LMFDB - Embedded newform 9075.2.a.cg.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9075,2,Mod(1,9075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9075, 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("9075.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9075 = 3 \cdot 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9075.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:	$72.4642398343$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	9075.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.17009 q^{2} +1.00000 q^{3} +2.70928 q^{4} -2.17009 q^{6} -3.70928 q^{7} -1.53919 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.17009 q^{2} +1.00000 q^{3} +2.70928 q^{4} -2.17009 q^{6} -3.70928 q^{7} -1.53919 q^{8} +1.00000 q^{9} +2.70928 q^{12} +1.70928 q^{13} +8.04945 q^{14} -2.07838 q^{16} +6.04945 q^{17} -2.17009 q^{18} +3.07838 q^{19} -3.70928 q^{21} +4.00000 q^{23} -1.53919 q^{24} -3.70928 q^{26} +1.00000 q^{27} -10.0494 q^{28} -5.26180 q^{29} -6.34017 q^{31} +7.58864 q^{32} -13.1278 q^{34} +2.70928 q^{36} -3.41855 q^{37} -6.68035 q^{38} +1.70928 q^{39} -9.57531 q^{41} +8.04945 q^{42} +3.12783 q^{43} -8.68035 q^{46} +2.73820 q^{47} -2.07838 q^{48} +6.75872 q^{49} +6.04945 q^{51} +4.63090 q^{52} +13.7587 q^{53} -2.17009 q^{54} +5.70928 q^{56} +3.07838 q^{57} +11.4186 q^{58} -3.60197 q^{59} +14.6803 q^{61} +13.7587 q^{62} -3.70928 q^{63} -12.3112 q^{64} +1.84324 q^{67} +16.3896 q^{68} +4.00000 q^{69} -7.23513 q^{71} -1.53919 q^{72} -6.38962 q^{73} +7.41855 q^{74} +8.34017 q^{76} -3.70928 q^{78} +7.44521 q^{79} +1.00000 q^{81} +20.7792 q^{82} +7.86603 q^{83} -10.0494 q^{84} -6.78765 q^{86} -5.26180 q^{87} -5.02052 q^{89} -6.34017 q^{91} +10.8371 q^{92} -6.34017 q^{93} -5.94214 q^{94} +7.58864 q^{96} +16.9939 q^{97} -14.6670 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 3 q^{3} + q^{4} - q^{6} - 4 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - q^2 + 3 * q^3 + q^4 - q^6 - 4 * q^7 - 3 * q^8 + 3 * q^9	$ 3 q - q^{2} + 3 q^{3} + q^{4} - q^{6} - 4 q^{7} - 3 q^{8} + 3 q^{9} + q^{12} - 2 q^{13} + 6 q^{14} - 3 q^{16} - q^{18} + 6 q^{19} - 4 q^{21} + 12 q^{23} - 3 q^{24} - 4 q^{26} + 3 q^{27} - 12 q^{28} - 8 q^{29} - 8 q^{31} + 3 q^{32} - 18 q^{34} + q^{36} + 4 q^{37} + 2 q^{38} - 2 q^{39} - 8 q^{41} + 6 q^{42} - 12 q^{43} - 4 q^{46} + 16 q^{47} - 3 q^{48} - 5 q^{49} + 10 q^{52} + 16 q^{53} - q^{54} + 10 q^{56} + 6 q^{57} + 20 q^{58} + 8 q^{59} + 22 q^{61} + 16 q^{62} - 4 q^{63} - 11 q^{64} + 12 q^{67} + 20 q^{68} + 12 q^{69} - 12 q^{71} - 3 q^{72} + 10 q^{73} + 8 q^{74} + 14 q^{76} - 4 q^{78} + 10 q^{79} + 3 q^{81} + 4 q^{82} + 10 q^{83} - 12 q^{84} - 10 q^{86} - 8 q^{87} + 18 q^{89} - 8 q^{91} + 4 q^{92} - 8 q^{93} + 12 q^{94} + 3 q^{96} + 16 q^{97} - 21 q^{98}+O(q^{100}) $ 3 * q - q^2 + 3 * q^3 + q^4 - q^6 - 4 * q^7 - 3 * q^8 + 3 * q^9 + q^12 - 2 * q^13 + 6 * q^14 - 3 * q^16 - q^18 + 6 * q^19 - 4 * q^21 + 12 * q^23 - 3 * q^24 - 4 * q^26 + 3 * q^27 - 12 * q^28 - 8 * q^29 - 8 * q^31 + 3 * q^32 - 18 * q^34 + q^36 + 4 * q^37 + 2 * q^38 - 2 * q^39 - 8 * q^41 + 6 * q^42 - 12 * q^43 - 4 * q^46 + 16 * q^47 - 3 * q^48 - 5 * q^49 + 10 * q^52 + 16 * q^53 - q^54 + 10 * q^56 + 6 * q^57 + 20 * q^58 + 8 * q^59 + 22 * q^61 + 16 * q^62 - 4 * q^63 - 11 * q^64 + 12 * q^67 + 20 * q^68 + 12 * q^69 - 12 * q^71 - 3 * q^72 + 10 * q^73 + 8 * q^74 + 14 * q^76 - 4 * q^78 + 10 * q^79 + 3 * q^81 + 4 * q^82 + 10 * q^83 - 12 * q^84 - 10 * q^86 - 8 * q^87 + 18 * q^89 - 8 * q^91 + 4 * q^92 - 8 * q^93 + 12 * q^94 + 3 * q^96 + 16 * q^97 - 21 * 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 $

$2$	−2.17009	−1.53448	−0.767241	−	0.641358i	$-0.778371\pi$
$2$	−2.17009	−1.53448	−0.767241	+	0.641358i	$0.778371\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.70928	1.35464
$5$	0	0
$5$	0	0
$6$	−2.17009	−0.885934
$7$	−3.70928	−1.40197	−0.700987	−	0.713174i	$-0.747257\pi$
$7$	−3.70928	−1.40197	−0.700987	+	0.713174i	$0.747257\pi$
$8$	−1.53919	−0.544185
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	2.70928	0.782100
$13$	1.70928	0.474068	0.237034	−	0.971501i	$-0.423825\pi$
$13$	1.70928	0.474068	0.237034	+	0.971501i	$0.423825\pi$
$14$	8.04945	2.15131
$15$	0	0
$16$	−2.07838	−0.519594
$17$	6.04945	1.46721	0.733603	−	0.679578i	$-0.237837\pi$
$17$	6.04945	1.46721	0.733603	+	0.679578i	$0.237837\pi$
$18$	−2.17009	−0.511494
$19$	3.07838	0.706228	0.353114	−	0.935580i	$-0.385123\pi$
$19$	3.07838	0.706228	0.353114	+	0.935580i	$0.385123\pi$
$20$	0	0
$21$	−3.70928	−0.809430
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	−1.53919	−0.314186
$25$	0	0
$26$	−3.70928	−0.727449
$27$	1.00000	0.192450
$28$	−10.0494	−1.89917
$29$	−5.26180	−0.977091	−0.488545	−	0.872538i	$-0.662472\pi$
$29$	−5.26180	−0.977091	−0.488545	+	0.872538i	$0.662472\pi$
$30$	0	0
$31$	−6.34017	−1.13873	−0.569364	−	0.822085i	$-0.692811\pi$
$31$	−6.34017	−1.13873	−0.569364	+	0.822085i	$0.692811\pi$
$32$	7.58864	1.34149
$33$	0	0
$34$	−13.1278	−2.25140
$35$	0	0
$36$	2.70928	0.451546
$37$	−3.41855	−0.562006	−0.281003	−	0.959707i	$-0.590667\pi$
$37$	−3.41855	−0.562006	−0.281003	+	0.959707i	$0.590667\pi$
$38$	−6.68035	−1.08370
$39$	1.70928	0.273703
$40$	0	0
$41$	−9.57531	−1.49541	−0.747706	−	0.664030i	$-0.768845\pi$
$41$	−9.57531	−1.49541	−0.747706	+	0.664030i	$0.768845\pi$
$42$	8.04945	1.24206
$43$	3.12783	0.476989	0.238495	−	0.971144i	$-0.423346\pi$
$43$	3.12783	0.476989	0.238495	+	0.971144i	$0.423346\pi$
$44$	0	0
$45$	0	0
$46$	−8.68035	−1.27985
$47$	2.73820	0.399408	0.199704	−	0.979856i	$-0.436002\pi$
$47$	2.73820	0.399408	0.199704	+	0.979856i	$0.436002\pi$
$48$	−2.07838	−0.299988
$49$	6.75872	0.965532
$50$	0	0
$51$	6.04945	0.847092
$52$	4.63090	0.642190
$53$	13.7587	1.88991	0.944953	−	0.327206i	$-0.106107\pi$
$53$	13.7587	1.88991	0.944953	+	0.327206i	$0.106107\pi$
$54$	−2.17009	−0.295311
$55$	0	0
$56$	5.70928	0.762934
$57$	3.07838	0.407741
$58$	11.4186	1.49933
$59$	−3.60197	−0.468936	−0.234468	−	0.972124i	$-0.575335\pi$
$59$	−3.60197	−0.468936	−0.234468	+	0.972124i	$0.575335\pi$
$60$	0	0
$61$	14.6803	1.87963	0.939813	−	0.341690i	$-0.110999\pi$
$61$	14.6803	1.87963	0.939813	+	0.341690i	$0.110999\pi$
$62$	13.7587	1.74736
$63$	−3.70928	−0.467325
$64$	−12.3112	−1.53891
$65$	0	0
$66$	0	0
$67$	1.84324	0.225188	0.112594	−	0.993641i	$-0.464084\pi$
$67$	1.84324	0.225188	0.112594	+	0.993641i	$0.464084\pi$
$68$	16.3896	1.98753
$69$	4.00000	0.481543
$70$	0	0
$71$	−7.23513	−0.858652	−0.429326	−	0.903150i	$-0.641249\pi$
$71$	−7.23513	−0.858652	−0.429326	+	0.903150i	$0.641249\pi$
$72$	−1.53919	−0.181395
$73$	−6.38962	−0.747849	−0.373924	−	0.927459i	$-0.621988\pi$
$73$	−6.38962	−0.747849	−0.373924	+	0.927459i	$0.621988\pi$
$74$	7.41855	0.862389
$75$	0	0
$76$	8.34017	0.956683
$77$	0	0
$78$	−3.70928	−0.419993
$79$	7.44521	0.837652	0.418826	−	0.908067i	$-0.362442\pi$
$79$	7.44521	0.837652	0.418826	+	0.908067i	$0.362442\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	20.7792	2.29468
$83$	7.86603	0.863409	0.431705	−	0.902015i	$-0.357912\pi$
$83$	7.86603	0.863409	0.431705	+	0.902015i	$0.357912\pi$
$84$	−10.0494	−1.09648
$85$	0	0
$86$	−6.78765	−0.731931
$87$	−5.26180	−0.564124
$88$	0	0
$89$	−5.02052	−0.532174	−0.266087	−	0.963949i	$-0.585731\pi$
$89$	−5.02052	−0.532174	−0.266087	+	0.963949i	$0.585731\pi$
$90$	0	0
$91$	−6.34017	−0.664631
$92$	10.8371	1.12985
$93$	−6.34017	−0.657445
$94$	−5.94214	−0.612885
$95$	0	0
$96$	7.58864	0.774512
$97$	16.9939	1.72546	0.862732	−	0.505661i	$-0.168751\pi$
$97$	16.9939	1.72546	0.862732	+	0.505661i	$0.168751\pi$
$98$	−14.6670	−1.48159
$99$	0	0
$100$	0	0
$101$	18.9360	1.88420	0.942101	−	0.335329i	$-0.108847\pi$
$101$	18.9360	1.88420	0.942101	+	0.335329i	$0.108847\pi$
$102$	−13.1278	−1.29985
$103$	−11.7854	−1.16125	−0.580624	−	0.814172i	$-0.697191\pi$
$103$	−11.7854	−1.16125	−0.580624	+	0.814172i	$0.697191\pi$
$104$	−2.63090	−0.257981
$105$	0	0
$106$	−29.8576	−2.90003
$107$	−11.2846	−1.09092	−0.545461	−	0.838136i	$-0.683645\pi$
$107$	−11.2846	−1.09092	−0.545461	+	0.838136i	$0.683645\pi$
$108$	2.70928	0.260700
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	−3.41855	−0.324474
$112$	7.70928	0.728458
$113$	−0.496928	−0.0467471	−0.0233735	−	0.999727i	$-0.507441\pi$
$113$	−0.496928	−0.0467471	−0.0233735	+	0.999727i	$0.507441\pi$
$114$	−6.68035	−0.625672
$115$	0	0
$116$	−14.2557	−1.32360
$117$	1.70928	0.158023
$118$	7.81658	0.719575
$119$	−22.4391	−2.05699
$120$	0	0
$121$	0	0
$122$	−31.8576	−2.88425
$123$	−9.57531	−0.863376
$124$	−17.1773	−1.54256
$125$	0	0
$126$	8.04945	0.717102
$127$	−2.81432	−0.249730	−0.124865	−	0.992174i	$-0.539850\pi$
$127$	−2.81432	−0.249730	−0.124865	+	0.992174i	$0.539850\pi$
$128$	11.5392	1.01993
$129$	3.12783	0.275390
$130$	0	0
$131$	−8.68035	−0.758405	−0.379203	−	0.925314i	$-0.623802\pi$
$131$	−8.68035	−0.758405	−0.379203	+	0.925314i	$0.623802\pi$
$132$	0	0
$133$	−11.4186	−0.990114
$134$	−4.00000	−0.345547
$135$	0	0
$136$	−9.31124	−0.798433
$137$	1.07838	0.0921320	0.0460660	−	0.998938i	$-0.485332\pi$
$137$	1.07838	0.0921320	0.0460660	+	0.998938i	$0.485332\pi$
$138$	−8.68035	−0.738920
$139$	−10.2823	−0.872135	−0.436067	−	0.899914i	$-0.643629\pi$
$139$	−10.2823	−0.872135	−0.436067	+	0.899914i	$0.643629\pi$
$140$	0	0
$141$	2.73820	0.230598
$142$	15.7009	1.31759
$143$	0	0
$144$	−2.07838	−0.173198
$145$	0	0
$146$	13.8660	1.14756
$147$	6.75872	0.557450
$148$	−9.26180	−0.761315
$149$	−11.4186	−0.935444	−0.467722	−	0.883876i	$-0.654925\pi$
$149$	−11.4186	−0.935444	−0.467722	+	0.883876i	$0.654925\pi$
$150$	0	0
$151$	4.92162	0.400516	0.200258	−	0.979743i	$-0.435822\pi$
$151$	4.92162	0.400516	0.200258	+	0.979743i	$0.435822\pi$
$152$	−4.73820	−0.384319
$153$	6.04945	0.489069
$154$	0	0
$155$	0	0
$156$	4.63090	0.370769
$157$	−3.41855	−0.272830	−0.136415	−	0.990652i	$-0.543558\pi$
$157$	−3.41855	−0.272830	−0.136415	+	0.990652i	$0.543558\pi$
$158$	−16.1568	−1.28536
$159$	13.7587	1.09114
$160$	0	0
$161$	−14.8371	−1.16933
$162$	−2.17009	−0.170498
$163$	9.26180	0.725440	0.362720	−	0.931898i	$-0.381848\pi$
$163$	9.26180	0.725440	0.362720	+	0.931898i	$0.381848\pi$
$164$	−25.9421	−2.02574
$165$	0	0
$166$	−17.0700	−1.32489
$167$	−7.55252	−0.584432	−0.292216	−	0.956352i	$-0.594393\pi$
$167$	−7.55252	−0.584432	−0.292216	+	0.956352i	$0.594393\pi$
$168$	5.70928	0.440480
$169$	−10.0784	−0.775260
$170$	0	0
$171$	3.07838	0.235409
$172$	8.47414	0.646147
$173$	−6.14834	−0.467450	−0.233725	−	0.972303i	$-0.575092\pi$
$173$	−6.14834	−0.467450	−0.233725	+	0.972303i	$0.575092\pi$
$174$	11.4186	0.865638
$175$	0	0
$176$	0	0
$177$	−3.60197	−0.270741
$178$	10.8950	0.816612
$179$	−6.15676	−0.460178	−0.230089	−	0.973170i	$-0.573902\pi$
$179$	−6.15676	−0.460178	−0.230089	+	0.973170i	$0.573902\pi$
$180$	0	0
$181$	14.5958	1.08490	0.542450	−	0.840088i	$-0.317497\pi$
$181$	14.5958	1.08490	0.542450	+	0.840088i	$0.317497\pi$
$182$	13.7587	1.01986
$183$	14.6803	1.08520
$184$	−6.15676	−0.453882
$185$	0	0
$186$	13.7587	1.00884
$187$	0	0
$188$	7.41855	0.541053
$189$	−3.70928	−0.269810
$190$	0	0
$191$	5.84324	0.422802	0.211401	−	0.977399i	$-0.432197\pi$
$191$	5.84324	0.422802	0.211401	+	0.977399i	$0.432197\pi$
$192$	−12.3112	−0.888487
$193$	−2.02279	−0.145603	−0.0728017	−	0.997346i	$-0.523194\pi$
$193$	−2.02279	−0.145603	−0.0728017	+	0.997346i	$0.523194\pi$
$194$	−36.8781	−2.64770
$195$	0	0
$196$	18.3112	1.30795
$197$	−17.8348	−1.27068	−0.635340	−	0.772233i	$-0.719140\pi$
$197$	−17.8348	−1.27068	−0.635340	+	0.772233i	$0.719140\pi$
$198$	0	0
$199$	25.6742	1.82000	0.909998	−	0.414613i	$-0.136083\pi$
$199$	25.6742	1.82000	0.909998	+	0.414613i	$0.136083\pi$
$200$	0	0
$201$	1.84324	0.130012
$202$	−41.0928	−2.89128
$203$	19.5174	1.36986
$204$	16.3896	1.14750
$205$	0	0
$206$	25.5753	1.78192
$207$	4.00000	0.278019
$208$	−3.55252	−0.246323
$209$	0	0
$210$	0	0
$211$	−8.43907	−0.580970	−0.290485	−	0.956880i	$-0.593817\pi$
$211$	−8.43907	−0.580970	−0.290485	+	0.956880i	$0.593817\pi$
$212$	37.2762	2.56014
$213$	−7.23513	−0.495743
$214$	24.4885	1.67400
$215$	0	0
$216$	−1.53919	−0.104729
$217$	23.5174	1.59647
$218$	−21.7009	−1.46977
$219$	−6.38962	−0.431771
$220$	0	0
$221$	10.3402	0.695555
$222$	7.41855	0.497901
$223$	12.5814	0.842516	0.421258	−	0.906941i	$-0.361589\pi$
$223$	12.5814	0.842516	0.421258	+	0.906941i	$0.361589\pi$
$224$	−28.1483	−1.88074
$225$	0	0
$226$	1.07838	0.0717326
$227$	−4.23287	−0.280945	−0.140473	−	0.990085i	$-0.544862\pi$
$227$	−4.23287	−0.280945	−0.140473	+	0.990085i	$0.544862\pi$
$228$	8.34017	0.552341
$229$	−26.1978	−1.73120	−0.865599	−	0.500737i	$-0.833062\pi$
$229$	−26.1978	−1.73120	−0.865599	+	0.500737i	$0.833062\pi$
$230$	0	0
$231$	0	0
$232$	8.09890	0.531719
$233$	18.6309	1.22055	0.610275	−	0.792189i	$-0.291059\pi$
$233$	18.6309	1.22055	0.610275	+	0.792189i	$0.291059\pi$
$234$	−3.70928	−0.242483
$235$	0	0
$236$	−9.75872	−0.635239
$237$	7.44521	0.483619
$238$	48.6947	3.15641
$239$	−22.3545	−1.44600	−0.722998	−	0.690850i	$-0.757236\pi$
$239$	−22.3545	−1.44600	−0.722998	+	0.690850i	$0.757236\pi$
$240$	0	0
$241$	−9.20394	−0.592878	−0.296439	−	0.955052i	$-0.595799\pi$
$241$	−9.20394	−0.592878	−0.296439	+	0.955052i	$0.595799\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	39.7731	2.54621
$245$	0	0
$246$	20.7792	1.32484
$247$	5.26180	0.334800
$248$	9.75872	0.619680
$249$	7.86603	0.498489
$250$	0	0
$251$	−22.1256	−1.39655	−0.698276	−	0.715828i	$-0.746049\pi$
$251$	−22.1256	−1.39655	−0.698276	+	0.715828i	$0.746049\pi$
$252$	−10.0494	−0.633056
$253$	0	0
$254$	6.10731	0.383207
$255$	0	0
$256$	−0.418551	−0.0261594
$257$	3.02052	0.188415	0.0942074	−	0.995553i	$-0.469968\pi$
$257$	3.02052	0.188415	0.0942074	+	0.995553i	$0.469968\pi$
$258$	−6.78765	−0.422581
$259$	12.6803	0.787918
$260$	0	0
$261$	−5.26180	−0.325697
$262$	18.8371	1.16376
$263$	−18.2907	−1.12785	−0.563927	−	0.825825i	$-0.690710\pi$
$263$	−18.2907	−1.12785	−0.563927	+	0.825825i	$0.690710\pi$
$264$	0	0
$265$	0	0
$266$	24.7792	1.51931
$267$	−5.02052	−0.307251
$268$	4.99386	0.305048
$269$	30.5646	1.86356	0.931779	−	0.363026i	$-0.118256\pi$
$269$	30.5646	1.86356	0.931779	+	0.363026i	$0.118256\pi$
$270$	0	0
$271$	20.0722	1.21930	0.609651	−	0.792670i	$-0.291310\pi$
$271$	20.0722	1.21930	0.609651	+	0.792670i	$0.291310\pi$
$272$	−12.5730	−0.762352
$273$	−6.34017	−0.383725
$274$	−2.34017	−0.141375
$275$	0	0
$276$	10.8371	0.652317
$277$	−0.760991	−0.0457235	−0.0228618	−	0.999739i	$-0.507278\pi$
$277$	−0.760991	−0.0457235	−0.0228618	+	0.999739i	$0.507278\pi$
$278$	22.3135	1.33828
$279$	−6.34017	−0.379576
$280$	0	0
$281$	−0.581449	−0.0346864	−0.0173432	−	0.999850i	$-0.505521\pi$
$281$	−0.581449	−0.0346864	−0.0173432	+	0.999850i	$0.505521\pi$
$282$	−5.94214	−0.353849
$283$	10.8143	0.642844	0.321422	−	0.946936i	$-0.395839\pi$
$283$	10.8143	0.642844	0.321422	+	0.946936i	$0.395839\pi$
$284$	−19.6020	−1.16316
$285$	0	0
$286$	0	0
$287$	35.5174	2.09653
$288$	7.58864	0.447165
$289$	19.5958	1.15270
$290$	0	0
$291$	16.9939	0.996198
$292$	−17.3112	−1.01306
$293$	−7.04331	−0.411474	−0.205737	−	0.978607i	$-0.565959\pi$
$293$	−7.04331	−0.411474	−0.205737	+	0.978607i	$0.565959\pi$
$294$	−14.6670	−0.855398
$295$	0	0
$296$	5.26180	0.305836
$297$	0	0
$298$	24.7792	1.43542
$299$	6.83710	0.395400
$300$	0	0
$301$	−11.6020	−0.668726
$302$	−10.6803	−0.614585
$303$	18.9360	1.08784
$304$	−6.39803	−0.366952
$305$	0	0
$306$	−13.1278	−0.750468
$307$	20.1750	1.15145	0.575724	−	0.817644i	$-0.304720\pi$
$307$	20.1750	1.15145	0.575724	+	0.817644i	$0.304720\pi$
$308$	0	0
$309$	−11.7854	−0.670447
$310$	0	0
$311$	21.2762	1.20646	0.603230	−	0.797567i	$-0.293880\pi$
$311$	21.2762	1.20646	0.603230	+	0.797567i	$0.293880\pi$
$312$	−2.63090	−0.148945
$313$	16.4657	0.930698	0.465349	−	0.885127i	$-0.345929\pi$
$313$	16.4657	0.930698	0.465349	+	0.885127i	$0.345929\pi$
$314$	7.41855	0.418653
$315$	0	0
$316$	20.1711	1.13471
$317$	−22.1711	−1.24525	−0.622627	−	0.782518i	$-0.713935\pi$
$317$	−22.1711	−1.24525	−0.622627	+	0.782518i	$0.713935\pi$
$318$	−29.8576	−1.67433
$319$	0	0
$320$	0	0
$321$	−11.2846	−0.629844
$322$	32.1978	1.79431
$323$	18.6225	1.03618
$324$	2.70928	0.150515
$325$	0	0
$326$	−20.0989	−1.11317
$327$	10.0000	0.553001
$328$	14.7382	0.813781
$329$	−10.1568	−0.559960
$330$	0	0
$331$	−6.34017	−0.348487	−0.174244	−	0.984703i	$-0.555748\pi$
$331$	−6.34017	−0.348487	−0.174244	+	0.984703i	$0.555748\pi$
$332$	21.3112	1.16961
$333$	−3.41855	−0.187335
$334$	16.3896	0.896800
$335$	0	0
$336$	7.70928	0.420575
$337$	−3.18568	−0.173535	−0.0867677	−	0.996229i	$-0.527654\pi$
$337$	−3.18568	−0.173535	−0.0867677	+	0.996229i	$0.527654\pi$
$338$	21.8710	1.18962
$339$	−0.496928	−0.0269894
$340$	0	0
$341$	0	0
$342$	−6.68035	−0.361232
$343$	0.894960	0.0483233
$344$	−4.81432	−0.259570
$345$	0	0
$346$	13.3424	0.717294
$347$	5.39576	0.289660	0.144830	−	0.989457i	$-0.453737\pi$
$347$	5.39576	0.289660	0.144830	+	0.989457i	$0.453737\pi$
$348$	−14.2557	−0.764183
$349$	15.6742	0.839021	0.419510	−	0.907751i	$-0.362202\pi$
$349$	15.6742	0.839021	0.419510	+	0.907751i	$0.362202\pi$
$350$	0	0
$351$	1.70928	0.0912344
$352$	0	0
$353$	−5.75872	−0.306506	−0.153253	−	0.988187i	$-0.548975\pi$
$353$	−5.75872	−0.306506	−0.153253	+	0.988187i	$0.548975\pi$
$354$	7.81658	0.415447
$355$	0	0
$356$	−13.6020	−0.720903
$357$	−22.4391	−1.18760
$358$	13.3607	0.706135
$359$	10.5236	0.555414	0.277707	−	0.960666i	$-0.410426\pi$
$359$	10.5236	0.555414	0.277707	+	0.960666i	$0.410426\pi$
$360$	0	0
$361$	−9.52359	−0.501242
$362$	−31.6742	−1.66476
$363$	0	0
$364$	−17.1773	−0.900334
$365$	0	0
$366$	−31.8576	−1.66522
$367$	8.89496	0.464313	0.232157	−	0.972678i	$-0.425422\pi$
$367$	8.89496	0.464313	0.232157	+	0.972678i	$0.425422\pi$
$368$	−8.31351	−0.433372
$369$	−9.57531	−0.498471
$370$	0	0
$371$	−51.0349	−2.64960
$372$	−17.1773	−0.890600
$373$	23.1689	1.19964	0.599819	−	0.800136i	$-0.295239\pi$
$373$	23.1689	1.19964	0.599819	+	0.800136i	$0.295239\pi$
$374$	0	0
$375$	0	0
$376$	−4.21461	−0.217352
$377$	−8.99386	−0.463207
$378$	8.04945	0.414019
$379$	−11.8310	−0.607716	−0.303858	−	0.952717i	$-0.598275\pi$
$379$	−11.8310	−0.607716	−0.303858	+	0.952717i	$0.598275\pi$
$380$	0	0
$381$	−2.81432	−0.144182
$382$	−12.6803	−0.648783
$383$	25.9421	1.32558	0.662791	−	0.748805i	$-0.269372\pi$
$383$	25.9421	1.32558	0.662791	+	0.748805i	$0.269372\pi$
$384$	11.5392	0.588857
$385$	0	0
$386$	4.38962	0.223426
$387$	3.12783	0.158996
$388$	46.0410	2.33738
$389$	1.00614	0.0510135	0.0255067	−	0.999675i	$-0.491880\pi$
$389$	1.00614	0.0510135	0.0255067	+	0.999675i	$0.491880\pi$
$390$	0	0
$391$	24.1978	1.22374
$392$	−10.4030	−0.525428
$393$	−8.68035	−0.437866
$394$	38.7031	1.94984
$395$	0	0
$396$	0	0
$397$	34.7214	1.74262	0.871308	−	0.490736i	$-0.163272\pi$
$397$	34.7214	1.74262	0.871308	+	0.490736i	$0.163272\pi$
$398$	−55.7152	−2.79275
$399$	−11.4186	−0.571643
$400$	0	0
$401$	−13.0205	−0.650214	−0.325107	−	0.945677i	$-0.605400\pi$
$401$	−13.0205	−0.650214	−0.325107	+	0.945677i	$0.605400\pi$
$402$	−4.00000	−0.199502
$403$	−10.8371	−0.539834
$404$	51.3028	2.55241
$405$	0	0
$406$	−42.3545	−2.10202
$407$	0	0
$408$	−9.31124	−0.460975
$409$	29.5174	1.45954	0.729772	−	0.683691i	$-0.239626\pi$
$409$	29.5174	1.45954	0.729772	+	0.683691i	$0.239626\pi$
$410$	0	0
$411$	1.07838	0.0531925
$412$	−31.9299	−1.57307
$413$	13.3607	0.657437
$414$	−8.68035	−0.426616
$415$	0	0
$416$	12.9711	0.635959
$417$	−10.2823	−0.503527
$418$	0	0
$419$	−6.15676	−0.300777	−0.150389	−	0.988627i	$-0.548052\pi$
$419$	−6.15676	−0.300777	−0.150389	+	0.988627i	$0.548052\pi$
$420$	0	0
$421$	9.96880	0.485850	0.242925	−	0.970045i	$-0.421893\pi$
$421$	9.96880	0.485850	0.242925	+	0.970045i	$0.421893\pi$
$422$	18.3135	0.891488
$423$	2.73820	0.133136
$424$	−21.1773	−1.02846
$425$	0	0
$426$	15.7009	0.760709
$427$	−54.4534	−2.63519
$428$	−30.5730	−1.47780
$429$	0	0
$430$	0	0
$431$	−8.68035	−0.418118	−0.209059	−	0.977903i	$-0.567040\pi$
$431$	−8.68035	−0.418118	−0.209059	+	0.977903i	$0.567040\pi$
$432$	−2.07838	−0.0999960
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	−51.0349	−2.44975
$435$	0	0
$436$	27.0928	1.29751
$437$	12.3135	0.589035
$438$	13.8660	0.662545
$439$	−3.07838	−0.146923	−0.0734615	−	0.997298i	$-0.523405\pi$
$439$	−3.07838	−0.146923	−0.0734615	+	0.997298i	$0.523405\pi$
$440$	0	0
$441$	6.75872	0.321844
$442$	−22.4391	−1.06732
$443$	29.2618	1.39027	0.695135	−	0.718879i	$-0.255345\pi$
$443$	29.2618	1.39027	0.695135	+	0.718879i	$0.255345\pi$
$444$	−9.26180	−0.439545
$445$	0	0
$446$	−27.3028	−1.29283
$447$	−11.4186	−0.540079
$448$	45.6658	2.15751
$449$	10.6947	0.504715	0.252358	−	0.967634i	$-0.418794\pi$
$449$	10.6947	0.504715	0.252358	+	0.967634i	$0.418794\pi$
$450$	0	0
$451$	0	0
$452$	−1.34632	−0.0633254
$453$	4.92162	0.231238
$454$	9.18568	0.431106
$455$	0	0
$456$	−4.73820	−0.221887
$457$	−22.8554	−1.06913	−0.534564	−	0.845128i	$-0.679524\pi$
$457$	−22.8554	−1.06913	−0.534564	+	0.845128i	$0.679524\pi$
$458$	56.8515	2.65650
$459$	6.04945	0.282364
$460$	0	0
$461$	21.7731	1.01407	0.507037	−	0.861924i	$-0.330741\pi$
$461$	21.7731	1.01407	0.507037	+	0.861924i	$0.330741\pi$
$462$	0	0
$463$	24.8950	1.15697	0.578483	−	0.815694i	$-0.303645\pi$
$463$	24.8950	1.15697	0.578483	+	0.815694i	$0.303645\pi$
$464$	10.9360	0.507691
$465$	0	0
$466$	−40.4307	−1.87291
$467$	−19.2039	−0.888652	−0.444326	−	0.895865i	$-0.646557\pi$
$467$	−19.2039	−0.888652	−0.444326	+	0.895865i	$0.646557\pi$
$468$	4.63090	0.214063
$469$	−6.83710	−0.315708
$470$	0	0
$471$	−3.41855	−0.157519
$472$	5.54411	0.255188
$473$	0	0
$474$	−16.1568	−0.742104
$475$	0	0
$476$	−60.7936	−2.78647
$477$	13.7587	0.629969
$478$	48.5113	2.21886
$479$	−9.47641	−0.432988	−0.216494	−	0.976284i	$-0.569462\pi$
$479$	−9.47641	−0.432988	−0.216494	+	0.976284i	$0.569462\pi$
$480$	0	0
$481$	−5.84324	−0.266429
$482$	19.9733	0.909761
$483$	−14.8371	−0.675111
$484$	0	0
$485$	0	0
$486$	−2.17009	−0.0984371
$487$	35.2039	1.59524	0.797621	−	0.603159i	$-0.206091\pi$
$487$	35.2039	1.59524	0.797621	+	0.603159i	$0.206091\pi$
$488$	−22.5958	−1.02286
$489$	9.26180	0.418833
$490$	0	0
$491$	8.00000	0.361035	0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000	0.361035	0.180517	+	0.983572i	$0.442223\pi$
$492$	−25.9421	−1.16956
$493$	−31.8310	−1.43359
$494$	−11.4186	−0.513745
$495$	0	0
$496$	13.1773	0.591677
$497$	26.8371	1.20381
$498$	−17.0700	−0.764924
$499$	26.1568	1.17094	0.585469	−	0.810695i	$-0.300911\pi$
$499$	26.1568	1.17094	0.585469	+	0.810695i	$0.300911\pi$
$500$	0	0
$501$	−7.55252	−0.337422
$502$	48.0144	2.14299
$503$	28.2784	1.26087	0.630437	−	0.776241i	$-0.282876\pi$
$503$	28.2784	1.26087	0.630437	+	0.776241i	$0.282876\pi$
$504$	5.70928	0.254311
$505$	0	0
$506$	0	0
$507$	−10.0784	−0.447596
$508$	−7.62475	−0.338294
$509$	8.47027	0.375438	0.187719	−	0.982223i	$-0.439891\pi$
$509$	8.47027	0.375438	0.187719	+	0.982223i	$0.439891\pi$
$510$	0	0
$511$	23.7009	1.04846
$512$	−22.1701	−0.979789
$513$	3.07838	0.135914
$514$	−6.55479	−0.289119
$515$	0	0
$516$	8.47414	0.373053
$517$	0	0
$518$	−27.5174	−1.20905
$519$	−6.14834	−0.269882
$520$	0	0
$521$	13.7009	0.600246	0.300123	−	0.953901i	$-0.402972\pi$
$521$	13.7009	0.600246	0.300123	+	0.953901i	$0.402972\pi$
$522$	11.4186	0.499776
$523$	−24.4885	−1.07081	−0.535404	−	0.844596i	$-0.679841\pi$
$523$	−24.4885	−1.07081	−0.535404	+	0.844596i	$0.679841\pi$
$524$	−23.5174	−1.02736
$525$	0	0
$526$	39.6925	1.73067
$527$	−38.3545	−1.67075
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−3.60197	−0.156312
$532$	−30.9360	−1.34125
$533$	−16.3668	−0.708926
$534$	10.8950	0.471471
$535$	0	0
$536$	−2.83710	−0.122544
$537$	−6.15676	−0.265684
$538$	−66.3279	−2.85960
$539$	0	0
$540$	0	0
$541$	−4.83710	−0.207963	−0.103982	−	0.994579i	$-0.533158\pi$
$541$	−4.83710	−0.207963	−0.103982	+	0.994579i	$0.533158\pi$
$542$	−43.5585	−1.87100
$543$	14.5958	0.626367
$544$	45.9071	1.96825
$545$	0	0
$546$	13.7587	0.588819
$547$	30.5464	1.30607	0.653034	−	0.757328i	$-0.273496\pi$
$547$	30.5464	1.30607	0.653034	+	0.757328i	$0.273496\pi$
$548$	2.92162	0.124806
$549$	14.6803	0.626542
$550$	0	0
$551$	−16.1978	−0.690049
$552$	−6.15676	−0.262049
$553$	−27.6163	−1.17437
$554$	1.65142	0.0701620
$555$	0	0
$556$	−27.8576	−1.18143
$557$	7.57918	0.321140	0.160570	−	0.987024i	$-0.448667\pi$
$557$	7.57918	0.321140	0.160570	+	0.987024i	$0.448667\pi$
$558$	13.7587	0.582453
$559$	5.34632	0.226125
$560$	0	0
$561$	0	0
$562$	1.26180	0.0532256
$563$	14.1750	0.597405	0.298703	−	0.954346i	$-0.403446\pi$
$563$	14.1750	0.597405	0.298703	+	0.954346i	$0.403446\pi$
$564$	7.41855	0.312377
$565$	0	0
$566$	−23.4680	−0.986434
$567$	−3.70928	−0.155775
$568$	11.1362	0.467266
$569$	14.7382	0.617858	0.308929	−	0.951085i	$-0.400030\pi$
$569$	14.7382	0.617858	0.308929	+	0.951085i	$0.400030\pi$
$570$	0	0
$571$	−1.23513	−0.0516887	−0.0258444	−	0.999666i	$-0.508227\pi$
$571$	−1.23513	−0.0516887	−0.0258444	+	0.999666i	$0.508227\pi$
$572$	0	0
$573$	5.84324	0.244105
$574$	−77.0759	−3.21709
$575$	0	0
$576$	−12.3112	−0.512968
$577$	28.9770	1.20633	0.603165	−	0.797617i	$-0.293906\pi$
$577$	28.9770	1.20633	0.603165	+	0.797617i	$0.293906\pi$
$578$	−42.5246	−1.76879
$579$	−2.02279	−0.0840641
$580$	0	0
$581$	−29.1773	−1.21048
$582$	−36.8781	−1.52865
$583$	0	0
$584$	9.83483	0.406968
$585$	0	0
$586$	15.2846	0.631400
$587$	−20.9939	−0.866509	−0.433255	−	0.901272i	$-0.642635\pi$
$587$	−20.9939	−0.866509	−0.433255	+	0.901272i	$0.642635\pi$
$588$	18.3112	0.755143
$589$	−19.5174	−0.804202
$590$	0	0
$591$	−17.8348	−0.733627
$592$	7.10504	0.292015
$593$	23.8927	0.981155	0.490578	−	0.871397i	$-0.336786\pi$
$593$	23.8927	0.981155	0.490578	+	0.871397i	$0.336786\pi$
$594$	0	0
$595$	0	0
$596$	−30.9360	−1.26719
$597$	25.6742	1.05078
$598$	−14.8371	−0.606734
$599$	−45.6742	−1.86620	−0.933099	−	0.359620i	$-0.882906\pi$
$599$	−45.6742	−1.86620	−0.933099	+	0.359620i	$0.882906\pi$
$600$	0	0
$601$	16.2101	0.661223	0.330611	−	0.943767i	$-0.392745\pi$
$601$	16.2101	0.661223	0.330611	+	0.943767i	$0.392745\pi$
$602$	25.1773	1.02615
$603$	1.84324	0.0750627
$604$	13.3340	0.542554
$605$	0	0
$606$	−41.0928	−1.66928
$607$	−20.8020	−0.844328	−0.422164	−	0.906519i	$-0.638729\pi$
$607$	−20.8020	−0.844328	−0.422164	+	0.906519i	$0.638729\pi$
$608$	23.3607	0.947401
$609$	19.5174	0.790887
$610$	0	0
$611$	4.68035	0.189347
$612$	16.3896	0.662511
$613$	−15.3835	−0.621333	−0.310666	−	0.950519i	$-0.600552\pi$
$613$	−15.3835	−0.621333	−0.310666	+	0.950519i	$0.600552\pi$
$614$	−43.7815	−1.76688
$615$	0	0
$616$	0	0
$617$	21.9733	0.884613	0.442307	−	0.896864i	$-0.354160\pi$
$617$	21.9733	0.884613	0.442307	+	0.896864i	$0.354160\pi$
$618$	25.5753	1.02879
$619$	10.3935	0.417750	0.208875	−	0.977942i	$-0.433020\pi$
$619$	10.3935	0.417750	0.208875	+	0.977942i	$0.433020\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	−46.1711	−1.85129
$623$	18.6225	0.746094
$624$	−3.55252	−0.142215
$625$	0	0
$626$	−35.7321	−1.42814
$627$	0	0
$628$	−9.26180	−0.369586
$629$	−20.6803	−0.824579
$630$	0	0
$631$	38.7214	1.54147	0.770737	−	0.637153i	$-0.219888\pi$
$631$	38.7214	1.54147	0.770737	+	0.637153i	$0.219888\pi$
$632$	−11.4596	−0.455838
$633$	−8.43907	−0.335423
$634$	48.1133	1.91082
$635$	0	0
$636$	37.2762	1.47810
$637$	11.5525	0.457728
$638$	0	0
$639$	−7.23513	−0.286217
$640$	0	0
$641$	24.2245	0.956808	0.478404	−	0.878140i	$-0.341215\pi$
$641$	24.2245	0.956808	0.478404	+	0.878140i	$0.341215\pi$
$642$	24.4885	0.966485
$643$	−25.8888	−1.02096	−0.510478	−	0.859891i	$-0.670531\pi$
$643$	−25.8888	−1.02096	−0.510478	+	0.859891i	$0.670531\pi$
$644$	−40.1978	−1.58401
$645$	0	0
$646$	−40.4124	−1.59000
$647$	−0.581449	−0.0228591	−0.0114296	−	0.999935i	$-0.503638\pi$
$647$	−0.581449	−0.0228591	−0.0114296	+	0.999935i	$0.503638\pi$
$648$	−1.53919	−0.0604650
$649$	0	0
$650$	0	0
$651$	23.5174	0.921721
$652$	25.0928	0.982708
$653$	37.0082	1.44824	0.724122	−	0.689672i	$-0.242245\pi$
$653$	37.0082	1.44824	0.724122	+	0.689672i	$0.242245\pi$
$654$	−21.7009	−0.848571
$655$	0	0
$656$	19.9011	0.777008
$657$	−6.38962	−0.249283
$658$	22.0410	0.859249
$659$	−30.5236	−1.18903	−0.594515	−	0.804084i	$-0.702656\pi$
$659$	−30.5236	−1.18903	−0.594515	+	0.804084i	$0.702656\pi$
$660$	0	0
$661$	5.88428	0.228872	0.114436	−	0.993431i	$-0.463494\pi$
$661$	5.88428	0.228872	0.114436	+	0.993431i	$0.463494\pi$
$662$	13.7587	0.534748
$663$	10.3402	0.401579
$664$	−12.1073	−0.469855
$665$	0	0
$666$	7.41855	0.287463
$667$	−21.0472	−0.814950
$668$	−20.4619	−0.791693
$669$	12.5814	0.486427
$670$	0	0
$671$	0	0
$672$	−28.1483	−1.08585
$673$	26.9711	1.03966	0.519829	−	0.854270i	$-0.325996\pi$
$673$	26.9711	1.03966	0.519829	+	0.854270i	$0.325996\pi$
$674$	6.91321	0.266287
$675$	0	0
$676$	−27.3051	−1.05020
$677$	17.9506	0.689896	0.344948	−	0.938622i	$-0.387897\pi$
$677$	17.9506	0.689896	0.344948	+	0.938622i	$0.387897\pi$
$678$	1.07838	0.0414148
$679$	−63.0349	−2.41906
$680$	0	0
$681$	−4.23287	−0.162204
$682$	0	0
$683$	8.77924	0.335928	0.167964	−	0.985793i	$-0.446281\pi$
$683$	8.77924	0.335928	0.167964	+	0.985793i	$0.446281\pi$
$684$	8.34017	0.318894
$685$	0	0
$686$	−1.94214	−0.0741513
$687$	−26.1978	−0.999508
$688$	−6.50080	−0.247841
$689$	23.5174	0.895943
$690$	0	0
$691$	−14.7214	−0.560028	−0.280014	−	0.959996i	$-0.590339\pi$
$691$	−14.7214	−0.560028	−0.280014	+	0.959996i	$0.590339\pi$
$692$	−16.6576	−0.633225
$693$	0	0
$694$	−11.7093	−0.444478
$695$	0	0
$696$	8.09890	0.306988
$697$	−57.9253	−2.19408
$698$	−34.0144	−1.28746
$699$	18.6309	0.704685
$700$	0	0
$701$	7.10504	0.268354	0.134177	−	0.990957i	$-0.457161\pi$
$701$	7.10504	0.268354	0.134177	+	0.990957i	$0.457161\pi$
$702$	−3.70928	−0.139998
$703$	−10.5236	−0.396905
$704$	0	0
$705$	0	0
$706$	12.4969	0.470328
$707$	−70.2388	−2.64160
$708$	−9.75872	−0.366755
$709$	−34.1666	−1.28315	−0.641577	−	0.767059i	$-0.721719\pi$
$709$	−34.1666	−1.28315	−0.641577	+	0.767059i	$0.721719\pi$
$710$	0	0
$711$	7.44521	0.279217
$712$	7.72753	0.289601
$713$	−25.3607	−0.949765
$714$	48.6947	1.82235
$715$	0	0
$716$	−16.6803	−0.623374
$717$	−22.3545	−0.834846
$718$	−22.8371	−0.852273
$719$	31.8310	1.18709	0.593547	−	0.804799i	$-0.297727\pi$
$719$	31.8310	1.18709	0.593547	+	0.804799i	$0.297727\pi$
$720$	0	0
$721$	43.7152	1.62804
$722$	20.6670	0.769147
$723$	−9.20394	−0.342298
$724$	39.5441	1.46965
$725$	0	0
$726$	0	0
$727$	5.16290	0.191481	0.0957407	−	0.995406i	$-0.469478\pi$
$727$	5.16290	0.191481	0.0957407	+	0.995406i	$0.469478\pi$
$728$	9.75872	0.361682
$729$	1.00000	0.0370370
$730$	0	0
$731$	18.9216	0.699841
$732$	39.7731	1.47006
$733$	36.4475	1.34622	0.673109	−	0.739543i	$-0.264958\pi$
$733$	36.4475	1.34622	0.673109	+	0.739543i	$0.264958\pi$
$734$	−19.3028	−0.712481
$735$	0	0
$736$	30.3545	1.11888
$737$	0	0
$738$	20.7792	0.764894
$739$	−25.4329	−0.935565	−0.467783	−	0.883844i	$-0.654947\pi$
$739$	−25.4329	−0.935565	−0.467783	+	0.883844i	$0.654947\pi$
$740$	0	0
$741$	5.26180	0.193297
$742$	110.750	4.06577
$743$	−10.1217	−0.371329	−0.185664	−	0.982613i	$-0.559444\pi$
$743$	−10.1217	−0.371329	−0.185664	+	0.982613i	$0.559444\pi$
$744$	9.75872	0.357772
$745$	0	0
$746$	−50.2784	−1.84082
$747$	7.86603	0.287803
$748$	0	0
$749$	41.8576	1.52944
$750$	0	0
$751$	10.4703	0.382065	0.191033	−	0.981584i	$-0.438816\pi$
$751$	10.4703	0.382065	0.191033	+	0.981584i	$0.438816\pi$
$752$	−5.69102	−0.207530
$753$	−22.1256	−0.806300
$754$	19.5174	0.710784
$755$	0	0
$756$	−10.0494	−0.365495
$757$	−6.73820	−0.244904	−0.122452	−	0.992474i	$-0.539076\pi$
$757$	−6.73820	−0.244904	−0.122452	+	0.992474i	$0.539076\pi$
$758$	25.6742	0.932529
$759$	0	0
$760$	0	0
$761$	5.57531	0.202105	0.101052	−	0.994881i	$-0.467779\pi$
$761$	5.57531	0.202105	0.101052	+	0.994881i	$0.467779\pi$
$762$	6.10731	0.221244
$763$	−37.0928	−1.34285
$764$	15.8310	0.572744
$765$	0	0
$766$	−56.2967	−2.03408
$767$	−6.15676	−0.222308
$768$	−0.418551	−0.0151031
$769$	11.5297	0.415773	0.207886	−	0.978153i	$-0.433342\pi$
$769$	11.5297	0.415773	0.207886	+	0.978153i	$0.433342\pi$
$770$	0	0
$771$	3.02052	0.108781
$772$	−5.48029	−0.197240
$773$	−28.7480	−1.03400	−0.516998	−	0.855987i	$-0.672950\pi$
$773$	−28.7480	−1.03400	−0.516998	+	0.855987i	$0.672950\pi$
$774$	−6.78765	−0.243977
$775$	0	0
$776$	−26.1568	−0.938973
$777$	12.6803	0.454905
$778$	−2.18342	−0.0782793
$779$	−29.4764	−1.05610
$780$	0	0
$781$	0	0
$782$	−52.5113	−1.87780
$783$	−5.26180	−0.188041
$784$	−14.0472	−0.501685
$785$	0	0
$786$	18.8371	0.671897
$787$	13.4536	0.479570	0.239785	−	0.970826i	$-0.422923\pi$
$787$	13.4536	0.479570	0.239785	+	0.970826i	$0.422923\pi$
$788$	−48.3195	−1.72131
$789$	−18.2907	−0.651167
$790$	0	0
$791$	1.84324	0.0655382
$792$	0	0
$793$	25.0928	0.891070
$794$	−75.3484	−2.67401
$795$	0	0
$796$	69.5585	2.46544
$797$	−48.3279	−1.71186	−0.855931	−	0.517090i	$-0.827015\pi$
$797$	−48.3279	−1.71186	−0.855931	+	0.517090i	$0.827015\pi$
$798$	24.7792	0.877176
$799$	16.5646	0.586014
$800$	0	0
$801$	−5.02052	−0.177391
$802$	28.2557	0.997742
$803$	0	0
$804$	4.99386	0.176120
$805$	0	0
$806$	23.5174	0.828367
$807$	30.5646	1.07593
$808$	−29.1461	−1.02536
$809$	43.8141	1.54042	0.770212	−	0.637789i	$-0.220151\pi$
$809$	43.8141	1.54042	0.770212	+	0.637789i	$0.220151\pi$
$810$	0	0
$811$	47.2762	1.66009	0.830045	−	0.557696i	$-0.188314\pi$
$811$	47.2762	1.66009	0.830045	+	0.557696i	$0.188314\pi$
$812$	52.8781	1.85566
$813$	20.0722	0.703964
$814$	0	0
$815$	0	0
$816$	−12.5730	−0.440144
$817$	9.62863	0.336863
$818$	−64.0554	−2.23965
$819$	−6.34017	−0.221544
$820$	0	0
$821$	3.30283	0.115270	0.0576348	−	0.998338i	$-0.481644\pi$
$821$	3.30283	0.115270	0.0576348	+	0.998338i	$0.481644\pi$
$822$	−2.34017	−0.0816229
$823$	11.0517	0.385239	0.192619	−	0.981274i	$-0.438302\pi$
$823$	11.0517	0.385239	0.192619	+	0.981274i	$0.438302\pi$
$824$	18.1399	0.631935
$825$	0	0
$826$	−28.9939	−1.00883
$827$	−5.12783	−0.178312	−0.0891560	−	0.996018i	$-0.528417\pi$
$827$	−5.12783	−0.178312	−0.0891560	+	0.996018i	$0.528417\pi$
$828$	10.8371	0.376615
$829$	−6.39803	−0.222213	−0.111106	−	0.993809i	$-0.535439\pi$
$829$	−6.39803	−0.222213	−0.111106	+	0.993809i	$0.535439\pi$
$830$	0	0
$831$	−0.760991	−0.0263985
$832$	−21.0433	−0.729545
$833$	40.8865	1.41663
$834$	22.3135	0.772654
$835$	0	0
$836$	0	0
$837$	−6.34017	−0.219148
$838$	13.3607	0.461537
$839$	−22.0722	−0.762018	−0.381009	−	0.924571i	$-0.624423\pi$
$839$	−22.0722	−0.762018	−0.381009	+	0.924571i	$0.624423\pi$
$840$	0	0
$841$	−1.31351	−0.0452935
$842$	−21.6332	−0.745528
$843$	−0.581449	−0.0200262
$844$	−22.8638	−0.787003
$845$	0	0
$846$	−5.94214	−0.204295
$847$	0	0
$848$	−28.5958	−0.981985
$849$	10.8143	0.371146
$850$	0	0
$851$	−13.6742	−0.468746
$852$	−19.6020	−0.671552
$853$	8.76099	0.299971	0.149985	−	0.988688i	$-0.452077\pi$
$853$	8.76099	0.299971	0.149985	+	0.988688i	$0.452077\pi$
$854$	118.169	4.04365
$855$	0	0
$856$	17.3691	0.593664
$857$	36.5730	1.24931	0.624656	−	0.780900i	$-0.285239\pi$
$857$	36.5730	1.24931	0.624656	+	0.780900i	$0.285239\pi$
$858$	0	0
$859$	−49.6886	−1.69535	−0.847676	−	0.530514i	$-0.821999\pi$
$859$	−49.6886	−1.69535	−0.847676	+	0.530514i	$0.821999\pi$
$860$	0	0
$861$	35.5174	1.21043
$862$	18.8371	0.641594
$863$	−34.1399	−1.16214	−0.581068	−	0.813855i	$-0.697365\pi$
$863$	−34.1399	−1.16214	−0.581068	+	0.813855i	$0.697365\pi$
$864$	7.58864	0.258171
$865$	0	0
$866$	34.7214	1.17988
$867$	19.5958	0.665509
$868$	63.7152	2.16264
$869$	0	0
$870$	0	0
$871$	3.15061	0.106754
$872$	−15.3919	−0.521235
$873$	16.9939	0.575155
$874$	−26.7214	−0.903864
$875$	0	0
$876$	−17.3112	−0.584893
$877$	43.5357	1.47010	0.735048	−	0.678015i	$-0.237160\pi$
$877$	43.5357	1.47010	0.735048	+	0.678015i	$0.237160\pi$
$878$	6.68035	0.225451
$879$	−7.04331	−0.237565
$880$	0	0
$881$	−8.52359	−0.287167	−0.143584	−	0.989638i	$-0.545863\pi$
$881$	−8.52359	−0.287167	−0.143584	+	0.989638i	$0.545863\pi$
$882$	−14.6670	−0.493864
$883$	43.0349	1.44824	0.724120	−	0.689674i	$-0.242246\pi$
$883$	43.0349	1.44824	0.724120	+	0.689674i	$0.242246\pi$
$884$	28.0144	0.942225
$885$	0	0
$886$	−63.5006	−2.13335
$887$	−17.0289	−0.571775	−0.285888	−	0.958263i	$-0.592288\pi$
$887$	−17.0289	−0.571775	−0.285888	+	0.958263i	$0.592288\pi$
$888$	5.26180	0.176574
$889$	10.4391	0.350115
$890$	0	0
$891$	0	0
$892$	34.0866	1.14130
$893$	8.42923	0.282073
$894$	24.7792	0.828742
$895$	0	0
$896$	−42.8020	−1.42992
$897$	6.83710	0.228284
$898$	−23.2085	−0.774477
$899$	33.3607	1.11264
$900$	0	0
$901$	83.2327	2.77288
$902$	0	0
$903$	−11.6020	−0.386089
$904$	0.764867	0.0254391
$905$	0	0
$906$	−10.6803	−0.354831
$907$	6.13993	0.203873	0.101937	−	0.994791i	$-0.467496\pi$
$907$	6.13993	0.203873	0.101937	+	0.994791i	$0.467496\pi$
$908$	−11.4680	−0.380579
$909$	18.9360	0.628067
$910$	0	0
$911$	53.8720	1.78486	0.892429	−	0.451187i	$-0.148999\pi$
$911$	53.8720	1.78486	0.892429	+	0.451187i	$0.148999\pi$
$912$	−6.39803	−0.211860
$913$	0	0
$914$	49.5981	1.64056
$915$	0	0
$916$	−70.9770	−2.34515
$917$	32.1978	1.06326
$918$	−13.1278	−0.433283
$919$	48.2700	1.59228	0.796141	−	0.605112i	$-0.206872\pi$
$919$	48.2700	1.59228	0.796141	+	0.605112i	$0.206872\pi$
$920$	0	0
$921$	20.1750	0.664789
$922$	−47.2495	−1.55608
$923$	−12.3668	−0.407059
$924$	0	0
$925$	0	0
$926$	−54.0242	−1.77535
$927$	−11.7854	−0.387083
$928$	−39.9299	−1.31076
$929$	4.10343	0.134629	0.0673146	−	0.997732i	$-0.478557\pi$
$929$	4.10343	0.134629	0.0673146	+	0.997732i	$0.478557\pi$
$930$	0	0
$931$	20.8059	0.681886
$932$	50.4762	1.65340
$933$	21.2762	0.696551
$934$	41.6742	1.36362
$935$	0	0
$936$	−2.63090	−0.0859936
$937$	38.5341	1.25885	0.629427	−	0.777060i	$-0.283290\pi$
$937$	38.5341	1.25885	0.629427	+	0.777060i	$0.283290\pi$
$938$	14.8371	0.484449
$939$	16.4657	0.537339
$940$	0	0
$941$	−14.6849	−0.478713	−0.239357	−	0.970932i	$-0.576937\pi$
$941$	−14.6849	−0.478713	−0.239357	+	0.970932i	$0.576937\pi$
$942$	7.41855	0.241709
$943$	−38.3012	−1.24726
$944$	7.48625	0.243657
$945$	0	0
$946$	0	0
$947$	6.05786	0.196854	0.0984270	−	0.995144i	$-0.468619\pi$
$947$	6.05786	0.196854	0.0984270	+	0.995144i	$0.468619\pi$
$948$	20.1711	0.655128
$949$	−10.9216	−0.354531
$950$	0	0
$951$	−22.1711	−0.718948
$952$	34.5380	1.11938
$953$	−40.1438	−1.30039	−0.650193	−	0.759769i	$-0.725312\pi$
$953$	−40.1438	−1.30039	−0.650193	+	0.759769i	$0.725312\pi$
$954$	−29.8576	−0.966676
$955$	0	0
$956$	−60.5646	−1.95880
$957$	0	0
$958$	20.5646	0.664413
$959$	−4.00000	−0.129167
$960$	0	0
$961$	9.19779	0.296703
$962$	12.6803	0.408831
$963$	−11.2846	−0.363641
$964$	−24.9360	−0.803134
$965$	0	0
$966$	32.1978	1.03595
$967$	50.6285	1.62810	0.814051	−	0.580794i	$-0.197258\pi$
$967$	50.6285	1.62810	0.814051	+	0.580794i	$0.197258\pi$
$968$	0	0
$969$	18.6225	0.598240
$970$	0	0
$971$	−6.69263	−0.214777	−0.107388	−	0.994217i	$-0.534249\pi$
$971$	−6.69263	−0.214777	−0.107388	+	0.994217i	$0.534249\pi$
$972$	2.70928	0.0869000
$973$	38.1399	1.22271
$974$	−76.3956	−2.44787
$975$	0	0
$976$	−30.5113	−0.976643
$977$	38.5835	1.23440	0.617198	−	0.786808i	$-0.288268\pi$
$977$	38.5835	1.23440	0.617198	+	0.786808i	$0.288268\pi$
$978$	−20.0989	−0.642692
$979$	0	0
$980$	0	0
$981$	10.0000	0.319275
$982$	−17.3607	−0.554002
$983$	−28.4657	−0.907916	−0.453958	−	0.891023i	$-0.649988\pi$
$983$	−28.4657	−0.907916	−0.453958	+	0.891023i	$0.649988\pi$
$984$	14.7382	0.469837
$985$	0	0
$986$	69.0759	2.19983
$987$	−10.1568	−0.323293
$988$	14.2557	0.453533
$989$	12.5113	0.397836
$990$	0	0
$991$	2.65368	0.0842970	0.0421485	−	0.999111i	$-0.486580\pi$
$991$	2.65368	0.0842970	0.0421485	+	0.999111i	$0.486580\pi$
$992$	−48.1133	−1.52760
$993$	−6.34017	−0.201199
$994$	−58.2388	−1.84722
$995$	0	0
$996$	21.3112	0.675273
$997$	8.08065	0.255917	0.127958	−	0.991780i	$-0.459158\pi$
$997$	8.08065	0.255917	0.127958	+	0.991780i	$0.459158\pi$
$998$	−56.7624	−1.79678
$999$	−3.41855	−0.108158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.cg.1.1		3
5.2	odd	4		1815.2.c.e.364.1		6
5.3	odd	4		1815.2.c.e.364.6		6
5.4	even	2		9075.2.a.ch.1.3		3
11.10	odd	2		825.2.a.l.1.3		3
33.32	even	2		2475.2.a.ba.1.1		3
55.32	even	4		165.2.c.b.34.6	yes	6
55.43	even	4		165.2.c.b.34.1	✓	6
55.54	odd	2		825.2.a.j.1.1		3
165.32	odd	4		495.2.c.e.199.1		6
165.98	odd	4		495.2.c.e.199.6		6
165.164	even	2		2475.2.a.bc.1.3		3
220.43	odd	4		2640.2.d.h.529.3		6
220.87	odd	4		2640.2.d.h.529.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.c.b.34.1	✓	6	55.43	even	4
165.2.c.b.34.6	yes	6	55.32	even	4
495.2.c.e.199.1		6	165.32	odd	4
495.2.c.e.199.6		6	165.98	odd	4
825.2.a.j.1.1		3	55.54	odd	2
825.2.a.l.1.3		3	11.10	odd	2
1815.2.c.e.364.1		6	5.2	odd	4
1815.2.c.e.364.6		6	5.3	odd	4
2475.2.a.ba.1.1		3	33.32	even	2
2475.2.a.bc.1.3		3	165.164	even	2
2640.2.d.h.529.3		6	220.43	odd	4
2640.2.d.h.529.6		6	220.87	odd	4
9075.2.a.cg.1.1		3	1.1	even	1	trivial
9075.2.a.ch.1.3		3	5.4	even	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:	\( N \)	\(=\)	\( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9075.a (trivial)

\(f(q)\)	\(=\)	\(q-2.17009 q^{2} +1.00000 q^{3} +2.70928 q^{4} -2.17009 q^{6} -3.70928 q^{7} -1.53919 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.17009 q^{2} +1.00000 q^{3} +2.70928 q^{4} -2.17009 q^{6} -3.70928 q^{7} -1.53919 q^{8} +1.00000 q^{9} +2.70928 q^{12} +1.70928 q^{13} +8.04945 q^{14} -2.07838 q^{16} +6.04945 q^{17} -2.17009 q^{18} +3.07838 q^{19} -3.70928 q^{21} +4.00000 q^{23} -1.53919 q^{24} -3.70928 q^{26} +1.00000 q^{27} -10.0494 q^{28} -5.26180 q^{29} -6.34017 q^{31} +7.58864 q^{32} -13.1278 q^{34} +2.70928 q^{36} -3.41855 q^{37} -6.68035 q^{38} +1.70928 q^{39} -9.57531 q^{41} +8.04945 q^{42} +3.12783 q^{43} -8.68035 q^{46} +2.73820 q^{47} -2.07838 q^{48} +6.75872 q^{49} +6.04945 q^{51} +4.63090 q^{52} +13.7587 q^{53} -2.17009 q^{54} +5.70928 q^{56} +3.07838 q^{57} +11.4186 q^{58} -3.60197 q^{59} +14.6803 q^{61} +13.7587 q^{62} -3.70928 q^{63} -12.3112 q^{64} +1.84324 q^{67} +16.3896 q^{68} +4.00000 q^{69} -7.23513 q^{71} -1.53919 q^{72} -6.38962 q^{73} +7.41855 q^{74} +8.34017 q^{76} -3.70928 q^{78} +7.44521 q^{79} +1.00000 q^{81} +20.7792 q^{82} +7.86603 q^{83} -10.0494 q^{84} -6.78765 q^{86} -5.26180 q^{87} -5.02052 q^{89} -6.34017 q^{91} +10.8371 q^{92} -6.34017 q^{93} -5.94214 q^{94} +7.58864 q^{96} +16.9939 q^{97} -14.6670 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 3 q^{3} + q^{4} - q^{6} - 4 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - q^2 + 3 * q^3 + q^4 - q^6 - 4 * q^7 - 3 * q^8 + 3 * q^9	\( 3 q - q^{2} + 3 q^{3} + q^{4} - q^{6} - 4 q^{7} - 3 q^{8} + 3 q^{9} + q^{12} - 2 q^{13} + 6 q^{14} - 3 q^{16} - q^{18} + 6 q^{19} - 4 q^{21} + 12 q^{23} - 3 q^{24} - 4 q^{26} + 3 q^{27} - 12 q^{28} - 8 q^{29} - 8 q^{31} + 3 q^{32} - 18 q^{34} + q^{36} + 4 q^{37} + 2 q^{38} - 2 q^{39} - 8 q^{41} + 6 q^{42} - 12 q^{43} - 4 q^{46} + 16 q^{47} - 3 q^{48} - 5 q^{49} + 10 q^{52} + 16 q^{53} - q^{54} + 10 q^{56} + 6 q^{57} + 20 q^{58} + 8 q^{59} + 22 q^{61} + 16 q^{62} - 4 q^{63} - 11 q^{64} + 12 q^{67} + 20 q^{68} + 12 q^{69} - 12 q^{71} - 3 q^{72} + 10 q^{73} + 8 q^{74} + 14 q^{76} - 4 q^{78} + 10 q^{79} + 3 q^{81} + 4 q^{82} + 10 q^{83} - 12 q^{84} - 10 q^{86} - 8 q^{87} + 18 q^{89} - 8 q^{91} + 4 q^{92} - 8 q^{93} + 12 q^{94} + 3 q^{96} + 16 q^{97} - 21 q^{98}+O(q^{100}) \) 3 * q - q^2 + 3 * q^3 + q^4 - q^6 - 4 * q^7 - 3 * q^8 + 3 * q^9 + q^12 - 2 * q^13 + 6 * q^14 - 3 * q^16 - q^18 + 6 * q^19 - 4 * q^21 + 12 * q^23 - 3 * q^24 - 4 * q^26 + 3 * q^27 - 12 * q^28 - 8 * q^29 - 8 * q^31 + 3 * q^32 - 18 * q^34 + q^36 + 4 * q^37 + 2 * q^38 - 2 * q^39 - 8 * q^41 + 6 * q^42 - 12 * q^43 - 4 * q^46 + 16 * q^47 - 3 * q^48 - 5 * q^49 + 10 * q^52 + 16 * q^53 - q^54 + 10 * q^56 + 6 * q^57 + 20 * q^58 + 8 * q^59 + 22 * q^61 + 16 * q^62 - 4 * q^63 - 11 * q^64 + 12 * q^67 + 20 * q^68 + 12 * q^69 - 12 * q^71 - 3 * q^72 + 10 * q^73 + 8 * q^74 + 14 * q^76 - 4 * q^78 + 10 * q^79 + 3 * q^81 + 4 * q^82 + 10 * q^83 - 12 * q^84 - 10 * q^86 - 8 * q^87 + 18 * q^89 - 8 * q^91 + 4 * q^92 - 8 * q^93 + 12 * q^94 + 3 * q^96 + 16 * q^97 - 21 * q^98

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