LMFDB - Embedded newform 3525.2.a.w.1.3

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:	$6$
Coefficient field:	6.6.414764096.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 11x^{4} + 20x^{3} + 29x^{2} - 42x - 11 $ x^6 - 2x^5 - 11x^4 + 20x^3 + 29x^2 - 42*x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 705)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.52844$ 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)

$f(q)$	$=$	$q-1.52844 q^{2} -1.00000 q^{3} +0.336137 q^{4} +1.52844 q^{6} -3.11578 q^{7} +2.54312 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.52844 q^{2} -1.00000 q^{3} +0.336137 q^{4} +1.52844 q^{6} -3.11578 q^{7} +2.54312 q^{8} +1.00000 q^{9} +2.66386 q^{11} -0.336137 q^{12} -5.03002 q^{13} +4.76229 q^{14} -4.55929 q^{16} +5.03002 q^{17} -1.52844 q^{18} +8.43772 q^{19} +3.11578 q^{21} -4.07156 q^{22} +3.11578 q^{23} -2.54312 q^{24} +7.68810 q^{26} -1.00000 q^{27} -1.04733 q^{28} +5.61965 q^{29} -9.86076 q^{31} +1.88237 q^{32} -2.66386 q^{33} -7.68810 q^{34} +0.336137 q^{36} -4.66386 q^{37} -12.8966 q^{38} +5.03002 q^{39} -9.06955 q^{41} -4.76229 q^{42} +0.895422 q^{44} -4.76229 q^{46} +1.00000 q^{47} +4.55929 q^{48} +2.70808 q^{49} -5.03002 q^{51} -1.69077 q^{52} +0.958459 q^{53} +1.52844 q^{54} -7.92380 q^{56} -8.43772 q^{57} -8.58931 q^{58} -3.70540 q^{59} +4.29192 q^{61} +15.0716 q^{62} -3.11578 q^{63} +6.24148 q^{64} +4.07156 q^{66} +15.0354 q^{67} +1.69077 q^{68} -3.11578 q^{69} -7.14690 q^{71} +2.54312 q^{72} +11.8608 q^{73} +7.12845 q^{74} +2.83623 q^{76} -8.30001 q^{77} -7.68810 q^{78} -10.9555 q^{79} +1.00000 q^{81} +13.8623 q^{82} -12.8754 q^{83} +1.04733 q^{84} -5.61965 q^{87} +6.77452 q^{88} +14.3632 q^{89} +15.6724 q^{91} +1.04733 q^{92} +9.86076 q^{93} -1.52844 q^{94} -1.88237 q^{96} -8.96387 q^{97} -4.13915 q^{98} +2.66386 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{2} - 6 q^{3} + 14 q^{4} + 2 q^{6} - 4 q^{7} - 6 q^{8} + 6 q^{9}+O(q^{10}) $ 6 * q - 2 * q^2 - 6 * q^3 + 14 * q^4 + 2 * q^6 - 4 * q^7 - 6 * q^8 + 6 * q^9	$ 6 q - 2 q^{2} - 6 q^{3} + 14 q^{4} + 2 q^{6} - 4 q^{7} - 6 q^{8} + 6 q^{9} + 4 q^{11} - 14 q^{12} - 6 q^{13} - 2 q^{14} + 26 q^{16} + 6 q^{17} - 2 q^{18} + 10 q^{19} + 4 q^{21} + 4 q^{22} + 4 q^{23} + 6 q^{24} + 8 q^{26} - 6 q^{27} - 4 q^{28} + 8 q^{29} + 8 q^{31} - 14 q^{32} - 4 q^{33} - 8 q^{34} + 14 q^{36} - 16 q^{37} + 12 q^{38} + 6 q^{39} - 12 q^{41} + 2 q^{42} - 36 q^{44} + 2 q^{46} + 6 q^{47} - 26 q^{48} + 18 q^{49} - 6 q^{51} + 10 q^{52} + 10 q^{53} + 2 q^{54} - 26 q^{56} - 10 q^{57} + 26 q^{58} - 6 q^{59} + 24 q^{61} + 44 q^{62} - 4 q^{63} + 54 q^{64} - 4 q^{66} + 8 q^{67} - 10 q^{68} - 4 q^{69} + 26 q^{71} - 6 q^{72} + 4 q^{73} + 30 q^{76} - 8 q^{77} - 8 q^{78} + 24 q^{79} + 6 q^{81} + 82 q^{82} + 4 q^{83} + 4 q^{84} - 8 q^{87} + 16 q^{88} - 20 q^{89} - 10 q^{91} + 4 q^{92} - 8 q^{93} - 2 q^{94} + 14 q^{96} - 24 q^{98} + 4 q^{99}+O(q^{100}) $ 6 * q - 2 * q^2 - 6 * q^3 + 14 * q^4 + 2 * q^6 - 4 * q^7 - 6 * q^8 + 6 * q^9 + 4 * q^11 - 14 * q^12 - 6 * q^13 - 2 * q^14 + 26 * q^16 + 6 * q^17 - 2 * q^18 + 10 * q^19 + 4 * q^21 + 4 * q^22 + 4 * q^23 + 6 * q^24 + 8 * q^26 - 6 * q^27 - 4 * q^28 + 8 * q^29 + 8 * q^31 - 14 * q^32 - 4 * q^33 - 8 * q^34 + 14 * q^36 - 16 * q^37 + 12 * q^38 + 6 * q^39 - 12 * q^41 + 2 * q^42 - 36 * q^44 + 2 * q^46 + 6 * q^47 - 26 * q^48 + 18 * q^49 - 6 * q^51 + 10 * q^52 + 10 * q^53 + 2 * q^54 - 26 * q^56 - 10 * q^57 + 26 * q^58 - 6 * q^59 + 24 * q^61 + 44 * q^62 - 4 * q^63 + 54 * q^64 - 4 * q^66 + 8 * q^67 - 10 * q^68 - 4 * q^69 + 26 * q^71 - 6 * q^72 + 4 * q^73 + 30 * q^76 - 8 * q^77 - 8 * q^78 + 24 * q^79 + 6 * q^81 + 82 * q^82 + 4 * q^83 + 4 * q^84 - 8 * q^87 + 16 * q^88 - 20 * q^89 - 10 * q^91 + 4 * q^92 - 8 * q^93 - 2 * q^94 + 14 * q^96 - 24 * q^98 + 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.52844	−1.08077	−0.540386	−	0.841417i	$-0.681722\pi$
$2$	−1.52844	−1.08077	−0.540386	+	0.841417i	$0.681722\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0.336137	0.168068
$5$	0	0
$5$	0	0
$6$	1.52844	0.623984
$7$	−3.11578	−1.17765	−0.588827	−	0.808259i	$-0.700410\pi$
$7$	−3.11578	−1.17765	−0.588827	+	0.808259i	$0.700410\pi$
$8$	2.54312	0.899129
$9$	1.00000	0.333333
$10$	0	0
$11$	2.66386	0.803185	0.401593	−	0.915818i	$-0.368457\pi$
$11$	2.66386	0.803185	0.401593	+	0.915818i	$0.368457\pi$
$12$	−0.336137	−0.0970343
$13$	−5.03002	−1.39508	−0.697538	−	0.716547i	$-0.745721\pi$
$13$	−5.03002	−1.39508	−0.697538	+	0.716547i	$0.745721\pi$
$14$	4.76229	1.27278
$15$	0	0
$16$	−4.55929	−1.13982
$17$	5.03002	1.21996	0.609980	−	0.792417i	$-0.291178\pi$
$17$	5.03002	1.21996	0.609980	+	0.792417i	$0.291178\pi$
$18$	−1.52844	−0.360257
$19$	8.43772	1.93575	0.967873	−	0.251440i	$-0.0809042\pi$
$19$	8.43772	1.93575	0.967873	+	0.251440i	$0.0809042\pi$
$20$	0	0
$21$	3.11578	0.679919
$22$	−4.07156	−0.868060
$23$	3.11578	0.649685	0.324842	−	0.945768i	$-0.394689\pi$
$23$	3.11578	0.649685	0.324842	+	0.945768i	$0.394689\pi$
$24$	−2.54312	−0.519112
$25$	0	0
$26$	7.68810	1.50776
$27$	−1.00000	−0.192450
$28$	−1.04733	−0.197926
$29$	5.61965	1.04354	0.521771	−	0.853086i	$-0.325272\pi$
$29$	5.61965	1.04354	0.521771	+	0.853086i	$0.325272\pi$
$30$	0	0
$31$	−9.86076	−1.77105	−0.885523	−	0.464596i	$-0.846199\pi$
$31$	−9.86076	−1.77105	−0.885523	+	0.464596i	$0.846199\pi$
$32$	1.88237	0.332759
$33$	−2.66386	−0.463719
$34$	−7.68810	−1.31850
$35$	0	0
$36$	0.336137	0.0560228
$37$	−4.66386	−0.766734	−0.383367	−	0.923596i	$-0.625236\pi$
$37$	−4.66386	−0.766734	−0.383367	+	0.923596i	$0.625236\pi$
$38$	−12.8966	−2.09210
$39$	5.03002	0.805448
$40$	0	0
$41$	−9.06955	−1.41643	−0.708213	−	0.705999i	$-0.750498\pi$
$41$	−9.06955	−1.41643	−0.708213	+	0.705999i	$0.750498\pi$
$42$	−4.76229	−0.734837
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0.895422	0.134990
$45$	0	0
$46$	−4.76229	−0.702161
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	4.55929	0.658076
$49$	2.70808	0.386869
$50$	0	0
$51$	−5.03002	−0.704344
$52$	−1.69077	−0.234468
$53$	0.958459	0.131654	0.0658272	−	0.997831i	$-0.479031\pi$
$53$	0.958459	0.131654	0.0658272	+	0.997831i	$0.479031\pi$
$54$	1.52844	0.207995
$55$	0	0
$56$	−7.92380	−1.05886
$57$	−8.43772	−1.11760
$58$	−8.58931	−1.12783
$59$	−3.70540	−0.482403	−0.241201	−	0.970475i	$-0.577541\pi$
$59$	−3.70540	−0.482403	−0.241201	+	0.970475i	$0.577541\pi$
$60$	0	0
$61$	4.29192	0.549524	0.274762	−	0.961512i	$-0.411401\pi$
$61$	4.29192	0.549524	0.274762	+	0.961512i	$0.411401\pi$
$62$	15.0716	1.91410
$63$	−3.11578	−0.392551
$64$	6.24148	0.780185
$65$	0	0
$66$	4.07156	0.501175
$67$	15.0354	1.83687	0.918435	−	0.395571i	$-0.129453\pi$
$67$	15.0354	1.83687	0.918435	+	0.395571i	$0.129453\pi$
$68$	1.69077	0.205036
$69$	−3.11578	−0.375096
$70$	0	0
$71$	−7.14690	−0.848181	−0.424091	−	0.905620i	$-0.639406\pi$
$71$	−7.14690	−0.848181	−0.424091	+	0.905620i	$0.639406\pi$
$72$	2.54312	0.299710
$73$	11.8608	1.38820	0.694099	−	0.719880i	$-0.255803\pi$
$73$	11.8608	1.38820	0.694099	+	0.719880i	$0.255803\pi$
$74$	7.12845	0.828665
$75$	0	0
$76$	2.83623	0.325337
$77$	−8.30001	−0.945874
$78$	−7.68810	−0.870506
$79$	−10.9555	−1.23259	−0.616293	−	0.787517i	$-0.711366\pi$
$79$	−10.9555	−1.23259	−0.616293	+	0.787517i	$0.711366\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	13.8623	1.53083
$83$	−12.8754	−1.41326	−0.706632	−	0.707582i	$-0.749786\pi$
$83$	−12.8754	−1.41326	−0.706632	+	0.707582i	$0.749786\pi$
$84$	1.04733	0.114273
$85$	0	0
$86$	0	0
$87$	−5.61965	−0.602489
$88$	6.77452	0.722167
$89$	14.3632	1.52249	0.761246	−	0.648463i	$-0.224588\pi$
$89$	14.3632	1.52249	0.761246	+	0.648463i	$0.224588\pi$
$90$	0	0
$91$	15.6724	1.64292
$92$	1.04733	0.109191
$93$	9.86076	1.02251
$94$	−1.52844	−0.157647
$95$	0	0
$96$	−1.88237	−0.192118
$97$	−8.96387	−0.910143	−0.455072	−	0.890455i	$-0.650386\pi$
$97$	−8.96387	−0.910143	−0.455072	+	0.890455i	$0.650386\pi$
$98$	−4.13915	−0.418117
$99$	2.66386	0.267728
$100$	0	0
$101$	−9.51306	−0.946585	−0.473292	−	0.880905i	$-0.656935\pi$
$101$	−9.51306	−0.946585	−0.473292	+	0.880905i	$0.656935\pi$
$102$	7.68810	0.761235
$103$	7.74494	0.763131	0.381566	−	0.924342i	$-0.375385\pi$
$103$	7.74494	0.763131	0.381566	+	0.924342i	$0.375385\pi$
$104$	−12.7919	−1.25435
$105$	0	0
$106$	−1.46495	−0.142288
$107$	0.319993	0.0309349	0.0154674	−	0.999880i	$-0.495076\pi$
$107$	0.319993	0.0309349	0.0154674	+	0.999880i	$0.495076\pi$
$108$	−0.336137	−0.0323448
$109$	−10.9470	−1.04853	−0.524266	−	0.851554i	$-0.675660\pi$
$109$	−10.9470	−1.04853	−0.524266	+	0.851554i	$0.675660\pi$
$110$	0	0
$111$	4.66386	0.442674
$112$	14.2057	1.34232
$113$	14.8670	1.39857	0.699286	−	0.714842i	$-0.253501\pi$
$113$	14.8670	1.39857	0.699286	+	0.714842i	$0.253501\pi$
$114$	12.8966	1.20787
$115$	0	0
$116$	1.88897	0.175386
$117$	−5.03002	−0.465026
$118$	5.66350	0.521367
$119$	−15.6724	−1.43669
$120$	0	0
$121$	−3.90383	−0.354894
$122$	−6.55995	−0.593910
$123$	9.06955	0.817774
$124$	−3.31456	−0.297657
$125$	0	0
$126$	4.76229	0.424258
$127$	−5.36988	−0.476500	−0.238250	−	0.971204i	$-0.576574\pi$
$127$	−5.36988	−0.476500	−0.238250	+	0.971204i	$0.576574\pi$
$128$	−13.3045	−1.17596
$129$	0	0
$130$	0	0
$131$	−3.06841	−0.268088	−0.134044	−	0.990975i	$-0.542796\pi$
$131$	−3.06841	−0.268088	−0.134044	+	0.990975i	$0.542796\pi$
$132$	−0.895422	−0.0779365
$133$	−26.2901	−2.27964
$134$	−22.9808	−1.98524
$135$	0	0
$136$	12.7919	1.09690
$137$	−8.42841	−0.720088	−0.360044	−	0.932935i	$-0.617238\pi$
$137$	−8.42841	−0.720088	−0.360044	+	0.932935i	$0.617238\pi$
$138$	4.76229	0.405393
$139$	3.51605	0.298228	0.149114	−	0.988820i	$-0.452358\pi$
$139$	3.51605	0.298228	0.149114	+	0.988820i	$0.452358\pi$
$140$	0	0
$141$	−1.00000	−0.0842152
$142$	10.9236	0.916691
$143$	−13.3993	−1.12050
$144$	−4.55929	−0.379940
$145$	0	0
$146$	−18.1285	−1.50032
$147$	−2.70808	−0.223359
$148$	−1.56770	−0.128864
$149$	7.20994	0.590661	0.295331	−	0.955395i	$-0.404570\pi$
$149$	7.20994	0.590661	0.295331	+	0.955395i	$0.404570\pi$
$150$	0	0
$151$	19.8792	1.61775	0.808874	−	0.587982i	$-0.200078\pi$
$151$	19.8792	1.61775	0.808874	+	0.587982i	$0.200078\pi$
$152$	21.4581	1.74048
$153$	5.03002	0.406653
$154$	12.6861	1.02227
$155$	0	0
$156$	1.69077	0.135370
$157$	−10.9292	−0.872243	−0.436121	−	0.899888i	$-0.643648\pi$
$157$	−10.9292	−0.872243	−0.436121	+	0.899888i	$0.643648\pi$
$158$	16.7448	1.33214
$159$	−0.958459	−0.0760107
$160$	0	0
$161$	−9.70808	−0.765104
$162$	−1.52844	−0.120086
$163$	−19.9046	−1.55905	−0.779525	−	0.626371i	$-0.784539\pi$
$163$	−19.9046	−1.55905	−0.779525	+	0.626371i	$0.784539\pi$
$164$	−3.04861	−0.238056
$165$	0	0
$166$	19.6794	1.52742
$167$	22.5373	1.74399	0.871994	−	0.489516i	$-0.162827\pi$
$167$	22.5373	1.74399	0.871994	+	0.489516i	$0.162827\pi$
$168$	7.92380	0.611334
$169$	12.3011	0.946239
$170$	0	0
$171$	8.43772	0.645249
$172$	0	0
$173$	−5.54687	−0.421720	−0.210860	−	0.977516i	$-0.567626\pi$
$173$	−5.54687	−0.421720	−0.210860	+	0.977516i	$0.567626\pi$
$174$	8.58931	0.651154
$175$	0	0
$176$	−12.1453	−0.915487
$177$	3.70540	0.278515
$178$	−21.9533	−1.64547
$179$	4.74389	0.354575	0.177287	−	0.984159i	$-0.443268\pi$
$179$	4.74389	0.354575	0.177287	+	0.984159i	$0.443268\pi$
$180$	0	0
$181$	4.43542	0.329682	0.164841	−	0.986320i	$-0.447289\pi$
$181$	4.43542	0.329682	0.164841	+	0.986320i	$0.447289\pi$
$182$	−23.9544	−1.77562
$183$	−4.29192	−0.317268
$184$	7.92380	0.584150
$185$	0	0
$186$	−15.0716	−1.10510
$187$	13.3993	0.979853
$188$	0.336137	0.0245153
$189$	3.11578	0.226640
$190$	0	0
$191$	−0.931595	−0.0674078	−0.0337039	−	0.999432i	$-0.510730\pi$
$191$	−0.931595	−0.0674078	−0.0337039	+	0.999432i	$0.510730\pi$
$192$	−6.24148	−0.450440
$193$	8.18076	0.588864	0.294432	−	0.955672i	$-0.404870\pi$
$193$	8.18076	0.588864	0.294432	+	0.955672i	$0.404870\pi$
$194$	13.7008	0.983658
$195$	0	0
$196$	0.910285	0.0650204
$197$	3.79293	0.270235	0.135118	−	0.990830i	$-0.456859\pi$
$197$	3.79293	0.270235	0.135118	+	0.990830i	$0.456859\pi$
$198$	−4.07156	−0.289353
$199$	17.3669	1.23111	0.615553	−	0.788095i	$-0.288933\pi$
$199$	17.3669	1.23111	0.615553	+	0.788095i	$0.288933\pi$
$200$	0	0
$201$	−15.0354	−1.06052
$202$	14.5402	1.02304
$203$	−17.5096	−1.22893
$204$	−1.69077	−0.118378
$205$	0	0
$206$	−11.8377	−0.824771
$207$	3.11578	0.216562
$208$	22.9333	1.59014
$209$	22.4769	1.55476
$210$	0	0
$211$	−16.6762	−1.14803	−0.574017	−	0.818843i	$-0.694616\pi$
$211$	−16.6762	−1.14803	−0.574017	+	0.818843i	$0.694616\pi$
$212$	0.322173	0.0221269
$213$	7.14690	0.489698
$214$	−0.489091	−0.0334335
$215$	0	0
$216$	−2.54312	−0.173037
$217$	30.7240	2.08568
$218$	16.7319	1.13322
$219$	−11.8608	−0.801476
$220$	0	0
$221$	−25.3011	−1.70194
$222$	−7.12845	−0.478430
$223$	15.4877	1.03713	0.518567	−	0.855037i	$-0.326466\pi$
$223$	15.4877	1.03713	0.518567	+	0.855037i	$0.326466\pi$
$224$	−5.86504	−0.391874
$225$	0	0
$226$	−22.7234	−1.51154
$227$	22.8120	1.51408	0.757042	−	0.653366i	$-0.226644\pi$
$227$	22.8120	1.51408	0.757042	+	0.653366i	$0.226644\pi$
$228$	−2.83623	−0.187834
$229$	22.7755	1.50505	0.752523	−	0.658566i	$-0.228837\pi$
$229$	22.7755	1.50505	0.752523	+	0.658566i	$0.228837\pi$
$230$	0	0
$231$	8.30001	0.546101
$232$	14.2914	0.938279
$233$	17.2393	1.12938	0.564692	−	0.825302i	$-0.308995\pi$
$233$	17.2393	1.12938	0.564692	+	0.825302i	$0.308995\pi$
$234$	7.68810	0.502587
$235$	0	0
$236$	−1.24552	−0.0810766
$237$	10.9555	0.711634
$238$	23.9544	1.55273
$239$	15.2115	0.983952	0.491976	−	0.870609i	$-0.336275\pi$
$239$	15.2115	0.983952	0.491976	+	0.870609i	$0.336275\pi$
$240$	0	0
$241$	18.7510	1.20786	0.603930	−	0.797038i	$-0.293601\pi$
$241$	18.7510	1.20786	0.603930	+	0.797038i	$0.293601\pi$
$242$	5.96678	0.383559
$243$	−1.00000	−0.0641500
$244$	1.44267	0.0923576
$245$	0	0
$246$	−13.8623	−0.883827
$247$	−42.4419	−2.70051
$248$	−25.0771	−1.59240
$249$	12.8754	0.815948
$250$	0	0
$251$	1.77476	0.112022	0.0560111	−	0.998430i	$-0.482162\pi$
$251$	1.77476	0.112022	0.0560111	+	0.998430i	$0.482162\pi$
$252$	−1.04733	−0.0659754
$253$	8.30001	0.521817
$254$	8.20756	0.514988
$255$	0	0
$256$	7.85217	0.490761
$257$	15.9731	0.996372	0.498186	−	0.867070i	$-0.334000\pi$
$257$	15.9731	0.996372	0.498186	+	0.867070i	$0.334000\pi$
$258$	0	0
$259$	14.5316	0.902948
$260$	0	0
$261$	5.61965	0.347847
$262$	4.68988	0.289742
$263$	−26.3400	−1.62420	−0.812098	−	0.583521i	$-0.801675\pi$
$263$	−26.3400	−1.62420	−0.812098	+	0.583521i	$0.801675\pi$
$264$	−6.77452	−0.416943
$265$	0	0
$266$	40.1829	2.46377
$267$	−14.3632	−0.879011
$268$	5.05396	0.308720
$269$	−13.5477	−0.826019	−0.413009	−	0.910727i	$-0.635522\pi$
$269$	−13.5477	−0.826019	−0.413009	+	0.910727i	$0.635522\pi$
$270$	0	0
$271$	4.05373	0.246246	0.123123	−	0.992391i	$-0.460709\pi$
$271$	4.05373	0.246246	0.123123	+	0.992391i	$0.460709\pi$
$272$	−22.9333	−1.39054
$273$	−15.6724	−0.948539
$274$	12.8823	0.778251
$275$	0	0
$276$	−1.04733	−0.0630417
$277$	2.57543	0.154743	0.0773713	−	0.997002i	$-0.475347\pi$
$277$	2.57543	0.154743	0.0773713	+	0.997002i	$0.475347\pi$
$278$	−5.37409	−0.322316
$279$	−9.86076	−0.590348
$280$	0	0
$281$	−20.8589	−1.24434	−0.622170	−	0.782882i	$-0.713749\pi$
$281$	−20.8589	−1.24434	−0.622170	+	0.782882i	$0.713749\pi$
$282$	1.52844	0.0910174
$283$	14.9101	0.886316	0.443158	−	0.896443i	$-0.353858\pi$
$283$	14.9101	0.886316	0.443158	+	0.896443i	$0.353858\pi$
$284$	−2.40234	−0.142552
$285$	0	0
$286$	20.4800	1.21101
$287$	28.2587	1.66806
$288$	1.88237	0.110920
$289$	8.30111	0.488301
$290$	0	0
$291$	8.96387	0.525472
$292$	3.98684	0.233312
$293$	−9.41310	−0.549919	−0.274960	−	0.961456i	$-0.588665\pi$
$293$	−9.41310	−0.549919	−0.274960	+	0.961456i	$0.588665\pi$
$294$	4.13915	0.241400
$295$	0	0
$296$	−11.8608	−0.689393
$297$	−2.66386	−0.154573
$298$	−11.0200	−0.638370
$299$	−15.6724	−0.906360
$300$	0	0
$301$	0	0
$302$	−30.3842	−1.74842
$303$	9.51306	0.546511
$304$	−38.4700	−2.20640
$305$	0	0
$306$	−7.68810	−0.439499
$307$	13.5134	0.771249	0.385625	−	0.922656i	$-0.373986\pi$
$307$	13.5134	0.771249	0.385625	+	0.922656i	$0.373986\pi$
$308$	−2.78994	−0.158971
$309$	−7.74494	−0.440594
$310$	0	0
$311$	24.9323	1.41378	0.706891	−	0.707322i	$-0.250097\pi$
$311$	24.9323	1.41378	0.706891	+	0.707322i	$0.250097\pi$
$312$	12.7919	0.724201
$313$	21.1922	1.19786	0.598928	−	0.800803i	$-0.295594\pi$
$313$	21.1922	1.19786	0.598928	+	0.800803i	$0.295594\pi$
$314$	16.7046	0.942695
$315$	0	0
$316$	−3.68253	−0.207159
$317$	−11.6193	−0.652606	−0.326303	−	0.945265i	$-0.605803\pi$
$317$	−11.6193	−0.652606	−0.326303	+	0.945265i	$0.605803\pi$
$318$	1.46495	0.0821503
$319$	14.9700	0.838157
$320$	0	0
$321$	−0.319993	−0.0178603
$322$	14.8382	0.826903
$323$	42.4419	2.36153
$324$	0.336137	0.0186743
$325$	0	0
$326$	30.4231	1.68498
$327$	10.9470	0.605371
$328$	−23.0650	−1.27355
$329$	−3.11578	−0.171778
$330$	0	0
$331$	14.0022	0.769631	0.384815	−	0.922994i	$-0.374265\pi$
$331$	14.0022	0.769631	0.384815	+	0.922994i	$0.374265\pi$
$332$	−4.32791	−0.237525
$333$	−4.66386	−0.255578
$334$	−34.4470	−1.88485
$335$	0	0
$336$	−14.2057	−0.774986
$337$	−30.1370	−1.64167	−0.820834	−	0.571167i	$-0.806491\pi$
$337$	−30.1370	−1.64167	−0.820834	+	0.571167i	$0.806491\pi$
$338$	−18.8015	−1.02267
$339$	−14.8670	−0.807466
$340$	0	0
$341$	−26.2677	−1.42248
$342$	−12.8966	−0.697367
$343$	13.3727	0.722057
$344$	0	0
$345$	0	0
$346$	8.47807	0.455784
$347$	−16.2010	−0.869713	−0.434857	−	0.900500i	$-0.643201\pi$
$347$	−16.2010	−0.869713	−0.434857	+	0.900500i	$0.643201\pi$
$348$	−1.88897	−0.101259
$349$	−4.26621	−0.228365	−0.114183	−	0.993460i	$-0.536425\pi$
$349$	−4.26621	−0.228365	−0.114183	+	0.993460i	$0.536425\pi$
$350$	0	0
$351$	5.03002	0.268483
$352$	5.01437	0.267267
$353$	−11.2178	−0.597066	−0.298533	−	0.954399i	$-0.596497\pi$
$353$	−11.2178	−0.597066	−0.298533	+	0.954399i	$0.596497\pi$
$354$	−5.66350	−0.301012
$355$	0	0
$356$	4.82798	0.255883
$357$	15.6724	0.829473
$358$	−7.25076	−0.383215
$359$	23.3092	1.23021	0.615106	−	0.788444i	$-0.289113\pi$
$359$	23.3092	1.23021	0.615106	+	0.788444i	$0.289113\pi$
$360$	0	0
$361$	52.1951	2.74711
$362$	−6.77928	−0.356311
$363$	3.90383	0.204898
$364$	5.26808	0.276122
$365$	0	0
$366$	6.55995	0.342894
$367$	24.2748	1.26713	0.633567	−	0.773688i	$-0.281590\pi$
$367$	24.2748	1.26713	0.633567	+	0.773688i	$0.281590\pi$
$368$	−14.2057	−0.740525
$369$	−9.06955	−0.472142
$370$	0	0
$371$	−2.98635	−0.155043
$372$	3.31456	0.171852
$373$	6.09003	0.315330	0.157665	−	0.987493i	$-0.449603\pi$
$373$	6.09003	0.315330	0.157665	+	0.987493i	$0.449603\pi$
$374$	−20.4800	−1.05900
$375$	0	0
$376$	2.54312	0.131151
$377$	−28.2669	−1.45582
$378$	−4.76229	−0.244946
$379$	17.3523	0.891329	0.445664	−	0.895200i	$-0.352967\pi$
$379$	17.3523	0.891329	0.445664	+	0.895200i	$0.352967\pi$
$380$	0	0
$381$	5.36988	0.275107
$382$	1.42389	0.0728525
$383$	16.4053	0.838273	0.419136	−	0.907923i	$-0.362333\pi$
$383$	16.4053	0.838273	0.419136	+	0.907923i	$0.362333\pi$
$384$	13.3045	0.678941
$385$	0	0
$386$	−12.5038	−0.636427
$387$	0	0
$388$	−3.01309	−0.152966
$389$	0.943355	0.0478300	0.0239150	−	0.999714i	$-0.492387\pi$
$389$	0.943355	0.0478300	0.0239150	+	0.999714i	$0.492387\pi$
$390$	0	0
$391$	15.6724	0.792589
$392$	6.88697	0.347845
$393$	3.06841	0.154781
$394$	−5.79728	−0.292063
$395$	0	0
$396$	0.895422	0.0449966
$397$	20.0762	1.00760	0.503798	−	0.863821i	$-0.331935\pi$
$397$	20.0762	1.00760	0.503798	+	0.863821i	$0.331935\pi$
$398$	−26.5443	−1.33055
$399$	26.2901	1.31615
$400$	0	0
$401$	1.42992	0.0714070	0.0357035	−	0.999362i	$-0.488633\pi$
$401$	1.42992	0.0714070	0.0357035	+	0.999362i	$0.488633\pi$
$402$	22.9808	1.14618
$403$	49.5998	2.47074
$404$	−3.19769	−0.159091
$405$	0	0
$406$	26.7624	1.32819
$407$	−12.4239	−0.615830
$408$	−12.7919	−0.633296
$409$	−34.8124	−1.72136	−0.860681	−	0.509145i	$-0.829962\pi$
$409$	−34.8124	−1.72136	−0.860681	+	0.509145i	$0.829962\pi$
$410$	0	0
$411$	8.42841	0.415743
$412$	2.60336	0.128258
$413$	11.5452	0.568103
$414$	−4.76229	−0.234054
$415$	0	0
$416$	−9.46834	−0.464224
$417$	−3.51605	−0.172182
$418$	−34.3547	−1.68034
$419$	31.0798	1.51835	0.759175	−	0.650887i	$-0.225603\pi$
$419$	31.0798	1.51835	0.759175	+	0.650887i	$0.225603\pi$
$420$	0	0
$421$	6.93452	0.337968	0.168984	−	0.985619i	$-0.445951\pi$
$421$	6.93452	0.337968	0.168984	+	0.985619i	$0.445951\pi$
$422$	25.4886	1.24076
$423$	1.00000	0.0486217
$424$	2.43748	0.118374
$425$	0	0
$426$	−10.9236	−0.529252
$427$	−13.3727	−0.647149
$428$	0.107561	0.00519917
$429$	13.3993	0.646924
$430$	0	0
$431$	36.1517	1.74137	0.870683	−	0.491844i	$-0.163677\pi$
$431$	36.1517	1.74137	0.870683	+	0.491844i	$0.163677\pi$
$432$	4.55929	0.219359
$433$	8.83317	0.424495	0.212247	−	0.977216i	$-0.431922\pi$
$433$	8.83317	0.424495	0.212247	+	0.977216i	$0.431922\pi$
$434$	−46.9598	−2.25414
$435$	0	0
$436$	−3.67969	−0.176225
$437$	26.2901	1.25762
$438$	18.1285	0.866213
$439$	−14.1162	−0.673729	−0.336865	−	0.941553i	$-0.609366\pi$
$439$	−14.1162	−0.673729	−0.336865	+	0.941553i	$0.609366\pi$
$440$	0	0
$441$	2.70808	0.128956
$442$	38.6713	1.83941
$443$	−21.3002	−1.01200	−0.506002	−	0.862533i	$-0.668877\pi$
$443$	−21.3002	−1.01200	−0.506002	+	0.862533i	$0.668877\pi$
$444$	1.56770	0.0743995
$445$	0	0
$446$	−23.6721	−1.12091
$447$	−7.20994	−0.341018
$448$	−19.4471	−0.918788
$449$	14.0828	0.664607	0.332303	−	0.943173i	$-0.392174\pi$
$449$	14.0828	0.664607	0.332303	+	0.943173i	$0.392174\pi$
$450$	0	0
$451$	−24.1601	−1.13765
$452$	4.99735	0.235056
$453$	−19.8792	−0.934007
$454$	−34.8668	−1.63638
$455$	0	0
$456$	−21.4581	−1.00487
$457$	−8.39150	−0.392538	−0.196269	−	0.980550i	$-0.562883\pi$
$457$	−8.39150	−0.392538	−0.196269	+	0.980550i	$0.562883\pi$
$458$	−34.8110	−1.62661
$459$	−5.03002	−0.234781
$460$	0	0
$461$	15.2745	0.711404	0.355702	−	0.934599i	$-0.384242\pi$
$461$	15.2745	0.711404	0.355702	+	0.934599i	$0.384242\pi$
$462$	−12.6861	−0.590210
$463$	−29.7434	−1.38229	−0.691146	−	0.722715i	$-0.742894\pi$
$463$	−29.7434	−1.38229	−0.691146	+	0.722715i	$0.742894\pi$
$464$	−25.6216	−1.18945
$465$	0	0
$466$	−26.3493	−1.22061
$467$	3.29265	0.152366	0.0761828	−	0.997094i	$-0.475727\pi$
$467$	3.29265	0.152366	0.0761828	+	0.997094i	$0.475727\pi$
$468$	−1.69077	−0.0781561
$469$	−46.8471	−2.16320
$470$	0	0
$471$	10.9292	0.503589
$472$	−9.42329	−0.433742
$473$	0	0
$474$	−16.7448	−0.769114
$475$	0	0
$476$	−5.26808	−0.241462
$477$	0.958459	0.0438848
$478$	−23.2499	−1.06343
$479$	24.6208	1.12496	0.562478	−	0.826812i	$-0.309848\pi$
$479$	24.6208	1.12496	0.562478	+	0.826812i	$0.309848\pi$
$480$	0	0
$481$	23.4593	1.06965
$482$	−28.6599	−1.30542
$483$	9.70808	0.441733
$484$	−1.31222	−0.0596464
$485$	0	0
$486$	1.52844	0.0693316
$487$	−29.4221	−1.33324	−0.666621	−	0.745396i	$-0.732260\pi$
$487$	−29.4221	−1.33324	−0.666621	+	0.745396i	$0.732260\pi$
$488$	10.9149	0.494093
$489$	19.9046	0.900118
$490$	0	0
$491$	−8.22405	−0.371146	−0.185573	−	0.982630i	$-0.559414\pi$
$491$	−8.22405	−0.371146	−0.185573	+	0.982630i	$0.559414\pi$
$492$	3.04861	0.137442
$493$	28.2669	1.27308
$494$	64.8700	2.91864
$495$	0	0
$496$	44.9580	2.01868
$497$	22.2682	0.998864
$498$	−19.6794	−0.881854
$499$	39.1301	1.75170	0.875852	−	0.482579i	$-0.160300\pi$
$499$	39.1301	1.75170	0.875852	+	0.482579i	$0.160300\pi$
$500$	0	0
$501$	−22.5373	−1.00689
$502$	−2.71263	−0.121070
$503$	18.2344	0.813029	0.406515	−	0.913644i	$-0.366744\pi$
$503$	18.2344	0.813029	0.406515	+	0.913644i	$0.366744\pi$
$504$	−7.92380	−0.352954
$505$	0	0
$506$	−12.6861	−0.563965
$507$	−12.3011	−0.546312
$508$	−1.80501	−0.0800845
$509$	−18.0666	−0.800787	−0.400394	−	0.916343i	$-0.631127\pi$
$509$	−18.0666	−0.800787	−0.400394	+	0.916343i	$0.631127\pi$
$510$	0	0
$511$	−36.9555	−1.63482
$512$	14.6074	0.645561
$513$	−8.43772	−0.372534
$514$	−24.4139	−1.07685
$515$	0	0
$516$	0	0
$517$	2.66386	0.117157
$518$	−22.2107	−0.975881
$519$	5.54687	0.243480
$520$	0	0
$521$	9.38242	0.411051	0.205526	−	0.978652i	$-0.434110\pi$
$521$	9.38242	0.411051	0.205526	+	0.978652i	$0.434110\pi$
$522$	−8.58931	−0.375944
$523$	6.06882	0.265371	0.132686	−	0.991158i	$-0.457640\pi$
$523$	6.06882	0.265371	0.132686	+	0.991158i	$0.457640\pi$
$524$	−1.03140	−0.0450571
$525$	0	0
$526$	40.2592	1.75539
$527$	−49.5998	−2.16060
$528$	12.1453	0.528557
$529$	−13.2919	−0.577910
$530$	0	0
$531$	−3.70540	−0.160801
$532$	−8.83706	−0.383135
$533$	45.6200	1.97602
$534$	21.9533	0.950011
$535$	0	0
$536$	38.2369	1.65158
$537$	−4.74389	−0.204714
$538$	20.7069	0.892738
$539$	7.21396	0.310727
$540$	0	0
$541$	−19.5419	−0.840173	−0.420086	−	0.907484i	$-0.638000\pi$
$541$	−19.5419	−0.840173	−0.420086	+	0.907484i	$0.638000\pi$
$542$	−6.19589	−0.266136
$543$	−4.43542	−0.190342
$544$	9.46834	0.405952
$545$	0	0
$546$	23.9544	1.02515
$547$	2.32009	0.0991997	0.0495999	−	0.998769i	$-0.484205\pi$
$547$	2.32009	0.0991997	0.0495999	+	0.998769i	$0.484205\pi$
$548$	−2.83310	−0.121024
$549$	4.29192	0.183175
$550$	0	0
$551$	47.4170	2.02003
$552$	−7.92380	−0.337259
$553$	34.1348	1.45156
$554$	−3.93640	−0.167241
$555$	0	0
$556$	1.18187	0.0501226
$557$	20.5653	0.871380	0.435690	−	0.900097i	$-0.356504\pi$
$557$	20.5653	0.871380	0.435690	+	0.900097i	$0.356504\pi$
$558$	15.0716	0.638032
$559$	0	0
$560$	0	0
$561$	−13.3993	−0.565718
$562$	31.8817	1.34485
$563$	22.8577	0.963335	0.481668	−	0.876354i	$-0.340031\pi$
$563$	22.8577	0.963335	0.481668	+	0.876354i	$0.340031\pi$
$564$	−0.336137	−0.0141539
$565$	0	0
$566$	−22.7893	−0.957906
$567$	−3.11578	−0.130850
$568$	−18.1754	−0.762624
$569$	−8.53537	−0.357821	−0.178911	−	0.983865i	$-0.557257\pi$
$569$	−8.53537	−0.357821	−0.178911	+	0.983865i	$0.557257\pi$
$570$	0	0
$571$	28.1926	1.17982	0.589911	−	0.807468i	$-0.299163\pi$
$571$	28.1926	1.17982	0.589911	+	0.807468i	$0.299163\pi$
$572$	−4.50399	−0.188321
$573$	0.931595	0.0389179
$574$	−43.1918	−1.80279
$575$	0	0
$576$	6.24148	0.260062
$577$	8.26475	0.344066	0.172033	−	0.985091i	$-0.444966\pi$
$577$	8.26475	0.344066	0.172033	+	0.985091i	$0.444966\pi$
$578$	−12.6878	−0.527742
$579$	−8.18076	−0.339981
$580$	0	0
$581$	40.1170	1.66433
$582$	−13.7008	−0.567915
$583$	2.55320	0.105743
$584$	30.1633	1.24817
$585$	0	0
$586$	14.3874	0.594338
$587$	25.3501	1.04631	0.523156	−	0.852237i	$-0.324755\pi$
$587$	25.3501	1.04631	0.523156	+	0.852237i	$0.324755\pi$
$588$	−0.910285	−0.0375395
$589$	−83.2024	−3.42829
$590$	0	0
$591$	−3.79293	−0.156020
$592$	21.2639	0.873940
$593$	−4.29599	−0.176415	−0.0882076	−	0.996102i	$-0.528114\pi$
$593$	−4.29599	−0.176415	−0.0882076	+	0.996102i	$0.528114\pi$
$594$	4.07156	0.167058
$595$	0	0
$596$	2.42352	0.0992714
$597$	−17.3669	−0.710779
$598$	23.9544	0.979569
$599$	−24.7662	−1.01192	−0.505959	−	0.862557i	$-0.668861\pi$
$599$	−24.7662	−1.01192	−0.505959	+	0.862557i	$0.668861\pi$
$600$	0	0
$601$	−23.1635	−0.944857	−0.472429	−	0.881369i	$-0.656623\pi$
$601$	−23.1635	−0.944857	−0.472429	+	0.881369i	$0.656623\pi$
$602$	0	0
$603$	15.0354	0.612290
$604$	6.68213	0.271892
$605$	0	0
$606$	−14.5402	−0.590654
$607$	−26.7755	−1.08679	−0.543393	−	0.839479i	$-0.682861\pi$
$607$	−26.7755	−1.08679	−0.543393	+	0.839479i	$0.682861\pi$
$608$	15.8829	0.644136
$609$	17.5096	0.709524
$610$	0	0
$611$	−5.03002	−0.203493
$612$	1.69077	0.0683455
$613$	−5.66244	−0.228704	−0.114352	−	0.993440i	$-0.536479\pi$
$613$	−5.66244	−0.228704	−0.114352	+	0.993440i	$0.536479\pi$
$614$	−20.6544	−0.833545
$615$	0	0
$616$	−21.1079	−0.850462
$617$	−4.44702	−0.179030	−0.0895151	−	0.995985i	$-0.528532\pi$
$617$	−4.44702	−0.179030	−0.0895151	+	0.995985i	$0.528532\pi$
$618$	11.8377	0.476182
$619$	−17.7123	−0.711917	−0.355958	−	0.934502i	$-0.615845\pi$
$619$	−17.7123	−0.711917	−0.355958	+	0.934502i	$0.615845\pi$
$620$	0	0
$621$	−3.11578	−0.125032
$622$	−38.1076	−1.52798
$623$	−44.7524	−1.79297
$624$	−22.9333	−0.918067
$625$	0	0
$626$	−32.3911	−1.29461
$627$	−22.4769	−0.897642
$628$	−3.67369	−0.146596
$629$	−23.4593	−0.935385
$630$	0	0
$631$	27.3177	1.08750	0.543751	−	0.839247i	$-0.317004\pi$
$631$	27.3177	1.08750	0.543751	+	0.839247i	$0.317004\pi$
$632$	−27.8611	−1.10825
$633$	16.6762	0.662818
$634$	17.7595	0.705319
$635$	0	0
$636$	−0.322173	−0.0127750
$637$	−13.6217	−0.539711
$638$	−22.8807	−0.905857
$639$	−7.14690	−0.282727
$640$	0	0
$641$	−30.3024	−1.19687	−0.598437	−	0.801170i	$-0.704211\pi$
$641$	−30.3024	−1.19687	−0.598437	+	0.801170i	$0.704211\pi$
$642$	0.489091	0.0193029
$643$	44.2436	1.74480	0.872399	−	0.488795i	$-0.162563\pi$
$643$	44.2436	1.74480	0.872399	+	0.488795i	$0.162563\pi$
$644$	−3.26324	−0.128590
$645$	0	0
$646$	−64.8700	−2.55228
$647$	−14.2147	−0.558839	−0.279420	−	0.960169i	$-0.590142\pi$
$647$	−14.2147	−0.558839	−0.279420	+	0.960169i	$0.590142\pi$
$648$	2.54312	0.0999032
$649$	−9.87069	−0.387459
$650$	0	0
$651$	−30.7240	−1.20417
$652$	−6.69067	−0.262027
$653$	30.6915	1.20105	0.600526	−	0.799605i	$-0.294958\pi$
$653$	30.6915	1.20105	0.600526	+	0.799605i	$0.294958\pi$
$654$	−16.7319	−0.654268
$655$	0	0
$656$	41.3507	1.61447
$657$	11.8608	0.462732
$658$	4.76229	0.185653
$659$	8.18820	0.318967	0.159484	−	0.987201i	$-0.449017\pi$
$659$	8.18820	0.318967	0.159484	+	0.987201i	$0.449017\pi$
$660$	0	0
$661$	−13.9503	−0.542606	−0.271303	−	0.962494i	$-0.587454\pi$
$661$	−13.9503	−0.542606	−0.271303	+	0.962494i	$0.587454\pi$
$662$	−21.4016	−0.831795
$663$	25.3011	0.982614
$664$	−32.7438	−1.27071
$665$	0	0
$666$	7.12845	0.276222
$667$	17.5096	0.677974
$668$	7.57561	0.293109
$669$	−15.4877	−0.598790
$670$	0	0
$671$	11.4331	0.441369
$672$	5.86504	0.226249
$673$	39.4082	1.51908	0.759538	−	0.650463i	$-0.225425\pi$
$673$	39.4082	1.51908	0.759538	+	0.650463i	$0.225425\pi$
$674$	46.0627	1.77427
$675$	0	0
$676$	4.13485	0.159033
$677$	−19.9254	−0.765797	−0.382898	−	0.923790i	$-0.625074\pi$
$677$	−19.9254	−0.765797	−0.382898	+	0.923790i	$0.625074\pi$
$678$	22.7234	0.872687
$679$	27.9295	1.07183
$680$	0	0
$681$	−22.8120	−0.874157
$682$	40.1487	1.53737
$683$	−18.4694	−0.706713	−0.353357	−	0.935489i	$-0.614960\pi$
$683$	−18.4694	−0.706713	−0.353357	+	0.935489i	$0.614960\pi$
$684$	2.83623	0.108446
$685$	0	0
$686$	−20.4394	−0.780379
$687$	−22.7755	−0.868939
$688$	0	0
$689$	−4.82107	−0.183668
$690$	0	0
$691$	−2.10226	−0.0799737	−0.0399869	−	0.999200i	$-0.512732\pi$
$691$	−2.10226	−0.0799737	−0.0399869	+	0.999200i	$0.512732\pi$
$692$	−1.86450	−0.0708778
$693$	−8.30001	−0.315291
$694$	24.7622	0.939962
$695$	0	0
$696$	−14.2914	−0.541715
$697$	−45.6200	−1.72798
$698$	6.52066	0.246811
$699$	−17.2393	−0.652050
$700$	0	0
$701$	−39.5251	−1.49284	−0.746421	−	0.665474i	$-0.768230\pi$
$701$	−39.5251	−1.49284	−0.746421	+	0.665474i	$0.768230\pi$
$702$	−7.68810	−0.290169
$703$	−39.3524	−1.48420
$704$	16.6265	0.626633
$705$	0	0
$706$	17.1458	0.645292
$707$	29.6406	1.11475
$708$	1.24552	0.0468096
$709$	18.6737	0.701307	0.350653	−	0.936505i	$-0.385960\pi$
$709$	18.6737	0.701307	0.350653	+	0.936505i	$0.385960\pi$
$710$	0	0
$711$	−10.9555	−0.410862
$712$	36.5272	1.36892
$713$	−30.7240	−1.15062
$714$	−23.9544	−0.896472
$715$	0	0
$716$	1.59459	0.0595928
$717$	−15.2115	−0.568085
$718$	−35.6268	−1.32958
$719$	−20.1777	−0.752500	−0.376250	−	0.926518i	$-0.622787\pi$
$719$	−20.1777	−0.752500	−0.376250	+	0.926518i	$0.622787\pi$
$720$	0	0
$721$	−24.1315	−0.898704
$722$	−79.7772	−2.96900
$723$	−18.7510	−0.697358
$724$	1.49091	0.0554091
$725$	0	0
$726$	−5.96678	−0.221448
$727$	36.0615	1.33745	0.668724	−	0.743511i	$-0.266841\pi$
$727$	36.0615	1.33745	0.668724	+	0.743511i	$0.266841\pi$
$728$	39.8569	1.47719
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	−1.44267	−0.0533227
$733$	−27.2648	−1.00705	−0.503525	−	0.863981i	$-0.667964\pi$
$733$	−27.2648	−1.00705	−0.503525	+	0.863981i	$0.667964\pi$
$734$	−37.1026	−1.36948
$735$	0	0
$736$	5.86504	0.216188
$737$	40.0523	1.47535
$738$	13.8623	0.510278
$739$	−1.62581	−0.0598066	−0.0299033	−	0.999553i	$-0.509520\pi$
$739$	−1.62581	−0.0598066	−0.0299033	+	0.999553i	$0.509520\pi$
$740$	0	0
$741$	42.4419	1.55914
$742$	4.56446	0.167567
$743$	20.8088	0.763401	0.381701	−	0.924286i	$-0.375339\pi$
$743$	20.8088	0.763401	0.381701	+	0.924286i	$0.375339\pi$
$744$	25.0771	0.919371
$745$	0	0
$746$	−9.30825	−0.340799
$747$	−12.8754	−0.471088
$748$	4.50399	0.164682
$749$	−0.997027	−0.0364306
$750$	0	0
$751$	−29.9762	−1.09385	−0.546923	−	0.837183i	$-0.684201\pi$
$751$	−29.9762	−1.09385	−0.546923	+	0.837183i	$0.684201\pi$
$752$	−4.55929	−0.166260
$753$	−1.77476	−0.0646760
$754$	43.2044	1.57341
$755$	0	0
$756$	1.04733	0.0380909
$757$	−4.79797	−0.174385	−0.0871925	−	0.996191i	$-0.527790\pi$
$757$	−4.79797	−0.174385	−0.0871925	+	0.996191i	$0.527790\pi$
$758$	−26.5220	−0.963323
$759$	−8.30001	−0.301271
$760$	0	0
$761$	−23.5039	−0.852014	−0.426007	−	0.904720i	$-0.640080\pi$
$761$	−23.5039	−0.852014	−0.426007	+	0.904720i	$0.640080\pi$
$762$	−8.20756	−0.297328
$763$	34.1084	1.23481
$764$	−0.313143	−0.0113291
$765$	0	0
$766$	−25.0746	−0.905982
$767$	18.6383	0.672989
$768$	−7.85217	−0.283341
$769$	−34.7708	−1.25387	−0.626933	−	0.779073i	$-0.715690\pi$
$769$	−34.7708	−1.25387	−0.626933	+	0.779073i	$0.715690\pi$
$770$	0	0
$771$	−15.9731	−0.575256
$772$	2.74985	0.0989693
$773$	−18.9562	−0.681807	−0.340903	−	0.940098i	$-0.610733\pi$
$773$	−18.9562	−0.681807	−0.340903	+	0.940098i	$0.610733\pi$
$774$	0	0
$775$	0	0
$776$	−22.7962	−0.818336
$777$	−14.5316	−0.521317
$778$	−1.44186	−0.0516933
$779$	−76.5263	−2.74184
$780$	0	0
$781$	−19.0384	−0.681246
$782$	−23.9544	−0.856608
$783$	−5.61965	−0.200830
$784$	−12.3469	−0.440961
$785$	0	0
$786$	−4.68988	−0.167283
$787$	42.9210	1.52997	0.764984	−	0.644050i	$-0.222747\pi$
$787$	42.9210	1.52997	0.764984	+	0.644050i	$0.222747\pi$
$788$	1.27494	0.0454180
$789$	26.3400	0.937730
$790$	0	0
$791$	−46.3224	−1.64703
$792$	6.77452	0.240722
$793$	−21.5884	−0.766628
$794$	−30.6854	−1.08898
$795$	0	0
$796$	5.83765	0.206910
$797$	−49.6483	−1.75863	−0.879317	−	0.476237i	$-0.842000\pi$
$797$	−49.6483	−1.75863	−0.879317	+	0.476237i	$0.842000\pi$
$798$	−40.1829	−1.42246
$799$	5.03002	0.177949
$800$	0	0
$801$	14.3632	0.507497
$802$	−2.18556	−0.0771747
$803$	31.5955	1.11498
$804$	−5.05396	−0.178239
$805$	0	0
$806$	−75.8105	−2.67031
$807$	13.5477	0.476902
$808$	−24.1928	−0.851101
$809$	−34.5440	−1.21450	−0.607251	−	0.794510i	$-0.707728\pi$
$809$	−34.5440	−1.21450	−0.607251	+	0.794510i	$0.707728\pi$
$810$	0	0
$811$	30.3409	1.06541	0.532707	−	0.846300i	$-0.321175\pi$
$811$	30.3409	1.06541	0.532707	+	0.846300i	$0.321175\pi$
$812$	−5.88561	−0.206544
$813$	−4.05373	−0.142170
$814$	18.9892	0.665571
$815$	0	0
$816$	22.9333	0.802826
$817$	0	0
$818$	53.2087	1.86040
$819$	15.6724	0.547639
$820$	0	0
$821$	45.7336	1.59611	0.798057	−	0.602582i	$-0.205861\pi$
$821$	45.7336	1.59611	0.798057	+	0.602582i	$0.205861\pi$
$822$	−12.8823	−0.449323
$823$	−5.32543	−0.185633	−0.0928164	−	0.995683i	$-0.529587\pi$
$823$	−5.32543	−0.185633	−0.0928164	+	0.995683i	$0.529587\pi$
$824$	19.6963	0.686153
$825$	0	0
$826$	−17.6462	−0.613990
$827$	−50.7215	−1.76376	−0.881880	−	0.471474i	$-0.843722\pi$
$827$	−50.7215	−1.76376	−0.881880	+	0.471474i	$0.843722\pi$
$828$	1.04733	0.0363971
$829$	40.3932	1.40291	0.701456	−	0.712713i	$-0.252534\pi$
$829$	40.3932	1.40291	0.701456	+	0.712713i	$0.252534\pi$
$830$	0	0
$831$	−2.57543	−0.0893406
$832$	−31.3948	−1.08842
$833$	13.6217	0.471964
$834$	5.37409	0.186089
$835$	0	0
$836$	7.55532	0.261306
$837$	9.86076	0.340838
$838$	−47.5038	−1.64099
$839$	52.2100	1.80249	0.901244	−	0.433311i	$-0.142655\pi$
$839$	52.2100	1.80249	0.901244	+	0.433311i	$0.142655\pi$
$840$	0	0
$841$	2.58042	0.0889801
$842$	−10.5990	−0.365266
$843$	20.8589	0.718420
$844$	−5.60547	−0.192948
$845$	0	0
$846$	−1.52844	−0.0525489
$847$	12.1635	0.417942
$848$	−4.36989	−0.150063
$849$	−14.9101	−0.511715
$850$	0	0
$851$	−14.5316	−0.498136
$852$	2.40234	0.0823026
$853$	17.3940	0.595559	0.297779	−	0.954635i	$-0.403754\pi$
$853$	17.3940	0.595559	0.297779	+	0.954635i	$0.403754\pi$
$854$	20.4394	0.699421
$855$	0	0
$856$	0.813780	0.0278144
$857$	3.12067	0.106600	0.0533000	−	0.998579i	$-0.483026\pi$
$857$	3.12067	0.106600	0.0533000	+	0.998579i	$0.483026\pi$
$858$	−20.4800	−0.699177
$859$	34.9054	1.19096	0.595478	−	0.803372i	$-0.296963\pi$
$859$	34.9054	1.19096	0.595478	+	0.803372i	$0.296963\pi$
$860$	0	0
$861$	−28.2587	−0.963055
$862$	−55.2558	−1.88202
$863$	15.0786	0.513282	0.256641	−	0.966507i	$-0.417384\pi$
$863$	15.0786	0.513282	0.256641	+	0.966507i	$0.417384\pi$
$864$	−1.88237	−0.0640394
$865$	0	0
$866$	−13.5010	−0.458782
$867$	−8.30111	−0.281921
$868$	10.3274	0.350536
$869$	−29.1839	−0.989995
$870$	0	0
$871$	−75.6286	−2.56258
$872$	−27.8395	−0.942766
$873$	−8.96387	−0.303381
$874$	−40.1829	−1.35921
$875$	0	0
$876$	−3.98684	−0.134703
$877$	−47.0622	−1.58918	−0.794588	−	0.607149i	$-0.792313\pi$
$877$	−47.0622	−1.58918	−0.794588	+	0.607149i	$0.792313\pi$
$878$	21.5758	0.728147
$879$	9.41310	0.317496
$880$	0	0
$881$	−15.5544	−0.524040	−0.262020	−	0.965062i	$-0.584389\pi$
$881$	−15.5544	−0.524040	−0.262020	+	0.965062i	$0.584389\pi$
$882$	−4.13915	−0.139372
$883$	0.575056	0.0193522	0.00967609	−	0.999953i	$-0.496920\pi$
$883$	0.575056	0.0193522	0.00967609	+	0.999953i	$0.496920\pi$
$884$	−8.50463	−0.286042
$885$	0	0
$886$	32.5561	1.09374
$887$	−58.0626	−1.94955	−0.974775	−	0.223189i	$-0.928353\pi$
$887$	−58.0626	−1.94955	−0.974775	+	0.223189i	$0.928353\pi$
$888$	11.8608	0.398021
$889$	16.7314	0.561152
$890$	0	0
$891$	2.66386	0.0892428
$892$	5.20599	0.174309
$893$	8.43772	0.282358
$894$	11.0200	0.368563
$895$	0	0
$896$	41.4538	1.38487
$897$	15.6724	0.523287
$898$	−21.5247	−0.718288
$899$	−55.4140	−1.84816
$900$	0	0
$901$	4.82107	0.160613
$902$	36.9272	1.22954
$903$	0	0
$904$	37.8086	1.25750
$905$	0	0
$906$	30.3842	1.00945
$907$	3.58681	0.119098	0.0595490	−	0.998225i	$-0.481034\pi$
$907$	3.58681	0.119098	0.0595490	+	0.998225i	$0.481034\pi$
$908$	7.66794	0.254469
$909$	−9.51306	−0.315528
$910$	0	0
$911$	24.5531	0.813482	0.406741	−	0.913544i	$-0.366665\pi$
$911$	24.5531	0.813482	0.406741	+	0.913544i	$0.366665\pi$
$912$	38.4700	1.27387
$913$	−34.2984	−1.13511
$914$	12.8259	0.424244
$915$	0	0
$916$	7.65567	0.252951
$917$	9.56047	0.315715
$918$	7.68810	0.253745
$919$	0.157692	0.00520177	0.00260089	−	0.999997i	$-0.499172\pi$
$919$	0.157692	0.00520177	0.00260089	+	0.999997i	$0.499172\pi$
$920$	0	0
$921$	−13.5134	−0.445281
$922$	−23.3462	−0.768866
$923$	35.9491	1.18328
$924$	2.78994	0.0917822
$925$	0	0
$926$	45.4610	1.49394
$927$	7.74494	0.254377
$928$	10.5782	0.347248
$929$	−30.8470	−1.01206	−0.506029	−	0.862516i	$-0.668887\pi$
$929$	−30.8470	−1.01206	−0.506029	+	0.862516i	$0.668887\pi$
$930$	0	0
$931$	22.8500	0.748879
$932$	5.79476	0.189814
$933$	−24.9323	−0.816248
$934$	−5.03262	−0.164672
$935$	0	0
$936$	−12.7919	−0.418118
$937$	40.0853	1.30953	0.654764	−	0.755833i	$-0.272768\pi$
$937$	40.0853	1.30953	0.654764	+	0.755833i	$0.272768\pi$
$938$	71.6031	2.33792
$939$	−21.1922	−0.691582
$940$	0	0
$941$	7.88673	0.257100	0.128550	−	0.991703i	$-0.458968\pi$
$941$	7.88673	0.257100	0.128550	+	0.991703i	$0.458968\pi$
$942$	−16.7046	−0.544265
$943$	−28.2587	−0.920231
$944$	16.8940	0.549853
$945$	0	0
$946$	0	0
$947$	41.1577	1.33745	0.668723	−	0.743512i	$-0.266841\pi$
$947$	41.1577	1.33745	0.668723	+	0.743512i	$0.266841\pi$
$948$	3.68253	0.119603
$949$	−59.6599	−1.93664
$950$	0	0
$951$	11.6193	0.376783
$952$	−39.8569	−1.29177
$953$	10.5683	0.342340	0.171170	−	0.985242i	$-0.445245\pi$
$953$	10.5683	0.342340	0.171170	+	0.985242i	$0.445245\pi$
$954$	−1.46495	−0.0474295
$955$	0	0
$956$	5.11315	0.165371
$957$	−14.9700	−0.483910
$958$	−37.6316	−1.21582
$959$	26.2611	0.848014
$960$	0	0
$961$	66.2346	2.13660
$962$	−35.8562	−1.15605
$963$	0.319993	0.0103116
$964$	6.30290	0.203003
$965$	0	0
$966$	−14.8382	−0.477413
$967$	10.1161	0.325311	0.162655	−	0.986683i	$-0.447994\pi$
$967$	10.1161	0.325311	0.162655	+	0.986683i	$0.447994\pi$
$968$	−9.92791	−0.319095
$969$	−42.4419	−1.36343
$970$	0	0
$971$	−8.70212	−0.279264	−0.139632	−	0.990203i	$-0.544592\pi$
$971$	−8.70212	−0.279264	−0.139632	+	0.990203i	$0.544592\pi$
$972$	−0.336137	−0.0107816
$973$	−10.9552	−0.351209
$974$	44.9700	1.44093
$975$	0	0
$976$	−19.5681	−0.626359
$977$	0.883840	0.0282766	0.0141383	−	0.999900i	$-0.495499\pi$
$977$	0.883840	0.0282766	0.0141383	+	0.999900i	$0.495499\pi$
$978$	−30.4231	−0.972822
$979$	38.2615	1.22284
$980$	0	0
$981$	−10.9470	−0.349511
$982$	12.5700	0.401124
$983$	4.23814	0.135176	0.0675878	−	0.997713i	$-0.478470\pi$
$983$	4.23814	0.135176	0.0675878	+	0.997713i	$0.478470\pi$
$984$	23.0650	0.735284
$985$	0	0
$986$	−43.2044	−1.37591
$987$	3.11578	0.0991763
$988$	−14.2663	−0.453871
$989$	0	0
$990$	0	0
$991$	−38.3971	−1.21972	−0.609862	−	0.792508i	$-0.708775\pi$
$991$	−38.3971	−1.21972	−0.609862	+	0.792508i	$0.708775\pi$
$992$	−18.5616	−0.589330
$993$	−14.0022	−0.444346
$994$	−34.0356	−1.07954
$995$	0	0
$996$	4.32791	0.137135
$997$	19.8902	0.629930	0.314965	−	0.949103i	$-0.398007\pi$
$997$	19.8902	0.629930	0.314965	+	0.949103i	$0.398007\pi$
$998$	−59.8081	−1.89319
$999$	4.66386	0.147558

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3525.2.a.w.1.3		6
5.4	even	2		705.2.a.m.1.4	✓	6
15.14	odd	2		2115.2.a.s.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
705.2.a.m.1.4	✓	6	5.4	even	2
2115.2.a.s.1.3		6	15.14	odd	2
3525.2.a.w.1.3		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3525.a (trivial)

\(f(q)\)	\(=\)	\(q-1.52844 q^{2} -1.00000 q^{3} +0.336137 q^{4} +1.52844 q^{6} -3.11578 q^{7} +2.54312 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.52844 q^{2} -1.00000 q^{3} +0.336137 q^{4} +1.52844 q^{6} -3.11578 q^{7} +2.54312 q^{8} +1.00000 q^{9} +2.66386 q^{11} -0.336137 q^{12} -5.03002 q^{13} +4.76229 q^{14} -4.55929 q^{16} +5.03002 q^{17} -1.52844 q^{18} +8.43772 q^{19} +3.11578 q^{21} -4.07156 q^{22} +3.11578 q^{23} -2.54312 q^{24} +7.68810 q^{26} -1.00000 q^{27} -1.04733 q^{28} +5.61965 q^{29} -9.86076 q^{31} +1.88237 q^{32} -2.66386 q^{33} -7.68810 q^{34} +0.336137 q^{36} -4.66386 q^{37} -12.8966 q^{38} +5.03002 q^{39} -9.06955 q^{41} -4.76229 q^{42} +0.895422 q^{44} -4.76229 q^{46} +1.00000 q^{47} +4.55929 q^{48} +2.70808 q^{49} -5.03002 q^{51} -1.69077 q^{52} +0.958459 q^{53} +1.52844 q^{54} -7.92380 q^{56} -8.43772 q^{57} -8.58931 q^{58} -3.70540 q^{59} +4.29192 q^{61} +15.0716 q^{62} -3.11578 q^{63} +6.24148 q^{64} +4.07156 q^{66} +15.0354 q^{67} +1.69077 q^{68} -3.11578 q^{69} -7.14690 q^{71} +2.54312 q^{72} +11.8608 q^{73} +7.12845 q^{74} +2.83623 q^{76} -8.30001 q^{77} -7.68810 q^{78} -10.9555 q^{79} +1.00000 q^{81} +13.8623 q^{82} -12.8754 q^{83} +1.04733 q^{84} -5.61965 q^{87} +6.77452 q^{88} +14.3632 q^{89} +15.6724 q^{91} +1.04733 q^{92} +9.86076 q^{93} -1.52844 q^{94} -1.88237 q^{96} -8.96387 q^{97} -4.13915 q^{98} +2.66386 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} - 6 q^{3} + 14 q^{4} + 2 q^{6} - 4 q^{7} - 6 q^{8} + 6 q^{9}+O(q^{10}) \) 6 * q - 2 * q^2 - 6 * q^3 + 14 * q^4 + 2 * q^6 - 4 * q^7 - 6 * q^8 + 6 * q^9	\( 6 q - 2 q^{2} - 6 q^{3} + 14 q^{4} + 2 q^{6} - 4 q^{7} - 6 q^{8} + 6 q^{9} + 4 q^{11} - 14 q^{12} - 6 q^{13} - 2 q^{14} + 26 q^{16} + 6 q^{17} - 2 q^{18} + 10 q^{19} + 4 q^{21} + 4 q^{22} + 4 q^{23} + 6 q^{24} + 8 q^{26} - 6 q^{27} - 4 q^{28} + 8 q^{29} + 8 q^{31} - 14 q^{32} - 4 q^{33} - 8 q^{34} + 14 q^{36} - 16 q^{37} + 12 q^{38} + 6 q^{39} - 12 q^{41} + 2 q^{42} - 36 q^{44} + 2 q^{46} + 6 q^{47} - 26 q^{48} + 18 q^{49} - 6 q^{51} + 10 q^{52} + 10 q^{53} + 2 q^{54} - 26 q^{56} - 10 q^{57} + 26 q^{58} - 6 q^{59} + 24 q^{61} + 44 q^{62} - 4 q^{63} + 54 q^{64} - 4 q^{66} + 8 q^{67} - 10 q^{68} - 4 q^{69} + 26 q^{71} - 6 q^{72} + 4 q^{73} + 30 q^{76} - 8 q^{77} - 8 q^{78} + 24 q^{79} + 6 q^{81} + 82 q^{82} + 4 q^{83} + 4 q^{84} - 8 q^{87} + 16 q^{88} - 20 q^{89} - 10 q^{91} + 4 q^{92} - 8 q^{93} - 2 q^{94} + 14 q^{96} - 24 q^{98} + 4 q^{99}+O(q^{100}) \) 6 * q - 2 * q^2 - 6 * q^3 + 14 * q^4 + 2 * q^6 - 4 * q^7 - 6 * q^8 + 6 * q^9 + 4 * q^11 - 14 * q^12 - 6 * q^13 - 2 * q^14 + 26 * q^16 + 6 * q^17 - 2 * q^18 + 10 * q^19 + 4 * q^21 + 4 * q^22 + 4 * q^23 + 6 * q^24 + 8 * q^26 - 6 * q^27 - 4 * q^28 + 8 * q^29 + 8 * q^31 - 14 * q^32 - 4 * q^33 - 8 * q^34 + 14 * q^36 - 16 * q^37 + 12 * q^38 + 6 * q^39 - 12 * q^41 + 2 * q^42 - 36 * q^44 + 2 * q^46 + 6 * q^47 - 26 * q^48 + 18 * q^49 - 6 * q^51 + 10 * q^52 + 10 * q^53 + 2 * q^54 - 26 * q^56 - 10 * q^57 + 26 * q^58 - 6 * q^59 + 24 * q^61 + 44 * q^62 - 4 * q^63 + 54 * q^64 - 4 * q^66 + 8 * q^67 - 10 * q^68 - 4 * q^69 + 26 * q^71 - 6 * q^72 + 4 * q^73 + 30 * q^76 - 8 * q^77 - 8 * q^78 + 24 * q^79 + 6 * q^81 + 82 * q^82 + 4 * q^83 + 4 * q^84 - 8 * q^87 + 16 * q^88 - 20 * q^89 - 10 * q^91 + 4 * q^92 - 8 * q^93 - 2 * q^94 + 14 * q^96 - 24 * q^98 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more