LMFDB - Embedded newform 2450.2.a.br.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2450,2,Mod(1,2450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2450 = 2 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2450.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$19.5633484952$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.64575$ of defining polynomial
Character	$\chi$	$=$	2450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.64575 q^{3} +1.00000 q^{4} -2.64575 q^{6} +1.00000 q^{8} +4.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.64575 q^{3} +1.00000 q^{4} -2.64575 q^{6} +1.00000 q^{8} +4.00000 q^{9} +5.00000 q^{11} -2.64575 q^{12} +5.29150 q^{13} +1.00000 q^{16} -2.64575 q^{17} +4.00000 q^{18} -7.93725 q^{19} +5.00000 q^{22} -4.00000 q^{23} -2.64575 q^{24} +5.29150 q^{26} -2.64575 q^{27} +6.00000 q^{29} +10.5830 q^{31} +1.00000 q^{32} -13.2288 q^{33} -2.64575 q^{34} +4.00000 q^{36} -4.00000 q^{37} -7.93725 q^{38} -14.0000 q^{39} +2.64575 q^{41} -8.00000 q^{43} +5.00000 q^{44} -4.00000 q^{46} -5.29150 q^{47} -2.64575 q^{48} +7.00000 q^{51} +5.29150 q^{52} +4.00000 q^{53} -2.64575 q^{54} +21.0000 q^{57} +6.00000 q^{58} -5.29150 q^{59} +5.29150 q^{61} +10.5830 q^{62} +1.00000 q^{64} -13.2288 q^{66} +5.00000 q^{67} -2.64575 q^{68} +10.5830 q^{69} +6.00000 q^{71} +4.00000 q^{72} +2.64575 q^{73} -4.00000 q^{74} -7.93725 q^{76} -14.0000 q^{78} +10.0000 q^{79} -5.00000 q^{81} +2.64575 q^{82} +2.64575 q^{83} -8.00000 q^{86} -15.8745 q^{87} +5.00000 q^{88} +13.2288 q^{89} -4.00000 q^{92} -28.0000 q^{93} -5.29150 q^{94} -2.64575 q^{96} +20.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 8 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 + 8 * q^9	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 8 q^{9} + 10 q^{11} + 2 q^{16} + 8 q^{18} + 10 q^{22} - 8 q^{23} + 12 q^{29} + 2 q^{32} + 8 q^{36} - 8 q^{37} - 28 q^{39} - 16 q^{43} + 10 q^{44} - 8 q^{46} + 14 q^{51} + 8 q^{53} + 42 q^{57} + 12 q^{58} + 2 q^{64} + 10 q^{67} + 12 q^{71} + 8 q^{72} - 8 q^{74} - 28 q^{78} + 20 q^{79} - 10 q^{81} - 16 q^{86} + 10 q^{88} - 8 q^{92} - 56 q^{93} + 40 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 + 8 * q^9 + 10 * q^11 + 2 * q^16 + 8 * q^18 + 10 * q^22 - 8 * q^23 + 12 * q^29 + 2 * q^32 + 8 * q^36 - 8 * q^37 - 28 * q^39 - 16 * q^43 + 10 * q^44 - 8 * q^46 + 14 * q^51 + 8 * q^53 + 42 * q^57 + 12 * q^58 + 2 * q^64 + 10 * q^67 + 12 * q^71 + 8 * q^72 - 8 * q^74 - 28 * q^78 + 20 * q^79 - 10 * q^81 - 16 * q^86 + 10 * q^88 - 8 * q^92 - 56 * q^93 + 40 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.64575	−1.52753	−0.763763	−	0.645497i	$-0.776650\pi$
$3$	−2.64575	−1.52753	−0.763763	+	0.645497i	$0.776650\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−2.64575	−1.08012
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	4.00000	1.33333
$10$	0	0
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	−2.64575	−0.763763
$13$	5.29150	1.46760	0.733799	−	0.679366i	$-0.237745\pi$
$13$	5.29150	1.46760	0.733799	+	0.679366i	$0.237745\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.64575	−0.641689	−0.320844	−	0.947132i	$-0.603967\pi$
$17$	−2.64575	−0.641689	−0.320844	+	0.947132i	$0.603967\pi$
$18$	4.00000	0.942809
$19$	−7.93725	−1.82093	−0.910465	−	0.413585i	$-0.864276\pi$
$19$	−7.93725	−1.82093	−0.910465	+	0.413585i	$0.864276\pi$
$20$	0	0
$21$	0	0
$22$	5.00000	1.06600
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	−2.64575	−0.540062
$25$	0	0
$26$	5.29150	1.03775
$27$	−2.64575	−0.509175
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	10.5830	1.90076	0.950382	−	0.311086i	$-0.100693\pi$
$31$	10.5830	1.90076	0.950382	+	0.311086i	$0.100693\pi$
$32$	1.00000	0.176777
$33$	−13.2288	−2.30283
$34$	−2.64575	−0.453743
$35$	0	0
$36$	4.00000	0.666667
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	−7.93725	−1.28759
$39$	−14.0000	−2.24179
$40$	0	0
$41$	2.64575	0.413197	0.206598	−	0.978426i	$-0.433761\pi$
$41$	2.64575	0.413197	0.206598	+	0.978426i	$0.433761\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	5.00000	0.753778
$45$	0	0
$46$	−4.00000	−0.589768
$47$	−5.29150	−0.771845	−0.385922	−	0.922531i	$-0.626117\pi$
$47$	−5.29150	−0.771845	−0.385922	+	0.922531i	$0.626117\pi$
$48$	−2.64575	−0.381881
$49$	0	0
$50$	0	0
$51$	7.00000	0.980196
$52$	5.29150	0.733799
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	−2.64575	−0.360041
$55$	0	0
$56$	0	0
$57$	21.0000	2.78152
$58$	6.00000	0.787839
$59$	−5.29150	−0.688895	−0.344447	−	0.938806i	$-0.611934\pi$
$59$	−5.29150	−0.688895	−0.344447	+	0.938806i	$0.611934\pi$
$60$	0	0
$61$	5.29150	0.677507	0.338754	−	0.940875i	$-0.389995\pi$
$61$	5.29150	0.677507	0.338754	+	0.940875i	$0.389995\pi$
$62$	10.5830	1.34404
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	−13.2288	−1.62835
$67$	5.00000	0.610847	0.305424	−	0.952217i	$-0.401202\pi$
$67$	5.00000	0.610847	0.305424	+	0.952217i	$0.401202\pi$
$68$	−2.64575	−0.320844
$69$	10.5830	1.27404
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	4.00000	0.471405
$73$	2.64575	0.309662	0.154831	−	0.987941i	$-0.450517\pi$
$73$	2.64575	0.309662	0.154831	+	0.987941i	$0.450517\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	−7.93725	−0.910465
$77$	0	0
$78$	−14.0000	−1.58519
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	2.64575	0.292174
$83$	2.64575	0.290409	0.145204	−	0.989402i	$-0.453616\pi$
$83$	2.64575	0.290409	0.145204	+	0.989402i	$0.453616\pi$
$84$	0	0
$85$	0	0
$86$	−8.00000	−0.862662
$87$	−15.8745	−1.70193
$88$	5.00000	0.533002
$89$	13.2288	1.40225	0.701123	−	0.713041i	$-0.252683\pi$
$89$	13.2288	1.40225	0.701123	+	0.713041i	$0.252683\pi$
$90$	0	0
$91$	0	0
$92$	−4.00000	−0.417029
$93$	−28.0000	−2.90346
$94$	−5.29150	−0.545777
$95$	0	0
$96$	−2.64575	−0.270031
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	0	0
$99$	20.0000	2.01008
$100$	0	0
$101$	10.5830	1.05305	0.526524	−	0.850160i	$-0.323495\pi$
$101$	10.5830	1.05305	0.526524	+	0.850160i	$0.323495\pi$
$102$	7.00000	0.693103
$103$	10.5830	1.04277	0.521387	−	0.853320i	$-0.325415\pi$
$103$	10.5830	1.04277	0.521387	+	0.853320i	$0.325415\pi$
$104$	5.29150	0.518875
$105$	0	0
$106$	4.00000	0.388514
$107$	15.0000	1.45010	0.725052	−	0.688694i	$-0.241816\pi$
$107$	15.0000	1.45010	0.725052	+	0.688694i	$0.241816\pi$
$108$	−2.64575	−0.254588
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	10.5830	1.00449
$112$	0	0
$113$	13.0000	1.22294	0.611469	−	0.791269i	$-0.290579\pi$
$113$	13.0000	1.22294	0.611469	+	0.791269i	$0.290579\pi$
$114$	21.0000	1.96683
$115$	0	0
$116$	6.00000	0.557086
$117$	21.1660	1.95680
$118$	−5.29150	−0.487122
$119$	0	0
$120$	0	0
$121$	14.0000	1.27273
$122$	5.29150	0.479070
$123$	−7.00000	−0.631169
$124$	10.5830	0.950382
$125$	0	0
$126$	0	0
$127$	6.00000	0.532414	0.266207	−	0.963916i	$-0.414230\pi$
$127$	6.00000	0.532414	0.266207	+	0.963916i	$0.414230\pi$
$128$	1.00000	0.0883883
$129$	21.1660	1.86356
$130$	0	0
$131$	−15.8745	−1.38696	−0.693481	−	0.720475i	$-0.743924\pi$
$131$	−15.8745	−1.38696	−0.693481	+	0.720475i	$0.743924\pi$
$132$	−13.2288	−1.15142
$133$	0	0
$134$	5.00000	0.431934
$135$	0	0
$136$	−2.64575	−0.226871
$137$	−17.0000	−1.45241	−0.726204	−	0.687479i	$-0.758717\pi$
$137$	−17.0000	−1.45241	−0.726204	+	0.687479i	$0.758717\pi$
$138$	10.5830	0.900885
$139$	−2.64575	−0.224410	−0.112205	−	0.993685i	$-0.535791\pi$
$139$	−2.64575	−0.224410	−0.112205	+	0.993685i	$0.535791\pi$
$140$	0	0
$141$	14.0000	1.17901
$142$	6.00000	0.503509
$143$	26.4575	2.21249
$144$	4.00000	0.333333
$145$	0	0
$146$	2.64575	0.218964
$147$	0	0
$148$	−4.00000	−0.328798
$149$	−4.00000	−0.327693	−0.163846	−	0.986486i	$-0.552390\pi$
$149$	−4.00000	−0.327693	−0.163846	+	0.986486i	$0.552390\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	−7.93725	−0.643796
$153$	−10.5830	−0.855585
$154$	0	0
$155$	0	0
$156$	−14.0000	−1.12090
$157$	21.1660	1.68923	0.844616	−	0.535373i	$-0.179829\pi$
$157$	21.1660	1.68923	0.844616	+	0.535373i	$0.179829\pi$
$158$	10.0000	0.795557
$159$	−10.5830	−0.839287
$160$	0	0
$161$	0	0
$162$	−5.00000	−0.392837
$163$	−13.0000	−1.01824	−0.509119	−	0.860696i	$-0.670029\pi$
$163$	−13.0000	−1.01824	−0.509119	+	0.860696i	$0.670029\pi$
$164$	2.64575	0.206598
$165$	0	0
$166$	2.64575	0.205350
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	15.0000	1.15385
$170$	0	0
$171$	−31.7490	−2.42791
$172$	−8.00000	−0.609994
$173$	15.8745	1.20692	0.603458	−	0.797395i	$-0.293789\pi$
$173$	15.8745	1.20692	0.603458	+	0.797395i	$0.293789\pi$
$174$	−15.8745	−1.20344
$175$	0	0
$176$	5.00000	0.376889
$177$	14.0000	1.05230
$178$	13.2288	0.991537
$179$	−17.0000	−1.27064	−0.635320	−	0.772249i	$-0.719132\pi$
$179$	−17.0000	−1.27064	−0.635320	+	0.772249i	$0.719132\pi$
$180$	0	0
$181$	5.29150	0.393314	0.196657	−	0.980472i	$-0.436991\pi$
$181$	5.29150	0.393314	0.196657	+	0.980472i	$0.436991\pi$
$182$	0	0
$183$	−14.0000	−1.03491
$184$	−4.00000	−0.294884
$185$	0	0
$186$	−28.0000	−2.05306
$187$	−13.2288	−0.967382
$188$	−5.29150	−0.385922
$189$	0	0
$190$	0	0
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	−2.64575	−0.190941
$193$	−9.00000	−0.647834	−0.323917	−	0.946085i	$-0.605000\pi$
$193$	−9.00000	−0.647834	−0.323917	+	0.946085i	$0.605000\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	20.0000	1.42134
$199$	−5.29150	−0.375105	−0.187552	−	0.982255i	$-0.560055\pi$
$199$	−5.29150	−0.375105	−0.187552	+	0.982255i	$0.560055\pi$
$200$	0	0
$201$	−13.2288	−0.933085
$202$	10.5830	0.744618
$203$	0	0
$204$	7.00000	0.490098
$205$	0	0
$206$	10.5830	0.737353
$207$	−16.0000	−1.11208
$208$	5.29150	0.366900
$209$	−39.6863	−2.74516
$210$	0	0
$211$	−19.0000	−1.30801	−0.654007	−	0.756489i	$-0.726913\pi$
$211$	−19.0000	−1.30801	−0.654007	+	0.756489i	$0.726913\pi$
$212$	4.00000	0.274721
$213$	−15.8745	−1.08770
$214$	15.0000	1.02538
$215$	0	0
$216$	−2.64575	−0.180021
$217$	0	0
$218$	−2.00000	−0.135457
$219$	−7.00000	−0.473016
$220$	0	0
$221$	−14.0000	−0.941742
$222$	10.5830	0.710285
$223$	−5.29150	−0.354345	−0.177173	−	0.984180i	$-0.556695\pi$
$223$	−5.29150	−0.354345	−0.177173	+	0.984180i	$0.556695\pi$
$224$	0	0
$225$	0	0
$226$	13.0000	0.864747
$227$	15.8745	1.05363	0.526814	−	0.849981i	$-0.323386\pi$
$227$	15.8745	1.05363	0.526814	+	0.849981i	$0.323386\pi$
$228$	21.0000	1.39076
$229$	26.4575	1.74836	0.874181	−	0.485601i	$-0.161399\pi$
$229$	26.4575	1.74836	0.874181	+	0.485601i	$0.161399\pi$
$230$	0	0
$231$	0	0
$232$	6.00000	0.393919
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	21.1660	1.38367
$235$	0	0
$236$	−5.29150	−0.344447
$237$	−26.4575	−1.71860
$238$	0	0
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	2.64575	0.170428	0.0852139	−	0.996363i	$-0.472843\pi$
$241$	2.64575	0.170428	0.0852139	+	0.996363i	$0.472843\pi$
$242$	14.0000	0.899954
$243$	21.1660	1.35780
$244$	5.29150	0.338754
$245$	0	0
$246$	−7.00000	−0.446304
$247$	−42.0000	−2.67240
$248$	10.5830	0.672022
$249$	−7.00000	−0.443607
$250$	0	0
$251$	−13.2288	−0.834992	−0.417496	−	0.908679i	$-0.637092\pi$
$251$	−13.2288	−0.834992	−0.417496	+	0.908679i	$0.637092\pi$
$252$	0	0
$253$	−20.0000	−1.25739
$254$	6.00000	0.376473
$255$	0	0
$256$	1.00000	0.0625000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	21.1660	1.31774
$259$	0	0
$260$	0	0
$261$	24.0000	1.48556
$262$	−15.8745	−0.980730
$263$	−30.0000	−1.84988	−0.924940	−	0.380114i	$-0.875885\pi$
$263$	−30.0000	−1.84988	−0.924940	+	0.380114i	$0.875885\pi$
$264$	−13.2288	−0.814174
$265$	0	0
$266$	0	0
$267$	−35.0000	−2.14197
$268$	5.00000	0.305424
$269$	5.29150	0.322629	0.161314	−	0.986903i	$-0.448427\pi$
$269$	5.29150	0.322629	0.161314	+	0.986903i	$0.448427\pi$
$270$	0	0
$271$	−5.29150	−0.321436	−0.160718	−	0.987000i	$-0.551381\pi$
$271$	−5.29150	−0.321436	−0.160718	+	0.987000i	$0.551381\pi$
$272$	−2.64575	−0.160422
$273$	0	0
$274$	−17.0000	−1.02701
$275$	0	0
$276$	10.5830	0.637022
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	−2.64575	−0.158682
$279$	42.3320	2.53435
$280$	0	0
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	14.0000	0.833688
$283$	2.64575	0.157274	0.0786368	−	0.996903i	$-0.474943\pi$
$283$	2.64575	0.157274	0.0786368	+	0.996903i	$0.474943\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	26.4575	1.56447
$287$	0	0
$288$	4.00000	0.235702
$289$	−10.0000	−0.588235
$290$	0	0
$291$	0	0
$292$	2.64575	0.154831
$293$	15.8745	0.927399	0.463699	−	0.885993i	$-0.346522\pi$
$293$	15.8745	0.927399	0.463699	+	0.885993i	$0.346522\pi$
$294$	0	0
$295$	0	0
$296$	−4.00000	−0.232495
$297$	−13.2288	−0.767610
$298$	−4.00000	−0.231714
$299$	−21.1660	−1.22406
$300$	0	0
$301$	0	0
$302$	12.0000	0.690522
$303$	−28.0000	−1.60856
$304$	−7.93725	−0.455233
$305$	0	0
$306$	−10.5830	−0.604990
$307$	34.3948	1.96301	0.981507	−	0.191429i	$-0.0613121\pi$
$307$	34.3948	1.96301	0.981507	+	0.191429i	$0.0613121\pi$
$308$	0	0
$309$	−28.0000	−1.59286
$310$	0	0
$311$	−5.29150	−0.300054	−0.150027	−	0.988682i	$-0.547936\pi$
$311$	−5.29150	−0.300054	−0.150027	+	0.988682i	$0.547936\pi$
$312$	−14.0000	−0.792594
$313$	−21.1660	−1.19637	−0.598187	−	0.801357i	$-0.704112\pi$
$313$	−21.1660	−1.19637	−0.598187	+	0.801357i	$0.704112\pi$
$314$	21.1660	1.19447
$315$	0	0
$316$	10.0000	0.562544
$317$	24.0000	1.34797	0.673987	−	0.738743i	$-0.264580\pi$
$317$	24.0000	1.34797	0.673987	+	0.738743i	$0.264580\pi$
$318$	−10.5830	−0.593465
$319$	30.0000	1.67968
$320$	0	0
$321$	−39.6863	−2.21507
$322$	0	0
$323$	21.0000	1.16847
$324$	−5.00000	−0.277778
$325$	0	0
$326$	−13.0000	−0.720003
$327$	5.29150	0.292621
$328$	2.64575	0.146087
$329$	0	0
$330$	0	0
$331$	17.0000	0.934405	0.467202	−	0.884150i	$-0.345262\pi$
$331$	17.0000	0.934405	0.467202	+	0.884150i	$0.345262\pi$
$332$	2.64575	0.145204
$333$	−16.0000	−0.876795
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−5.00000	−0.272367	−0.136184	−	0.990684i	$-0.543484\pi$
$337$	−5.00000	−0.272367	−0.136184	+	0.990684i	$0.543484\pi$
$338$	15.0000	0.815892
$339$	−34.3948	−1.86807
$340$	0	0
$341$	52.9150	2.86551
$342$	−31.7490	−1.71679
$343$	0	0
$344$	−8.00000	−0.431331
$345$	0	0
$346$	15.8745	0.853419
$347$	5.00000	0.268414	0.134207	−	0.990953i	$-0.457151\pi$
$347$	5.00000	0.268414	0.134207	+	0.990953i	$0.457151\pi$
$348$	−15.8745	−0.850963
$349$	−21.1660	−1.13299	−0.566495	−	0.824065i	$-0.691701\pi$
$349$	−21.1660	−1.13299	−0.566495	+	0.824065i	$0.691701\pi$
$350$	0	0
$351$	−14.0000	−0.747265
$352$	5.00000	0.266501
$353$	−31.7490	−1.68983	−0.844915	−	0.534901i	$-0.820349\pi$
$353$	−31.7490	−1.68983	−0.844915	+	0.534901i	$0.820349\pi$
$354$	14.0000	0.744092
$355$	0	0
$356$	13.2288	0.701123
$357$	0	0
$358$	−17.0000	−0.898478
$359$	10.0000	0.527780	0.263890	−	0.964553i	$-0.414994\pi$
$359$	10.0000	0.527780	0.263890	+	0.964553i	$0.414994\pi$
$360$	0	0
$361$	44.0000	2.31579
$362$	5.29150	0.278115
$363$	−37.0405	−1.94412
$364$	0	0
$365$	0	0
$366$	−14.0000	−0.731792
$367$	−10.5830	−0.552428	−0.276214	−	0.961096i	$-0.589080\pi$
$367$	−10.5830	−0.552428	−0.276214	+	0.961096i	$0.589080\pi$
$368$	−4.00000	−0.208514
$369$	10.5830	0.550929
$370$	0	0
$371$	0	0
$372$	−28.0000	−1.45173
$373$	24.0000	1.24267	0.621336	−	0.783544i	$-0.286590\pi$
$373$	24.0000	1.24267	0.621336	+	0.783544i	$0.286590\pi$
$374$	−13.2288	−0.684043
$375$	0	0
$376$	−5.29150	−0.272888
$377$	31.7490	1.63516
$378$	0	0
$379$	−1.00000	−0.0513665	−0.0256833	−	0.999670i	$-0.508176\pi$
$379$	−1.00000	−0.0513665	−0.0256833	+	0.999670i	$0.508176\pi$
$380$	0	0
$381$	−15.8745	−0.813276
$382$	−6.00000	−0.306987
$383$	−26.4575	−1.35192	−0.675958	−	0.736940i	$-0.736270\pi$
$383$	−26.4575	−1.35192	−0.675958	+	0.736940i	$0.736270\pi$
$384$	−2.64575	−0.135015
$385$	0	0
$386$	−9.00000	−0.458088
$387$	−32.0000	−1.62665
$388$	0	0
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	0	0
$391$	10.5830	0.535206
$392$	0	0
$393$	42.0000	2.11862
$394$	6.00000	0.302276
$395$	0	0
$396$	20.0000	1.00504
$397$	−26.4575	−1.32786	−0.663932	−	0.747793i	$-0.731114\pi$
$397$	−26.4575	−1.32786	−0.663932	+	0.747793i	$0.731114\pi$
$398$	−5.29150	−0.265239
$399$	0	0
$400$	0	0
$401$	−11.0000	−0.549314	−0.274657	−	0.961542i	$-0.588564\pi$
$401$	−11.0000	−0.549314	−0.274657	+	0.961542i	$0.588564\pi$
$402$	−13.2288	−0.659790
$403$	56.0000	2.78956
$404$	10.5830	0.526524
$405$	0	0
$406$	0	0
$407$	−20.0000	−0.991363
$408$	7.00000	0.346552
$409$	2.64575	0.130824	0.0654120	−	0.997858i	$-0.479164\pi$
$409$	2.64575	0.130824	0.0654120	+	0.997858i	$0.479164\pi$
$410$	0	0
$411$	44.9778	2.21859
$412$	10.5830	0.521387
$413$	0	0
$414$	−16.0000	−0.786357
$415$	0	0
$416$	5.29150	0.259437
$417$	7.00000	0.342791
$418$	−39.6863	−1.94112
$419$	7.93725	0.387760	0.193880	−	0.981025i	$-0.437893\pi$
$419$	7.93725	0.387760	0.193880	+	0.981025i	$0.437893\pi$
$420$	0	0
$421$	20.0000	0.974740	0.487370	−	0.873195i	$-0.337956\pi$
$421$	20.0000	0.974740	0.487370	+	0.873195i	$0.337956\pi$
$422$	−19.0000	−0.924906
$423$	−21.1660	−1.02913
$424$	4.00000	0.194257
$425$	0	0
$426$	−15.8745	−0.769122
$427$	0	0
$428$	15.0000	0.725052
$429$	−70.0000	−3.37963
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	−2.64575	−0.127294
$433$	13.2288	0.635733	0.317867	−	0.948135i	$-0.397034\pi$
$433$	13.2288	0.635733	0.317867	+	0.948135i	$0.397034\pi$
$434$	0	0
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	31.7490	1.51876
$438$	−7.00000	−0.334473
$439$	−21.1660	−1.01020	−0.505099	−	0.863061i	$-0.668544\pi$
$439$	−21.1660	−1.01020	−0.505099	+	0.863061i	$0.668544\pi$
$440$	0	0
$441$	0	0
$442$	−14.0000	−0.665912
$443$	1.00000	0.0475114	0.0237557	−	0.999718i	$-0.492438\pi$
$443$	1.00000	0.0475114	0.0237557	+	0.999718i	$0.492438\pi$
$444$	10.5830	0.502247
$445$	0	0
$446$	−5.29150	−0.250560
$447$	10.5830	0.500559
$448$	0	0
$449$	−15.0000	−0.707894	−0.353947	−	0.935266i	$-0.615161\pi$
$449$	−15.0000	−0.707894	−0.353947	+	0.935266i	$0.615161\pi$
$450$	0	0
$451$	13.2288	0.622918
$452$	13.0000	0.611469
$453$	−31.7490	−1.49170
$454$	15.8745	0.745028
$455$	0	0
$456$	21.0000	0.983415
$457$	3.00000	0.140334	0.0701670	−	0.997535i	$-0.477647\pi$
$457$	3.00000	0.140334	0.0701670	+	0.997535i	$0.477647\pi$
$458$	26.4575	1.23628
$459$	7.00000	0.326732
$460$	0	0
$461$	15.8745	0.739350	0.369675	−	0.929161i	$-0.379469\pi$
$461$	15.8745	0.739350	0.369675	+	0.929161i	$0.379469\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	10.0000	0.463241
$467$	−26.4575	−1.22431	−0.612154	−	0.790739i	$-0.709697\pi$
$467$	−26.4575	−1.22431	−0.612154	+	0.790739i	$0.709697\pi$
$468$	21.1660	0.978399
$469$	0	0
$470$	0	0
$471$	−56.0000	−2.58034
$472$	−5.29150	−0.243561
$473$	−40.0000	−1.83920
$474$	−26.4575	−1.21523
$475$	0	0
$476$	0	0
$477$	16.0000	0.732590
$478$	16.0000	0.731823
$479$	−37.0405	−1.69242	−0.846212	−	0.532846i	$-0.821123\pi$
$479$	−37.0405	−1.69242	−0.846212	+	0.532846i	$0.821123\pi$
$480$	0	0
$481$	−21.1660	−0.965087
$482$	2.64575	0.120511
$483$	0	0
$484$	14.0000	0.636364
$485$	0	0
$486$	21.1660	0.960110
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	5.29150	0.239535
$489$	34.3948	1.55539
$490$	0	0
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	−7.00000	−0.315584
$493$	−15.8745	−0.714952
$494$	−42.0000	−1.88967
$495$	0	0
$496$	10.5830	0.475191
$497$	0	0
$498$	−7.00000	−0.313678
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	0	0
$502$	−13.2288	−0.590428
$503$	15.8745	0.707809	0.353905	−	0.935282i	$-0.384854\pi$
$503$	15.8745	0.707809	0.353905	+	0.935282i	$0.384854\pi$
$504$	0	0
$505$	0	0
$506$	−20.0000	−0.889108
$507$	−39.6863	−1.76253
$508$	6.00000	0.266207
$509$	−15.8745	−0.703625	−0.351813	−	0.936070i	$-0.614435\pi$
$509$	−15.8745	−0.703625	−0.351813	+	0.936070i	$0.614435\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	21.0000	0.927173
$514$	0	0
$515$	0	0
$516$	21.1660	0.931782
$517$	−26.4575	−1.16360
$518$	0	0
$519$	−42.0000	−1.84360
$520$	0	0
$521$	23.8118	1.04321	0.521606	−	0.853186i	$-0.325333\pi$
$521$	23.8118	1.04321	0.521606	+	0.853186i	$0.325333\pi$
$522$	24.0000	1.05045
$523$	13.2288	0.578453	0.289227	−	0.957261i	$-0.406602\pi$
$523$	13.2288	0.578453	0.289227	+	0.957261i	$0.406602\pi$
$524$	−15.8745	−0.693481
$525$	0	0
$526$	−30.0000	−1.30806
$527$	−28.0000	−1.21970
$528$	−13.2288	−0.575708
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−21.1660	−0.918527
$532$	0	0
$533$	14.0000	0.606407
$534$	−35.0000	−1.51460
$535$	0	0
$536$	5.00000	0.215967
$537$	44.9778	1.94093
$538$	5.29150	0.228133
$539$	0	0
$540$	0	0
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	−5.29150	−0.227289
$543$	−14.0000	−0.600798
$544$	−2.64575	−0.113436
$545$	0	0
$546$	0	0
$547$	13.0000	0.555840	0.277920	−	0.960604i	$-0.410355\pi$
$547$	13.0000	0.555840	0.277920	+	0.960604i	$0.410355\pi$
$548$	−17.0000	−0.726204
$549$	21.1660	0.903343
$550$	0	0
$551$	−47.6235	−2.02883
$552$	10.5830	0.450443
$553$	0	0
$554$	−10.0000	−0.424859
$555$	0	0
$556$	−2.64575	−0.112205
$557$	12.0000	0.508456	0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000	0.508456	0.254228	+	0.967144i	$0.418179\pi$
$558$	42.3320	1.79206
$559$	−42.3320	−1.79045
$560$	0	0
$561$	35.0000	1.47770
$562$	−22.0000	−0.928014
$563$	−15.8745	−0.669031	−0.334515	−	0.942390i	$-0.608573\pi$
$563$	−15.8745	−0.669031	−0.334515	+	0.942390i	$0.608573\pi$
$564$	14.0000	0.589506
$565$	0	0
$566$	2.64575	0.111209
$567$	0	0
$568$	6.00000	0.251754
$569$	−11.0000	−0.461144	−0.230572	−	0.973055i	$-0.574060\pi$
$569$	−11.0000	−0.461144	−0.230572	+	0.973055i	$0.574060\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	26.4575	1.10624
$573$	15.8745	0.663167
$574$	0	0
$575$	0	0
$576$	4.00000	0.166667
$577$	34.3948	1.43187	0.715936	−	0.698165i	$-0.246000\pi$
$577$	34.3948	1.43187	0.715936	+	0.698165i	$0.246000\pi$
$578$	−10.0000	−0.415945
$579$	23.8118	0.989583
$580$	0	0
$581$	0	0
$582$	0	0
$583$	20.0000	0.828315
$584$	2.64575	0.109482
$585$	0	0
$586$	15.8745	0.655770
$587$	−34.3948	−1.41962	−0.709812	−	0.704391i	$-0.751220\pi$
$587$	−34.3948	−1.41962	−0.709812	+	0.704391i	$0.751220\pi$
$588$	0	0
$589$	−84.0000	−3.46116
$590$	0	0
$591$	−15.8745	−0.652990
$592$	−4.00000	−0.164399
$593$	2.64575	0.108648	0.0543240	−	0.998523i	$-0.482700\pi$
$593$	2.64575	0.108648	0.0543240	+	0.998523i	$0.482700\pi$
$594$	−13.2288	−0.542782
$595$	0	0
$596$	−4.00000	−0.163846
$597$	14.0000	0.572982
$598$	−21.1660	−0.865543
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	0	0
$601$	−44.9778	−1.83468	−0.917341	−	0.398103i	$-0.869669\pi$
$601$	−44.9778	−1.83468	−0.917341	+	0.398103i	$0.869669\pi$
$602$	0	0
$603$	20.0000	0.814463
$604$	12.0000	0.488273
$605$	0	0
$606$	−28.0000	−1.13742
$607$	−10.5830	−0.429551	−0.214775	−	0.976663i	$-0.568902\pi$
$607$	−10.5830	−0.429551	−0.214775	+	0.976663i	$0.568902\pi$
$608$	−7.93725	−0.321898
$609$	0	0
$610$	0	0
$611$	−28.0000	−1.13276
$612$	−10.5830	−0.427793
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	34.3948	1.38806
$615$	0	0
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	−28.0000	−1.12633
$619$	15.8745	0.638050	0.319025	−	0.947746i	$-0.396645\pi$
$619$	15.8745	0.638050	0.319025	+	0.947746i	$0.396645\pi$
$620$	0	0
$621$	10.5830	0.424681
$622$	−5.29150	−0.212170
$623$	0	0
$624$	−14.0000	−0.560449
$625$	0	0
$626$	−21.1660	−0.845964
$627$	105.000	4.19330
$628$	21.1660	0.844616
$629$	10.5830	0.421972
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	10.0000	0.397779
$633$	50.2693	1.99802
$634$	24.0000	0.953162
$635$	0	0
$636$	−10.5830	−0.419643
$637$	0	0
$638$	30.0000	1.18771
$639$	24.0000	0.949425
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	−39.6863	−1.56629
$643$	15.8745	0.626029	0.313015	−	0.949748i	$-0.398661\pi$
$643$	15.8745	0.626029	0.313015	+	0.949748i	$0.398661\pi$
$644$	0	0
$645$	0	0
$646$	21.0000	0.826234
$647$	10.5830	0.416061	0.208030	−	0.978122i	$-0.433295\pi$
$647$	10.5830	0.416061	0.208030	+	0.978122i	$0.433295\pi$
$648$	−5.00000	−0.196419
$649$	−26.4575	−1.03855
$650$	0	0
$651$	0	0
$652$	−13.0000	−0.509119
$653$	8.00000	0.313064	0.156532	−	0.987673i	$-0.449969\pi$
$653$	8.00000	0.313064	0.156532	+	0.987673i	$0.449969\pi$
$654$	5.29150	0.206914
$655$	0	0
$656$	2.64575	0.103299
$657$	10.5830	0.412882
$658$	0	0
$659$	23.0000	0.895953	0.447976	−	0.894045i	$-0.352145\pi$
$659$	23.0000	0.895953	0.447976	+	0.894045i	$0.352145\pi$
$660$	0	0
$661$	−37.0405	−1.44071	−0.720355	−	0.693606i	$-0.756021\pi$
$661$	−37.0405	−1.44071	−0.720355	+	0.693606i	$0.756021\pi$
$662$	17.0000	0.660724
$663$	37.0405	1.43853
$664$	2.64575	0.102675
$665$	0	0
$666$	−16.0000	−0.619987
$667$	−24.0000	−0.929284
$668$	0	0
$669$	14.0000	0.541271
$670$	0	0
$671$	26.4575	1.02138
$672$	0	0
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	−5.00000	−0.192593
$675$	0	0
$676$	15.0000	0.576923
$677$	−10.5830	−0.406738	−0.203369	−	0.979102i	$-0.565189\pi$
$677$	−10.5830	−0.406738	−0.203369	+	0.979102i	$0.565189\pi$
$678$	−34.3948	−1.32092
$679$	0	0
$680$	0	0
$681$	−42.0000	−1.60944
$682$	52.9150	2.02622
$683$	−9.00000	−0.344375	−0.172188	−	0.985064i	$-0.555084\pi$
$683$	−9.00000	−0.344375	−0.172188	+	0.985064i	$0.555084\pi$
$684$	−31.7490	−1.21395
$685$	0	0
$686$	0	0
$687$	−70.0000	−2.67067
$688$	−8.00000	−0.304997
$689$	21.1660	0.806361
$690$	0	0
$691$	−18.5203	−0.704544	−0.352272	−	0.935898i	$-0.614591\pi$
$691$	−18.5203	−0.704544	−0.352272	+	0.935898i	$0.614591\pi$
$692$	15.8745	0.603458
$693$	0	0
$694$	5.00000	0.189797
$695$	0	0
$696$	−15.8745	−0.601722
$697$	−7.00000	−0.265144
$698$	−21.1660	−0.801145
$699$	−26.4575	−1.00072
$700$	0	0
$701$	16.0000	0.604312	0.302156	−	0.953259i	$-0.402294\pi$
$701$	16.0000	0.604312	0.302156	+	0.953259i	$0.402294\pi$
$702$	−14.0000	−0.528396
$703$	31.7490	1.19744
$704$	5.00000	0.188445
$705$	0	0
$706$	−31.7490	−1.19489
$707$	0	0
$708$	14.0000	0.526152
$709$	36.0000	1.35201	0.676004	−	0.736898i	$-0.263710\pi$
$709$	36.0000	1.35201	0.676004	+	0.736898i	$0.263710\pi$
$710$	0	0
$711$	40.0000	1.50012
$712$	13.2288	0.495769
$713$	−42.3320	−1.58535
$714$	0	0
$715$	0	0
$716$	−17.0000	−0.635320
$717$	−42.3320	−1.58092
$718$	10.0000	0.373197
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	44.0000	1.63751
$723$	−7.00000	−0.260333
$724$	5.29150	0.196657
$725$	0	0
$726$	−37.0405	−1.37470
$727$	−26.4575	−0.981255	−0.490627	−	0.871369i	$-0.663232\pi$
$727$	−26.4575	−0.981255	−0.490627	+	0.871369i	$0.663232\pi$
$728$	0	0
$729$	−41.0000	−1.51852
$730$	0	0
$731$	21.1660	0.782853
$732$	−14.0000	−0.517455
$733$	31.7490	1.17268	0.586338	−	0.810066i	$-0.300569\pi$
$733$	31.7490	1.17268	0.586338	+	0.810066i	$0.300569\pi$
$734$	−10.5830	−0.390626
$735$	0	0
$736$	−4.00000	−0.147442
$737$	25.0000	0.920887
$738$	10.5830	0.389566
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	0	0
$741$	111.122	4.08215
$742$	0	0
$743$	−12.0000	−0.440237	−0.220119	−	0.975473i	$-0.570644\pi$
$743$	−12.0000	−0.440237	−0.220119	+	0.975473i	$0.570644\pi$
$744$	−28.0000	−1.02653
$745$	0	0
$746$	24.0000	0.878702
$747$	10.5830	0.387212
$748$	−13.2288	−0.483691
$749$	0	0
$750$	0	0
$751$	22.0000	0.802791	0.401396	−	0.915905i	$-0.368525\pi$
$751$	22.0000	0.802791	0.401396	+	0.915905i	$0.368525\pi$
$752$	−5.29150	−0.192961
$753$	35.0000	1.27547
$754$	31.7490	1.15623
$755$	0	0
$756$	0	0
$757$	34.0000	1.23575	0.617876	−	0.786276i	$-0.287994\pi$
$757$	34.0000	1.23575	0.617876	+	0.786276i	$0.287994\pi$
$758$	−1.00000	−0.0363216
$759$	52.9150	1.92069
$760$	0	0
$761$	−39.6863	−1.43863	−0.719313	−	0.694686i	$-0.755543\pi$
$761$	−39.6863	−1.43863	−0.719313	+	0.694686i	$0.755543\pi$
$762$	−15.8745	−0.575073
$763$	0	0
$764$	−6.00000	−0.217072
$765$	0	0
$766$	−26.4575	−0.955949
$767$	−28.0000	−1.01102
$768$	−2.64575	−0.0954703
$769$	−13.2288	−0.477041	−0.238521	−	0.971137i	$-0.576662\pi$
$769$	−13.2288	−0.477041	−0.238521	+	0.971137i	$0.576662\pi$
$770$	0	0
$771$	0	0
$772$	−9.00000	−0.323917
$773$	31.7490	1.14193	0.570966	−	0.820973i	$-0.306569\pi$
$773$	31.7490	1.14193	0.570966	+	0.820973i	$0.306569\pi$
$774$	−32.0000	−1.15022
$775$	0	0
$776$	0	0
$777$	0	0
$778$	12.0000	0.430221
$779$	−21.0000	−0.752403
$780$	0	0
$781$	30.0000	1.07348
$782$	10.5830	0.378447
$783$	−15.8745	−0.567309
$784$	0	0
$785$	0	0
$786$	42.0000	1.49809
$787$	15.8745	0.565865	0.282933	−	0.959140i	$-0.408693\pi$
$787$	15.8745	0.565865	0.282933	+	0.959140i	$0.408693\pi$
$788$	6.00000	0.213741
$789$	79.3725	2.82574
$790$	0	0
$791$	0	0
$792$	20.0000	0.710669
$793$	28.0000	0.994309
$794$	−26.4575	−0.938942
$795$	0	0
$796$	−5.29150	−0.187552
$797$	5.29150	0.187435	0.0937173	−	0.995599i	$-0.470125\pi$
$797$	5.29150	0.187435	0.0937173	+	0.995599i	$0.470125\pi$
$798$	0	0
$799$	14.0000	0.495284
$800$	0	0
$801$	52.9150	1.86966
$802$	−11.0000	−0.388424
$803$	13.2288	0.466833
$804$	−13.2288	−0.466542
$805$	0	0
$806$	56.0000	1.97252
$807$	−14.0000	−0.492823
$808$	10.5830	0.372309
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	0	0
$811$	37.0405	1.30067	0.650334	−	0.759648i	$-0.274629\pi$
$811$	37.0405	1.30067	0.650334	+	0.759648i	$0.274629\pi$
$812$	0	0
$813$	14.0000	0.491001
$814$	−20.0000	−0.701000
$815$	0	0
$816$	7.00000	0.245049
$817$	63.4980	2.22151
$818$	2.64575	0.0925065
$819$	0	0
$820$	0	0
$821$	−48.0000	−1.67521	−0.837606	−	0.546275i	$-0.816045\pi$
$821$	−48.0000	−1.67521	−0.837606	+	0.546275i	$0.816045\pi$
$822$	44.9778	1.56878
$823$	26.0000	0.906303	0.453152	−	0.891434i	$-0.350300\pi$
$823$	26.0000	0.906303	0.453152	+	0.891434i	$0.350300\pi$
$824$	10.5830	0.368676
$825$	0	0
$826$	0	0
$827$	−43.0000	−1.49526	−0.747628	−	0.664117i	$-0.768807\pi$
$827$	−43.0000	−1.49526	−0.747628	+	0.664117i	$0.768807\pi$
$828$	−16.0000	−0.556038
$829$	−42.3320	−1.47025	−0.735126	−	0.677931i	$-0.762877\pi$
$829$	−42.3320	−1.47025	−0.735126	+	0.677931i	$0.762877\pi$
$830$	0	0
$831$	26.4575	0.917801
$832$	5.29150	0.183450
$833$	0	0
$834$	7.00000	0.242390
$835$	0	0
$836$	−39.6863	−1.37258
$837$	−28.0000	−0.967822
$838$	7.93725	0.274188
$839$	15.8745	0.548049	0.274024	−	0.961723i	$-0.411645\pi$
$839$	15.8745	0.548049	0.274024	+	0.961723i	$0.411645\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	20.0000	0.689246
$843$	58.2065	2.00474
$844$	−19.0000	−0.654007
$845$	0	0
$846$	−21.1660	−0.727702
$847$	0	0
$848$	4.00000	0.137361
$849$	−7.00000	−0.240239
$850$	0	0
$851$	16.0000	0.548473
$852$	−15.8745	−0.543852
$853$	5.29150	0.181178	0.0905888	−	0.995888i	$-0.471125\pi$
$853$	5.29150	0.181178	0.0905888	+	0.995888i	$0.471125\pi$
$854$	0	0
$855$	0	0
$856$	15.0000	0.512689
$857$	−39.6863	−1.35566	−0.677829	−	0.735220i	$-0.737079\pi$
$857$	−39.6863	−1.35566	−0.677829	+	0.735220i	$0.737079\pi$
$858$	−70.0000	−2.38976
$859$	−2.64575	−0.0902719	−0.0451359	−	0.998981i	$-0.514372\pi$
$859$	−2.64575	−0.0902719	−0.0451359	+	0.998981i	$0.514372\pi$
$860$	0	0
$861$	0	0
$862$	−24.0000	−0.817443
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	−2.64575	−0.0900103
$865$	0	0
$866$	13.2288	0.449531
$867$	26.4575	0.898544
$868$	0	0
$869$	50.0000	1.69613
$870$	0	0
$871$	26.4575	0.896479
$872$	−2.00000	−0.0677285
$873$	0	0
$874$	31.7490	1.07393
$875$	0	0
$876$	−7.00000	−0.236508
$877$	−48.0000	−1.62084	−0.810422	−	0.585846i	$-0.800762\pi$
$877$	−48.0000	−1.62084	−0.810422	+	0.585846i	$0.800762\pi$
$878$	−21.1660	−0.714318
$879$	−42.0000	−1.41662
$880$	0	0
$881$	−42.3320	−1.42620	−0.713101	−	0.701061i	$-0.752710\pi$
$881$	−42.3320	−1.42620	−0.713101	+	0.701061i	$0.752710\pi$
$882$	0	0
$883$	−29.0000	−0.975928	−0.487964	−	0.872864i	$-0.662260\pi$
$883$	−29.0000	−0.975928	−0.487964	+	0.872864i	$0.662260\pi$
$884$	−14.0000	−0.470871
$885$	0	0
$886$	1.00000	0.0335957
$887$	26.4575	0.888356	0.444178	−	0.895938i	$-0.353496\pi$
$887$	26.4575	0.888356	0.444178	+	0.895938i	$0.353496\pi$
$888$	10.5830	0.355142
$889$	0	0
$890$	0	0
$891$	−25.0000	−0.837532
$892$	−5.29150	−0.177173
$893$	42.0000	1.40548
$894$	10.5830	0.353949
$895$	0	0
$896$	0	0
$897$	56.0000	1.86979
$898$	−15.0000	−0.500556
$899$	63.4980	2.11778
$900$	0	0
$901$	−10.5830	−0.352571
$902$	13.2288	0.440469
$903$	0	0
$904$	13.0000	0.432374
$905$	0	0
$906$	−31.7490	−1.05479
$907$	4.00000	0.132818	0.0664089	−	0.997792i	$-0.478846\pi$
$907$	4.00000	0.132818	0.0664089	+	0.997792i	$0.478846\pi$
$908$	15.8745	0.526814
$909$	42.3320	1.40406
$910$	0	0
$911$	−8.00000	−0.265052	−0.132526	−	0.991180i	$-0.542309\pi$
$911$	−8.00000	−0.265052	−0.132526	+	0.991180i	$0.542309\pi$
$912$	21.0000	0.695379
$913$	13.2288	0.437808
$914$	3.00000	0.0992312
$915$	0	0
$916$	26.4575	0.874181
$917$	0	0
$918$	7.00000	0.231034
$919$	38.0000	1.25350	0.626752	−	0.779219i	$-0.284384\pi$
$919$	38.0000	1.25350	0.626752	+	0.779219i	$0.284384\pi$
$920$	0	0
$921$	−91.0000	−2.99855
$922$	15.8745	0.522799
$923$	31.7490	1.04503
$924$	0	0
$925$	0	0
$926$	16.0000	0.525793
$927$	42.3320	1.39037
$928$	6.00000	0.196960
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	0	0
$932$	10.0000	0.327561
$933$	14.0000	0.458339
$934$	−26.4575	−0.865716
$935$	0	0
$936$	21.1660	0.691833
$937$	7.93725	0.259299	0.129649	−	0.991560i	$-0.458615\pi$
$937$	7.93725	0.259299	0.129649	+	0.991560i	$0.458615\pi$
$938$	0	0
$939$	56.0000	1.82749
$940$	0	0
$941$	−42.3320	−1.37998	−0.689992	−	0.723817i	$-0.742386\pi$
$941$	−42.3320	−1.37998	−0.689992	+	0.723817i	$0.742386\pi$
$942$	−56.0000	−1.82458
$943$	−10.5830	−0.344630
$944$	−5.29150	−0.172224
$945$	0	0
$946$	−40.0000	−1.30051
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	−26.4575	−0.859300
$949$	14.0000	0.454459
$950$	0	0
$951$	−63.4980	−2.05906
$952$	0	0
$953$	41.0000	1.32812	0.664060	−	0.747679i	$-0.268832\pi$
$953$	41.0000	1.32812	0.664060	+	0.747679i	$0.268832\pi$
$954$	16.0000	0.518019
$955$	0	0
$956$	16.0000	0.517477
$957$	−79.3725	−2.56575
$958$	−37.0405	−1.19672
$959$	0	0
$960$	0	0
$961$	81.0000	2.61290
$962$	−21.1660	−0.682420
$963$	60.0000	1.93347
$964$	2.64575	0.0852139
$965$	0	0
$966$	0	0
$967$	2.00000	0.0643157	0.0321578	−	0.999483i	$-0.489762\pi$
$967$	2.00000	0.0643157	0.0321578	+	0.999483i	$0.489762\pi$
$968$	14.0000	0.449977
$969$	−55.5608	−1.78487
$970$	0	0
$971$	−29.1033	−0.933968	−0.466984	−	0.884266i	$-0.654659\pi$
$971$	−29.1033	−0.933968	−0.466984	+	0.884266i	$0.654659\pi$
$972$	21.1660	0.678900
$973$	0	0
$974$	−2.00000	−0.0640841
$975$	0	0
$976$	5.29150	0.169377
$977$	−9.00000	−0.287936	−0.143968	−	0.989582i	$-0.545986\pi$
$977$	−9.00000	−0.287936	−0.143968	+	0.989582i	$0.545986\pi$
$978$	34.3948	1.09982
$979$	66.1438	2.11396
$980$	0	0
$981$	−8.00000	−0.255420
$982$	−36.0000	−1.14881
$983$	−42.3320	−1.35018	−0.675091	−	0.737735i	$-0.735896\pi$
$983$	−42.3320	−1.35018	−0.675091	+	0.737735i	$0.735896\pi$
$984$	−7.00000	−0.223152
$985$	0	0
$986$	−15.8745	−0.505547
$987$	0	0
$988$	−42.0000	−1.33620
$989$	32.0000	1.01754
$990$	0	0
$991$	12.0000	0.381193	0.190596	−	0.981669i	$-0.438958\pi$
$991$	12.0000	0.381193	0.190596	+	0.981669i	$0.438958\pi$
$992$	10.5830	0.336011
$993$	−44.9778	−1.42733
$994$	0	0
$995$	0	0
$996$	−7.00000	−0.221803
$997$	−47.6235	−1.50825	−0.754126	−	0.656730i	$-0.771939\pi$
$997$	−47.6235	−1.50825	−0.754126	+	0.656730i	$0.771939\pi$
$998$	−20.0000	−0.633089
$999$	10.5830	0.334831

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.2.a.br.1.1	yes	2
5.2	odd	4		2450.2.c.u.99.4		4
5.3	odd	4		2450.2.c.u.99.1		4
5.4	even	2		2450.2.a.bm.1.2	yes	2
7.6	odd	2	inner	2450.2.a.br.1.2	yes	2
35.13	even	4		2450.2.c.u.99.2		4
35.27	even	4		2450.2.c.u.99.3		4
35.34	odd	2		2450.2.a.bm.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2450.2.a.bm.1.1	✓	2	35.34	odd	2
2450.2.a.bm.1.2	yes	2	5.4	even	2
2450.2.a.br.1.1	yes	2	1.1	even	1	trivial
2450.2.a.br.1.2	yes	2	7.6	odd	2	inner
2450.2.c.u.99.1		4	5.3	odd	4
2450.2.c.u.99.2		4	35.13	even	4
2450.2.c.u.99.3		4	35.27	even	4
2450.2.c.u.99.4		4	5.2	odd	4

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:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2450.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.64575 q^{3} +1.00000 q^{4} -2.64575 q^{6} +1.00000 q^{8} +4.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.64575 q^{3} +1.00000 q^{4} -2.64575 q^{6} +1.00000 q^{8} +4.00000 q^{9} +5.00000 q^{11} -2.64575 q^{12} +5.29150 q^{13} +1.00000 q^{16} -2.64575 q^{17} +4.00000 q^{18} -7.93725 q^{19} +5.00000 q^{22} -4.00000 q^{23} -2.64575 q^{24} +5.29150 q^{26} -2.64575 q^{27} +6.00000 q^{29} +10.5830 q^{31} +1.00000 q^{32} -13.2288 q^{33} -2.64575 q^{34} +4.00000 q^{36} -4.00000 q^{37} -7.93725 q^{38} -14.0000 q^{39} +2.64575 q^{41} -8.00000 q^{43} +5.00000 q^{44} -4.00000 q^{46} -5.29150 q^{47} -2.64575 q^{48} +7.00000 q^{51} +5.29150 q^{52} +4.00000 q^{53} -2.64575 q^{54} +21.0000 q^{57} +6.00000 q^{58} -5.29150 q^{59} +5.29150 q^{61} +10.5830 q^{62} +1.00000 q^{64} -13.2288 q^{66} +5.00000 q^{67} -2.64575 q^{68} +10.5830 q^{69} +6.00000 q^{71} +4.00000 q^{72} +2.64575 q^{73} -4.00000 q^{74} -7.93725 q^{76} -14.0000 q^{78} +10.0000 q^{79} -5.00000 q^{81} +2.64575 q^{82} +2.64575 q^{83} -8.00000 q^{86} -15.8745 q^{87} +5.00000 q^{88} +13.2288 q^{89} -4.00000 q^{92} -28.0000 q^{93} -5.29150 q^{94} -2.64575 q^{96} +20.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 8 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 + 8 * q^9	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 8 q^{9} + 10 q^{11} + 2 q^{16} + 8 q^{18} + 10 q^{22} - 8 q^{23} + 12 q^{29} + 2 q^{32} + 8 q^{36} - 8 q^{37} - 28 q^{39} - 16 q^{43} + 10 q^{44} - 8 q^{46} + 14 q^{51} + 8 q^{53} + 42 q^{57} + 12 q^{58} + 2 q^{64} + 10 q^{67} + 12 q^{71} + 8 q^{72} - 8 q^{74} - 28 q^{78} + 20 q^{79} - 10 q^{81} - 16 q^{86} + 10 q^{88} - 8 q^{92} - 56 q^{93} + 40 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 + 8 * q^9 + 10 * q^11 + 2 * q^16 + 8 * q^18 + 10 * q^22 - 8 * q^23 + 12 * q^29 + 2 * q^32 + 8 * q^36 - 8 * q^37 - 28 * q^39 - 16 * q^43 + 10 * q^44 - 8 * q^46 + 14 * q^51 + 8 * q^53 + 42 * q^57 + 12 * q^58 + 2 * q^64 + 10 * q^67 + 12 * q^71 + 8 * q^72 - 8 * q^74 - 28 * q^78 + 20 * q^79 - 10 * q^81 - 16 * q^86 + 10 * q^88 - 8 * q^92 - 56 * q^93 + 40 * q^99

Introduction

L-functions

Modular forms

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