LMFDB - Embedded newform 225.12.b.f.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,12,Mod(199,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 12, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("225.199");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	225.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$172.877215626$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{151})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 75x^{2} + 1444 $ x^4 - 75*x^2 + 1444
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 5)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.1
Root			$-6.14410 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	225.199
Dual form			225.12.b.f.199.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-83.7292i q^{2} -4962.58 q^{4} -15973.7i q^{7} +244036. i q^{8} +O(q^{10})$	$q-83.7292i q^{2} -4962.58 q^{4} -15973.7i q^{7} +244036. i q^{8} -339729. q^{11} +2.02328e6i q^{13} -1.33746e6 q^{14} +1.02696e7 q^{16} -2.45063e6i q^{17} +4.08504e6 q^{19} +2.84453e7i q^{22} -2.86497e7i q^{23} +1.69408e8 q^{26} +7.92706e7i q^{28} -9.41230e6 q^{29} +2.99399e8 q^{31} -3.60078e8i q^{32} -2.05190e8 q^{34} +4.57279e8i q^{37} -3.42037e8i q^{38} -1.83814e8 q^{41} +6.56811e8i q^{43} +1.68594e9 q^{44} -2.39882e9 q^{46} -1.97090e8i q^{47} +1.72217e9 q^{49} -1.00407e10i q^{52} -5.15890e9i q^{53} +3.89815e9 q^{56} +7.88085e8i q^{58} -6.62200e8 q^{59} +5.58296e8 q^{61} -2.50684e10i q^{62} -9.11694e9 q^{64} -1.01206e10i q^{67} +1.21615e10i q^{68} -1.78161e10 q^{71} -2.33380e10i q^{73} +3.82876e10 q^{74} -2.02724e10 q^{76} +5.42672e9i q^{77} -1.24957e10 q^{79} +1.53906e10i q^{82} -3.37037e10i q^{83} +5.49943e10 q^{86} -8.29062e10i q^{88} +2.94282e10 q^{89} +3.23192e10 q^{91} +1.42177e11i q^{92} -1.65022e10 q^{94} +1.13262e11i q^{97} -1.44196e11i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 13952 q^{4}+O(q^{10}) $ 4 * q - 13952 * q^4	$ 4 q - 13952 q^{4} + 1236352 q^{11} + 2668944 q^{14} + 12011264 q^{16} - 10650640 q^{19} + 161523472 q^{26} + 188280760 q^{29} + 489086928 q^{31} - 890716144 q^{34} + 1491486632 q^{41} - 485451776 q^{44} - 936389712 q^{46} + 3883354828 q^{49} + 7981107840 q^{56} + 14635031120 q^{59} - 3032851352 q^{61} - 1639063552 q^{64} - 65876943088 q^{71} + 137536396144 q^{74} - 2650785280 q^{76} + 6605646240 q^{79} + 113412187792 q^{86} - 25349541720 q^{89} + 181275718128 q^{91} + 120579531056 q^{94}+O(q^{100}) $ 4 * q - 13952 * q^4 + 1236352 * q^11 + 2668944 * q^14 + 12011264 * q^16 - 10650640 * q^19 + 161523472 * q^26 + 188280760 * q^29 + 489086928 * q^31 - 890716144 * q^34 + 1491486632 * q^41 - 485451776 * q^44 - 936389712 * q^46 + 3883354828 * q^49 + 7981107840 * q^56 + 14635031120 * q^59 - 3032851352 * q^61 - 1639063552 * q^64 - 65876943088 * q^71 + 137536396144 * q^74 - 2650785280 * q^76 + 6605646240 * q^79 + 113412187792 * q^86 - 25349541720 * q^89 + 181275718128 * q^91 + 120579531056 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 83.7292i	− 1.85017i	−0.379757	−	0.925086i	$-0.623993\pi$
$2$	− 83.7292i	− 1.85017i	0.379757	−	0.925086i	$-0.376007\pi$
$3$	0	0
$3$	0	0
$4$	−4962.58	−2.42314
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 15973.7i	− 0.359224i	−0.983738	−	0.179612i	$-0.942516\pi$
$7$	− 15973.7i	− 0.359224i	0.983738	−	0.179612i	$-0.0574842\pi$
$8$	244036.i	2.63305i
$9$	0	0
$10$	0	0
$11$	−339729.	−0.636024	−0.318012	−	0.948087i	$-0.603015\pi$
$11$	−339729.	−0.636024	−0.318012	+	0.948087i	$0.603015\pi$
$12$	0	0
$13$	2.02328e6i	1.51136i	0.654941	+	0.755680i	$0.272693\pi$
$13$	2.02328e6i	1.51136i	−0.654941	+	0.755680i	$0.727307\pi$
$14$	−1.33746e6	−0.664626
$15$	0	0
$16$	1.02696e7	2.44846
$17$	− 2.45063e6i	− 0.418610i	−0.977850	−	0.209305i	$-0.932880\pi$
$17$	− 2.45063e6i	− 0.418610i	0.977850	−	0.209305i	$-0.0671201\pi$
$18$	0	0
$19$	4.08504e6	0.378487	0.189244	−	0.981930i	$-0.439396\pi$
$19$	4.08504e6	0.378487	0.189244	+	0.981930i	$0.439396\pi$
$20$	0	0
$21$	0	0
$22$	2.84453e7i	1.17675i
$23$	− 2.86497e7i	− 0.928149i	−0.885796	−	0.464074i	$-0.846387\pi$
$23$	− 2.86497e7i	− 0.928149i	0.885796	−	0.464074i	$-0.153613\pi$
$24$	0	0
$25$	0	0
$26$	1.69408e8	2.79628
$27$	0	0
$28$	7.92706e7i	0.870448i
$29$	−9.41230e6	−0.0852132	−0.0426066	−	0.999092i	$-0.513566\pi$
$29$	−9.41230e6	−0.0852132	−0.0426066	+	0.999092i	$0.513566\pi$
$30$	0	0
$31$	2.99399e8	1.87828	0.939141	−	0.343532i	$-0.111623\pi$
$31$	2.99399e8	1.87828	0.939141	+	0.343532i	$0.111623\pi$
$32$	− 3.60078e8i	− 1.89702i
$33$	0	0
$34$	−2.05190e8	−0.774500
$35$	0	0
$36$	0	0
$37$	4.57279e8i	1.08411i	0.840344	+	0.542053i	$0.182353\pi$
$37$	4.57279e8i	1.08411i	−0.840344	+	0.542053i	$0.817647\pi$
$38$	− 3.42037e8i	− 0.700267i
$39$	0	0
$40$	0	0
$41$	−1.83814e8	−0.247780	−0.123890	−	0.992296i	$-0.539537\pi$
$41$	−1.83814e8	−0.247780	−0.123890	+	0.992296i	$0.539537\pi$
$42$	0	0
$43$	6.56811e8i	0.681340i	0.940183	+	0.340670i	$0.110654\pi$
$43$	6.56811e8i	0.681340i	−0.940183	+	0.340670i	$0.889346\pi$
$44$	1.68594e9	1.54117
$45$	0	0
$46$	−2.39882e9	−1.71724
$47$	− 1.97090e8i	− 0.125350i	−0.998034	−	0.0626752i	$-0.980037\pi$
$47$	− 1.97090e8i	− 0.125350i	0.998034	−	0.0626752i	$-0.0199632\pi$
$48$	0	0
$49$	1.72217e9	0.870958
$50$	0	0
$51$	0	0
$52$	− 1.00407e10i	− 3.66223i
$53$	− 5.15890e9i	− 1.69449i	−0.531199	−	0.847247i	$-0.678258\pi$
$53$	− 5.15890e9i	− 1.69449i	0.531199	−	0.847247i	$-0.321742\pi$
$54$	0	0
$55$	0	0
$56$	3.89815e9	0.945854
$57$	0	0
$58$	7.88085e8i	0.157659i
$59$	−6.62200e8	−0.120588	−0.0602939	−	0.998181i	$-0.519204\pi$
$59$	−6.62200e8	−0.120588	−0.0602939	+	0.998181i	$0.519204\pi$
$60$	0	0
$61$	5.58296e8	0.0846351	0.0423176	−	0.999104i	$-0.486526\pi$
$61$	5.58296e8	0.0846351	0.0423176	+	0.999104i	$0.486526\pi$
$62$	− 2.50684e10i	− 3.47515i
$63$	0	0
$64$	−9.11694e9	−1.06135
$65$	0	0
$66$	0	0
$67$	− 1.01206e10i	− 0.915787i	−0.889007	−	0.457894i	$-0.848604\pi$
$67$	− 1.01206e10i	− 0.915787i	0.889007	−	0.457894i	$-0.151396\pi$
$68$	1.21615e10i	1.01435i
$69$	0	0
$70$	0	0
$71$	−1.78161e10	−1.17190	−0.585952	−	0.810346i	$-0.699279\pi$
$71$	−1.78161e10	−1.17190	−0.585952	+	0.810346i	$0.699279\pi$
$72$	0	0
$73$	− 2.33380e10i	− 1.31761i	−0.752312	−	0.658807i	$-0.771061\pi$
$73$	− 2.33380e10i	− 1.31761i	0.752312	−	0.658807i	$-0.228939\pi$
$74$	3.82876e10	2.00578
$75$	0	0
$76$	−2.02724e10	−0.917127
$77$	5.42672e9i	0.228475i
$78$	0	0
$79$	−1.24957e10	−0.456888	−0.228444	−	0.973557i	$-0.573364\pi$
$79$	−1.24957e10	−0.456888	−0.228444	+	0.973557i	$0.573364\pi$
$80$	0	0
$81$	0	0
$82$	1.53906e10i	0.458436i
$83$	− 3.37037e10i	− 0.939179i	−0.882885	−	0.469589i	$-0.844402\pi$
$83$	− 3.37037e10i	− 0.939179i	0.882885	−	0.469589i	$-0.155598\pi$
$84$	0	0
$85$	0	0
$86$	5.49943e10	1.26060
$87$	0	0
$88$	− 8.29062e10i	− 1.67468i
$89$	2.94282e10	0.558623	0.279311	−	0.960201i	$-0.409894\pi$
$89$	2.94282e10	0.558623	0.279311	+	0.960201i	$0.409894\pi$
$90$	0	0
$91$	3.23192e10	0.542916
$92$	1.42177e11i	2.24903i
$93$	0	0
$94$	−1.65022e10	−0.231920
$95$	0	0
$96$	0	0
$97$	1.13262e11i	1.33918i	0.742732	+	0.669588i	$0.233529\pi$
$97$	1.13262e11i	1.33918i	−0.742732	+	0.669588i	$0.766471\pi$
$98$	− 1.44196e11i	− 1.61142i
$99$	0	0
$100$	0	0
$101$	−1.19564e11	−1.13196	−0.565982	−	0.824417i	$-0.691503\pi$
$101$	−1.19564e11	−1.13196	−0.565982	+	0.824417i	$0.691503\pi$
$102$	0	0
$103$	− 9.46874e10i	− 0.804799i	−0.915464	−	0.402399i	$-0.868176\pi$
$103$	− 9.46874e10i	− 0.804799i	0.915464	−	0.402399i	$-0.131824\pi$
$104$	−4.93753e11	−3.97948
$105$	0	0
$106$	−4.31951e11	−3.13511
$107$	1.52769e11i	1.05299i	0.850177	+	0.526497i	$0.176495\pi$
$107$	1.52769e11i	1.05299i	−0.850177	+	0.526497i	$0.823505\pi$
$108$	0	0
$109$	−2.85078e11	−1.77467	−0.887337	−	0.461122i	$-0.847447\pi$
$109$	−2.85078e11	−1.77467	−0.887337	+	0.461122i	$0.847447\pi$
$110$	0	0
$111$	0	0
$112$	− 1.64043e11i	− 0.879544i
$113$	2.35563e11i	1.20275i	0.798968	+	0.601374i	$0.205380\pi$
$113$	2.35563e11i	1.20275i	−0.798968	+	0.601374i	$0.794620\pi$
$114$	0	0
$115$	0	0
$116$	4.67093e10	0.206483
$117$	0	0
$118$	5.54455e10i	0.223108i
$119$	−3.91456e10	−0.150375
$120$	0	0
$121$	−1.69896e11	−0.595474
$122$	− 4.67457e10i	− 0.156590i
$123$	0	0
$124$	−1.48579e12	−4.55133
$125$	0	0
$126$	0	0
$127$	1.25786e11i	0.337840i	0.985630	+	0.168920i	$0.0540279\pi$
$127$	1.25786e11i	0.337840i	−0.985630	+	0.168920i	$0.945972\pi$
$128$	2.59156e10i	0.0666664i
$129$	0	0
$130$	0	0
$131$	2.97347e11	0.673397	0.336698	−	0.941613i	$-0.390690\pi$
$131$	2.97347e11	0.673397	0.336698	+	0.941613i	$0.390690\pi$
$132$	0	0
$133$	− 6.52530e10i	− 0.135962i
$134$	−8.47389e11	−1.69436
$135$	0	0
$136$	5.98043e11	1.10222
$137$	− 6.29203e10i	− 0.111385i	−0.998448	−	0.0556926i	$-0.982263\pi$
$137$	− 6.29203e10i	− 0.111385i	0.998448	−	0.0556926i	$-0.0177367\pi$
$138$	0	0
$139$	5.11416e11	0.835974	0.417987	−	0.908453i	$-0.362736\pi$
$139$	5.11416e11	0.835974	0.417987	+	0.908453i	$0.362736\pi$
$140$	0	0
$141$	0	0
$142$	1.49173e12i	2.16823i
$143$	− 6.87368e11i	− 0.961260i
$144$	0	0
$145$	0	0
$146$	−1.95407e12	−2.43781
$147$	0	0
$148$	− 2.26929e12i	− 2.62694i
$149$	7.94154e11	0.885891	0.442945	−	0.896549i	$-0.353934\pi$
$149$	7.94154e11	0.885891	0.442945	+	0.896549i	$0.353934\pi$
$150$	0	0
$151$	1.24629e12	1.29195	0.645974	−	0.763360i	$-0.276452\pi$
$151$	1.24629e12	1.29195	0.645974	+	0.763360i	$0.276452\pi$
$152$	9.96896e11i	0.996576i
$153$	0	0
$154$	4.54375e11	0.422718
$155$	0	0
$156$	0	0
$157$	− 9.73519e11i	− 0.814510i	−0.913315	−	0.407255i	$-0.866486\pi$
$157$	− 9.73519e11i	− 0.814510i	0.913315	−	0.407255i	$-0.133514\pi$
$158$	1.04625e12i	0.845322i
$159$	0	0
$160$	0	0
$161$	−4.57641e11	−0.333413
$162$	0	0
$163$	1.26084e12i	0.858281i	0.903238	+	0.429140i	$0.141183\pi$
$163$	1.26084e12i	0.858281i	−0.903238	+	0.429140i	$0.858817\pi$
$164$	9.12190e11	0.600405
$165$	0	0
$166$	−2.82199e12	−1.73764
$167$	− 3.01398e12i	− 1.79556i	−0.440443	−	0.897780i	$-0.645179\pi$
$167$	− 3.01398e12i	− 1.79556i	0.440443	−	0.897780i	$-0.354821\pi$
$168$	0	0
$169$	−2.30151e12	−1.28421
$170$	0	0
$171$	0	0
$172$	− 3.25948e12i	− 1.65098i
$173$	3.07482e11i	0.150857i	0.997151	+	0.0754285i	$0.0240325\pi$
$173$	3.07482e11i	0.150857i	−0.997151	+	0.0754285i	$0.975968\pi$
$174$	0	0
$175$	0	0
$176$	−3.48887e12	−1.55728
$177$	0	0
$178$	− 2.46400e12i	− 1.03355i
$179$	−1.91470e12	−0.778769	−0.389384	−	0.921075i	$-0.627312\pi$
$179$	−1.91470e12	−0.778769	−0.389384	+	0.921075i	$0.627312\pi$
$180$	0	0
$181$	1.10300e11	0.0422032	0.0211016	−	0.999777i	$-0.493283\pi$
$181$	1.10300e11	0.0422032	0.0211016	+	0.999777i	$0.493283\pi$
$182$	− 2.70606e12i	− 1.00449i
$183$	0	0
$184$	6.99157e12	2.44386
$185$	0	0
$186$	0	0
$187$	8.32552e11i	0.266246i
$188$	9.78074e11i	0.303741i
$189$	0	0
$190$	0	0
$191$	4.43991e12	1.26384	0.631918	−	0.775035i	$-0.282268\pi$
$191$	4.43991e12	1.26384	0.631918	+	0.775035i	$0.282268\pi$
$192$	0	0
$193$	3.15713e12i	0.848647i	0.905511	+	0.424324i	$0.139488\pi$
$193$	3.15713e12i	0.848647i	−0.905511	+	0.424324i	$0.860512\pi$
$194$	9.48330e12	2.47771
$195$	0	0
$196$	−8.54641e12	−2.11045
$197$	− 3.58626e12i	− 0.861147i	−0.902555	−	0.430574i	$-0.858311\pi$
$197$	− 3.58626e12i	− 0.861147i	0.902555	−	0.430574i	$-0.141689\pi$
$198$	0	0
$199$	−4.14823e12	−0.942260	−0.471130	−	0.882064i	$-0.656154\pi$
$199$	−4.14823e12	−0.942260	−0.471130	+	0.882064i	$0.656154\pi$
$200$	0	0
$201$	0	0
$202$	1.00110e13i	2.09433i
$203$	1.50349e11i	0.0306106i
$204$	0	0
$205$	0	0
$206$	−7.92810e12	−1.48902
$207$	0	0
$208$	2.07782e13i	3.70050i
$209$	−1.38781e12	−0.240727
$210$	0	0
$211$	1.14275e12	0.188104	0.0940519	−	0.995567i	$-0.470018\pi$
$211$	1.14275e12	0.188104	0.0940519	+	0.995567i	$0.470018\pi$
$212$	2.56015e13i	4.10599i
$213$	0	0
$214$	1.27913e13	1.94822
$215$	0	0
$216$	0	0
$217$	− 4.78249e12i	− 0.674724i
$218$	2.38694e13i	3.28345i
$219$	0	0
$220$	0	0
$221$	4.95832e12	0.632670
$222$	0	0
$223$	3.54889e12i	0.430939i	0.976511	+	0.215469i	$0.0691282\pi$
$223$	3.54889e12i	0.430939i	−0.976511	+	0.215469i	$0.930872\pi$
$224$	−5.75175e12	−0.681454
$225$	0	0
$226$	1.97235e13	2.22529
$227$	3.90543e12i	0.430058i	0.976608	+	0.215029i	$0.0689846\pi$
$227$	3.90543e12i	0.430058i	−0.976608	+	0.215029i	$0.931015\pi$
$228$	0	0
$229$	4.85126e12	0.509048	0.254524	−	0.967066i	$-0.418081\pi$
$229$	4.85126e12	0.509048	0.254524	+	0.967066i	$0.418081\pi$
$230$	0	0
$231$	0	0
$232$	− 2.29694e12i	− 0.224370i
$233$	− 1.03473e13i	− 0.987115i	−0.869713	−	0.493557i	$-0.835696\pi$
$233$	− 1.03473e13i	− 0.987115i	0.869713	−	0.493557i	$-0.164304\pi$
$234$	0	0
$235$	0	0
$236$	3.28622e12	0.292201
$237$	0	0
$238$	3.27763e12i	0.278219i
$239$	1.85140e12	0.153572	0.0767859	−	0.997048i	$-0.475534\pi$
$239$	1.85140e12	0.153572	0.0767859	+	0.997048i	$0.475534\pi$
$240$	0	0
$241$	2.73018e12	0.216321	0.108160	−	0.994133i	$-0.465504\pi$
$241$	2.73018e12	0.216321	0.108160	+	0.994133i	$0.465504\pi$
$242$	1.42252e13i	1.10173i
$243$	0	0
$244$	−2.77059e12	−0.205083
$245$	0	0
$246$	0	0
$247$	8.26518e12i	0.572031i
$248$	7.30641e13i	4.94561i
$249$	0	0
$250$	0	0
$251$	−1.18976e13	−0.753796	−0.376898	−	0.926255i	$-0.623009\pi$
$251$	−1.18976e13	−0.753796	−0.376898	+	0.926255i	$0.623009\pi$
$252$	0	0
$253$	9.73316e12i	0.590324i
$254$	1.05319e13	0.625062
$255$	0	0
$256$	−1.65016e13	−0.938007
$257$	3.53878e10i	0.00196889i	1.00000		0.000984443i	$0.000313358\pi$
$257$	3.53878e10i	0.00196889i	−1.00000		0.000984443i	$0.999687\pi$
$258$	0	0
$259$	7.30442e12	0.389437
$260$	0	0
$261$	0	0
$262$	− 2.48966e13i	− 1.24590i
$263$	1.19904e13i	0.587592i	0.955868	+	0.293796i	$0.0949187\pi$
$263$	1.19904e13i	0.587592i	−0.955868	+	0.293796i	$0.905081\pi$
$264$	0	0
$265$	0	0
$266$	−5.46358e12	−0.251553
$267$	0	0
$268$	5.02243e13i	2.21908i
$269$	−3.81149e13	−1.64990	−0.824948	−	0.565208i	$-0.808796\pi$
$269$	−3.81149e13	−1.64990	−0.824948	+	0.565208i	$0.808796\pi$
$270$	0	0
$271$	−1.13510e13	−0.471740	−0.235870	−	0.971785i	$-0.575794\pi$
$271$	−1.13510e13	−0.471740	−0.235870	+	0.971785i	$0.575794\pi$
$272$	− 2.51670e13i	− 1.02495i
$273$	0	0
$274$	−5.26827e12	−0.206082
$275$	0	0
$276$	0	0
$277$	− 3.02533e13i	− 1.11464i	−0.830298	−	0.557320i	$-0.811830\pi$
$277$	− 3.02533e13i	− 1.11464i	0.830298	−	0.557320i	$-0.188170\pi$
$278$	− 4.28205e13i	− 1.54670i
$279$	0	0
$280$	0	0
$281$	−4.72047e13	−1.60731	−0.803657	−	0.595093i	$-0.797115\pi$
$281$	−4.72047e13	−1.60731	−0.803657	+	0.595093i	$0.797115\pi$
$282$	0	0
$283$	− 8.68805e12i	− 0.284510i	−0.989830	−	0.142255i	$-0.954565\pi$
$283$	− 8.68805e12i	− 0.284510i	0.989830	−	0.142255i	$-0.0454353\pi$
$284$	8.84140e13	2.83969
$285$	0	0
$286$	−5.75528e13	−1.77850
$287$	2.93617e12i	0.0890085i
$288$	0	0
$289$	2.82663e13	0.824766
$290$	0	0
$291$	0	0
$292$	1.15817e14i	3.19276i
$293$	− 2.17869e13i	− 0.589417i	−0.955587	−	0.294709i	$-0.904777\pi$
$293$	− 2.17869e13i	− 0.589417i	0.955587	−	0.294709i	$-0.0952226\pi$
$294$	0	0
$295$	0	0
$296$	−1.11593e14	−2.85450
$297$	0	0
$298$	− 6.64939e13i	− 1.63905i
$299$	5.79665e13	1.40277
$300$	0	0
$301$	1.04917e13	0.244754
$302$	− 1.04351e14i	− 2.39033i
$303$	0	0
$304$	4.19516e13	0.926710
$305$	0	0
$306$	0	0
$307$	− 2.05380e13i	− 0.429829i	−0.976633	−	0.214915i	$-0.931053\pi$
$307$	− 2.05380e13i	− 0.429829i	0.976633	−	0.214915i	$-0.0689473\pi$
$308$	− 2.69305e13i	− 0.553626i
$309$	0	0
$310$	0	0
$311$	7.37156e13	1.43674	0.718369	−	0.695663i	$-0.244889\pi$
$311$	7.37156e13	1.43674	0.718369	+	0.695663i	$0.244889\pi$
$312$	0	0
$313$	− 2.82026e13i	− 0.530634i	−0.964161	−	0.265317i	$-0.914523\pi$
$313$	− 2.82026e13i	− 0.530634i	0.964161	−	0.265317i	$-0.0854765\pi$
$314$	−8.15120e13	−1.50698
$315$	0	0
$316$	6.20108e13	1.10710
$317$	2.56938e13i	0.450819i	0.974264	+	0.225410i	$0.0723720\pi$
$317$	2.56938e13i	0.450819i	−0.974264	+	0.225410i	$0.927628\pi$
$318$	0	0
$319$	3.19763e12	0.0541976
$320$	0	0
$321$	0	0
$322$	3.83179e13i	0.616872i
$323$	− 1.00109e13i	− 0.158439i
$324$	0	0
$325$	0	0
$326$	1.05569e14	1.58797
$327$	0	0
$328$	− 4.48571e13i	− 0.652417i
$329$	−3.14824e12	−0.0450288
$330$	0	0
$331$	−6.14309e13	−0.849831	−0.424916	−	0.905233i	$-0.639696\pi$
$331$	−6.14309e13	−0.849831	−0.424916	+	0.905233i	$0.639696\pi$
$332$	1.67258e14i	2.27576i
$333$	0	0
$334$	−2.52358e14	−3.32210
$335$	0	0
$336$	0	0
$337$	− 1.35566e14i	− 1.69898i	−0.527608	−	0.849488i	$-0.676911\pi$
$337$	− 1.35566e14i	− 1.69898i	0.527608	−	0.849488i	$-0.323089\pi$
$338$	1.92703e14i	2.37600i
$339$	0	0
$340$	0	0
$341$	−1.01715e14	−1.19463
$342$	0	0
$343$	− 5.90945e13i	− 0.672093i
$344$	−1.60285e14	−1.79400
$345$	0	0
$346$	2.57452e13	0.279111
$347$	− 2.88020e13i	− 0.307334i	−0.988123	−	0.153667i	$-0.950892\pi$
$347$	− 2.88020e13i	− 0.307334i	0.988123	−	0.153667i	$-0.0491083\pi$
$348$	0	0
$349$	−3.70742e13	−0.383294	−0.191647	−	0.981464i	$-0.561383\pi$
$349$	−3.70742e13	−0.383294	−0.191647	+	0.981464i	$0.561383\pi$
$350$	0	0
$351$	0	0
$352$	1.22329e14i	1.20655i
$353$	5.64627e13i	0.548278i	0.961690	+	0.274139i	$0.0883928\pi$
$353$	5.64627e13i	0.548278i	−0.961690	+	0.274139i	$0.911607\pi$
$354$	0	0
$355$	0	0
$356$	−1.46040e14	−1.35362
$357$	0	0
$358$	1.60316e14i	1.44086i
$359$	−1.96868e14	−1.74243	−0.871217	−	0.490899i	$-0.836668\pi$
$359$	−1.96868e14	−1.74243	−0.871217	+	0.490899i	$0.836668\pi$
$360$	0	0
$361$	−9.98027e13	−0.856747
$362$	− 9.23537e12i	− 0.0780831i
$363$	0	0
$364$	−1.60387e14	−1.31556
$365$	0	0
$366$	0	0
$367$	1.00444e14i	0.787514i	0.919214	+	0.393757i	$0.128825\pi$
$367$	1.00444e14i	0.787514i	−0.919214	+	0.393757i	$0.871175\pi$
$368$	− 2.94221e14i	− 2.27253i
$369$	0	0
$370$	0	0
$371$	−8.24065e13	−0.608703
$372$	0	0
$373$	− 7.09849e13i	− 0.509058i	−0.967065	−	0.254529i	$-0.918079\pi$
$373$	− 7.09849e13i	− 0.509058i	0.967065	−	0.254529i	$-0.0819205\pi$
$374$	6.97090e13	0.492600
$375$	0	0
$376$	4.80970e13	0.330054
$377$	− 1.90437e13i	− 0.128788i
$378$	0	0
$379$	−6.79976e13	−0.446661	−0.223330	−	0.974743i	$-0.571693\pi$
$379$	−6.79976e13	−0.446661	−0.223330	+	0.974743i	$0.571693\pi$
$380$	0	0
$381$	0	0
$382$	− 3.71750e14i	− 2.33831i
$383$	− 1.89007e14i	− 1.17188i	−0.810353	−	0.585942i	$-0.800725\pi$
$383$	− 1.89007e14i	− 1.17188i	0.810353	−	0.585942i	$-0.199275\pi$
$384$	0	0
$385$	0	0
$386$	2.64344e14	1.57014
$387$	0	0
$388$	− 5.62070e14i	− 3.24501i
$389$	−2.90163e14	−1.65165	−0.825826	−	0.563924i	$-0.809291\pi$
$389$	−2.90163e14	−1.65165	−0.825826	+	0.563924i	$0.809291\pi$
$390$	0	0
$391$	−7.02100e13	−0.388532
$392$	4.20271e14i	2.29328i
$393$	0	0
$394$	−3.00275e14	−1.59327
$395$	0	0
$396$	0	0
$397$	5.15115e13i	0.262154i	0.991372	+	0.131077i	$0.0418435\pi$
$397$	5.15115e13i	0.262154i	−0.991372	+	0.131077i	$0.958157\pi$
$398$	3.47328e14i	1.74334i
$399$	0	0
$400$	0	0
$401$	1.11012e14	0.534657	0.267329	−	0.963605i	$-0.413859\pi$
$401$	1.11012e14	0.534657	0.267329	+	0.963605i	$0.413859\pi$
$402$	0	0
$403$	6.05768e14i	2.83876i
$404$	5.93346e14	2.74291
$405$	0	0
$406$	1.25886e13	0.0566349
$407$	− 1.55351e14i	− 0.689517i
$408$	0	0
$409$	1.92951e14	0.833623	0.416811	−	0.908993i	$-0.363148\pi$
$409$	1.92951e14	0.833623	0.416811	+	0.908993i	$0.363148\pi$
$410$	0	0
$411$	0	0
$412$	4.69894e14i	1.95014i
$413$	1.05778e13i	0.0433180i
$414$	0	0
$415$	0	0
$416$	7.28538e14	2.86707
$417$	0	0
$418$	1.16200e14i	0.445386i
$419$	2.28750e14	0.865336	0.432668	−	0.901553i	$-0.357572\pi$
$419$	2.28750e14	0.865336	0.432668	+	0.901553i	$0.357572\pi$
$420$	0	0
$421$	−4.00332e14	−1.47526	−0.737630	−	0.675205i	$-0.764055\pi$
$421$	−4.00332e14	−1.47526	−0.737630	+	0.675205i	$0.764055\pi$
$422$	− 9.56816e13i	− 0.348024i
$423$	0	0
$424$	1.25896e15	4.46169
$425$	0	0
$426$	0	0
$427$	− 8.91803e12i	− 0.0304030i
$428$	− 7.58132e14i	− 2.55155i
$429$	0	0
$430$	0	0
$431$	−1.29757e14	−0.420250	−0.210125	−	0.977675i	$-0.567387\pi$
$431$	−1.29757e14	−0.420250	−0.210125	+	0.977675i	$0.567387\pi$
$432$	0	0
$433$	− 3.09354e14i	− 0.976725i	−0.872641	−	0.488363i	$-0.837594\pi$
$433$	− 3.09354e14i	− 0.976725i	0.872641	−	0.488363i	$-0.162406\pi$
$434$	−4.00435e14	−1.24835
$435$	0	0
$436$	1.41473e15	4.30028
$437$	− 1.17035e14i	− 0.351293i
$438$	0	0
$439$	6.15652e13	0.180211	0.0901054	−	0.995932i	$-0.471280\pi$
$439$	6.15652e13	0.180211	0.0901054	+	0.995932i	$0.471280\pi$
$440$	0	0
$441$	0	0
$442$	− 4.15156e14i	− 1.17055i
$443$	2.42894e13i	0.0676389i	0.999428	+	0.0338194i	$0.0107671\pi$
$443$	2.42894e13i	0.0676389i	−0.999428	+	0.0338194i	$0.989233\pi$
$444$	0	0
$445$	0	0
$446$	2.97146e14	0.797311
$447$	0	0
$448$	1.45631e14i	0.381263i
$449$	6.65728e14	1.72164	0.860820	−	0.508910i	$-0.169951\pi$
$449$	6.65728e14	1.72164	0.860820	+	0.508910i	$0.169951\pi$
$450$	0	0
$451$	6.24468e13	0.157594
$452$	− 1.16900e15i	− 2.91442i
$453$	0	0
$454$	3.26999e14	0.795681
$455$	0	0
$456$	0	0
$457$	− 6.99716e14i	− 1.64204i	−0.570902	−	0.821018i	$-0.693406\pi$
$457$	− 6.99716e14i	− 1.64204i	0.570902	−	0.821018i	$-0.306594\pi$
$458$	− 4.06192e14i	− 0.941827i
$459$	0	0
$460$	0	0
$461$	−3.43320e14	−0.767971	−0.383985	−	0.923339i	$-0.625449\pi$
$461$	−3.43320e14	−0.767971	−0.383985	+	0.923339i	$0.625449\pi$
$462$	0	0
$463$	3.77708e14i	0.825012i	0.910955	+	0.412506i	$0.135346\pi$
$463$	3.77708e14i	0.825012i	−0.910955	+	0.412506i	$0.864654\pi$
$464$	−9.66603e13	−0.208641
$465$	0	0
$466$	−8.66368e14	−1.82633
$467$	− 5.01733e14i	− 1.04527i	−0.852555	−	0.522637i	$-0.824948\pi$
$467$	− 5.01733e14i	− 1.04527i	0.852555	−	0.522637i	$-0.175052\pi$
$468$	0	0
$469$	−1.61663e14	−0.328972
$470$	0	0
$471$	0	0
$472$	− 1.61601e14i	− 0.317513i
$473$	− 2.23138e14i	− 0.433348i
$474$	0	0
$475$	0	0
$476$	1.94263e14	0.364378
$477$	0	0
$478$	− 1.55016e14i	− 0.284134i
$479$	9.56649e14	1.73343	0.866717	−	0.498800i	$-0.166226\pi$
$479$	9.56649e14	1.73343	0.866717	+	0.498800i	$0.166226\pi$
$480$	0	0
$481$	−9.25204e14	−1.63847
$482$	− 2.28596e14i	− 0.400231i
$483$	0	0
$484$	8.43122e14	1.44292
$485$	0	0
$486$	0	0
$487$	− 1.81856e14i	− 0.300828i	−0.988623	−	0.150414i	$-0.951939\pi$
$487$	− 1.81856e14i	− 0.300828i	0.988623	−	0.150414i	$-0.0480606\pi$
$488$	1.36244e14i	0.222848i
$489$	0	0
$490$	0	0
$491$	−9.28536e14	−1.46842	−0.734210	−	0.678922i	$-0.762447\pi$
$491$	−9.28536e14	−1.46842	−0.734210	+	0.678922i	$0.762447\pi$
$492$	0	0
$493$	2.30661e13i	0.0356711i
$494$	6.92037e14	1.05835
$495$	0	0
$496$	3.07470e15	4.59889
$497$	2.84589e14i	0.420976i
$498$	0	0
$499$	−1.10604e15	−1.60036	−0.800178	−	0.599763i	$-0.795261\pi$
$499$	−1.10604e15	−1.60036	−0.800178	+	0.599763i	$0.795261\pi$
$500$	0	0
$501$	0	0
$502$	9.96177e14i	1.39465i
$503$	1.07467e15i	1.48816i	0.668089	+	0.744081i	$0.267112\pi$
$503$	1.07467e15i	1.48816i	−0.668089	+	0.744081i	$0.732888\pi$
$504$	0	0
$505$	0	0
$506$	8.14950e14	1.09220
$507$	0	0
$508$	− 6.24222e14i	− 0.818632i
$509$	−1.50116e15	−1.94750	−0.973752	−	0.227610i	$-0.926909\pi$
$509$	−1.50116e15	−1.94750	−0.973752	+	0.227610i	$0.926909\pi$
$510$	0	0
$511$	−3.72793e14	−0.473318
$512$	1.43474e15i	1.80214i
$513$	0	0
$514$	2.96299e12	0.00364278
$515$	0	0
$516$	0	0
$517$	6.69571e13i	0.0797258i
$518$	− 6.11593e14i	− 0.720525i
$519$	0	0
$520$	0	0
$521$	8.31060e13	0.0948473	0.0474237	−	0.998875i	$-0.484899\pi$
$521$	8.31060e13	0.0948473	0.0474237	+	0.998875i	$0.484899\pi$
$522$	0	0
$523$	− 9.42223e14i	− 1.05292i	−0.850201	−	0.526459i	$-0.823519\pi$
$523$	− 9.42223e14i	− 1.05292i	0.850201	−	0.526459i	$-0.176481\pi$
$524$	−1.47561e15	−1.63173
$525$	0	0
$526$	1.00394e15	1.08715
$527$	− 7.33717e14i	− 0.786267i
$528$	0	0
$529$	1.32002e14	0.138540
$530$	0	0
$531$	0	0
$532$	3.23824e14i	0.329454i
$533$	− 3.71906e14i	− 0.374485i
$534$	0	0
$535$	0	0
$536$	2.46979e15	2.41131
$537$	0	0
$538$	3.19133e15i	3.05259i
$539$	−5.85071e14	−0.553950
$540$	0	0
$541$	1.18229e15	1.09683	0.548416	−	0.836206i	$-0.315231\pi$
$541$	1.18229e15	1.09683	0.548416	+	0.836206i	$0.315231\pi$
$542$	9.50410e14i	0.872800i
$543$	0	0
$544$	−8.82418e14	−0.794110
$545$	0	0
$546$	0	0
$547$	− 1.57701e15i	− 1.37691i	−0.725281	−	0.688453i	$-0.758290\pi$
$547$	− 1.57701e15i	− 1.37691i	0.725281	−	0.688453i	$-0.241710\pi$
$548$	3.12247e14i	0.269902i
$549$	0	0
$550$	0	0
$551$	−3.84496e13	−0.0322521
$552$	0	0
$553$	1.99601e14i	0.164125i
$554$	−2.53309e15	−2.06228
$555$	0	0
$556$	−2.53795e15	−2.02568
$557$	4.53070e14i	0.358065i	0.983843	+	0.179033i	$0.0572968\pi$
$557$	4.53070e14i	0.358065i	−0.983843	+	0.179033i	$0.942703\pi$
$558$	0	0
$559$	−1.32891e15	−1.02975
$560$	0	0
$561$	0	0
$562$	3.95241e15i	2.97381i
$563$	− 4.23450e13i	− 0.0315505i	−0.999876	−	0.0157752i	$-0.994978\pi$
$563$	− 4.23450e13i	− 0.0315505i	0.999876	−	0.0157752i	$-0.00502163\pi$
$564$	0	0
$565$	0	0
$566$	−7.27444e14	−0.526392
$567$	0	0
$568$	− 4.34777e15i	− 3.08568i
$569$	−9.72594e14	−0.683619	−0.341810	−	0.939769i	$-0.611040\pi$
$569$	−9.72594e14	−0.683619	−0.341810	+	0.939769i	$0.611040\pi$
$570$	0	0
$571$	−2.07663e15	−1.43173	−0.715863	−	0.698241i	$-0.753966\pi$
$571$	−2.07663e15	−1.43173	−0.715863	+	0.698241i	$0.753966\pi$
$572$	3.41112e15i	2.32927i
$573$	0	0
$574$	2.45844e14	0.164681
$575$	0	0
$576$	0	0
$577$	− 2.62999e15i	− 1.71194i	−0.517028	−	0.855968i	$-0.672962\pi$
$577$	− 2.62999e15i	− 1.71194i	0.517028	−	0.855968i	$-0.327038\pi$
$578$	− 2.36671e15i	− 1.52596i
$579$	0	0
$580$	0	0
$581$	−5.38371e14	−0.337375
$582$	0	0
$583$	1.75263e15i	1.07774i
$584$	5.69531e15	3.46934
$585$	0	0
$586$	−1.82420e15	−1.09052
$587$	− 2.89275e15i	− 1.71317i	−0.516003	−	0.856587i	$-0.672581\pi$
$587$	− 2.89275e15i	− 1.71317i	0.516003	−	0.856587i	$-0.327419\pi$
$588$	0	0
$589$	1.22306e15	0.710906
$590$	0	0
$591$	0	0
$592$	4.69606e15i	2.65439i
$593$	1.15734e15i	0.648125i	0.946036	+	0.324063i	$0.105049\pi$
$593$	1.15734e15i	0.648125i	−0.946036	+	0.324063i	$0.894951\pi$
$594$	0	0
$595$	0	0
$596$	−3.94106e15	−2.14663
$597$	0	0
$598$	− 4.85349e15i	− 2.59536i
$599$	2.42056e14	0.128253	0.0641265	−	0.997942i	$-0.479574\pi$
$599$	2.42056e14	0.128253	0.0641265	+	0.997942i	$0.479574\pi$
$600$	0	0
$601$	−3.08465e15	−1.60471	−0.802354	−	0.596848i	$-0.796419\pi$
$601$	−3.08465e15	−1.60471	−0.802354	+	0.596848i	$0.796419\pi$
$602$	− 8.78460e14i	− 0.452836i
$603$	0	0
$604$	−6.18480e15	−3.13057
$605$	0	0
$606$	0	0
$607$	− 2.71223e15i	− 1.33595i	−0.744186	−	0.667973i	$-0.767162\pi$
$607$	− 2.71223e15i	− 1.33595i	0.744186	−	0.667973i	$-0.232838\pi$
$608$	− 1.47093e15i	− 0.717997i
$609$	0	0
$610$	0	0
$611$	3.98768e14	0.189449
$612$	0	0
$613$	− 1.80466e15i	− 0.842097i	−0.907038	−	0.421049i	$-0.861662\pi$
$613$	− 1.80466e15i	− 0.842097i	0.907038	−	0.421049i	$-0.138338\pi$
$614$	−1.71963e15	−0.795258
$615$	0	0
$616$	−1.32431e15	−0.601585
$617$	4.28183e14i	0.192780i	0.995344	+	0.0963898i	$0.0307295\pi$
$617$	4.28183e14i	0.192780i	−0.995344	+	0.0963898i	$0.969270\pi$
$618$	0	0
$619$	−2.39110e15	−1.05754	−0.528772	−	0.848764i	$-0.677347\pi$
$619$	−2.39110e15	−1.05754	−0.528772	+	0.848764i	$0.677347\pi$
$620$	0	0
$621$	0	0
$622$	− 6.17215e15i	− 2.65821i
$623$	− 4.70076e14i	− 0.200671i
$624$	0	0
$625$	0	0
$626$	−2.36138e15	−0.981763
$627$	0	0
$628$	4.83117e15i	1.97367i
$629$	1.12062e15	0.453818
$630$	0	0
$631$	3.26028e15	1.29746	0.648730	−	0.761019i	$-0.275301\pi$
$631$	3.26028e15	1.29746	0.648730	+	0.761019i	$0.275301\pi$
$632$	− 3.04939e15i	− 1.20301i
$633$	0	0
$634$	2.15132e15	0.834093
$635$	0	0
$636$	0	0
$637$	3.48443e15i	1.31633i
$638$	− 2.67735e14i	− 0.100275i
$639$	0	0
$640$	0	0
$641$	2.32104e15	0.847157	0.423579	−	0.905859i	$-0.360774\pi$
$641$	2.32104e15	0.847157	0.423579	+	0.905859i	$0.360774\pi$
$642$	0	0
$643$	− 2.55397e15i	− 0.916336i	−0.888866	−	0.458168i	$-0.848506\pi$
$643$	− 2.55397e15i	− 0.916336i	0.888866	−	0.458168i	$-0.151494\pi$
$644$	2.27108e15	0.807906
$645$	0	0
$646$	−8.38208e14	−0.293139
$647$	− 1.70408e15i	− 0.590902i	−0.955358	−	0.295451i	$-0.904530\pi$
$647$	− 1.70408e15i	− 0.590902i	0.955358	−	0.295451i	$-0.0954699\pi$
$648$	0	0
$649$	2.24969e14	0.0766966
$650$	0	0
$651$	0	0
$652$	− 6.25704e15i	− 2.07973i
$653$	2.20439e15i	0.726552i	0.931682	+	0.363276i	$0.118342\pi$
$653$	2.20439e15i	0.726552i	−0.931682	+	0.363276i	$0.881658\pi$
$654$	0	0
$655$	0	0
$656$	−1.88769e15	−0.606679
$657$	0	0
$658$	2.63600e14i	0.0833111i
$659$	1.97759e14	0.0619821	0.0309910	−	0.999520i	$-0.490134\pi$
$659$	1.97759e14	0.0619821	0.0309910	+	0.999520i	$0.490134\pi$
$660$	0	0
$661$	−3.43414e15	−1.05855	−0.529274	−	0.848451i	$-0.677536\pi$
$661$	−3.43414e15	−1.05855	−0.529274	+	0.848451i	$0.677536\pi$
$662$	5.14356e15i	1.57233i
$663$	0	0
$664$	8.22492e15	2.47290
$665$	0	0
$666$	0	0
$667$	2.69660e14i	0.0790905i
$668$	1.49571e16i	4.35089i
$669$	0	0
$670$	0	0
$671$	−1.89670e14	−0.0538299
$672$	0	0
$673$	2.01250e15i	0.561892i	0.959724	+	0.280946i	$0.0906481\pi$
$673$	2.01250e15i	0.561892i	−0.959724	+	0.280946i	$0.909352\pi$
$674$	−1.13509e16	−3.14340
$675$	0	0
$676$	1.14214e16	3.11181
$677$	− 1.01903e15i	− 0.275392i	−0.990475	−	0.137696i	$-0.956030\pi$
$677$	− 1.01903e15i	− 0.275392i	0.990475	−	0.137696i	$-0.0439696\pi$
$678$	0	0
$679$	1.80920e15	0.481064
$680$	0	0
$681$	0	0
$682$	8.51648e15i	2.21027i
$683$	− 2.86885e15i	− 0.738575i	−0.929315	−	0.369287i	$-0.879602\pi$
$683$	− 2.86885e15i	− 0.738575i	0.929315	−	0.369287i	$-0.120398\pi$
$684$	0	0
$685$	0	0
$686$	−4.94793e15	−1.24349
$687$	0	0
$688$	6.74517e15i	1.66823i
$689$	1.04379e16	2.56099
$690$	0	0
$691$	−1.79931e15	−0.434487	−0.217243	−	0.976117i	$-0.569707\pi$
$691$	−1.79931e15	−0.434487	−0.217243	+	0.976117i	$0.569707\pi$
$692$	− 1.52590e15i	− 0.365547i
$693$	0	0
$694$	−2.41157e15	−0.568621
$695$	0	0
$696$	0	0
$697$	4.50460e14i	0.103723i
$698$	3.10420e15i	0.709160i
$699$	0	0
$700$	0	0
$701$	−2.99337e15	−0.667899	−0.333949	−	0.942591i	$-0.608381\pi$
$701$	−2.99337e15	−0.667899	−0.333949	+	0.942591i	$0.608381\pi$
$702$	0	0
$703$	1.86800e15i	0.410321i
$704$	3.09729e15	0.675045
$705$	0	0
$706$	4.72758e15	1.01441
$707$	1.90987e15i	0.406629i
$708$	0	0
$709$	−2.74187e15	−0.574769	−0.287384	−	0.957815i	$-0.592786\pi$
$709$	−2.74187e15	−0.574769	−0.287384	+	0.957815i	$0.592786\pi$
$710$	0	0
$711$	0	0
$712$	7.18154e15i	1.47088i
$713$	− 8.57770e15i	− 1.74333i
$714$	0	0
$715$	0	0
$716$	9.50185e15	1.88706
$717$	0	0
$718$	1.64836e16i	3.22380i
$719$	−4.38021e15	−0.850131	−0.425065	−	0.905163i	$-0.639749\pi$
$719$	−4.38021e15	−0.850131	−0.425065	+	0.905163i	$0.639749\pi$
$720$	0	0
$721$	−1.51250e15	−0.289103
$722$	8.35640e15i	1.58513i
$723$	0	0
$724$	−5.47375e14	−0.102264
$725$	0	0
$726$	0	0
$727$	− 6.60205e15i	− 1.20570i	−0.797854	−	0.602851i	$-0.794031\pi$
$727$	− 6.60205e15i	− 1.20570i	0.797854	−	0.602851i	$-0.205969\pi$
$728$	7.88704e15i	1.42952i
$729$	0	0
$730$	0	0
$731$	1.60960e15	0.285216
$732$	0	0
$733$	− 8.22795e15i	− 1.43622i	−0.695932	−	0.718108i	$-0.745008\pi$
$733$	− 8.22795e15i	− 1.43622i	0.695932	−	0.718108i	$-0.254992\pi$
$734$	8.41006e15	1.45704
$735$	0	0
$736$	−1.03161e16	−1.76071
$737$	3.43826e15i	0.582462i
$738$	0	0
$739$	1.10770e16	1.84875	0.924376	−	0.381483i	$-0.124586\pi$
$739$	1.10770e16	1.84875	0.924376	+	0.381483i	$0.124586\pi$
$740$	0	0
$741$	0	0
$742$	6.89984e15i	1.12620i
$743$	3.86160e15i	0.625646i	0.949811	+	0.312823i	$0.101275\pi$
$743$	3.86160e15i	0.625646i	−0.949811	+	0.312823i	$0.898725\pi$
$744$	0	0
$745$	0	0
$746$	−5.94352e15	−0.941846
$747$	0	0
$748$	− 4.13161e15i	− 0.645150i
$749$	2.44029e15	0.378260
$750$	0	0
$751$	1.05947e15	0.161834	0.0809168	−	0.996721i	$-0.474215\pi$
$751$	1.05947e15	0.161834	0.0809168	+	0.996721i	$0.474215\pi$
$752$	− 2.02403e15i	− 0.306915i
$753$	0	0
$754$	−1.59452e15	−0.238280
$755$	0	0
$756$	0	0
$757$	7.04375e15i	1.02986i	0.857233	+	0.514928i	$0.172182\pi$
$757$	7.04375e15i	1.02986i	−0.857233	+	0.514928i	$0.827818\pi$
$758$	5.69339e15i	0.826400i
$759$	0	0
$760$	0	0
$761$	−3.63598e15	−0.516424	−0.258212	−	0.966088i	$-0.583133\pi$
$761$	−3.63598e15	−0.516424	−0.258212	+	0.966088i	$0.583133\pi$
$762$	0	0
$763$	4.55374e15i	0.637505i
$764$	−2.20334e16	−3.06245
$765$	0	0
$766$	−1.58254e16	−2.16819
$767$	− 1.33982e15i	− 0.182251i
$768$	0	0
$769$	5.34829e15	0.717167	0.358583	−	0.933498i	$-0.383260\pi$
$769$	5.34829e15	0.717167	0.358583	+	0.933498i	$0.383260\pi$
$770$	0	0
$771$	0	0
$772$	− 1.56675e16i	− 2.05639i
$773$	− 8.06669e15i	− 1.05126i	−0.850715	−	0.525628i	$-0.823831\pi$
$773$	− 8.06669e15i	− 1.05126i	0.850715	−	0.525628i	$-0.176169\pi$
$774$	0	0
$775$	0	0
$776$	−2.76399e16	−3.52612
$777$	0	0
$778$	2.42951e16i	3.05584i
$779$	−7.50886e14	−0.0937816
$780$	0	0
$781$	6.05266e15	0.745359
$782$	5.87863e15i	0.718851i
$783$	0	0
$784$	1.76859e16	2.13250
$785$	0	0
$786$	0	0
$787$	8.56068e15i	1.01076i	0.862897	+	0.505379i	$0.168647\pi$
$787$	8.56068e15i	1.01076i	−0.862897	+	0.505379i	$0.831353\pi$
$788$	1.77971e16i	2.08668i
$789$	0	0
$790$	0	0
$791$	3.76279e15	0.432056
$792$	0	0
$793$	1.12959e15i	0.127914i
$794$	4.31302e15	0.485030
$795$	0	0
$796$	2.05859e16	2.28323
$797$	− 1.47001e16i	− 1.61919i	−0.586987	−	0.809596i	$-0.699686\pi$
$797$	− 1.47001e16i	− 1.61919i	0.586987	−	0.809596i	$-0.300314\pi$
$798$	0	0
$799$	−4.82995e14	−0.0524729
$800$	0	0
$801$	0	0
$802$	− 9.29494e15i	− 0.989208i
$803$	7.92861e15i	0.838033i
$804$	0	0
$805$	0	0
$806$	5.07205e16	5.25219
$807$	0	0
$808$	− 2.91779e16i	− 2.98052i
$809$	5.20792e15	0.528382	0.264191	−	0.964470i	$-0.414895\pi$
$809$	5.20792e15	0.528382	0.264191	+	0.964470i	$0.414895\pi$
$810$	0	0
$811$	1.95571e16	1.95745	0.978724	−	0.205183i	$-0.0657789\pi$
$811$	1.95571e16	1.95745	0.978724	+	0.205183i	$0.0657789\pi$
$812$	− 7.46119e14i	− 0.0741737i
$813$	0	0
$814$	−1.30074e16	−1.27573
$815$	0	0
$816$	0	0
$817$	2.68310e15i	0.257879i
$818$	− 1.61557e16i	− 1.54235i
$819$	0	0
$820$	0	0
$821$	−1.02440e16	−0.958476	−0.479238	−	0.877685i	$-0.659087\pi$
$821$	−1.02440e16	−0.958476	−0.479238	+	0.877685i	$0.659087\pi$
$822$	0	0
$823$	− 1.40372e16i	− 1.29593i	−0.761671	−	0.647964i	$-0.775621\pi$
$823$	− 1.40372e16i	− 1.29593i	0.761671	−	0.647964i	$-0.224379\pi$
$824$	2.31071e16	2.11907
$825$	0	0
$826$	8.85667e14	0.0801457
$827$	8.80160e14i	0.0791191i	0.999217	+	0.0395596i	$0.0125955\pi$
$827$	8.80160e14i	0.0791191i	−0.999217	+	0.0395596i	$0.987405\pi$
$828$	0	0
$829$	−4.54642e14	−0.0403292	−0.0201646	−	0.999797i	$-0.506419\pi$
$829$	−4.54642e14	−0.0403292	−0.0201646	+	0.999797i	$0.506419\pi$
$830$	0	0
$831$	0	0
$832$	− 1.84461e16i	− 1.60408i
$833$	− 4.22041e15i	− 0.364592i
$834$	0	0
$835$	0	0
$836$	6.88711e15	0.583314
$837$	0	0
$838$	− 1.91531e16i	− 1.60102i
$839$	1.32346e16	1.09906	0.549529	−	0.835474i	$-0.314807\pi$
$839$	1.32346e16	1.09906	0.549529	+	0.835474i	$0.314807\pi$
$840$	0	0
$841$	−1.21119e16	−0.992739
$842$	3.35195e16i	2.72949i
$843$	0	0
$844$	−5.67099e15	−0.455801
$845$	0	0
$846$	0	0
$847$	2.71386e15i	0.213908i
$848$	− 5.29797e16i	− 4.14889i
$849$	0	0
$850$	0	0
$851$	1.31009e16	1.00621
$852$	0	0
$853$	− 2.60074e15i	− 0.197186i	−0.995128	−	0.0985932i	$-0.968566\pi$
$853$	− 2.60074e15i	− 0.197186i	0.995128	−	0.0985932i	$-0.0314343\pi$
$854$	−7.46700e14	−0.0562507
$855$	0	0
$856$	−3.72812e16	−2.77258
$857$	− 1.68895e16i	− 1.24802i	−0.781415	−	0.624011i	$-0.785502\pi$
$857$	− 1.68895e16i	− 1.24802i	0.781415	−	0.624011i	$-0.214498\pi$
$858$	0	0
$859$	−8.98241e15	−0.655285	−0.327643	−	0.944802i	$-0.606254\pi$
$859$	−8.98241e15	−0.655285	−0.327643	+	0.944802i	$0.606254\pi$
$860$	0	0
$861$	0	0
$862$	1.08645e16i	0.777534i
$863$	1.16178e16i	0.826164i	0.910694	+	0.413082i	$0.135548\pi$
$863$	1.16178e16i	0.826164i	−0.910694	+	0.413082i	$0.864452\pi$
$864$	0	0
$865$	0	0
$866$	−2.59020e16	−1.80711
$867$	0	0
$868$	2.37335e16i	1.63495i
$869$	4.24514e15	0.290592
$870$	0	0
$871$	2.04768e16	1.38408
$872$	− 6.95694e16i	− 4.67280i
$873$	0	0
$874$	−9.79928e15	−0.649952
$875$	0	0
$876$	0	0
$877$	− 4.73202e15i	− 0.307999i	−0.988071	−	0.153999i	$-0.950785\pi$
$877$	− 4.73202e15i	− 0.307999i	0.988071	−	0.153999i	$-0.0492154\pi$
$878$	− 5.15481e15i	− 0.333421i
$879$	0	0
$880$	0	0
$881$	−2.81982e15	−0.179000	−0.0895002	−	0.995987i	$-0.528527\pi$
$881$	−2.81982e15	−0.179000	−0.0895002	+	0.995987i	$0.528527\pi$
$882$	0	0
$883$	1.91741e16i	1.20208i	0.799221	+	0.601038i	$0.205246\pi$
$883$	1.91741e16i	1.20208i	−0.799221	+	0.601038i	$0.794754\pi$
$884$	−2.46061e16	−1.53305
$885$	0	0
$886$	2.03373e15	0.125144
$887$	2.08389e15i	0.127437i	0.997968	+	0.0637183i	$0.0202959\pi$
$887$	2.08389e15i	0.127437i	−0.997968	+	0.0637183i	$0.979704\pi$
$888$	0	0
$889$	2.00926e15	0.121360
$890$	0	0
$891$	0	0
$892$	− 1.76117e16i	− 1.04422i
$893$	− 8.05119e14i	− 0.0474435i
$894$	0	0
$895$	0	0
$896$	4.13967e14	0.0239481
$897$	0	0
$898$	− 5.57409e16i	− 3.18533i
$899$	−2.81803e15	−0.160054
$900$	0	0
$901$	−1.26426e16	−0.709332
$902$	− 5.22863e15i	− 0.291576i
$903$	0	0
$904$	−5.74857e16	−3.16689
$905$	0	0
$906$	0	0
$907$	− 3.20443e16i	− 1.73345i	−0.498791	−	0.866723i	$-0.666222\pi$
$907$	− 3.20443e16i	− 1.73345i	0.498791	−	0.866723i	$-0.333778\pi$
$908$	− 1.93810e16i	− 1.04209i
$909$	0	0
$910$	0	0
$911$	8.44282e15	0.445797	0.222898	−	0.974842i	$-0.428448\pi$
$911$	8.44282e15	0.445797	0.222898	+	0.974842i	$0.428448\pi$
$912$	0	0
$913$	1.14501e16i	0.597340i
$914$	−5.85867e16	−3.03805
$915$	0	0
$916$	−2.40748e16	−1.23349
$917$	− 4.74971e15i	− 0.241900i
$918$	0	0
$919$	1.08277e16	0.544880	0.272440	−	0.962173i	$-0.412169\pi$
$919$	1.08277e16	0.544880	0.272440	+	0.962173i	$0.412169\pi$
$920$	0	0
$921$	0	0
$922$	2.87460e16i	1.42088i
$923$	− 3.60470e16i	− 1.77117i
$924$	0	0
$925$	0	0
$926$	3.16252e16	1.52641
$927$	0	0
$928$	3.38916e15i	0.161651i
$929$	−1.30338e16	−0.617994	−0.308997	−	0.951063i	$-0.599993\pi$
$929$	−1.30338e16	−0.617994	−0.308997	+	0.951063i	$0.599993\pi$
$930$	0	0
$931$	7.03513e15	0.329647
$932$	5.13491e16i	2.39191i
$933$	0	0
$934$	−4.20098e16	−1.93394
$935$	0	0
$936$	0	0
$937$	2.06679e16i	0.934819i	0.884041	+	0.467410i	$0.154813\pi$
$937$	2.06679e16i	0.934819i	−0.884041	+	0.467410i	$0.845187\pi$
$938$	1.35359e16i	0.608656i
$939$	0	0
$940$	0	0
$941$	3.48692e16	1.54063	0.770317	−	0.637661i	$-0.220098\pi$
$941$	3.48692e16	1.54063	0.770317	+	0.637661i	$0.220098\pi$
$942$	0	0
$943$	5.26621e15i	0.229977i
$944$	−6.80051e15	−0.295254
$945$	0	0
$946$	−1.86832e16	−0.801769
$947$	− 2.53080e15i	− 0.107977i	−0.998542	−	0.0539887i	$-0.982807\pi$
$947$	− 2.53080e15i	− 0.107977i	0.998542	−	0.0539887i	$-0.0171935\pi$
$948$	0	0
$949$	4.72194e16	1.99139
$950$	0	0
$951$	0	0
$952$	− 9.55293e15i	− 0.395944i
$953$	1.68411e16i	0.693999i	0.937865	+	0.347000i	$0.112799\pi$
$953$	1.68411e16i	0.693999i	−0.937865	+	0.347000i	$0.887201\pi$
$954$	0	0
$955$	0	0
$956$	−9.18773e15	−0.372126
$957$	0	0
$958$	− 8.00995e16i	− 3.20715i
$959$	−1.00507e15	−0.0400122
$960$	0	0
$961$	6.42312e16	2.52794
$962$	7.74666e16i	3.03146i
$963$	0	0
$964$	−1.35488e16	−0.524175
$965$	0	0
$966$	0	0
$967$	2.82761e16i	1.07541i	0.843133	+	0.537705i	$0.180708\pi$
$967$	2.82761e16i	1.07541i	−0.843133	+	0.537705i	$0.819292\pi$
$968$	− 4.14607e16i	− 1.56791i
$969$	0	0
$970$	0	0
$971$	−2.55089e16	−0.948386	−0.474193	−	0.880421i	$-0.657260\pi$
$971$	−2.55089e16	−0.948386	−0.474193	+	0.880421i	$0.657260\pi$
$972$	0	0
$973$	− 8.16918e15i	− 0.300302i
$974$	−1.52266e16	−0.556583
$975$	0	0
$976$	5.73346e15	0.207225
$977$	5.16659e16i	1.85688i	0.371480	+	0.928441i	$0.378850\pi$
$977$	5.16659e16i	1.85688i	−0.371480	+	0.928441i	$0.621150\pi$
$978$	0	0
$979$	−9.99762e15	−0.355297
$980$	0	0
$981$	0	0
$982$	7.77456e16i	2.71683i
$983$	− 1.06975e15i	− 0.0371740i	−0.999827	−	0.0185870i	$-0.994083\pi$
$983$	− 1.06975e15i	− 0.0371740i	0.999827	−	0.0185870i	$-0.00591677\pi$
$984$	0	0
$985$	0	0
$986$	1.93131e15	0.0659976
$987$	0	0
$988$	− 4.10167e16i	− 1.38611i
$989$	1.88175e16	0.632385
$990$	0	0
$991$	2.05333e16	0.682423	0.341211	−	0.939987i	$-0.389163\pi$
$991$	2.05333e16	0.682423	0.341211	+	0.939987i	$0.389163\pi$
$992$	− 1.07807e17i	− 3.56313i
$993$	0	0
$994$	2.38284e16	0.778878
$995$	0	0
$996$	0	0
$997$	1.61931e16i	0.520603i	0.965527	+	0.260301i	$0.0838219\pi$
$997$	1.61931e16i	0.520603i	−0.965527	+	0.260301i	$0.916178\pi$
$998$	9.26076e16i	2.96093i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.12.b.f.199.1		4
3.2	odd	2		25.12.b.c.24.4		4
5.2	odd	4		45.12.a.d.1.2		2
5.3	odd	4		225.12.a.h.1.1		2
5.4	even	2	inner	225.12.b.f.199.4		4
15.2	even	4		5.12.a.b.1.1	✓	2
15.8	even	4		25.12.a.c.1.2		2
15.14	odd	2		25.12.b.c.24.1		4
60.47	odd	4		80.12.a.j.1.2		2
105.62	odd	4		245.12.a.b.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.12.a.b.1.1	✓	2	15.2	even	4
25.12.a.c.1.2		2	15.8	even	4
25.12.b.c.24.1		4	15.14	odd	2
25.12.b.c.24.4		4	3.2	odd	2
45.12.a.d.1.2		2	5.2	odd	4
80.12.a.j.1.2		2	60.47	odd	4
225.12.a.h.1.1		2	5.3	odd	4
225.12.b.f.199.1		4	1.1	even	1	trivial
225.12.b.f.199.4		4	5.4	even	2	inner
245.12.a.b.1.1		2	105.62	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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	225.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-83.7292i q^{2} -4962.58 q^{4} -15973.7i q^{7} +244036. i q^{8} +O(q^{10})\)	\(q-83.7292i q^{2} -4962.58 q^{4} -15973.7i q^{7} +244036. i q^{8} -339729. q^{11} +2.02328e6i q^{13} -1.33746e6 q^{14} +1.02696e7 q^{16} -2.45063e6i q^{17} +4.08504e6 q^{19} +2.84453e7i q^{22} -2.86497e7i q^{23} +1.69408e8 q^{26} +7.92706e7i q^{28} -9.41230e6 q^{29} +2.99399e8 q^{31} -3.60078e8i q^{32} -2.05190e8 q^{34} +4.57279e8i q^{37} -3.42037e8i q^{38} -1.83814e8 q^{41} +6.56811e8i q^{43} +1.68594e9 q^{44} -2.39882e9 q^{46} -1.97090e8i q^{47} +1.72217e9 q^{49} -1.00407e10i q^{52} -5.15890e9i q^{53} +3.89815e9 q^{56} +7.88085e8i q^{58} -6.62200e8 q^{59} +5.58296e8 q^{61} -2.50684e10i q^{62} -9.11694e9 q^{64} -1.01206e10i q^{67} +1.21615e10i q^{68} -1.78161e10 q^{71} -2.33380e10i q^{73} +3.82876e10 q^{74} -2.02724e10 q^{76} +5.42672e9i q^{77} -1.24957e10 q^{79} +1.53906e10i q^{82} -3.37037e10i q^{83} +5.49943e10 q^{86} -8.29062e10i q^{88} +2.94282e10 q^{89} +3.23192e10 q^{91} +1.42177e11i q^{92} -1.65022e10 q^{94} +1.13262e11i q^{97} -1.44196e11i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 13952 q^{4}+O(q^{10}) \) 4 * q - 13952 * q^4	\( 4 q - 13952 q^{4} + 1236352 q^{11} + 2668944 q^{14} + 12011264 q^{16} - 10650640 q^{19} + 161523472 q^{26} + 188280760 q^{29} + 489086928 q^{31} - 890716144 q^{34} + 1491486632 q^{41} - 485451776 q^{44} - 936389712 q^{46} + 3883354828 q^{49} + 7981107840 q^{56} + 14635031120 q^{59} - 3032851352 q^{61} - 1639063552 q^{64} - 65876943088 q^{71} + 137536396144 q^{74} - 2650785280 q^{76} + 6605646240 q^{79} + 113412187792 q^{86} - 25349541720 q^{89} + 181275718128 q^{91} + 120579531056 q^{94}+O(q^{100}) \) 4 * q - 13952 * q^4 + 1236352 * q^11 + 2668944 * q^14 + 12011264 * q^16 - 10650640 * q^19 + 161523472 * q^26 + 188280760 * q^29 + 489086928 * q^31 - 890716144 * q^34 + 1491486632 * q^41 - 485451776 * q^44 - 936389712 * q^46 + 3883354828 * q^49 + 7981107840 * q^56 + 14635031120 * q^59 - 3032851352 * q^61 - 1639063552 * q^64 - 65876943088 * q^71 + 137536396144 * q^74 - 2650785280 * q^76 + 6605646240 * q^79 + 113412187792 * q^86 - 25349541720 * q^89 + 181275718128 * q^91 + 120579531056 * q^94

Introduction

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