LMFDB - Embedded newform 400.6.c.h.49.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,6,Mod(49,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("400.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	400.c (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:	$64.1535279252$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	400.49
Dual form			400.6.c.h.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.00000i q^{3} +108.000i q^{7} +179.000 q^{9} +O(q^{10})$	$q-8.00000i q^{3} +108.000i q^{7} +179.000 q^{9} +604.000 q^{11} +306.000i q^{13} +930.000i q^{17} -1324.00 q^{19} +864.000 q^{21} -852.000i q^{23} -3376.00i q^{27} -5902.00 q^{29} +3320.00 q^{31} -4832.00i q^{33} +10774.0i q^{37} +2448.00 q^{39} -17958.0 q^{41} +9264.00i q^{43} +9796.00i q^{47} +5143.00 q^{49} +7440.00 q^{51} +31434.0i q^{53} +10592.0i q^{57} +33228.0 q^{59} -40210.0 q^{61} +19332.0i q^{63} -58864.0i q^{67} -6816.00 q^{69} +55312.0 q^{71} -27258.0i q^{73} +65232.0i q^{77} +31456.0 q^{79} +16489.0 q^{81} +24552.0i q^{83} +47216.0i q^{87} +90854.0 q^{89} -33048.0 q^{91} -26560.0i q^{93} +154706. i q^{97} +108116. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 358 q^{9}+O(q^{10}) $ 2 * q + 358 * q^9	$ 2 q + 358 q^{9} + 1208 q^{11} - 2648 q^{19} + 1728 q^{21} - 11804 q^{29} + 6640 q^{31} + 4896 q^{39} - 35916 q^{41} + 10286 q^{49} + 14880 q^{51} + 66456 q^{59} - 80420 q^{61} - 13632 q^{69} + 110624 q^{71} + 62912 q^{79} + 32978 q^{81} + 181708 q^{89} - 66096 q^{91} + 216232 q^{99}+O(q^{100}) $ 2 * q + 358 * q^9 + 1208 * q^11 - 2648 * q^19 + 1728 * q^21 - 11804 * q^29 + 6640 * q^31 + 4896 * q^39 - 35916 * q^41 + 10286 * q^49 + 14880 * q^51 + 66456 * q^59 - 80420 * q^61 - 13632 * q^69 + 110624 * q^71 + 62912 * q^79 + 32978 * q^81 + 181708 * q^89 - 66096 * q^91 + 216232 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	− 8.00000i	− 0.513200i	−0.966518	−	0.256600i	$-0.917398\pi$
$3$	− 8.00000i	− 0.513200i	0.966518	−	0.256600i	$-0.0826023\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	108.000i	0.833065i	0.909121	+	0.416532i	$0.136755\pi$
$7$	108.000i	0.833065i	−0.909121	+	0.416532i	$0.863245\pi$
$8$	0	0
$9$	179.000	0.736626
$10$	0	0
$11$	604.000	1.50506	0.752532	−	0.658555i	$-0.228832\pi$
$11$	604.000	1.50506	0.752532	+	0.658555i	$0.228832\pi$
$12$	0	0
$13$	306.000i	0.502184i	0.967963	+	0.251092i	$0.0807897\pi$
$13$	306.000i	0.502184i	−0.967963	+	0.251092i	$0.919210\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	930.000i	0.780478i	0.920714	+	0.390239i	$0.127608\pi$
$17$	930.000i	0.780478i	−0.920714	+	0.390239i	$0.872392\pi$
$18$	0	0
$19$	−1324.00	−0.841403	−0.420701	−	0.907199i	$-0.638216\pi$
$19$	−1324.00	−0.841403	−0.420701	+	0.907199i	$0.638216\pi$
$20$	0	0
$21$	864.000	0.427529
$22$	0	0
$23$	− 852.000i	− 0.335830i	−0.985801	−	0.167915i	$-0.946297\pi$
$23$	− 852.000i	− 0.335830i	0.985801	−	0.167915i	$-0.0537035\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 3376.00i	− 0.891237i
$28$	0	0
$29$	−5902.00	−1.30318	−0.651590	−	0.758572i	$-0.725898\pi$
$29$	−5902.00	−1.30318	−0.651590	+	0.758572i	$0.725898\pi$
$30$	0	0
$31$	3320.00	0.620489	0.310244	−	0.950657i	$-0.399589\pi$
$31$	3320.00	0.620489	0.310244	+	0.950657i	$0.399589\pi$
$32$	0	0
$33$	− 4832.00i	− 0.772400i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10774.0i	1.29382i	0.762568	+	0.646908i	$0.223938\pi$
$37$	10774.0i	1.29382i	−0.762568	+	0.646908i	$0.776062\pi$
$38$	0	0
$39$	2448.00	0.257721
$40$	0	0
$41$	−17958.0	−1.66839	−0.834196	−	0.551467i	$-0.814068\pi$
$41$	−17958.0	−1.66839	−0.834196	+	0.551467i	$0.814068\pi$
$42$	0	0
$43$	9264.00i	0.764060i	0.924150	+	0.382030i	$0.124775\pi$
$43$	9264.00i	0.764060i	−0.924150	+	0.382030i	$0.875225\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9796.00i	0.646851i	0.946254	+	0.323425i	$0.104834\pi$
$47$	9796.00i	0.646851i	−0.946254	+	0.323425i	$0.895166\pi$
$48$	0	0
$49$	5143.00	0.306003
$50$	0	0
$51$	7440.00	0.400541
$52$	0	0
$53$	31434.0i	1.53713i	0.639773	+	0.768564i	$0.279028\pi$
$53$	31434.0i	1.53713i	−0.639773	+	0.768564i	$0.720972\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	10592.0i	0.431808i
$58$	0	0
$59$	33228.0	1.24272	0.621361	−	0.783524i	$-0.286580\pi$
$59$	33228.0	1.24272	0.621361	+	0.783524i	$0.286580\pi$
$60$	0	0
$61$	−40210.0	−1.38360	−0.691798	−	0.722091i	$-0.743181\pi$
$61$	−40210.0	−1.38360	−0.691798	+	0.722091i	$0.743181\pi$
$62$	0	0
$63$	19332.0i	0.613657i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 58864.0i	− 1.60200i	−0.598664	−	0.801000i	$-0.704301\pi$
$67$	− 58864.0i	− 1.60200i	0.598664	−	0.801000i	$-0.295699\pi$
$68$	0	0
$69$	−6816.00	−0.172348
$70$	0	0
$71$	55312.0	1.30219	0.651094	−	0.758997i	$-0.274310\pi$
$71$	55312.0	1.30219	0.651094	+	0.758997i	$0.274310\pi$
$72$	0	0
$73$	− 27258.0i	− 0.598669i	−0.954148	−	0.299335i	$-0.903235\pi$
$73$	− 27258.0i	− 0.598669i	0.954148	−	0.299335i	$-0.0967647\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	65232.0i	1.25382i
$78$	0	0
$79$	31456.0	0.567069	0.283534	−	0.958962i	$-0.408493\pi$
$79$	31456.0	0.567069	0.283534	+	0.958962i	$0.408493\pi$
$80$	0	0
$81$	16489.0	0.279243
$82$	0	0
$83$	24552.0i	0.391194i	0.980684	+	0.195597i	$0.0626644\pi$
$83$	24552.0i	0.391194i	−0.980684	+	0.195597i	$0.937336\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	47216.0i	0.668792i
$88$	0	0
$89$	90854.0	1.21582	0.607910	−	0.794006i	$-0.292008\pi$
$89$	90854.0	1.21582	0.607910	+	0.794006i	$0.292008\pi$
$90$	0	0
$91$	−33048.0	−0.418352
$92$	0	0
$93$	− 26560.0i	− 0.318435i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	154706.i	1.66947i	0.550654	+	0.834733i	$0.314378\pi$
$97$	154706.i	1.66947i	−0.550654	+	0.834733i	$0.685622\pi$
$98$	0	0
$99$	108116.	1.10867
$100$	0	0
$101$	−72714.0	−0.709275	−0.354637	−	0.935004i	$-0.615396\pi$
$101$	−72714.0	−0.709275	−0.354637	+	0.935004i	$0.615396\pi$
$102$	0	0
$103$	− 129396.i	− 1.20179i	−0.799329	−	0.600894i	$-0.794811\pi$
$103$	− 129396.i	− 1.20179i	0.799329	−	0.600894i	$-0.205189\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	206680.i	1.74518i	0.488458	+	0.872588i	$0.337560\pi$
$107$	206680.i	1.74518i	−0.488458	+	0.872588i	$0.662440\pi$
$108$	0	0
$109$	70146.0	0.565505	0.282753	−	0.959193i	$-0.408752\pi$
$109$	70146.0	0.565505	0.282753	+	0.959193i	$0.408752\pi$
$110$	0	0
$111$	86192.0	0.663987
$112$	0	0
$113$	151854.i	1.11874i	0.828917	+	0.559371i	$0.188957\pi$
$113$	151854.i	1.11874i	−0.828917	+	0.559371i	$0.811043\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	54774.0i	0.369922i
$118$	0	0
$119$	−100440.	−0.650189
$120$	0	0
$121$	203765.	1.26522
$122$	0	0
$123$	143664.i	0.856220i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	336596.i	1.85182i	0.377740	+	0.925912i	$0.376701\pi$
$127$	336596.i	1.85182i	−0.377740	+	0.925912i	$0.623299\pi$
$128$	0	0
$129$	74112.0	0.392116
$130$	0	0
$131$	−275308.	−1.40165	−0.700827	−	0.713332i	$-0.747185\pi$
$131$	−275308.	−1.40165	−0.700827	+	0.713332i	$0.747185\pi$
$132$	0	0
$133$	− 142992.i	− 0.700943i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 228502.i	− 1.04013i	−0.854126	−	0.520066i	$-0.825907\pi$
$137$	− 228502.i	− 1.04013i	0.854126	−	0.520066i	$-0.174093\pi$
$138$	0	0
$139$	224284.	0.984603	0.492302	−	0.870425i	$-0.336156\pi$
$139$	224284.	0.984603	0.492302	+	0.870425i	$0.336156\pi$
$140$	0	0
$141$	78368.0	0.331964
$142$	0	0
$143$	184824.i	0.755820i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 41144.0i	− 0.157041i
$148$	0	0
$149$	183802.	0.678242	0.339121	−	0.940743i	$-0.389870\pi$
$149$	183802.	0.678242	0.339121	+	0.940743i	$0.389870\pi$
$150$	0	0
$151$	−296032.	−1.05657	−0.528283	−	0.849069i	$-0.677164\pi$
$151$	−296032.	−1.05657	−0.528283	+	0.849069i	$0.677164\pi$
$152$	0	0
$153$	166470.i	0.574920i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	134766.i	0.436346i	0.975910	+	0.218173i	$0.0700097\pi$
$157$	134766.i	0.436346i	−0.975910	+	0.218173i	$0.929990\pi$
$158$	0	0
$159$	251472.	0.788854
$160$	0	0
$161$	92016.0	0.279768
$162$	0	0
$163$	− 60248.0i	− 0.177613i	−0.996049	−	0.0888063i	$-0.971695\pi$
$163$	− 60248.0i	− 0.177613i	0.996049	−	0.0888063i	$-0.0283052\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	62012.0i	0.172062i	0.996292	+	0.0860309i	$0.0274184\pi$
$167$	62012.0i	0.172062i	−0.996292	+	0.0860309i	$0.972582\pi$
$168$	0	0
$169$	277657.	0.747811
$170$	0	0
$171$	−236996.	−0.619799
$172$	0	0
$173$	591682.i	1.50305i	0.659705	+	0.751524i	$0.270681\pi$
$173$	591682.i	1.50305i	−0.659705	+	0.751524i	$0.729319\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 265824.i	− 0.637766i
$178$	0	0
$179$	−241404.	−0.563134	−0.281567	−	0.959542i	$-0.590854\pi$
$179$	−241404.	−0.563134	−0.281567	+	0.959542i	$0.590854\pi$
$180$	0	0
$181$	187622.	0.425684	0.212842	−	0.977087i	$-0.431728\pi$
$181$	187622.	0.425684	0.212842	+	0.977087i	$0.431728\pi$
$182$	0	0
$183$	321680.i	0.710062i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	561720.i	1.17467i
$188$	0	0
$189$	364608.	0.742458
$190$	0	0
$191$	−37560.0	−0.0744976	−0.0372488	−	0.999306i	$-0.511859\pi$
$191$	−37560.0	−0.0744976	−0.0372488	+	0.999306i	$0.511859\pi$
$192$	0	0
$193$	− 164434.i	− 0.317759i	−0.987298	−	0.158880i	$-0.949212\pi$
$193$	− 164434.i	− 0.317759i	0.987298	−	0.158880i	$-0.0507882\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	360518.i	0.661853i	0.943657	+	0.330926i	$0.107361\pi$
$197$	360518.i	0.661853i	−0.943657	+	0.330926i	$0.892639\pi$
$198$	0	0
$199$	−654168.	−1.17100	−0.585500	−	0.810673i	$-0.699102\pi$
$199$	−654168.	−1.17100	−0.585500	+	0.810673i	$0.699102\pi$
$200$	0	0
$201$	−470912.	−0.822147
$202$	0	0
$203$	− 637416.i	− 1.08563i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 152508.i	− 0.247381i
$208$	0	0
$209$	−799696.	−1.26637
$210$	0	0
$211$	693156.	1.07183	0.535914	−	0.844273i	$-0.319967\pi$
$211$	693156.	1.07183	0.535914	+	0.844273i	$0.319967\pi$
$212$	0	0
$213$	− 442496.i	− 0.668283i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	358560.i	0.516907i
$218$	0	0
$219$	−218064.	−0.307237
$220$	0	0
$221$	−284580.	−0.391944
$222$	0	0
$223$	494756.i	0.666237i	0.942885	+	0.333119i	$0.108101\pi$
$223$	494756.i	0.666237i	−0.942885	+	0.333119i	$0.891899\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 907088.i	− 1.16838i	−0.811616	−	0.584191i	$-0.801412\pi$
$227$	− 907088.i	− 1.16838i	0.811616	−	0.584191i	$-0.198588\pi$
$228$	0	0
$229$	−1.08949e6	−1.37289	−0.686446	−	0.727181i	$-0.740830\pi$
$229$	−1.08949e6	−1.37289	−0.686446	+	0.727181i	$0.740830\pi$
$230$	0	0
$231$	521856.	0.643459
$232$	0	0
$233$	− 499706.i	− 0.603010i	−0.953465	−	0.301505i	$-0.902511\pi$
$233$	− 499706.i	− 0.603010i	0.953465	−	0.301505i	$-0.0974891\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 251648.i	− 0.291020i
$238$	0	0
$239$	1.62038e6	1.83495	0.917473	−	0.397799i	$-0.130226\pi$
$239$	1.62038e6	1.83495	0.917473	+	0.397799i	$0.130226\pi$
$240$	0	0
$241$	1.00122e6	1.11042	0.555208	−	0.831711i	$-0.312638\pi$
$241$	1.00122e6	1.11042	0.555208	+	0.831711i	$0.312638\pi$
$242$	0	0
$243$	− 952280.i	− 1.03454i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 405144.i	− 0.422539i
$248$	0	0
$249$	196416.	0.200761
$250$	0	0
$251$	−368980.	−0.369674	−0.184837	−	0.982769i	$-0.559176\pi$
$251$	−368980.	−0.369674	−0.184837	+	0.982769i	$0.559176\pi$
$252$	0	0
$253$	− 514608.i	− 0.505447i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	279010.i	0.263504i	0.991283	+	0.131752i	$0.0420602\pi$
$257$	279010.i	0.263504i	−0.991283	+	0.131752i	$0.957940\pi$
$258$	0	0
$259$	−1.16359e6	−1.07783
$260$	0	0
$261$	−1.05646e6	−0.959955
$262$	0	0
$263$	811740.i	0.723648i	0.932246	+	0.361824i	$0.117846\pi$
$263$	811740.i	0.723648i	−0.932246	+	0.361824i	$0.882154\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 726832.i	− 0.623959i
$268$	0	0
$269$	−353214.	−0.297617	−0.148808	−	0.988866i	$-0.547544\pi$
$269$	−353214.	−0.297617	−0.148808	+	0.988866i	$0.547544\pi$
$270$	0	0
$271$	1.71622e6	1.41954	0.709772	−	0.704432i	$-0.248798\pi$
$271$	1.71622e6	1.41954	0.709772	+	0.704432i	$0.248798\pi$
$272$	0	0
$273$	264384.i	0.214698i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 245882.i	− 0.192543i	−0.995355	−	0.0962714i	$-0.969308\pi$
$277$	− 245882.i	− 0.192543i	0.995355	−	0.0962714i	$-0.0306917\pi$
$278$	0	0
$279$	594280.	0.457068
$280$	0	0
$281$	−1.67618e6	−1.26635	−0.633177	−	0.774007i	$-0.718250\pi$
$281$	−1.67618e6	−1.26635	−0.633177	+	0.774007i	$0.718250\pi$
$282$	0	0
$283$	− 1.25882e6i	− 0.934321i	−0.884173	−	0.467161i	$-0.845277\pi$
$283$	− 1.25882e6i	− 0.934321i	0.884173	−	0.467161i	$-0.154723\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1.93946e6i	− 1.38988i
$288$	0	0
$289$	554957.	0.390854
$290$	0	0
$291$	1.23765e6	0.856771
$292$	0	0
$293$	− 719158.i	− 0.489390i	−0.969600	−	0.244695i	$-0.921312\pi$
$293$	− 719158.i	− 0.489390i	0.969600	−	0.244695i	$-0.0786879\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 2.03910e6i	− 1.34137i
$298$	0	0
$299$	260712.	0.168649
$300$	0	0
$301$	−1.00051e6	−0.636511
$302$	0	0
$303$	581712.i	0.364000i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 1.86013e6i	− 1.12641i	−0.826317	−	0.563206i	$-0.809568\pi$
$307$	− 1.86013e6i	− 1.12641i	0.826317	−	0.563206i	$-0.190432\pi$
$308$	0	0
$309$	−1.03517e6	−0.616758
$310$	0	0
$311$	−278384.	−0.163209	−0.0816043	−	0.996665i	$-0.526004\pi$
$311$	−278384.	−0.163209	−0.0816043	+	0.996665i	$0.526004\pi$
$312$	0	0
$313$	474182.i	0.273580i	0.990600	+	0.136790i	$0.0436785\pi$
$313$	474182.i	0.273580i	−0.990600	+	0.136790i	$0.956321\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1.83738e6i	− 1.02695i	−0.858104	−	0.513476i	$-0.828357\pi$
$317$	− 1.83738e6i	− 1.02695i	0.858104	−	0.513476i	$-0.171643\pi$
$318$	0	0
$319$	−3.56481e6	−1.96137
$320$	0	0
$321$	1.65344e6	0.895624
$322$	0	0
$323$	− 1.23132e6i	− 0.656696i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 561168.i	− 0.290217i
$328$	0	0
$329$	−1.05797e6	−0.538868
$330$	0	0
$331$	−2.99743e6	−1.50376	−0.751880	−	0.659299i	$-0.770853\pi$
$331$	−2.99743e6	−1.50376	−0.751880	+	0.659299i	$0.770853\pi$
$332$	0	0
$333$	1.92855e6i	0.953058i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.87531e6i	0.899496i	0.893155	+	0.449748i	$0.148486\pi$
$337$	1.87531e6i	0.899496i	−0.893155	+	0.449748i	$0.851514\pi$
$338$	0	0
$339$	1.21483e6	0.574139
$340$	0	0
$341$	2.00528e6	0.933876
$342$	0	0
$343$	2.37060e6i	1.08799i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 180312.i	− 0.0803898i	−0.999192	−	0.0401949i	$-0.987202\pi$
$347$	− 180312.i	− 0.0803898i	0.999192	−	0.0401949i	$-0.0127979\pi$
$348$	0	0
$349$	87058.0	0.0382600	0.0191300	−	0.999817i	$-0.493910\pi$
$349$	87058.0	0.0382600	0.0191300	+	0.999817i	$0.493910\pi$
$350$	0	0
$351$	1.03306e6	0.447565
$352$	0	0
$353$	− 2.65901e6i	− 1.13575i	−0.823114	−	0.567876i	$-0.807765\pi$
$353$	− 2.65901e6i	− 1.13575i	0.823114	−	0.567876i	$-0.192235\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	803520.i	0.333677i
$358$	0	0
$359$	−2.14937e6	−0.880186	−0.440093	−	0.897952i	$-0.645055\pi$
$359$	−2.14937e6	−0.880186	−0.440093	+	0.897952i	$0.645055\pi$
$360$	0	0
$361$	−723123.	−0.292041
$362$	0	0
$363$	− 1.63012e6i	− 0.649311i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.08258e6i	1.19467i	0.801991	+	0.597337i	$0.203774\pi$
$367$	3.08258e6i	1.19467i	−0.801991	+	0.597337i	$0.796226\pi$
$368$	0	0
$369$	−3.21448e6	−1.22898
$370$	0	0
$371$	−3.39487e6	−1.28053
$372$	0	0
$373$	− 2.28727e6i	− 0.851227i	−0.904905	−	0.425613i	$-0.860058\pi$
$373$	− 2.28727e6i	− 0.851227i	0.904905	−	0.425613i	$-0.139942\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 1.80601e6i	− 0.654436i
$378$	0	0
$379$	−1.30154e6	−0.465435	−0.232718	−	0.972544i	$-0.574762\pi$
$379$	−1.30154e6	−0.465435	−0.232718	+	0.972544i	$0.574762\pi$
$380$	0	0
$381$	2.69277e6	0.950356
$382$	0	0
$383$	− 2.03276e6i	− 0.708093i	−0.935228	−	0.354046i	$-0.884806\pi$
$383$	− 2.03276e6i	− 0.708093i	0.935228	−	0.354046i	$-0.115194\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.65826e6i	0.562826i
$388$	0	0
$389$	−94230.0	−0.0315730	−0.0157865	−	0.999875i	$-0.505025\pi$
$389$	−94230.0	−0.0315730	−0.0157865	+	0.999875i	$0.505025\pi$
$390$	0	0
$391$	792360.	0.262108
$392$	0	0
$393$	2.20246e6i	0.719329i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 5.55551e6i	− 1.76908i	−0.466465	−	0.884540i	$-0.654473\pi$
$397$	− 5.55551e6i	− 1.76908i	0.466465	−	0.884540i	$-0.345527\pi$
$398$	0	0
$399$	−1.14394e6	−0.359724
$400$	0	0
$401$	−784814.	−0.243728	−0.121864	−	0.992547i	$-0.538887\pi$
$401$	−784814.	−0.243728	−0.121864	+	0.992547i	$0.538887\pi$
$402$	0	0
$403$	1.01592e6i	0.311600i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.50750e6i	1.94728i
$408$	0	0
$409$	4.59401e6	1.35795	0.678974	−	0.734162i	$-0.262425\pi$
$409$	4.59401e6	1.35795	0.678974	+	0.734162i	$0.262425\pi$
$410$	0	0
$411$	−1.82802e6	−0.533796
$412$	0	0
$413$	3.58862e6i	1.03527i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 1.79427e6i	− 0.505299i
$418$	0	0
$419$	1.41301e6	0.393198	0.196599	−	0.980484i	$-0.437010\pi$
$419$	1.41301e6	0.393198	0.196599	+	0.980484i	$0.437010\pi$
$420$	0	0
$421$	5.94556e6	1.63489	0.817443	−	0.576010i	$-0.195391\pi$
$421$	5.94556e6	1.63489	0.817443	+	0.576010i	$0.195391\pi$
$422$	0	0
$423$	1.75348e6i	0.476487i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 4.34268e6i	− 1.15263i
$428$	0	0
$429$	1.47859e6	0.387887
$430$	0	0
$431$	6.48114e6	1.68058	0.840289	−	0.542139i	$-0.182385\pi$
$431$	6.48114e6	1.68058	0.840289	+	0.542139i	$0.182385\pi$
$432$	0	0
$433$	− 4.05597e6i	− 1.03962i	−0.854282	−	0.519810i	$-0.826003\pi$
$433$	− 4.05597e6i	− 1.03962i	0.854282	−	0.519810i	$-0.173997\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.12805e6i	0.282569i
$438$	0	0
$439$	−1.21450e6	−0.300772	−0.150386	−	0.988627i	$-0.548052\pi$
$439$	−1.21450e6	−0.300772	−0.150386	+	0.988627i	$0.548052\pi$
$440$	0	0
$441$	920597.	0.225410
$442$	0	0
$443$	− 5.53154e6i	− 1.33917i	−0.742734	−	0.669586i	$-0.766472\pi$
$443$	− 5.53154e6i	− 1.33917i	0.742734	−	0.669586i	$-0.233528\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 1.47042e6i	− 0.348074i
$448$	0	0
$449$	−2.20111e6	−0.515258	−0.257629	−	0.966244i	$-0.582941\pi$
$449$	−2.20111e6	−0.515258	−0.257629	+	0.966244i	$0.582941\pi$
$450$	0	0
$451$	−1.08466e7	−2.51104
$452$	0	0
$453$	2.36826e6i	0.542229i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.29835e6i	0.738764i	0.929278	+	0.369382i	$0.120431\pi$
$457$	3.29835e6i	0.738764i	−0.929278	+	0.369382i	$0.879569\pi$
$458$	0	0
$459$	3.13968e6	0.695591
$460$	0	0
$461$	−3.94266e6	−0.864046	−0.432023	−	0.901863i	$-0.642200\pi$
$461$	−3.94266e6	−0.864046	−0.432023	+	0.901863i	$0.642200\pi$
$462$	0	0
$463$	8.82040e6i	1.91221i	0.293021	+	0.956106i	$0.405339\pi$
$463$	8.82040e6i	1.91221i	−0.293021	+	0.956106i	$0.594661\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.28709e6i	0.273096i	0.990633	+	0.136548i	$0.0436009\pi$
$467$	1.28709e6i	0.273096i	−0.990633	+	0.136548i	$0.956399\pi$
$468$	0	0
$469$	6.35731e6	1.33457
$470$	0	0
$471$	1.07813e6	0.223933
$472$	0	0
$473$	5.59546e6i	1.14996i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5.62669e6i	1.13229i
$478$	0	0
$479$	6.51179e6	1.29677	0.648383	−	0.761314i	$-0.275445\pi$
$479$	6.51179e6	1.29677	0.648383	+	0.761314i	$0.275445\pi$
$480$	0	0
$481$	−3.29684e6	−0.649734
$482$	0	0
$483$	− 736128.i	− 0.143577i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	5.79523e6i	1.10726i	0.832764	+	0.553628i	$0.186757\pi$
$487$	5.79523e6i	1.10726i	−0.832764	+	0.553628i	$0.813243\pi$
$488$	0	0
$489$	−481984.	−0.0911508
$490$	0	0
$491$	−990276.	−0.185376	−0.0926878	−	0.995695i	$-0.529546\pi$
$491$	−990276.	−0.185376	−0.0926878	+	0.995695i	$0.529546\pi$
$492$	0	0
$493$	− 5.48886e6i	− 1.01710i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.97370e6i	1.08481i
$498$	0	0
$499$	2.91500e6	0.524067	0.262033	−	0.965059i	$-0.415607\pi$
$499$	2.91500e6	0.524067	0.262033	+	0.965059i	$0.415607\pi$
$500$	0	0
$501$	496096.	0.0883022
$502$	0	0
$503$	2.47872e6i	0.436824i	0.975857	+	0.218412i	$0.0700877\pi$
$503$	2.47872e6i	0.436824i	−0.975857	+	0.218412i	$0.929912\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 2.22126e6i	− 0.383777i
$508$	0	0
$509$	6.75807e6	1.15619	0.578093	−	0.815971i	$-0.303797\pi$
$509$	6.75807e6	1.15619	0.578093	+	0.815971i	$0.303797\pi$
$510$	0	0
$511$	2.94386e6	0.498730
$512$	0	0
$513$	4.46982e6i	0.749889i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	5.91678e6i	0.973552i
$518$	0	0
$519$	4.73346e6	0.771365
$520$	0	0
$521$	−6.33903e6	−1.02312	−0.511562	−	0.859246i	$-0.670933\pi$
$521$	−6.33903e6	−1.02312	−0.511562	+	0.859246i	$0.670933\pi$
$522$	0	0
$523$	231920.i	0.0370752i	0.999828	+	0.0185376i	$0.00590105\pi$
$523$	231920.i	0.0370752i	−0.999828	+	0.0185376i	$0.994099\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.08760e6i	0.484278i
$528$	0	0
$529$	5.71044e6	0.887218
$530$	0	0
$531$	5.94781e6	0.915421
$532$	0	0
$533$	− 5.49515e6i	− 0.837841i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.93123e6i	0.289001i
$538$	0	0
$539$	3.10637e6	0.460555
$540$	0	0
$541$	−9.44440e6	−1.38733	−0.693667	−	0.720295i	$-0.744006\pi$
$541$	−9.44440e6	−1.38733	−0.693667	+	0.720295i	$0.744006\pi$
$542$	0	0
$543$	− 1.50098e6i	− 0.218461i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 3.10162e6i	− 0.443220i	−0.975135	−	0.221610i	$-0.928869\pi$
$547$	− 3.10162e6i	− 0.443220i	0.975135	−	0.221610i	$-0.0711312\pi$
$548$	0	0
$549$	−7.19759e6	−1.01919
$550$	0	0
$551$	7.81425e6	1.09650
$552$	0	0
$553$	3.39725e6i	0.472405i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 1.22330e6i	− 0.167068i	−0.996505	−	0.0835342i	$-0.973379\pi$
$557$	− 1.22330e6i	− 0.167068i	0.996505	−	0.0835342i	$-0.0266208\pi$
$558$	0	0
$559$	−2.83478e6	−0.383699
$560$	0	0
$561$	4.49376e6	0.602841
$562$	0	0
$563$	− 1.40896e7i	− 1.87339i	−0.350151	−	0.936693i	$-0.613870\pi$
$563$	− 1.40896e7i	− 1.87339i	0.350151	−	0.936693i	$-0.386130\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.78081e6i	0.232627i
$568$	0	0
$569$	−1.48468e6	−0.192244	−0.0961220	−	0.995370i	$-0.530644\pi$
$569$	−1.48468e6	−0.192244	−0.0961220	+	0.995370i	$0.530644\pi$
$570$	0	0
$571$	2.86470e6	0.367696	0.183848	−	0.982955i	$-0.441145\pi$
$571$	2.86470e6	0.367696	0.183848	+	0.982955i	$0.441145\pi$
$572$	0	0
$573$	300480.i	0.0382322i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	4.21728e6i	0.527343i	0.964613	+	0.263671i	$0.0849335\pi$
$577$	4.21728e6i	0.527343i	−0.964613	+	0.263671i	$0.915067\pi$
$578$	0	0
$579$	−1.31547e6	−0.163074
$580$	0	0
$581$	−2.65162e6	−0.325889
$582$	0	0
$583$	1.89861e7i	2.31348i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.01047e6i	0.240826i	0.992724	+	0.120413i	$0.0384218\pi$
$587$	2.01047e6i	0.240826i	−0.992724	+	0.120413i	$0.961578\pi$
$588$	0	0
$589$	−4.39568e6	−0.522081
$590$	0	0
$591$	2.88414e6	0.339663
$592$	0	0
$593$	− 7.33691e6i	− 0.856795i	−0.903590	−	0.428397i	$-0.859078\pi$
$593$	− 7.33691e6i	− 0.856795i	0.903590	−	0.428397i	$-0.140922\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	5.23334e6i	0.600957i
$598$	0	0
$599$	1.14884e6	0.130826	0.0654128	−	0.997858i	$-0.479164\pi$
$599$	1.14884e6	0.130826	0.0654128	+	0.997858i	$0.479164\pi$
$600$	0	0
$601$	1.16409e7	1.31462	0.657312	−	0.753618i	$-0.271693\pi$
$601$	1.16409e7	1.31462	0.657312	+	0.753618i	$0.271693\pi$
$602$	0	0
$603$	− 1.05367e7i	− 1.18007i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	155540.i	0.0171345i	0.999963	+	0.00856723i	$0.00272707\pi$
$607$	155540.i	0.0171345i	−0.999963	+	0.00856723i	$0.997273\pi$
$608$	0	0
$609$	−5.09933e6	−0.557147
$610$	0	0
$611$	−2.99758e6	−0.324838
$612$	0	0
$613$	1.18137e7i	1.26980i	0.772595	+	0.634899i	$0.218958\pi$
$613$	1.18137e7i	1.26980i	−0.772595	+	0.634899i	$0.781042\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.42252e6i	0.679192i	0.940571	+	0.339596i	$0.110290\pi$
$617$	6.42252e6i	0.679192i	−0.940571	+	0.339596i	$0.889710\pi$
$618$	0	0
$619$	3.85252e6	0.404128	0.202064	−	0.979372i	$-0.435235\pi$
$619$	3.85252e6	0.404128	0.202064	+	0.979372i	$0.435235\pi$
$620$	0	0
$621$	−2.87635e6	−0.299304
$622$	0	0
$623$	9.81223e6i	1.01286i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	6.39757e6i	0.649899i
$628$	0	0
$629$	−1.00198e7	−1.00980
$630$	0	0
$631$	6.75136e6	0.675022	0.337511	−	0.941322i	$-0.390415\pi$
$631$	6.75136e6	0.675022	0.337511	+	0.941322i	$0.390415\pi$
$632$	0	0
$633$	− 5.54525e6i	− 0.550062i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.57376e6i	0.153670i
$638$	0	0
$639$	9.90085e6	0.959224
$640$	0	0
$641$	−7.35493e6	−0.707022	−0.353511	−	0.935430i	$-0.615012\pi$
$641$	−7.35493e6	−0.707022	−0.353511	+	0.935430i	$0.615012\pi$
$642$	0	0
$643$	1.59694e7i	1.52322i	0.648036	+	0.761610i	$0.275591\pi$
$643$	1.59694e7i	1.52322i	−0.648036	+	0.761610i	$0.724409\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 1.72667e7i	− 1.62162i	−0.585311	−	0.810809i	$-0.699028\pi$
$647$	− 1.72667e7i	− 1.62162i	0.585311	−	0.810809i	$-0.300972\pi$
$648$	0	0
$649$	2.00697e7	1.87038
$650$	0	0
$651$	2.86848e6	0.265277
$652$	0	0
$653$	1.36251e6i	0.125043i	0.998044	+	0.0625213i	$0.0199141\pi$
$653$	1.36251e6i	0.125043i	−0.998044	+	0.0625213i	$0.980086\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 4.87918e6i	− 0.440995i
$658$	0	0
$659$	−8.81808e6	−0.790971	−0.395485	−	0.918472i	$-0.629424\pi$
$659$	−8.81808e6	−0.790971	−0.395485	+	0.918472i	$0.629424\pi$
$660$	0	0
$661$	−1.52035e6	−0.135344	−0.0676720	−	0.997708i	$-0.521557\pi$
$661$	−1.52035e6	−0.135344	−0.0676720	+	0.997708i	$0.521557\pi$
$662$	0	0
$663$	2.27664e6i	0.201146i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5.02850e6i	0.437647i
$668$	0	0
$669$	3.95805e6	0.341913
$670$	0	0
$671$	−2.42868e7	−2.08240
$672$	0	0
$673$	315086.i	0.0268158i	0.999910	+	0.0134079i	$0.00426800\pi$
$673$	315086.i	0.0268158i	−0.999910	+	0.0134079i	$0.995732\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1.74092e6i	− 0.145985i	−0.997332	−	0.0729924i	$-0.976745\pi$
$677$	− 1.74092e6i	− 0.145985i	0.997332	−	0.0729924i	$-0.0232549\pi$
$678$	0	0
$679$	−1.67082e7	−1.39077
$680$	0	0
$681$	−7.25670e6	−0.599614
$682$	0	0
$683$	− 1.98935e7i	− 1.63177i	−0.578214	−	0.815885i	$-0.696250\pi$
$683$	− 1.98935e7i	− 1.63177i	0.578214	−	0.815885i	$-0.303750\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	8.71595e6i	0.704568i
$688$	0	0
$689$	−9.61880e6	−0.771921
$690$	0	0
$691$	−2.01519e7	−1.60554	−0.802770	−	0.596289i	$-0.796641\pi$
$691$	−2.01519e7	−1.60554	−0.802770	+	0.596289i	$0.796641\pi$
$692$	0	0
$693$	1.16765e7i	0.923593i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 1.67009e7i	− 1.30214i
$698$	0	0
$699$	−3.99765e6	−0.309465
$700$	0	0
$701$	8.10766e6	0.623161	0.311581	−	0.950220i	$-0.399142\pi$
$701$	8.10766e6	0.623161	0.311581	+	0.950220i	$0.399142\pi$
$702$	0	0
$703$	− 1.42648e7i	− 1.08862i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 7.85311e6i	− 0.590872i
$708$	0	0
$709$	1.35613e7	1.01317	0.506587	−	0.862189i	$-0.330907\pi$
$709$	1.35613e7	1.01317	0.506587	+	0.862189i	$0.330907\pi$
$710$	0	0
$711$	5.63062e6	0.417717
$712$	0	0
$713$	− 2.82864e6i	− 0.208379i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 1.29631e7i	− 0.941695i
$718$	0	0
$719$	4.28314e6	0.308987	0.154493	−	0.987994i	$-0.450625\pi$
$719$	4.28314e6	0.308987	0.154493	+	0.987994i	$0.450625\pi$
$720$	0	0
$721$	1.39748e7	1.00117
$722$	0	0
$723$	− 8.00974e6i	− 0.569866i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 1.12084e7i	− 0.786515i	−0.919428	−	0.393258i	$-0.871348\pi$
$727$	− 1.12084e7i	− 0.786515i	0.919428	−	0.393258i	$-0.128652\pi$
$728$	0	0
$729$	−3.61141e6	−0.251686
$730$	0	0
$731$	−8.61552e6	−0.596332
$732$	0	0
$733$	− 4.70549e6i	− 0.323478i	−0.986834	−	0.161739i	$-0.948290\pi$
$733$	− 4.70549e6i	− 0.323478i	0.986834	−	0.161739i	$-0.0517102\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 3.55539e7i	− 2.41112i
$738$	0	0
$739$	−2.31099e7	−1.55663	−0.778317	−	0.627872i	$-0.783926\pi$
$739$	−2.31099e7	−1.55663	−0.778317	+	0.627872i	$0.783926\pi$
$740$	0	0
$741$	−3.24115e6	−0.216847
$742$	0	0
$743$	5.75294e6i	0.382312i	0.981560	+	0.191156i	$0.0612236\pi$
$743$	5.75294e6i	0.382312i	−0.981560	+	0.191156i	$0.938776\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	4.39481e6i	0.288163i
$748$	0	0
$749$	−2.23214e7	−1.45384
$750$	0	0
$751$	1.92424e7	1.24497	0.622485	−	0.782632i	$-0.286123\pi$
$751$	1.92424e7	1.24497	0.622485	+	0.782632i	$0.286123\pi$
$752$	0	0
$753$	2.95184e6i	0.189717i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	4.49210e6i	0.284911i	0.989801	+	0.142456i	$0.0454999\pi$
$757$	4.49210e6i	0.284911i	−0.989801	+	0.142456i	$0.954500\pi$
$758$	0	0
$759$	−4.11686e6	−0.259395
$760$	0	0
$761$	7.33500e6	0.459133	0.229567	−	0.973293i	$-0.426269\pi$
$761$	7.33500e6	0.459133	0.229567	+	0.973293i	$0.426269\pi$
$762$	0	0
$763$	7.57577e6i	0.471102i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.01678e7i	0.624076i
$768$	0	0
$769$	5.85526e6	0.357051	0.178526	−	0.983935i	$-0.442867\pi$
$769$	5.85526e6	0.357051	0.178526	+	0.983935i	$0.442867\pi$
$770$	0	0
$771$	2.23208e6	0.135230
$772$	0	0
$773$	− 1.34558e7i	− 0.809952i	−0.914327	−	0.404976i	$-0.867280\pi$
$773$	− 1.34558e7i	− 0.809952i	0.914327	−	0.404976i	$-0.132720\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	9.30874e6i	0.553144i
$778$	0	0
$779$	2.37764e7	1.40379
$780$	0	0
$781$	3.34084e7	1.95988
$782$	0	0
$783$	1.99252e7i	1.16144i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 1.00706e7i	− 0.579587i	−0.957089	−	0.289794i	$-0.906413\pi$
$787$	− 1.00706e7i	− 0.579587i	0.957089	−	0.289794i	$-0.0935867\pi$
$788$	0	0
$789$	6.49392e6	0.371377
$790$	0	0
$791$	−1.64002e7	−0.931985
$792$	0	0
$793$	− 1.23043e7i	− 0.694820i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1.18844e7i	− 0.662723i	−0.943504	−	0.331362i	$-0.892492\pi$
$797$	− 1.18844e7i	− 0.662723i	0.943504	−	0.331362i	$-0.107508\pi$
$798$	0	0
$799$	−9.11028e6	−0.504853
$800$	0	0
$801$	1.62629e7	0.895604
$802$	0	0
$803$	− 1.64638e7i	− 0.901036i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.82571e6i	0.152737i
$808$	0	0
$809$	−1.06053e7	−0.569705	−0.284852	−	0.958571i	$-0.591945\pi$
$809$	−1.06053e7	−0.569705	−0.284852	+	0.958571i	$0.591945\pi$
$810$	0	0
$811$	1.38944e6	0.0741799	0.0370900	−	0.999312i	$-0.488191\pi$
$811$	1.38944e6	0.0741799	0.0370900	+	0.999312i	$0.488191\pi$
$812$	0	0
$813$	− 1.37297e7i	− 0.728510i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 1.22655e7i	− 0.642882i
$818$	0	0
$819$	−5.91559e6	−0.308169
$820$	0	0
$821$	−1.12661e7	−0.583334	−0.291667	−	0.956520i	$-0.594210\pi$
$821$	−1.12661e7	−0.583334	−0.291667	+	0.956520i	$0.594210\pi$
$822$	0	0
$823$	− 2.77093e7i	− 1.42602i	−0.701152	−	0.713011i	$-0.747331\pi$
$823$	− 2.77093e7i	− 1.42602i	0.701152	−	0.713011i	$-0.252669\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1.23662e7i	− 0.628740i	−0.949300	−	0.314370i	$-0.898207\pi$
$827$	− 1.23662e7i	− 0.628740i	0.949300	−	0.314370i	$-0.101793\pi$
$828$	0	0
$829$	−1.23182e7	−0.622530	−0.311265	−	0.950323i	$-0.600753\pi$
$829$	−1.23182e7	−0.622530	−0.311265	+	0.950323i	$0.600753\pi$
$830$	0	0
$831$	−1.96706e6	−0.0988130
$832$	0	0
$833$	4.78299e6i	0.238829i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 1.12083e7i	− 0.553002i
$838$	0	0
$839$	−1.17277e7	−0.575183	−0.287592	−	0.957753i	$-0.592855\pi$
$839$	−1.17277e7	−0.575183	−0.287592	+	0.957753i	$0.592855\pi$
$840$	0	0
$841$	1.43225e7	0.698277
$842$	0	0
$843$	1.34095e7i	0.649894i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.20066e7i	1.05401i
$848$	0	0
$849$	−1.00705e7	−0.479494
$850$	0	0
$851$	9.17945e6	0.434503
$852$	0	0
$853$	− 1.57059e7i	− 0.739077i	−0.929215	−	0.369538i	$-0.879516\pi$
$853$	− 1.57059e7i	− 0.739077i	0.929215	−	0.369538i	$-0.120484\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.52390e7i	1.17387i	0.809634	+	0.586935i	$0.199666\pi$
$857$	2.52390e7i	1.17387i	−0.809634	+	0.586935i	$0.800334\pi$
$858$	0	0
$859$	3.66248e6	0.169353	0.0846763	−	0.996409i	$-0.473014\pi$
$859$	3.66248e6	0.169353	0.0846763	+	0.996409i	$0.473014\pi$
$860$	0	0
$861$	−1.55157e7	−0.713286
$862$	0	0
$863$	− 4.17938e7i	− 1.91023i	−0.296243	−	0.955113i	$-0.595734\pi$
$863$	− 4.17938e7i	− 1.91023i	0.296243	−	0.955113i	$-0.404266\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 4.43966e6i	− 0.200586i
$868$	0	0
$869$	1.89994e7	0.853475
$870$	0	0
$871$	1.80124e7	0.804500
$872$	0	0
$873$	2.76924e7i	1.22977i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.08990e7i	0.478505i	0.970957	+	0.239253i	$0.0769024\pi$
$877$	1.08990e7i	0.478505i	−0.970957	+	0.239253i	$0.923098\pi$
$878$	0	0
$879$	−5.75326e6	−0.251155
$880$	0	0
$881$	−3.04336e7	−1.32103	−0.660517	−	0.750811i	$-0.729663\pi$
$881$	−3.04336e7	−1.32103	−0.660517	+	0.750811i	$0.729663\pi$
$882$	0	0
$883$	6.09028e6i	0.262867i	0.991325	+	0.131433i	$0.0419579\pi$
$883$	6.09028e6i	0.262867i	−0.991325	+	0.131433i	$0.958042\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 2.77908e7i	− 1.18602i	−0.805195	−	0.593010i	$-0.797940\pi$
$887$	− 2.77908e7i	− 1.18602i	0.805195	−	0.593010i	$-0.202060\pi$
$888$	0	0
$889$	−3.63524e7	−1.54269
$890$	0	0
$891$	9.95936e6	0.420278
$892$	0	0
$893$	− 1.29699e7i	− 0.544262i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 2.08570e6i	− 0.0865506i
$898$	0	0
$899$	−1.95946e7	−0.808608
$900$	0	0
$901$	−2.92336e7	−1.19969
$902$	0	0
$903$	8.00410e6i	0.326658i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 3.71510e7i	− 1.49952i	−0.661709	−	0.749761i	$-0.730169\pi$
$907$	− 3.71510e7i	− 1.49952i	0.661709	−	0.749761i	$-0.269831\pi$
$908$	0	0
$909$	−1.30158e7	−0.522470
$910$	0	0
$911$	7.85959e6	0.313765	0.156882	−	0.987617i	$-0.449856\pi$
$911$	7.85959e6	0.313765	0.156882	+	0.987617i	$0.449856\pi$
$912$	0	0
$913$	1.48294e7i	0.588772i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 2.97333e7i	− 1.16767i
$918$	0	0
$919$	−1.62693e7	−0.635448	−0.317724	−	0.948183i	$-0.602919\pi$
$919$	−1.62693e7	−0.635448	−0.317724	+	0.948183i	$0.602919\pi$
$920$	0	0
$921$	−1.48810e7	−0.578074
$922$	0	0
$923$	1.69255e7i	0.653938i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 2.31619e7i	− 0.885268i
$928$	0	0
$929$	3.69365e7	1.40416	0.702079	−	0.712099i	$-0.252255\pi$
$929$	3.69365e7	1.40416	0.702079	+	0.712099i	$0.252255\pi$
$930$	0	0
$931$	−6.80933e6	−0.257472
$932$	0	0
$933$	2.22707e6i	0.0837587i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	4.89705e6i	0.182216i	0.995841	+	0.0911078i	$0.0290408\pi$
$937$	4.89705e6i	0.182216i	−0.995841	+	0.0911078i	$0.970959\pi$
$938$	0	0
$939$	3.79346e6	0.140401
$940$	0	0
$941$	−6.83943e6	−0.251794	−0.125897	−	0.992043i	$-0.540181\pi$
$941$	−6.83943e6	−0.251794	−0.125897	+	0.992043i	$0.540181\pi$
$942$	0	0
$943$	1.53002e7i	0.560297i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1.03790e7i	− 0.376082i	−0.982161	−	0.188041i	$-0.939786\pi$
$947$	− 1.03790e7i	− 0.376082i	0.982161	−	0.188041i	$-0.0602137\pi$
$948$	0	0
$949$	8.34095e6	0.300642
$950$	0	0
$951$	−1.46990e7	−0.527032
$952$	0	0
$953$	− 2.59587e7i	− 0.925873i	−0.886391	−	0.462937i	$-0.846796\pi$
$953$	− 2.59587e7i	− 0.925873i	0.886391	−	0.462937i	$-0.153204\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.85185e7i	1.00658i
$958$	0	0
$959$	2.46782e7	0.866497
$960$	0	0
$961$	−1.76068e7	−0.614994
$962$	0	0
$963$	3.69957e7i	1.28554i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	3.92120e7i	1.34851i	0.738501	+	0.674253i	$0.235534\pi$
$967$	3.92120e7i	1.34851i	−0.738501	+	0.674253i	$0.764466\pi$
$968$	0	0
$969$	−9.85056e6	−0.337017
$970$	0	0
$971$	1.06876e7	0.363774	0.181887	−	0.983319i	$-0.441779\pi$
$971$	1.06876e7	0.363774	0.181887	+	0.983319i	$0.441779\pi$
$972$	0	0
$973$	2.42227e7i	0.820238i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.77266e7i	0.929308i	0.885492	+	0.464654i	$0.153821\pi$
$977$	2.77266e7i	0.929308i	−0.885492	+	0.464654i	$0.846179\pi$
$978$	0	0
$979$	5.48758e7	1.82989
$980$	0	0
$981$	1.25561e7	0.416566
$982$	0	0
$983$	− 9.49272e6i	− 0.313334i	−0.987652	−	0.156667i	$-0.949925\pi$
$983$	− 9.49272e6i	− 0.313334i	0.987652	−	0.156667i	$-0.0500749\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	8.46374e6i	0.276547i
$988$	0	0
$989$	7.89293e6	0.256595
$990$	0	0
$991$	2.03243e7	0.657403	0.328702	−	0.944434i	$-0.393389\pi$
$991$	2.03243e7	0.657403	0.328702	+	0.944434i	$0.393389\pi$
$992$	0	0
$993$	2.39794e7i	0.771730i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	4.70508e7i	1.49909i	0.661951	+	0.749547i	$0.269729\pi$
$997$	4.70508e7i	1.49909i	−0.661951	+	0.749547i	$0.730271\pi$
$998$	0	0
$999$	3.63730e7	1.15310

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.c.h.49.1		2
4.3	odd	2		200.6.c.c.49.2		2
5.2	odd	4		400.6.a.f.1.1		1
5.3	odd	4		80.6.a.f.1.1		1
5.4	even	2	inner	400.6.c.h.49.2		2
15.8	even	4		720.6.a.h.1.1		1
20.3	even	4		40.6.a.b.1.1	✓	1
20.7	even	4		200.6.a.c.1.1		1
20.19	odd	2		200.6.c.c.49.1		2
40.3	even	4		320.6.a.l.1.1		1
40.13	odd	4		320.6.a.e.1.1		1
60.23	odd	4		360.6.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.a.b.1.1	✓	1	20.3	even	4
80.6.a.f.1.1		1	5.3	odd	4
200.6.a.c.1.1		1	20.7	even	4
200.6.c.c.49.1		2	20.19	odd	2
200.6.c.c.49.2		2	4.3	odd	2
320.6.a.e.1.1		1	40.13	odd	4
320.6.a.l.1.1		1	40.3	even	4
360.6.a.b.1.1		1	60.23	odd	4
400.6.a.f.1.1		1	5.2	odd	4
400.6.c.h.49.1		2	1.1	even	1	trivial
400.6.c.h.49.2		2	5.4	even	2	inner
720.6.a.h.1.1		1	15.8	even	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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	400.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-8.00000i q^{3} +108.000i q^{7} +179.000 q^{9} +O(q^{10})\)	\(q-8.00000i q^{3} +108.000i q^{7} +179.000 q^{9} +604.000 q^{11} +306.000i q^{13} +930.000i q^{17} -1324.00 q^{19} +864.000 q^{21} -852.000i q^{23} -3376.00i q^{27} -5902.00 q^{29} +3320.00 q^{31} -4832.00i q^{33} +10774.0i q^{37} +2448.00 q^{39} -17958.0 q^{41} +9264.00i q^{43} +9796.00i q^{47} +5143.00 q^{49} +7440.00 q^{51} +31434.0i q^{53} +10592.0i q^{57} +33228.0 q^{59} -40210.0 q^{61} +19332.0i q^{63} -58864.0i q^{67} -6816.00 q^{69} +55312.0 q^{71} -27258.0i q^{73} +65232.0i q^{77} +31456.0 q^{79} +16489.0 q^{81} +24552.0i q^{83} +47216.0i q^{87} +90854.0 q^{89} -33048.0 q^{91} -26560.0i q^{93} +154706. i q^{97} +108116. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 358 q^{9}+O(q^{10}) \) 2 * q + 358 * q^9	\( 2 q + 358 q^{9} + 1208 q^{11} - 2648 q^{19} + 1728 q^{21} - 11804 q^{29} + 6640 q^{31} + 4896 q^{39} - 35916 q^{41} + 10286 q^{49} + 14880 q^{51} + 66456 q^{59} - 80420 q^{61} - 13632 q^{69} + 110624 q^{71} + 62912 q^{79} + 32978 q^{81} + 181708 q^{89} - 66096 q^{91} + 216232 q^{99}+O(q^{100}) \) 2 * q + 358 * q^9 + 1208 * q^11 - 2648 * q^19 + 1728 * q^21 - 11804 * q^29 + 6640 * q^31 + 4896 * q^39 - 35916 * q^41 + 10286 * q^49 + 14880 * q^51 + 66456 * q^59 - 80420 * q^61 - 13632 * q^69 + 110624 * q^71 + 62912 * q^79 + 32978 * q^81 + 181708 * q^89 - 66096 * q^91 + 216232 * q^99

Introduction

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