LMFDB - Embedded newform 2475.2.a.bc.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.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:	$19.7629745003$
Analytic rank:	$1$
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			1.3
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.17009 q^{2} +2.70928 q^{4} -3.70928 q^{7} +1.53919 q^{8} +O(q^{10})$	$q+2.17009 q^{2} +2.70928 q^{4} -3.70928 q^{7} +1.53919 q^{8} -1.00000 q^{11} +1.70928 q^{13} -8.04945 q^{14} -2.07838 q^{16} -6.04945 q^{17} -3.07838 q^{19} -2.17009 q^{22} +4.00000 q^{23} +3.70928 q^{26} -10.0494 q^{28} -5.26180 q^{29} -6.34017 q^{31} -7.58864 q^{32} -13.1278 q^{34} +3.41855 q^{37} -6.68035 q^{38} -9.57531 q^{41} +3.12783 q^{43} -2.70928 q^{44} +8.68035 q^{46} +2.73820 q^{47} +6.75872 q^{49} +4.63090 q^{52} +13.7587 q^{53} -5.70928 q^{56} -11.4186 q^{58} +3.60197 q^{59} -14.6803 q^{61} -13.7587 q^{62} -12.3112 q^{64} -1.84324 q^{67} -16.3896 q^{68} +7.23513 q^{71} -6.38962 q^{73} +7.41855 q^{74} -8.34017 q^{76} +3.70928 q^{77} -7.44521 q^{79} -20.7792 q^{82} -7.86603 q^{83} +6.78765 q^{86} -1.53919 q^{88} +5.02052 q^{89} -6.34017 q^{91} +10.8371 q^{92} +5.94214 q^{94} -16.9939 q^{97} +14.6670 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + q^{4} - 4 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + q^2 + q^4 - 4 * q^7 + 3 * q^8	$ 3 q + q^{2} + q^{4} - 4 q^{7} + 3 q^{8} - 3 q^{11} - 2 q^{13} - 6 q^{14} - 3 q^{16} - 6 q^{19} - q^{22} + 12 q^{23} + 4 q^{26} - 12 q^{28} - 8 q^{29} - 8 q^{31} - 3 q^{32} - 18 q^{34} - 4 q^{37} + 2 q^{38} - 8 q^{41} - 12 q^{43} - q^{44} + 4 q^{46} + 16 q^{47} - 5 q^{49} + 10 q^{52} + 16 q^{53} - 10 q^{56} - 20 q^{58} - 8 q^{59} - 22 q^{61} - 16 q^{62} - 11 q^{64} - 12 q^{67} - 20 q^{68} + 12 q^{71} + 10 q^{73} + 8 q^{74} - 14 q^{76} + 4 q^{77} - 10 q^{79} - 4 q^{82} - 10 q^{83} + 10 q^{86} - 3 q^{88} - 18 q^{89} - 8 q^{91} + 4 q^{92} - 12 q^{94} - 16 q^{97} + 21 q^{98}+O(q^{100}) $ 3 * q + q^2 + q^4 - 4 * q^7 + 3 * q^8 - 3 * q^11 - 2 * q^13 - 6 * q^14 - 3 * q^16 - 6 * q^19 - q^22 + 12 * q^23 + 4 * q^26 - 12 * q^28 - 8 * q^29 - 8 * q^31 - 3 * q^32 - 18 * q^34 - 4 * q^37 + 2 * q^38 - 8 * q^41 - 12 * q^43 - q^44 + 4 * q^46 + 16 * q^47 - 5 * q^49 + 10 * q^52 + 16 * q^53 - 10 * q^56 - 20 * q^58 - 8 * q^59 - 22 * q^61 - 16 * q^62 - 11 * q^64 - 12 * q^67 - 20 * q^68 + 12 * q^71 + 10 * q^73 + 8 * q^74 - 14 * q^76 + 4 * q^77 - 10 * q^79 - 4 * q^82 - 10 * q^83 + 10 * q^86 - 3 * q^88 - 18 * q^89 - 8 * q^91 + 4 * q^92 - 12 * q^94 - 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	$-0.221629\pi$
$2$	2.17009	1.53448	0.767241	+	0.641358i	$0.221629\pi$
$3$	0	0
$3$	0	0
$4$	2.70928	1.35464
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−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
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	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	$-0.762163\pi$
$17$	−6.04945	−1.46721	−0.733603	+	0.679578i	$0.762163\pi$
$18$	0	0
$19$	−3.07838	−0.706228	−0.353114	−	0.935580i	$-0.614877\pi$
$19$	−3.07838	−0.706228	−0.353114	+	0.935580i	$0.614877\pi$
$20$	0	0
$21$	0	0
$22$	−2.17009	−0.462664
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	0	0
$26$	3.70928	0.727449
$27$	0	0
$28$	−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
$36$	0	0
$37$	3.41855	0.562006	0.281003	−	0.959707i	$-0.409333\pi$
$37$	3.41855	0.562006	0.281003	+	0.959707i	$0.409333\pi$
$38$	−6.68035	−1.08370
$39$	0	0
$40$	0	0
$41$	−9.57531	−1.49541	−0.747706	−	0.664030i	$-0.768845\pi$
$41$	−9.57531	−1.49541	−0.747706	+	0.664030i	$0.768845\pi$
$42$	0	0
$43$	3.12783	0.476989	0.238495	−	0.971144i	$-0.423346\pi$
$43$	3.12783	0.476989	0.238495	+	0.971144i	$0.423346\pi$
$44$	−2.70928	−0.408439
$45$	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	$0.436002\pi$
$48$	0	0
$49$	6.75872	0.965532
$50$	0	0
$51$	0	0
$52$	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	$0.106107\pi$
$54$	0	0
$55$	0	0
$56$	−5.70928	−0.762934
$57$	0	0
$58$	−11.4186	−1.49933
$59$	3.60197	0.468936	0.234468	−	0.972124i	$-0.424665\pi$
$59$	3.60197	0.468936	0.234468	+	0.972124i	$0.424665\pi$
$60$	0	0
$61$	−14.6803	−1.87963	−0.939813	−	0.341690i	$-0.889001\pi$
$61$	−14.6803	−1.87963	−0.939813	+	0.341690i	$0.889001\pi$
$62$	−13.7587	−1.74736
$63$	0	0
$64$	−12.3112	−1.53891
$65$	0	0
$66$	0	0
$67$	−1.84324	−0.225188	−0.112594	−	0.993641i	$-0.535916\pi$
$67$	−1.84324	−0.225188	−0.112594	+	0.993641i	$0.535916\pi$
$68$	−16.3896	−1.98753
$69$	0	0
$70$	0	0
$71$	7.23513	0.858652	0.429326	−	0.903150i	$-0.358751\pi$
$71$	7.23513	0.858652	0.429326	+	0.903150i	$0.358751\pi$
$72$	0	0
$73$	−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
$77$	3.70928	0.422711
$78$	0	0
$79$	−7.44521	−0.837652	−0.418826	−	0.908067i	$-0.637558\pi$
$79$	−7.44521	−0.837652	−0.418826	+	0.908067i	$0.637558\pi$
$80$	0	0
$81$	0	0
$82$	−20.7792	−2.29468
$83$	−7.86603	−0.863409	−0.431705	−	0.902015i	$-0.642088\pi$
$83$	−7.86603	−0.863409	−0.431705	+	0.902015i	$0.642088\pi$
$84$	0	0
$85$	0	0
$86$	6.78765	0.731931
$87$	0	0
$88$	−1.53919	−0.164078
$89$	5.02052	0.532174	0.266087	−	0.963949i	$-0.414269\pi$
$89$	5.02052	0.532174	0.266087	+	0.963949i	$0.414269\pi$
$90$	0	0
$91$	−6.34017	−0.664631
$92$	10.8371	1.12985
$93$	0	0
$94$	5.94214	0.612885
$95$	0	0
$96$	0	0
$97$	−16.9939	−1.72546	−0.862732	−	0.505661i	$-0.831249\pi$
$97$	−16.9939	−1.72546	−0.862732	+	0.505661i	$0.831249\pi$
$98$	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	$0.108847\pi$
$102$	0	0
$103$	11.7854	1.16125	0.580624	−	0.814172i	$-0.302809\pi$
$103$	11.7854	1.16125	0.580624	+	0.814172i	$0.302809\pi$
$104$	2.63090	0.257981
$105$	0	0
$106$	29.8576	2.90003
$107$	11.2846	1.09092	0.545461	−	0.838136i	$-0.316355\pi$
$107$	11.2846	1.09092	0.545461	+	0.838136i	$0.316355\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	0	0
$112$	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	$0.507441\pi$
$114$	0	0
$115$	0	0
$116$	−14.2557	−1.32360
$117$	0	0
$118$	7.81658	0.719575
$119$	22.4391	2.05699
$120$	0	0
$121$	1.00000	0.0909091
$122$	−31.8576	−2.88425
$123$	0	0
$124$	−17.1773	−1.54256
$125$	0	0
$126$	0	0
$127$	−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
$129$	0	0
$130$	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	$0.485332\pi$
$138$	0	0
$139$	10.2823	0.872135	0.436067	−	0.899914i	$-0.356371\pi$
$139$	10.2823	0.872135	0.436067	+	0.899914i	$0.356371\pi$
$140$	0	0
$141$	0	0
$142$	15.7009	1.31759
$143$	−1.70928	−0.142937
$144$	0	0
$145$	0	0
$146$	−13.8660	−1.14756
$147$	0	0
$148$	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	$-0.564178\pi$
$151$	−4.92162	−0.400516	−0.200258	+	0.979743i	$0.564178\pi$
$152$	−4.73820	−0.384319
$153$	0	0
$154$	8.04945	0.648643
$155$	0	0
$156$	0	0
$157$	3.41855	0.272830	0.136415	−	0.990652i	$-0.456442\pi$
$157$	3.41855	0.272830	0.136415	+	0.990652i	$0.456442\pi$
$158$	−16.1568	−1.28536
$159$	0	0
$160$	0	0
$161$	−14.8371	−1.16933
$162$	0	0
$163$	−9.26180	−0.725440	−0.362720	−	0.931898i	$-0.618152\pi$
$163$	−9.26180	−0.725440	−0.362720	+	0.931898i	$0.618152\pi$
$164$	−25.9421	−2.02574
$165$	0	0
$166$	−17.0700	−1.32489
$167$	7.55252	0.584432	0.292216	−	0.956352i	$-0.405607\pi$
$167$	7.55252	0.584432	0.292216	+	0.956352i	$0.405607\pi$
$168$	0	0
$169$	−10.0784	−0.775260
$170$	0	0
$171$	0	0
$172$	8.47414	0.646147
$173$	6.14834	0.467450	0.233725	−	0.972303i	$-0.424908\pi$
$173$	6.14834	0.467450	0.233725	+	0.972303i	$0.424908\pi$
$174$	0	0
$175$	0	0
$176$	2.07838	0.156664
$177$	0	0
$178$	10.8950	0.816612
$179$	6.15676	0.460178	0.230089	−	0.973170i	$-0.426098\pi$
$179$	6.15676	0.460178	0.230089	+	0.973170i	$0.426098\pi$
$180$	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
$183$	0	0
$184$	6.15676	0.453882
$185$	0	0
$186$	0	0
$187$	6.04945	0.442379
$188$	7.41855	0.541053
$189$	0	0
$190$	0	0
$191$	−5.84324	−0.422802	−0.211401	−	0.977399i	$-0.567803\pi$
$191$	−5.84324	−0.422802	−0.211401	+	0.977399i	$0.567803\pi$
$192$	0	0
$193$	−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	$-0.280860\pi$
$197$	17.8348	1.27068	0.635340	+	0.772233i	$0.280860\pi$
$198$	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
$201$	0	0
$202$	41.0928	2.89128
$203$	19.5174	1.36986
$204$	0	0
$205$	0	0
$206$	25.5753	1.78192
$207$	0	0
$208$	−3.55252	−0.246323
$209$	3.07838	0.212936
$210$	0	0
$211$	8.43907	0.580970	0.290485	−	0.956880i	$-0.406183\pi$
$211$	8.43907	0.580970	0.290485	+	0.956880i	$0.406183\pi$
$212$	37.2762	2.56014
$213$	0	0
$214$	24.4885	1.67400
$215$	0	0
$216$	0	0
$217$	23.5174	1.59647
$218$	−21.7009	−1.46977
$219$	0	0
$220$	0	0
$221$	−10.3402	−0.695555
$222$	0	0
$223$	−12.5814	−0.842516	−0.421258	−	0.906941i	$-0.638411\pi$
$223$	−12.5814	−0.842516	−0.421258	+	0.906941i	$0.638411\pi$
$224$	28.1483	1.88074
$225$	0	0
$226$	−1.07838	−0.0717326
$227$	4.23287	0.280945	0.140473	−	0.990085i	$-0.455138\pi$
$227$	4.23287	0.280945	0.140473	+	0.990085i	$0.455138\pi$
$228$	0	0
$229$	−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	$-0.708941\pi$
$233$	−18.6309	−1.22055	−0.610275	+	0.792189i	$0.708941\pi$
$234$	0	0
$235$	0	0
$236$	9.75872	0.635239
$237$	0	0
$238$	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	$-0.404201\pi$
$241$	9.20394	0.592878	0.296439	+	0.955052i	$0.404201\pi$
$242$	2.17009	0.139498
$243$	0	0
$244$	−39.7731	−2.54621
$245$	0	0
$246$	0	0
$247$	−5.26180	−0.334800
$248$	−9.75872	−0.619680
$249$	0	0
$250$	0	0
$251$	22.1256	1.39655	0.698276	−	0.715828i	$-0.253951\pi$
$251$	22.1256	1.39655	0.698276	+	0.715828i	$0.253951\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	−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	$0.469968\pi$
$258$	0	0
$259$	−12.6803	−0.787918
$260$	0	0
$261$	0	0
$262$	−18.8371	−1.16376
$263$	18.2907	1.12785	0.563927	−	0.825825i	$-0.309290\pi$
$263$	18.2907	1.12785	0.563927	+	0.825825i	$0.309290\pi$
$264$	0	0
$265$	0	0
$266$	24.7792	1.51931
$267$	0	0
$268$	−4.99386	−0.305048
$269$	−30.5646	−1.86356	−0.931779	−	0.363026i	$-0.881744\pi$
$269$	−30.5646	−1.86356	−0.931779	+	0.363026i	$0.881744\pi$
$270$	0	0
$271$	−20.0722	−1.21930	−0.609651	−	0.792670i	$-0.708690\pi$
$271$	−20.0722	−1.21930	−0.609651	+	0.792670i	$0.708690\pi$
$272$	12.5730	0.762352
$273$	0	0
$274$	2.34017	0.141375
$275$	0	0
$276$	0	0
$277$	−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
$279$	0	0
$280$	0	0
$281$	−0.581449	−0.0346864	−0.0173432	−	0.999850i	$-0.505521\pi$
$281$	−0.581449	−0.0346864	−0.0173432	+	0.999850i	$0.505521\pi$
$282$	0	0
$283$	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
$286$	−3.70928	−0.219334
$287$	35.5174	2.09653
$288$	0	0
$289$	19.5958	1.15270
$290$	0	0
$291$	0	0
$292$	−17.3112	−1.01306
$293$	7.04331	0.411474	0.205737	−	0.978607i	$-0.434041\pi$
$293$	7.04331	0.411474	0.205737	+	0.978607i	$0.434041\pi$
$294$	0	0
$295$	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
$303$	0	0
$304$	6.39803	0.366952
$305$	0	0
$306$	0	0
$307$	20.1750	1.15145	0.575724	−	0.817644i	$-0.304720\pi$
$307$	20.1750	1.15145	0.575724	+	0.817644i	$0.304720\pi$
$308$	10.0494	0.572620
$309$	0	0
$310$	0	0
$311$	−21.2762	−1.20646	−0.603230	−	0.797567i	$-0.706120\pi$
$311$	−21.2762	−1.20646	−0.603230	+	0.797567i	$0.706120\pi$
$312$	0	0
$313$	−16.4657	−0.930698	−0.465349	−	0.885127i	$-0.654071\pi$
$313$	−16.4657	−0.930698	−0.465349	+	0.885127i	$0.654071\pi$
$314$	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	$0.713935\pi$
$318$	0	0
$319$	5.26180	0.294604
$320$	0	0
$321$	0	0
$322$	−32.1978	−1.79431
$323$	18.6225	1.03618
$324$	0	0
$325$	0	0
$326$	−20.0989	−1.11317
$327$	0	0
$328$	−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
$333$	0	0
$334$	16.3896	0.896800
$335$	0	0
$336$	0	0
$337$	−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	0
$340$	0	0
$341$	6.34017	0.343340
$342$	0	0
$343$	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	$-0.546263\pi$
$347$	−5.39576	−0.289660	−0.144830	+	0.989457i	$0.546263\pi$
$348$	0	0
$349$	−15.6742	−0.839021	−0.419510	−	0.907751i	$-0.637798\pi$
$349$	−15.6742	−0.839021	−0.419510	+	0.907751i	$0.637798\pi$
$350$	0	0
$351$	0	0
$352$	7.58864	0.404476
$353$	−5.75872	−0.306506	−0.153253	−	0.988187i	$-0.548975\pi$
$353$	−5.75872	−0.306506	−0.153253	+	0.988187i	$0.548975\pi$
$354$	0	0
$355$	0	0
$356$	13.6020	0.720903
$357$	0	0
$358$	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
$366$	0	0
$367$	−8.89496	−0.464313	−0.232157	−	0.972678i	$-0.574578\pi$
$367$	−8.89496	−0.464313	−0.232157	+	0.972678i	$0.574578\pi$
$368$	−8.31351	−0.433372
$369$	0	0
$370$	0	0
$371$	−51.0349	−2.64960
$372$	0	0
$373$	23.1689	1.19964	0.599819	−	0.800136i	$-0.295239\pi$
$373$	23.1689	1.19964	0.599819	+	0.800136i	$0.295239\pi$
$374$	13.1278	0.678824
$375$	0	0
$376$	4.21461	0.217352
$377$	−8.99386	−0.463207
$378$	0	0
$379$	−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
$381$	0	0
$382$	−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	$0.269372\pi$
$384$	0	0
$385$	0	0
$386$	−4.38962	−0.223426
$387$	0	0
$388$	−46.0410	−2.33738
$389$	−1.00614	−0.0510135	−0.0255067	−	0.999675i	$-0.508120\pi$
$389$	−1.00614	−0.0510135	−0.0255067	+	0.999675i	$0.508120\pi$
$390$	0	0
$391$	−24.1978	−1.22374
$392$	10.4030	0.525428
$393$	0	0
$394$	38.7031	1.94984
$395$	0	0
$396$	0	0
$397$	−34.7214	−1.74262	−0.871308	−	0.490736i	$-0.836728\pi$
$397$	−34.7214	−1.74262	−0.871308	+	0.490736i	$0.836728\pi$
$398$	55.7152	2.79275
$399$	0	0
$400$	0	0
$401$	13.0205	0.650214	0.325107	−	0.945677i	$-0.394600\pi$
$401$	13.0205	0.650214	0.325107	+	0.945677i	$0.394600\pi$
$402$	0	0
$403$	−10.8371	−0.539834
$404$	51.3028	2.55241
$405$	0	0
$406$	42.3545	2.10202
$407$	−3.41855	−0.169451
$408$	0	0
$409$	−29.5174	−1.45954	−0.729772	−	0.683691i	$-0.760374\pi$
$409$	−29.5174	−1.45954	−0.729772	+	0.683691i	$0.760374\pi$
$410$	0	0
$411$	0	0
$412$	31.9299	1.57307
$413$	−13.3607	−0.657437
$414$	0	0
$415$	0	0
$416$	−12.9711	−0.635959
$417$	0	0
$418$	6.68035	0.326746
$419$	6.15676	0.300777	0.150389	−	0.988627i	$-0.451948\pi$
$419$	6.15676	0.300777	0.150389	+	0.988627i	$0.451948\pi$
$420$	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
$423$	0	0
$424$	21.1773	1.02846
$425$	0	0
$426$	0	0
$427$	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	$0.567040\pi$
$432$	0	0
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	51.0349	2.44975
$435$	0	0
$436$	−27.0928	−1.29751
$437$	−12.3135	−0.589035
$438$	0	0
$439$	3.07838	0.146923	0.0734615	−	0.997298i	$-0.476595\pi$
$439$	3.07838	0.146923	0.0734615	+	0.997298i	$0.476595\pi$
$440$	0	0
$441$	0	0
$442$	−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	$0.255345\pi$
$444$	0	0
$445$	0	0
$446$	−27.3028	−1.29283
$447$	0	0
$448$	45.6658	2.15751
$449$	−10.6947	−0.504715	−0.252358	−	0.967634i	$-0.581206\pi$
$449$	−10.6947	−0.504715	−0.252358	+	0.967634i	$0.581206\pi$
$450$	0	0
$451$	9.57531	0.450884
$452$	−1.34632	−0.0633254
$453$	0	0
$454$	9.18568	0.431106
$455$	0	0
$456$	0	0
$457$	−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
$459$	0	0
$460$	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	$-0.696355\pi$
$463$	−24.8950	−1.15697	−0.578483	+	0.815694i	$0.696355\pi$
$464$	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	$0.646557\pi$
$468$	0	0
$469$	6.83710	0.315708
$470$	0	0
$471$	0	0
$472$	5.54411	0.255188
$473$	−3.12783	−0.143818
$474$	0	0
$475$	0	0
$476$	60.7936	2.78647
$477$	0	0
$478$	−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
$483$	0	0
$484$	2.70928	0.123149
$485$	0	0
$486$	0	0
$487$	−35.2039	−1.59524	−0.797621	−	0.603159i	$-0.793909\pi$
$487$	−35.2039	−1.59524	−0.797621	+	0.603159i	$0.793909\pi$
$488$	−22.5958	−1.02286
$489$	0	0
$490$	0	0
$491$	8.00000	0.361035	0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000	0.361035	0.180517	+	0.983572i	$0.442223\pi$
$492$	0	0
$493$	31.8310	1.43359
$494$	−11.4186	−0.513745
$495$	0	0
$496$	13.1773	0.591677
$497$	−26.8371	−1.20381
$498$	0	0
$499$	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
$501$	0	0
$502$	48.0144	2.14299
$503$	−28.2784	−1.26087	−0.630437	−	0.776241i	$-0.717124\pi$
$503$	−28.2784	−1.26087	−0.630437	+	0.776241i	$0.717124\pi$
$504$	0	0
$505$	0	0
$506$	−8.68035	−0.385888
$507$	0	0
$508$	−7.62475	−0.338294
$509$	−8.47027	−0.375438	−0.187719	−	0.982223i	$-0.560109\pi$
$509$	−8.47027	−0.375438	−0.187719	+	0.982223i	$0.560109\pi$
$510$	0	0
$511$	23.7009	1.04846
$512$	22.1701	0.979789
$513$	0	0
$514$	6.55479	0.289119
$515$	0	0
$516$	0	0
$517$	−2.73820	−0.120426
$518$	−27.5174	−1.20905
$519$	0	0
$520$	0	0
$521$	−13.7009	−0.600246	−0.300123	−	0.953901i	$-0.597028\pi$
$521$	−13.7009	−0.600246	−0.300123	+	0.953901i	$0.597028\pi$
$522$	0	0
$523$	−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
$531$	0	0
$532$	30.9360	1.34125
$533$	−16.3668	−0.708926
$534$	0	0
$535$	0	0
$536$	−2.83710	−0.122544
$537$	0	0
$538$	−66.3279	−2.85960
$539$	−6.75872	−0.291119
$540$	0	0
$541$	4.83710	0.207963	0.103982	−	0.994579i	$-0.466842\pi$
$541$	4.83710	0.207963	0.103982	+	0.994579i	$0.466842\pi$
$542$	−43.5585	−1.87100
$543$	0	0
$544$	45.9071	1.96825
$545$	0	0
$546$	0	0
$547$	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
$549$	0	0
$550$	0	0
$551$	16.1978	0.690049
$552$	0	0
$553$	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	$-0.551333\pi$
$557$	−7.57918	−0.321140	−0.160570	+	0.987024i	$0.551333\pi$
$558$	0	0
$559$	5.34632	0.226125
$560$	0	0
$561$	0	0
$562$	−1.26180	−0.0532256
$563$	−14.1750	−0.597405	−0.298703	−	0.954346i	$-0.596554\pi$
$563$	−14.1750	−0.597405	−0.298703	+	0.954346i	$0.596554\pi$
$564$	0	0
$565$	0	0
$566$	23.4680	0.986434
$567$	0	0
$568$	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	$-0.491773\pi$
$571$	1.23513	0.0516887	0.0258444	+	0.999666i	$0.491773\pi$
$572$	−4.63090	−0.193628
$573$	0	0
$574$	77.0759	3.21709
$575$	0	0
$576$	0	0
$577$	−28.9770	−1.20633	−0.603165	−	0.797617i	$-0.706094\pi$
$577$	−28.9770	−1.20633	−0.603165	+	0.797617i	$0.706094\pi$
$578$	42.5246	1.76879
$579$	0	0
$580$	0	0
$581$	29.1773	1.21048
$582$	0	0
$583$	−13.7587	−0.569828
$584$	−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	$0.642635\pi$
$588$	0	0
$589$	19.5174	0.804202
$590$	0	0
$591$	0	0
$592$	−7.10504	−0.292015
$593$	−23.8927	−0.981155	−0.490578	−	0.871397i	$-0.663214\pi$
$593$	−23.8927	−0.981155	−0.490578	+	0.871397i	$0.663214\pi$
$594$	0	0
$595$	0	0
$596$	−30.9360	−1.26719
$597$	0	0
$598$	14.8371	0.606734
$599$	45.6742	1.86620	0.933099	−	0.359620i	$-0.117094\pi$
$599$	45.6742	1.86620	0.933099	+	0.359620i	$0.117094\pi$
$600$	0	0
$601$	−16.2101	−0.661223	−0.330611	−	0.943767i	$-0.607255\pi$
$601$	−16.2101	−0.661223	−0.330611	+	0.943767i	$0.607255\pi$
$602$	−25.1773	−1.02615
$603$	0	0
$604$	−13.3340	−0.542554
$605$	0	0
$606$	0	0
$607$	−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
$609$	0	0
$610$	0	0
$611$	4.68035	0.189347
$612$	0	0
$613$	−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
$616$	5.70928	0.230033
$617$	21.9733	0.884613	0.442307	−	0.896864i	$-0.354160\pi$
$617$	21.9733	0.884613	0.442307	+	0.896864i	$0.354160\pi$
$618$	0	0
$619$	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
$621$	0	0
$622$	−46.1711	−1.85129
$623$	−18.6225	−0.746094
$624$	0	0
$625$	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
$633$	0	0
$634$	−48.1133	−1.91082
$635$	0	0
$636$	0	0
$637$	11.5525	0.457728
$638$	11.4186	0.452065
$639$	0	0
$640$	0	0
$641$	−24.2245	−0.956808	−0.478404	−	0.878140i	$-0.658785\pi$
$641$	−24.2245	−0.956808	−0.478404	+	0.878140i	$0.658785\pi$
$642$	0	0
$643$	25.8888	1.02096	0.510478	−	0.859891i	$-0.329469\pi$
$643$	25.8888	1.02096	0.510478	+	0.859891i	$0.329469\pi$
$644$	−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	$0.503638\pi$
$648$	0	0
$649$	−3.60197	−0.141390
$650$	0	0
$651$	0	0
$652$	−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	$0.242245\pi$
$654$	0	0
$655$	0	0
$656$	19.9011	0.777008
$657$	0	0
$658$	−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
$663$	0	0
$664$	−12.1073	−0.469855
$665$	0	0
$666$	0	0
$667$	−21.0472	−0.814950
$668$	20.4619	0.791693
$669$	0	0
$670$	0	0
$671$	14.6803	0.566728
$672$	0	0
$673$	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	$-0.612103\pi$
$677$	−17.9506	−0.689896	−0.344948	+	0.938622i	$0.612103\pi$
$678$	0	0
$679$	63.0349	2.41906
$680$	0	0
$681$	0	0
$682$	13.7587	0.526849
$683$	8.77924	0.335928	0.167964	−	0.985793i	$-0.446281\pi$
$683$	8.77924	0.335928	0.167964	+	0.985793i	$0.446281\pi$
$684$	0	0
$685$	0	0
$686$	1.94214	0.0741513
$687$	0	0
$688$	−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
$696$	0	0
$697$	57.9253	2.19408
$698$	−34.0144	−1.28746
$699$	0	0
$700$	0	0
$701$	7.10504	0.268354	0.134177	−	0.990957i	$-0.457161\pi$
$701$	7.10504	0.268354	0.134177	+	0.990957i	$0.457161\pi$
$702$	0	0
$703$	−10.5236	−0.396905
$704$	12.3112	0.463997
$705$	0	0
$706$	−12.4969	−0.470328
$707$	−70.2388	−2.64160
$708$	0	0
$709$	−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
$711$	0	0
$712$	7.72753	0.289601
$713$	−25.3607	−0.949765
$714$	0	0
$715$	0	0
$716$	16.6803	0.623374
$717$	0	0
$718$	22.8371	0.852273
$719$	−31.8310	−1.18709	−0.593547	−	0.804799i	$-0.702273\pi$
$719$	−31.8310	−1.18709	−0.593547	+	0.804799i	$0.702273\pi$
$720$	0	0
$721$	−43.7152	−1.62804
$722$	−20.6670	−0.769147
$723$	0	0
$724$	39.5441	1.46965
$725$	0	0
$726$	0	0
$727$	−5.16290	−0.191481	−0.0957407	−	0.995406i	$-0.530522\pi$
$727$	−5.16290	−0.191481	−0.0957407	+	0.995406i	$0.530522\pi$
$728$	−9.75872	−0.361682
$729$	0	0
$730$	0	0
$731$	−18.9216	−0.699841
$732$	0	0
$733$	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
$737$	1.84324	0.0678968
$738$	0	0
$739$	25.4329	0.935565	0.467783	−	0.883844i	$-0.345053\pi$
$739$	25.4329	0.935565	0.467783	+	0.883844i	$0.345053\pi$
$740$	0	0
$741$	0	0
$742$	−110.750	−4.06577
$743$	10.1217	0.371329	0.185664	−	0.982613i	$-0.440556\pi$
$743$	10.1217	0.371329	0.185664	+	0.982613i	$0.440556\pi$
$744$	0	0
$745$	0	0
$746$	50.2784	1.84082
$747$	0	0
$748$	16.3896	0.599264
$749$	−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
$753$	0	0
$754$	−19.5174	−0.710784
$755$	0	0
$756$	0	0
$757$	6.73820	0.244904	0.122452	−	0.992474i	$-0.460924\pi$
$757$	6.73820	0.244904	0.122452	+	0.992474i	$0.460924\pi$
$758$	−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	$0.467779\pi$
$762$	0	0
$763$	37.0928	1.34285
$764$	−15.8310	−0.572744
$765$	0	0
$766$	56.2967	2.03408
$767$	6.15676	0.222308
$768$	0	0
$769$	−11.5297	−0.415773	−0.207886	−	0.978153i	$-0.566658\pi$
$769$	−11.5297	−0.415773	−0.207886	+	0.978153i	$0.566658\pi$
$770$	0	0
$771$	0	0
$772$	−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	$0.672950\pi$
$774$	0	0
$775$	0	0
$776$	−26.1568	−0.938973
$777$	0	0
$778$	−2.18342	−0.0782793
$779$	29.4764	1.05610
$780$	0	0
$781$	−7.23513	−0.258893
$782$	−52.5113	−1.87780
$783$	0	0
$784$	−14.0472	−0.501685
$785$	0	0
$786$	0	0
$787$	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
$789$	0	0
$790$	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	$0.827015\pi$
$798$	0	0
$799$	−16.5646	−0.586014
$800$	0	0
$801$	0	0
$802$	28.2557	0.997742
$803$	6.38962	0.225485
$804$	0	0
$805$	0	0
$806$	−23.5174	−0.828367
$807$	0	0
$808$	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	$-0.811686\pi$
$811$	−47.2762	−1.66009	−0.830045	+	0.557696i	$0.811686\pi$
$812$	52.8781	1.85566
$813$	0	0
$814$	−7.41855	−0.260020
$815$	0	0
$816$	0	0
$817$	−9.62863	−0.336863
$818$	−64.0554	−2.23965
$819$	0	0
$820$	0	0
$821$	3.30283	0.115270	0.0576348	−	0.998338i	$-0.481644\pi$
$821$	3.30283	0.115270	0.0576348	+	0.998338i	$0.481644\pi$
$822$	0	0
$823$	−11.0517	−0.385239	−0.192619	−	0.981274i	$-0.561698\pi$
$823$	−11.0517	−0.385239	−0.192619	+	0.981274i	$0.561698\pi$
$824$	18.1399	0.631935
$825$	0	0
$826$	−28.9939	−1.00883
$827$	5.12783	0.178312	0.0891560	−	0.996018i	$-0.471583\pi$
$827$	5.12783	0.178312	0.0891560	+	0.996018i	$0.471583\pi$
$828$	0	0
$829$	−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	0
$832$	−21.0433	−0.729545
$833$	−40.8865	−1.41663
$834$	0	0
$835$	0	0
$836$	8.34017	0.288451
$837$	0	0
$838$	13.3607	0.461537
$839$	22.0722	0.762018	0.381009	−	0.924571i	$-0.375577\pi$
$839$	22.0722	0.762018	0.381009	+	0.924571i	$0.375577\pi$
$840$	0	0
$841$	−1.31351	−0.0452935
$842$	21.6332	0.745528
$843$	0	0
$844$	22.8638	0.787003
$845$	0	0
$846$	0	0
$847$	−3.70928	−0.127452
$848$	−28.5958	−0.981985
$849$	0	0
$850$	0	0
$851$	13.6742	0.468746
$852$	0	0
$853$	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	$-0.714761\pi$
$857$	−36.5730	−1.24931	−0.624656	+	0.780900i	$0.714761\pi$
$858$	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
$861$	0	0
$862$	−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	$0.697365\pi$
$864$	0	0
$865$	0	0
$866$	34.7214	1.17988
$867$	0	0
$868$	63.7152	2.16264
$869$	7.44521	0.252562
$870$	0	0
$871$	−3.15061	−0.106754
$872$	−15.3919	−0.521235
$873$	0	0
$874$	−26.7214	−0.903864
$875$	0	0
$876$	0	0
$877$	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
$879$	0	0
$880$	0	0
$881$	8.52359	0.287167	0.143584	−	0.989638i	$-0.454137\pi$
$881$	8.52359	0.287167	0.143584	+	0.989638i	$0.454137\pi$
$882$	0	0
$883$	−43.0349	−1.44824	−0.724120	−	0.689674i	$-0.757754\pi$
$883$	−43.0349	−1.44824	−0.724120	+	0.689674i	$0.757754\pi$
$884$	−28.0144	−0.942225
$885$	0	0
$886$	63.5006	2.13335
$887$	17.0289	0.571775	0.285888	−	0.958263i	$-0.407712\pi$
$887$	17.0289	0.571775	0.285888	+	0.958263i	$0.407712\pi$
$888$	0	0
$889$	10.4391	0.350115
$890$	0	0
$891$	0	0
$892$	−34.0866	−1.14130
$893$	−8.42923	−0.282073
$894$	0	0
$895$	0	0
$896$	42.8020	1.42992
$897$	0	0
$898$	−23.2085	−0.774477
$899$	33.3607	1.11264
$900$	0	0
$901$	−83.2327	−2.77288
$902$	20.7792	0.691873
$903$	0	0
$904$	−0.764867	−0.0254391
$905$	0	0
$906$	0	0
$907$	−6.13993	−0.203873	−0.101937	−	0.994791i	$-0.532504\pi$
$907$	−6.13993	−0.203873	−0.101937	+	0.994791i	$0.532504\pi$
$908$	11.4680	0.380579
$909$	0	0
$910$	0	0
$911$	−53.8720	−1.78486	−0.892429	−	0.451187i	$-0.851001\pi$
$911$	−53.8720	−1.78486	−0.892429	+	0.451187i	$0.851001\pi$
$912$	0	0
$913$	7.86603	0.260328
$914$	−49.5981	−1.64056
$915$	0	0
$916$	−70.9770	−2.34515
$917$	32.1978	1.06326
$918$	0	0
$919$	−48.2700	−1.59228	−0.796141	−	0.605112i	$-0.793128\pi$
$919$	−48.2700	−1.59228	−0.796141	+	0.605112i	$0.793128\pi$
$920$	0	0
$921$	0	0
$922$	47.2495	1.55608
$923$	12.3668	0.407059
$924$	0	0
$925$	0	0
$926$	−54.0242	−1.77535
$927$	0	0
$928$	39.9299	1.31076
$929$	−4.10343	−0.134629	−0.0673146	−	0.997732i	$-0.521443\pi$
$929$	−4.10343	−0.134629	−0.0673146	+	0.997732i	$0.521443\pi$
$930$	0	0
$931$	−20.8059	−0.681886
$932$	−50.4762	−1.65340
$933$	0	0
$934$	−41.6742	−1.36362
$935$	0	0
$936$	0	0
$937$	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
$939$	0	0
$940$	0	0
$941$	−14.6849	−0.478713	−0.239357	−	0.970932i	$-0.576937\pi$
$941$	−14.6849	−0.478713	−0.239357	+	0.970932i	$0.576937\pi$
$942$	0	0
$943$	−38.3012	−1.24726
$944$	−7.48625	−0.243657
$945$	0	0
$946$	−6.78765	−0.220686
$947$	6.05786	0.196854	0.0984270	−	0.995144i	$-0.468619\pi$
$947$	6.05786	0.196854	0.0984270	+	0.995144i	$0.468619\pi$
$948$	0	0
$949$	−10.9216	−0.354531
$950$	0	0
$951$	0	0
$952$	34.5380	1.11938
$953$	40.1438	1.30039	0.650193	−	0.759769i	$-0.274688\pi$
$953$	40.1438	1.30039	0.650193	+	0.759769i	$0.274688\pi$
$954$	0	0
$955$	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
$963$	0	0
$964$	24.9360	0.803134
$965$	0	0
$966$	0	0
$967$	50.6285	1.62810	0.814051	−	0.580794i	$-0.197258\pi$
$967$	50.6285	1.62810	0.814051	+	0.580794i	$0.197258\pi$
$968$	1.53919	0.0494714
$969$	0	0
$970$	0	0
$971$	6.69263	0.214777	0.107388	−	0.994217i	$-0.465751\pi$
$971$	6.69263	0.214777	0.107388	+	0.994217i	$0.465751\pi$
$972$	0	0
$973$	−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	$0.288268\pi$
$978$	0	0
$979$	−5.02052	−0.160456
$980$	0	0
$981$	0	0
$982$	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	$0.649988\pi$
$984$	0	0
$985$	0	0
$986$	69.0759	2.19983
$987$	0	0
$988$	−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
$993$	0	0
$994$	−58.2388	−1.84722
$995$	0	0
$996$	0	0
$997$	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
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.c.b.34.1	✓	6	15.2	even	4
165.2.c.b.34.6	yes	6	15.8	even	4
495.2.c.e.199.1		6	5.3	odd	4
495.2.c.e.199.6		6	5.2	odd	4
825.2.a.j.1.1		3	3.2	odd	2
825.2.a.l.1.3		3	15.14	odd	2
1815.2.c.e.364.1		6	165.98	odd	4
1815.2.c.e.364.6		6	165.32	odd	4
2475.2.a.ba.1.1		3	5.4	even	2
2475.2.a.bc.1.3		3	1.1	even	1	trivial
2640.2.d.h.529.3		6	60.47	odd	4
2640.2.d.h.529.6		6	60.23	odd	4
9075.2.a.cg.1.1		3	165.164	even	2
9075.2.a.ch.1.3		3	33.32	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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q+2.17009 q^{2} +2.70928 q^{4} -3.70928 q^{7} +1.53919 q^{8} +O(q^{10})\)	\(q+2.17009 q^{2} +2.70928 q^{4} -3.70928 q^{7} +1.53919 q^{8} -1.00000 q^{11} +1.70928 q^{13} -8.04945 q^{14} -2.07838 q^{16} -6.04945 q^{17} -3.07838 q^{19} -2.17009 q^{22} +4.00000 q^{23} +3.70928 q^{26} -10.0494 q^{28} -5.26180 q^{29} -6.34017 q^{31} -7.58864 q^{32} -13.1278 q^{34} +3.41855 q^{37} -6.68035 q^{38} -9.57531 q^{41} +3.12783 q^{43} -2.70928 q^{44} +8.68035 q^{46} +2.73820 q^{47} +6.75872 q^{49} +4.63090 q^{52} +13.7587 q^{53} -5.70928 q^{56} -11.4186 q^{58} +3.60197 q^{59} -14.6803 q^{61} -13.7587 q^{62} -12.3112 q^{64} -1.84324 q^{67} -16.3896 q^{68} +7.23513 q^{71} -6.38962 q^{73} +7.41855 q^{74} -8.34017 q^{76} +3.70928 q^{77} -7.44521 q^{79} -20.7792 q^{82} -7.86603 q^{83} +6.78765 q^{86} -1.53919 q^{88} +5.02052 q^{89} -6.34017 q^{91} +10.8371 q^{92} +5.94214 q^{94} -16.9939 q^{97} +14.6670 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + q^{4} - 4 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + q^2 + q^4 - 4 * q^7 + 3 * q^8	\( 3 q + q^{2} + q^{4} - 4 q^{7} + 3 q^{8} - 3 q^{11} - 2 q^{13} - 6 q^{14} - 3 q^{16} - 6 q^{19} - q^{22} + 12 q^{23} + 4 q^{26} - 12 q^{28} - 8 q^{29} - 8 q^{31} - 3 q^{32} - 18 q^{34} - 4 q^{37} + 2 q^{38} - 8 q^{41} - 12 q^{43} - q^{44} + 4 q^{46} + 16 q^{47} - 5 q^{49} + 10 q^{52} + 16 q^{53} - 10 q^{56} - 20 q^{58} - 8 q^{59} - 22 q^{61} - 16 q^{62} - 11 q^{64} - 12 q^{67} - 20 q^{68} + 12 q^{71} + 10 q^{73} + 8 q^{74} - 14 q^{76} + 4 q^{77} - 10 q^{79} - 4 q^{82} - 10 q^{83} + 10 q^{86} - 3 q^{88} - 18 q^{89} - 8 q^{91} + 4 q^{92} - 12 q^{94} - 16 q^{97} + 21 q^{98}+O(q^{100}) \) 3 * q + q^2 + q^4 - 4 * q^7 + 3 * q^8 - 3 * q^11 - 2 * q^13 - 6 * q^14 - 3 * q^16 - 6 * q^19 - q^22 + 12 * q^23 + 4 * q^26 - 12 * q^28 - 8 * q^29 - 8 * q^31 - 3 * q^32 - 18 * q^34 - 4 * q^37 + 2 * q^38 - 8 * q^41 - 12 * q^43 - q^44 + 4 * q^46 + 16 * q^47 - 5 * q^49 + 10 * q^52 + 16 * q^53 - 10 * q^56 - 20 * q^58 - 8 * q^59 - 22 * q^61 - 16 * q^62 - 11 * q^64 - 12 * q^67 - 20 * q^68 + 12 * q^71 + 10 * q^73 + 8 * q^74 - 14 * q^76 + 4 * q^77 - 10 * q^79 - 4 * q^82 - 10 * q^83 + 10 * q^86 - 3 * q^88 - 18 * q^89 - 8 * q^91 + 4 * q^92 - 12 * q^94 - 16 * q^97 + 21 * q^98

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