	1.44767	0.723836	−	0.689972i	$-0.242377\pi$
$227$	21.8114	1.44767	0.723836	+	0.689972i	$0.242377\pi$
$228$	0	0
$229$	24.1623	1.59669	0.798344	−	0.602202i	$-0.205710\pi$
$229$	24.1623	1.59669	0.798344	+	0.602202i	$0.205710\pi$
$230$	0	0
$231$	0	0
$232$	8.16228	0.535880
$233$	3.48683	0.228430	0.114215	−	0.993456i	$-0.463565\pi$
$233$	3.48683	0.228430	0.114215	+	0.993456i	$0.463565\pi$
$234$	0	0
$235$	0	0
$236$	12.3246	0.802260
$237$	0	0
$238$	−5.32456	−0.345140
$239$	−5.81139	−0.375907	−0.187954	−	0.982178i	$-0.560186\pi$
$239$	−5.81139	−0.375907	−0.187954	+	0.982178i	$0.560186\pi$
$240$	0	0
$241$	−4.51317	−0.290719	−0.145359	−	0.989379i	$-0.546434\pi$
$241$	−4.51317	−0.290719	−0.145359	+	0.989379i	$0.546434\pi$
$242$	−7.00000	−0.449977
$243$	0	0
$244$	−10.1623	−0.650573
$245$	0	0
$246$	0	0
$247$	−6.48683	−0.412747
$248$	6.32456	0.401610
$249$	0	0
$250$	0	0
$251$	−16.3246	−1.03040	−0.515198	−	0.857071i	$-0.672282\pi$
$251$	−16.3246	−1.03040	−0.515198	+	0.857071i	$0.672282\pi$
$252$	0	0
$253$	−10.3246	−0.649099
$254$	14.4868	0.908985
$255$	0	0
$256$	1.00000	0.0625000
$257$	−19.0000	−1.18519	−0.592594	−	0.805502i	$-0.701896\pi$
$257$	−19.0000	−1.18519	−0.592594	+	0.805502i	$0.701896\pi$
$258$	0	0
$259$	−11.4868	−0.713757
$260$	0	0
$261$	0	0
$262$	13.4868	0.833219
$263$	3.67544	0.226638	0.113319	−	0.993559i	$-0.463852\pi$
$263$	3.67544	0.226638	0.113319	+	0.993559i	$0.463852\pi$
$264$	0	0
$265$	0	0
$266$	3.00000	0.183942
$267$	0	0
$268$	15.4868	0.946009
$269$	−2.64911	−0.161519	−0.0807596	−	0.996734i	$-0.525735\pi$
$269$	−2.64911	−0.161519	−0.0807596	+	0.996734i	$0.525735\pi$
$270$	0	0
$271$	10.3246	0.627172	0.313586	−	0.949560i	$-0.398470\pi$
$271$	10.3246	0.627172	0.313586	+	0.949560i	$0.398470\pi$
$272$	−5.32456	−0.322849
$273$	0	0
$274$	−21.4868	−1.29807
$275$	0	0
$276$	0	0
$277$	−4.32456	−0.259837	−0.129919	−	0.991525i	$-0.541472\pi$
$277$	−4.32456	−0.259837	−0.129919	+	0.991525i	$0.541472\pi$
$278$	−0.324555	−0.0194655
$279$	0	0
$280$	0	0
$281$	15.1623	0.904506	0.452253	−	0.891890i	$-0.350620\pi$
$281$	15.1623	0.904506	0.452253	+	0.891890i	$0.350620\pi$
$282$	0	0
$283$	3.64911	0.216917	0.108459	−	0.994101i	$-0.465409\pi$
$283$	3.64911	0.216917	0.108459	+	0.994101i	$0.465409\pi$
$284$	2.32456	0.137937
$285$	0	0
$286$	−4.32456	−0.255716
$287$	4.00000	0.236113
$288$	0	0
$289$	11.3509	0.667699
$290$	0	0
$291$	0	0
$292$	8.32456	0.487158
$293$	−20.1359	−1.17635	−0.588177	−	0.808732i	$-0.700154\pi$
$293$	−20.1359	−1.17635	−0.588177	+	0.808732i	$0.700154\pi$
$294$	0	0
$295$	0	0
$296$	−11.4868	−0.667659
$297$	0	0
$298$	−0.486833	−0.0282015
$299$	11.1623	0.645531
$300$	0	0
$301$	−3.16228	−0.182271
$302$	−1.83772	−0.105749
$303$	0	0
$304$	3.00000	0.172062
$305$	0	0
$306$	0	0
$307$	−18.6491	−1.06436	−0.532180	−	0.846631i	$-0.678627\pi$
$307$	−18.6491	−1.06436	−0.532180	+	0.846631i	$0.678627\pi$
$308$	2.00000	0.113961
$309$	0	0
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	−1.16228	−0.0656958	−0.0328479	−	0.999460i	$-0.510458\pi$
$313$	−1.16228	−0.0656958	−0.0328479	+	0.999460i	$0.510458\pi$
$314$	17.8377	1.00664
$315$	0	0
$316$	4.48683	0.252404
$317$	−28.1623	−1.58175	−0.790876	−	0.611977i	$-0.790374\pi$
$317$	−28.1623	−1.58175	−0.790876	+	0.611977i	$0.790374\pi$
$318$	0	0
$319$	16.3246	0.914000
$320$	0	0
$321$	0	0
$322$	−5.16228	−0.287682
$323$	−15.9737	−0.888799
$324$	0	0
$325$	0	0
$326$	−12.3246	−0.682594
$327$	0	0
$328$	4.00000	0.220863
$329$	8.48683	0.467894
$330$	0	0
$331$	−33.8114	−1.85844	−0.929221	−	0.369524i	$-0.879521\pi$
$331$	−33.8114	−1.85844	−0.929221	+	0.369524i	$0.879521\pi$
$332$	−3.16228	−0.173553
$333$	0	0
$334$	−12.0000	−0.656611
$335$	0	0
$336$	0	0
$337$	0.675445	0.0367938	0.0183969	−	0.999831i	$-0.494144\pi$
$337$	0.675445	0.0367938	0.0183969	+	0.999831i	$0.494144\pi$
$338$	−8.32456	−0.452796
$339$	0	0
$340$	0	0
$341$	12.6491	0.684988
$342$	0	0
$343$	1.00000	0.0539949
$344$	−3.16228	−0.170499
$345$	0	0
$346$	19.1623	1.03017
$347$	−23.0000	−1.23470	−0.617352	−	0.786687i	$-0.711795\pi$
$347$	−23.0000	−1.23470	−0.617352	+	0.786687i	$0.711795\pi$
$348$	0	0
$349$	20.8114	1.11401	0.557004	−	0.830510i	$-0.311951\pi$
$349$	20.8114	1.11401	0.557004	+	0.830510i	$0.311951\pi$
$350$	0	0
$351$	0	0
$352$	2.00000	0.106600
$353$	24.0000	1.27739	0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000	1.27739	0.638696	+	0.769460i	$0.279474\pi$
$354$	0	0
$355$	0	0
$356$	5.00000	0.264999
$357$	0	0
$358$	25.3246	1.33844
$359$	22.0000	1.16112	0.580558	−	0.814219i	$-0.302835\pi$
$359$	22.0000	1.16112	0.580558	+	0.814219i	$0.302835\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	−16.1623	−0.849470
$363$	0	0
$364$	−2.16228	−0.113334
$365$	0	0
$366$	0	0
$367$	−26.4605	−1.38123	−0.690613	−	0.723224i	$-0.742659\pi$
$367$	−26.4605	−1.38123	−0.690613	+	0.723224i	$0.742659\pi$
$368$	−5.16228	−0.269102
$369$	0	0
$370$	0	0
$371$	6.16228	0.319930
$372$	0	0
$373$	8.32456	0.431029	0.215515	−	0.976501i	$-0.430857\pi$
$373$	8.32456	0.431029	0.215515	+	0.976501i	$0.430857\pi$
$374$	−10.6491	−0.550652
$375$	0	0
$376$	8.48683	0.437675
$377$	−17.6491	−0.908975
$378$	0	0
$379$	−11.1623	−0.573368	−0.286684	−	0.958025i	$-0.592553\pi$
$379$	−11.1623	−0.573368	−0.286684	+	0.958025i	$0.592553\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	13.6754	0.698783	0.349391	−	0.936977i	$-0.386388\pi$
$383$	13.6754	0.698783	0.349391	+	0.936977i	$0.386388\pi$
$384$	0	0
$385$	0	0
$386$	−27.3246	−1.39078
$387$	0	0
$388$	7.48683	0.380086
$389$	−7.51317	−0.380933	−0.190466	−	0.981694i	$-0.561000\pi$
$389$	−7.51317	−0.380933	−0.190466	+	0.981694i	$0.561000\pi$
$390$	0	0
$391$	27.4868	1.39007
$392$	1.00000	0.0505076
$393$	0	0
$394$	18.8114	0.947704
$395$	0	0
$396$	0	0
$397$	12.8114	0.642985	0.321493	−	0.946912i	$-0.395815\pi$
$397$	12.8114	0.642985	0.321493	+	0.946912i	$0.395815\pi$
$398$	20.1359	1.00932
$399$	0	0
$400$	0	0
$401$	2.83772	0.141709	0.0708545	−	0.997487i	$-0.477427\pi$
$401$	2.83772	0.141709	0.0708545	+	0.997487i	$0.477427\pi$
$402$	0	0
$403$	−13.6754	−0.681222
$404$	−3.48683	−0.173476
$405$	0	0
$406$	8.16228	0.405087
$407$	−22.9737	−1.13876
$408$	0	0
$409$	−24.1359	−1.19345	−0.596723	−	0.802447i	$-0.703531\pi$
$409$	−24.1359	−1.19345	−0.596723	+	0.802447i	$0.703531\pi$
$410$	0	0
$411$	0	0
$412$	6.32456	0.311588
$413$	12.3246	0.606452
$414$	0	0
$415$	0	0
$416$	−2.16228	−0.106014
$417$	0	0
$418$	6.00000	0.293470
$419$	24.1359	1.17912	0.589559	−	0.807725i	$-0.299302\pi$
$419$	24.1359	1.17912	0.589559	+	0.807725i	$0.299302\pi$
$420$	0	0
$421$	−30.6491	−1.49375	−0.746873	−	0.664967i	$-0.768446\pi$
$421$	−30.6491	−1.49375	−0.746873	+	0.664967i	$0.768446\pi$
$422$	−12.3246	−0.599950
$423$	0	0
$424$	6.16228	0.299267
$425$	0	0
$426$	0	0
$427$	−10.1623	−0.491787
$428$	−4.32456	−0.209035
$429$	0	0
$430$	0	0
$431$	−6.97367	−0.335910	−0.167955	−	0.985795i	$-0.553716\pi$
$431$	−6.97367	−0.335910	−0.167955	+	0.985795i	$0.553716\pi$
$432$	0	0
$433$	17.6754	0.849428	0.424714	−	0.905328i	$-0.360375\pi$
$433$	17.6754	0.849428	0.424714	+	0.905328i	$0.360375\pi$
$434$	6.32456	0.303588
$435$	0	0
$436$	−3.48683	−0.166989
$437$	−15.4868	−0.740836
$438$	0	0
$439$	2.18861	0.104457	0.0522284	−	0.998635i	$-0.483368\pi$
$439$	2.18861	0.104457	0.0522284	+	0.998635i	$0.483368\pi$
$440$	0	0
$441$	0	0
$442$	11.5132	0.547626
$443$	34.6228	1.64498	0.822489	−	0.568781i	$-0.192585\pi$
$443$	34.6228	1.64498	0.822489	+	0.568781i	$0.192585\pi$
$444$	0	0
$445$	0	0
$446$	7.16228	0.339144
$447$	0	0
$448$	1.00000	0.0472456
$449$	18.8377	0.889007	0.444504	−	0.895777i	$-0.353380\pi$
$449$	18.8377	0.889007	0.444504	+	0.895777i	$0.353380\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	10.6491	0.500892
$453$	0	0
$454$	21.8114	1.02366
$455$	0	0
$456$	0	0
$457$	32.6228	1.52603	0.763015	−	0.646381i	$-0.223718\pi$
$457$	32.6228	1.52603	0.763015	+	0.646381i	$0.223718\pi$
$458$	24.1623	1.12903
$459$	0	0
$460$	0	0
$461$	1.16228	0.0541327	0.0270663	−	0.999634i	$-0.491383\pi$
$461$	1.16228	0.0541327	0.0270663	+	0.999634i	$0.491383\pi$
$462$	0	0
$463$	−1.51317	−0.0703228	−0.0351614	−	0.999382i	$-0.511195\pi$
$463$	−1.51317	−0.0703228	−0.0351614	+	0.999382i	$0.511195\pi$
$464$	8.16228	0.378924
$465$	0	0
$466$	3.48683	0.161524
$467$	−2.32456	−0.107568	−0.0537838	−	0.998553i	$-0.517128\pi$
$467$	−2.32456	−0.107568	−0.0537838	+	0.998553i	$0.517128\pi$
$468$	0	0
$469$	15.4868	0.715116
$470$	0	0
$471$	0	0
$472$	12.3246	0.567284
$473$	−6.32456	−0.290803
$474$	0	0
$475$	0	0
$476$	−5.32456	−0.244051
$477$	0	0
$478$	−5.81139	−0.265807
$479$	2.16228	0.0987970	0.0493985	−	0.998779i	$-0.484270\pi$
$479$	2.16228	0.0987970	0.0493985	+	0.998779i	$0.484270\pi$
$480$	0	0
$481$	24.8377	1.13250
$482$	−4.51317	−0.205569
$483$	0	0
$484$	−7.00000	−0.318182
$485$	0	0
$486$	0	0
$487$	10.9737	0.497264	0.248632	−	0.968598i	$-0.420019\pi$
$487$	10.9737	0.497264	0.248632	+	0.968598i	$0.420019\pi$
$488$	−10.1623	−0.460025
$489$	0	0
$490$	0	0
$491$	−4.67544	−0.211000	−0.105500	−	0.994419i	$-0.533644\pi$
$491$	−4.67544	−0.211000	−0.105500	+	0.994419i	$0.533644\pi$
$492$	0	0
$493$	−43.4605	−1.95736
$494$	−6.48683	−0.291856
$495$	0	0
$496$	6.32456	0.283981
$497$	2.32456	0.104271
$498$	0	0
$499$	11.1623	0.499692	0.249846	−	0.968286i	$-0.419620\pi$
$499$	11.1623	0.499692	0.249846	+	0.968286i	$0.419620\pi$
$500$	0	0
$501$	0	0
$502$	−16.3246	−0.728601
$503$	30.0000	1.33763	0.668817	−	0.743427i	$-0.266801\pi$
$503$	30.0000	1.33763	0.668817	+	0.743427i	$0.266801\pi$
$504$	0	0
$505$	0	0
$506$	−10.3246	−0.458982
$507$	0	0
$508$	14.4868	0.642749
$509$	−0.513167	−0.0227457	−0.0113729	−	0.999935i	$-0.503620\pi$
$509$	−0.513167	−0.0227457	−0.0113729	+	0.999935i	$0.503620\pi$
$510$	0	0
$511$	8.32456	0.368257
$512$	1.00000	0.0441942
$513$	0	0
$514$	−19.0000	−0.838054
$515$	0	0
$516$	0	0
$517$	16.9737	0.746501
$518$	−11.4868	−0.504702
$519$	0	0
$520$	0	0
$521$	−18.6754	−0.818186	−0.409093	−	0.912493i	$-0.634155\pi$
$521$	−18.6754	−0.818186	−0.409093	+	0.912493i	$0.634155\pi$
$522$	0	0
$523$	−14.6754	−0.641712	−0.320856	−	0.947128i	$-0.603971\pi$
$523$	−14.6754	−0.641712	−0.320856	+	0.947128i	$0.603971\pi$
$524$	13.4868	0.589175
$525$	0	0
$526$	3.67544	0.160257
$527$	−33.6754	−1.46693
$528$	0	0
$529$	3.64911	0.158657
$530$	0	0
$531$	0	0
$532$	3.00000	0.130066
$533$	−8.64911	−0.374635
$534$	0	0
$535$	0	0
$536$	15.4868	0.668929
$537$	0	0
$538$	−2.64911	−0.114211
$539$	2.00000	0.0861461
$540$	0	0
$541$	13.8114	0.593798	0.296899	−	0.954909i	$-0.404048\pi$
$541$	13.8114	0.593798	0.296899	+	0.954909i	$0.404048\pi$
$542$	10.3246	0.443478
$543$	0	0
$544$	−5.32456	−0.228288
$545$	0	0
$546$	0	0
$547$	17.2982	0.739619	0.369809	−	0.929108i	$-0.379423\pi$
$547$	17.2982	0.739619	0.369809	+	0.929108i	$0.379423\pi$
$548$	−21.4868	−0.917872
$549$	0	0
$550$	0	0
$551$	24.4868	1.04317
$552$	0	0
$553$	4.48683	0.190800
$554$	−4.32456	−0.183733
$555$	0	0
$556$	−0.324555	−0.0137642
$557$	32.8114	1.39026	0.695132	−	0.718883i	$-0.255346\pi$
$557$	32.8114	1.39026	0.695132	+	0.718883i	$0.255346\pi$
$558$	0	0
$559$	6.83772	0.289205
$560$	0	0
$561$	0	0
$562$	15.1623	0.639582
$563$	42.4605	1.78950	0.894748	−	0.446571i	$-0.147355\pi$
$563$	42.4605	1.78950	0.894748	+	0.446571i	$0.147355\pi$
$564$	0	0
$565$	0	0
$566$	3.64911	0.153384
$567$	0	0
$568$	2.32456	0.0975362
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	−11.2982	−0.472816	−0.236408	−	0.971654i	$-0.575970\pi$
$571$	−11.2982	−0.472816	−0.236408	+	0.971654i	$0.575970\pi$
$572$	−4.32456	−0.180819
$573$	0	0
$574$	4.00000	0.166957
$575$	0	0
$576$	0	0
$577$	−28.0000	−1.16566	−0.582828	−	0.812596i	$-0.698054\pi$
$577$	−28.0000	−1.16566	−0.582828	+	0.812596i	$0.698054\pi$
$578$	11.3509	0.472135
$579$	0	0
$580$	0	0
$581$	−3.16228	−0.131193
$582$	0	0
$583$	12.3246	0.510431
$584$	8.32456	0.344473
$585$	0	0
$586$	−20.1359	−0.831808
$587$	25.1623	1.03856	0.519279	−	0.854605i	$-0.326200\pi$
$587$	25.1623	1.03856	0.519279	+	0.854605i	$0.326200\pi$
$588$	0	0
$589$	18.9737	0.781796
$590$	0	0
$591$	0	0
$592$	−11.4868	−0.472106
$593$	13.3246	0.547174	0.273587	−	0.961847i	$-0.411790\pi$
$593$	13.3246	0.547174	0.273587	+	0.961847i	$0.411790\pi$
$594$	0	0
$595$	0	0
$596$	−0.486833	−0.0199415
$597$	0	0
$598$	11.1623	0.456459
$599$	−24.1359	−0.986168	−0.493084	−	0.869982i	$-0.664130\pi$
$599$	−24.1359	−0.986168	−0.493084	+	0.869982i	$0.664130\pi$
$600$	0	0
$601$	−34.3246	−1.40013	−0.700064	−	0.714080i	$-0.746845\pi$
$601$	−34.3246	−1.40013	−0.700064	+	0.714080i	$0.746845\pi$
$602$	−3.16228	−0.128885
$603$	0	0
$604$	−1.83772	−0.0747759
$605$	0	0
$606$	0	0
$607$	−27.1623	−1.10248	−0.551241	−	0.834346i	$-0.685846\pi$
$607$	−27.1623	−1.10248	−0.551241	+	0.834346i	$0.685846\pi$
$608$	3.00000	0.121666
$609$	0	0
$610$	0	0
$611$	−18.3509	−0.742398
$612$	0	0
$613$	19.8114	0.800174	0.400087	−	0.916477i	$-0.368980\pi$
$613$	19.8114	0.800174	0.400087	+	0.916477i	$0.368980\pi$
$614$	−18.6491	−0.752617
$615$	0	0
$616$	2.00000	0.0805823
$617$	−18.5132	−0.745312	−0.372656	−	0.927970i	$-0.621553\pi$
$617$	−18.5132	−0.745312	−0.372656	+	0.927970i	$0.621553\pi$
$618$	0	0
$619$	−28.9737	−1.16455	−0.582275	−	0.812992i	$-0.697837\pi$
$619$	−28.9737	−1.16455	−0.582275	+	0.812992i	$0.697837\pi$
$620$	0	0
$621$	0	0
$622$	18.0000	0.721734
$623$	5.00000	0.200321
$624$	0	0
$625$	0	0
$626$	−1.16228	−0.0464540
$627$	0	0
$628$	17.8377	0.711803
$629$	61.1623	2.43870
$630$	0	0
$631$	−30.1623	−1.20074	−0.600371	−	0.799722i	$-0.704980\pi$
$631$	−30.1623	−1.20074	−0.600371	+	0.799722i	$0.704980\pi$
$632$	4.48683	0.178477
$633$	0	0
$634$	−28.1623	−1.11847
$635$	0	0
$636$	0	0
$637$	−2.16228	−0.0856726
$638$	16.3246	0.646295
$639$	0	0
$640$	0	0
$641$	2.32456	0.0918144	0.0459072	−	0.998946i	$-0.485382\pi$
$641$	2.32456	0.0918144	0.0459072	+	0.998946i	$0.485382\pi$
$642$	0	0
$643$	−27.6491	−1.09037	−0.545187	−	0.838314i	$-0.683541\pi$
$643$	−27.6491	−1.09037	−0.545187	+	0.838314i	$0.683541\pi$
$644$	−5.16228	−0.203422
$645$	0	0
$646$	−15.9737	−0.628475
$647$	43.1359	1.69585	0.847924	−	0.530117i	$-0.177852\pi$
$647$	43.1359	1.69585	0.847924	+	0.530117i	$0.177852\pi$
$648$	0	0
$649$	24.6491	0.967562
$650$	0	0
$651$	0	0
$652$	−12.3246	−0.482667
$653$	−40.4868	−1.58437	−0.792186	−	0.610280i	$-0.791057\pi$
$653$	−40.4868	−1.58437	−0.792186	+	0.610280i	$0.791057\pi$
$654$	0	0
$655$	0	0
$656$	4.00000	0.156174
$657$	0	0
$658$	8.48683	0.330851
$659$	31.6491	1.23287	0.616437	−	0.787404i	$-0.288575\pi$
$659$	31.6491	1.23287	0.616437	+	0.787404i	$0.288575\pi$
$660$	0	0
$661$	−22.6491	−0.880948	−0.440474	−	0.897765i	$-0.645190\pi$
$661$	−22.6491	−0.880948	−0.440474	+	0.897765i	$0.645190\pi$
$662$	−33.8114	−1.31412
$663$	0	0
$664$	−3.16228	−0.122720
$665$	0	0
$666$	0	0
$667$	−42.1359	−1.63151
$668$	−12.0000	−0.464294
$669$	0	0
$670$	0	0
$671$	−20.3246	−0.784621
$672$	0	0
$673$	2.97367	0.114626	0.0573132	−	0.998356i	$-0.481747\pi$
$673$	2.97367	0.114626	0.0573132	+	0.998356i	$0.481747\pi$
$674$	0.675445	0.0260172
$675$	0	0
$676$	−8.32456	−0.320175
$677$	−16.3246	−0.627404	−0.313702	−	0.949522i	$-0.601569\pi$
$677$	−16.3246	−0.627404	−0.313702	+	0.949522i	$0.601569\pi$
$678$	0	0
$679$	7.48683	0.287318
$680$	0	0
$681$	0	0
$682$	12.6491	0.484359
$683$	20.0000	0.765279	0.382639	−	0.923898i	$-0.375015\pi$
$683$	20.0000	0.765279	0.382639	+	0.923898i	$0.375015\pi$
$684$	0	0
$685$	0	0
$686$	1.00000	0.0381802
$687$	0	0
$688$	−3.16228	−0.120561
$689$	−13.3246	−0.507625
$690$	0	0
$691$	27.2982	1.03847	0.519237	−	0.854631i	$-0.326216\pi$
$691$	27.2982	1.03847	0.519237	+	0.854631i	$0.326216\pi$
$692$	19.1623	0.728440
$693$	0	0
$694$	−23.0000	−0.873068
$695$	0	0
$696$	0	0
$697$	−21.2982	−0.806728
$698$	20.8114	0.787723
$699$	0	0
$700$	0	0
$701$	3.67544	0.138820	0.0694098	−	0.997588i	$-0.477888\pi$
$701$	3.67544	0.138820	0.0694098	+	0.997588i	$0.477888\pi$
$702$	0	0
$703$	−34.4605	−1.29970
$704$	2.00000	0.0753778
$705$	0	0
$706$	24.0000	0.903252
$707$	−3.48683	−0.131136
$708$	0	0
$709$	−6.32456	−0.237524	−0.118762	−	0.992923i	$-0.537892\pi$
$709$	−6.32456	−0.237524	−0.118762	+	0.992923i	$0.537892\pi$
$710$	0	0
$711$	0	0
$712$	5.00000	0.187383
$713$	−32.6491	−1.22272
$714$	0	0
$715$	0	0
$716$	25.3246	0.946423
$717$	0	0
$718$	22.0000	0.821033
$719$	−8.16228	−0.304402	−0.152201	−	0.988350i	$-0.548636\pi$
$719$	−8.16228	−0.304402	−0.152201	+	0.988350i	$0.548636\pi$
$720$	0	0
$721$	6.32456	0.235539
$722$	−10.0000	−0.372161
$723$	0	0
$724$	−16.1623	−0.600666
$725$	0	0
$726$	0	0
$727$	−18.8377	−0.698652	−0.349326	−	0.937001i	$-0.613589\pi$
$727$	−18.8377	−0.698652	−0.349326	+	0.937001i	$0.613589\pi$
$728$	−2.16228	−0.0801393
$729$	0	0
$730$	0	0
$731$	16.8377	0.622766
$732$	0	0
$733$	−19.1359	−0.706802	−0.353401	−	0.935472i	$-0.614975\pi$
$733$	−19.1359	−0.706802	−0.353401	+	0.935472i	$0.614975\pi$
$734$	−26.4605	−0.976675
$735$	0	0
$736$	−5.16228	−0.190284
$737$	30.9737	1.14093
$738$	0	0
$739$	18.6491	0.686019	0.343009	−	0.939332i	$-0.388554\pi$
$739$	18.6491	0.686019	0.343009	+	0.939332i	$0.388554\pi$
$740$	0	0
$741$	0	0
$742$	6.16228	0.226224
$743$	11.2982	0.414492	0.207246	−	0.978289i	$-0.433550\pi$
$743$	11.2982	0.414492	0.207246	+	0.978289i	$0.433550\pi$
$744$	0	0
$745$	0	0
$746$	8.32456	0.304784
$747$	0	0
$748$	−10.6491	−0.389370
$749$	−4.32456	−0.158016
$750$	0	0
$751$	45.9473	1.67664	0.838321	−	0.545177i	$-0.183538\pi$
$751$	45.9473	1.67664	0.838321	+	0.545177i	$0.183538\pi$
$752$	8.48683	0.309483
$753$	0	0
$754$	−17.6491	−0.642743
$755$	0	0
$756$	0	0
$757$	−29.1096	−1.05801	−0.529003	−	0.848620i	$-0.677434\pi$
$757$	−29.1096	−1.05801	−0.529003	+	0.848620i	$0.677434\pi$
$758$	−11.1623	−0.405432
$759$	0	0
$760$	0	0
$761$	−34.9473	−1.26684	−0.633420	−	0.773808i	$-0.718349\pi$
$761$	−34.9473	−1.26684	−0.633420	+	0.773808i	$0.718349\pi$
$762$	0	0
$763$	−3.48683	−0.126232
$764$	0	0
$765$	0	0
$766$	13.6754	0.494114
$767$	−26.6491	−0.962244
$768$	0	0
$769$	16.0000	0.576975	0.288487	−	0.957484i	$-0.406848\pi$
$769$	16.0000	0.576975	0.288487	+	0.957484i	$0.406848\pi$
$770$	0	0
$771$	0	0
$772$	−27.3246	−0.983432
$773$	36.0000	1.29483	0.647415	−	0.762138i	$-0.275850\pi$
$773$	36.0000	1.29483	0.647415	+	0.762138i	$0.275850\pi$
$774$	0	0
$775$	0	0
$776$	7.48683	0.268762
$777$	0	0
$778$	−7.51317	−0.269360
$779$	12.0000	0.429945
$780$	0	0
$781$	4.64911	0.166358
$782$	27.4868	0.982927
$783$	0	0
$784$	1.00000	0.0357143
$785$	0	0
$786$	0	0
$787$	20.9737	0.747630	0.373815	−	0.927503i	$-0.378049\pi$
$787$	20.9737	0.747630	0.373815	+	0.927503i	$0.378049\pi$
$788$	18.8114	0.670128
$789$	0	0
$790$	0	0
$791$	10.6491	0.378639
$792$	0	0
$793$	21.9737	0.780308
$794$	12.8114	0.454659
$795$	0	0
$796$	20.1359	0.713700
$797$	−21.4868	−0.761103	−0.380551	−	0.924760i	$-0.624266\pi$
$797$	−21.4868	−0.761103	−0.380551	+	0.924760i	$0.624266\pi$
$798$	0	0
$799$	−45.1886	−1.59866
$800$	0	0
$801$	0	0
$802$	2.83772	0.100203
$803$	16.6491	0.587534
$804$	0	0
$805$	0	0
$806$	−13.6754	−0.481697
$807$	0	0
$808$	−3.48683	−0.122666
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	3.67544	0.129062	0.0645312	−	0.997916i	$-0.479445\pi$
$811$	3.67544	0.129062	0.0645312	+	0.997916i	$0.479445\pi$
$812$	8.16228	0.286440
$813$	0	0
$814$	−22.9737	−0.805227
$815$	0	0
$816$	0	0
$817$	−9.48683	−0.331902
$818$	−24.1359	−0.843893
$819$	0	0
$820$	0	0
$821$	13.4605	0.469775	0.234887	−	0.972023i	$-0.424528\pi$
$821$	13.4605	0.469775	0.234887	+	0.972023i	$0.424528\pi$
$822$	0	0
$823$	−51.2982	−1.78814	−0.894072	−	0.447924i	$-0.852164\pi$
$823$	−51.2982	−1.78814	−0.894072	+	0.447924i	$0.852164\pi$
$824$	6.32456	0.220326
$825$	0	0
$826$	12.3246	0.428826
$827$	−10.6228	−0.369390	−0.184695	−	0.982796i	$-0.559130\pi$
$827$	−10.6228	−0.369390	−0.184695	+	0.982796i	$0.559130\pi$
$828$	0	0
$829$	−50.1096	−1.74038	−0.870189	−	0.492717i	$-0.836004\pi$
$829$	−50.1096	−1.74038	−0.870189	+	0.492717i	$0.836004\pi$
$830$	0	0
$831$	0	0
$832$	−2.16228	−0.0749635
$833$	−5.32456	−0.184485
$834$	0	0
$835$	0	0
$836$	6.00000	0.207514
$837$	0	0
$838$	24.1359	0.833762
$839$	−19.7851	−0.683056	−0.341528	−	0.939872i	$-0.610944\pi$
$839$	−19.7851	−0.683056	−0.341528	+	0.939872i	$0.610944\pi$
$840$	0	0
$841$	37.6228	1.29734
$842$	−30.6491	−1.05624
$843$	0	0
$844$	−12.3246	−0.424229
$845$	0	0
$846$	0	0
$847$	−7.00000	−0.240523
$848$	6.16228	0.211613
$849$	0	0
$850$	0	0
$851$	59.2982	2.03272
$852$	0	0
$853$	19.6228	0.671871	0.335936	−	0.941885i	$-0.390948\pi$
$853$	19.6228	0.671871	0.335936	+	0.941885i	$0.390948\pi$
$854$	−10.1623	−0.347746
$855$	0	0
$856$	−4.32456	−0.147810
$857$	−43.5964	−1.48923	−0.744613	−	0.667496i	$-0.767366\pi$
$857$	−43.5964	−1.48923	−0.744613	+	0.667496i	$0.767366\pi$
$858$	0	0
$859$	24.0000	0.818869	0.409435	−	0.912339i	$-0.365726\pi$
$859$	24.0000	0.818869	0.409435	+	0.912339i	$0.365726\pi$
$860$	0	0
$861$	0	0
$862$	−6.97367	−0.237524
$863$	−39.8114	−1.35520	−0.677598	−	0.735433i	$-0.736979\pi$
$863$	−39.8114	−1.35520	−0.677598	+	0.735433i	$0.736979\pi$
$864$	0	0
$865$	0	0
$866$	17.6754	0.600636
$867$	0	0
$868$	6.32456	0.214669
$869$	8.97367	0.304411
$870$	0	0
$871$	−33.4868	−1.13466
$872$	−3.48683	−0.118079
$873$	0	0
$874$	−15.4868	−0.523850
$875$	0	0
$876$	0	0
$877$	22.9737	0.775766	0.387883	−	0.921709i	$-0.373207\pi$
$877$	22.9737	0.775766	0.387883	+	0.921709i	$0.373207\pi$
$878$	2.18861	0.0738621
$879$	0	0
$880$	0	0
$881$	−47.9737	−1.61627	−0.808137	−	0.588995i	$-0.799524\pi$
$881$	−47.9737	−1.61627	−0.808137	+	0.588995i	$0.799524\pi$
$882$	0	0
$883$	−17.0263	−0.572982	−0.286491	−	0.958083i	$-0.592489\pi$
$883$	−17.0263	−0.572982	−0.286491	+	0.958083i	$0.592489\pi$
$884$	11.5132	0.387230
$885$	0	0
$886$	34.6228	1.16317
$887$	−17.6754	−0.593483	−0.296742	−	0.954958i	$-0.595900\pi$
$887$	−17.6754	−0.593483	−0.296742	+	0.954958i	$0.595900\pi$
$888$	0	0
$889$	14.4868	0.485873
$890$	0	0
$891$	0	0
$892$	7.16228	0.239811
$893$	25.4605	0.852003
$894$	0	0
$895$	0	0
$896$	1.00000	0.0334077
$897$	0	0
$898$	18.8377	0.628623
$899$	51.6228	1.72172
$900$	0	0
$901$	−32.8114	−1.09311
$902$	8.00000	0.266371
$903$	0	0
$904$	10.6491	0.354184
$905$	0	0
$906$	0	0
$907$	−14.9737	−0.497192	−0.248596	−	0.968607i	$-0.579969\pi$
$907$	−14.9737	−0.497192	−0.248596	+	0.968607i	$0.579969\pi$
$908$	21.8114	0.723836
$909$	0	0
$910$	0	0
$911$	−5.29822	−0.175538	−0.0877690	−	0.996141i	$-0.527974\pi$
$911$	−5.29822	−0.175538	−0.0877690	+	0.996141i	$0.527974\pi$
$912$	0	0
$913$	−6.32456	−0.209312
$914$	32.6228	1.07907
$915$	0	0
$916$	24.1623	0.798344
$917$	13.4868	0.445374
$918$	0	0
$919$	−27.2982	−0.900485	−0.450243	−	0.892906i	$-0.648663\pi$
$919$	−27.2982	−0.900485	−0.450243	+	0.892906i	$0.648663\pi$
$920$	0	0
$921$	0	0
$922$	1.16228	0.0382776
$923$	−5.02633	−0.165444
$924$	0	0
$925$	0	0
$926$	−1.51317	−0.0497258
$927$	0	0
$928$	8.16228	0.267940
$929$	−36.2982	−1.19091	−0.595453	−	0.803390i	$-0.703027\pi$
$929$	−36.2982	−1.19091	−0.595453	+	0.803390i	$0.703027\pi$
$930$	0	0
$931$	3.00000	0.0983210
$932$	3.48683	0.114215
$933$	0	0
$934$	−2.32456	−0.0760618
$935$	0	0
$936$	0	0
$937$	−32.1359	−1.04984	−0.524918	−	0.851153i	$-0.675904\pi$
$937$	−32.1359	−1.04984	−0.524918	+	0.851153i	$0.675904\pi$
$938$	15.4868	0.505663
$939$	0	0
$940$	0	0
$941$	54.7851	1.78594	0.892971	−	0.450114i	$-0.148617\pi$
$941$	54.7851	1.78594	0.892971	+	0.450114i	$0.148617\pi$
$942$	0	0
$943$	−20.6491	−0.672428
$944$	12.3246	0.401130
$945$	0	0
$946$	−6.32456	−0.205629
$947$	−46.0000	−1.49480	−0.747400	−	0.664375i	$-0.768698\pi$
$947$	−46.0000	−1.49480	−0.747400	+	0.664375i	$0.768698\pi$
$948$	0	0
$949$	−18.0000	−0.584305
$950$	0	0
$951$	0	0
$952$	−5.32456	−0.172570
$953$	44.7851	1.45073	0.725365	−	0.688364i	$-0.241671\pi$
$953$	44.7851	1.45073	0.725365	+	0.688364i	$0.241671\pi$
$954$	0	0
$955$	0	0
$956$	−5.81139	−0.187954
$957$	0	0
$958$	2.16228	0.0698600
$959$	−21.4868	−0.693846
$960$	0	0
$961$	9.00000	0.290323
$962$	24.8377	0.800800
$963$	0	0
$964$	−4.51317	−0.145359
$965$	0	0
$966$	0	0
$967$	−3.51317	−0.112976	−0.0564879	−	0.998403i	$-0.517990\pi$
$967$	−3.51317	−0.112976	−0.0564879	+	0.998403i	$0.517990\pi$
$968$	−7.00000	−0.224989
$969$	0	0
$970$	0	0
$971$	−10.4605	−0.335693	−0.167847	−	0.985813i	$-0.553681\pi$
$971$	−10.4605	−0.335693	−0.167847	+	0.985813i	$0.553681\pi$
$972$	0	0
$973$	−0.324555	−0.0104048
$974$	10.9737	0.351619
$975$	0	0
$976$	−10.1623	−0.325287
$977$	16.4605	0.526618	0.263309	−	0.964712i	$-0.415186\pi$
$977$	16.4605	0.526618	0.263309	+	0.964712i	$0.415186\pi$
$978$	0	0
$979$	10.0000	0.319601
$980$	0	0
$981$	0	0
$982$	−4.67544	−0.149199
$983$	−36.4342	−1.16207	−0.581035	−	0.813879i	$-0.697352\pi$
$983$	−36.4342	−1.16207	−0.581035	+	0.813879i	$0.697352\pi$
$984$	0	0
$985$	0	0
$986$	−43.4605	−1.38406
$987$	0	0
$988$	−6.48683	−0.206374
$989$	16.3246	0.519091
$990$	0	0
$991$	39.5132	1.25518	0.627589	−	0.778545i	$-0.284042\pi$
$991$	39.5132	1.25518	0.627589	+	0.778545i	$0.284042\pi$
$992$	6.32456	0.200805
$993$	0	0
$994$	2.32456	0.0737304
$995$	0	0
$996$	0	0
$997$	28.9737	0.917605	0.458803	−	0.888538i	$-0.348279\pi$
$997$	28.9737	0.917605	0.458803	+	0.888538i	$0.348279\pi$
$998$	11.1623	0.353336
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9450.2.a.ex.1.1	yes	2
3.2	odd	2		9450.2.a.eh.1.1	yes	2
5.4	even	2		9450.2.a.ee.1.2	✓	2
15.14	odd	2		9450.2.a.em.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9450.2.a.ee.1.2	✓	2	5.4	even	2
9450.2.a.eh.1.1	yes	2	3.2	odd	2
9450.2.a.em.1.2	yes	2	15.14	odd	2
9450.2.a.ex.1.1	yes	2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9450 = 2 \cdot 3^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9450.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} +1.00000 q^{8} +2.00000 q^{11} -2.16228 q^{13} +1.00000 q^{14} +1.00000 q^{16} -5.32456 q^{17} +3.00000 q^{19} +2.00000 q^{22} -5.16228 q^{23} -2.16228 q^{26} +1.00000 q^{28} +8.16228 q^{29} +6.32456 q^{31} +1.00000 q^{32} -5.32456 q^{34} -11.4868 q^{37} +3.00000 q^{38} +4.00000 q^{41} -3.16228 q^{43} +2.00000 q^{44} -5.16228 q^{46} +8.48683 q^{47} +1.00000 q^{49} -2.16228 q^{52} +6.16228 q^{53} +1.00000 q^{56} +8.16228 q^{58} +12.3246 q^{59} -10.1623 q^{61} +6.32456 q^{62} +1.00000 q^{64} +15.4868 q^{67} -5.32456 q^{68} +2.32456 q^{71} +8.32456 q^{73} -11.4868 q^{74} +3.00000 q^{76} +2.00000 q^{77} +4.48683 q^{79} +4.00000 q^{82} -3.16228 q^{83} -3.16228 q^{86} +2.00000 q^{88} +5.00000 q^{89} -2.16228 q^{91} -5.16228 q^{92} +8.48683 q^{94} +7.48683 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8} + 4 q^{11} + 2 q^{13} + 2 q^{14} + 2 q^{16} + 2 q^{17} + 6 q^{19} + 4 q^{22} - 4 q^{23} + 2 q^{26} + 2 q^{28} + 10 q^{29} + 2 q^{32} + 2 q^{34} - 4 q^{37} + 6 q^{38} + 8 q^{41} + 4 q^{44} - 4 q^{46} - 2 q^{47} + 2 q^{49} + 2 q^{52} + 6 q^{53} + 2 q^{56} + 10 q^{58} + 12 q^{59} - 14 q^{61} + 2 q^{64} + 12 q^{67} + 2 q^{68} - 8 q^{71} + 4 q^{73} - 4 q^{74} + 6 q^{76} + 4 q^{77} - 10 q^{79} + 8 q^{82} + 4 q^{88} + 10 q^{89} + 2 q^{91} - 4 q^{92} - 2 q^{94} - 4 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^7 + 2 * q^8 + 4 * q^11 + 2 * q^13 + 2 * q^14 + 2 * q^16 + 2 * q^17 + 6 * q^19 + 4 * q^22 - 4 * q^23 + 2 * q^26 + 2 * q^28 + 10 * q^29 + 2 * q^32 + 2 * q^34 - 4 * q^37 + 6 * q^38 + 8 * q^41 + 4 * q^44 - 4 * q^46 - 2 * q^47 + 2 * q^49 + 2 * q^52 + 6 * q^53 + 2 * q^56 + 10 * q^58 + 12 * q^59 - 14 * q^61 + 2 * q^64 + 12 * q^67 + 2 * q^68 - 8 * q^71 + 4 * q^73 - 4 * q^74 + 6 * q^76 + 4 * q^77 - 10 * q^79 + 8 * q^82 + 4 * q^88 + 10 * q^89 + 2 * q^91 - 4 * q^92 - 2 * q^94 - 4 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more