LMFDB - Embedded newform 3525.2.a.z.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3525,2,Mod(1,3525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3525, 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("3525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3525 = 3 \cdot 5^{2} \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3525.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:	$28.1472667125$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 9x^{5} + 10x^{4} + 16x^{3} - 15x^{2} - 6x + 5 $ x^7 - x^6 - 9x^5 + 10x^4 + 16x^3 - 15x^2 - 6*x + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-2.68265$ of defining polynomial
Character	$\chi$	$=$	3525.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.68265 q^{2} +1.00000 q^{3} +5.19661 q^{4} +2.68265 q^{6} +3.41850 q^{7} +8.57540 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.68265 q^{2} +1.00000 q^{3} +5.19661 q^{4} +2.68265 q^{6} +3.41850 q^{7} +8.57540 q^{8} +1.00000 q^{9} +0.785508 q^{11} +5.19661 q^{12} +0.625067 q^{13} +9.17064 q^{14} +12.6116 q^{16} -1.30190 q^{17} +2.68265 q^{18} -7.31191 q^{19} +3.41850 q^{21} +2.10724 q^{22} -4.48799 q^{23} +8.57540 q^{24} +1.67684 q^{26} +1.00000 q^{27} +17.7646 q^{28} +0.982324 q^{29} -1.96472 q^{31} +16.6816 q^{32} +0.785508 q^{33} -3.49256 q^{34} +5.19661 q^{36} -4.40601 q^{37} -19.6153 q^{38} +0.625067 q^{39} +4.01158 q^{41} +9.17064 q^{42} -5.44380 q^{43} +4.08198 q^{44} -12.0397 q^{46} +1.00000 q^{47} +12.6116 q^{48} +4.68614 q^{49} -1.30190 q^{51} +3.24823 q^{52} -12.9830 q^{53} +2.68265 q^{54} +29.3150 q^{56} -7.31191 q^{57} +2.63523 q^{58} -6.08274 q^{59} -0.112157 q^{61} -5.27067 q^{62} +3.41850 q^{63} +19.5279 q^{64} +2.10724 q^{66} +9.41810 q^{67} -6.76550 q^{68} -4.48799 q^{69} +6.78837 q^{71} +8.57540 q^{72} -9.51807 q^{73} -11.8198 q^{74} -37.9972 q^{76} +2.68526 q^{77} +1.67684 q^{78} -8.50134 q^{79} +1.00000 q^{81} +10.7617 q^{82} +14.0784 q^{83} +17.7646 q^{84} -14.6038 q^{86} +0.982324 q^{87} +6.73604 q^{88} +13.6300 q^{89} +2.13679 q^{91} -23.3223 q^{92} -1.96472 q^{93} +2.68265 q^{94} +16.6816 q^{96} +13.5982 q^{97} +12.5713 q^{98} +0.785508 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{2} + 7 q^{3} + 5 q^{4} - q^{6} + 7 q^{7} + 6 q^{8} + 7 q^{9}+O(q^{10}) $ 7 * q - q^2 + 7 * q^3 + 5 * q^4 - q^6 + 7 * q^7 + 6 * q^8 + 7 * q^9	$ 7 q - q^{2} + 7 q^{3} + 5 q^{4} - q^{6} + 7 q^{7} + 6 q^{8} + 7 q^{9} + 5 q^{12} + 5 q^{13} + 7 q^{14} + 9 q^{16} + 2 q^{17} - q^{18} - 13 q^{19} + 7 q^{21} - 14 q^{22} + 6 q^{23} + 6 q^{24} + 7 q^{27} + 30 q^{28} + 9 q^{29} + 5 q^{31} + 26 q^{32} - 8 q^{34} + 5 q^{36} - 5 q^{37} - 2 q^{38} + 5 q^{39} + 18 q^{41} + 7 q^{42} + 14 q^{43} + 17 q^{44} - 27 q^{46} + 7 q^{47} + 9 q^{48} + 14 q^{49} + 2 q^{51} - 3 q^{52} + 20 q^{53} - q^{54} + 17 q^{56} - 13 q^{57} + 37 q^{58} + 10 q^{59} - 8 q^{61} - 6 q^{62} + 7 q^{63} + 18 q^{64} - 14 q^{66} + 4 q^{67} + 10 q^{68} + 6 q^{69} + 12 q^{71} + 6 q^{72} + 4 q^{73} - 25 q^{74} - 66 q^{76} + 6 q^{77} - 5 q^{79} + 7 q^{81} - 29 q^{82} + 52 q^{83} + 30 q^{84} - 17 q^{86} + 9 q^{87} + 26 q^{88} + 32 q^{89} - 26 q^{91} - 17 q^{92} + 5 q^{93} - q^{94} + 26 q^{96} - 12 q^{97} + 40 q^{98}+O(q^{100}) $ 7 * q - q^2 + 7 * q^3 + 5 * q^4 - q^6 + 7 * q^7 + 6 * q^8 + 7 * q^9 + 5 * q^12 + 5 * q^13 + 7 * q^14 + 9 * q^16 + 2 * q^17 - q^18 - 13 * q^19 + 7 * q^21 - 14 * q^22 + 6 * q^23 + 6 * q^24 + 7 * q^27 + 30 * q^28 + 9 * q^29 + 5 * q^31 + 26 * q^32 - 8 * q^34 + 5 * q^36 - 5 * q^37 - 2 * q^38 + 5 * q^39 + 18 * q^41 + 7 * q^42 + 14 * q^43 + 17 * q^44 - 27 * q^46 + 7 * q^47 + 9 * q^48 + 14 * q^49 + 2 * q^51 - 3 * q^52 + 20 * q^53 - q^54 + 17 * q^56 - 13 * q^57 + 37 * q^58 + 10 * q^59 - 8 * q^61 - 6 * q^62 + 7 * q^63 + 18 * q^64 - 14 * q^66 + 4 * q^67 + 10 * q^68 + 6 * q^69 + 12 * q^71 + 6 * q^72 + 4 * q^73 - 25 * q^74 - 66 * q^76 + 6 * q^77 - 5 * q^79 + 7 * q^81 - 29 * q^82 + 52 * q^83 + 30 * q^84 - 17 * q^86 + 9 * q^87 + 26 * q^88 + 32 * q^89 - 26 * q^91 - 17 * q^92 + 5 * q^93 - q^94 + 26 * q^96 - 12 * q^97 + 40 * 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 $

$2$	2.68265	1.89692	0.948460	−	0.316896i	$-0.102641\pi$
$2$	2.68265	1.89692	0.948460	+	0.316896i	$0.102641\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	5.19661	2.59831
$5$	0	0
$5$	0	0
$6$	2.68265	1.09519
$7$	3.41850	1.29207	0.646036	−	0.763307i	$-0.276426\pi$
$7$	3.41850	1.29207	0.646036	+	0.763307i	$0.276426\pi$
$8$	8.57540	3.03186
$9$	1.00000	0.333333
$10$	0	0
$11$	0.785508	0.236839	0.118420	−	0.992964i	$-0.462217\pi$
$11$	0.785508	0.236839	0.118420	+	0.992964i	$0.462217\pi$
$12$	5.19661	1.50013
$13$	0.625067	0.173362	0.0866812	−	0.996236i	$-0.472374\pi$
$13$	0.625067	0.173362	0.0866812	+	0.996236i	$0.472374\pi$
$14$	9.17064	2.45096
$15$	0	0
$16$	12.6116	3.15289
$17$	−1.30190	−0.315758	−0.157879	−	0.987458i	$-0.550466\pi$
$17$	−1.30190	−0.315758	−0.157879	+	0.987458i	$0.550466\pi$
$18$	2.68265	0.632307
$19$	−7.31191	−1.67747	−0.838734	−	0.544542i	$-0.816703\pi$
$19$	−7.31191	−1.67747	−0.838734	+	0.544542i	$0.816703\pi$
$20$	0	0
$21$	3.41850	0.745978
$22$	2.10724	0.449266
$23$	−4.48799	−0.935810	−0.467905	−	0.883779i	$-0.654991\pi$
$23$	−4.48799	−0.935810	−0.467905	+	0.883779i	$0.654991\pi$
$24$	8.57540	1.75045
$25$	0	0
$26$	1.67684	0.328855
$27$	1.00000	0.192450
$28$	17.7646	3.35720
$29$	0.982324	0.182413	0.0912064	−	0.995832i	$-0.470928\pi$
$29$	0.982324	0.182413	0.0912064	+	0.995832i	$0.470928\pi$
$30$	0	0
$31$	−1.96472	−0.352875	−0.176438	−	0.984312i	$-0.556457\pi$
$31$	−1.96472	−0.352875	−0.176438	+	0.984312i	$0.556457\pi$
$32$	16.6816	2.94893
$33$	0.785508	0.136739
$34$	−3.49256	−0.598968
$35$	0	0
$36$	5.19661	0.866102
$37$	−4.40601	−0.724343	−0.362172	−	0.932111i	$-0.617965\pi$
$37$	−4.40601	−0.724343	−0.362172	+	0.932111i	$0.617965\pi$
$38$	−19.6153	−3.18202
$39$	0.625067	0.100091
$40$	0	0
$41$	4.01158	0.626504	0.313252	−	0.949670i	$-0.398582\pi$
$41$	4.01158	0.626504	0.313252	+	0.949670i	$0.398582\pi$
$42$	9.17064	1.41506
$43$	−5.44380	−0.830171	−0.415086	−	0.909782i	$-0.636248\pi$
$43$	−5.44380	−0.830171	−0.415086	+	0.909782i	$0.636248\pi$
$44$	4.08198	0.615382
$45$	0	0
$46$	−12.0397	−1.77516
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	12.6116	1.82032
$49$	4.68614	0.669448
$50$	0	0
$51$	−1.30190	−0.182303
$52$	3.24823	0.450449
$53$	−12.9830	−1.78335	−0.891674	−	0.452678i	$-0.850469\pi$
$53$	−12.9830	−1.78335	−0.891674	+	0.452678i	$0.850469\pi$
$54$	2.68265	0.365063
$55$	0	0
$56$	29.3150	3.91738
$57$	−7.31191	−0.968486
$58$	2.63523	0.346023
$59$	−6.08274	−0.791905	−0.395953	−	0.918271i	$-0.629586\pi$
$59$	−6.08274	−0.791905	−0.395953	+	0.918271i	$0.629586\pi$
$60$	0	0
$61$	−0.112157	−0.0143602	−0.00718009	−	0.999974i	$-0.502286\pi$
$61$	−0.112157	−0.0143602	−0.00718009	+	0.999974i	$0.502286\pi$
$62$	−5.27067	−0.669376
$63$	3.41850	0.430690
$64$	19.5279	2.44099
$65$	0	0
$66$	2.10724	0.259384
$67$	9.41810	1.15060	0.575302	−	0.817941i	$-0.304885\pi$
$67$	9.41810	1.15060	0.575302	+	0.817941i	$0.304885\pi$
$68$	−6.76550	−0.820437
$69$	−4.48799	−0.540290
$70$	0	0
$71$	6.78837	0.805631	0.402815	−	0.915281i	$-0.368032\pi$
$71$	6.78837	0.805631	0.402815	+	0.915281i	$0.368032\pi$
$72$	8.57540	1.01062
$73$	−9.51807	−1.11401	−0.557003	−	0.830511i	$-0.688049\pi$
$73$	−9.51807	−1.11401	−0.557003	+	0.830511i	$0.688049\pi$
$74$	−11.8198	−1.37402
$75$	0	0
$76$	−37.9972	−4.35858
$77$	2.68526	0.306013
$78$	1.67684	0.189864
$79$	−8.50134	−0.956476	−0.478238	−	0.878230i	$-0.658724\pi$
$79$	−8.50134	−0.956476	−0.478238	+	0.878230i	$0.658724\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	10.7617	1.18843
$83$	14.0784	1.54530	0.772651	−	0.634831i	$-0.218930\pi$
$83$	14.0784	1.54530	0.772651	+	0.634831i	$0.218930\pi$
$84$	17.7646	1.93828
$85$	0	0
$86$	−14.6038	−1.57477
$87$	0.982324	0.105316
$88$	6.73604	0.718065
$89$	13.6300	1.44478	0.722388	−	0.691488i	$-0.243045\pi$
$89$	13.6300	1.44478	0.722388	+	0.691488i	$0.243045\pi$
$90$	0	0
$91$	2.13679	0.223997
$92$	−23.3223	−2.43152
$93$	−1.96472	−0.203732
$94$	2.68265	0.276694
$95$	0	0
$96$	16.6816	1.70256
$97$	13.5982	1.38069	0.690343	−	0.723482i	$-0.257460\pi$
$97$	13.5982	1.38069	0.690343	+	0.723482i	$0.257460\pi$
$98$	12.5713	1.26989
$99$	0.785508	0.0789465
$100$	0	0
$101$	−7.97706	−0.793747	−0.396873	−	0.917873i	$-0.629905\pi$
$101$	−7.97706	−0.793747	−0.396873	+	0.917873i	$0.629905\pi$
$102$	−3.49256	−0.345814
$103$	14.4220	1.42104	0.710521	−	0.703676i	$-0.248459\pi$
$103$	14.4220	1.42104	0.710521	+	0.703676i	$0.248459\pi$
$104$	5.36020	0.525611
$105$	0	0
$106$	−34.8288	−3.38287
$107$	−9.34413	−0.903331	−0.451666	−	0.892187i	$-0.649170\pi$
$107$	−9.34413	−0.903331	−0.451666	+	0.892187i	$0.649170\pi$
$108$	5.19661	0.500044
$109$	−1.66559	−0.159535	−0.0797674	−	0.996814i	$-0.525418\pi$
$109$	−1.66559	−0.159535	−0.0797674	+	0.996814i	$0.525418\pi$
$110$	0	0
$111$	−4.40601	−0.418200
$112$	43.1127	4.07376
$113$	−16.6157	−1.56308	−0.781538	−	0.623857i	$-0.785565\pi$
$113$	−16.6157	−1.56308	−0.781538	+	0.623857i	$0.785565\pi$
$114$	−19.6153	−1.83714
$115$	0	0
$116$	5.10476	0.473965
$117$	0.625067	0.0577875
$118$	−16.3179	−1.50218
$119$	−4.45056	−0.407982
$120$	0	0
$121$	−10.3830	−0.943907
$122$	−0.300877	−0.0272401
$123$	4.01158	0.361712
$124$	−10.2099	−0.916878
$125$	0	0
$126$	9.17064	0.816985
$127$	−16.1242	−1.43079	−0.715396	−	0.698719i	$-0.753754\pi$
$127$	−16.1242	−1.43079	−0.715396	+	0.698719i	$0.753754\pi$
$128$	19.0232	1.68143
$129$	−5.44380	−0.479299
$130$	0	0
$131$	−1.37297	−0.119957	−0.0599786	−	0.998200i	$-0.519103\pi$
$131$	−1.37297	−0.119957	−0.0599786	+	0.998200i	$0.519103\pi$
$132$	4.08198	0.355291
$133$	−24.9958	−2.16741
$134$	25.2655	2.18260
$135$	0	0
$136$	−11.1644	−0.957335
$137$	8.16454	0.697544	0.348772	−	0.937208i	$-0.386599\pi$
$137$	8.16454	0.697544	0.348772	+	0.937208i	$0.386599\pi$
$138$	−12.0397	−1.02489
$139$	20.7156	1.75708	0.878538	−	0.477673i	$-0.158520\pi$
$139$	20.7156	1.75708	0.878538	+	0.477673i	$0.158520\pi$
$140$	0	0
$141$	1.00000	0.0842152
$142$	18.2108	1.52822
$143$	0.490995	0.0410591
$144$	12.6116	1.05096
$145$	0	0
$146$	−25.5337	−2.11318
$147$	4.68614	0.386506
$148$	−22.8963	−1.88207
$149$	20.7653	1.70116	0.850581	−	0.525844i	$-0.176250\pi$
$149$	20.7653	1.70116	0.850581	+	0.525844i	$0.176250\pi$
$150$	0	0
$151$	5.78722	0.470957	0.235479	−	0.971880i	$-0.424334\pi$
$151$	5.78722	0.470957	0.235479	+	0.971880i	$0.424334\pi$
$152$	−62.7026	−5.08585
$153$	−1.30190	−0.105253
$154$	7.20361	0.580483
$155$	0	0
$156$	3.24823	0.260067
$157$	20.6607	1.64891	0.824453	−	0.565930i	$-0.191483\pi$
$157$	20.6607	1.64891	0.824453	+	0.565930i	$0.191483\pi$
$158$	−22.8061	−1.81436
$159$	−12.9830	−1.02962
$160$	0	0
$161$	−15.3422	−1.20913
$162$	2.68265	0.210769
$163$	−3.88528	−0.304319	−0.152159	−	0.988356i	$-0.548623\pi$
$163$	−3.88528	−0.304319	−0.152159	+	0.988356i	$0.548623\pi$
$164$	20.8467	1.62785
$165$	0	0
$166$	37.7673	2.93131
$167$	17.9593	1.38973	0.694866	−	0.719139i	$-0.255464\pi$
$167$	17.9593	1.38973	0.694866	+	0.719139i	$0.255464\pi$
$168$	29.3150	2.26170
$169$	−12.6093	−0.969945
$170$	0	0
$171$	−7.31191	−0.559156
$172$	−28.2893	−2.15704
$173$	23.8409	1.81259	0.906295	−	0.422645i	$-0.138898\pi$
$173$	23.8409	1.81259	0.906295	+	0.422645i	$0.138898\pi$
$174$	2.63523	0.199776
$175$	0	0
$176$	9.90649	0.746730
$177$	−6.08274	−0.457207
$178$	36.5645	2.74062
$179$	23.1984	1.73393	0.866966	−	0.498367i	$-0.166067\pi$
$179$	23.1984	1.73393	0.866966	+	0.498367i	$0.166067\pi$
$180$	0	0
$181$	−3.22758	−0.239904	−0.119952	−	0.992780i	$-0.538274\pi$
$181$	−3.22758	−0.239904	−0.119952	+	0.992780i	$0.538274\pi$
$182$	5.73227	0.424904
$183$	−0.112157	−0.00829085
$184$	−38.4863	−2.83725
$185$	0	0
$186$	−5.27067	−0.386464
$187$	−1.02266	−0.0747840
$188$	5.19661	0.379002
$189$	3.41850	0.248659
$190$	0	0
$191$	1.26548	0.0915672	0.0457836	−	0.998951i	$-0.485422\pi$
$191$	1.26548	0.0915672	0.0457836	+	0.998951i	$0.485422\pi$
$192$	19.5279	1.40930
$193$	−14.6377	−1.05364	−0.526822	−	0.849976i	$-0.676616\pi$
$193$	−14.6377	−1.05364	−0.526822	+	0.849976i	$0.676616\pi$
$194$	36.4792	2.61905
$195$	0	0
$196$	24.3520	1.73943
$197$	−4.99971	−0.356215	−0.178107	−	0.984011i	$-0.556997\pi$
$197$	−4.99971	−0.356215	−0.178107	+	0.984011i	$0.556997\pi$
$198$	2.10724	0.149755
$199$	19.5397	1.38513	0.692566	−	0.721354i	$-0.256480\pi$
$199$	19.5397	1.38513	0.692566	+	0.721354i	$0.256480\pi$
$200$	0	0
$201$	9.41810	0.664301
$202$	−21.3997	−1.50567
$203$	3.35807	0.235690
$204$	−6.76550	−0.473679
$205$	0	0
$206$	38.6892	2.69560
$207$	−4.48799	−0.311937
$208$	7.88308	0.546593
$209$	−5.74356	−0.397290
$210$	0	0
$211$	−2.16107	−0.148774	−0.0743870	−	0.997229i	$-0.523700\pi$
$211$	−2.16107	−0.148774	−0.0743870	+	0.997229i	$0.523700\pi$
$212$	−67.4675	−4.63369
$213$	6.78837	0.465131
$214$	−25.0670	−1.71355
$215$	0	0
$216$	8.57540	0.583482
$217$	−6.71641	−0.455940
$218$	−4.46820	−0.302625
$219$	−9.51807	−0.643172
$220$	0	0
$221$	−0.813778	−0.0547406
$222$	−11.8198	−0.793292
$223$	−23.5058	−1.57407	−0.787034	−	0.616910i	$-0.788384\pi$
$223$	−23.5058	−1.57407	−0.787034	+	0.616910i	$0.788384\pi$
$224$	57.0262	3.81022
$225$	0	0
$226$	−44.5742	−2.96503
$227$	13.6710	0.907376	0.453688	−	0.891161i	$-0.350108\pi$
$227$	13.6710	0.907376	0.453688	+	0.891161i	$0.350108\pi$
$228$	−37.9972	−2.51642
$229$	−0.0354618	−0.00234338	−0.00117169	−	0.999999i	$-0.500373\pi$
$229$	−0.0354618	−0.00234338	−0.00117169	+	0.999999i	$0.500373\pi$
$230$	0	0
$231$	2.68526	0.176677
$232$	8.42382	0.553051
$233$	17.1370	1.12268	0.561340	−	0.827585i	$-0.310286\pi$
$233$	17.1370	1.12268	0.561340	+	0.827585i	$0.310286\pi$
$234$	1.67684	0.109618
$235$	0	0
$236$	−31.6097	−2.05761
$237$	−8.50134	−0.552222
$238$	−11.9393	−0.773910
$239$	−13.2193	−0.855085	−0.427542	−	0.903995i	$-0.640621\pi$
$239$	−13.2193	−0.855085	−0.427542	+	0.903995i	$0.640621\pi$
$240$	0	0
$241$	4.50304	0.290067	0.145033	−	0.989427i	$-0.453671\pi$
$241$	4.50304	0.290067	0.145033	+	0.989427i	$0.453671\pi$
$242$	−27.8539	−1.79052
$243$	1.00000	0.0641500
$244$	−0.582834	−0.0373121
$245$	0	0
$246$	10.7617	0.686140
$247$	−4.57044	−0.290810
$248$	−16.8483	−1.06987
$249$	14.0784	0.892180
$250$	0	0
$251$	−12.7694	−0.805999	−0.403000	−	0.915200i	$-0.632032\pi$
$251$	−12.7694	−0.805999	−0.403000	+	0.915200i	$0.632032\pi$
$252$	17.7646	1.11907
$253$	−3.52535	−0.221637
$254$	−43.2556	−2.71410
$255$	0	0
$256$	11.9768	0.748552
$257$	5.27702	0.329171	0.164586	−	0.986363i	$-0.447371\pi$
$257$	5.27702	0.329171	0.164586	+	0.986363i	$0.447371\pi$
$258$	−14.6038	−0.909193
$259$	−15.0619	−0.935903
$260$	0	0
$261$	0.982324	0.0608043
$262$	−3.68321	−0.227549
$263$	−0.627820	−0.0387130	−0.0193565	−	0.999813i	$-0.506162\pi$
$263$	−0.627820	−0.0387130	−0.0193565	+	0.999813i	$0.506162\pi$
$264$	6.73604	0.414575
$265$	0	0
$266$	−67.0549	−4.11140
$267$	13.6300	0.834142
$268$	48.9422	2.98962
$269$	2.79478	0.170401	0.0852005	−	0.996364i	$-0.472847\pi$
$269$	2.79478	0.170401	0.0852005	+	0.996364i	$0.472847\pi$
$270$	0	0
$271$	−21.0626	−1.27946	−0.639730	−	0.768599i	$-0.720954\pi$
$271$	−21.0626	−1.27946	−0.639730	+	0.768599i	$0.720954\pi$
$272$	−16.4191	−0.995552
$273$	2.13679	0.129325
$274$	21.9026	1.32318
$275$	0	0
$276$	−23.3223	−1.40384
$277$	2.77237	0.166576	0.0832878	−	0.996526i	$-0.473458\pi$
$277$	2.77237	0.166576	0.0832878	+	0.996526i	$0.473458\pi$
$278$	55.5728	3.33303
$279$	−1.96472	−0.117625
$280$	0	0
$281$	6.81901	0.406788	0.203394	−	0.979097i	$-0.434803\pi$
$281$	6.81901	0.406788	0.203394	+	0.979097i	$0.434803\pi$
$282$	2.68265	0.159750
$283$	3.72423	0.221382	0.110691	−	0.993855i	$-0.464694\pi$
$283$	3.72423	0.221382	0.110691	+	0.993855i	$0.464694\pi$
$284$	35.2765	2.09328
$285$	0	0
$286$	1.31717	0.0778858
$287$	13.7136	0.809488
$288$	16.6816	0.982976
$289$	−15.3050	−0.900297
$290$	0	0
$291$	13.5982	0.797139
$292$	−49.4617	−2.89453
$293$	25.8384	1.50950	0.754748	−	0.656015i	$-0.227759\pi$
$293$	25.8384	1.50950	0.754748	+	0.656015i	$0.227759\pi$
$294$	12.5713	0.733171
$295$	0	0
$296$	−37.7833	−2.19611
$297$	0.785508	0.0455798
$298$	55.7061	3.22697
$299$	−2.80529	−0.162234
$300$	0	0
$301$	−18.6096	−1.07264
$302$	15.5251	0.893368
$303$	−7.97706	−0.458270
$304$	−92.2147	−5.28888
$305$	0	0
$306$	−3.49256	−0.199656
$307$	9.60254	0.548046	0.274023	−	0.961723i	$-0.411646\pi$
$307$	9.60254	0.548046	0.274023	+	0.961723i	$0.411646\pi$
$308$	13.9542	0.795117
$309$	14.4220	0.820439
$310$	0	0
$311$	−18.4716	−1.04743	−0.523713	−	0.851895i	$-0.675454\pi$
$311$	−18.4716	−1.04743	−0.523713	+	0.851895i	$0.675454\pi$
$312$	5.36020	0.303462
$313$	−18.2068	−1.02911	−0.514555	−	0.857457i	$-0.672043\pi$
$313$	−18.2068	−1.02911	−0.514555	+	0.857457i	$0.672043\pi$
$314$	55.4255	3.12784
$315$	0	0
$316$	−44.1782	−2.48522
$317$	−10.1705	−0.571230	−0.285615	−	0.958344i	$-0.592198\pi$
$317$	−10.1705	−0.571230	−0.285615	+	0.958344i	$0.592198\pi$
$318$	−34.8288	−1.95310
$319$	0.771623	0.0432026
$320$	0	0
$321$	−9.34413	−0.521539
$322$	−41.1577	−2.29363
$323$	9.51941	0.529674
$324$	5.19661	0.288701
$325$	0	0
$326$	−10.4229	−0.577269
$327$	−1.66559	−0.0921074
$328$	34.4009	1.89947
$329$	3.41850	0.188468
$330$	0	0
$331$	7.07159	0.388690	0.194345	−	0.980933i	$-0.437742\pi$
$331$	7.07159	0.388690	0.194345	+	0.980933i	$0.437742\pi$
$332$	73.1598	4.01517
$333$	−4.40601	−0.241448
$334$	48.1785	2.63621
$335$	0	0
$336$	43.1127	2.35199
$337$	−7.71240	−0.420121	−0.210061	−	0.977688i	$-0.567366\pi$
$337$	−7.71240	−0.420121	−0.210061	+	0.977688i	$0.567366\pi$
$338$	−33.8263	−1.83991
$339$	−16.6157	−0.902443
$340$	0	0
$341$	−1.54331	−0.0835747
$342$	−19.6153	−1.06067
$343$	−7.90994	−0.427097
$344$	−46.6827	−2.51696
$345$	0	0
$346$	63.9568	3.43834
$347$	21.6311	1.16122	0.580610	−	0.814182i	$-0.302814\pi$
$347$	21.6311	1.16122	0.580610	+	0.814182i	$0.302814\pi$
$348$	5.10476	0.273644
$349$	−35.1347	−1.88072	−0.940360	−	0.340181i	$-0.889512\pi$
$349$	−35.1347	−1.88072	−0.940360	+	0.340181i	$0.889512\pi$
$350$	0	0
$351$	0.625067	0.0333636
$352$	13.1036	0.698422
$353$	−25.7885	−1.37258	−0.686291	−	0.727327i	$-0.740762\pi$
$353$	−25.7885	−1.37258	−0.686291	+	0.727327i	$0.740762\pi$
$354$	−16.3179	−0.867285
$355$	0	0
$356$	70.8298	3.75397
$357$	−4.45056	−0.235549
$358$	62.2333	3.28913
$359$	−24.6584	−1.30142	−0.650711	−	0.759325i	$-0.725529\pi$
$359$	−24.6584	−1.30142	−0.650711	+	0.759325i	$0.725529\pi$
$360$	0	0
$361$	34.4640	1.81390
$362$	−8.65846	−0.455079
$363$	−10.3830	−0.544965
$364$	11.1041	0.582012
$365$	0	0
$366$	−0.300877	−0.0157271
$367$	3.03093	0.158213	0.0791066	−	0.996866i	$-0.474793\pi$
$367$	3.03093	0.158213	0.0791066	+	0.996866i	$0.474793\pi$
$368$	−56.6006	−2.95051
$369$	4.01158	0.208835
$370$	0	0
$371$	−44.3823	−2.30421
$372$	−10.2099	−0.529360
$373$	36.5177	1.89082	0.945408	−	0.325890i	$-0.105664\pi$
$373$	36.5177	1.89082	0.945408	+	0.325890i	$0.105664\pi$
$374$	−2.74343	−0.141859
$375$	0	0
$376$	8.57540	0.442243
$377$	0.614018	0.0316236
$378$	9.17064	0.471687
$379$	−34.5749	−1.77599	−0.887996	−	0.459851i	$-0.847903\pi$
$379$	−34.5749	−1.77599	−0.887996	+	0.459851i	$0.847903\pi$
$380$	0	0
$381$	−16.1242	−0.826069
$382$	3.39485	0.173696
$383$	15.6247	0.798382	0.399191	−	0.916868i	$-0.369291\pi$
$383$	15.6247	0.798382	0.399191	+	0.916868i	$0.369291\pi$
$384$	19.0232	0.970774
$385$	0	0
$386$	−39.2678	−1.99868
$387$	−5.44380	−0.276724
$388$	70.6645	3.58745
$389$	6.29760	0.319301	0.159650	−	0.987174i	$-0.448963\pi$
$389$	6.29760	0.319301	0.159650	+	0.987174i	$0.448963\pi$
$390$	0	0
$391$	5.84293	0.295490
$392$	40.1855	2.02967
$393$	−1.37297	−0.0692573
$394$	−13.4125	−0.675711
$395$	0	0
$396$	4.08198	0.205127
$397$	22.9148	1.15006	0.575031	−	0.818132i	$-0.304990\pi$
$397$	22.9148	1.15006	0.575031	+	0.818132i	$0.304990\pi$
$398$	52.4182	2.62749
$399$	−24.9958	−1.25135
$400$	0	0
$401$	5.32057	0.265697	0.132848	−	0.991136i	$-0.457588\pi$
$401$	5.32057	0.265697	0.132848	+	0.991136i	$0.457588\pi$
$402$	25.2655	1.26013
$403$	−1.22809	−0.0611753
$404$	−41.4537	−2.06240
$405$	0	0
$406$	9.00853	0.447086
$407$	−3.46095	−0.171553
$408$	−11.1644	−0.552718
$409$	−35.1977	−1.74042	−0.870208	−	0.492685i	$-0.836016\pi$
$409$	−35.1977	−1.74042	−0.870208	+	0.492685i	$0.836016\pi$
$410$	0	0
$411$	8.16454	0.402727
$412$	74.9456	3.69230
$413$	−20.7938	−1.02320
$414$	−12.0397	−0.591719
$415$	0	0
$416$	10.4272	0.511233
$417$	20.7156	1.01445
$418$	−15.4080	−0.753628
$419$	10.1779	0.497224	0.248612	−	0.968603i	$-0.420026\pi$
$419$	10.1779	0.497224	0.248612	+	0.968603i	$0.420026\pi$
$420$	0	0
$421$	13.5002	0.657960	0.328980	−	0.944337i	$-0.393295\pi$
$421$	13.5002	0.657960	0.328980	+	0.944337i	$0.393295\pi$
$422$	−5.79739	−0.282213
$423$	1.00000	0.0486217
$424$	−111.334	−5.40686
$425$	0	0
$426$	18.2108	0.882317
$427$	−0.383407	−0.0185544
$428$	−48.5578	−2.34713
$429$	0.490995	0.0237055
$430$	0	0
$431$	−7.63497	−0.367763	−0.183882	−	0.982948i	$-0.558866\pi$
$431$	−7.63497	−0.367763	−0.183882	+	0.982948i	$0.558866\pi$
$432$	12.6116	0.606775
$433$	−33.1629	−1.59371	−0.796854	−	0.604172i	$-0.793504\pi$
$433$	−33.1629	−1.59371	−0.796854	+	0.604172i	$0.793504\pi$
$434$	−18.0178	−0.864881
$435$	0	0
$436$	−8.65544	−0.414520
$437$	32.8158	1.56979
$438$	−25.5337	−1.22005
$439$	−18.7275	−0.893813	−0.446907	−	0.894581i	$-0.647474\pi$
$439$	−18.7275	−0.893813	−0.446907	+	0.894581i	$0.647474\pi$
$440$	0	0
$441$	4.68614	0.223149
$442$	−2.18308	−0.103839
$443$	−9.53277	−0.452915	−0.226458	−	0.974021i	$-0.572715\pi$
$443$	−9.53277	−0.452915	−0.226458	+	0.974021i	$0.572715\pi$
$444$	−22.8963	−1.08661
$445$	0	0
$446$	−63.0579	−2.98588
$447$	20.7653	0.982166
$448$	66.7561	3.15393
$449$	−18.9640	−0.894966	−0.447483	−	0.894292i	$-0.647680\pi$
$449$	−18.9640	−0.894966	−0.447483	+	0.894292i	$0.647680\pi$
$450$	0	0
$451$	3.15113	0.148381
$452$	−86.3456	−4.06135
$453$	5.78722	0.271907
$454$	36.6745	1.72122
$455$	0	0
$456$	−62.7026	−2.93632
$457$	39.4326	1.84458	0.922290	−	0.386498i	$-0.126315\pi$
$457$	39.4326	1.84458	0.922290	+	0.386498i	$0.126315\pi$
$458$	−0.0951317	−0.00444521
$459$	−1.30190	−0.0607677
$460$	0	0
$461$	3.66147	0.170531	0.0852657	−	0.996358i	$-0.472826\pi$
$461$	3.66147	0.170531	0.0852657	+	0.996358i	$0.472826\pi$
$462$	7.20361	0.335142
$463$	−9.05802	−0.420962	−0.210481	−	0.977598i	$-0.567503\pi$
$463$	−9.05802	−0.420962	−0.210481	+	0.977598i	$0.567503\pi$
$464$	12.3886	0.575128
$465$	0	0
$466$	45.9725	2.12964
$467$	28.1293	1.30167	0.650835	−	0.759219i	$-0.274419\pi$
$467$	28.1293	1.30167	0.650835	+	0.759219i	$0.274419\pi$
$468$	3.24823	0.150150
$469$	32.1958	1.48666
$470$	0	0
$471$	20.6607	0.951997
$472$	−52.1619	−2.40095
$473$	−4.27614	−0.196617
$474$	−22.8061	−1.04752
$475$	0	0
$476$	−23.1278	−1.06006
$477$	−12.9830	−0.594449
$478$	−35.4627	−1.62203
$479$	−26.2233	−1.19817	−0.599087	−	0.800684i	$-0.704470\pi$
$479$	−26.2233	−1.19817	−0.599087	+	0.800684i	$0.704470\pi$
$480$	0	0
$481$	−2.75405	−0.125574
$482$	12.0801	0.550233
$483$	−15.3422	−0.698094
$484$	−53.9563	−2.45256
$485$	0	0
$486$	2.68265	0.121688
$487$	1.75207	0.0793940	0.0396970	−	0.999212i	$-0.487361\pi$
$487$	1.75207	0.0793940	0.0396970	+	0.999212i	$0.487361\pi$
$488$	−0.961787	−0.0435381
$489$	−3.88528	−0.175699
$490$	0	0
$491$	−30.5341	−1.37799	−0.688993	−	0.724768i	$-0.741947\pi$
$491$	−30.5341	−1.37799	−0.688993	+	0.724768i	$0.741947\pi$
$492$	20.8467	0.939840
$493$	−1.27889	−0.0575984
$494$	−12.2609	−0.551643
$495$	0	0
$496$	−24.7783	−1.11258
$497$	23.2060	1.04093
$498$	37.7673	1.69239
$499$	−32.5199	−1.45579	−0.727894	−	0.685689i	$-0.759501\pi$
$499$	−32.5199	−1.45579	−0.727894	+	0.685689i	$0.759501\pi$
$500$	0	0
$501$	17.9593	0.802362
$502$	−34.2559	−1.52892
$503$	−7.81328	−0.348377	−0.174188	−	0.984712i	$-0.555730\pi$
$503$	−7.81328	−0.348377	−0.174188	+	0.984712i	$0.555730\pi$
$504$	29.3150	1.30579
$505$	0	0
$506$	−9.45728	−0.420427
$507$	−12.6093	−0.559998
$508$	−83.7913	−3.71764
$509$	−22.4051	−0.993090	−0.496545	−	0.868011i	$-0.665398\pi$
$509$	−22.4051	−0.993090	−0.496545	+	0.868011i	$0.665398\pi$
$510$	0	0
$511$	−32.5375	−1.43937
$512$	−5.91675	−0.261486
$513$	−7.31191	−0.322829
$514$	14.1564	0.624412
$515$	0	0
$516$	−28.2893	−1.24537
$517$	0.785508	0.0345466
$518$	−40.4059	−1.77533
$519$	23.8409	1.04650
$520$	0	0
$521$	36.7908	1.61183	0.805916	−	0.592029i	$-0.201673\pi$
$521$	36.7908	1.61183	0.805916	+	0.592029i	$0.201673\pi$
$522$	2.63523	0.115341
$523$	−4.31962	−0.188884	−0.0944418	−	0.995530i	$-0.530107\pi$
$523$	−4.31962	−0.188884	−0.0944418	+	0.995530i	$0.530107\pi$
$524$	−7.13481	−0.311686
$525$	0	0
$526$	−1.68422	−0.0734356
$527$	2.55788	0.111423
$528$	9.90649	0.431125
$529$	−2.85796	−0.124259
$530$	0	0
$531$	−6.08274	−0.263968
$532$	−129.893	−5.63159
$533$	2.50751	0.108612
$534$	36.5645	1.58230
$535$	0	0
$536$	80.7640	3.48847
$537$	23.1984	1.00109
$538$	7.49743	0.323237
$539$	3.68100	0.158552
$540$	0	0
$541$	−25.4095	−1.09244	−0.546220	−	0.837642i	$-0.683934\pi$
$541$	−25.4095	−1.09244	−0.546220	+	0.837642i	$0.683934\pi$
$542$	−56.5036	−2.42704
$543$	−3.22758	−0.138509
$544$	−21.7179	−0.931148
$545$	0	0
$546$	5.73227	0.245318
$547$	39.3602	1.68292	0.841461	−	0.540318i	$-0.181696\pi$
$547$	39.3602	1.68292	0.841461	+	0.540318i	$0.181696\pi$
$548$	42.4280	1.81243
$549$	−0.112157	−0.00478672
$550$	0	0
$551$	−7.18266	−0.305992
$552$	−38.4863	−1.63809
$553$	−29.0618	−1.23584
$554$	7.43730	0.315981
$555$	0	0
$556$	107.651	4.56542
$557$	−6.19079	−0.262312	−0.131156	−	0.991362i	$-0.541869\pi$
$557$	−6.19079	−0.262312	−0.131156	+	0.991362i	$0.541869\pi$
$558$	−5.27067	−0.223125
$559$	−3.40274	−0.143921
$560$	0	0
$561$	−1.02266	−0.0431766
$562$	18.2930	0.771645
$563$	12.8348	0.540921	0.270461	−	0.962731i	$-0.412824\pi$
$563$	12.8348	0.540921	0.270461	+	0.962731i	$0.412824\pi$
$564$	5.19661	0.218817
$565$	0	0
$566$	9.99079	0.419944
$567$	3.41850	0.143563
$568$	58.2130	2.44256
$569$	31.7537	1.33119	0.665593	−	0.746315i	$-0.268179\pi$
$569$	31.7537	1.33119	0.665593	+	0.746315i	$0.268179\pi$
$570$	0	0
$571$	22.7986	0.954092	0.477046	−	0.878878i	$-0.341708\pi$
$571$	22.7986	0.954092	0.477046	+	0.878878i	$0.341708\pi$
$572$	2.55151	0.106684
$573$	1.26548	0.0528663
$574$	36.7888	1.53553
$575$	0	0
$576$	19.5279	0.813662
$577$	9.16496	0.381542	0.190771	−	0.981635i	$-0.438901\pi$
$577$	9.16496	0.381542	0.190771	+	0.981635i	$0.438901\pi$
$578$	−41.0581	−1.70779
$579$	−14.6377	−0.608321
$580$	0	0
$581$	48.1269	1.99664
$582$	36.4792	1.51211
$583$	−10.1982	−0.422367
$584$	−81.6213	−3.37751
$585$	0	0
$586$	69.3154	2.86339
$587$	−37.9276	−1.56544	−0.782719	−	0.622375i	$-0.786168\pi$
$587$	−37.9276	−1.56544	−0.782719	+	0.622375i	$0.786168\pi$
$588$	24.3520	1.00426
$589$	14.3659	0.591936
$590$	0	0
$591$	−4.99971	−0.205661
$592$	−55.5667	−2.28378
$593$	−9.32773	−0.383044	−0.191522	−	0.981488i	$-0.561342\pi$
$593$	−9.32773	−0.383044	−0.191522	+	0.981488i	$0.561342\pi$
$594$	2.10724	0.0864612
$595$	0	0
$596$	107.909	4.42014
$597$	19.5397	0.799706
$598$	−7.52563	−0.307746
$599$	20.5208	0.838459	0.419229	−	0.907880i	$-0.362300\pi$
$599$	20.5208	0.838459	0.419229	+	0.907880i	$0.362300\pi$
$600$	0	0
$601$	11.1198	0.453585	0.226792	−	0.973943i	$-0.427176\pi$
$601$	11.1198	0.453585	0.226792	+	0.973943i	$0.427176\pi$
$602$	−49.9231	−2.03471
$603$	9.41810	0.383535
$604$	30.0739	1.22369
$605$	0	0
$606$	−21.3997	−0.869302
$607$	−13.6821	−0.555340	−0.277670	−	0.960676i	$-0.589562\pi$
$607$	−13.6821	−0.555340	−0.277670	+	0.960676i	$0.589562\pi$
$608$	−121.975	−4.94673
$609$	3.35807	0.136076
$610$	0	0
$611$	0.625067	0.0252875
$612$	−6.76550	−0.273479
$613$	19.3905	0.783174	0.391587	−	0.920141i	$-0.371926\pi$
$613$	19.3905	0.783174	0.391587	+	0.920141i	$0.371926\pi$
$614$	25.7603	1.03960
$615$	0	0
$616$	23.0272	0.927791
$617$	−16.6923	−0.672006	−0.336003	−	0.941861i	$-0.609075\pi$
$617$	−16.6923	−0.672006	−0.336003	+	0.941861i	$0.609075\pi$
$618$	38.6892	1.55631
$619$	20.8874	0.839537	0.419769	−	0.907631i	$-0.362111\pi$
$619$	20.8874	0.839537	0.419769	+	0.907631i	$0.362111\pi$
$620$	0	0
$621$	−4.48799	−0.180097
$622$	−49.5527	−1.98688
$623$	46.5941	1.86675
$624$	7.88308	0.315576
$625$	0	0
$626$	−48.8426	−1.95214
$627$	−5.74356	−0.229376
$628$	107.366	4.28437
$629$	5.73620	0.228717
$630$	0	0
$631$	4.10147	0.163277	0.0816384	−	0.996662i	$-0.473985\pi$
$631$	4.10147	0.163277	0.0816384	+	0.996662i	$0.473985\pi$
$632$	−72.9024	−2.89990
$633$	−2.16107	−0.0858947
$634$	−27.2838	−1.08358
$635$	0	0
$636$	−67.4675	−2.67526
$637$	2.92915	0.116057
$638$	2.06999	0.0819518
$639$	6.78837	0.268544
$640$	0	0
$641$	−46.4449	−1.83446	−0.917232	−	0.398353i	$-0.869582\pi$
$641$	−46.4449	−1.83446	−0.917232	+	0.398353i	$0.869582\pi$
$642$	−25.0670	−0.989317
$643$	−35.3812	−1.39530	−0.697648	−	0.716440i	$-0.745770\pi$
$643$	−35.3812	−1.39530	−0.697648	+	0.716440i	$0.745770\pi$
$644$	−79.7274	−3.14170
$645$	0	0
$646$	25.5372	1.00475
$647$	38.3669	1.50836	0.754180	−	0.656668i	$-0.228035\pi$
$647$	38.3669	1.50836	0.754180	+	0.656668i	$0.228035\pi$
$648$	8.57540	0.336874
$649$	−4.77804	−0.187554
$650$	0	0
$651$	−6.71641	−0.263237
$652$	−20.1903	−0.790714
$653$	35.2466	1.37931	0.689653	−	0.724140i	$-0.257763\pi$
$653$	35.2466	1.37931	0.689653	+	0.724140i	$0.257763\pi$
$654$	−4.46820	−0.174720
$655$	0	0
$656$	50.5924	1.97530
$657$	−9.51807	−0.371335
$658$	9.17064	0.357509
$659$	6.35357	0.247500	0.123750	−	0.992313i	$-0.460508\pi$
$659$	6.35357	0.247500	0.123750	+	0.992313i	$0.460508\pi$
$660$	0	0
$661$	24.1313	0.938601	0.469300	−	0.883039i	$-0.344506\pi$
$661$	24.1313	0.938601	0.469300	+	0.883039i	$0.344506\pi$
$662$	18.9706	0.737313
$663$	−0.813778	−0.0316045
$664$	120.728	4.68514
$665$	0	0
$666$	−11.8198	−0.458007
$667$	−4.40866	−0.170704
$668$	93.3275	3.61095
$669$	−23.5058	−0.908788
$670$	0	0
$671$	−0.0880998	−0.00340106
$672$	57.0262	2.19983
$673$	37.5042	1.44568	0.722839	−	0.691016i	$-0.242837\pi$
$673$	37.5042	1.44568	0.722839	+	0.691016i	$0.242837\pi$
$674$	−20.6897	−0.796937
$675$	0	0
$676$	−65.5256	−2.52022
$677$	−28.7908	−1.10652	−0.553260	−	0.833008i	$-0.686617\pi$
$677$	−28.7908	−1.10652	−0.553260	+	0.833008i	$0.686617\pi$
$678$	−44.5742	−1.71186
$679$	46.4854	1.78394
$680$	0	0
$681$	13.6710	0.523874
$682$	−4.14015	−0.158535
$683$	9.18982	0.351639	0.175819	−	0.984422i	$-0.443743\pi$
$683$	9.18982	0.351639	0.175819	+	0.984422i	$0.443743\pi$
$684$	−37.9972	−1.45286
$685$	0	0
$686$	−21.2196	−0.810168
$687$	−0.0354618	−0.00135295
$688$	−68.6548	−2.61744
$689$	−8.11523	−0.309166
$690$	0	0
$691$	−23.9691	−0.911827	−0.455913	−	0.890024i	$-0.650687\pi$
$691$	−23.9691	−0.911827	−0.455913	+	0.890024i	$0.650687\pi$
$692$	123.892	4.70967
$693$	2.68526	0.102004
$694$	58.0287	2.20274
$695$	0	0
$696$	8.42382	0.319304
$697$	−5.22270	−0.197824
$698$	−94.2543	−3.56758
$699$	17.1370	0.648180
$700$	0	0
$701$	10.3354	0.390363	0.195181	−	0.980767i	$-0.437470\pi$
$701$	10.3354	0.390363	0.195181	+	0.980767i	$0.437470\pi$
$702$	1.67684	0.0632881
$703$	32.2163	1.21506
$704$	15.3393	0.578122
$705$	0	0
$706$	−69.1814	−2.60368
$707$	−27.2696	−1.02558
$708$	−31.6097	−1.18796
$709$	−15.5543	−0.584153	−0.292077	−	0.956395i	$-0.594346\pi$
$709$	−15.5543	−0.584153	−0.292077	+	0.956395i	$0.594346\pi$
$710$	0	0
$711$	−8.50134	−0.318825
$712$	116.883	4.38036
$713$	8.81766	0.330224
$714$	−11.9393	−0.446817
$715$	0	0
$716$	120.553	4.50529
$717$	−13.2193	−0.493683
$718$	−66.1500	−2.46870
$719$	−12.1834	−0.454362	−0.227181	−	0.973853i	$-0.572951\pi$
$719$	−12.1834	−0.454362	−0.227181	+	0.973853i	$0.572951\pi$
$720$	0	0
$721$	49.3016	1.83609
$722$	92.4549	3.44082
$723$	4.50304	0.167470
$724$	−16.7725	−0.623344
$725$	0	0
$726$	−27.8539	−1.03376
$727$	39.7198	1.47313	0.736563	−	0.676369i	$-0.236448\pi$
$727$	39.7198	1.47313	0.736563	+	0.676369i	$0.236448\pi$
$728$	18.3238	0.679127
$729$	1.00000	0.0370370
$730$	0	0
$731$	7.08730	0.262133
$732$	−0.582834	−0.0215422
$733$	2.76292	0.102051	0.0510254	−	0.998697i	$-0.483751\pi$
$733$	2.76292	0.102051	0.0510254	+	0.998697i	$0.483751\pi$
$734$	8.13092	0.300118
$735$	0	0
$736$	−74.8671	−2.75964
$737$	7.39799	0.272508
$738$	10.7617	0.396143
$739$	8.20070	0.301668	0.150834	−	0.988559i	$-0.451804\pi$
$739$	8.20070	0.301668	0.150834	+	0.988559i	$0.451804\pi$
$740$	0	0
$741$	−4.57044	−0.167899
$742$	−119.062	−4.37091
$743$	45.1864	1.65773	0.828864	−	0.559450i	$-0.188988\pi$
$743$	45.1864	1.65773	0.828864	+	0.559450i	$0.188988\pi$
$744$	−16.8483	−0.617689
$745$	0	0
$746$	97.9643	3.58673
$747$	14.0784	0.515100
$748$	−5.31435	−0.194312
$749$	−31.9429	−1.16717
$750$	0	0
$751$	13.9655	0.509608	0.254804	−	0.966993i	$-0.417989\pi$
$751$	13.9655	0.509608	0.254804	+	0.966993i	$0.417989\pi$
$752$	12.6116	0.459897
$753$	−12.7694	−0.465344
$754$	1.64720	0.0599874
$755$	0	0
$756$	17.7646	0.646093
$757$	−30.6025	−1.11227	−0.556133	−	0.831093i	$-0.687716\pi$
$757$	−30.6025	−1.11227	−0.556133	+	0.831093i	$0.687716\pi$
$758$	−92.7523	−3.36892
$759$	−3.52535	−0.127962
$760$	0	0
$761$	−37.9892	−1.37711	−0.688553	−	0.725186i	$-0.741754\pi$
$761$	−37.9892	−1.37711	−0.688553	+	0.725186i	$0.741754\pi$
$762$	−43.2556	−1.56699
$763$	−5.69382	−0.206130
$764$	6.57623	0.237920
$765$	0	0
$766$	41.9155	1.51447
$767$	−3.80212	−0.137287
$768$	11.9768	0.432177
$769$	27.5690	0.994163	0.497082	−	0.867704i	$-0.334405\pi$
$769$	27.5690	0.994163	0.497082	+	0.867704i	$0.334405\pi$
$770$	0	0
$771$	5.27702	0.190047
$772$	−76.0664	−2.73769
$773$	4.06786	0.146311	0.0731555	−	0.997321i	$-0.476693\pi$
$773$	4.06786	0.146311	0.0731555	+	0.997321i	$0.476693\pi$
$774$	−14.6038	−0.524923
$775$	0	0
$776$	116.610	4.18605
$777$	−15.0619	−0.540344
$778$	16.8943	0.605688
$779$	−29.3323	−1.05094
$780$	0	0
$781$	5.33231	0.190805
$782$	15.6745	0.560521
$783$	0.982324	0.0351054
$784$	59.0996	2.11070
$785$	0	0
$786$	−3.68321	−0.131376
$787$	25.2049	0.898456	0.449228	−	0.893417i	$-0.351699\pi$
$787$	25.2049	0.898456	0.449228	+	0.893417i	$0.351699\pi$
$788$	−25.9816	−0.925555
$789$	−0.627820	−0.0223510
$790$	0	0
$791$	−56.8009	−2.01961
$792$	6.73604	0.239355
$793$	−0.0701054	−0.00248952
$794$	61.4725	2.18158
$795$	0	0
$796$	101.540	3.59900
$797$	19.1180	0.677194	0.338597	−	0.940932i	$-0.390048\pi$
$797$	19.1180	0.677194	0.338597	+	0.940932i	$0.390048\pi$
$798$	−67.0549	−2.37372
$799$	−1.30190	−0.0460581
$800$	0	0
$801$	13.6300	0.481592
$802$	14.2732	0.504005
$803$	−7.47652	−0.263841
$804$	48.9422	1.72606
$805$	0	0
$806$	−3.29452	−0.116045
$807$	2.79478	0.0983811
$808$	−68.4065	−2.40653
$809$	−29.7515	−1.04601	−0.523003	−	0.852331i	$-0.675188\pi$
$809$	−29.7515	−1.04601	−0.523003	+	0.852331i	$0.675188\pi$
$810$	0	0
$811$	−28.6089	−1.00460	−0.502298	−	0.864695i	$-0.667512\pi$
$811$	−28.6089	−1.00460	−0.502298	+	0.864695i	$0.667512\pi$
$812$	17.4506	0.612396
$813$	−21.0626	−0.738697
$814$	−9.28453	−0.325423
$815$	0	0
$816$	−16.4191	−0.574782
$817$	39.8045	1.39258
$818$	−94.4232	−3.30143
$819$	2.13679	0.0746656
$820$	0	0
$821$	30.5052	1.06464	0.532319	−	0.846544i	$-0.321321\pi$
$821$	30.5052	1.06464	0.532319	+	0.846544i	$0.321321\pi$
$822$	21.9026	0.763941
$823$	4.98020	0.173599	0.0867994	−	0.996226i	$-0.472336\pi$
$823$	4.98020	0.173599	0.0867994	+	0.996226i	$0.472336\pi$
$824$	123.674	4.30840
$825$	0	0
$826$	−55.7826	−1.94093
$827$	−54.2755	−1.88735	−0.943673	−	0.330881i	$-0.892654\pi$
$827$	−54.2755	−1.88735	−0.943673	+	0.330881i	$0.892654\pi$
$828$	−23.3223	−0.810508
$829$	26.8494	0.932517	0.466258	−	0.884649i	$-0.345602\pi$
$829$	26.8494	0.932517	0.466258	+	0.884649i	$0.345602\pi$
$830$	0	0
$831$	2.77237	0.0961725
$832$	12.2062	0.423175
$833$	−6.10090	−0.211384
$834$	55.5728	1.92433
$835$	0	0
$836$	−29.8471	−1.03228
$837$	−1.96472	−0.0679108
$838$	27.3038	0.943194
$839$	32.5823	1.12487	0.562433	−	0.826843i	$-0.309866\pi$
$839$	32.5823	1.12487	0.562433	+	0.826843i	$0.309866\pi$
$840$	0	0
$841$	−28.0350	−0.966726
$842$	36.2163	1.24810
$843$	6.81901	0.234859
$844$	−11.2302	−0.386561
$845$	0	0
$846$	2.68265	0.0922314
$847$	−35.4942	−1.21960
$848$	−163.736	−5.62271
$849$	3.72423	0.127815
$850$	0	0
$851$	19.7741	0.677848
$852$	35.2765	1.20855
$853$	−7.62559	−0.261095	−0.130548	−	0.991442i	$-0.541674\pi$
$853$	−7.62559	−0.261095	−0.130548	+	0.991442i	$0.541674\pi$
$854$	−1.02855	−0.0351962
$855$	0	0
$856$	−80.1297	−2.73878
$857$	−2.08585	−0.0712511	−0.0356256	−	0.999365i	$-0.511342\pi$
$857$	−2.08585	−0.0712511	−0.0356256	+	0.999365i	$0.511342\pi$
$858$	1.31717	0.0449674
$859$	19.5799	0.668059	0.334030	−	0.942563i	$-0.391591\pi$
$859$	19.5799	0.668059	0.334030	+	0.942563i	$0.391591\pi$
$860$	0	0
$861$	13.7136	0.467358
$862$	−20.4819	−0.697618
$863$	13.7052	0.466532	0.233266	−	0.972413i	$-0.425059\pi$
$863$	13.7052	0.466532	0.233266	+	0.972413i	$0.425059\pi$
$864$	16.6816	0.567521
$865$	0	0
$866$	−88.9646	−3.02314
$867$	−15.3050	−0.519787
$868$	−34.9026	−1.18467
$869$	−6.67787	−0.226531
$870$	0	0
$871$	5.88695	0.199472
$872$	−14.2831	−0.483687
$873$	13.5982	0.460229
$874$	88.0332	2.97777
$875$	0	0
$876$	−49.4617	−1.67116
$877$	−6.60196	−0.222932	−0.111466	−	0.993768i	$-0.535555\pi$
$877$	−6.60196	−0.222932	−0.111466	+	0.993768i	$0.535555\pi$
$878$	−50.2392	−1.69549
$879$	25.8384	0.871507
$880$	0	0
$881$	−33.5373	−1.12990	−0.564951	−	0.825125i	$-0.691105\pi$
$881$	−33.5373	−1.12990	−0.564951	+	0.825125i	$0.691105\pi$
$882$	12.5713	0.423297
$883$	39.1642	1.31798	0.658991	−	0.752151i	$-0.270984\pi$
$883$	39.1642	1.31798	0.658991	+	0.752151i	$0.270984\pi$
$884$	−4.22889	−0.142233
$885$	0	0
$886$	−25.5731	−0.859145
$887$	−5.56536	−0.186866	−0.0934332	−	0.995626i	$-0.529784\pi$
$887$	−5.56536	−0.186866	−0.0934332	+	0.995626i	$0.529784\pi$
$888$	−37.7833	−1.26792
$889$	−55.1206	−1.84869
$890$	0	0
$891$	0.785508	0.0263155
$892$	−122.151	−4.08991
$893$	−7.31191	−0.244684
$894$	55.7061	1.86309
$895$	0	0
$896$	65.0308	2.17253
$897$	−2.80529	−0.0936661
$898$	−50.8738	−1.69768
$899$	−1.93000	−0.0643690
$900$	0	0
$901$	16.9026	0.563107
$902$	8.45338	0.281467
$903$	−18.6096	−0.619289
$904$	−142.487	−4.73903
$905$	0	0
$906$	15.5251	0.515786
$907$	−5.67737	−0.188514	−0.0942569	−	0.995548i	$-0.530048\pi$
$907$	−5.67737	−0.188514	−0.0942569	+	0.995548i	$0.530048\pi$
$908$	71.0429	2.35764
$909$	−7.97706	−0.264582
$910$	0	0
$911$	3.81590	0.126426	0.0632132	−	0.998000i	$-0.479865\pi$
$911$	3.81590	0.126426	0.0632132	+	0.998000i	$0.479865\pi$
$912$	−92.2147	−3.05353
$913$	11.0587	0.365988
$914$	105.784	3.49902
$915$	0	0
$916$	−0.184282	−0.00608883
$917$	−4.69351	−0.154993
$918$	−3.49256	−0.115271
$919$	2.91210	0.0960614	0.0480307	−	0.998846i	$-0.484705\pi$
$919$	2.91210	0.0960614	0.0480307	+	0.998846i	$0.484705\pi$
$920$	0	0
$921$	9.60254	0.316415
$922$	9.82244	0.323485
$923$	4.24319	0.139666
$924$	13.9542	0.459061
$925$	0	0
$926$	−24.2995	−0.798531
$927$	14.4220	0.473681
$928$	16.3868	0.537922
$929$	17.9972	0.590469	0.295235	−	0.955425i	$-0.404602\pi$
$929$	17.9972	0.590469	0.295235	+	0.955425i	$0.404602\pi$
$930$	0	0
$931$	−34.2646	−1.12298
$932$	89.0542	2.91707
$933$	−18.4716	−0.604732
$934$	75.4612	2.46917
$935$	0	0
$936$	5.36020	0.175204
$937$	43.8185	1.43149	0.715744	−	0.698362i	$-0.246088\pi$
$937$	43.8185	1.43149	0.715744	+	0.698362i	$0.246088\pi$
$938$	86.3700	2.82008
$939$	−18.2068	−0.594157
$940$	0	0
$941$	33.3306	1.08655	0.543273	−	0.839556i	$-0.317185\pi$
$941$	33.3306	1.08655	0.543273	+	0.839556i	$0.317185\pi$
$942$	55.4255	1.80586
$943$	−18.0039	−0.586289
$944$	−76.7129	−2.49679
$945$	0	0
$946$	−11.4714	−0.372967
$947$	44.5443	1.44750	0.723748	−	0.690065i	$-0.242418\pi$
$947$	44.5443	1.44750	0.723748	+	0.690065i	$0.242418\pi$
$948$	−44.1782	−1.43484
$949$	−5.94943	−0.193127
$950$	0	0
$951$	−10.1705	−0.329800
$952$	−38.1653	−1.23695
$953$	34.8165	1.12782	0.563909	−	0.825837i	$-0.309297\pi$
$953$	34.8165	1.12782	0.563909	+	0.825837i	$0.309297\pi$
$954$	−34.8288	−1.12762
$955$	0	0
$956$	−68.6956	−2.22177
$957$	0.771623	0.0249430
$958$	−70.3480	−2.27284
$959$	27.9105	0.901276
$960$	0	0
$961$	−27.1399	−0.875479
$962$	−7.38816	−0.238204
$963$	−9.34413	−0.301110
$964$	23.4006	0.753682
$965$	0	0
$966$	−41.1577	−1.32423
$967$	−30.7160	−0.987758	−0.493879	−	0.869531i	$-0.664422\pi$
$967$	−30.7160	−0.987758	−0.493879	+	0.869531i	$0.664422\pi$
$968$	−89.0382	−2.86180
$969$	9.51941	0.305807
$970$	0	0
$971$	28.0889	0.901415	0.450708	−	0.892672i	$-0.351172\pi$
$971$	28.0889	0.901415	0.450708	+	0.892672i	$0.351172\pi$
$972$	5.19661	0.166681
$973$	70.8163	2.27027
$974$	4.70020	0.150604
$975$	0	0
$976$	−1.41447	−0.0452761
$977$	−29.4088	−0.940870	−0.470435	−	0.882435i	$-0.655903\pi$
$977$	−29.4088	−0.940870	−0.470435	+	0.882435i	$0.655903\pi$
$978$	−10.4229	−0.333286
$979$	10.7065	0.342180
$980$	0	0
$981$	−1.66559	−0.0531783
$982$	−81.9123	−2.61393
$983$	18.6200	0.593886	0.296943	−	0.954895i	$-0.404033\pi$
$983$	18.6200	0.593886	0.296943	+	0.954895i	$0.404033\pi$
$984$	34.4009	1.09666
$985$	0	0
$986$	−3.43082	−0.109260
$987$	3.41850	0.108812
$988$	−23.7508	−0.755613
$989$	24.4317	0.776883
$990$	0	0
$991$	20.2343	0.642764	0.321382	−	0.946950i	$-0.395853\pi$
$991$	20.2343	0.642764	0.321382	+	0.946950i	$0.395853\pi$
$992$	−32.7749	−1.04060
$993$	7.07159	0.224410
$994$	62.2537	1.97457
$995$	0	0
$996$	73.1598	2.31816
$997$	−15.4447	−0.489138	−0.244569	−	0.969632i	$-0.578646\pi$
$997$	−15.4447	−0.489138	−0.244569	+	0.969632i	$0.578646\pi$
$998$	−87.2394	−2.76152
$999$	−4.40601	−0.139400

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3525.2.a.z.1.7	✓	7
5.4	even	2		3525.2.a.ba.1.1	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3525.2.a.z.1.7	✓	7	1.1	even	1	trivial
3525.2.a.ba.1.1	yes	7	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:	\( N \)	\(=\)	\( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3525.a (trivial)

\(f(q)\)	\(=\)	\(q+2.68265 q^{2} +1.00000 q^{3} +5.19661 q^{4} +2.68265 q^{6} +3.41850 q^{7} +8.57540 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.68265 q^{2} +1.00000 q^{3} +5.19661 q^{4} +2.68265 q^{6} +3.41850 q^{7} +8.57540 q^{8} +1.00000 q^{9} +0.785508 q^{11} +5.19661 q^{12} +0.625067 q^{13} +9.17064 q^{14} +12.6116 q^{16} -1.30190 q^{17} +2.68265 q^{18} -7.31191 q^{19} +3.41850 q^{21} +2.10724 q^{22} -4.48799 q^{23} +8.57540 q^{24} +1.67684 q^{26} +1.00000 q^{27} +17.7646 q^{28} +0.982324 q^{29} -1.96472 q^{31} +16.6816 q^{32} +0.785508 q^{33} -3.49256 q^{34} +5.19661 q^{36} -4.40601 q^{37} -19.6153 q^{38} +0.625067 q^{39} +4.01158 q^{41} +9.17064 q^{42} -5.44380 q^{43} +4.08198 q^{44} -12.0397 q^{46} +1.00000 q^{47} +12.6116 q^{48} +4.68614 q^{49} -1.30190 q^{51} +3.24823 q^{52} -12.9830 q^{53} +2.68265 q^{54} +29.3150 q^{56} -7.31191 q^{57} +2.63523 q^{58} -6.08274 q^{59} -0.112157 q^{61} -5.27067 q^{62} +3.41850 q^{63} +19.5279 q^{64} +2.10724 q^{66} +9.41810 q^{67} -6.76550 q^{68} -4.48799 q^{69} +6.78837 q^{71} +8.57540 q^{72} -9.51807 q^{73} -11.8198 q^{74} -37.9972 q^{76} +2.68526 q^{77} +1.67684 q^{78} -8.50134 q^{79} +1.00000 q^{81} +10.7617 q^{82} +14.0784 q^{83} +17.7646 q^{84} -14.6038 q^{86} +0.982324 q^{87} +6.73604 q^{88} +13.6300 q^{89} +2.13679 q^{91} -23.3223 q^{92} -1.96472 q^{93} +2.68265 q^{94} +16.6816 q^{96} +13.5982 q^{97} +12.5713 q^{98} +0.785508 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{2} + 7 q^{3} + 5 q^{4} - q^{6} + 7 q^{7} + 6 q^{8} + 7 q^{9}+O(q^{10}) \) 7 * q - q^2 + 7 * q^3 + 5 * q^4 - q^6 + 7 * q^7 + 6 * q^8 + 7 * q^9	\( 7 q - q^{2} + 7 q^{3} + 5 q^{4} - q^{6} + 7 q^{7} + 6 q^{8} + 7 q^{9} + 5 q^{12} + 5 q^{13} + 7 q^{14} + 9 q^{16} + 2 q^{17} - q^{18} - 13 q^{19} + 7 q^{21} - 14 q^{22} + 6 q^{23} + 6 q^{24} + 7 q^{27} + 30 q^{28} + 9 q^{29} + 5 q^{31} + 26 q^{32} - 8 q^{34} + 5 q^{36} - 5 q^{37} - 2 q^{38} + 5 q^{39} + 18 q^{41} + 7 q^{42} + 14 q^{43} + 17 q^{44} - 27 q^{46} + 7 q^{47} + 9 q^{48} + 14 q^{49} + 2 q^{51} - 3 q^{52} + 20 q^{53} - q^{54} + 17 q^{56} - 13 q^{57} + 37 q^{58} + 10 q^{59} - 8 q^{61} - 6 q^{62} + 7 q^{63} + 18 q^{64} - 14 q^{66} + 4 q^{67} + 10 q^{68} + 6 q^{69} + 12 q^{71} + 6 q^{72} + 4 q^{73} - 25 q^{74} - 66 q^{76} + 6 q^{77} - 5 q^{79} + 7 q^{81} - 29 q^{82} + 52 q^{83} + 30 q^{84} - 17 q^{86} + 9 q^{87} + 26 q^{88} + 32 q^{89} - 26 q^{91} - 17 q^{92} + 5 q^{93} - q^{94} + 26 q^{96} - 12 q^{97} + 40 q^{98}+O(q^{100}) \) 7 * q - q^2 + 7 * q^3 + 5 * q^4 - q^6 + 7 * q^7 + 6 * q^8 + 7 * q^9 + 5 * q^12 + 5 * q^13 + 7 * q^14 + 9 * q^16 + 2 * q^17 - q^18 - 13 * q^19 + 7 * q^21 - 14 * q^22 + 6 * q^23 + 6 * q^24 + 7 * q^27 + 30 * q^28 + 9 * q^29 + 5 * q^31 + 26 * q^32 - 8 * q^34 + 5 * q^36 - 5 * q^37 - 2 * q^38 + 5 * q^39 + 18 * q^41 + 7 * q^42 + 14 * q^43 + 17 * q^44 - 27 * q^46 + 7 * q^47 + 9 * q^48 + 14 * q^49 + 2 * q^51 - 3 * q^52 + 20 * q^53 - q^54 + 17 * q^56 - 13 * q^57 + 37 * q^58 + 10 * q^59 - 8 * q^61 - 6 * q^62 + 7 * q^63 + 18 * q^64 - 14 * q^66 + 4 * q^67 + 10 * q^68 + 6 * q^69 + 12 * q^71 + 6 * q^72 + 4 * q^73 - 25 * q^74 - 66 * q^76 + 6 * q^77 - 5 * q^79 + 7 * q^81 - 29 * q^82 + 52 * q^83 + 30 * q^84 - 17 * q^86 + 9 * q^87 + 26 * q^88 + 32 * q^89 - 26 * q^91 - 17 * q^92 + 5 * q^93 - q^94 + 26 * q^96 - 12 * q^97 + 40 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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