LMFDB - Embedded newform 3525.2.a.z.1.1

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			1.1
Root			$2.30790$ 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)

$f(q)$	$=$	$q-2.30790 q^{2} +1.00000 q^{3} +3.32640 q^{4} -2.30790 q^{6} +0.860272 q^{7} -3.06120 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.30790 q^{2} +1.00000 q^{3} +3.32640 q^{4} -2.30790 q^{6} +0.860272 q^{7} -3.06120 q^{8} +1.00000 q^{9} +2.86972 q^{11} +3.32640 q^{12} +4.01667 q^{13} -1.98542 q^{14} +0.412149 q^{16} +6.25340 q^{17} -2.30790 q^{18} -7.86109 q^{19} +0.860272 q^{21} -6.62302 q^{22} +1.67752 q^{23} -3.06120 q^{24} -9.27007 q^{26} +1.00000 q^{27} +2.86161 q^{28} -1.38052 q^{29} +7.32196 q^{31} +5.17121 q^{32} +2.86972 q^{33} -14.4322 q^{34} +3.32640 q^{36} +7.22336 q^{37} +18.1426 q^{38} +4.01667 q^{39} +7.55591 q^{41} -1.98542 q^{42} -6.88207 q^{43} +9.54584 q^{44} -3.87155 q^{46} +1.00000 q^{47} +0.412149 q^{48} -6.25993 q^{49} +6.25340 q^{51} +13.3611 q^{52} +10.3790 q^{53} -2.30790 q^{54} -2.63347 q^{56} -7.86109 q^{57} +3.18610 q^{58} -8.17819 q^{59} -7.18776 q^{61} -16.8984 q^{62} +0.860272 q^{63} -12.7589 q^{64} -6.62302 q^{66} -5.56732 q^{67} +20.8013 q^{68} +1.67752 q^{69} +11.3542 q^{71} -3.06120 q^{72} -3.97123 q^{73} -16.6708 q^{74} -26.1492 q^{76} +2.46874 q^{77} -9.27007 q^{78} +16.6332 q^{79} +1.00000 q^{81} -17.4383 q^{82} +6.31404 q^{83} +2.86161 q^{84} +15.8831 q^{86} -1.38052 q^{87} -8.78479 q^{88} -8.21367 q^{89} +3.45543 q^{91} +5.58011 q^{92} +7.32196 q^{93} -2.30790 q^{94} +5.17121 q^{96} -13.1992 q^{97} +14.4473 q^{98} +2.86972 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.30790	−1.63193	−0.815966	−	0.578100i	$-0.803794\pi$
$2$	−2.30790	−1.63193	−0.815966	+	0.578100i	$0.803794\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	3.32640	1.66320
$5$	0	0
$5$	0	0
$6$	−2.30790	−0.942196
$7$	0.860272	0.325152	0.162576	−	0.986696i	$-0.448020\pi$
$7$	0.860272	0.325152	0.162576	+	0.986696i	$0.448020\pi$
$8$	−3.06120	−1.08230
$9$	1.00000	0.333333
$10$	0	0
$11$	2.86972	0.865253	0.432626	−	0.901573i	$-0.357587\pi$
$11$	2.86972	0.865253	0.432626	+	0.901573i	$0.357587\pi$
$12$	3.32640	0.960250
$13$	4.01667	1.11402	0.557012	−	0.830505i	$-0.311948\pi$
$13$	4.01667	1.11402	0.557012	+	0.830505i	$0.311948\pi$
$14$	−1.98542	−0.530626
$15$	0	0
$16$	0.412149	0.103037
$17$	6.25340	1.51667	0.758336	−	0.651864i	$-0.226013\pi$
$17$	6.25340	1.51667	0.758336	+	0.651864i	$0.226013\pi$
$18$	−2.30790	−0.543977
$19$	−7.86109	−1.80346	−0.901729	−	0.432301i	$-0.857702\pi$
$19$	−7.86109	−1.80346	−0.901729	+	0.432301i	$0.857702\pi$
$20$	0	0
$21$	0.860272	0.187727
$22$	−6.62302	−1.41203
$23$	1.67752	0.349787	0.174894	−	0.984587i	$-0.444042\pi$
$23$	1.67752	0.349787	0.174894	+	0.984587i	$0.444042\pi$
$24$	−3.06120	−0.624866
$25$	0	0
$26$	−9.27007	−1.81801
$27$	1.00000	0.192450
$28$	2.86161	0.540794
$29$	−1.38052	−0.256356	−0.128178	−	0.991751i	$-0.540913\pi$
$29$	−1.38052	−0.256356	−0.128178	+	0.991751i	$0.540913\pi$
$30$	0	0
$31$	7.32196	1.31506	0.657532	−	0.753427i	$-0.271601\pi$
$31$	7.32196	1.31506	0.657532	+	0.753427i	$0.271601\pi$
$32$	5.17121	0.914150
$33$	2.86972	0.499554
$34$	−14.4322	−2.47511
$35$	0	0
$36$	3.32640	0.554400
$37$	7.22336	1.18751	0.593756	−	0.804645i	$-0.297644\pi$
$37$	7.22336	1.18751	0.593756	+	0.804645i	$0.297644\pi$
$38$	18.1426	2.94312
$39$	4.01667	0.643182
$40$	0	0
$41$	7.55591	1.18004	0.590018	−	0.807390i	$-0.299121\pi$
$41$	7.55591	1.18004	0.590018	+	0.807390i	$0.299121\pi$
$42$	−1.98542	−0.306357
$43$	−6.88207	−1.04951	−0.524753	−	0.851255i	$-0.675842\pi$
$43$	−6.88207	−1.04951	−0.524753	+	0.851255i	$0.675842\pi$
$44$	9.54584	1.43909
$45$	0	0
$46$	−3.87155	−0.570829
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0.412149	0.0594885
$49$	−6.25993	−0.894276
$50$	0	0
$51$	6.25340	0.875651
$52$	13.3611	1.85285
$53$	10.3790	1.42567	0.712836	−	0.701331i	$-0.247410\pi$
$53$	10.3790	1.42567	0.712836	+	0.701331i	$0.247410\pi$
$54$	−2.30790	−0.314065
$55$	0	0
$56$	−2.63347	−0.351912
$57$	−7.86109	−1.04123
$58$	3.18610	0.418355
$59$	−8.17819	−1.06471	−0.532355	−	0.846521i	$-0.678693\pi$
$59$	−8.17819	−1.06471	−0.532355	+	0.846521i	$0.678693\pi$
$60$	0	0
$61$	−7.18776	−0.920298	−0.460149	−	0.887842i	$-0.652204\pi$
$61$	−7.18776	−0.920298	−0.460149	+	0.887842i	$0.652204\pi$
$62$	−16.8984	−2.14609
$63$	0.860272	0.108384
$64$	−12.7589	−1.59487
$65$	0	0
$66$	−6.62302	−0.815238
$67$	−5.56732	−0.680156	−0.340078	−	0.940397i	$-0.610454\pi$
$67$	−5.56732	−0.680156	−0.340078	+	0.940397i	$0.610454\pi$
$68$	20.8013	2.52253
$69$	1.67752	0.201950
$70$	0	0
$71$	11.3542	1.34750	0.673749	−	0.738960i	$-0.264683\pi$
$71$	11.3542	1.34750	0.673749	+	0.738960i	$0.264683\pi$
$72$	−3.06120	−0.360766
$73$	−3.97123	−0.464797	−0.232398	−	0.972621i	$-0.574657\pi$
$73$	−3.97123	−0.464797	−0.232398	+	0.972621i	$0.574657\pi$
$74$	−16.6708	−1.93794
$75$	0	0
$76$	−26.1492	−2.99951
$77$	2.46874	0.281339
$78$	−9.27007	−1.04963
$79$	16.6332	1.87138	0.935688	−	0.352828i	$-0.114780\pi$
$79$	16.6332	1.87138	0.935688	+	0.352828i	$0.114780\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−17.4383	−1.92574
$83$	6.31404	0.693056	0.346528	−	0.938040i	$-0.387361\pi$
$83$	6.31404	0.693056	0.346528	+	0.938040i	$0.387361\pi$
$84$	2.86161	0.312227
$85$	0	0
$86$	15.8831	1.71272
$87$	−1.38052	−0.148007
$88$	−8.78479	−0.936462
$89$	−8.21367	−0.870648	−0.435324	−	0.900274i	$-0.643366\pi$
$89$	−8.21367	−0.870648	−0.435324	+	0.900274i	$0.643366\pi$
$90$	0	0
$91$	3.45543	0.362227
$92$	5.58011	0.581767
$93$	7.32196	0.759252
$94$	−2.30790	−0.238042
$95$	0	0
$96$	5.17121	0.527785
$97$	−13.1992	−1.34018	−0.670088	−	0.742282i	$-0.733744\pi$
$97$	−13.1992	−1.34018	−0.670088	+	0.742282i	$0.733744\pi$
$98$	14.4473	1.45940
$99$	2.86972	0.288418
$100$	0	0
$101$	−18.7067	−1.86139	−0.930695	−	0.365797i	$-0.880797\pi$
$101$	−18.7067	−1.86139	−0.930695	+	0.365797i	$0.880797\pi$
$102$	−14.4322	−1.42900
$103$	6.78830	0.668871	0.334436	−	0.942419i	$-0.391454\pi$
$103$	6.78830	0.668871	0.334436	+	0.942419i	$0.391454\pi$
$104$	−12.2958	−1.20571
$105$	0	0
$106$	−23.9538	−2.32660
$107$	12.4206	1.20075	0.600373	−	0.799720i	$-0.295019\pi$
$107$	12.4206	1.20075	0.600373	+	0.799720i	$0.295019\pi$
$108$	3.32640	0.320083
$109$	4.30272	0.412126	0.206063	−	0.978539i	$-0.433935\pi$
$109$	4.30272	0.412126	0.206063	+	0.978539i	$0.433935\pi$
$110$	0	0
$111$	7.22336	0.685611
$112$	0.354560	0.0335028
$113$	−9.47563	−0.891392	−0.445696	−	0.895184i	$-0.647044\pi$
$113$	−9.47563	−0.891392	−0.445696	+	0.895184i	$0.647044\pi$
$114$	18.1426	1.69921
$115$	0	0
$116$	−4.59216	−0.426371
$117$	4.01667	0.371341
$118$	18.8745	1.73753
$119$	5.37963	0.493149
$120$	0	0
$121$	−2.76472	−0.251338
$122$	16.5886	1.50186
$123$	7.55591	0.681294
$124$	24.3558	2.18721
$125$	0	0
$126$	−1.98542	−0.176875
$127$	12.4618	1.10581	0.552903	−	0.833246i	$-0.313520\pi$
$127$	12.4618	1.10581	0.552903	+	0.833246i	$0.313520\pi$
$128$	19.1039	1.68856
$129$	−6.88207	−0.605932
$130$	0	0
$131$	14.3498	1.25375	0.626873	−	0.779122i	$-0.284335\pi$
$131$	14.3498	1.25375	0.626873	+	0.779122i	$0.284335\pi$
$132$	9.54584	0.830859
$133$	−6.76268	−0.586399
$134$	12.8488	1.10997
$135$	0	0
$136$	−19.1429	−1.64149
$137$	−3.74430	−0.319897	−0.159948	−	0.987125i	$-0.551133\pi$
$137$	−3.74430	−0.319897	−0.159948	+	0.987125i	$0.551133\pi$
$138$	−3.87155	−0.329568
$139$	0.537864	0.0456210	0.0228105	−	0.999740i	$-0.492739\pi$
$139$	0.537864	0.0456210	0.0228105	+	0.999740i	$0.492739\pi$
$140$	0	0
$141$	1.00000	0.0842152
$142$	−26.2044	−2.19902
$143$	11.5267	0.963912
$144$	0.412149	0.0343457
$145$	0	0
$146$	9.16519	0.758517
$147$	−6.25993	−0.516311
$148$	24.0278	1.97507
$149$	13.9660	1.14414	0.572068	−	0.820206i	$-0.306141\pi$
$149$	13.9660	1.14414	0.572068	+	0.820206i	$0.306141\pi$
$150$	0	0
$151$	−18.5462	−1.50926	−0.754632	−	0.656148i	$-0.772185\pi$
$151$	−18.5462	−1.50926	−0.754632	+	0.656148i	$0.772185\pi$
$152$	24.0644	1.95188
$153$	6.25340	0.505558
$154$	−5.69760	−0.459126
$155$	0	0
$156$	13.3611	1.06974
$157$	−10.6512	−0.850060	−0.425030	−	0.905179i	$-0.639737\pi$
$157$	−10.6512	−0.850060	−0.425030	+	0.905179i	$0.639737\pi$
$158$	−38.3877	−3.05396
$159$	10.3790	0.823112
$160$	0	0
$161$	1.44312	0.113734
$162$	−2.30790	−0.181326
$163$	9.61416	0.753039	0.376520	−	0.926409i	$-0.377121\pi$
$163$	9.61416	0.753039	0.376520	+	0.926409i	$0.377121\pi$
$164$	25.1340	1.96264
$165$	0	0
$166$	−14.5722	−1.13102
$167$	−5.59100	−0.432644	−0.216322	−	0.976322i	$-0.569406\pi$
$167$	−5.59100	−0.432644	−0.216322	+	0.976322i	$0.569406\pi$
$168$	−2.63347	−0.203176
$169$	3.13363	0.241048
$170$	0	0
$171$	−7.86109	−0.601153
$172$	−22.8925	−1.74554
$173$	12.2528	0.931562	0.465781	−	0.884900i	$-0.345774\pi$
$173$	12.2528	0.931562	0.465781	+	0.884900i	$0.345774\pi$
$174$	3.18610	0.241537
$175$	0	0
$176$	1.18275	0.0891532
$177$	−8.17819	−0.614711
$178$	18.9563	1.42084
$179$	−26.0992	−1.95075	−0.975373	−	0.220562i	$-0.929211\pi$
$179$	−26.0992	−1.95075	−0.975373	+	0.220562i	$0.929211\pi$
$180$	0	0
$181$	11.0691	0.822761	0.411380	−	0.911464i	$-0.365047\pi$
$181$	11.0691	0.822761	0.411380	+	0.911464i	$0.365047\pi$
$182$	−7.97478	−0.591130
$183$	−7.18776	−0.531334
$184$	−5.13524	−0.378575
$185$	0	0
$186$	−16.8984	−1.23905
$187$	17.9455	1.31230
$188$	3.32640	0.242603
$189$	0.860272	0.0625756
$190$	0	0
$191$	−1.50035	−0.108561	−0.0542806	−	0.998526i	$-0.517287\pi$
$191$	−1.50035	−0.108561	−0.0542806	+	0.998526i	$0.517287\pi$
$192$	−12.7589	−0.920797
$193$	−17.4657	−1.25721	−0.628606	−	0.777724i	$-0.716374\pi$
$193$	−17.4657	−1.25721	−0.628606	+	0.777724i	$0.716374\pi$
$194$	30.4624	2.18708
$195$	0	0
$196$	−20.8231	−1.48736
$197$	−21.9598	−1.56457	−0.782285	−	0.622920i	$-0.785946\pi$
$197$	−21.9598	−1.56457	−0.782285	+	0.622920i	$0.785946\pi$
$198$	−6.62302	−0.470678
$199$	−10.8996	−0.772652	−0.386326	−	0.922362i	$-0.626256\pi$
$199$	−10.8996	−0.772652	−0.386326	+	0.922362i	$0.626256\pi$
$200$	0	0
$201$	−5.56732	−0.392688
$202$	43.1733	3.03766
$203$	−1.18762	−0.0833546
$204$	20.8013	1.45638
$205$	0	0
$206$	−15.6667	−1.09155
$207$	1.67752	0.116596
$208$	1.65546	0.114786
$209$	−22.5591	−1.56045
$210$	0	0
$211$	−13.5183	−0.930640	−0.465320	−	0.885142i	$-0.654061\pi$
$211$	−13.5183	−0.930640	−0.465320	+	0.885142i	$0.654061\pi$
$212$	34.5249	2.37118
$213$	11.3542	0.777978
$214$	−28.6655	−1.95954
$215$	0	0
$216$	−3.06120	−0.208289
$217$	6.29888	0.427596
$218$	−9.93024	−0.672561
$219$	−3.97123	−0.268351
$220$	0	0
$221$	25.1178	1.68961
$222$	−16.6708	−1.11887
$223$	3.42274	0.229204	0.114602	−	0.993412i	$-0.463441\pi$
$223$	3.42274	0.229204	0.114602	+	0.993412i	$0.463441\pi$
$224$	4.44865	0.297238
$225$	0	0
$226$	21.8688	1.45469
$227$	26.8967	1.78519	0.892597	−	0.450856i	$-0.148881\pi$
$227$	26.8967	1.78519	0.892597	+	0.450856i	$0.148881\pi$
$228$	−26.1492	−1.73177
$229$	10.5653	0.698173	0.349086	−	0.937091i	$-0.386492\pi$
$229$	10.5653	0.698173	0.349086	+	0.937091i	$0.386492\pi$
$230$	0	0
$231$	2.46874	0.162431
$232$	4.22605	0.277454
$233$	−5.05163	−0.330944	−0.165472	−	0.986215i	$-0.552915\pi$
$233$	−5.05163	−0.330944	−0.165472	+	0.986215i	$0.552915\pi$
$234$	−9.27007	−0.606003
$235$	0	0
$236$	−27.2040	−1.77083
$237$	16.6332	1.08044
$238$	−12.4156	−0.804786
$239$	6.73903	0.435912	0.217956	−	0.975959i	$-0.430061\pi$
$239$	6.73903	0.435912	0.217956	+	0.975959i	$0.430061\pi$
$240$	0	0
$241$	−21.3842	−1.37748	−0.688740	−	0.725009i	$-0.741836\pi$
$241$	−21.3842	−1.37748	−0.688740	+	0.725009i	$0.741836\pi$
$242$	6.38069	0.410166
$243$	1.00000	0.0641500
$244$	−23.9094	−1.53064
$245$	0	0
$246$	−17.4383	−1.11182
$247$	−31.5754	−2.00910
$248$	−22.4140	−1.42329
$249$	6.31404	0.400136
$250$	0	0
$251$	15.7999	0.997281	0.498641	−	0.866809i	$-0.333833\pi$
$251$	15.7999	0.997281	0.498641	+	0.866809i	$0.333833\pi$
$252$	2.86161	0.180265
$253$	4.81401	0.302654
$254$	−28.7606	−1.80460
$255$	0	0
$256$	−18.5721	−1.16075
$257$	−8.34387	−0.520477	−0.260238	−	0.965544i	$-0.583801\pi$
$257$	−8.34387	−0.520477	−0.260238	+	0.965544i	$0.583801\pi$
$258$	15.8831	0.988840
$259$	6.21405	0.386122
$260$	0	0
$261$	−1.38052	−0.0854519
$262$	−33.1178	−2.04603
$263$	−10.0710	−0.621003	−0.310502	−	0.950573i	$-0.600497\pi$
$263$	−10.0710	−0.621003	−0.310502	+	0.950573i	$0.600497\pi$
$264$	−8.78479	−0.540667
$265$	0	0
$266$	15.6076	0.956962
$267$	−8.21367	−0.502669
$268$	−18.5191	−1.13124
$269$	13.1491	0.801717	0.400859	−	0.916140i	$-0.368712\pi$
$269$	13.1491	0.801717	0.400859	+	0.916140i	$0.368712\pi$
$270$	0	0
$271$	7.77052	0.472026	0.236013	−	0.971750i	$-0.424159\pi$
$271$	7.77052	0.472026	0.236013	+	0.971750i	$0.424159\pi$
$272$	2.57733	0.156274
$273$	3.45543	0.209132
$274$	8.64146	0.522050
$275$	0	0
$276$	5.58011	0.335883
$277$	3.34256	0.200835	0.100417	−	0.994945i	$-0.467982\pi$
$277$	3.34256	0.200835	0.100417	+	0.994945i	$0.467982\pi$
$278$	−1.24134	−0.0744504
$279$	7.32196	0.438354
$280$	0	0
$281$	13.1042	0.781729	0.390864	−	0.920448i	$-0.372176\pi$
$281$	13.1042	0.781729	0.390864	+	0.920448i	$0.372176\pi$
$282$	−2.30790	−0.137433
$283$	31.3949	1.86623	0.933115	−	0.359578i	$-0.117079\pi$
$283$	31.3949	1.86623	0.933115	+	0.359578i	$0.117079\pi$
$284$	37.7687	2.24116
$285$	0	0
$286$	−26.6025	−1.57304
$287$	6.50014	0.383691
$288$	5.17121	0.304717
$289$	22.1050	1.30030
$290$	0	0
$291$	−13.1992	−0.773751
$292$	−13.2099	−0.773051
$293$	16.1948	0.946110	0.473055	−	0.881033i	$-0.343151\pi$
$293$	16.1948	0.946110	0.473055	+	0.881033i	$0.343151\pi$
$294$	14.4473	0.842584
$295$	0	0
$296$	−22.1122	−1.28524
$297$	2.86972	0.166518
$298$	−32.2321	−1.86715
$299$	6.73805	0.389671
$300$	0	0
$301$	−5.92045	−0.341249
$302$	42.8027	2.46302
$303$	−18.7067	−1.07467
$304$	−3.23994	−0.185823
$305$	0	0
$306$	−14.4322	−0.825035
$307$	25.8625	1.47605	0.738026	−	0.674772i	$-0.235758\pi$
$307$	25.8625	1.47605	0.738026	+	0.674772i	$0.235758\pi$
$308$	8.21202	0.467923
$309$	6.78830	0.386173
$310$	0	0
$311$	10.5509	0.598288	0.299144	−	0.954208i	$-0.403299\pi$
$311$	10.5509	0.598288	0.299144	+	0.954208i	$0.403299\pi$
$312$	−12.2958	−0.696115
$313$	13.3525	0.754728	0.377364	−	0.926065i	$-0.376831\pi$
$313$	13.3525	0.754728	0.377364	+	0.926065i	$0.376831\pi$
$314$	24.5820	1.38724
$315$	0	0
$316$	55.3286	3.11248
$317$	2.87288	0.161357	0.0806786	−	0.996740i	$-0.474291\pi$
$317$	2.87288	0.161357	0.0806786	+	0.996740i	$0.474291\pi$
$318$	−23.9538	−1.34326
$319$	−3.96170	−0.221812
$320$	0	0
$321$	12.4206	0.693251
$322$	−3.33059	−0.185606
$323$	−49.1586	−2.73526
$324$	3.32640	0.184800
$325$	0	0
$326$	−22.1885	−1.22891
$327$	4.30272	0.237941
$328$	−23.1302	−1.27715
$329$	0.860272	0.0474283
$330$	0	0
$331$	26.8952	1.47829	0.739147	−	0.673544i	$-0.235229\pi$
$331$	26.8952	1.47829	0.739147	+	0.673544i	$0.235229\pi$
$332$	21.0030	1.15269
$333$	7.22336	0.395838
$334$	12.9035	0.706046
$335$	0	0
$336$	0.354560	0.0193428
$337$	5.56374	0.303076	0.151538	−	0.988451i	$-0.451577\pi$
$337$	5.56374	0.303076	0.151538	+	0.988451i	$0.451577\pi$
$338$	−7.23210	−0.393374
$339$	−9.47563	−0.514646
$340$	0	0
$341$	21.0120	1.13786
$342$	18.1426	0.981040
$343$	−11.4071	−0.615928
$344$	21.0674	1.13588
$345$	0	0
$346$	−28.2782	−1.52024
$347$	−23.1678	−1.24371	−0.621855	−	0.783132i	$-0.713621\pi$
$347$	−23.1678	−1.24371	−0.621855	+	0.783132i	$0.713621\pi$
$348$	−4.59216	−0.246166
$349$	0.135140	0.00723388	0.00361694	−	0.999993i	$-0.498849\pi$
$349$	0.135140	0.00723388	0.00361694	+	0.999993i	$0.498849\pi$
$350$	0	0
$351$	4.01667	0.214394
$352$	14.8399	0.790970
$353$	−28.9889	−1.54293	−0.771463	−	0.636274i	$-0.780475\pi$
$353$	−28.9889	−1.54293	−0.771463	+	0.636274i	$0.780475\pi$
$354$	18.8745	1.00317
$355$	0	0
$356$	−27.3220	−1.44806
$357$	5.37963	0.284720
$358$	60.2344	3.18348
$359$	−13.1094	−0.691885	−0.345943	−	0.938256i	$-0.612441\pi$
$359$	−13.1094	−0.691885	−0.345943	+	0.938256i	$0.612441\pi$
$360$	0	0
$361$	42.7968	2.25246
$362$	−25.5464	−1.34269
$363$	−2.76472	−0.145110
$364$	11.4941	0.602457
$365$	0	0
$366$	16.5886	0.867101
$367$	−6.48436	−0.338481	−0.169240	−	0.985575i	$-0.554131\pi$
$367$	−6.48436	−0.338481	−0.169240	+	0.985575i	$0.554131\pi$
$368$	0.691388	0.0360411
$369$	7.55591	0.393345
$370$	0	0
$371$	8.92880	0.463560
$372$	24.3558	1.26279
$373$	−23.6810	−1.22616	−0.613078	−	0.790022i	$-0.710069\pi$
$373$	−23.6810	−1.22616	−0.613078	+	0.790022i	$0.710069\pi$
$374$	−41.4164	−2.14159
$375$	0	0
$376$	−3.06120	−0.157870
$377$	−5.54508	−0.285586
$378$	−1.98542	−0.102119
$379$	7.41124	0.380690	0.190345	−	0.981717i	$-0.439039\pi$
$379$	7.41124	0.380690	0.190345	+	0.981717i	$0.439039\pi$
$380$	0	0
$381$	12.4618	0.638437
$382$	3.46265	0.177165
$383$	−8.40592	−0.429523	−0.214761	−	0.976667i	$-0.568897\pi$
$383$	−8.40592	−0.429523	−0.214761	+	0.976667i	$0.568897\pi$
$384$	19.1039	0.974893
$385$	0	0
$386$	40.3092	2.05168
$387$	−6.88207	−0.349835
$388$	−43.9059	−2.22898
$389$	−15.2122	−0.771291	−0.385646	−	0.922647i	$-0.626021\pi$
$389$	−15.2122	−0.771291	−0.385646	+	0.922647i	$0.626021\pi$
$390$	0	0
$391$	10.4902	0.530513
$392$	19.1629	0.967874
$393$	14.3498	0.723850
$394$	50.6810	2.55327
$395$	0	0
$396$	9.54584	0.479696
$397$	15.8445	0.795211	0.397605	−	0.917557i	$-0.369841\pi$
$397$	15.8445	0.795211	0.397605	+	0.917557i	$0.369841\pi$
$398$	25.1552	1.26092
$399$	−6.76268	−0.338557
$400$	0	0
$401$	26.4884	1.32277	0.661385	−	0.750047i	$-0.269969\pi$
$401$	26.4884	1.32277	0.661385	+	0.750047i	$0.269969\pi$
$402$	12.8488	0.640841
$403$	29.4099	1.46501
$404$	−62.2261	−3.09587
$405$	0	0
$406$	2.74091	0.136029
$407$	20.7290	1.02750
$408$	−19.1429	−0.947717
$409$	−14.7515	−0.729415	−0.364707	−	0.931122i	$-0.618831\pi$
$409$	−14.7515	−0.729415	−0.364707	+	0.931122i	$0.618831\pi$
$410$	0	0
$411$	−3.74430	−0.184693
$412$	22.5806	1.11247
$413$	−7.03547	−0.346193
$414$	−3.87155	−0.190276
$415$	0	0
$416$	20.7710	1.01838
$417$	0.537864	0.0263393
$418$	52.0642	2.54654
$419$	7.31296	0.357261	0.178631	−	0.983916i	$-0.442833\pi$
$419$	7.31296	0.357261	0.178631	+	0.983916i	$0.442833\pi$
$420$	0	0
$421$	−1.65488	−0.0806541	−0.0403271	−	0.999187i	$-0.512840\pi$
$421$	−1.65488	−0.0806541	−0.0403271	+	0.999187i	$0.512840\pi$
$422$	31.1990	1.51874
$423$	1.00000	0.0486217
$424$	−31.7724	−1.54300
$425$	0	0
$426$	−26.2044	−1.26961
$427$	−6.18342	−0.299237
$428$	41.3160	1.99708
$429$	11.5267	0.556515
$430$	0	0
$431$	8.61351	0.414898	0.207449	−	0.978246i	$-0.433484\pi$
$431$	8.61351	0.414898	0.207449	+	0.978246i	$0.433484\pi$
$432$	0.412149	0.0198295
$433$	0.474268	0.0227919	0.0113959	−	0.999935i	$-0.496372\pi$
$433$	0.474268	0.0227919	0.0113959	+	0.999935i	$0.496372\pi$
$434$	−14.5372	−0.697807
$435$	0	0
$436$	14.3126	0.685448
$437$	−13.1872	−0.630827
$438$	9.16519	0.437930
$439$	21.7771	1.03936	0.519682	−	0.854360i	$-0.326050\pi$
$439$	21.7771	1.03936	0.519682	+	0.854360i	$0.326050\pi$
$440$	0	0
$441$	−6.25993	−0.298092
$442$	−57.9695	−2.75733
$443$	30.9632	1.47111	0.735553	−	0.677468i	$-0.236923\pi$
$443$	30.9632	1.47111	0.735553	+	0.677468i	$0.236923\pi$
$444$	24.0278	1.14031
$445$	0	0
$446$	−7.89934	−0.374045
$447$	13.9660	0.660568
$448$	−10.9762	−0.518575
$449$	19.3567	0.913501	0.456751	−	0.889595i	$-0.349013\pi$
$449$	19.3567	0.913501	0.456751	+	0.889595i	$0.349013\pi$
$450$	0	0
$451$	21.6833	1.02103
$452$	−31.5198	−1.48256
$453$	−18.5462	−0.871374
$454$	−62.0748	−2.91331
$455$	0	0
$456$	24.0644	1.12692
$457$	−13.4013	−0.626888	−0.313444	−	0.949607i	$-0.601483\pi$
$457$	−13.4013	−0.626888	−0.313444	+	0.949607i	$0.601483\pi$
$458$	−24.3836	−1.13937
$459$	6.25340	0.291884
$460$	0	0
$461$	16.6488	0.775413	0.387707	−	0.921783i	$-0.373267\pi$
$461$	16.6488	0.775413	0.387707	+	0.921783i	$0.373267\pi$
$462$	−5.69760	−0.265076
$463$	−6.09512	−0.283264	−0.141632	−	0.989919i	$-0.545235\pi$
$463$	−6.09512	−0.283264	−0.141632	+	0.989919i	$0.545235\pi$
$464$	−0.568979	−0.0264142
$465$	0	0
$466$	11.6587	0.540077
$467$	−11.0692	−0.512223	−0.256112	−	0.966647i	$-0.582441\pi$
$467$	−11.0692	−0.512223	−0.256112	+	0.966647i	$0.582441\pi$
$468$	13.3611	0.617615
$469$	−4.78941	−0.221154
$470$	0	0
$471$	−10.6512	−0.490783
$472$	25.0351	1.15234
$473$	−19.7496	−0.908088
$474$	−38.3877	−1.76320
$475$	0	0
$476$	17.8948	0.820207
$477$	10.3790	0.475224
$478$	−15.5530	−0.711378
$479$	−9.72149	−0.444186	−0.222093	−	0.975025i	$-0.571289\pi$
$479$	−9.72149	−0.444186	−0.222093	+	0.975025i	$0.571289\pi$
$480$	0	0
$481$	29.0138	1.32292
$482$	49.3527	2.24795
$483$	1.44312	0.0656644
$484$	−9.19657	−0.418026
$485$	0	0
$486$	−2.30790	−0.104688
$487$	−22.1252	−1.00259	−0.501294	−	0.865277i	$-0.667143\pi$
$487$	−22.1252	−1.00259	−0.501294	+	0.865277i	$0.667143\pi$
$488$	22.0032	0.996038
$489$	9.61416	0.434767
$490$	0	0
$491$	−2.80744	−0.126698	−0.0633490	−	0.997991i	$-0.520178\pi$
$491$	−2.80744	−0.126698	−0.0633490	+	0.997991i	$0.520178\pi$
$492$	25.1340	1.13313
$493$	−8.63293	−0.388808
$494$	72.8729	3.27871
$495$	0	0
$496$	3.01774	0.135500
$497$	9.76771	0.438142
$498$	−14.5722	−0.652995
$499$	35.0285	1.56809	0.784045	−	0.620703i	$-0.213153\pi$
$499$	35.0285	1.56809	0.784045	+	0.620703i	$0.213153\pi$
$500$	0	0
$501$	−5.59100	−0.249787
$502$	−36.4646	−1.62749
$503$	11.5993	0.517187	0.258593	−	0.965986i	$-0.416741\pi$
$503$	11.5993	0.517187	0.258593	+	0.965986i	$0.416741\pi$
$504$	−2.63347	−0.117304
$505$	0	0
$506$	−11.1103	−0.493911
$507$	3.13363	0.139169
$508$	41.4530	1.83918
$509$	29.3580	1.30127	0.650636	−	0.759390i	$-0.274502\pi$
$509$	29.3580	1.30127	0.650636	+	0.759390i	$0.274502\pi$
$510$	0	0
$511$	−3.41633	−0.151130
$512$	4.65465	0.205708
$513$	−7.86109	−0.347076
$514$	19.2568	0.849382
$515$	0	0
$516$	−22.8925	−1.00779
$517$	2.86972	0.126210
$518$	−14.3414	−0.630125
$519$	12.2528	0.537837
$520$	0	0
$521$	−22.8903	−1.00284	−0.501421	−	0.865203i	$-0.667189\pi$
$521$	−22.8903	−1.00284	−0.501421	+	0.865203i	$0.667189\pi$
$522$	3.18610	0.139452
$523$	34.8070	1.52200	0.761001	−	0.648751i	$-0.224708\pi$
$523$	34.8070	1.52200	0.761001	+	0.648751i	$0.224708\pi$
$524$	47.7331	2.08523
$525$	0	0
$526$	23.2428	1.01343
$527$	45.7872	1.99452
$528$	1.18275	0.0514726
$529$	−20.1859	−0.877649
$530$	0	0
$531$	−8.17819	−0.354903
$532$	−22.4954	−0.975299
$533$	30.3496	1.31459
$534$	18.9563	0.820321
$535$	0	0
$536$	17.0427	0.736132
$537$	−26.0992	−1.12626
$538$	−30.3469	−1.30835
$539$	−17.9642	−0.773775
$540$	0	0
$541$	−25.3915	−1.09167	−0.545833	−	0.837894i	$-0.683787\pi$
$541$	−25.3915	−1.09167	−0.545833	+	0.837894i	$0.683787\pi$
$542$	−17.9336	−0.770314
$543$	11.0691	0.475021
$544$	32.3377	1.38647
$545$	0	0
$546$	−7.97478	−0.341289
$547$	−16.5226	−0.706457	−0.353228	−	0.935537i	$-0.614916\pi$
$547$	−16.5226	−0.706457	−0.353228	+	0.935537i	$0.614916\pi$
$548$	−12.4550	−0.532053
$549$	−7.18776	−0.306766
$550$	0	0
$551$	10.8524	0.462327
$552$	−5.13524	−0.218570
$553$	14.3090	0.608482
$554$	−7.71429	−0.327749
$555$	0	0
$556$	1.78915	0.0758769
$557$	44.1449	1.87048	0.935240	−	0.354014i	$-0.115183\pi$
$557$	44.1449	1.87048	0.935240	+	0.354014i	$0.115183\pi$
$558$	−16.8984	−0.715364
$559$	−27.6430	−1.16917
$560$	0	0
$561$	17.9455	0.757660
$562$	−30.2431	−1.27573
$563$	23.5370	0.991968	0.495984	−	0.868332i	$-0.334807\pi$
$563$	23.5370	0.991968	0.495984	+	0.868332i	$0.334807\pi$
$564$	3.32640	0.140067
$565$	0	0
$566$	−72.4562	−3.04556
$567$	0.860272	0.0361280
$568$	−34.7576	−1.45840
$569$	−20.0449	−0.840326	−0.420163	−	0.907449i	$-0.638027\pi$
$569$	−20.0449	−0.840326	−0.420163	+	0.907449i	$0.638027\pi$
$570$	0	0
$571$	−5.33472	−0.223251	−0.111625	−	0.993750i	$-0.535606\pi$
$571$	−5.33472	−0.223251	−0.111625	+	0.993750i	$0.535606\pi$
$572$	38.3425	1.60318
$573$	−1.50035	−0.0626779
$574$	−15.0017	−0.626158
$575$	0	0
$576$	−12.7589	−0.531622
$577$	32.3381	1.34625	0.673125	−	0.739528i	$-0.264951\pi$
$577$	32.3381	1.34625	0.673125	+	0.739528i	$0.264951\pi$
$578$	−51.0162	−2.12199
$579$	−17.4657	−0.725852
$580$	0	0
$581$	5.43179	0.225349
$582$	30.4624	1.26271
$583$	29.7849	1.23357
$584$	12.1567	0.503049
$585$	0	0
$586$	−37.3760	−1.54399
$587$	−10.7947	−0.445544	−0.222772	−	0.974871i	$-0.571511\pi$
$587$	−10.7947	−0.445544	−0.222772	+	0.974871i	$0.571511\pi$
$588$	−20.8231	−0.858728
$589$	−57.5586	−2.37166
$590$	0	0
$591$	−21.9598	−0.903305
$592$	2.97710	0.122358
$593$	14.7449	0.605501	0.302751	−	0.953070i	$-0.402095\pi$
$593$	14.7449	0.605501	0.302751	+	0.953070i	$0.402095\pi$
$594$	−6.62302	−0.271746
$595$	0	0
$596$	46.4564	1.90293
$597$	−10.8996	−0.446091
$598$	−15.5507	−0.635917
$599$	−20.3969	−0.833395	−0.416698	−	0.909045i	$-0.636813\pi$
$599$	−20.3969	−0.833395	−0.416698	+	0.909045i	$0.636813\pi$
$600$	0	0
$601$	−34.2300	−1.39627	−0.698135	−	0.715966i	$-0.745986\pi$
$601$	−34.2300	−1.39627	−0.698135	+	0.715966i	$0.745986\pi$
$602$	13.6638	0.556895
$603$	−5.56732	−0.226719
$604$	−61.6920	−2.51021
$605$	0	0
$606$	43.1733	1.75379
$607$	48.2028	1.95649	0.978245	−	0.207451i	$-0.0665168\pi$
$607$	48.2028	1.95649	0.978245	+	0.207451i	$0.0665168\pi$
$608$	−40.6514	−1.64863
$609$	−1.18762	−0.0481248
$610$	0	0
$611$	4.01667	0.162497
$612$	20.8013	0.840844
$613$	20.9737	0.847119	0.423559	−	0.905868i	$-0.360781\pi$
$613$	20.9737	0.847119	0.423559	+	0.905868i	$0.360781\pi$
$614$	−59.6881	−2.40882
$615$	0	0
$616$	−7.55731	−0.304493
$617$	22.6650	0.912458	0.456229	−	0.889862i	$-0.349200\pi$
$617$	22.6650	0.912458	0.456229	+	0.889862i	$0.349200\pi$
$618$	−15.6667	−0.630208
$619$	21.2811	0.855360	0.427680	−	0.903930i	$-0.359331\pi$
$619$	21.2811	0.855360	0.427680	+	0.903930i	$0.359331\pi$
$620$	0	0
$621$	1.67752	0.0673166
$622$	−24.3505	−0.976366
$623$	−7.06599	−0.283093
$624$	1.65546	0.0662716
$625$	0	0
$626$	−30.8162	−1.23166
$627$	−22.5591	−0.900925
$628$	−35.4303	−1.41382
$629$	45.1706	1.80107
$630$	0	0
$631$	−39.3678	−1.56721	−0.783603	−	0.621261i	$-0.786621\pi$
$631$	−39.3678	−1.56721	−0.783603	+	0.621261i	$0.786621\pi$
$632$	−50.9175	−2.02539
$633$	−13.5183	−0.537305
$634$	−6.63033	−0.263324
$635$	0	0
$636$	34.5249	1.36900
$637$	−25.1441	−0.996244
$638$	9.14320	0.361983
$639$	11.3542	0.449166
$640$	0	0
$641$	25.7640	1.01762	0.508808	−	0.860880i	$-0.330086\pi$
$641$	25.7640	1.01762	0.508808	+	0.860880i	$0.330086\pi$
$642$	−28.6655	−1.13134
$643$	−18.9260	−0.746368	−0.373184	−	0.927757i	$-0.621734\pi$
$643$	−18.9260	−0.746368	−0.373184	+	0.927757i	$0.621734\pi$
$644$	4.80041	0.189163
$645$	0	0
$646$	113.453	4.46375
$647$	25.3575	0.996908	0.498454	−	0.866916i	$-0.333901\pi$
$647$	25.3575	0.996908	0.498454	+	0.866916i	$0.333901\pi$
$648$	−3.06120	−0.120255
$649$	−23.4691	−0.921243
$650$	0	0
$651$	6.29888	0.246872
$652$	31.9806	1.25246
$653$	−4.84804	−0.189718	−0.0948592	−	0.995491i	$-0.530240\pi$
$653$	−4.84804	−0.189718	−0.0948592	+	0.995491i	$0.530240\pi$
$654$	−9.93024	−0.388303
$655$	0	0
$656$	3.11416	0.121587
$657$	−3.97123	−0.154932
$658$	−1.98542	−0.0773998
$659$	19.1497	0.745966	0.372983	−	0.927838i	$-0.378335\pi$
$659$	19.1497	0.745966	0.372983	+	0.927838i	$0.378335\pi$
$660$	0	0
$661$	21.6815	0.843313	0.421656	−	0.906756i	$-0.361449\pi$
$661$	21.6815	0.843313	0.421656	+	0.906756i	$0.361449\pi$
$662$	−62.0715	−2.41248
$663$	25.1178	0.975496
$664$	−19.3286	−0.750094
$665$	0	0
$666$	−16.6708	−0.645980
$667$	−2.31585	−0.0896700
$668$	−18.5979	−0.719575
$669$	3.42274	0.132331
$670$	0	0
$671$	−20.6268	−0.796290
$672$	4.44865	0.171610
$673$	−3.54925	−0.136813	−0.0684067	−	0.997658i	$-0.521792\pi$
$673$	−3.54925	−0.136813	−0.0684067	+	0.997658i	$0.521792\pi$
$674$	−12.8406	−0.494600
$675$	0	0
$676$	10.4237	0.400912
$677$	3.34037	0.128381	0.0641905	−	0.997938i	$-0.479553\pi$
$677$	3.34037	0.128381	0.0641905	+	0.997938i	$0.479553\pi$
$678$	21.8688	0.839866
$679$	−11.3549	−0.435761
$680$	0	0
$681$	26.8967	1.03068
$682$	−48.4935	−1.85691
$683$	27.5303	1.05342	0.526708	−	0.850046i	$-0.323426\pi$
$683$	27.5303	1.05342	0.526708	+	0.850046i	$0.323426\pi$
$684$	−26.1492	−0.999838
$685$	0	0
$686$	26.3266	1.00515
$687$	10.5653	0.403090
$688$	−2.83644	−0.108138
$689$	41.6892	1.58823
$690$	0	0
$691$	−28.9214	−1.10022	−0.550111	−	0.835092i	$-0.685414\pi$
$691$	−28.9214	−1.10022	−0.550111	+	0.835092i	$0.685414\pi$
$692$	40.7577	1.54937
$693$	2.46874	0.0937796
$694$	53.4689	2.02965
$695$	0	0
$696$	4.22605	0.160188
$697$	47.2501	1.78973
$698$	−0.311890	−0.0118052
$699$	−5.05163	−0.191070
$700$	0	0
$701$	−25.5099	−0.963496	−0.481748	−	0.876310i	$-0.659998\pi$
$701$	−25.5099	−0.963496	−0.481748	+	0.876310i	$0.659998\pi$
$702$	−9.27007	−0.349876
$703$	−56.7835	−2.14163
$704$	−36.6145	−1.37996
$705$	0	0
$706$	66.9036	2.51795
$707$	−16.0929	−0.605235
$708$	−27.2040	−1.02239
$709$	−21.7987	−0.818667	−0.409334	−	0.912385i	$-0.634239\pi$
$709$	−21.7987	−0.818667	−0.409334	+	0.912385i	$0.634239\pi$
$710$	0	0
$711$	16.6332	0.623792
$712$	25.1437	0.942301
$713$	12.2827	0.459993
$714$	−12.4156	−0.464644
$715$	0	0
$716$	−86.8165	−3.24448
$717$	6.73903	0.251674
$718$	30.2551	1.12911
$719$	−5.07665	−0.189327	−0.0946636	−	0.995509i	$-0.530178\pi$
$719$	−5.07665	−0.189327	−0.0946636	+	0.995509i	$0.530178\pi$
$720$	0	0
$721$	5.83979	0.217485
$722$	−98.7707	−3.67587
$723$	−21.3842	−0.795288
$724$	36.8203	1.36842
$725$	0	0
$726$	6.38069	0.236810
$727$	−46.8976	−1.73934	−0.869668	−	0.493638i	$-0.835667\pi$
$727$	−46.8976	−1.73934	−0.869668	+	0.493638i	$0.835667\pi$
$728$	−10.5778	−0.392038
$729$	1.00000	0.0370370
$730$	0	0
$731$	−43.0363	−1.59176
$732$	−23.9094	−0.883716
$733$	8.55200	0.315875	0.157938	−	0.987449i	$-0.449516\pi$
$733$	8.55200	0.315875	0.157938	+	0.987449i	$0.449516\pi$
$734$	14.9653	0.552378
$735$	0	0
$736$	8.67482	0.319758
$737$	−15.9766	−0.588507
$738$	−17.4383	−0.641912
$739$	33.0654	1.21633	0.608166	−	0.793810i	$-0.291906\pi$
$739$	33.0654	1.21633	0.608166	+	0.793810i	$0.291906\pi$
$740$	0	0
$741$	−31.5754	−1.15995
$742$	−20.6068	−0.756499
$743$	−16.2284	−0.595362	−0.297681	−	0.954665i	$-0.596213\pi$
$743$	−16.2284	−0.595362	−0.297681	+	0.954665i	$0.596213\pi$
$744$	−22.4140	−0.821738
$745$	0	0
$746$	54.6534	2.00100
$747$	6.31404	0.231019
$748$	59.6939	2.18263
$749$	10.6851	0.390425
$750$	0	0
$751$	−29.3312	−1.07031	−0.535155	−	0.844754i	$-0.679747\pi$
$751$	−29.3312	−1.07031	−0.535155	+	0.844754i	$0.679747\pi$
$752$	0.412149	0.0150295
$753$	15.7999	0.575781
$754$	12.7975	0.466057
$755$	0	0
$756$	2.86161	0.104076
$757$	−18.9100	−0.687295	−0.343647	−	0.939099i	$-0.611662\pi$
$757$	−18.9100	−0.687295	−0.343647	+	0.939099i	$0.611662\pi$
$758$	−17.1044	−0.621259
$759$	4.81401	0.174738
$760$	0	0
$761$	39.2493	1.42279	0.711393	−	0.702794i	$-0.248065\pi$
$761$	39.2493	1.42279	0.711393	+	0.702794i	$0.248065\pi$
$762$	−28.7606	−1.04189
$763$	3.70151	0.134004
$764$	−4.99076	−0.180559
$765$	0	0
$766$	19.4000	0.700952
$767$	−32.8491	−1.18611
$768$	−18.5721	−0.670162
$769$	−38.5999	−1.39195	−0.695973	−	0.718068i	$-0.745027\pi$
$769$	−38.5999	−1.39195	−0.695973	+	0.718068i	$0.745027\pi$
$770$	0	0
$771$	−8.34387	−0.300497
$772$	−58.0981	−2.09100
$773$	−51.5701	−1.85485	−0.927424	−	0.374012i	$-0.877982\pi$
$773$	−51.5701	−1.85485	−0.927424	+	0.374012i	$0.877982\pi$
$774$	15.8831	0.570907
$775$	0	0
$776$	40.4055	1.45047
$777$	6.21405	0.222928
$778$	35.1083	1.25869
$779$	−59.3977	−2.12814
$780$	0	0
$781$	32.5834	1.16593
$782$	−24.2104	−0.865761
$783$	−1.38052	−0.0493357
$784$	−2.58002	−0.0921437
$785$	0	0
$786$	−33.1178	−1.18127
$787$	5.02336	0.179064	0.0895318	−	0.995984i	$-0.471463\pi$
$787$	5.02336	0.179064	0.0895318	+	0.995984i	$0.471463\pi$
$788$	−73.0471	−2.60220
$789$	−10.0710	−0.358536
$790$	0	0
$791$	−8.15162	−0.289838
$792$	−8.78479	−0.312154
$793$	−28.8708	−1.02523
$794$	−36.5674	−1.29773
$795$	0	0
$796$	−36.2565	−1.28508
$797$	−44.4893	−1.57589	−0.787945	−	0.615745i	$-0.788855\pi$
$797$	−44.4893	−1.57589	−0.787945	+	0.615745i	$0.788855\pi$
$798$	15.6076	0.552503
$799$	6.25340	0.221229
$800$	0	0
$801$	−8.21367	−0.290216
$802$	−61.1327	−2.15867
$803$	−11.3963	−0.402167
$804$	−18.5191	−0.653120
$805$	0	0
$806$	−67.8751	−2.39080
$807$	13.1491	0.462872
$808$	57.2651	2.01458
$809$	−52.3087	−1.83908	−0.919538	−	0.393002i	$-0.871437\pi$
$809$	−52.3087	−1.83908	−0.919538	+	0.393002i	$0.871437\pi$
$810$	0	0
$811$	20.6701	0.725825	0.362912	−	0.931823i	$-0.381782\pi$
$811$	20.6701	0.725825	0.362912	+	0.931823i	$0.381782\pi$
$812$	−3.95051	−0.138636
$813$	7.77052	0.272524
$814$	−47.8405	−1.67681
$815$	0	0
$816$	2.57733	0.0902246
$817$	54.1006	1.89274
$818$	34.0450	1.19035
$819$	3.45543	0.120742
$820$	0	0
$821$	2.78712	0.0972713	0.0486356	−	0.998817i	$-0.484513\pi$
$821$	2.78712	0.0972713	0.0486356	+	0.998817i	$0.484513\pi$
$822$	8.64146	0.301406
$823$	32.9272	1.14777	0.573886	−	0.818935i	$-0.305435\pi$
$823$	32.9272	1.14777	0.573886	+	0.818935i	$0.305435\pi$
$824$	−20.7804	−0.723919
$825$	0	0
$826$	16.2372	0.564963
$827$	−15.0311	−0.522681	−0.261341	−	0.965247i	$-0.584165\pi$
$827$	−15.0311	−0.522681	−0.261341	+	0.965247i	$0.584165\pi$
$828$	5.58011	0.193922
$829$	−42.7779	−1.48574	−0.742869	−	0.669436i	$-0.766536\pi$
$829$	−42.7779	−1.48574	−0.742869	+	0.669436i	$0.766536\pi$
$830$	0	0
$831$	3.34256	0.115952
$832$	−51.2484	−1.77672
$833$	−39.1459	−1.35632
$834$	−1.24134	−0.0429839
$835$	0	0
$836$	−75.0407	−2.59534
$837$	7.32196	0.253084
$838$	−16.8776	−0.583026
$839$	−40.9986	−1.41543	−0.707714	−	0.706499i	$-0.750273\pi$
$839$	−40.9986	−1.41543	−0.707714	+	0.706499i	$0.750273\pi$
$840$	0	0
$841$	−27.0942	−0.934282
$842$	3.81931	0.131622
$843$	13.1042	0.451331
$844$	−44.9674	−1.54784
$845$	0	0
$846$	−2.30790	−0.0793472
$847$	−2.37841	−0.0817231
$848$	4.27771	0.146897
$849$	31.3949	1.07747
$850$	0	0
$851$	12.1173	0.415377
$852$	37.7687	1.29393
$853$	3.77117	0.129122	0.0645611	−	0.997914i	$-0.479435\pi$
$853$	3.77117	0.129122	0.0645611	+	0.997914i	$0.479435\pi$
$854$	14.2707	0.488334
$855$	0	0
$856$	−38.0220	−1.29957
$857$	−22.5822	−0.771395	−0.385697	−	0.922625i	$-0.626039\pi$
$857$	−22.5822	−0.771395	−0.385697	+	0.922625i	$0.626039\pi$
$858$	−26.6025	−0.908194
$859$	35.9250	1.22574	0.612872	−	0.790182i	$-0.290014\pi$
$859$	35.9250	1.22574	0.612872	+	0.790182i	$0.290014\pi$
$860$	0	0
$861$	6.50014	0.221524
$862$	−19.8791	−0.677085
$863$	−17.8731	−0.608409	−0.304204	−	0.952607i	$-0.598391\pi$
$863$	−17.8731	−0.608409	−0.304204	+	0.952607i	$0.598391\pi$
$864$	5.17121	0.175928
$865$	0	0
$866$	−1.09456	−0.0371948
$867$	22.1050	0.750726
$868$	20.9526	0.711178
$869$	47.7325	1.61921
$870$	0	0
$871$	−22.3621	−0.757710
$872$	−13.1715	−0.446043
$873$	−13.1992	−0.446725
$874$	30.4346	1.02947
$875$	0	0
$876$	−13.2099	−0.446321
$877$	5.96001	0.201255	0.100628	−	0.994924i	$-0.467915\pi$
$877$	5.96001	0.201255	0.100628	+	0.994924i	$0.467915\pi$
$878$	−50.2594	−1.69617
$879$	16.1948	0.546237
$880$	0	0
$881$	12.9463	0.436173	0.218087	−	0.975929i	$-0.430018\pi$
$881$	12.9463	0.436173	0.218087	+	0.975929i	$0.430018\pi$
$882$	14.4473	0.486466
$883$	42.3233	1.42429	0.712146	−	0.702031i	$-0.247723\pi$
$883$	42.3233	1.42429	0.712146	+	0.702031i	$0.247723\pi$
$884$	83.5520	2.81016
$885$	0	0
$886$	−71.4599	−2.40074
$887$	7.08289	0.237820	0.118910	−	0.992905i	$-0.462060\pi$
$887$	7.08289	0.237820	0.118910	+	0.992905i	$0.462060\pi$
$888$	−22.1122	−0.742036
$889$	10.7205	0.359555
$890$	0	0
$891$	2.86972	0.0961392
$892$	11.3854	0.381212
$893$	−7.86109	−0.263061
$894$	−32.2321	−1.07800
$895$	0	0
$896$	16.4346	0.549040
$897$	6.73805	0.224977
$898$	−44.6734	−1.49077
$899$	−10.1081	−0.337124
$900$	0	0
$901$	64.9043	2.16228
$902$	−50.0430	−1.66625
$903$	−5.92045	−0.197020
$904$	29.0068	0.964753
$905$	0	0
$906$	42.8027	1.42202
$907$	−20.5590	−0.682652	−0.341326	−	0.939945i	$-0.610876\pi$
$907$	−20.5590	−0.682652	−0.341326	+	0.939945i	$0.610876\pi$
$908$	89.4691	2.96914
$909$	−18.7067	−0.620463
$910$	0	0
$911$	−55.6550	−1.84393	−0.921966	−	0.387271i	$-0.873418\pi$
$911$	−55.6550	−1.84393	−0.921966	+	0.387271i	$0.873418\pi$
$912$	−3.23994	−0.107285
$913$	18.1195	0.599669
$914$	30.9289	1.02304
$915$	0	0
$916$	35.1444	1.16120
$917$	12.3447	0.407658
$918$	−14.4322	−0.476334
$919$	−51.2264	−1.68980	−0.844901	−	0.534922i	$-0.820341\pi$
$919$	−51.2264	−1.68980	−0.844901	+	0.534922i	$0.820341\pi$
$920$	0	0
$921$	25.8625	0.852199
$922$	−38.4238	−1.26542
$923$	45.6061	1.50114
$924$	8.21202	0.270155
$925$	0	0
$926$	14.0669	0.462268
$927$	6.78830	0.222957
$928$	−7.13895	−0.234348
$929$	5.49756	0.180369	0.0901846	−	0.995925i	$-0.471254\pi$
$929$	5.49756	0.180369	0.0901846	+	0.995925i	$0.471254\pi$
$930$	0	0
$931$	49.2099	1.61279
$932$	−16.8038	−0.550426
$933$	10.5509	0.345422
$934$	25.5467	0.835914
$935$	0	0
$936$	−12.2958	−0.401902
$937$	−9.57490	−0.312798	−0.156399	−	0.987694i	$-0.549989\pi$
$937$	−9.57490	−0.312798	−0.156399	+	0.987694i	$0.549989\pi$
$938$	11.0535	0.360909
$939$	13.3525	0.435742
$940$	0	0
$941$	17.6504	0.575387	0.287694	−	0.957722i	$-0.407111\pi$
$941$	17.6504	0.575387	0.287694	+	0.957722i	$0.407111\pi$
$942$	24.5820	0.800924
$943$	12.6752	0.412761
$944$	−3.37063	−0.109705
$945$	0	0
$946$	45.5801	1.48194
$947$	6.29609	0.204595	0.102298	−	0.994754i	$-0.467381\pi$
$947$	6.29609	0.204595	0.102298	+	0.994754i	$0.467381\pi$
$948$	55.3286	1.79699
$949$	−15.9511	−0.517795
$950$	0	0
$951$	2.87288	0.0931596
$952$	−16.4681	−0.533735
$953$	−18.1746	−0.588734	−0.294367	−	0.955692i	$-0.595109\pi$
$953$	−18.1746	−0.588734	−0.294367	+	0.955692i	$0.595109\pi$
$954$	−23.9538	−0.775533
$955$	0	0
$956$	22.4167	0.725009
$957$	−3.96170	−0.128064
$958$	22.4362	0.724881
$959$	−3.22111	−0.104015
$960$	0	0
$961$	22.6111	0.729391
$962$	−66.9610	−2.15891
$963$	12.4206	0.400249
$964$	−71.1326	−2.29103
$965$	0	0
$966$	−3.33059	−0.107160
$967$	−1.90365	−0.0612173	−0.0306086	−	0.999531i	$-0.509745\pi$
$967$	−1.90365	−0.0612173	−0.0306086	+	0.999531i	$0.509745\pi$
$968$	8.46337	0.272023
$969$	−49.1586	−1.57920
$970$	0	0
$971$	−38.9716	−1.25066	−0.625328	−	0.780362i	$-0.715035\pi$
$971$	−38.9716	−1.25066	−0.625328	+	0.780362i	$0.715035\pi$
$972$	3.32640	0.106694
$973$	0.462709	0.0148338
$974$	51.0627	1.63616
$975$	0	0
$976$	−2.96242	−0.0948249
$977$	21.3069	0.681669	0.340834	−	0.940123i	$-0.389290\pi$
$977$	21.3069	0.681669	0.340834	+	0.940123i	$0.389290\pi$
$978$	−22.1885	−0.709511
$979$	−23.5709	−0.753330
$980$	0	0
$981$	4.30272	0.137375
$982$	6.47929	0.206763
$983$	−20.2703	−0.646523	−0.323262	−	0.946310i	$-0.604779\pi$
$983$	−20.2703	−0.646523	−0.323262	+	0.946310i	$0.604779\pi$
$984$	−23.1302	−0.737364
$985$	0	0
$986$	19.9239	0.634508
$987$	0.860272	0.0273828
$988$	−105.033	−3.34153
$989$	−11.5448	−0.367104
$990$	0	0
$991$	−35.8010	−1.13726	−0.568628	−	0.822595i	$-0.692526\pi$
$991$	−35.8010	−1.13726	−0.568628	+	0.822595i	$0.692526\pi$
$992$	37.8634	1.20216
$993$	26.8952	0.853494
$994$	−22.5429	−0.715018
$995$	0	0
$996$	21.0030	0.665507
$997$	−42.5205	−1.34664	−0.673319	−	0.739352i	$-0.735132\pi$
$997$	−42.5205	−1.34664	−0.673319	+	0.739352i	$0.735132\pi$
$998$	−80.8423	−2.55902
$999$	7.22336	0.228537

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3525.2.a.z.1.1	✓	7	1.1	even	1	trivial
3525.2.a.ba.1.7	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)

\(f(q)\)	\(=\)	\(q-2.30790 q^{2} +1.00000 q^{3} +3.32640 q^{4} -2.30790 q^{6} +0.860272 q^{7} -3.06120 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.30790 q^{2} +1.00000 q^{3} +3.32640 q^{4} -2.30790 q^{6} +0.860272 q^{7} -3.06120 q^{8} +1.00000 q^{9} +2.86972 q^{11} +3.32640 q^{12} +4.01667 q^{13} -1.98542 q^{14} +0.412149 q^{16} +6.25340 q^{17} -2.30790 q^{18} -7.86109 q^{19} +0.860272 q^{21} -6.62302 q^{22} +1.67752 q^{23} -3.06120 q^{24} -9.27007 q^{26} +1.00000 q^{27} +2.86161 q^{28} -1.38052 q^{29} +7.32196 q^{31} +5.17121 q^{32} +2.86972 q^{33} -14.4322 q^{34} +3.32640 q^{36} +7.22336 q^{37} +18.1426 q^{38} +4.01667 q^{39} +7.55591 q^{41} -1.98542 q^{42} -6.88207 q^{43} +9.54584 q^{44} -3.87155 q^{46} +1.00000 q^{47} +0.412149 q^{48} -6.25993 q^{49} +6.25340 q^{51} +13.3611 q^{52} +10.3790 q^{53} -2.30790 q^{54} -2.63347 q^{56} -7.86109 q^{57} +3.18610 q^{58} -8.17819 q^{59} -7.18776 q^{61} -16.8984 q^{62} +0.860272 q^{63} -12.7589 q^{64} -6.62302 q^{66} -5.56732 q^{67} +20.8013 q^{68} +1.67752 q^{69} +11.3542 q^{71} -3.06120 q^{72} -3.97123 q^{73} -16.6708 q^{74} -26.1492 q^{76} +2.46874 q^{77} -9.27007 q^{78} +16.6332 q^{79} +1.00000 q^{81} -17.4383 q^{82} +6.31404 q^{83} +2.86161 q^{84} +15.8831 q^{86} -1.38052 q^{87} -8.78479 q^{88} -8.21367 q^{89} +3.45543 q^{91} +5.58011 q^{92} +7.32196 q^{93} -2.30790 q^{94} +5.17121 q^{96} -13.1992 q^{97} +14.4473 q^{98} +2.86972 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{2} + 7 q^{3} + 5 q^{4} - q^{6} + 7 q^{7} + 6 q^{8} + 7 q^{9}+O(q^{10}) \) 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

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