LMFDB - Embedded newform 225.6.b.i.199.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,6,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, 6, names="a")

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

chi := DirichletCharacter("225.199");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
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:	$36.0863594579$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{241})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 121x^{2} + 3600 $ x^4 + 121*x^2 + 3600
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 5^{2} $
Twist minimal:	no (minimal twist has level 25)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$-7.26209i$ of defining polynomial
Character	$\chi$	$=$	225.199
Dual form			225.6.b.i.199.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.26209i q^{2} +4.31044 q^{4} -131.048i q^{7} -191.069i q^{8} +O(q^{10})$	$q-5.26209i q^{2} +4.31044 q^{4} -131.048i q^{7} -191.069i q^{8} -290.104 q^{11} +68.3868i q^{13} -689.588 q^{14} -867.486 q^{16} -310.644i q^{17} +2133.35 q^{19} +1526.55i q^{22} -873.145i q^{23} +359.857 q^{26} -564.876i q^{28} -2580.97 q^{29} -9086.30 q^{31} -1549.41i q^{32} -1634.64 q^{34} -3990.64i q^{37} -11225.9i q^{38} -16981.8 q^{41} -18017.7i q^{43} -1250.48 q^{44} -4594.57 q^{46} +24864.7i q^{47} -366.670 q^{49} +294.777i q^{52} +7652.91i q^{53} -25039.2 q^{56} +13581.3i q^{58} -9233.69 q^{59} +3326.17 q^{61} +47812.9i q^{62} -35912.7 q^{64} +32340.7i q^{67} -1339.01i q^{68} +35885.9 q^{71} -26513.6i q^{73} -20999.1 q^{74} +9195.65 q^{76} +38017.7i q^{77} -71705.7 q^{79} +89359.9i q^{82} -39630.1i q^{83} -94810.8 q^{86} +55429.9i q^{88} -117441. q^{89} +8961.98 q^{91} -3763.64i q^{92} +130840. q^{94} -21878.3i q^{97} +1929.45i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 138 q^{4}+O(q^{10}) $ 4 * q - 138 * q^4	$ 4 q - 138 q^{4} + 392 q^{11} + 36 q^{14} + 2274 q^{16} + 6360 q^{19} + 9512 q^{26} - 7840 q^{29} - 2192 q^{31} - 40226 q^{34} - 55508 q^{41} - 73774 q^{44} + 4908 q^{46} + 23372 q^{49} - 108540 q^{56} + 23920 q^{59} - 48792 q^{61} - 87298 q^{64} + 174592 q^{71} - 82444 q^{74} - 135070 q^{76} - 130960 q^{79} - 497848 q^{86} - 145620 q^{89} - 41152 q^{91} + 243304 q^{94}+O(q^{100}) $ 4 * q - 138 * q^4 + 392 * q^11 + 36 * q^14 + 2274 * q^16 + 6360 * q^19 + 9512 * q^26 - 7840 * q^29 - 2192 * q^31 - 40226 * q^34 - 55508 * q^41 - 73774 * q^44 + 4908 * q^46 + 23372 * q^49 - 108540 * q^56 + 23920 * q^59 - 48792 * q^61 - 87298 * q^64 + 174592 * q^71 - 82444 * q^74 - 135070 * q^76 - 130960 * q^79 - 497848 * q^86 - 145620 * q^89 - 41152 * q^91 + 243304 * 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 $

$2$	− 5.26209i	− 0.930214i	−0.885254	−	0.465107i	$-0.846016\pi$
$2$	− 5.26209i	− 0.930214i	0.885254	−	0.465107i	$-0.153984\pi$
$3$	0	0
$3$	0	0
$4$	4.31044	0.134701
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 131.048i	− 1.01085i	−0.862871	−	0.505425i	$-0.831336\pi$
$7$	− 131.048i	− 1.01085i	0.862871	−	0.505425i	$-0.168664\pi$
$8$	− 191.069i	− 1.05552i
$9$	0	0
$10$	0	0
$11$	−290.104	−0.722891	−0.361445	−	0.932393i	$-0.617717\pi$
$11$	−290.104	−0.722891	−0.361445	+	0.932393i	$0.617717\pi$
$12$	0	0
$13$	68.3868i	0.112231i	0.998424	+	0.0561156i	$0.0178715\pi$
$13$	68.3868i	0.112231i	−0.998424	+	0.0561156i	$0.982128\pi$
$14$	−689.588	−0.940307
$15$	0	0
$16$	−867.486	−0.847154
$17$	− 310.644i	− 0.260700i	−0.991468	−	0.130350i	$-0.958390\pi$
$17$	− 310.644i	− 0.260700i	0.991468	−	0.130350i	$-0.0416100\pi$
$18$	0	0
$19$	2133.35	1.35574	0.677871	−	0.735180i	$-0.262903\pi$
$19$	2133.35	1.35574	0.677871	+	0.735180i	$0.262903\pi$
$20$	0	0
$21$	0	0
$22$	1526.55i	0.672443i
$23$	− 873.145i	− 0.344165i	−0.985083	−	0.172083i	$-0.944950\pi$
$23$	− 873.145i	− 0.344165i	0.985083	−	0.172083i	$-0.0550496\pi$
$24$	0	0
$25$	0	0
$26$	359.857	0.104399
$27$	0	0
$28$	− 564.876i	− 0.136163i
$29$	−2580.97	−0.569885	−0.284943	−	0.958545i	$-0.591975\pi$
$29$	−2580.97	−0.569885	−0.284943	+	0.958545i	$0.591975\pi$
$30$	0	0
$31$	−9086.30	−1.69818	−0.849088	−	0.528252i	$-0.822848\pi$
$31$	−9086.30	−1.69818	−0.849088	+	0.528252i	$0.822848\pi$
$32$	− 1549.41i	− 0.267480i
$33$	0	0
$34$	−1634.64	−0.242507
$35$	0	0
$36$	0	0
$37$	− 3990.64i	− 0.479224i	−0.970869	−	0.239612i	$-0.922980\pi$
$37$	− 3990.64i	− 0.479224i	0.970869	−	0.239612i	$-0.0770202\pi$
$38$	− 11225.9i	− 1.26113i
$39$	0	0
$40$	0	0
$41$	−16981.8	−1.57770	−0.788851	−	0.614584i	$-0.789324\pi$
$41$	−16981.8	−1.57770	−0.788851	+	0.614584i	$0.789324\pi$
$42$	0	0
$43$	− 18017.7i	− 1.48603i	−0.669273	−	0.743017i	$-0.733394\pi$
$43$	− 18017.7i	− 1.48603i	0.669273	−	0.743017i	$-0.266606\pi$
$44$	−1250.48	−0.0973742
$45$	0	0
$46$	−4594.57	−0.320147
$47$	24864.7i	1.64187i	0.571024	+	0.820933i	$0.306546\pi$
$47$	24864.7i	1.64187i	−0.571024	+	0.820933i	$0.693454\pi$
$48$	0	0
$49$	−366.670	−0.0218165
$50$	0	0
$51$	0	0
$52$	294.777i	0.0151177i
$53$	7652.91i	0.374229i	0.982338	+	0.187114i	$0.0599135\pi$
$53$	7652.91i	0.374229i	−0.982338	+	0.187114i	$0.940087\pi$
$54$	0	0
$55$	0	0
$56$	−25039.2	−1.06697
$57$	0	0
$58$	13581.3i	0.530116i
$59$	−9233.69	−0.345339	−0.172669	−	0.984980i	$-0.555239\pi$
$59$	−9233.69	−0.345339	−0.172669	+	0.984980i	$0.555239\pi$
$60$	0	0
$61$	3326.17	0.114451	0.0572256	−	0.998361i	$-0.481775\pi$
$61$	3326.17	0.114451	0.0572256	+	0.998361i	$0.481775\pi$
$62$	47812.9i	1.57967i
$63$	0	0
$64$	−35912.7	−1.09597
$65$	0	0
$66$	0	0
$67$	32340.7i	0.880161i	0.897958	+	0.440080i	$0.145050\pi$
$67$	32340.7i	0.880161i	−0.897958	+	0.440080i	$0.854950\pi$
$68$	− 1339.01i	− 0.0351165i
$69$	0	0
$70$	0	0
$71$	35885.9	0.844847	0.422424	−	0.906399i	$-0.361179\pi$
$71$	35885.9	0.844847	0.422424	+	0.906399i	$0.361179\pi$
$72$	0	0
$73$	− 26513.6i	− 0.582319i	−0.956675	−	0.291159i	$-0.905959\pi$
$73$	− 26513.6i	− 0.582319i	0.956675	−	0.291159i	$-0.0940410\pi$
$74$	−20999.1	−0.445781
$75$	0	0
$76$	9195.65	0.182620
$77$	38017.7i	0.730733i
$78$	0	0
$79$	−71705.7	−1.29266	−0.646332	−	0.763056i	$-0.723698\pi$
$79$	−71705.7	−1.29266	−0.646332	+	0.763056i	$0.723698\pi$
$80$	0	0
$81$	0	0
$82$	89359.9i	1.46760i
$83$	− 39630.1i	− 0.631437i	−0.948853	−	0.315719i	$-0.897754\pi$
$83$	− 39630.1i	− 0.631437i	0.948853	−	0.315719i	$-0.102246\pi$
$84$	0	0
$85$	0	0
$86$	−94810.8	−1.38233
$87$	0	0
$88$	55429.9i	0.763022i
$89$	−117441.	−1.57161	−0.785806	−	0.618473i	$-0.787752\pi$
$89$	−117441.	−1.57161	−0.785806	+	0.618473i	$0.787752\pi$
$90$	0	0
$91$	8961.98	0.113449
$92$	− 3763.64i	− 0.0463594i
$93$	0	0
$94$	130840.	1.52729
$95$	0	0
$96$	0	0
$97$	− 21878.3i	− 0.236093i	−0.993008	−	0.118047i	$-0.962337\pi$
$97$	− 21878.3i	− 0.236093i	0.993008	−	0.118047i	$-0.0376633\pi$
$98$	1929.45i	0.0202940i
$99$	0	0
$100$	0	0
$101$	75072.1	0.732276	0.366138	−	0.930561i	$-0.380680\pi$
$101$	75072.1	0.732276	0.366138	+	0.930561i	$0.380680\pi$
$102$	0	0
$103$	47928.6i	0.445145i	0.974916	+	0.222573i	$0.0714455\pi$
$103$	47928.6i	0.445145i	−0.974916	+	0.222573i	$0.928555\pi$
$104$	13066.6	0.118462
$105$	0	0
$106$	40270.3	0.348113
$107$	92012.3i	0.776938i	0.921462	+	0.388469i	$0.126996\pi$
$107$	92012.3i	0.776938i	−0.921462	+	0.388469i	$0.873004\pi$
$108$	0	0
$109$	10647.5	0.0858387	0.0429194	−	0.999079i	$-0.486334\pi$
$109$	10647.5	0.0858387	0.0429194	+	0.999079i	$0.486334\pi$
$110$	0	0
$111$	0	0
$112$	113683.i	0.856346i
$113$	− 87373.9i	− 0.643703i	−0.946790	−	0.321852i	$-0.895695\pi$
$113$	− 87373.9i	− 0.643703i	0.946790	−	0.321852i	$-0.104305\pi$
$114$	0	0
$115$	0	0
$116$	−11125.1	−0.0767642
$117$	0	0
$118$	48588.5i	0.321239i
$119$	−40709.4	−0.263528
$120$	0	0
$121$	−76890.5	−0.477429
$122$	− 17502.6i	− 0.106464i
$123$	0	0
$124$	−39165.9	−0.228746
$125$	0	0
$126$	0	0
$127$	− 197379.i	− 1.08591i	−0.839763	−	0.542953i	$-0.817306\pi$
$127$	− 197379.i	− 1.08591i	0.839763	−	0.542953i	$-0.182694\pi$
$128$	139395.i	0.752005i
$129$	0	0
$130$	0	0
$131$	118490.	0.603258	0.301629	−	0.953425i	$-0.402470\pi$
$131$	118490.	0.603258	0.301629	+	0.953425i	$0.402470\pi$
$132$	0	0
$133$	− 279571.i	− 1.37045i
$134$	170179.	0.818738
$135$	0	0
$136$	−59354.3	−0.275173
$137$	− 302570.i	− 1.37728i	−0.725101	−	0.688642i	$-0.758207\pi$
$137$	− 302570.i	− 1.37728i	0.725101	−	0.688642i	$-0.241793\pi$
$138$	0	0
$139$	−157190.	−0.690062	−0.345031	−	0.938591i	$-0.612132\pi$
$139$	−157190.	−0.690062	−0.345031	+	0.938591i	$0.612132\pi$
$140$	0	0
$141$	0	0
$142$	− 188835.i	− 0.785889i
$143$	− 19839.3i	− 0.0811309i
$144$	0	0
$145$	0	0
$146$	−139517.	−0.541681
$147$	0	0
$148$	− 17201.4i	− 0.0645520i
$149$	526340.	1.94223	0.971115	−	0.238612i	$-0.0766923\pi$
$149$	526340.	1.94223	0.971115	+	0.238612i	$0.0766923\pi$
$150$	0	0
$151$	1849.08	0.00659954	0.00329977	−	0.999995i	$-0.498950\pi$
$151$	1849.08	0.00659954	0.00329977	+	0.999995i	$0.498950\pi$
$152$	− 407616.i	− 1.43101i
$153$	0	0
$154$	200052.	0.679739
$155$	0	0
$156$	0	0
$157$	− 343342.i	− 1.11167i	−0.831292	−	0.555837i	$-0.812398\pi$
$157$	− 343342.i	− 1.11167i	0.831292	−	0.555837i	$-0.187602\pi$
$158$	377322.i	1.20246i
$159$	0	0
$160$	0	0
$161$	−114424.	−0.347899
$162$	0	0
$163$	− 267463.i	− 0.788487i	−0.919006	−	0.394243i	$-0.871007\pi$
$163$	− 267463.i	− 0.788487i	0.919006	−	0.394243i	$-0.128993\pi$
$164$	−73199.1	−0.212518
$165$	0	0
$166$	−208537.	−0.587372
$167$	122968.i	0.341193i	0.985341	+	0.170596i	$0.0545694\pi$
$167$	122968.i	0.341193i	−0.985341	+	0.170596i	$0.945431\pi$
$168$	0	0
$169$	366616.	0.987404
$170$	0	0
$171$	0	0
$172$	− 77664.3i	− 0.200170i
$173$	− 288020.i	− 0.731657i	−0.930682	−	0.365829i	$-0.880786\pi$
$173$	− 288020.i	− 0.731657i	0.930682	−	0.365829i	$-0.119214\pi$
$174$	0	0
$175$	0	0
$176$	251662.	0.612400
$177$	0	0
$178$	617986.i	1.46194i
$179$	246177.	0.574268	0.287134	−	0.957890i	$-0.407298\pi$
$179$	246177.	0.574268	0.287134	+	0.957890i	$0.407298\pi$
$180$	0	0
$181$	433120.	0.982678	0.491339	−	0.870968i	$-0.336508\pi$
$181$	433120.	0.982678	0.491339	+	0.870968i	$0.336508\pi$
$182$	− 47158.7i	− 0.105532i
$183$	0	0
$184$	−166831.	−0.363272
$185$	0	0
$186$	0	0
$187$	90119.1i	0.188457i
$188$	107178.i	0.221161i
$189$	0	0
$190$	0	0
$191$	701011.	1.39040	0.695202	−	0.718814i	$-0.255315\pi$
$191$	701011.	1.39040	0.695202	+	0.718814i	$0.255315\pi$
$192$	0	0
$193$	215730.i	0.416887i	0.978034	+	0.208443i	$0.0668397\pi$
$193$	215730.i	0.416887i	−0.978034	+	0.208443i	$0.933160\pi$
$194$	−115125.	−0.219618
$195$	0	0
$196$	−1580.51	−0.00293871
$197$	− 700484.i	− 1.28598i	−0.765876	−	0.642988i	$-0.777695\pi$
$197$	− 700484.i	− 1.28598i	0.765876	−	0.642988i	$-0.222305\pi$
$198$	0	0
$199$	−22097.5	−0.0395558	−0.0197779	−	0.999804i	$-0.506296\pi$
$199$	−22097.5	−0.0395558	−0.0197779	+	0.999804i	$0.506296\pi$
$200$	0	0
$201$	0	0
$202$	− 395036.i	− 0.681174i
$203$	338231.i	0.576068i
$204$	0	0
$205$	0	0
$206$	252205.	0.414081
$207$	0	0
$208$	− 59324.6i	− 0.0950772i
$209$	−618893.	−0.980054
$210$	0	0
$211$	910782.	1.40834	0.704172	−	0.710030i	$-0.251319\pi$
$211$	910782.	1.40834	0.704172	+	0.710030i	$0.251319\pi$
$212$	32987.4i	0.0504090i
$213$	0	0
$214$	484177.	0.722719
$215$	0	0
$216$	0	0
$217$	1.19074e6i	1.71660i
$218$	− 56028.3i	− 0.0798484i
$219$	0	0
$220$	0	0
$221$	21243.9	0.0292587
$222$	0	0
$223$	132745.i	0.178754i	0.995998	+	0.0893768i	$0.0284875\pi$
$223$	132745.i	0.178754i	−0.995998	+	0.0893768i	$0.971512\pi$
$224$	−203048.	−0.270382
$225$	0	0
$226$	−459769.	−0.598782
$227$	− 354321.i	− 0.456386i	−0.973616	−	0.228193i	$-0.926718\pi$
$227$	− 354321.i	− 0.456386i	0.973616	−	0.228193i	$-0.0732817\pi$
$228$	0	0
$229$	−366643.	−0.462013	−0.231007	−	0.972952i	$-0.574202\pi$
$229$	−366643.	−0.462013	−0.231007	+	0.972952i	$0.574202\pi$
$230$	0	0
$231$	0	0
$232$	493142.i	0.601523i
$233$	− 1.02388e6i	− 1.23555i	−0.786355	−	0.617776i	$-0.788034\pi$
$233$	− 1.02388e6i	− 1.23555i	0.786355	−	0.617776i	$-0.211966\pi$
$234$	0	0
$235$	0	0
$236$	−39801.2	−0.0465175
$237$	0	0
$238$	214216.i	0.245138i
$239$	1.19966e6	1.35852	0.679258	−	0.733899i	$-0.262302\pi$
$239$	1.19966e6	1.35852	0.679258	+	0.733899i	$0.262302\pi$
$240$	0	0
$241$	−94967.5	−0.105325	−0.0526626	−	0.998612i	$-0.516771\pi$
$241$	−94967.5	−0.105325	−0.0526626	+	0.998612i	$0.516771\pi$
$242$	404604.i	0.444112i
$243$	0	0
$244$	14337.3	0.0154167
$245$	0	0
$246$	0	0
$247$	145893.i	0.152157i
$248$	1.73611e6i	1.79245i
$249$	0	0
$250$	0	0
$251$	−418053.	−0.418839	−0.209419	−	0.977826i	$-0.567157\pi$
$251$	−418053.	−0.418839	−0.209419	+	0.977826i	$0.567157\pi$
$252$	0	0
$253$	253303.i	0.248794i
$254$	−1.03863e6	−1.01012
$255$	0	0
$256$	−415700.	−0.396442
$257$	2.04586e6i	1.93216i	0.258246	+	0.966079i	$0.416856\pi$
$257$	2.04586e6i	1.93216i	−0.258246	+	0.966079i	$0.583144\pi$
$258$	0	0
$259$	−522967.	−0.484423
$260$	0	0
$261$	0	0
$262$	− 623505.i	− 0.561160i
$263$	− 1.64024e6i	− 1.46224i	−0.682250	−	0.731119i	$-0.738998\pi$
$263$	− 1.64024e6i	− 1.46224i	0.682250	−	0.731119i	$-0.261002\pi$
$264$	0	0
$265$	0	0
$266$	−1.47113e6	−1.27481
$267$	0	0
$268$	139402.i	0.118559i
$269$	−720582.	−0.607160	−0.303580	−	0.952806i	$-0.598182\pi$
$269$	−720582.	−0.607160	−0.303580	+	0.952806i	$0.598182\pi$
$270$	0	0
$271$	1.14186e6	0.944477	0.472238	−	0.881471i	$-0.343446\pi$
$271$	1.14186e6	0.944477	0.472238	+	0.881471i	$0.343446\pi$
$272$	269479.i	0.220853i
$273$	0	0
$274$	−1.59215e6	−1.28117
$275$	0	0
$276$	0	0
$277$	− 377028.i	− 0.295239i	−0.989044	−	0.147620i	$-0.952839\pi$
$277$	− 377028.i	− 0.295239i	0.989044	−	0.147620i	$-0.0471612\pi$
$278$	827148.i	0.641906i
$279$	0	0
$280$	0	0
$281$	617249.	0.466331	0.233166	−	0.972437i	$-0.425092\pi$
$281$	617249.	0.466331	0.233166	+	0.972437i	$0.425092\pi$
$282$	0	0
$283$	− 1.25311e6i	− 0.930087i	−0.885288	−	0.465044i	$-0.846039\pi$
$283$	− 1.25311e6i	− 0.930087i	0.885288	−	0.465044i	$-0.153961\pi$
$284$	154684.	0.113802
$285$	0	0
$286$	−104396.	−0.0754692
$287$	2.22544e6i	1.59482i
$288$	0	0
$289$	1.32336e6	0.932036
$290$	0	0
$291$	0	0
$292$	− 114285.i	− 0.0784390i
$293$	− 818972.i	− 0.557314i	−0.960391	−	0.278657i	$-0.910111\pi$
$293$	− 818972.i	− 0.557314i	0.960391	−	0.278657i	$-0.0898893\pi$
$294$	0	0
$295$	0	0
$296$	−762487.	−0.505828
$297$	0	0
$298$	− 2.76965e6i	− 1.80669i
$299$	59711.6	0.0386261
$300$	0	0
$301$	−2.36119e6	−1.50216
$302$	− 9730.03i	− 0.00613899i
$303$	0	0
$304$	−1.85065e6	−1.14852
$305$	0	0
$306$	0	0
$307$	136224.i	0.0824915i	0.999149	+	0.0412458i	$0.0131327\pi$
$307$	136224.i	0.0824915i	−0.999149	+	0.0412458i	$0.986867\pi$
$308$	163873.i	0.0984306i
$309$	0	0
$310$	0	0
$311$	−2.62886e6	−1.54122	−0.770612	−	0.637304i	$-0.780050\pi$
$311$	−2.62886e6	−1.54122	−0.770612	+	0.637304i	$0.780050\pi$
$312$	0	0
$313$	− 218161.i	− 0.125868i	−0.998018	−	0.0629341i	$-0.979954\pi$
$313$	− 218161.i	− 0.125868i	0.998018	−	0.0629341i	$-0.0200458\pi$
$314$	−1.80669e6	−1.03409
$315$	0	0
$316$	−309083.	−0.174123
$317$	− 1.25865e6i	− 0.703491i	−0.936096	−	0.351745i	$-0.885588\pi$
$317$	− 1.25865e6i	− 0.703491i	0.936096	−	0.351745i	$-0.114412\pi$
$318$	0	0
$319$	748750.	0.411965
$320$	0	0
$321$	0	0
$322$	602110.i	0.323621i
$323$	− 662711.i	− 0.353442i
$324$	0	0
$325$	0	0
$326$	−1.40741e6	−0.733462
$327$	0	0
$328$	3.24470e6i	1.66529i
$329$	3.25847e6	1.65968
$330$	0	0
$331$	−3.21863e6	−1.61473	−0.807366	−	0.590051i	$-0.799108\pi$
$331$	−3.21863e6	−1.61473	−0.807366	+	0.590051i	$0.799108\pi$
$332$	− 170823.i	− 0.0850553i
$333$	0	0
$334$	647067.	0.317382
$335$	0	0
$336$	0	0
$337$	1.63574e6i	0.784585i	0.919840	+	0.392293i	$0.128318\pi$
$337$	1.63574e6i	0.784585i	−0.919840	+	0.392293i	$0.871682\pi$
$338$	− 1.92917e6i	− 0.918498i
$339$	0	0
$340$	0	0
$341$	2.63597e6	1.22760
$342$	0	0
$343$	− 2.15448e6i	− 0.988796i
$344$	−3.44262e6	−1.56853
$345$	0	0
$346$	−1.51559e6	−0.680598
$347$	1.83815e6i	0.819514i	0.912195	+	0.409757i	$0.134386\pi$
$347$	1.83815e6i	0.819514i	−0.912195	+	0.409757i	$0.865614\pi$
$348$	0	0
$349$	2.53806e6	1.11542	0.557710	−	0.830036i	$-0.311680\pi$
$349$	2.53806e6	1.11542	0.557710	+	0.830036i	$0.311680\pi$
$350$	0	0
$351$	0	0
$352$	449491.i	0.193359i
$353$	− 1.88471e6i	− 0.805023i	−0.915415	−	0.402511i	$-0.868137\pi$
$353$	− 1.88471e6i	− 0.805023i	0.915415	−	0.402511i	$-0.131863\pi$
$354$	0	0
$355$	0	0
$356$	−506223.	−0.211698
$357$	0	0
$358$	− 1.29540e6i	− 0.534192i
$359$	305057.	0.124924	0.0624619	−	0.998047i	$-0.480105\pi$
$359$	305057.	0.124924	0.0624619	+	0.998047i	$0.480105\pi$
$360$	0	0
$361$	2.07507e6	0.838039
$362$	− 2.27911e6i	− 0.914102i
$363$	0	0
$364$	38630.0	0.0152817
$365$	0	0
$366$	0	0
$367$	− 727834.i	− 0.282077i	−0.990004	−	0.141038i	$-0.954956\pi$
$367$	− 727834.i	− 0.282077i	0.990004	−	0.141038i	$-0.0450441\pi$
$368$	757441.i	0.291561i
$369$	0	0
$370$	0	0
$371$	1.00290e6	0.378289
$372$	0	0
$373$	4.77676e6i	1.77771i	0.458188	+	0.888855i	$0.348499\pi$
$373$	4.77676e6i	1.77771i	−0.458188	+	0.888855i	$0.651501\pi$
$374$	474215.	0.175306
$375$	0	0
$376$	4.75086e6	1.73302
$377$	− 176504.i	− 0.0639590i
$378$	0	0
$379$	701558.	0.250880	0.125440	−	0.992101i	$-0.459966\pi$
$379$	701558.	0.250880	0.125440	+	0.992101i	$0.459966\pi$
$380$	0	0
$381$	0	0
$382$	− 3.68878e6i	− 1.29337i
$383$	4.01069e6i	1.39708i	0.715570	+	0.698541i	$0.246167\pi$
$383$	4.01069e6i	1.39708i	−0.715570	+	0.698541i	$0.753833\pi$
$384$	0	0
$385$	0	0
$386$	1.13519e6	0.387794
$387$	0	0
$388$	− 94305.0i	− 0.0318021i
$389$	−4.45952e6	−1.49422	−0.747108	−	0.664702i	$-0.768558\pi$
$389$	−4.45952e6	−1.49422	−0.747108	+	0.664702i	$0.768558\pi$
$390$	0	0
$391$	−271237.	−0.0897237
$392$	70059.1i	0.0230276i
$393$	0	0
$394$	−3.68601e6	−1.19623
$395$	0	0
$396$	0	0
$397$	3.36993e6i	1.07311i	0.843865	+	0.536555i	$0.180275\pi$
$397$	3.36993e6i	1.07311i	−0.843865	+	0.536555i	$0.819725\pi$
$398$	116279.i	0.0367953i
$399$	0	0
$400$	0	0
$401$	3.00679e6	0.933775	0.466888	−	0.884317i	$-0.345375\pi$
$401$	3.00679e6	0.933775	0.466888	+	0.884317i	$0.345375\pi$
$402$	0	0
$403$	− 621383.i	− 0.190588i
$404$	323593.	0.0986384
$405$	0	0
$406$	1.77980e6	0.535867
$407$	1.15770e6i	0.346426i
$408$	0	0
$409$	998012.	0.295004	0.147502	−	0.989062i	$-0.452877\pi$
$409$	998012.	0.295004	0.147502	+	0.989062i	$0.452877\pi$
$410$	0	0
$411$	0	0
$412$	206593.i	0.0599616i
$413$	1.21006e6i	0.349085i
$414$	0	0
$415$	0	0
$416$	105959.	0.0300196
$417$	0	0
$418$	3.25667e6i	0.911660i
$419$	5.53743e6	1.54090	0.770448	−	0.637503i	$-0.220033\pi$
$419$	5.53743e6	1.54090	0.770448	+	0.637503i	$0.220033\pi$
$420$	0	0
$421$	1.98635e6	0.546198	0.273099	−	0.961986i	$-0.411951\pi$
$421$	1.98635e6	0.546198	0.273099	+	0.961986i	$0.411951\pi$
$422$	− 4.79262e6i	− 1.31006i
$423$	0	0
$424$	1.46223e6	0.395004
$425$	0	0
$426$	0	0
$427$	− 435890.i	− 0.115693i
$428$	396613.i	0.104654i
$429$	0	0
$430$	0	0
$431$	−116512.	−0.0302118	−0.0151059	−	0.999886i	$-0.504809\pi$
$431$	−116512.	−0.0302118	−0.0151059	+	0.999886i	$0.504809\pi$
$432$	0	0
$433$	− 4.56166e6i	− 1.16924i	−0.811308	−	0.584619i	$-0.801244\pi$
$433$	− 4.56166e6i	− 1.16924i	0.811308	−	0.584619i	$-0.198756\pi$
$434$	6.26580e6	1.59681
$435$	0	0
$436$	45895.6	0.0115626
$437$	− 1.86272e6i	− 0.466599i
$438$	0	0
$439$	−2.92172e6	−0.723565	−0.361782	−	0.932263i	$-0.617832\pi$
$439$	−2.92172e6	−0.723565	−0.361782	+	0.932263i	$0.617832\pi$
$440$	0	0
$441$	0	0
$442$	− 111787.i	− 0.0272168i
$443$	1.59752e6i	0.386756i	0.981124	+	0.193378i	$0.0619444\pi$
$443$	1.59752e6i	0.386756i	−0.981124	+	0.193378i	$0.938056\pi$
$444$	0	0
$445$	0	0
$446$	698514.	0.166279
$447$	0	0
$448$	4.70630e6i	1.10786i
$449$	3.11073e6	0.728193	0.364096	−	0.931361i	$-0.381378\pi$
$449$	3.11073e6	0.728193	0.364096	+	0.931361i	$0.381378\pi$
$450$	0	0
$451$	4.92650e6	1.14051
$452$	− 376620.i	− 0.0867076i
$453$	0	0
$454$	−1.86447e6	−0.424537
$455$	0	0
$456$	0	0
$457$	− 6.47145e6i	− 1.44948i	−0.689025	−	0.724738i	$-0.741961\pi$
$457$	− 6.47145e6i	− 1.44948i	0.689025	−	0.724738i	$-0.258039\pi$
$458$	1.92931e6i	0.429771i
$459$	0	0
$460$	0	0
$461$	5.47864e6	1.20066	0.600330	−	0.799752i	$-0.295036\pi$
$461$	5.47864e6	1.20066	0.600330	+	0.799752i	$0.295036\pi$
$462$	0	0
$463$	2.35489e6i	0.510526i	0.966872	+	0.255263i	$0.0821621\pi$
$463$	2.35489e6i	0.510526i	−0.966872	+	0.255263i	$0.917838\pi$
$464$	2.23895e6	0.482781
$465$	0	0
$466$	−5.38776e6	−1.14933
$467$	− 4.56027e6i	− 0.967606i	−0.875177	−	0.483803i	$-0.839255\pi$
$467$	− 4.56027e6i	− 0.967606i	0.875177	−	0.483803i	$-0.160745\pi$
$468$	0	0
$469$	4.23819e6	0.889710
$470$	0	0
$471$	0	0
$472$	1.76427e6i	0.364510i
$473$	5.22702e6i	1.07424i
$474$	0	0
$475$	0	0
$476$	−175475.	−0.0354975
$477$	0	0
$478$	− 6.31274e6i	− 1.26371i
$479$	−1.88004e6	−0.374394	−0.187197	−	0.982322i	$-0.559940\pi$
$479$	−1.88004e6	−0.374394	−0.187197	+	0.982322i	$0.559940\pi$
$480$	0	0
$481$	272907.	0.0537839
$482$	499727.i	0.0979751i
$483$	0	0
$484$	−331431.	−0.0643103
$485$	0	0
$486$	0	0
$487$	− 1.69396e6i	− 0.323654i	−0.986819	−	0.161827i	$-0.948261\pi$
$487$	− 1.69396e6i	− 0.323654i	0.986819	−	0.161827i	$-0.0517386\pi$
$488$	− 635528.i	− 0.120805i
$489$	0	0
$490$	0	0
$491$	−1.48645e6	−0.278258	−0.139129	−	0.990274i	$-0.544430\pi$
$491$	−1.48645e6	−0.278258	−0.139129	+	0.990274i	$0.544430\pi$
$492$	0	0
$493$	801762.i	0.148569i
$494$	767700.	0.141538
$495$	0	0
$496$	7.88224e6	1.43862
$497$	− 4.70279e6i	− 0.854013i
$498$	0	0
$499$	−7.09934e6	−1.27634	−0.638170	−	0.769896i	$-0.720308\pi$
$499$	−7.09934e6	−1.27634	−0.638170	+	0.769896i	$0.720308\pi$
$500$	0	0
$501$	0	0
$502$	2.19983e6i	0.389610i
$503$	− 9.24224e6i	− 1.62876i	−0.580331	−	0.814381i	$-0.697077\pi$
$503$	− 9.24224e6i	− 1.62876i	0.580331	−	0.814381i	$-0.302923\pi$
$504$	0	0
$505$	0	0
$506$	1.33290e6	0.231431
$507$	0	0
$508$	− 850790.i	− 0.146273i
$509$	−8.12506e6	−1.39006	−0.695028	−	0.718983i	$-0.744608\pi$
$509$	−8.12506e6	−1.39006	−0.695028	+	0.718983i	$0.744608\pi$
$510$	0	0
$511$	−3.47456e6	−0.588637
$512$	6.64807e6i	1.12078i
$513$	0	0
$514$	1.07655e7	1.79732
$515$	0	0
$516$	0	0
$517$	− 7.21335e6i	− 1.18689i
$518$	2.75190e6i	0.450617i
$519$	0	0
$520$	0	0
$521$	−5.06245e6	−0.817084	−0.408542	−	0.912740i	$-0.633963\pi$
$521$	−5.06245e6	−0.817084	−0.408542	+	0.912740i	$0.633963\pi$
$522$	0	0
$523$	− 4.76222e6i	− 0.761299i	−0.924719	−	0.380649i	$-0.875700\pi$
$523$	− 4.76222e6i	− 0.761299i	0.924719	−	0.380649i	$-0.124300\pi$
$524$	510744.	0.0812596
$525$	0	0
$526$	−8.63109e6	−1.36019
$527$	2.82260e6i	0.442714i
$528$	0	0
$529$	5.67396e6	0.881550
$530$	0	0
$531$	0	0
$532$	− 1.20508e6i	− 0.184601i
$533$	− 1.16133e6i	− 0.177067i
$534$	0	0
$535$	0	0
$536$	6.17929e6	0.929023
$537$	0	0
$538$	3.79177e6i	0.564789i
$539$	106373.	0.0157709
$540$	0	0
$541$	−2.89920e6	−0.425877	−0.212939	−	0.977066i	$-0.568303\pi$
$541$	−2.89920e6	−0.425877	−0.212939	+	0.977066i	$0.568303\pi$
$542$	− 6.00859e6i	− 0.878566i
$543$	0	0
$544$	−481315.	−0.0697320
$545$	0	0
$546$	0	0
$547$	5.74434e6i	0.820866i	0.911891	+	0.410433i	$0.134622\pi$
$547$	5.74434e6i	0.820866i	−0.911891	+	0.410433i	$0.865378\pi$
$548$	− 1.30421e6i	− 0.185522i
$549$	0	0
$550$	0	0
$551$	−5.50610e6	−0.772618
$552$	0	0
$553$	9.39691e6i	1.30669i
$554$	−1.98395e6	−0.274636
$555$	0	0
$556$	−677558.	−0.0929522
$557$	− 7.29174e6i	− 0.995848i	−0.867221	−	0.497924i	$-0.834096\pi$
$557$	− 7.29174e6i	− 0.995848i	0.867221	−	0.497924i	$-0.165904\pi$
$558$	0	0
$559$	1.23217e6	0.166779
$560$	0	0
$561$	0	0
$562$	− 3.24802e6i	− 0.433788i
$563$	− 6.65348e6i	− 0.884663i	−0.896852	−	0.442331i	$-0.854151\pi$
$563$	− 6.65348e6i	− 0.884663i	0.896852	−	0.442331i	$-0.145849\pi$
$564$	0	0
$565$	0	0
$566$	−6.59398e6	−0.865181
$567$	0	0
$568$	− 6.85667e6i	− 0.891749i
$569$	5.78715e6	0.749349	0.374675	−	0.927156i	$-0.377754\pi$
$569$	5.78715e6	0.749349	0.374675	+	0.927156i	$0.377754\pi$
$570$	0	0
$571$	−1.22059e7	−1.56667	−0.783336	−	0.621599i	$-0.786483\pi$
$571$	−1.22059e7	−1.56667	−0.783336	+	0.621599i	$0.786483\pi$
$572$	− 85516.1i	− 0.0109284i
$573$	0	0
$574$	1.17105e7	1.48352
$575$	0	0
$576$	0	0
$577$	1.02981e7i	1.28771i	0.765149	+	0.643853i	$0.222665\pi$
$577$	1.02981e7i	1.28771i	−0.765149	+	0.643853i	$0.777335\pi$
$578$	− 6.96362e6i	− 0.866993i
$579$	0	0
$580$	0	0
$581$	−5.19346e6	−0.638288
$582$	0	0
$583$	− 2.22014e6i	− 0.270526i
$584$	−5.06591e6	−0.614647
$585$	0	0
$586$	−4.30950e6	−0.518422
$587$	1.30519e7i	1.56343i	0.623636	+	0.781715i	$0.285655\pi$
$587$	1.30519e7i	1.56343i	−0.623636	+	0.781715i	$0.714345\pi$
$588$	0	0
$589$	−1.93842e7	−2.30229
$590$	0	0
$591$	0	0
$592$	3.46183e6i	0.405977i
$593$	6.43920e6i	0.751961i	0.926628	+	0.375980i	$0.122694\pi$
$593$	6.43920e6i	0.751961i	−0.926628	+	0.375980i	$0.877306\pi$
$594$	0	0
$595$	0	0
$596$	2.26876e6	0.261621
$597$	0	0
$598$	− 314208.i	− 0.0359305i
$599$	−1.00760e7	−1.14741	−0.573707	−	0.819061i	$-0.694495\pi$
$599$	−1.00760e7	−1.14741	−0.573707	+	0.819061i	$0.694495\pi$
$600$	0	0
$601$	1.57050e6	0.177358	0.0886791	−	0.996060i	$-0.471735\pi$
$601$	1.57050e6	0.177358	0.0886791	+	0.996060i	$0.471735\pi$
$602$	1.24248e7i	1.39733i
$603$	0	0
$604$	7970.35	0.000888966 0
$605$	0	0
$606$	0	0
$607$	− 7.31039e6i	− 0.805321i	−0.915349	−	0.402660i	$-0.868086\pi$
$607$	− 7.31039e6i	− 0.805321i	0.915349	−	0.402660i	$-0.131914\pi$
$608$	− 3.30543e6i	− 0.362634i
$609$	0	0
$610$	0	0
$611$	−1.70041e6	−0.184269
$612$	0	0
$613$	− 1.31997e7i	− 1.41878i	−0.704817	−	0.709389i	$-0.748971\pi$
$613$	− 1.31997e7i	− 1.41878i	0.704817	−	0.709389i	$-0.251029\pi$
$614$	716825.	0.0767348
$615$	0	0
$616$	7.26399e6	0.771300
$617$	− 1.02423e7i	− 1.08314i	−0.840655	−	0.541570i	$-0.817830\pi$
$617$	− 1.02423e7i	− 1.08314i	0.840655	−	0.541570i	$-0.182170\pi$
$618$	0	0
$619$	−1.05614e7	−1.10788	−0.553942	−	0.832555i	$-0.686877\pi$
$619$	−1.05614e7	−1.10788	−0.553942	+	0.832555i	$0.686877\pi$
$620$	0	0
$621$	0	0
$622$	1.38333e7i	1.43367i
$623$	1.53905e7i	1.58866i
$624$	0	0
$625$	0	0
$626$	−1.14798e6	−0.117084
$627$	0	0
$628$	− 1.47995e6i	− 0.149744i
$629$	−1.23967e6	−0.124933
$630$	0	0
$631$	1.90535e7	1.90503	0.952513	−	0.304497i	$-0.0984883\pi$
$631$	1.90535e7	1.90503	0.952513	+	0.304497i	$0.0984883\pi$
$632$	1.37007e7i	1.36443i
$633$	0	0
$634$	−6.62315e6	−0.654397
$635$	0	0
$636$	0	0
$637$	− 25075.4i	− 0.00244849i
$638$	− 3.93999e6i	− 0.383216i
$639$	0	0
$640$	0	0
$641$	−8.56937e6	−0.823766	−0.411883	−	0.911237i	$-0.635129\pi$
$641$	−8.56937e6	−0.823766	−0.411883	+	0.911237i	$0.635129\pi$
$642$	0	0
$643$	1.79513e7i	1.71226i	0.516761	+	0.856130i	$0.327138\pi$
$643$	1.79513e7i	1.71226i	−0.516761	+	0.856130i	$0.672862\pi$
$644$	−493218.	−0.0468624
$645$	0	0
$646$	−3.48724e6	−0.328777
$647$	1.05470e7i	0.990534i	0.868741	+	0.495267i	$0.164930\pi$
$647$	1.05470e7i	0.990534i	−0.868741	+	0.495267i	$0.835070\pi$
$648$	0	0
$649$	2.67873e6	0.249642
$650$	0	0
$651$	0	0
$652$	− 1.15288e6i	− 0.106210i
$653$	− 1.00324e7i	− 0.920712i	−0.887734	−	0.460356i	$-0.847722\pi$
$653$	− 1.00324e7i	− 0.920712i	0.887734	−	0.460356i	$-0.152278\pi$
$654$	0	0
$655$	0	0
$656$	1.47315e7	1.33656
$657$	0	0
$658$	− 1.71464e7i	− 1.54386i
$659$	−8.99161e6	−0.806536	−0.403268	−	0.915082i	$-0.632126\pi$
$659$	−8.99161e6	−0.806536	−0.403268	+	0.915082i	$0.632126\pi$
$660$	0	0
$661$	2.39297e6	0.213027	0.106513	−	0.994311i	$-0.466031\pi$
$661$	2.39297e6	0.213027	0.106513	+	0.994311i	$0.466031\pi$
$662$	1.69367e7i	1.50205i
$663$	0	0
$664$	−7.57208e6	−0.666492
$665$	0	0
$666$	0	0
$667$	2.25356e6i	0.196135i
$668$	530044.i	0.0459590i
$669$	0	0
$670$	0	0
$671$	−964938.	−0.0827357
$672$	0	0
$673$	1.53612e7i	1.30733i	0.756783	+	0.653666i	$0.226770\pi$
$673$	1.53612e7i	1.30733i	−0.756783	+	0.653666i	$0.773230\pi$
$674$	8.60742e6	0.729833
$675$	0	0
$676$	1.58028e6	0.133004
$677$	1.16026e7i	0.972934i	0.873699	+	0.486467i	$0.161715\pi$
$677$	1.16026e7i	0.972934i	−0.873699	+	0.486467i	$0.838285\pi$
$678$	0	0
$679$	−2.86711e6	−0.238655
$680$	0	0
$681$	0	0
$682$	− 1.38707e7i	− 1.14193i
$683$	1.20315e7i	0.986890i	0.869777	+	0.493445i	$0.164263\pi$
$683$	1.20315e7i	0.986890i	−0.869777	+	0.493445i	$0.835737\pi$
$684$	0	0
$685$	0	0
$686$	−1.13371e7	−0.919792
$687$	0	0
$688$	1.56301e7i	1.25890i
$689$	−523358.	−0.0420001
$690$	0	0
$691$	5.18616e6	0.413191	0.206595	−	0.978426i	$-0.433762\pi$
$691$	5.18616e6	0.413191	0.206595	+	0.978426i	$0.433762\pi$
$692$	− 1.24149e6i	− 0.0985550i
$693$	0	0
$694$	9.67248e6	0.762323
$695$	0	0
$696$	0	0
$697$	5.27530e6i	0.411306i
$698$	− 1.33555e7i	− 1.03758i
$699$	0	0
$700$	0	0
$701$	−6.00859e6	−0.461825	−0.230913	−	0.972974i	$-0.574171\pi$
$701$	−6.00859e6	−0.461825	−0.230913	+	0.972974i	$0.574171\pi$
$702$	0	0
$703$	− 8.51342e6i	− 0.649704i
$704$	1.04184e7	0.792265
$705$	0	0
$706$	−9.91752e6	−0.748844
$707$	− 9.83807e6i	− 0.740221i
$708$	0	0
$709$	−5.90083e6	−0.440857	−0.220429	−	0.975403i	$-0.570746\pi$
$709$	−5.90083e6	−0.440857	−0.220429	+	0.975403i	$0.570746\pi$
$710$	0	0
$711$	0	0
$712$	2.24393e7i	1.65886i
$713$	7.93365e6i	0.584453i
$714$	0	0
$715$	0	0
$716$	1.06113e6	0.0773545
$717$	0	0
$718$	− 1.60524e6i	− 0.116206i
$719$	−1.36592e7	−0.985382	−0.492691	−	0.870204i	$-0.663987\pi$
$719$	−1.36592e7	−0.985382	−0.492691	+	0.870204i	$0.663987\pi$
$720$	0	0
$721$	6.28097e6	0.449975
$722$	− 1.09192e7i	− 0.779556i
$723$	0	0
$724$	1.86693e6	0.132368
$725$	0	0
$726$	0	0
$727$	1.11594e7i	0.783079i	0.920161	+	0.391539i	$0.128057\pi$
$727$	1.11594e7i	0.783079i	−0.920161	+	0.391539i	$0.871943\pi$
$728$	− 1.71235e6i	− 0.119747i
$729$	0	0
$730$	0	0
$731$	−5.59710e6	−0.387409
$732$	0	0
$733$	− 1.52510e7i	− 1.04843i	−0.851586	−	0.524215i	$-0.824359\pi$
$733$	− 1.52510e7i	− 1.04843i	0.851586	−	0.524215i	$-0.175641\pi$
$734$	−3.82993e6	−0.262392
$735$	0	0
$736$	−1.35286e6	−0.0920573
$737$	− 9.38217e6i	− 0.636260i
$738$	0	0
$739$	1.11820e7	0.753196	0.376598	−	0.926377i	$-0.377094\pi$
$739$	1.11820e7	0.753196	0.376598	+	0.926377i	$0.377094\pi$
$740$	0	0
$741$	0	0
$742$	− 5.27735e6i	− 0.351890i
$743$	− 7.71450e6i	− 0.512667i	−0.966588	−	0.256334i	$-0.917485\pi$
$743$	− 7.71450e6i	− 0.512667i	0.966588	−	0.256334i	$-0.0825146\pi$
$744$	0	0
$745$	0	0
$746$	2.51357e7	1.65365
$747$	0	0
$748$	388453.i	0.0253854i
$749$	1.20581e7	0.785367
$750$	0	0
$751$	−2.23973e7	−1.44909	−0.724545	−	0.689228i	$-0.757950\pi$
$751$	−2.23973e7	−1.44909	−0.724545	+	0.689228i	$0.757950\pi$
$752$	− 2.15698e7i	− 1.39091i
$753$	0	0
$754$	−928780.	−0.0594955
$755$	0	0
$756$	0	0
$757$	− 2.57267e7i	− 1.63171i	−0.578254	−	0.815857i	$-0.696266\pi$
$757$	− 2.57267e7i	− 1.63171i	0.578254	−	0.815857i	$-0.303734\pi$
$758$	− 3.69166e6i	− 0.233372i
$759$	0	0
$760$	0	0
$761$	1.48340e7	0.928533	0.464267	−	0.885696i	$-0.346318\pi$
$761$	1.48340e7	0.928533	0.464267	+	0.885696i	$0.346318\pi$
$762$	0	0
$763$	− 1.39534e6i	− 0.0867700i
$764$	3.02166e6	0.187289
$765$	0	0
$766$	2.11046e7	1.29959
$767$	− 631463.i	− 0.0387578i
$768$	0	0
$769$	5.57112e6	0.339724	0.169862	−	0.985468i	$-0.445668\pi$
$769$	5.57112e6	0.339724	0.169862	+	0.985468i	$0.445668\pi$
$770$	0	0
$771$	0	0
$772$	929892.i	0.0561551i
$773$	1.58230e7i	0.952447i	0.879324	+	0.476224i	$0.157995\pi$
$773$	1.58230e7i	0.952447i	−0.879324	+	0.476224i	$0.842005\pi$
$774$	0	0
$775$	0	0
$776$	−4.18026e6	−0.249200
$777$	0	0
$778$	2.34664e7i	1.38994i
$779$	−3.62281e7	−2.13896
$780$	0	0
$781$	−1.04107e7	−0.610732
$782$	1.42727e6i	0.0834623i
$783$	0	0
$784$	318081.	0.0184819
$785$	0	0
$786$	0	0
$787$	1.31529e7i	0.756981i	0.925605	+	0.378491i	$0.123557\pi$
$787$	1.31529e7i	0.756981i	−0.925605	+	0.378491i	$0.876443\pi$
$788$	− 3.01939e6i	− 0.173222i
$789$	0	0
$790$	0	0
$791$	−1.14502e7	−0.650687
$792$	0	0
$793$	227466.i	0.0128450i
$794$	1.77328e7	0.998222
$795$	0	0
$796$	−95249.7	−0.00532821
$797$	− 2.58443e7i	− 1.44118i	−0.693361	−	0.720590i	$-0.743871\pi$
$797$	− 2.58443e7i	− 1.44118i	0.693361	−	0.720590i	$-0.256129\pi$
$798$	0	0
$799$	7.72406e6	0.428034
$800$	0	0
$801$	0	0
$802$	− 1.58220e7i	− 0.868611i
$803$	7.69170e6i	0.420953i
$804$	0	0
$805$	0	0
$806$	−3.26977e6	−0.177288
$807$	0	0
$808$	− 1.43439e7i	− 0.772929i
$809$	1.78857e7	0.960804	0.480402	−	0.877048i	$-0.340491\pi$
$809$	1.78857e7	0.960804	0.480402	+	0.877048i	$0.340491\pi$
$810$	0	0
$811$	−1.41608e7	−0.756026	−0.378013	−	0.925800i	$-0.623393\pi$
$811$	−1.41608e7	−0.756026	−0.378013	+	0.925800i	$0.623393\pi$
$812$	1.45793e6i	0.0775970i
$813$	0	0
$814$	6.09193e6	0.322251
$815$	0	0
$816$	0	0
$817$	− 3.84380e7i	− 2.01468i
$818$	− 5.25162e6i	− 0.274417i
$819$	0	0
$820$	0	0
$821$	3.46248e7	1.79279	0.896394	−	0.443258i	$-0.146177\pi$
$821$	3.46248e7	1.79279	0.896394	+	0.443258i	$0.146177\pi$
$822$	0	0
$823$	2.13360e7i	1.09803i	0.835813	+	0.549015i	$0.184997\pi$
$823$	2.13360e7i	1.09803i	−0.835813	+	0.549015i	$0.815003\pi$
$824$	9.15766e6	0.469858
$825$	0	0
$826$	6.36744e6	0.324724
$827$	− 1.59813e6i	− 0.0812548i	−0.999174	−	0.0406274i	$-0.987064\pi$
$827$	− 1.59813e6i	− 0.0812548i	0.999174	−	0.0406274i	$-0.0129357\pi$
$828$	0	0
$829$	2.53923e7	1.28327	0.641633	−	0.767012i	$-0.278257\pi$
$829$	2.53923e7	1.28327	0.641633	+	0.767012i	$0.278257\pi$
$830$	0	0
$831$	0	0
$832$	− 2.45595e6i	− 0.123002i
$833$	113904.i	0.00568755i
$834$	0	0
$835$	0	0
$836$	−2.66770e6	−0.132014
$837$	0	0
$838$	− 2.91384e7i	− 1.43336i
$839$	1.98528e7	0.973681	0.486841	−	0.873491i	$-0.338149\pi$
$839$	1.98528e7	0.973681	0.486841	+	0.873491i	$0.338149\pi$
$840$	0	0
$841$	−1.38498e7	−0.675231
$842$	− 1.04523e7i	− 0.508081i
$843$	0	0
$844$	3.92587e6	0.189705
$845$	0	0
$846$	0	0
$847$	1.00764e7i	0.482609i
$848$	− 6.63879e6i	− 0.317029i
$849$	0	0
$850$	0	0
$851$	−3.48441e6	−0.164932
$852$	0	0
$853$	1.59794e7i	0.751948i	0.926630	+	0.375974i	$0.122692\pi$
$853$	1.59794e7i	0.751948i	−0.926630	+	0.375974i	$0.877308\pi$
$854$	−2.29369e6	−0.107619
$855$	0	0
$856$	1.75807e7	0.820070
$857$	− 7.00157e6i	− 0.325644i	−0.986655	−	0.162822i	$-0.947940\pi$
$857$	− 7.00157e6i	− 0.325644i	0.986655	−	0.162822i	$-0.0520597\pi$
$858$	0	0
$859$	7.28414e6	0.336818	0.168409	−	0.985717i	$-0.446137\pi$
$859$	7.28414e6	0.336818	0.168409	+	0.985717i	$0.446137\pi$
$860$	0	0
$861$	0	0
$862$	613094.i	0.0281034i
$863$	− 1.76361e7i	− 0.806075i	−0.915184	−	0.403037i	$-0.867954\pi$
$863$	− 1.76361e7i	− 0.806075i	0.915184	−	0.403037i	$-0.132046\pi$
$864$	0	0
$865$	0	0
$866$	−2.40038e7	−1.08764
$867$	0	0
$868$	5.13263e6i	0.231228i
$869$	2.08021e7	0.934455
$870$	0	0
$871$	−2.21167e6	−0.0987816
$872$	− 2.03441e6i	− 0.0906041i
$873$	0	0
$874$	−9.80180e6	−0.434037
$875$	0	0
$876$	0	0
$877$	− 2.69004e7i	− 1.18102i	−0.807029	−	0.590512i	$-0.798926\pi$
$877$	− 2.69004e7i	− 1.18102i	0.807029	−	0.590512i	$-0.201074\pi$
$878$	1.53743e7i	0.673070i
$879$	0	0
$880$	0	0
$881$	−2.51911e7	−1.09347	−0.546735	−	0.837306i	$-0.684130\pi$
$881$	−2.51911e7	−1.09347	−0.546735	+	0.837306i	$0.684130\pi$
$882$	0	0
$883$	− 3.22126e7i	− 1.39035i	−0.718840	−	0.695175i	$-0.755327\pi$
$883$	− 3.22126e7i	− 1.39035i	0.718840	−	0.695175i	$-0.244673\pi$
$884$	91570.7	0.00394117
$885$	0	0
$886$	8.40629e6	0.359766
$887$	8.96139e6i	0.382443i	0.981547	+	0.191221i	$0.0612448\pi$
$887$	8.96139e6i	0.382443i	−0.981547	+	0.191221i	$0.938755\pi$
$888$	0	0
$889$	−2.58662e7	−1.09769
$890$	0	0
$891$	0	0
$892$	572187.i	0.0240783i
$893$	5.30449e7i	2.22595i
$894$	0	0
$895$	0	0
$896$	1.82674e7	0.760164
$897$	0	0
$898$	− 1.63689e7i	− 0.677376i
$899$	2.34514e7	0.967765
$900$	0	0
$901$	2.37733e6	0.0975613
$902$	− 2.59237e7i	− 1.06092i
$903$	0	0
$904$	−1.66944e7	−0.679439
$905$	0	0
$906$	0	0
$907$	5.81689e6i	0.234786i	0.993086	+	0.117393i	$0.0374537\pi$
$907$	5.81689e6i	0.234786i	−0.993086	+	0.117393i	$0.962546\pi$
$908$	− 1.52728e6i	− 0.0614757i
$909$	0	0
$910$	0	0
$911$	1.96435e7	0.784192	0.392096	−	0.919924i	$-0.371750\pi$
$911$	1.96435e7	0.784192	0.392096	+	0.919924i	$0.371750\pi$
$912$	0	0
$913$	1.14969e7i	0.456460i
$914$	−3.40533e7	−1.34832
$915$	0	0
$916$	−1.58039e6	−0.0622337
$917$	− 1.55279e7i	− 0.609803i
$918$	0	0
$919$	89962.4	0.00351376	0.00175688	−	0.999998i	$-0.499441\pi$
$919$	89962.4	0.00351376	0.00175688	+	0.999998i	$0.499441\pi$
$920$	0	0
$921$	0	0
$922$	− 2.88291e7i	− 1.11687i
$923$	2.45412e6i	0.0948183i
$924$	0	0
$925$	0	0
$926$	1.23916e7	0.474899
$927$	0	0
$928$	3.99898e6i	0.152433i
$929$	3.65192e7	1.38830	0.694149	−	0.719832i	$-0.255781\pi$
$929$	3.65192e7	1.38830	0.694149	+	0.719832i	$0.255781\pi$
$930$	0	0
$931$	−782234.	−0.0295776
$932$	− 4.41338e6i	− 0.166430i
$933$	0	0
$934$	−2.39966e7	−0.900081
$935$	0	0
$936$	0	0
$937$	3.58659e7i	1.33454i	0.744814	+	0.667272i	$0.232538\pi$
$937$	3.58659e7i	1.33454i	−0.744814	+	0.667272i	$0.767462\pi$
$938$	− 2.23017e7i	− 0.827621i
$939$	0	0
$940$	0	0
$941$	3.19693e7	1.17695	0.588476	−	0.808515i	$-0.299728\pi$
$941$	3.19693e7	1.17695	0.588476	+	0.808515i	$0.299728\pi$
$942$	0	0
$943$	1.48276e7i	0.542990i
$944$	8.01010e6	0.292555
$945$	0	0
$946$	2.75050e7	0.999273
$947$	4.71846e7i	1.70972i	0.518858	+	0.854861i	$0.326357\pi$
$947$	4.71846e7i	1.70972i	−0.518858	+	0.854861i	$0.673643\pi$
$948$	0	0
$949$	1.81318e6	0.0653544
$950$	0	0
$951$	0	0
$952$	7.77829e6i	0.278158i
$953$	− 1.65226e6i	− 0.0589315i	−0.999566	−	0.0294657i	$-0.990619\pi$
$953$	− 1.65226e6i	− 0.0589315i	0.999566	−	0.0294657i	$-0.00938059\pi$
$954$	0	0
$955$	0	0
$956$	5.17108e6	0.182994
$957$	0	0
$958$	9.89295e6i	0.348267i
$959$	−3.96512e7	−1.39223
$960$	0	0
$961$	5.39316e7	1.88380
$962$	− 1.43606e6i	− 0.0500306i
$963$	0	0
$964$	−409351.	−0.0141874
$965$	0	0
$966$	0	0
$967$	3.23040e7i	1.11094i	0.831537	+	0.555470i	$0.187462\pi$
$967$	3.23040e7i	1.11094i	−0.831537	+	0.555470i	$0.812538\pi$
$968$	1.46914e7i	0.503934i
$969$	0	0
$970$	0	0
$971$	−1.15927e7	−0.394582	−0.197291	−	0.980345i	$-0.563214\pi$
$971$	−1.15927e7	−0.394582	−0.197291	+	0.980345i	$0.563214\pi$
$972$	0	0
$973$	2.05995e7i	0.697549i
$974$	−8.91377e6	−0.301068
$975$	0	0
$976$	−2.88541e6	−0.0969579
$977$	2.58947e7i	0.867909i	0.900935	+	0.433954i	$0.142882\pi$
$977$	2.58947e7i	0.867909i	−0.900935	+	0.433954i	$0.857118\pi$
$978$	0	0
$979$	3.40702e7	1.13610
$980$	0	0
$981$	0	0
$982$	7.82184e6i	0.258839i
$983$	− 3.46040e7i	− 1.14220i	−0.820880	−	0.571101i	$-0.806517\pi$
$983$	− 3.46040e7i	− 1.14220i	0.820880	−	0.571101i	$-0.193483\pi$
$984$	0	0
$985$	0	0
$986$	4.21894e6	0.138201
$987$	0	0
$988$	628861.i	0.0204957i
$989$	−1.57321e7	−0.511441
$990$	0	0
$991$	−3.71464e7	−1.20152	−0.600762	−	0.799428i	$-0.705136\pi$
$991$	−3.71464e7	−1.20152	−0.600762	+	0.799428i	$0.705136\pi$
$992$	1.40784e7i	0.454228i
$993$	0	0
$994$	−2.47465e7	−0.794415
$995$	0	0
$996$	0	0
$997$	2.47350e7i	0.788086i	0.919092	+	0.394043i	$0.128924\pi$
$997$	2.47350e7i	0.788086i	−0.919092	+	0.394043i	$0.871076\pi$
$998$	3.73573e7i	1.18727i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.6.b.i.199.2		4
3.2	odd	2		25.6.b.b.24.3		4
5.2	odd	4		225.6.a.l.1.2		2
5.3	odd	4		225.6.a.s.1.1		2
5.4	even	2	inner	225.6.b.i.199.3		4
12.11	even	2		400.6.c.n.49.1		4
15.2	even	4		25.6.a.d.1.1	yes	2
15.8	even	4		25.6.a.b.1.2	✓	2
15.14	odd	2		25.6.b.b.24.2		4
60.23	odd	4		400.6.a.w.1.2		2
60.47	odd	4		400.6.a.o.1.1		2
60.59	even	2		400.6.c.n.49.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.a.b.1.2	✓	2	15.8	even	4
25.6.a.d.1.1	yes	2	15.2	even	4
25.6.b.b.24.2		4	15.14	odd	2
25.6.b.b.24.3		4	3.2	odd	2
225.6.a.l.1.2		2	5.2	odd	4
225.6.a.s.1.1		2	5.3	odd	4
225.6.b.i.199.2		4	1.1	even	1	trivial
225.6.b.i.199.3		4	5.4	even	2	inner
400.6.a.o.1.1		2	60.47	odd	4
400.6.a.w.1.2		2	60.23	odd	4
400.6.c.n.49.1		4	12.11	even	2
400.6.c.n.49.4		4	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	225.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.26209i q^{2} +4.31044 q^{4} -131.048i q^{7} -191.069i q^{8} +O(q^{10})\)	\(q-5.26209i q^{2} +4.31044 q^{4} -131.048i q^{7} -191.069i q^{8} -290.104 q^{11} +68.3868i q^{13} -689.588 q^{14} -867.486 q^{16} -310.644i q^{17} +2133.35 q^{19} +1526.55i q^{22} -873.145i q^{23} +359.857 q^{26} -564.876i q^{28} -2580.97 q^{29} -9086.30 q^{31} -1549.41i q^{32} -1634.64 q^{34} -3990.64i q^{37} -11225.9i q^{38} -16981.8 q^{41} -18017.7i q^{43} -1250.48 q^{44} -4594.57 q^{46} +24864.7i q^{47} -366.670 q^{49} +294.777i q^{52} +7652.91i q^{53} -25039.2 q^{56} +13581.3i q^{58} -9233.69 q^{59} +3326.17 q^{61} +47812.9i q^{62} -35912.7 q^{64} +32340.7i q^{67} -1339.01i q^{68} +35885.9 q^{71} -26513.6i q^{73} -20999.1 q^{74} +9195.65 q^{76} +38017.7i q^{77} -71705.7 q^{79} +89359.9i q^{82} -39630.1i q^{83} -94810.8 q^{86} +55429.9i q^{88} -117441. q^{89} +8961.98 q^{91} -3763.64i q^{92} +130840. q^{94} -21878.3i q^{97} +1929.45i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 138 q^{4}+O(q^{10}) \) 4 * q - 138 * q^4	\( 4 q - 138 q^{4} + 392 q^{11} + 36 q^{14} + 2274 q^{16} + 6360 q^{19} + 9512 q^{26} - 7840 q^{29} - 2192 q^{31} - 40226 q^{34} - 55508 q^{41} - 73774 q^{44} + 4908 q^{46} + 23372 q^{49} - 108540 q^{56} + 23920 q^{59} - 48792 q^{61} - 87298 q^{64} + 174592 q^{71} - 82444 q^{74} - 135070 q^{76} - 130960 q^{79} - 497848 q^{86} - 145620 q^{89} - 41152 q^{91} + 243304 q^{94}+O(q^{100}) \) 4 * q - 138 * q^4 + 392 * q^11 + 36 * q^14 + 2274 * q^16 + 6360 * q^19 + 9512 * q^26 - 7840 * q^29 - 2192 * q^31 - 40226 * q^34 - 55508 * q^41 - 73774 * q^44 + 4908 * q^46 + 23372 * q^49 - 108540 * q^56 + 23920 * q^59 - 48792 * q^61 - 87298 * q^64 + 174592 * q^71 - 82444 * q^74 - 135070 * q^76 - 130960 * q^79 - 497848 * q^86 - 145620 * q^89 - 41152 * q^91 + 243304 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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